The mariners magazine, or, Sturmy's mathematical and practical arts containing the
description and use of the scale of scales, it being a mathematical ruler, that resolves
most mathematical conclusions, and likewise the making and use of the crostaff, quadrant,
and the quadrat, nocturnals, and other most useful instruments for all artists and
navigators : the art of navigation, resolved geometrically, instrumentally, and by
calculation, and by that late excellent invention of logarithms, in the three principal
kinds of sailing : with new tables of the longitude and latitude of the most eminent
places ... : together with a discourse of the practick part of navigation ..., a new
way of surveying land ..., the art of gauging all sorts of vessels ..., the art of
dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties
and forfeitures, by acts of parliaments appointed, relating to the customs and navigation
: also a compendium of fortification, both geometrically and instrumentally / by Capt.
Samuel Sturmy.
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THE Mariners Magazine; OR, STURMY'S Mathematical and Practical ARTS. CONTAINING, The
Description and Use of the SCALE of SCALES; it being a Mathematical Ruler, that resolves
most Mathematical Conclusions: And likewise the Making and Use of the Crostaff, Quadrant, and the Quadrat, Nocturnals, and other most Useful Instruments for all Artists and Navigators. The ART of NAVIGATION,
Resolved Geometrically, Instrumentally, and by Calculation, and by that late Excellent
Invention of Logarithms, in the Three Principal kinds of Sailing; with New Tables of the Longitude and Latitude of the most Eminent Places round the World, from the Meridian of the Lizard: And New Exact Tables of the Sun's Declination, newly Calculated; and of the Longitude and Latitude, Declination and Right Ascension of some Eminent Fixed Stars. TOGETHER WITH A Discourse of the Practick Part of NAVIGATION,
in Working a Ship in all Weathers and Conditions at Sea. A New Way of Surveying of
Land, by the Mariners Azimuth or Amplitude Compass; very easie and delightful to all
sorts of Navigators, Mariners, or others. The ART of GAUGING all Sorts of VESSELS;
and the Measuring of Timber, Glass, Board, Stone, Walls, Cielings, and Tylings. The
ART of GUNNERY, Geometrically, Instrumentally, and by a most easie way of Calculation,
by the Logarithm Tables, by Addition and Substraction, in the place of other Mens
way of Arithmetick, of Multiplication and Division: Also, Artificial Fire-Works for
Sea and Land Service, as also for Recreation and Delight, with Figures. ASTRONOMY,
Geometrical, Instrumental, and by Calculation. The ART of DIALLING by a Gnomonical
Scale, and likewise by Calculation; Making all sorts of Dials both without Doors and
within, upon any Wall, Cieling, or Floor, be they never so Irregular, wheresoever
the Direct or Reflect Beams of the Sun may come. WHEREUNTO IS ANNEXED, An Abridgment
of the Penalties and Forfeitures, by Acts of Parliaments appointed, relating to the
Customs and Navigation. ALSO, A COMPENDIUM of FORTIFICATION, both Geometrically and
Instrumentally. By Capt. SAMUEL STURMY.
LONDON, Printed by E. Cotes for G. Hurlock, W. Fisher, E. Thomas, and D. Page; and are to be sold at their Shops over against St. Magnus Church neer London-bridge, at the Postern-gate neer Tower-hill, at the Adam and Eve in Little Britain, and at the Anchor and Mariner in East-Smithfield. MDCLXIX.
To the most August and Most Serene MAJESTY OF CHARLES II. King of Great-Britain, France, and Ireland, Defender of the Faith, &c.
Most Gracious Sovereign,
YOur Majesties Royal Grandfather King James, the Peace-maker, was by the Wisest of His Age justly reputed worthy to be entituled
Great Britain's Solomon, for His Wisdom, Learning, and Prosperous Government.
Your Pious and Judicious Father King Charles the First His esteem of the more Ingenuous and Politer Arts, Sculpture, Picture, and whatever else was Ancient or Excellent, drew hither into His own Galleries and
Cabinets, and other Noble Palaces of this happy Island, the best Monuments of Old
Greece, and of Modern Rome: And His own Incomparable Writings will demonstrate to succeeding Ages, how richly
he was furnished with the best kinds of Knowledge, and with the truly Celestial and
Divine Arts, which appeared in the whole [Page] Course of His Life, but most of all in His last Sufferings.
And now Your Majesty hath not only an Hereditary Claim to Your Ancestors Virtues and
Blessings, being by God's miraculous Providence and Protection restored, and redeemed
from Your Fathers Calamities; but herein You have infinitely excelled them all, and
that in the main, in that You are the first Founder of a Royal Institution for the
Advancement and Propagation of all Noble Arts and Beneficial Inventions: So that all
the Kings in the Christian World do emulate Your Great Example, and do value themselves
in nothing more, than that some of their wisest Subjects do follow and adore the Foot-steps
of Your Royal Society. And by this that Everlasting Renown is already spread over the face of the whole
Earth; it hath surrounded the Globe by Sea and Land; and we who are Borderers on the
Shores, and all they who inhabite on the deep Seas, are witnesses of the perpetual
Ecchoes of your Fame, with this Averment, That the better Spirits in all the Universities
of Christendom are allured by Your Majesties Lustre to embrace and rejoyce in this Brighter Light,
which can never henceforth be extinguished. And this constrains as well all Your
Loyal and True-hearted Subjects, as all Intelligent and Friendly Foreiners, to offer
their Applauses, Veneration, and Admiration.
But it is far above the reach of my Abilities [Page] to make just report of these Great Designs, and of all Your Majesties more than Heroical
Accomplishments. And yet I have a more Obliging and Personal Concernment: For as
the Amplitude and Prosperousness of Your Majesties Dominions are chiefly maintained
by Your Majesties Countenance upon the Mathematical Arts, and especially of Navigation; so Your Majesty doth apparently excel all the Kings that ever were, in this Noble
kind of Knowledge of the Naval Arts and Sciences. And for these, and many other good
Reasons, I hold it my bounden duty to lay these Essays of my best Endeavours at Your
Royal Feet; and with the Protestation of a tryed and trusty Sea-man, to avouch, That
not only my best Skill in Arts, but my Heart and Life also, are entirely devoted
for Your Majesties Service: And all my fellow Sea-men, and all true-hearted Englishmen, do joyn in this one Voice, God save King Charles the Second.
Most Gracious Sovereign,
Your Majesties most Dutiful and most Obedient Subject, SAMUEL STURMY.
To the Honourable SOCIETY of MERCHANT-ADVENTURERS Of the CITY of BRISTOL. TO THE MASTER,
WARDENS, and ASSISTANTS OF THE SAID SOCIETY. To my Honoured Friends Sir Robert Cann and Sir Robert Yeomans, Knights and Baronets, Sir Humfry Hooke, Sir Henry Creswick, Sir John Knight, and Sir Thomas Langton, Knights, John Willoughby and John Knight, Esquires.
Honourable Gentlemen,
OƲt of the great Respect I bear to you all of this Society, in regard from my Youth
I have lived among you, and am a Burgess of your City, and have been commanded, and
a Commander, and have sailed to several Parts of the World out of this Port, I cannot
commit these Productions of my Pen to the wide Ocean of fluctuating Opinions, without
the assurance of the Protection of some Honourable Patronage, as your selves; without
the which I might haply appear misshapen, and monstrous to the eye of the World,
and uneasily escape submersion; since that as the Year consisting of more foul than
fair Days, so the World in truth affords more bitter Blasts and virulent Censures
of Detraction, than candid, serene, unprejudiced Judgment. Therefore I must repair
to you, as to Seth's Pillar, which in despight of whole Torrents of Opposers, and Cataracts of Zoilus his furious Offspring, [Page] will (media inter praelia) secure these my poor Labours, and perpetuate them as indelible as the Stars, or your
deserved Honours, who of your Function are the Pillars of a Country, supporting
all manner of Trade and Commerce in the World, and, like the industrious Bee, bringing
Honey to every ones Hive, in adventuring your Estates to several Parts of the World,
for the Good of many thousands, which live and depend upon your Prosperity. I could,
with many others, desire, That this City and Society had in all respects the same
Laws, and Customs in all Maritime Affairs, as the Honourable City of London: then would much Loss, Charge, and Damage be prevented, that many times befals your
Ships and Goods; a competent Company of Sea-men, for half Pay, would attend on board
at the Ships going out, and coming home, until discharged; and that Pay would relieve
themselves, their Wives, and Children, and fit them for the Voyage, and be great satisfaction
to them for their attendance on your Service: for by the Rules of Charity, The Labourer is worthy of his Hire; and none deserve it more than Sea-men. Then would your Sea-mens Courage be fortified,
Honesty incouraged, and deserving Men rewarded, and the Reformers of an ill Custom
be had in everlasting memory, for the Good they did in their Generation. I humbly
beg pardon for this Digression, and humbly desire that you would take it into consideration.
For my part I ever did, and shall to the utmost of my poor abilities endeauour to
honour this City, and will adventure my Life to serve this Society in any Parts of
the World.
Accept therefore this Off-spring of some spare Hours, improved more with an intent
for the Publick Good, than for any Private Benefit.
I shall conclude this my Humble Address with a Temporal and Spiritual wish, viz. That the Encrease of your Treasures may answer your Hazards and Desires, and that
your Virtues and Graces may exceed your Treasures in this Life; and in that to come,
may your Glories as far transcend both, as Heaven doth Earth: And for this you have
not only the earnest Wishes, but the cordial Prayers of,
Your Honours most humble Servitor to be Commanded, SAMUEL STURMY.
To my much Honoured Patron, Sir JOHN SHAW Knight and Baronet: And to the rest of the Honourable FARMERS OF HIS MAJESTIES CUSTOMS,
Sir John Wolstenholme, Sir Robert Vyner Knights and Baronets; Sir Edmond Turner Knight, Edward Backwell and Francis Millington Esquires. External, Internal, and Eternal Happiness be wished.
HONOURABLE SIRS,
MAN had at the first, and so have all Souls before their entrance into the Body, an
Explicite Methodical Knowledge; but they are no sooner Vessel d, but that Liberty
is lost, and nothing remains but a vast confused Notion of the Creature. Thus had
I only a Capacity without Power, and a Will to do that which was far enough above
me. In this perplexity I studied several Arts, and put them into practice: For my
own sullen Fate hath forc d me to several Courses of Life; but I find not one hither
to which ends not in Surfeits or Saciety, and all [Page] the Fortunes of this Life are Follies. Thus I rambled over all these Mathematical
Inventions or Sciences, Wherefore (Honourable Sirs) I having Composed, out of my poor Studies, this Miscellany, and considering there was nothing in it more useful for your Service than the Art
of Surveying, knowing that you bestow Surveyors Offices upon many, but divers may be wanting in
the Knowledge and Labour of the Art; my self once enjoyed both in your Honours Service, by the kind thankful Remembrance of my much Honoured Patron: but Envy soon eclipsed my Office without desert, and left me only my Art: Therefore
(as St. Paul saith in another Case) I will wait with Patience until my Change will come; and in the mean time I am your Honours humble Servant, and humbly prostrate these
my poor Labours at your Honours Feet, begging your Patronage thereof; knowing, That
there was never any thing so well contrived by the Wit of Man, that hath not been
subject to the Censure and Misconstruction of the Envious: Nor do I at this time
(in the Production of this) expect Immunity from the Censorious Criticks of this present
Age; yet for such was this Work never intended, but only for the Judicious, whose
candid Censures I dare abide: But yet not without labour and difficulty can Books
have passage into the World; therefore to the end the placid Fruits of these my Labours
(now grown up among the wild Grapes of the Field) may be cherished and preserved from
the turbulent Storms of discontented Spirits, it [Page] now being come to its Maturity and Perfection, must humbly implore the Protection
of some Honourable Persons to defend it: And being well assured (Sirs) of your Honours most Heroick and Candid Dispositions, I humbly cast this into the
Arms of your Humanity for Shelter and Protection; not doubting but that your refulgent
Rays shining thereon, will be sufficient to annihilate and dispel the most dark and
misty Clouds ascending our Horizon; which will not a little strengthen both my present
and future Ʋndertakings for the Publick Good, and excite the Author to a grateful
Acknowledgment of your Memorable Virtues, and to echo forth the Praises due to your
Names and Eminencies. All that I have endeavoured, is to profit others, and to make
my diligent and studious Reader and Practicioner able to benefit his Country (which,
certes, is no more than the Common Law of Humanity requires at all our Hands) and not, like
some, to bury their Treasures in the Ashes of Oblivion; which puts me in mind of
that excellent Saying of Tullie, Non nobis solum nati sumus. Wherefore (Honourable Sirs) I have endeavoured, in as plain and succinct a Method as I could imagine, to lay
down the Art of Surveying, a new way, by the Mariners Sea-Compass, which is the best Instrument for their use
and purpose, it agreeing so neer their Traverse Rules at Sea, that there is very little
difference. And likewise I have shewed the Sea-men the Land-man's Art of Surveying and Gauging all sorts of Vessels, and plain Superficies [Page] and Solids; the Art of Gunnery, Artificial Fire-works, and Astronomy, in the following Book; and so furnished the whole Work with such Theorems and Problems, Geometrically, Instrumentally, and by Calculation, as are most necessary and subservient
thereunto: And therefore (Honourable Sirs) you being able to protect it, I most humbly commit it to your gracious Protection,
resting,
Sirs,
Your Honours most humble and faithful Servant, SAMUEL STƲRMY.
St. Georges, or the Pill, neer Bristol,Mar. 25. Anno 1669.
MAthematical Studies have for these many Years been much neglected, if not contemned;
yet have there been so many rare Inventions found, even by Men of our own Nation,
that nothing now seems almost possible to be added more. As in other Studies, so
we may say in these, Nil dictum quod non dictum prius: We at the least must needs acknowledge, That in this we have presented thee with
nothing new, nothing that is our own. Ex integra graeca, integram Comoediam hodie sum acturus,Qui edesie en publick place, Faict maison trop haut ou trop basse. saith Terence, that most excellent Comedian, in his Heautontimorumenon. Translation was his Apology; Transcription, Collection, and Composition, ours. This only we have endeavoured, That the first Principles and Foundations of
those Studies (which were not to be known until now,Who builds i'th' way where all pass by, Shall make his house too low or high. but by being acquainted with many Books) might in a due method, and a perspicuous
manner, be as it were at once presented to thy view.
The Matter, being Mathematical and Practical Arts of my own practice, I can the better
avouch the ease and truth of them to all ingenuous Practicioners, and unto such as
have as yet learned nothing but Arithmetick.
To that purpose, we have at first laid down such Propositions, as all young Seamen
are or should be perfect in, concerning the Compass, and the Moon's Motion, Instrumentally
and Arithmetically; and by it, in the same manner, how to know the Rules of the Ebbing
and Flowing of the Sea, with the Rules of Time of Flood and High-water in any Port
in the World; with a Discourse of the Practick Part of Navigation, in working of a Ship in all Cases and Conditions of Weather at Sea, to the best of
my Experience.
And the ABC of Geometry, its Definitions and Geometrical Problems, out of Euclid and others, as must be known to such as would know the Nature and Mensuration of
Triangles. Next, We have proceeded to the Descriptions of all the most useful Instruments for
Artists and Navigators; as the Scale of Scales, which is a Mathematical Ruler, that resolves all Mathematical Rules whatsoever:
And we our selves have fitted Tables and Diagrams in that manner, as we presume has
not been done in that plainness, and so easie to be understood, by any Man before.
There is the Diagrams and Tables together, both Natural and Artificial; and the Scale, and its Making and Use follows. Secondly, The Making and Use of the Traverse-Scale of Artificial Points and Quarters; The Making of the Quadrant and Index, and their ready Use in Astronomy and Navigation; and the Protractor; The Projection and Use of the Nocturnal, and new Tables of the North Stars Declination. And on the back-side are 32 of the most useful Stars in the Heaven for Navigators, and its Use; [Page] with Tables of the Longitude and Latitude, Right Ascension and Declination; The Description and Use of the Forestaff, Davis his Quadrant; as also a new Quadrant and Quadrat, that I use my self at Land and Sea; A Constant Kalendar, joyned with the Tables of the Suns Declination, for 32 years to come. And, Thirdly, The Nature and Quality of Triangles. And, Fourthly, Of Sailing by the Plain Sea-Chart, and the Uncertainty thereof; and of Navigation by Mercator, or Mr. Wright's Projection, and by the neerest way of Sailing by the Globe, or Arch of a Great Circle; with the making of the True Sea-Chart, Geometrically, Arithmetically, and Instrumentally, as the true way of keeping a Sea-Journal
at Sea, very easie at once by Plain, Mercator, or Great Circle Sailing; with new Tables of Longitude and Latitude round the World, from the Meridian of the Lizard, terminating at 180 deg. East and West of that Meridian.
The Fifth Book, The Art of Surveying of Land by the Sea Azimuth or Amplitude Compass, very easie and useful for Sea-men: The Art of Gauging of all sorts of Vessels, and Measuring of Timber, Stone, and Glass, and Ships, Geometrically, Instrumentally,
and Arithmetically; and a most excellent Gunners Scale, with the easiest way of Gunnery that hath been writ by any: For what Nathaniel Nye hath done by Arithmetick, by the Square and Cube, and their Roots, which is the hardest sort of Arithmetick, by Maltiplication and Division, I have done by the Logarithme Tables, by Addition and Substraction, and likewise Geometrically and Instrumentally. The Scale shews at once, in a moment, the ready Dimensions of twenty sorts of most useful
Ordnance, from a Base to a Cannon-Royal, their length, and weight of the Gun, Powder, and Shot, and Tables of weight of Shot
of Lead, Iron, and Stone; with a Table of Right Ranges and Point Blanks; with a Plain Scale and Dialling Scale, Quadrant, and Quadrat, for taking of Heights and Distances; with a Line of Inches and Numbers, for the ready working all other Proportions of Solids, or otherwise; being a most
useful Instrument for all Land and Sea Gunners: But most especially I do advise all Sea Gunners to carry one of those most useful Instruments in his Pocket, and by our Directions
learn the Use perfectly of them. I am ashamed to hear how senslesly many Sea Gunners will talk of rhe Art, and know little or nothing therein, but only how to spunge,
lade, and fire a Gun at Random, without any Rules of finding the Dispart, thickness
of the Mettle in all places, and proportion any Charge of Powder thereto, and other
Rules which should be known. Herein how many of them are defective? And to supply
that defect, I have taken this pains in the Art, to the end to help all such as are
ingenuous, and willing to learn: As also, all manner of Artificial Fire-works and
Rockets, with their Figures and Fiery Arrows, Granadoes, and Pots.
The Sixth Book is the Art of Astronomy, containing the Definition of the Circle of the Spheres, with the manner how to resolve all the most necessary Propositions thereto belonging,
by a Line of Chords and Sines, and Chords and Tangents, and half Tangents, Geometrically and by Calculation, by the Logarithme Tables of Artificial Sines and Tangents: And all useful Astronomical Propositions appertaning to the First Motion; and Tables for finding always the Suns true place; being all of extraordinary use,
and made plain to the meanest Capacity.
Seventhly, The Seventh Book is the Art of Dialling by the Gnomonical [Page] Scale, with the Diagram and making of the said Scales, with Tables also in the Second Book described; with the Fundamental Diagram of all Scales on the Ruler, as also by Calculation, shewing the making of all sorts of Dials, both within doors and without, upon any Wall, Ceiling, or Floor, be they never so
irregular, wheresoever the direct or reflected Beams of the Sun may come, for any
Latitude; and how to find the true Hour of the Night by the Moon and Stars; and how to Colour, Guild, and Paint Dials; and how to fasten the Gnomon in Stone or Wood. We have insisted the more upon it, and by our Explanation have
endeavoured to make it plain and easie, it being all our own Practical Arts; so that
nothing may be wanting, which either former Ages or our own (by Gods Blessing and
their Industry) have afforded to us. We have to the Artificial Canon added out of Mr. Wing's Harmonicon Coeleste, page 263. the Rules to be taken and observed in use of Mr. Gunter's Canon of Artificial Sines and Tangents, and Mr. Brigs his Canon of Logarithme Numbers, as in that Form, and in this Work, we have made use of his Directions in the Astronomical
Calculation, and the Demonstration by our own Rules; and of Mr. Norwood's Advice in Navigation, and by Demonstration our own (the way of our usual practice at Sea in keeping our
Journal) and for the Longitude and Latitude of Places, we have had the best Experience we could procure from the ablest Pilots
and Masters that have been in the several Places of the World, and likewise of our
own Observation of several Places in the West-Indies, and other Parts of the World; together comparing of them with several Tables, formerly made and lately corrected, and fitted for a Meridian of our own Country, and the principal Cape of this Land, for thy ease; the Lizard being the farewel Cape to most Ships that sail out of the British Seas, any way to the South or West, and likewise the first Land made at their return home; and therefore it must needs
be very useful for all Northern and Southern Navigators in their Voyages, with great
ease and exactness.
It's nothing new, nor does it come by chance,
That Art is envy'd still by Ignorance.
For the Art of Gauging, I have conferred with Mr. Philip Staynred, Mathematician and Gager in Bristol; and all the Rules that have been laid down in the following Treatise, are most exact
and easie to the meanest Capacity of such as are skilful in Arithmetick; but with a great deal of Labour, Study, Care, and Charge, in the Tryal of the Practice
of them by our self: which may be considered by the Ingenuous Practicioners, though
much more abused by ignorant Momus and his Mates, who make it their business to scoff, deride, affront, and abuse all
such as are Ingenious, and pretend to have any thing more than themselves; and you
shall know them by their railing Discourse of any Ingenious Work or Artist. For such
Loiterers there is a pair of Stocks fitted in Hell by the Devil, where for their malice,
abuses, cursed railings, and villainous revilings of those that Study the Honour of
their Country to Posterity, the harmless study of Virtue, and praise-worthy commendation
of all good honest-minded men; I say, such Momusses will have their Heads in such Stocks, and their Tails lash'd by the Devils for ever,
for their malice and envy, if [Page] they give it not over, and repent of it in time. But for all honest-minded men that
love Arts and Sciences, Theorical and Practical, God doth give them his Spirit to
guide them in all Lawful Arts, to the knowledge thereof, according to their desire
of him.
Others that have either spent more time, or made a farther progress in these ravishing
Studies, might (if they would have taken the pains) have haply presented thee with
more in a less room; but the most of this was at the first collected for our private
use, and direction of our three Brothers and Son; but now published for the good
of others. Nevertheless, I am not ignorant, how that never any man living, in his
writing, could please the phansie of all men, neither do I expect to be the first.
To please the envious, I cannot; for they are resolute: To content the scornful, I
will not attempt it: To flatter the haughty, were much folly: To disswade the capricious,
were needless; and to perswade the courteous, were unnecessary. Let every one do
as his Genius doth best dispose him, take where he pleaseth, read what he liketh, and leave what
he liketh not. For my own part, I have with much diligence and industry waded through
many aenigmatical Difficulties, and have removed and drawn back the Curtain of Darkness
from off our English Horizon, in our Mathematical and Practical Arts following.
Lastly, I desire the Judicious Reader, if he chance to meet with any Errata (as some may happen in a Work of this Nature) that he would courteously amend them,
and not with cavillation ungratefully requite my painful Labours. Haply, if this find
acceptance, it may encourage me to publish some other thing, which perhaps may give
thee much satisfaction, and be commodious to my Country-men of England. Vale.
AN INDEX, SHEWING, The CONTENTS of the SEVEN BOOKS OF THE MARINER'S MAGAZINE.
BOOK I.
THe Description of Navigation in general. Page 1
Of what is needful first to be known in the Practick Part: And of the Compass; and
how to divide the Circles and Parts. Page 3
The Moon's Motion, and the Ebbing and Flowing of the Sea. Page 6
The making a most useful Instrument for the Moon. ibid.
The Variation Compass, and the Ʋse thereof, in 10 Propositions. ibid.
The finding the Golden Number, or Prime, and Epact, according to the English Accompt, and all other things relating to the Moon and Tides, Arithmetically. Page 9
The Practick Part of Navigation, in Working a Ship in all Weathers and Conditions
at Sea. Page 15, to 22
Geometrical Definitions. Page 22
Geometrical Problems. Page 28, to 43
BOOK II.
THe Argument, and Description of Instruments in general. Page 45
A Description of the Fundamental Diagram, and Tables for the making of the Lines of
Chords and Rumbs, as also of Sines, Tangents, and Secants Natural, on the Scale; and
of what Instruments you must be provided with before you can make Instruments for
Mathematical Ʋses. Page 47
The Explanation of the other half of the former Semicircle, being a Description of
the Fundamental Diagram of the Dialling Scale on the Mathematical Ruler; with a Table
for the dividing of the Hours and Minutes on the same; a Table for the dividing of
the Gnomon-line on the Scale, called by some a Line of Latitudes, as also a Table of Tangents for five Hours, to every five Minutes of an Hour, for
the inlarging the Hour-line Scale. Page 55
The Scales or Lines on the back-side of the Mathematical Ruler, viz. A Line of Artificial Numbers, with a Table how to make the same; A Table of Artificial
Tangents, and how to make the Line; as also a Table of Artificial [Page] Sines; The making of Mercator's Meridian Line, by our Table of Meridional Parts, in Leagues and 10 parts of a League: And the Aequinoctial is the Line of Equal parts, by which the
Table of Numbers were taken out, and the Lines made by. Page 58
How to calculate and make a Table for the Division of the Scale of Reduction, and
the use thereof. Page 62
A Table for the division and making of the Artificial Rumbs, or Points, Halfs, and
Quarters, on the Traverse Scale. Page 63
How to make a Quadrant, which will resolve many Questions in Astronomy by the help
of an Index, and also very useful in Navigation; with the Ʋse thereof in Astronomy
and Navigation, in seven Sections. Page 64
To find how many Leagues do answer to every Rumb and Quarter, in six Propositions
in Navigation. Page 70
How to make a useful Protractor. Page 71
The Projection of the Nocturnal, and the Ʋse thereof by the North Star. Page 73
How to use the Pole Stars Declination, and thereby to get the Latitude, with the Table.
Page 74
How to make a most useful Instrument of the Stars on the back side of the Nocturnal,
and by it to know most readily when any of 31 of the most notable Stars will come to the Meridian, what Hour of the Night at any
time of the Year, at the first sight; with a Table of the Longitude and Latitude from
the beginning af the Year 1671; with the Right Ascension and Declination of 31 of the most notable fixed Stars, Calculated from Tycho his Tables, Rectified from the Year of our Lord 1671. Page 76
The Ʋse of the most useful Instrument of the Stars, how to know the Hour of the Night
any Star comes to the Meridian in any Latitude; and how to know what Stars are in
Course at any Time or Day of the Year. Page 77
The Description and Ʋse of 3 Stars called the Crosiers. Page 78
A Description of the making of the Cross-staff, and how to use the same fully. Page 79, to 85
A Description and Ʋse of the Quadrant or Back-staff, in six Propositions, declaring
the Ʋse thereof in all Observations. Page 85
The Description and Ʋse of the most useful Quadrant for the taking of Altitudes of
the Sun or Stars, on Land or Sea, backwards or forwards, or any other Altitude of
Hills, Trees, Castles, or Things whatsoever. Page 92
A Constant Kalendar or Almanack for 300 Years; but more exactly serving for 19 Years, being the Circle of the Moon, or the Golden Number; with new exact Tables
of the Suns Declination, rectified by the best Hypothesis until the Leap-years, and
the Ʋse thereof. Page 101, to 122
BOOK III.
CHAP. I. OF the Nature and Quality of Triangles. Page 123
CHAP. II. Containing the Doctrine of the Dimensions of Right-lined Triangles, whether Right-angled
or Oblique-angled; and the several Cases therein resolved, both by Tables, and also
by the Lines of Artificial Numbers, Sines, and Tangents. Page 125
CASE I. In a Right-angled Plain Triangle, the Base and the Angle at the Base being given,
to find the Perpendicular, Page 126
[Page]CASE II. The Base aad the Angle at the Base being given, to find the Hypothenusal. Page 127
CASE III. The Hypothenusal and Angle at the Base being given, to find the Perpendicular. Page 128
CASE IV. The Hypothenusal and Angle at the Base being given, to find the Base. Page 129
CASE V. Let the Perpendicular be the Difference of Latitude 253 Leagues, and the Angle at C be S W b W 1 deg. 45 min. Westerly, or 58 deg. let it be given to find the Hypothenusal. Page 129
CASE VI. The Hypothenusal or Distance sailed, the Perpendicular of Difference of Latitude
given, to find the Rumb. Page 130
CASE VII. The Hypothenusal, and the Parallel of Longitude, and the Radius given, to find the
Rumb or Course sailed. ibid.
CASE VIII. Having two Angles and a Side opposite to one of them given, to find the Side opposite
to the other. Page 131
CASE IX. Two Sides and an Angle opposite to one of them being given, to find the Angle opposite
unto the other. Page 132
CASE X. Having two Sides and the Angle contained by them given, to find either of the other
Angles. ibid.
CASE XI. Two Sides and their contained Angle given, to find the third Side, Page 134
CASE XII. Three Sides of an Oblique Triangle being given, to find the Angles. ibid.
BOOK IV.
CHAP I. OF Sailing by the Plain Chart, and the Ʋncertainties thereof; and of Navigation,
with its parts. Page 137
Questions of Sailing by the ordinary Sea-Chart. Page 140
CHAP. II. Declaring what must be observed by all that keep Accompt of a Ships way; and to find
the true Point of the Ship at any time, according to the Plain Chart. Page 144
Directions how we do keep our Reckonings at Sea by the Log-board, and also by our
Journal Book. Page 145
CHAP. III. A formal and exact way of setting down and perfecting a Sea-Reckoning. Page 147
A Traverse-Table for every Point, Half-Point, and Quarter-Point of the Compass, to
the hundredth part of a League or Mile, which gives the Difference of Latitude and
Departure from the Meridian. Page 149
Examples and Ʋse of the Tables, with a Journal from Lundy to Barbadoes by the Plain Chart. Page 153
The Plain Sea-Chart, and how to make it, and the Ʋse thereof. Page 156
CHAP. IV. How to correct the Accompt when the Dead Latitude differs from the Latitude by Observation.
Page 157
CHAP. V. How to allow for known Currents, in estimating the Ships Course and Distance. Page 159; 160
CHAP. VI. Curious Questions in Navigation, and how to resolve them Geometrically and by Calculation.
Page 161
CHAP. VII. The disagreement betwixt the ordinary Sea-Chart and the [Page] Globe; and the agreement betwixt the Globe and the True Sea-Chart, made after Mercator's way, or Mr. Edward Wright's Projection; with the use thereof. Page 166
A Table of Meridional Parts to the tenth part of a League, and for every 10 Minutes of Latitude, from the Aequinoctial to the Poles, with the use thereof in
Mercator's Sailing, Geometrically and by Calculation. Page 169
CHAP. VIII. How to divide a Particular Sea-Chart according to Mercator and Mr. Wright's Projection. Page 174
CHAP. IX. The Projection of the Meridian-line by Geometry; and how to make a Scale of Leagues
for to measure Distances in any Latitude. Page 184, 185
CHAP. X. The way of Sailing by a Great Circle. Page 176
CHAP. XI. How to find the true distance of Places, one of them having no Latitude, the other
having Latitude and Difference of Longitude less than 180 deg. To find (1) Their Distance in a Great Circle, (2) The direct Position of the first Place from the second, (3) And the second Place from the first. Page 179
CHAP. XII. The Description of the Globe in Plano, and the several Conclusions wrought thereby. Page 189
CHAP. XIII. To Calculate the Arch of a Great Circle for every fifth or tenth Degree of Latitude
or Longitude. Page 192
CHAP. XIV. How by the Scale of Tangents to make a part of the Globe in Plano, whereby you may trace out the Latitudes to every Degree of Longitude, or every 5 or 10 Degrees, as neer as you will desire, without Calculation. Page 193
CHAP. XV. By the Latitude, and Difference of Longitude from the Obliquity, to find the true
Great Circles Distance. Page 196
CHAP. XVI. How to make the Truest Sea-Chart, and the Ʋse thereof in Mercator's and Great Circle Sailing, called a General Chart. Page 200
CHAP. XVII. How to keep a true and perfect Sea-Journal by Plain Sailing and the True Sea-Chart,
together with the Explanation thereof. Page 202
CHAP. XVIII. A Description of the Table of the Latitude and Longitude of Places, and the way how
to find both. Page 206
BOOK V.
CHAP. I. THe Art of Surveying Land by the Azimuth or Amplitude Compass, with the Description
thereof; as also the Staff and Chain, with the Ʋse thereof. Page 1
How to measure a Square piece of Ground. Page 4
To measure a Long Square piece of Ground by the Line of Numbers and Arithmetick.
Page 5
How to measure a Triangular piece of Ground. Page 6
How to measure a piece of Ground of four unequal Sides, called a Trapezia. ibid.
How to measure a piece of Ground being a perfect Circle. Page 7
How to measure an Oval piece of Ground. Page 8
How to measure a piece of Ground lying in form of a Sector. ibid.
To measure a piece of Ground being a Segment or part of a Circle. Page 9
Having the Content of a piece of Ground in Acres, to find how many Perch of that Scale
was contained in one Inch whereby it was Plotted. ibid.
[Page]CHAP. II. How to take the Plot of a Field at one station, taken in the middle thereof, by
the Compass. Page 11
CHAP. III. How to take the Plot of a Field at one station, taken at any Angle thereof. Page 13
CHAP. IV. How to measure an Irregular piece of Ground, by reducing the Sides into Triangles
and Trapezia's, and how to lay it down in your Field-Book. Page 14
CHAP. V. How to take the Height of Tenariffe, or any other Island or Mountain. Page 18
CHAP. VI. How to find the distance of a Fort or Castle, or the breadth of a River, by two stations,
with the Quantity of the Angle at each station. Page 21
CHAP. VII. How to take the distance of divers Places one from another, and to protract as it
were a Map thereof by the Compass and Plain Scale. Page 24
CHAP. VIII. The Art of Gauging of Vessels by the Line of Numbers, and the Lines on the Gauging
Rod or Staff, and by Arithmetick. Page 26
The true Content of a solid Measure being known, to find the Gauge-point of the same
Measure. ibid.
The Description of the Gauging Rod or Staff. Page 27
The Description of Symbols of Words for brevity in Arithmetick. ibid.
How to measure a Cubical Vessel. Page 28
How to measure any square Vessel. ibid.
How to measure a Cylinder Vessel. Page 29
How to measure a Vessel in form of a Globe. ibid.
How to measure a Barrel, Pipe, But, Puncheon, Hogshead, or small Cask. ibid.
How to find the Quantity of Liquor in a Cask that is part full. Page 31
How to measure a Brewers Tun or Mash-vat. Page 32
How to measure a Cone Vessel. Page 33
How to measure a Segment of a Globe or Sphere. Page 34
How to reduce Ale-measure into wine, and likewise to reduce Wine-gallons into Ale.
ibid.
How to measure a Brewers Oval Tun. ibid.
CHAP. IX. Wherein is shewed, How to measure exactly all kinds of Plain Superficies, both by
Arithmetick and Instrumentally. Page 36
How to measure a wall of an House in form of a long Square. ibid.
How to measure Boards, Glass, Pavement, Wainscot, and the like. Page 38
How to measure solid Bodies, as Timber and Stone. ibid.
To find how many Inches in length will make one Foot of Timber, being alike in the
Squares. Page 39
How to measure a Cylinder or a Tree whose Diameters at the ends be equal. Page 40
How to measure a round piece of taper Timber. Page 41
How to measure a Pyramidal piece of Timber. ibid.
How to measure a Conical piece of Timber. Page 42
How to find the Burden of a Ship. Page 43
The Ʋse of the Line of Numbers in Reduction and the Rule of Three. Page 44
CHAP. X.
Sect. 1. The Art of Gunnery, by a New-invented, Ʋseful, and Portable Scale. Page 45
Sect. 2. The Qualifications each Gunner ought to have, and his Duty and Office. ibid.
Sect. 3. The Description and Ʋse of the Gunners Scale on both Sides. Page 47
[Page]Sect. 4. The Ʋse of the Line of Numbers on the edge of the Scale, for the help of such as
cannot extract the Square and Cube Roots. Page 49
Sect. 5. As likewise, how by the Logarithm Tables and Addition and Substraction, to Resolve
with wonderful ease all Conclusions in the Art of Gunnery. Page 50
Sect. 6. The Geometrical finding the Diameter for the weight of any Shot assigned. Page 51
Sect. 7. How to find what Proportion is between Bullets of Iron, Lead, and Stone; by knowing
the weight of one Shot of Iron, to find the weight of another Shot of Lead, Brass,
or Stone, of the like Diameter. Page 53
Sect. 8. How by knowing the weight of one Piece or Ordnance, to find the weight of another
Piece of the same shape, and the same Metal, or any other Metal. Page 54
Sect. 9. How to make a Shot of Lead and Stone in the same Mold, of the same Diameter as the
Iron Shot is of. Page 55
Sect. 10. How by knowing what Quantity of Powder will load one Piece of Ordnance, to know how
much will load any other Piece whatsoever. Page 56
How to make the true Dispart of any true boared Piece of Ordnance, or otherwise,
to know whether the Piece be Chamber-boared. Page 57
To know what Diameters every Shot must be of to fit any Piece of Ordnance. Page 58
To find what Flaws, Cracks, or Honey-combs are in any Piece of Ordnance; and likewise
to find whether a Piece of Ordnance be true boared, or no. ibid.
Of Iron Ordnance, what Quantity of Powder to allow for their Loading; and what Powder
to allow for Ordnance not true boared. Page 61
How Molds, Forms, and Cartrages are to be made for any sort of Ordnance. Page 63
How to make Ladles, Rammers, Sponges, for all sorts of Ordnance; and how the Carriage
of a Piece should be made. ibid.
How much Rope will make Britching and Tackle for any Piece. Page 64
What Powder is allowed for Proof, and what for Action. ibid.
The difference between common Legitimate Pieces, and Bastard Pieces. ibid.
How Powder is made, and the several ways to know when it is decaying. Page 65
How to make excellent good Match; and how to make Powder that it shall not waste with
Time, and how to make good that which is bad, and how to make Powder of divers Colours.
Page 66
Several sorts of Saltpetre, and how to make an excellent sort, very easie, and less
Charge; and how to load and fire a Piece of Ordnance like an Artist. Page 67
The difference of shooting by the Metal, and by a Dispart, by Right Ranges, and at
Random; with the Figures thereof, Page 68
How to make a good Shot to any place assigned; out of any Gun. Page 70
How to make an effectual Shot out of a Piece of Ordnance at Random. Page 72
How to find the Right Line or Range of any Shot discharged out of any Piece, for every
Elevation, by one Right or Dead Range given for the Piece assigned: And to know how
much of the Horizontal Line is contained under the Right Line of any Shot made out
of any Piece, at any Elevation. Page 74
Of the violent, crooked, and natural Motion or Course of a Shot, discharged out of
any Piece of Ordnance assigned. Page 75
How to make a Gunners Ruler, and how to divide the same, by the help of a Table, fitting
it for any Piece; and how to give Level with the Gunners Ruler at any Degree of Random.
Page 76
How to give Level to a Piece or Ordnance without the Gunners Rule. Page 78
How to make a Shot at the Enemies Light in the Night. Page 79
[Page]How to shoot perfectly at a Company of Foot or Horse, or a Ship under sail. P. 79
How the same Powder in weight shall carry the Shot more close or scattering: And how
a Shot that sticketh fast within the Concavity of the Piece, that cannot be driven
home, may be shot out without any harm to the Gunner; and what difference there is
in shooting out of one Piece several Shots together. ibid.
Sect. 46. How to weigh Ships that are sunk, or Ordnance under Water; or to know what empty
Cask will carry any sort of Ordnance over a River. Page 80
How many Oxen, Horses, or Men, will serve to draw a Piece of Ordnance. Page 81
How Gunners may take a Plot of their Garrison, and every Object therein, or neer it.
ibid.
ARTIFICIAL FIRE-WORKS.
A Description of the Mortar-piece: How to make one of Wood and Paste-board: How to
fit and prepare Granadoes for the Mortar-piece: How to make Fuces. Page 83
How to make Granadoes of Canvas for the Mortar-piece; and how Granadoes are to be
charged in a Mortar-piece, and fired. Page 84
How to make Hand-granadoes, to heave by Hand. Page 85
How to make Fiery Arrows or Darts, like Death-Arrows Heads. ibid.
How to make Fiery Pots of Clay, and Powder Chests. Page 86
How to make Artificial Fire-works for Recreation and Delight. ibid.
How to make Composition for Rockets of any size, and how to fire them. Page 87
How to make Fiery Serpents and Rockets that will run upon a Line, and return again;
and how to make Fire-wheels, as some call them, Girondels. Page 88
How to make divers Compositions for Stars, and the Ʋse of them. Page 89
How to represent divers sorts of Figures in the Air with Rockets. ibid.
How to make Silver and Golden Rain, Fire-Lances, and Balloons for the Mortar-piece;
and the Figures of the most useful sorts of Fire-works, and the Explanation thereof.
Page 90
Most Precious Salves for Burning by Fire. Page 91
BOOK VI.
THe Projection of the Sphere by Tangents and half Tangents. Page 94
The Rudiments of Astronomy put into plain Rimes. Page 95
The Definitions of the Circles of the Sphere, and Imaginary Circles, which are not
described in a Material Sphere or Globe. Page 97
The Projection of the Sphere in Plano, represented by the Analemna; and the Points and Circles before described in a Convex and a Concave Sphere, by
Chords and Sines, and likewise resolved by Chords and Tangents. Page 101
How to Calculate the Suns true Place, and the Table of his mean Motion. Page 105
Probl. 2. The Suns distance from the next Aequinoctial Point, and his greatest Declination
being given, to find the Declination of any Point required. Page 107
Probl. 3. Having the Suns greatest Declination, and his distance from the next Aequinoctial
Point, to find his Right Ascension. Page 108
Probl. 4. The Elevation of the Pole and Declination of the Sun being given, to find the Ascensional
Difference. Page 109
[Page]Probl. 5. The Suns Right Ascension, and his Ascensional Difference being given, to find his
Oblique Ascension and Descension. Page 110
Probl. 6. To find the time of Sun-rising and setting, with the length of the Day and Night.
ibid.
Probl. 7. The Elevation of the Pole and the Declination of the Sun being given, to find his
Amplitude, and by it to know the Variation. Page 111
Probl. 8. Having the Latitude of the Place and the Suns Declination, to find when the Sun comes
to the due East and West. Page 112
Probl. 9. The Elevation of the Pole and the Declination of the Sun being given, to find the
Suns Altitude when he comes due East and West. ibid.
Probl. 10. The same being given, to find his Altitude at the hour of six. Page 113
Probl. 11. The same being given, to find his Azimuth at the hour of six. ib.
Probl. 12. Having the Latitude of the Place, and the Suns Declination and his distance from
the Meridian being given, to find the Suns Altitude at any time assigned. Page 114
Probl. 13. The Latitude of the Place, and the Suns Altitude and Declination being given, to
find the Suns Azimuth, and by it how to find the Variation. Page 118
Probl. 15. How to find the Altitude of the Sun by the Shadow of a Gnomon set perpendicular to
the Horizon, by Scale and Compass, as also by Calculation. Page 122
Probl. 16. Having the Latitude of the Place, the Declination of the Sun, and the Suns Altitude,
to find the hour of the day, Page 123
Probl. 17. Having the Azimuth of the Sun, and his Altitude, to find the hour of the day. Page 124
Probl. 18. How to find the Right Ascension of a Star, and the Declination of a Star, having
the Longitude and Latitude of the Star given. ibid.
Probl. 19. Having the Declination and Right Ascension of a Star, to find the Longitude and Latitude
thereof. Page 126
Probl. 20. Having the Meridian Altitude of an unknown Star, and the distance thereof from a
known Star, to find the Longitude and Latitude of the unknown Star. Page 128
Probl. 21. To find the Parallax of Altitude of the Sun, Moon, and Stars. Page 131
BOOK VII.
THe Fundamental Diagram of the Dialling Scale, and the Argument. P. 1
The Preface of the kinds of Dials, and Theorems premised. Page 2
How to make the Polar or Aequinoctial Dial, and how to place it. Page 5
How to make the Aequinoctial Dial, or Polar Plane, Geometrically, and by Calculation.
Page 8
How to make the East Aequinoctial Dial, or the West, Lat. 51 d. 30 m. Page 9
How to make a Vertical Horizontal Dial. Page 11
How to make a South inclining 23 deg. in Latitude 51 d. 30 m. Page 14
How to observe the Declination of any Declining Plane. Page 15
How to take the Declination of any Wall or Plane, without the help of a Needle or
Load-stone. Page 16
How to make a Declining Horizontal Dial, or South erect declining from the South Eastward.
Page 17
[Page]To find how much time the Substiler is distant from the Meridian, or Inclination
of Meridians, Geometrically and by Calculation. Page 18
How to draw the Hour lines in a Declining Horizontal Plane, or South Erect declining
32 d. 30 m. from the South Eastward. Page 19
How to observe the Reclination or Inclination of any Plane. Page 21
How to draw Hour-lines in all Declining Reclining Inclining Planes. ibid.
How to describe the Sphere or Diagram. Page 22
How to make a North or South Reclining Dial Page 23
How to make an East and West Reclining or Inclining Dial. Page 25
How to find the Arches and Angles that are requisite for the making of the Reclining
Declining Dials. Page 27
How to draw the Reclining Declining Dial. Page 30
How to find the Horary distance of a Reclining Declining Dial. Page 31
How to know in what Country any Declining Dial shall serve for a Vertical. Page 33
How to find the Arches and Angles which are requisite in a North Decliner Recliner,
and a South Decliner Incliner. ibid.
How to draw the Declining Inclining Dial. Page 36
How to know the sundry sorts of Dials in the Fundamental Diagram of the Sphere. Page 37
How other Circles upon the Sphere may be described upon Dials, besides the Meridians.
Page 38
How to describe on any Dial the proper Azimuths and Almicantars of the Plane. Page 39
How to deal with Declining, Reclining, or Inclining Planes, where the Pole is but
of small Elevation. ibid.
How to inlarge the Hours of any Plane. Page 40
How to make a Vertical Dial upon the Cieling of a Floor within doors, where the direct
Beams of the Sun never come. Page 42
A Table for the Altitude of the Sun in the beginning of each Sign, for all the Hours
of the Day, for the Latitude of 51 d. 30 m. Page 44
How to make an Ʋniversal Dial on a Globe, and how to cover it if need requires. Page 45
How to make a North Dial for the Cape of Good Hope, in South Latitude 35 deg. and Longitude 32 deg. 54 min. to the Eastward of the Meridian of the Lizard. Page 46
How to find the Hour of the Night by the Moon shining upon a Sun dial. Page 48
How to find the Hour of the Day or Night by a Gold Ring and a Silver Bowl, or Brass,
Glass, or Iron Vessel. ibid.
How to Paint the Dials that you make, and fasten the Gnomons in Wood or Stone. Page 49
The Ʋse of the Tables of Artificial Sines and Tangents. Page 50
The Ʋse of the Logarithme Numbers. Page 51
Next follow the Tables.
After them is an Abridgment of Custom-Laws in Navigation.
And last of all is annexed, A Compendium of Fortification both Geometrically and Instrumentally.
TO THE Truly Industrious, and Highly Deserving of English-men Captain SAMUEL STURMY. On his Excellent and Elaborate Treatise, Entituled, THE MARINERS MAGAZINE, &c.
A FRIENDLY ADVERTISEMENT TO THE Navigators and Mariners of ENGLAND.
BRETHREN,
THe Duties of a Friend and the Properties of a Flatterer do differ so greatly, that
a Man cannot perform the Office of the one, but he must renounce the Practice of the
other; And a very Fountain it is, from whence many Mischiefs do spring and overflow
the wretched Life of Mankind, that the true dealing of Friends is most commonly unpleasant
and hateful; but the soothing of Flatterers is become plausible, and much set by:
In resemblance they bear many times like shew; but in purposes they always differ.
A true Friend will sometimes commend and praise divers things in his Friend; and so
will also the Flatterer, in those whom he followeth. The one commendeth that which
in Judgment he thinketh commendable, to the end that his Friend should still proceed
in Actions worthy of Commendations; the other commendeth even those things many times
which in his heart he doth detest, to the end that he may sooth up the Humour of the
Party. A faithful Friend, what he disallows in his Judgment of his Friend, he will
be earnest with him to see the fault, to the end the Party may amend, and give no
advantage to his Enemy: The Flatterer sometimes, though seldom, will also discommend,
but evermore trifling matters; fearing to offend the Party, if he should touch him;
so counterfeiting sincere Love (the Badge only of true Friendship) and so leaveth
the Party, thus abused, to the scorn and reproach of the Adversary, reaping the Commodity
which he looked for, as the only end of his desire.
I do not think that there is any Man, that either regardeth Gods Glory, or esteemeth
of Humane Society, but holdeth our Art worthy to be numbred with the most excellent
that are exercised among men; and therefore it is great reason the Practicers of it
should be had in greater reputation than they be now adays. Neither is there any other
Art wherein God sheweth his Divine Power so manifestly, as in ours; permitting unto
us certain Rules to work by, and increasing of them from time to time, growing still
onwards towards perfection, as the World doth towards its end; and yet reserveth still
unto himself the managing of the whole, that when we have done what we can, according
to the Skill we have already, or may have by any thing that we may learn hereafter,
yet always will God make it manifest, That he alone is Lord and Ruler of Sea and Land;
That all Storms and Tempests do but fulfil his will and pleasure, who oftentimes administreth
many helps, beyond all expectation, when the Art of Man utterly faileth; which is
lively expressed in Psalm 107. where is nothing omitted which is necessary, nor any thing affirmed but that
which the continual experience of our daily dangers do proclaim to be true.
O that we that see his wondrous Works in the Deep, would therefore praise the Lord
for his Mercies, and shew forth his Wonders before the Children of Men; that we might
once learn, That the Fear of the Lord is the beginning of Wisdom. Most undoubtedly then would our Art flourish, our Voyages prosper, and have better
success; yea, our selves would be more esteemed and honoured of all men. Whereas now
the profane Lives, and brutish Behaviour of too too many of our Trade, doth somewhat
eclipse the Glory of the Profession it self. Besides other manifold punishments, God
striketh some of us with the Spirit of Blindness, as no men living, of any Trade whatsoever,
are to be found so ignorant as many of us are: so senseless are we in our own defects,
[Page] so little desirous to amend them, Yea, and some of us of the greatest Skill and Practice
are so loth to give God his due Glory, that many times labouring to suppress it, we
make Shipwreck of our own Credits and Reputations, which otherwise of right might
accrew unto us. When we have performed a long Voyage, of great difficulties, wherein
many a time and oft we have been at our wits end, and knew not which way in the World
to turn our selves, God delivering us beyond our expectation, as our Consciences can
witness; yet when the danger is once past, and that home we be come, we take it as
a blemish of our Estimations, and a great Impeachment to our Credits, to give God
the Praise, and yield him Thanks; imagining that would derogate too much from the
Admiration which we so greedily hunt after among Men. But let me give you one Example
of this Ingratitude to God, on a Voyage from the West-Indies, in the Society of Topsham, a Ship that I had command of. It pleased God by a violent Storm and Sea, 500 Leagues
from England, we lost all our Masts, and were several times like to founder our Ship: It pleased
God that little Provision we made for Sail, and the mischievous Storm continuing,
turned to our good; for the Wind was fair, but the Sea so dangerous and grone, that
we could not Scud or Sail but sometimes; but in good time it brought us safe into
the foresaid Harbour of Topsham. And in our distress our Men were very mindful of Prayer, as all are; but coming to
our desired Port, I desired them to return our gracious God Thanks, with me, for our
great Deliverance: Some were willing, but two refused; whereupon I told them, That when they were next in distress, it may be God would refuse them help or deliverance: And so it fell out; for William Witheridge of Kenton in Devonshire was drowned at Bilboa the next Voyage following, and the other was drowned at London. Therefore let me advise all, to have a care not to be ungrateful to God.
We of this Nation are too much given to admire Strangers, and contemn our own Country-men.There are many men that perform long Voyages God knoweth how, but not they themselves;
yet will swear and stare, crack and boast. That they have done all things according
to Art; and tell a Tale to Strangers at home, of such Gulfs and swift Currents, more
than ever God made, to shadow their Ignorance, and rob God of his Praise: But yet
for the Navigators and Mariners of England, I do hope and verily believe in my Conscience, That divers of them do fear God unfeignedly,
and do as much dislike the dissolute course of the common sort, as any men can: And
I do nothing doubt, although the number of such are too few in our Nation, yet are
they more than any Nation in the World can shew besides. However, two things are greatly
wish'd by all our Well-willers; An Increase in us of the true Fear of God, and a careful
Diligence in us in things belonging to our Art. Where the Fear of God is not, no Art
can serve the turn; for that were to make of Art an Idol: And yet all those that fear
God, must take heed that they do not tempt God; and therefore ought they to use Art,
as the means that God hath ordained for their benefit, and be thankful unto him for
it. Farewel.
Yours, SAMƲEL STƲRMY.
From my House and Study at St. Georges, or Pill, neer Bristol, November 12. 1667.
IN a Work of this Nature it is impossible to escape Mistakes, the which Man-kind could
never totally evade since the first lapse of his Great-grandfather Adam. I hope I shall obtain your pardon therefore, though this present Tract hath some
few Typographical Errors, which yet are not many, nor considerable, though the Author
were far remote from the Press all the while. I have here given thee notice of as
many as I could here readily espy; If thou findest any other, I desire thy favourable
Censure, and that thou wouldest correct these in manner following.
In the First, Second, Third, and Fourth Books.
Page 16. line 3. for when I at Sea, read were I at Sea. l. 35. for haft of r. hawl aft, p. 17. l. 22. for all r. hawl l. 35. for therefore r. the fore, l. 41. for Mast r. must, p. 18. l. 5. for laught r. taught, l. 45. for laught r. taught, p. 19. l. 3. for Private r. Privateer, p. 20. l. 17. for Guns r. Gunner, p. 34. l. 6. for flat r slat. p. 54. l. 10. for inexed r. invexed, p. 61. l. 10. in the Table of Sines against 40 m. put in 112, p. 140. l. 4. for first r. fifth, p. 145. the last Course but one upon the Log-board, for SE, B r. SE, L; the last Course on the Log-board, for SE 9 knots r. SE o knots, p. 146. l. 47. for the wind at WSW and ESE r. the wind at SSE and SSW the Ship made Leeward way, p. 147. in the Table, at the fifth Course, for N by W r. NNW, p. 160. l. 52. for NS r. NE, p. 162. l. 3. for Anabically r. Analytically, p. 174. l 40. for 51 deg. r. 51 parts, p. 117. l. 4. for legree r. degree, p. 195. l. 46. for 20 deg. of Longitude r. 20 deg. 30 m. of Latitude, p. 200. l. 38. for let r. set.
In the Fifth and Sixth Books.
P. 2. l, 44. for Veracter r. Peractor, p. 4. the Square ABCD should be a Geometrical Square of equal sides, p. 7, 8, 9. some figures in the Sums
of Division are set out of order, p. 24. l 11. for rotract r. protract, p. 80. l. 23. in marg. for having done r. hanging down, p. 103. l. 1. for convex r. concave.
In the Seventh Book.
In tit. p. for Gnomical r. Gnomonical, a. 1. l. 5. for Gnomical r. Gnomonical. p. 8. l. 23. for CH r. CG, p. 12. in the Dial there wants the letter E toward the top of the Stile, p. 13. this should be added to Chap. 8. for the North
and South erect Dial.
To Calculate the Hour-Lines.
As the Radius,
To the Stiles Height 38 d. 30.
So the Tangent of the Hour,
To the distance from the Meridian.
Hours.
Stile 38 deg. 30 min.
Angle of the Hour.
Arch on the Plane.
d. m.
d. m.
12
0 0
0 0
1
15 0
9 28
2
30 0
19 54
3
45 0
31 54
4
60 0
47 9
5
75 0
66 42
6
90 0
90 0
p. 18. in the Figure, C is wanting in the Center, p. 19. l. 7. for B [...]. H, and some small Letters are wanting in the Hour-Scale. p. 23. l. 7. for FW r. EW, l. 8. for continue r. contain, l. 21. for third Point r. three Points, p 28. l. 1. for FLc r. FLe, p. 30 P is wanting at the edge of the Stile, p. 31. l. 34. for os r. as, for FORCQ r. FOReQ. p. 34. in the Dial, for e r. C, p. 40. in the Dial, the Letter I is wanting at the Intersection of the Line FBA with the Hour-line of 5. p. 40. l. 14. for R r. little r. l. 16. for ER r. E r. p. 41. in the Dial, for K make B. p. 43. in the Dial, for 4 make q.
I thought Good to advertise those that have occasion for any Instruments mentioned
in this Book, or any other for the Mathematical Practice, either in Silver, Brass,
or Wood, they may be exactly furnished by Mr. Walter Hayes at the Cross-daggers in Moor-fields, next door to the Popes Head Tavern, where they may be furnished with all sorts of Carpenters Rules, Post and
Pocket Dials for any Latitude, at reasonable Rates.
Edward Fage at the Suger Loaf in Hosier Lane London Fecit, who makes all sorts of
Mathematcall Instruments.
The SCALE of SCALES
These Naturall and Artificiall Scales are described at large with their Fundamentall
Diagrams in the Second Booke of this Tretise, and their Use exemplified in the Resolution
of all Mathematicall conclusions, with many other Instruments, the makeing and necessary
use whereof is Demonstrate.
The Compleat MARINER, OR NAVIGATOR. The First Book.
CHAP. I. The Argument or Description of the Art of Navigation in general.
NAVIGATION of all Arts and Sciences (setting Divinity aside) hath much reason to have the preheminence, it being of such necessary and
publick Concernment; and what use there is made of it by Seamen at this present, as well as hath been in times past, All men know, to whom the Countries
are beholden for their good Service, whose Courage hath kept Great Britain, Queen and Regent of the Sea, and deserves it well, in respect of the Skill and Valour of her Mariners,
and Goodness and Number of her Ships. I wish as long as the Sun and Moon endures,
That they may maintain their Courage, and improve their Art, as they ever have, against
all Nations that have been England's Enemies; and ever may they crown their Undertakings with everlasting Credit.
The Art of Navigation being such, I think I may be bold to affirm without presumption, This Art is more
necessary for the well-being and honour of our Nation, than any other Art or Science
Mathematical, which is more carefully kept in the Universities. Look upon Grammar, Rhetorick, and Logick, these are but Introductions to other Arts; Musick is but of little use.
The chief Professions now in the Universities are Physick and Law. Without envy be it spoken, we may as well live as the ancient Romans without Physicians, and as honest Neighbours without Lawyers, better than without
skilful Seamen, which are the chief Importers of our Wealth, and Supporters of our Warfare.
Besides that, of all Mathematical Sciences and Arts professed in the Universities,
of this Art of Navigation is made the most general and profitable use; for what can the Scholar make of his
Geometry, with all the nice and notional Problems thereof: or of Astronomy, with all his curious Speculations about the motion of the Planets, without they be
applied to some more Mechanical and Practical Arts, as Cosmography, Geography, Surveying, Dyalling, Architecture, Military Employments, which shall in some measure (sufficient for the help of Mariners) be shewn in the
following Treatise, wherein it will appear, That the Art of Navigation comprehends them all in the use thereof?
And those that will be compleat Sea-Ariists, had need to endeavour to have some skill and understanding in most of these Arts,
namely, the Theorick and Practick parts, whereby they may be fully informed of the
Composition of the Sphere in general; and in particular for the Figure, Number, and Motion made in the Heavens by [Page 2] the highest Moveable called Primum Mobile, and likewise of the first, fourth, eighth, and ninth Heavens. It will also inform
them how the Elements are disposed, with their quantities and scituations, especially
in the Composition of the Sphere of the World, which is commonly understood to be
the whole Globe of the Heavens, with all that therein is contained; which is divided
into two parts, Elemental and Coelestial. The Elemental hath again four parts, viz. The first is the Earth, which together with the Water, as the second, maketh a perfect Round Globe, whereupon we dwell; therefore the Nature
and Circles which are supposed to be contained in that Sphere, are fit to be known. The next is the Air, comprehending the Earth; and the fourth the Fire, which according to the opinion of Philosophers containeth the space which is between
the Air and the Heavens, or Circle of the Moon. Out of these Elements, which are the
beginning of all things that are subject to change, together with the warmth of the
Heavens, all things do come forth, and decay, as we see and find upon the Earth, by
the continual Change and Motion of the One into the Other.
The Coelestial part (containing within the concavity thereof the Elemental) is transparent and perspicuous, shining, severed, and free from all mutability;
and is divided into eight Spheres, or round hollow Globes, which are called Heavens, whereof the greatest doth contain
the next unto it Globe-like; the seven Inferior have in each of them but one Star
or Planet only, whereof the first (the next to the Earth) is the Heaven of the Moon; The second of Mercury; The third of Venus; The fourth of the Sun; The fifth of Mars; The sixth of Jupiter; The leventh of Saturn; And the eighth of all the Fixed Stars. The number of these Heavens are known by their Courses round about the Poles of the Zodiack. The Moon runneth through her Heaven by her Natural Course from the West to the East in 27 days 8 ho. Mercury, Venus, and the Sun, their course in a year, and some less than a year; Mars his course in two years, Jupiter in 12, and Saturn in 30 years; The Eighth Heaven, according to the Observation of Tycho Brahe, in 25400 years.
These Heavens are carried all together in 24 Hours upon the Poles, about the Axletree of the World
thorow the ninth Heaven, by vertue of the Primum Mobile, that is, the First Moveable; by which Motion to our appearance is caused the Day
and Night, and the daily Rising and Setting of the Coelestial Lights; But more of
this in another place, for here I have made a Digression. So that no Art is more capacious;
and were the Excellency well understood, and put in practice, as it might be (as Mr.
Philips saith in the like case) no Employment would be more honourable and advantageous for
the most generous Gentleman, and Learned Student, than this of Navigation; thus it was in esteem in the days of Queen Elizabeth,
When Drake and Candish Sayl'd the World about,
And many Hero's found new Countries out,
To Britain's Glory, and their lasting Fame;
Were we like-minded, we might do the same.
The Practick part of Navigation is properly placed in making and using of Instruments, which is shewn in the second
Book. Yet there is a certain Composition in the Practick, more rare than all the rest,
in the compleat Sea-Artist; and that is the right Words and Phrases used in guiding,
governing, and constraining, to perform the expert Navigator's pleasure in the Sea;
In ruling the unparallell'd Fabrick of a gallant Ship, which hath been omitted by
most men that have writ of this Art; therefore I shall explain it with my Pen, because
I know with proper Phrases how to perform it, not hindring any other, as they not
me, to shew truly and lively their Skill in controlling, guiding, and working a Ship,
according to all Weathers at Sea; although it be of no use to Sea-men, that have been
all their life-time at Sea: but for Gentlemen on Shore to read for their Recreation,
the Words of Command at Sea, which may be delightful unto them. But for experienced
Sea-men, they have all those things imprinted in them, and make use thereof according
as their business shall fall out at Sea, but after the same manner.
In regard all Arts and Sciences are divided into two principal parts, that is, the
Theorick and Practick, I tooke upon me to demonstrate according to my ability, [Page 3] which will give the most reasonable men satisfaction; for the unreasonable, I care
not a fig for them; for I know it to be impossible for any man to be a compleat Seaman,
wherein this Knowledge is wanting, they being both inseparable Companions which always
wait upon Perfection. I shall draw out the Description in as small a compass as it can be, and hasten to
the most material Practice.
CHAP. II. Of what is needful first to be known in the Practick Part of the Compass, and how to divide the Circles and Parts.
THE principal Hand-maids that expert Sea-men are furnished with, that their Undertakings
may be crowned with everlasting Credit, are these, viz. Arithmetick, Astronomy, Geometry, and the Mathematiques. By the operation of these loving Sisters, and excellent Arts, as hath been said,
Navigation is daily practiced by expert Sea-men: but much abused by hundreds of ignorant Asses,
that know nothing what belongs to them, yet do undertake Voyages, to direct a Ship
navigable upon the Terrestrial Globe, resting wholly upon favourable Fortune, which
hath made some of them famous; but many times disasterous Periods have ended their
Undertakings, with the loss of many mens Goods and Lives; which yet I must confess
have and do happen to the best, but not so often as to them by great difference.
But to come to the Substance of what is here intended, I would have it to be understood,
That he that intendeth the Art of Navigation, hath Arithmetick in readiness. If he want it, he may be instructed by divers Books now extant, as
Record, Blundevill, and Mr. William Leybourn's Arithmeticks. As for the Mathematical and Astronomical Knowledge, so much as is
useful for Sea-men, will be shewn in the Projection and use of divers Instruments,
which will after follow in its due place. In this Treatise we will come to the Sea-Compass,
that we may proceed in a regular form. The knowledge of it is the root of that famous
Art we chiefly treat of, and presents himself as the first Principle framed by God
in the Operation and Nature of the Magnet, which being in its quality beyond our capacities, yet it is the first thing to be
learned and understood, it being the foundation to all the following Conclusions,
and is first taught to our Youths and Boys which are intended for Navigators. They
are taught first to know the Point on the Card, and by what Name it is called, and
to say it perfectly backwards and forwards; and to know that to every Point of the
Compass there is allowed for Time ¾ of an Hour, which is 11 Degrees 15 Minutes; and
how to number the Hours from the North and South, either Eastward or Westward, readily
to answer as soon as demanded: As also to know how the Ship Capes; that is, to know
the Point of the Compass that looks straight forwards to the Head of the Ship: As
likewise to know upon what point of the Compass the Wind blows over; that is, if the
Wind be at North, it blows over the Flower de Luce toward the South; and so o [...] the rest. So we teach them to know what Point the Sun is on; That in England a South-east Sun on the Aequator makes 9. 24 of the Clock; and when he is South, makes 12 of the Clock; and South-west,
2. 36 of the Clock. As also they learn to set the Moon in the same manner on the Full
and Change-days, to know the Tides by, as shall be shewed.
The Compass we Steer our Ships by, is only a Circle of some 8 inches diameter; and
is divided into 32 Points, which have several denominations, as you may see expressed
in the Figure. The whole Circle is divided into 360 degrees, and 24 hours: The Compass
contains also 16 distinct Rhombs or Courses; for each several Course hath two Points
of the Compass, by which it is expressed. As for example, Where there is any place
scituated South-east, in respect of another place, we say the Rhomb or Course that
runneth betwixt them, is South-east and North-west: or if it bear South or North,
so we call it: or if West, we say West and East. The Compass swings in the Boxes,
the Wyers first well touched with a good Load-stone, and the Chard swimming well on
the Pin perpendicular in the middle of the Box; it represents [Page 4] the plain Superficies or Horizon as we call it, when looking round about us at Sea, and see the Heavens make intersection
with the Waters, sheweth that you are in the Center, and that every place in the Horizon is equally distant from you: So that when you espy any Island, Rocks, Ships, or Cape-Lands,
by looking straight upon the Compass, you shall know upon what Point of the Compass
the Object beareth from you. But we will haste to shew the young Practitioners the
Sea-Compass, with the 32 Points, expressed by the Letters upon each Line, and also
how to make it, as followeth.
The COMPASS.
How to divide the Circles of the Mariners Compass.
FIrst draw a Line at pleasure, and cross it in the midst with another Line at right
Angles; Then in the crossing of these two Lines set one foot of your Compass, and
open the other to what distance you please, and with that distance draw the Circle,
which by the cross Lines of East and West, North and South, are divided into four Quadrants and equal parts, each of them containing 6 hours
a piece; set VI at East and VI at West, XII at North, and XII at South, so have you the four first Divisions of your Figure: Then keeping your Compass at
the same distance as you draw'd the Circle, set one foot in the crossing of the Line
and the Circle at East 6, with the other make two marks, one of II, and X. Then set one Foot in the West at 6, on the other side mark out the hours of II and X, as before; keeping the Compasses
still at the same distance, set one Foot at South XII, and with the other you shall
mark out the Hours of VIII and IIII. Then set one Foot of your Compasses at North at XII, and in the same manner mark out the Hours of VIII and IIII. Thus the Circle
is divided into 12 equal parts, and each of them contains 2 hour [Page 5] piece; so that it will be easie for you to divide each of these into two parts; which
done, you have the 24 hours. Lastly, you may divide each hour into 4 equal parts,
which will be quarters of an Houre, as you may see in the Figures.
To divide a Circle into 360 equal parts, is a thing very necessary; for in all Questions
in Astronomy, and in the Calculation of all Triangles, these parts are the measure of the Angles: so that in respect of this, every Arch
is supposed to be divided into 360 equal parts or Degrees; and every Degree is supposed
to be divided into 60 lesser parts, called Minutes. To divide a Circle after this
manner, draw a Line at pleasure, and cross it at right Angles with another Line, and
draw a Circle as before. Keep your Compasses at the same distance, and divide the
Circle from the 4 Quarters into 12 equal parts. Then closing your Compasses, divide
each of these into 3; so you have in all 36 parts. Then you may easily divide with
your Pen each of these parts into 10 little parts, as you may see in the middle Circle
of the Figure, which are Degrees.
For the 32 Points of the Compass, draw the Line of North and South, and cross it at right Angles with the Line of East and West, and draw the Circle, as before; and with the same distance, set one Foot of your
Compasses at East, and with the other draw a small Arch at A and B, and cross it from North to South with the same distance; the like do from the West Point to C and D. Then laying your Rule cross-ways to these Crosses, draw the Line
BD and AC; so is your Circle divided in 8 equal parts. Then closing your Compasses,
you may easily divide these 8 parts into 4; divide one, and that distance which will
divide all the rest into equal parts, if you have followed Directions. And so you
have the 32 Rhombs or Points of the Compass; and so you may subdivide these Points
into halves and quarters, as you may see in the Figure. So have you made the Mariner's
Sea-Compass. The Use shall be shew'd in its place.
The Figure of the Compass, and the Traverse Quadrat, joyned both together.
The Traverse Quadrat sheweth the making of the Traverse Table, in Chap. 3. Of Sayling by the plain Sea-Chart.
The Moons Motion, and the Ebbing and Flowing of the Sea.
THe Practitioner in Navigation, is next to learn to know the certain time of the Flowing and Ebbing of the Sea; In
all Ports called by Sea-men the shifting of Tydes, which is governed by the Moon's Motion, as it is found by experience.
Wherefore I will first shew the use of a small Instrument, which is here framed, whereby
the meanest Capacity (which is void of Arithmetick) shall be able to know the Age of the Moon, with what Flood and Ebb it maketh in
all Channels, and in every Port and Creek at High-water; and also be able to know
what a Clock it is at any time of the Night; and divers other Questions, only by moving
the Instrument, according as shall be directed.
I shall also shew you, how you may do all these Questions of the Tyde by the Moon
Arithmetically: But first by your Instrument it must be projected according to the
following Figure, which you may make of three pieces of Board, well planed, and exactly
divided, according as you see it formed in the Figure. The outward Circle, being the
biggest Board, hath 32 Points of the Compass; The inward Circle on the same Board
is divided into 24 Hours, being the thickest Board. The second Circle must be divided
into 30 equal parts, representing the distance 30 times 24 Hours, or 30 natural days,
attributed to the Sun. The uppermost Circle of the three, is attributed to the Moon,
with an Index as that of the Sun, and to be turned and applied to either the 30 days,
containing the Computation of the Moon betwixt Change and Change, or the 24 hours;
as likewise to the Points of the Compass. And so may the Index of the Sun be applied
either to Time, or the points of the Compass, which shall be made plain by the following
Questions; which will appear delightful and easie; and the illiterate man will find
in most useful; and he that hath better Knowledge, will sometime use the Instrument
for variety sake. First, for the Figure of the Instrument.
A Ʋseful Variation-Compass.
UPon the two upper Circles of the Instrument I have set a most useful Variation-Compass, easie to be understood, and as exact as any Instrument whatsoever for that purpose.
You shall have full direction how to use them in the following Discourse, when we
come to treat of the Variation of the Compass. But this observe, The midle Compass
representeth the Compass you steer your Ship by, which is subject to Variation; but
the upper Compass and Circle representeth the true Compass, that never varieth; and
by it you may very readily know the Variation of the Steering Compass, how much it
varieth from the true Point. The inward Circle of the middle Compass is divided into
the 32 Points, with their halves and quarters: and likewise the outward Circle of
the smaller or upper Compass. This is too hard for Practitioners at first to know
how to use this Instrument, to rectifie the variation of the Compass; therefore I
shall be no longer on this Discourse, and proceed to what was promised, and shew the
farther use of it afterward.
PROPOSITION I. The Moon being 16 days old,Here the Variation Compass is to be placed. I demand upon what Point of the Compass she will be at 8 of the Clock at night.
In all Questions of this nature you have the Hour and Time given, and the Moons Age,
to find the Point of the Compass. To answer these Questions, place the middle Index
of the Sun on 8 of the Clock at night; then bring the upper Index of the Moon right
over the 16th. day of her Age, in the middle Circle of the Sun, and the Index of the
Moon or upper Circle points to E. b. S. half a point Southerly, the true Point of the Compass the Moon will be, when she
is 16 days old, at 8 of the Clock at night.
that Point of the Compass, by turning the Index of the Moon, as before was shewed:
So you may be sure to have the Hour always under the Index, on the Change-day, throughout
all the Points of the Compass; and so we shall proceed to Examples.
PROPOSITION VII. The Moon being 16 days old, I demand, What a Clock it will be Full-Sea at Bristol, Start-point, Waterford, where an East-by-South Moon on the Change-day makes the Full-Sea?
You are to consider the Point of the Compass the Moon is upon in these Ports, when
it is Full-Sea on the Change-day (as in all other Ports) which in these Ports is found
by observation to be always East-by-South Moon (which is 6 hours 45 minutes) Then
consider whether it be the Day or the Night-Tyde you would know the Time of Full-Sea;
if it be the Morning-Tyde, bring the Index of the Moon to the West-by-North Point,
staying it there; bring the 16th. day of her Age under the Index of the Moon, and
the Index of the Sun will point you to 7 of the Clock and 33 Minutes, the time of
Full-Sea in the Morning. If it be the Evening Tyde, bring the Index of Luna to the East-by-South, and stay it there, until you have brought 16 Days and half
under the Index of Luna, and the Index of Sol will point directly upon 8 of the Clock at Night, the time of Full-Sea in the aforesaid
Ports: Thus you see there is 27 Minutes difference in every Tyde in these Ports. So
you may know in every other Port in the same manner, if you do as before-directed,
and allowing half a day more for the Night-Tyde, by turning of it half a day further.
And take this for a Rule, That the Moon betwixt Change and Full is ever to the Eastward
of the Sun, and riseth by day, still separating it self from the Sun until she be
at the Full: Then after the Full, in regard she hath gone more Degrees in her separation
than is contained in a Semicircle, she is gotten to the Westward of the Sun (rising
by night) and applieth towards the Sun again until the Change-day, which you may see
plainly demonstrated by the Instrument.
PROPOSITION VIII. The Moon being 16 days old, I desire to know at what hour it will be Full Sea at London, Tinmouth, Amsterdam, and Rotterdam, where a S. W. and N. E. Moon makes a Full-Sea upon the Change-day.
It is found by observation, That the S. W. and N. E. Moon makes Full-Sea in all the
aforesaid Ports. You may know the Moon is to the Eastward of the Sun that is before
him; Bring the Index of the Moon to the S. W. Point; then turn the 16 day of her Age
under the Moons Index, and the Index of the Sun answereth the Question, That it is
Full-Sea at all the aforesaid Ports at 3 of the clock and 48 minutes in the morning,
the Moon being 16 days old.
PROPOSITION IX. At Yarmouth, Dover, and Harwich, where a S. S. E. Moon maketh Full-Sea on the Change-day, the Moon being 9 days old, I demand the time or hour of Full-Sea that day in the aforesaid Places.
Here it hath been found by experience, That a S. S. E. Moon makes Full-Sea on the
Change-day, in the aforesaid places; therefore bring the Index of the Moon to the
S. S. E. Point, keep it there fast directly on the Point, and bring the Moons Age
to cut the edge of the Moons Index, and the Index of the Sun will shew you, That the
time of Full-Sea in the aforesaid Ports, will be at 5 a Clock and 42 minutes in the
morning.
PROPOSITION X. At St. Andrews, Dundee, Lisbone, and St. Lucas, where a South-West-and-by-South Moon makes High-Water or Full-Sea on the Change-day;
The Moon being 28 days old, I demand the time of Full-Sea that day in these Places.
You have here given you the S. W. b. S. Point of the Compass; therefore bring the Index of the Moon, and stay it on that
Point, and bring the 28 day of her Age under the edge of the Index of the Moon, and
the Index of the Sun will point you out the time of Full-Sea, which is at 39 minutes
past 12 of the clock at noon, in the aforesaid places. And so are all Questions of
this nature answered. And so I will conclude the Use of this Instrument, for finding
the Ebbing and Flowing of the Tyde, and so will proceed to shew you Arithmetically
how to find the Golden Number or Prime without a Table, the Epact, and Full, Change, and Qu [...]rters of the Moon, and how to know her Age for ever; and what Sign and Degree she is in the Zodiack, how long the Moon shineth, and what time of the day or night it is High-Water or Full-Sea in any Port;
and also the Moon's Motion, as far as it is useful for Mariners.
I How to find the Golden Number or Prime, according to the Julian, English, or Old Account.
YOu may observe this, That the Prime or Golden Number is the space of 19 years, in which the Moon performeth all her Motions with the Sun: At the expiration of which Term, she beginneth again in the same Sign and Degree of the Zodiack, that she was 19 years before; and always finisheth her Course with the Sun, and never exceedeth that Term. To find this useful Number, you must do thus; Always
in what year you would know what is the Prime Number, add 1 to the date thereof, and then divide it by 19, and that which remaineth upon
the Division, and cometh not into the Quotient, is the Number required. As for example,—
In the year of our Lord 1665. I demand what is the Prime Number. Add to the year of our Lord always 1, which makes in this Question 1666. Then divide
that sum by 19, the remain is the Prime or Golden Number, as you may see by the Work, which answereth the Prime or Golden Number for this present [...] year to be 13, it being left out of the Division that cometh not into the Quotient.
Thus you see it is very easie to do it for any other year. Observe, That when you
find nothing remaining upon the Division, that is the last year of the Moon's Revolution, and may conclude, that 19 is the Prime for that year. Note, The Prime beginneth always in January, and the Epact in March.
II How to find the Epact, according to the Julian, English, or Old Account, and what it proceedeth from.
THe Epact is a Number that proceedeth from the difference which is made in the space of one
whole year, between the Sun and the Moon. Note, The Solar year doth contain 365 days, 5 hours, 48 minutes; and the Lunar year, allowing 12 Moons, there being 29 days, 12 h. 44 min. between Change and Change, doth contain but 354
days, 8 hours, 48 min. So that there is almost 11 days difference between the Revolution
of the Sun and Moon, at every years end; which difference makes the Epact. Therefore to find the Epact for any year, first you must know the Prime Number for that year, which we found before for the year 1665. to be 13. Then you must multiply
this Prime Number 13. by the difference 11. and it will make 143. which divide by 30. and there remaineth
of the Division, that cometh not into the Quotient, 23. which is the Epact for the year required. [...] So I make no Question but that you understand how to find the Prime and the Epact for any year past, present, or to come. Therefore I hold this sufficient to express
so facile a thing as this is. I have told you already, That the Epact always beginneth in March; but I shall make a small Table for those that are ignorant in Arithmetick, and cannot find these two Golden Numbers, as I may call them, for 45 years to [Page 10] come, where any one may find the Prime and Epact most readily in any year you shall desire.
III A Rule to find the Change, Full, and Quarters of the Moon.
ADd unto the Epact of the year proposed all the Months from March, including the Month of March, and substract that sum from 30. the remain sheweth the day of the Change: But if the Epact be above 26. there this Rule faileth a day at the least; but at other times it will
be no great difference: Therefore it may serve for the following Conclusions.
As for Example, I desire to know the New-Moon in October, 1665. The Epact is 23. the Months from March are 8. which added makes 31. from it 30 substracted, remains 1. which taken from
30. one whole Moon, there remains 29. So that the 29th. day of October is the day of her Change, or New-Moon, which by exact Calculation it is at 58 min. past 4 in the morning.
Having thus found the time of the New-Moon, you may from thence reckon the Age of the Moon, and so find the Quarters, or Full-Moon.
Thus the Moon's Age is
Days
Hours
Min.
At the First Quarter
07
09
11
At the Full Moon
14
18
22
At the Last Quarter
22
03
33
At the Whole Moon
29
12
44
IV How to finde the Age of the Moon at any time for ever.
ADd to the days of the Month you are in, the Epact, and as many days more as are Months from March, including March for one; and if these 3 Numbers added together exceed 30, take 30 from it as often
as you can, and the remain is her Age: But if the Numbers added be under 30, that's her Age; As for Example, 1665. the Epact is 23. I demand, What Age the Moon is the 21th. day of September? From March to September is 7 Months, the Epact 23, and the day of the Month is 21. Added together, makes 51. From it substract 29,
because the Month hath but 30 days in it, and the remain is 22, the Age of the Moon that day. Had it been the 22th. of August, and added them together, it would have made 51. Then to have taken 30 out, there
had remained 21 for the Moon's Age the 22 day of August.
V To finde what Sign the Moon is in, by which is gathered, what the Moon differeth from the Sun.
MUltiply the Age of the Moon by 4. divide the Augment or Sum by 10, the Quotient sheweth the Sign the Moon differeth from the Sun; the Remain multiplied by 4, giveth the Degrees to be added. As for Example,—The Moon 22 days old, I demand what she differeth from the Sun? [...] Multiply 22 by 4, and the Product is 88. That divide by 10, and in the Quotient is
8, and 8 remaineth upon the Division: That multiplied by 4, is 32; from which take
30, the number of Degrees in a Sign, and add the 8 Signs in the Quotient, it makes 9 Signs. The odd 2, multiplied by 4, make 8 Degrees; to which add the Sun's Motion from his entrance into the Sign ♎ which was the 14 day, to the 21, make 7 days or Degrees to be added, to the 8 Degrees make 9 Signs 15; which counted after this manner, from ♎, saying, ♏ 1, ♐ 2, ♑ 3, ♒ 4, ♓ 5, ♈ 6,
♉ 7, ♊ 8, ♋ 9 Sign, and the odd 15 Degrees is 15 Degrees of Cancer. So the Question is answered, That the Moon is 9 Signs 15 Degrees from the Sun at 22 days old; which note, She differeth but 4 Degrees from her true Motion by the Tables, which is near enough for the Mariner to answer any Man.
How to find what Sign the Moon is in more exact; with the Moon's Motion for every day of her Age.
AStronomers divide the Compass of the Heavens into 12 Signs, which they set forth by these Names and Characters, which you must be a little acquainted
with, and the place of the Sun in the Zodiack. Each of these Signes you have them as followeth.
A Table shewing the Moon's Motion according to her Age.
D. Age.
S.
D.
M.
1
00
13
11
2
00
26
21
3
01
09
32
4
01
22
42
5
02
05
53
6
02
19
04
7
03
02
04
8
03
15
26
9
03
28
35
10
04
11
46
11
04
24
56
12
05
08
07
13
05
21
18
14
06
04
28
15
06
17
39
16
07
00
49
17
07
14
00
18
07
27
11
19
08
10
21
20
08
23
32
21
09
06
42
22
09
19
53
23
10
03
03
24
10
16
14
25
10
29
25
26
11
12
35
27
11
25
46
28
12
08
56
29
12
22
07
30
13
05
17
First know, That the Sun entreth the first Sign ♈ the 11th of March, ♉ the 11th of April, ♊ the 12th of May, ♋ the 12th of June, ♌ the 14th of July, ♍ the 14th of August, ♎ the 14th of September, ♏ the 14th of October, ♐ the 13th of November, ♑ the 12th of December, ♒ the 11th of January, ♓ the 10th of February. This known, the place of the Sun is well found, adding for every day past any of these, 1 Degree.
Thus you see, the Sun runs through these 12 Signs but once in a year; The Moon in less than a Month, viz. in 27 days, 7 hours, 43 minutes. Note, That every New-Moon, the Sun and Moon are in one Sign and Degree; but the Moon hath a Motion of about 13 Degrees every day, as is shewed in this Table. Therefore according to the Age of the Moon, add the Signs and Degrees of the Moon's Motion, to the place of the Sun at the New Moon, and so you shall have the Sign and Degree which the Moon is in at any time desired.
Thus for Example, A New-Moon 1665. the 26th. November, and the Sun and Moon are both in 14 Degrees of ♐. Now upon the 11th of December, the Moon being 14 days old, I would know what Sign the Moon is in. This Table shews, for the 14 days of the Moon's Motion, you must add 6 S, 4 D, 28 Min. to the said 14 Degrees of ♐.
Now counting those 6 Signs upon your Fingers, reckoning the Names of the Signs in order from Sagittarius, ♑ 1, ♒ 2, ♓ 3, ♈ 4, ♉ 5, ♊ 6, it falls upon the Sign Gemini. Lastly, adding the odd 14 Degrees unto the 4 Deg. of the Moon's Motion together, shews the place of the Moon to be in 18 Degrees of Gemini.
There is much use made of the Moon's being in such and such Signs, in Physick and Husbandry, of which I shall say nothing; but give you one Conclusion which much depends hereon;
that is,
To know the time of the Moon's Rising, Southing, and Setting.
FOr her Rising (know this) having found the place, or what Sign she is in, seek out in the following Kalendar what time the Sun is in this Sign and Degree, and there you shall find the true time of the Sun-Setting, being in that place: This is half the continuance of the Sun above the Horizon in that Sign and Degree. Add this to the time of the Moon's coming to the South, it shews the time of her Setting; and substracted from it, shews the time of her Rising.
Thus upon the 11th of September, as before, the Moon being 14 days old, and in the 18 Degree of Gemini, I desire to know the time of the Moon's Rising and Setting.
First multiply 14, the Moon's Age, by 4. Divide the Product by 5. In the Quotient will be 11 a Clock, and the one Unite
upon the Division is Min. 12, that the Moon will be South that night. Secondly, The Sun is in this Sign and Degree about the first day of June, and then sets at 8 a Clock 10 minutes past. This substracted, shews the Rising of the Moon to be at 3 of the Clock 2 minutes in the afternoon. The said 8 hours, 10 being added,
makes 19 hours 22 min. which by casting away 12, the remain shews the Moon's Setting to be at 7 of the Clock, and 22 min. past in the morning, which answers the Question
desired; which is as neer as can be for your use.
PROP. I. How to find when it is Full-Sea in any Port, Rode, Creek, or River.
I have shewed you already how to find the Prime, Epact, and Age of the Moon, at any time desired. Now we will proceed to shew you the finding of Full-Sea in any Place; as in manner following.— First, Carefully watch the time of High-Water, and what Point of the Compass the Moon is upon, on her Change-day, in that Port or Place where you would know the time of the Full-Sea, or find by the Table what Moon makes a Full-Sea in the said Port. Secondly, Consider the Age of the Moon; then by Arithmetick resolve it in this manner. Multiply the Moon's Age by 4, divide the Product by 5, the Quotient shews the Moon's being South. If any thing remaineth upon the Division, for every Unite you must add 12 Min. If
it was 4 remaining, it would be 48 Minutes to be added. Then add the hour that it
Flows on the Change-day to it, and the Total is the hour of Full-Sea. If it exceed 12, substract 12 from it, the remain is the hour of the day or night
of Full-Sea, in any Port, River, or Creek. Which I will make plain by some Examples, (viz.)
PROP. II. The Moon 16 days old, I demand, What a Clock it will be Full-Sea at Bristol, Start-point, and Waterford, where E. b. S. Moon maketh Full-Sea on Change-day?
Consider here an E. b. S. Moon maketh 6 hours 45 min. and the Age of the Moon is 16 days old: Therefore multiply the Age by 4, and it makes 64; divide that by 5, and it is 12, and 4 remaineth, which is
48 min. To it add 6 hours 45 min. E. b. S. it makes 19 ho. 33 min. Therefore substract 12 hours from it▪ there remaineth 7 a
Clock 33 minutes, the time of Full-Sea in the morning at the aforesaid Ports; which you may compare with your Instrument, and find it very well agree. [...]
PROP. III. The Moon being 25 days old, I demand, What a Clock it will be Full-Sea at London, Tinmouth, Amsterdam, and Rotterdam, where it flows S. W?
Consider that at these Places on the Change-days a S. W. Moon maketh Full-Sea, which is 3 hours. Therefore multiply 25, the Moon's Age, by 4, it makes 100. That divide by 5, in the Quotient will be 20, and nothing remain.
To it add 3 ho. S. W. and it makes 23 hours. From it substract 12, and the Remainder shews you, That it
will be Full-Sea at all the aforesaid Places, at 11 of the Clock in the morning. So you will find
it agree with your Instrument. [...]
PROP. IV. The Moon being 9 days old, I desire to know the hour of Full-Sea at Quinborough, Southam. and Portsmouth.
Note, That a South-Moon on the Change-day, maketh Full-Sea at these Places. Therefore multiply the Moon's Age by 4, it makes 36. That divide by 5, and the Quotient is 7 of the Clock; and 1 remaineth,
which is 12 minutes, the time of Full-Sea at the aforesaid Places, the Moon's Age being nine days. Note, If a North or South Moon makes Full-Sea on the Change-day, there is nothing to be added to the Quotient; but the Quotient is the hour of the
day, and the Remainder is the min. as before directed. One Example more shall suffice.
[...]
PROP. V. The Moon 5 days old, I demand the time of Full-Sea at Rochester, Malden, Blacktail, where S. b. W. Moon is Full-Sea.
Here you may note, That on the Change-day at these Places it flows S. b. W. which is but one Point from the South, being but ¾ of an hour, or 45 min. And it had been all one if it had been North-by-East. Multiply by 4, divide by 5, and the Quotient will be 4; to it add 45 min. S. b. W. shews you it will be Full-Sea at the aforesaid Places at 4 a Clok and 45 min. in the morning. But note, Had it
been S. b. E. or N. b. W. it had been 11 ho. 15 min. By this time I hope I have made the Practitioner able to know the time of Full-Sea in any Port, by Instrument and Arithmetick: Therefore I will leave him a small Table for his use.
A TIDE-TABLE.
H.
M.
Rye, Winchelsey, Culshot, a S. b. E. Moon.
11
15
Rochester, Malden, Blacktaile, S. b. W.
0
45
Yarmouth, Dover, Harwich, S. S. E.
10
30
Gravesend, Downs, Blackness, Silly, S. S. W.
1
30
Needles, Orford, South and North Fore-land, S. E. b. S.
9
45
Dundee, St. Andrews, Lisbone, St. Lucas, S. W. b. S.
2
15
Poole, Isles of Man, Dunbar, Diepe, S. E.
9
00
London, Tinmouth, Amsterdam, Rotterdam, S. W.
3
00
Portland, Hartflew, Dublin, S. E. b. E.
8
15
Barwick, Flushing, Hamborough, S. W. b. W.
3
45
Milford, Bridgewater, Lands-end, E. S. E.
7
30
Baltimore, Corke, Severn, Calice, W. S. W.
4
30
Bristol, Start-point, Waterford, E. b. S.
6
45
Falmouth, Humber, Newcastle, W. b. S.
5
15
Plimouth, Hull, Lyn, St. Davids, W. & E.
6
00
Quinborough, Southam. Portsmouth, N. & S.
0
00
Add any two Numbers together of the foregoing Table, and they shall be 12 hours; Except the two last, N, S. and E. W. So that you may perceive, what hath been said from the South, either Eastward or Westward, the same answereth to the North, either Westward or Eastward. And so much for the Tydes. But we will know the Moon's Motion, and the Proportion between Tyde and Tyde.
PROP. VI. The Motion of the Moon, and the Proportion of Time betwixt Tyde and Tyde.
After all this, I will shew you in brief the Motion of the Moon, and the reason of the difference between Tyde and Tyde.
You must note, the Motion of the Moon is twofold. First, A violent Motion, which is from East to West, caused by the diurnal swiftness of the Primum Mobile. Secondly, A natural Motion from West to East, which is the reason the Moon requireth 27 days and 8 hours 8 min. to come into the same minute of the Zodiack from whence [Page 14] she departed. But coming to the same Point and Degree where she was in Conjunction
with the Sun last, she is short of the Sun, by reason the Sun's Motion every day is natural East, 1 Degree, or 60 Minutes, which maketh so much difference, that the Moon must go longer 2 days, 4 ho. 36 min. nearest, more than her natural Motion, before she can fetch up the Sun, to come into Conjunction with her: So that betwixt Change and Change is 29 days, 12 hours, 44 min. by my account. The Mariners always allow just 30 days between the Changes, by reason he will not be troubled with small Fractions of Time, in this Account of
Tydes, which breedeth no great error: Experience therefore must needs shew me this, That I must allow the some Proportion to the Moon in every 24 hours, to depart from the Sun 12 Degrees, which is 48 min. of time, untill her full East; but then having performed her Natural Motion above half the Globe, she is to the VVest, as we may know by Reason. Now if the Moon move in 24 hours, 48 min. then in 12 hours she must move 24 min. and in six hours,
12 min. By this proportion each hour she moveth 2 min. So the Tydes differeth as the time differeth.
I will add one old approved Experience for the Mariners use, though it is impertinent in this place; that is, to cut Hair, the Moon in ♉, ♑, ♎: Cutting, shaving, clipping in the Wane, causeth baldness; but the best time in the Wane, is in ♋, ♏, or ♓. So I hope I have satisfied the Learner concerning the Moon.
THE Mariners Magazine; OR, STURMY's Mathematical and Practical ARTS. The Practick Part of Navigation, in working of a Ship in all Weathers at Sea.
WE have been shewing the Practitioner all this while, the Course and Motion of the Moon, and so by it to know how to shift the Tydes, or time of High-Water, in any Port, Road, Harbour, or Creek, Instrumentally and Arithmetically. The next thing to be observed by a Learner, is the Words of Command, with readiness to answer and obey, which is the most excellent Ornament that can
be in a Compleat Navigator, or Mariner. And as Captains Exercise their Men on Shore, that their Souldiers may understand
the Postures of War, and to execute it when the Word of Command is given by their
Commander; In like manner are Seamen ond Mariners brought up in Practical Knowledge of Navigation at Sea, in working a Ship in all Weathers. Although the Rules here demonstrated, are but
of little benefit to him, that hath been brought up all his Life-time at Sea; and less to those that be altogether ignorant in Marine Affairs: But that the Practick may be delivered in proper Sea-Phrases, according to each several Material that belongs to a Ship compleatly rigg'd, with the Use of the several Ropes in working and trimming of Sails
at Sea on all Occasions, cannot be denied by those that know these things perfectly: Therefore
it is impossible for any Man to be a Compleat Mariner or Navigator, without he hath attained to the true Knowledge of Theorick and Practick, being both Sisters and inseparable Companions, that makes them perfect Navigators; Therefore I could not let this scape my Pen.
And to explain my self, that I may prevent the Censures of all such that will be curious,
inquiring whether I am not lame, or incapable of that, and like themselves appear
imperfect; I may speak it with trouble to my self, and shame to others, That there
was never more lame and decrepit Fellows preferred by Favour and Fortune, as also by Kindred, and by Serving for Under-Wages (which a deserving Man might and would have) as is
now adays. Let a man go aboard the best Ship at Sea, and it will be very rare to find
Ignorance out of the Officers Cabins, and commonly able Mariners and more sufficient Men before the Mast, which are first to hawl a bowling, through
the averseness of their Fates, which is great pity. I should be glad to live to see
a more equaller Balance among Sea-men, and their Imployers, to further the industrious, and encourage the deserving Men; for if this partiality
should continue long, it is to be feared, in some short time, the Compleat Mariner wil be hardly found aboard any Ship, to the great disparagement of our English Nation, which hath from time to time so long deservingly had the Superiority over all other parts of the World, for breeding the most famous Navigators. The Hollander to his Loss knows it right well, that there are none like English for Courage at Sea; but that many of them out-strip us in the Art of Navigation, which proceeds from the former [Page 16] unequal Balance, which makes our expert Saylers to seek if Fortune will be favourable amongst them, They had not at this day been High and Mighty, and in such a flourishing Condition as now they are. Therefore I hope to see and
hear, That the English Mariner will make better use of swift-stealing Time, that he may redeem what is lost, and attain to such perfection, as that he may parallel
his Art with his V [...]lour and Courage; And that Imployers will use more Equity, in placing deserving men according to their merit. I shall
not draw out my Digression to any longer Discourse; for I know my plain Rhetorick will not rellish in some mens ears, though it may in others: Therefore I shall draw
to a Conclusion, desiring that no man will censure me, before he knows what is in
me, or is able to mend this. For some there are, being a little touched (as the common
Saying is) that if they had me at Sea, they would put me to seek all my prescribed
Rules; but I would have such to know, That when I at Sea, I shall work the Ship in
all Assays as well as ever they did, and can as often as I shall be called thereunto,
after this manner, (viz.)
PROP. I. The Wind is fair.
The Wind is fair, though but little; he comes well, as if he would stand; therefore
up a hand and loose fore Top-sail in the Top, that the Ships may see we will Sail;
Bring Cable to the Capston, have up your Anchor, loose your Fore sail in the Brailes;
put abroad our Colours, loose the Misne in the Brailes. Is all our men on board? Those
that be on Shore may have a Towe, and be blest with a Ruther; [...]f [...]r we will stay for no man. Come my Hearts, have up your Anchor, that we may ha [...] a good Prize. Come, Who say Amen? One and all. Oh brave Hearts, the Anc [...]or [...] a Peck; have out fore Top-sail, have out main Top-sail, hawl home the Top-sa [...] Sheets. The Anchor is away, let fall your Fore-sail, hoise up your fore Top-sail, hoise up your main Top-sail;
up and loose the Main-sail, and set him; loose Sprit-sail, and Sprit-sail Top-sail.
A brave Gale. Bring the fore-Tack to the Cat-head, and trim our Sails quartering;
hoise up our small Sails; have out the Misne Top-sail and set him. Now we are clear,
and the Wind like to stand; hoise in our Boats before it is too much Sea; aboard Main-tack,
aboard Fore-tack, a Lee the Helmne handsomly, and bring her to easily, that she may
not stay. Breace the Fore-sail and Fore-top-sail to the Mast, and hawl up the Lee-Bowlings,
that the Ship may not stay; pass Ropes for the Boot on the Lee-side, and be ready
to clap on your Tackle, and hoise them in; stow them fast. Let go the Lee-Bowling
of Fore-sail, and Weather-Braces. Right your Helmnes, hale aft the Fore-sheet, trim
the Sails quartering as before: Let go the Sprit sail Breales, and haft of the Sheets;
and hoise up the Sprit-sail Top-sail, and other small Sails. Set the main Stay-sail,
and fore Top-sail, Stay-sail and Misne Stay-sail, and main Top-sail, Stay-sail and
leese in your Boonets, that we may make most of our way. To our Station, and clear
your Ropes. Come, get up our steering Sails. The Lee steering Sails of Main-sail,
and Main-top-sail, Fore-sail and Fore-top-sail only; for they will set fairest, and
draw most away. I have on purpose omitted several Words, by reason I would not trouble
the Reader with such indifferent things as is conceived by all Mariners to be done; as Cooning the Ship, Breasing, Vereing, and haleing aft, and hoising,
looring, and the like: but it is to be supposed all to be done at the same time.
Thus have you a brave Ship under all her Sails and Canvas, in her swiftest way of
Sailing upon the Sea. Now let us have her right before the Wind.
Right afore the Wind, and a fresh Gale.
The Wind is vered right aft, take in your Fore and Fore-top-sail, Steering-sail, and
Fore-top-sail, and Main and Main-top-sail, Stay-sails; for they are becalmed by the
after Sails, and will only beat or rub out. The Wind blows a fresh Gale, round aft
the Main-sheets, and Fore-sheets, brasse. Square your Yards, take in your Main and
Main-top-sail, Steering-sails. Unlease your Bonnets. Take in your Main and Fore-top-gallant-sails,
In the Sprit-sail, and Misne Top-sail, let go the Sheets, hale from [Page 17] the Cholyens, cast off Top-gallant Bowlings. Thus you have all the small Sails in,
and furled, when it blows too hard a Wind to bear them.
The wind vereth forward, and scanteth.
The Wind scanteth, vere out some of your Fore and Main-sheets, and Sprisle-sheets,
and let go your Weather Braces; tope your Sprit-sail Yard. The Wind still vereth forward;
Get aboard the Fore and Main-Tack; cast off your Weather-sheets and Braces: The Sails
are in the Wind, hawle off Main and Fore-sheets; the Wind is sharp, hawl forward the
main Bowline, and hawl up the Main top-sail, and Fore top-sail Bowline, and set in
your Lee-braces, and keep her as near as she will lie. Thus have you all the Sails
trimm'd sharp, full, and by a Wind.
The wind blows Frisking.
The Wind blows hard; settle our fore and main Top-sails two thirds of the Mast down.
It is more Wind, come, hawl down both Top-sails close. Come, stand by, take in our
Top-sails; Let go the Top-sail Bowlines, and Lee-Braces; let go the Lee-Sheets, set
in your Weather Braces, spill the Sails, hawl home the Top-sail Clue-lines, square
the Yeard. Now the Sail is furled, and you have the Ship in all her low Sails, or
Courses at such a time.
It bloweth a Storm.
It is like to over-blow; Take in your Sprit-sail, stand by to hand the Fore-sail.
Cast off the Top-sail Sheets, Clue-garnets, Leech-lines, Bunt-lines; stand by the
Sheet, and brace; loure the Yeard, and furl the Sail (here is like to be very much
Wind) See that your main Hallyards be clear, and all the rest of your Geer clear and
cast off. (It is all clear.) Loure the main Yard. All down upon your doone hall; now
the Yard is down, hawl up the Clue-garnets, Lifts, Leach-lines, and Bunt-lines, and
furle the Sail fast, and fasten the Yards, that they may not travers and gall. Thus
have you the Ship a trije under a Mizen.
A very hollow grown Sea.
We make foul weather, look the Guns be all fast, come hand the Mizen. The Ship lies
very broad off; it is better spooning before the Sea, than trying or hulling; go reefe
the Fore-sail, and set him; hawl aft the Fore-sheet; The Helmne is hard a weather,
mind at Helmne what is said to you carefully. The Ship wears bravely study, she is
before it, and the Sheets are afle and braces; belay the fore doon hall, that the
Yard may not turn up; it is done. The Sail is split; go hawl down the Yeard, and get
the Sail into the Ship, and unbind all things clear of it, and bring too the Fore-bonnet;
make all clear, and hoise up the Fore-yard; hawl aft the Sheets, get aft on the Quarter-Deck,
therefore Braces.
Starb [...]ard; hard up, right your Helmne Port. Port hard, more hands, he cannot put up the Helmne.
A very fierce Storm. The Sea breaks strange and dangerous; stand by to hawl off above the Lennerd of the
Whipstaff, and help the man at Helmne, and mind what is said to you. Shall we get
down our Top-masts? No, let all stand; yet we may have occasion to spoon before the
Sea with our Powles. As we mast, get down the Fore-yeard— She scuds before the Sea
very well; the Top-mast being aloft the Ship is the holsomest, and maketh better way
through the Sea, seeing we have Sea-Room. I would advise none in our condition to
strike their Top-masts, before the Sea or under. Thus you see the Ship handled in
fair weather and foul, by and learge. Now let us see how we can turn to windward.
The Storm is over, set Fore-sail and Main sail; bring our Ship too; set the Misne,
and Main Top-sail, and Fore Top-sail. Our Course is E. S. E. the Wind is at South: Get the Starboard-Tacks aboard, cast off our Weather Braces and Lifts; Set in the
Lee-Braces, and hawl forward by the Weather Bowlines, and hawl them laught and belaye
them, and hawl over the Mizen Tack to Winerd; keep her full, and by as near as she
will lye. How Wind you? East. A bad quade Wind. (No near) hard, no near. The Wind vereth to the Eastward still. How Wind you? N. E. hard, no near. The Wind is right in our teeth; no near still. How wind you? N. W. b. N. The Wind will be Northerly, make ready to go about; we shall lay our Course another way, no near, give the Ship
way, that she may stay: ready, ready a Lee the Helmne. Vare out there Fore-sheet,
cast off your Lee-Braces, brace in upon your Weather Braces. The Fore-sails is a
back stays, hawl Main-sail, hawl, about, let rise the main Tack, cast off your Larboard-Braces,
let go main Bowline, and main Top-sail Bowline; brace about the Yard, hawl forward
by the Larboard-Bowlines; get the main Tack close down, in the Cheese-tree: hawl up
the Weather Bowline, and set the Lee Brace of Main and Main Top-sail Yards, and the
Sheets is close aft; hawl, of all; hawl; get to fore-Tack, let go fore-Bowline, and
fore-Top-sail Bowline; hawl afle the fore-Sheet, hawl taught, the main Bowline, and
main Top-sail Bowline; shift the Mizen, tack, hawl bout fore Bowline, and fore Top-sail
Bowline; set in the Lee-Braces taught, fore and aft, keep her as near as she will
lie.— No near, How Wind you? N. N. E. thus werr no more; no near, keep her full. The Wind is at N. N. E. thus werr no more. (How Wind you?) E. N. E. The Wind is at N. keep her away her Course E. S. E. Cast off the Lee-Braces, and Weather-Bowlines, and set in your Weather-Braces. Vere
out the main Sheet, and fore Sheet, loose the Sprit-sail, and Spit-sail, Top-sail,
and Mizen Top-sail, and Top-gallant-sails; hoise them up, the Wind vears afle still;
let rise the fore-Tack: the Wind's quartering, hawl aft the fore-Sheet, bring him
down to the Cat-head with a pass-a-ree; studdy in your Weather Braces; the Wind stands,
here. Thus you have the Ship as at first, steering under all her Canvas, quarter Wind,
as she did at first, setting Sail. She hath been wrought with all manner of Weather,
and all sorts of Winds. Therefore we will draw to the last with a Man of War in Chase
and taking of her Prize, and so leave this Practick Part to your Censure.
The Man of War in her Station.
Now we are in our Station, and a good Latitude, hand your Top-sails, and furle your Main-sail and Fore-sail, and brail up the Mizen,
and let her lie at Hull, until Fortune appear within our Horizon. Up alaft to the Top-mast-head, and look abroad, young-men; look well to the Westward, if you can see any Ships that have been nipt with the last Easterly Winds. A Sail, a Sail. Where? Fair by us. How stands she? To the Eastward, and is two Points upon her Weather Bow, and hath her Larboard-Tacks aboard. O then
she lies close by a wind; we see her upon the Decks plainly. A good man to Helmne.
Up young-men, and loose the Fore-sail, Main-sail, and Mizen. Get the Larboard Tacks
aboard; have out the Main top-sail, and Fore-top-sail, and loose the Sprit-sail, aloofe.
Keep her as neer as she will lie; hawl aft the Sheets, and hawl up your Bowlines laught.
Do you see your Chase? How Wind you? E. N. E. Then the Wind is at N. Hoise up your Top-sails as high as you can; have out Sprit-sail, Top-sail, and Mizen
Top-sail; hawl home the Sheets, and hoise them up: A young-man and loose the Main-top-gallant-sail,
and Fore-top-gallant-sail; hawl home the Sheets, and hoise them up; hoise up Main
Stay-sail, and Mizen Stay-sail, and loose the Main-top-sail, and Fore-top-sail; Stay-sail,
and set them. It blows a brave chaseing Gale of Wind; The Ship makes brave way through
the Sea; we rise her apace; if she keep her Course, we shall be up with her in three
Glasses. No near, keep the Chase open with little of the Fore-sail. So, thus, keep
her thus. Come ofle all hands, the Ship will Stear the better when you sit all [Page 19] quiet; by, by her small Sails, for she is too much by the Head. The Chase is a lusty
brave Ship. So much the better, she hath the more Goods in her Hold. The Ship hath
a great many Guns (no force) it may be she is a Private [...]Port, the Chase is about, come fetch her wack, and we will be about after her. We sayl
far better than she; we have her Wack; a Lee the Helmn, about Ship, vere out Fore-sheet.
Every man stand handsomly to his business, and mind the Bowlines and Braces, Tacks
and Sheets; hawl Main-sail, hawl about. Let go Main Bowline, Top Bowline, Top-gallant
Bowline; Hawl off all, hawl Fore-sail, about, shifts the-Helmn; bring her too, Hawl
the Main-sheets close afle, and fore-sheet. Set in the Lee-Braces, hawl too the Bowlines.
Thus the Chase keeps close to the Wind; keep her open under our Lee. Gunner, see that you have all things in readiness, and that the Guns be clear; and that nothing
pester our decks.—Down with all Hammocks and Cabins that may hinder and hurt us. Gunner, is all our Geer ready? Is the store of Cartrages ready fill'd, all manner of Shot
at the Main-mast? Is there Rammers, Sponges, Ladles, Priming-Irons, and Horns, Lyntstocks,
Wads, and Water at their several quarters sufficient for them? Be sure that none of
our Guns be cloy'd; and when we are in Fight, be sure to load our Guns with Cross-bar
and Langrel. Always observe to give Fire when the Word is given. See that there be
Half-Pikes and Javelings in a readiness, and all our Small-shot well furnished, and
all their Bandaliers fill'd with Powder, and Shot in their Pooches.
See that our Murtherers and Stockfowlers have their Chambers fill'd with good Powder,
and Bags of small Shot to load them, that if we should be laid aboard, we might clear
our Decks.—Starboard, the Chase pays away more room, Starboard hard; Vere out some
of the Main-sheet and Fore-sheet; Cast of the Larboard-Braces, (Steady) Keep her
thus: Well Steer'd; the Chase goes away room, her Sheets are both aft, she is right
before the Wind: Stereboard hard; Let rise main Tack, let rise fore-Tack; Hawl afle
Main-sheet, hawl afle fore-Sheet. We have a stearn-Chase, but we shall be up with
her presently, for we fetch upon her hand-going. The Chase hawls up his Main-sail
and furles it; she puts aboard her Waste-clothes; she will fight us. Come up from
alow young men, and furle our Main-sail; sling our Main Yard, with the Chains in the
Main-top; sling our Fore-yard, put aboard our Waste-Cloaths; he will fight us before
the Winde I see; She is full of Men; It is a hot Ship, but deep and foul. Come chearly
my Hearts, It is a Prize worth fighting for; The Chase takes in her small Sails; Up
aloft and take in our Top-gallant-sails, Sprit-sails, Top-sails, Mizen Top-sail, and
furle the Sprit-sail, and get the Yard alongst under the Bowsprit. She puts abroad
her Colours, It is Red, White, and Blew; they are Dutch Colours; no force, the worst of Enemies. Boy, up and put abroad St. George's Colours in our Main-top; step oft a hand, and put abroad our bloody Ancient; Call
all hands aloft; Come up aloft all hands. They are all up Captain.
Gentlemen, We are here employed and maintained by his Majesty King CHARLES and our Country, to do our Endeavours to keep this Coast from Pyracy and Robbers, and his Majesties Enemies; whereof it is our Fortune to meet this Ship at this time:
Therefore I desire you in his Majesties Name, and for the Sake of our Country, and
the Honour of our English Nation, and our selves, for every man to behave himself courageous like Englishmen; and not to have the least shew of a Coward: but to observe the Words of Command, and
do his utmost endeavour against this barbarous and inhumane Enemy the Dutch, who have treacherously and inhumanely murthered so many of our English Nation, in the East-Indies and other Parts, whose Blood cries for vengeance. Therefore our Quarrel is just, and
into Gods hands we commit our Cause, and our selves. So every man to his Quarters,
and shew his Courage, and God be with you.
She settles her Top-sails, we are within shot; let all our Guns be loose, In the Tackles
and the Ports, all knockt open, that we may be ready to run out our Guns when the
Word is given. Up noise of Trumpets, and hail our Prize; she answereth again with
her Trumpet: Hold fast Gunner, do not fire till we hail them with our Voices. (Haye, Hooe) Amain for King CHARLES. (Port) edge towards him, he fires his Broad-side upon us. (What chear my Hearts?) Is all
well betwixt [Page 20] Decks? Yea, Yea, only he rackt us through and through. No force, it is his turn next; but give not
Fire until we are within Pistol-shot. Port, edge towards him. He plies his small Shot; hold fast Gunner. Port, right your Helmne. We will run up his Side. Starboard a little; Give fire, Gunner.
(That was well done.) This Broad-side hath made their Deck thin, but the small Shot
at first did gaul us. Clap in some Case-shot in the Guns you are now a loading; Brace
too the Fore-top-sail, that we may not shoot a head; He lies broad off to the Southward,
to bring his other Broad-side to bear upon us. (Starboard hard.) Get to Larboard Fore-tack;
trim your Top-sails, run out your Larboard Guns. He fires his Sterboard Broad-side
upon us; He pours in his small Shot. Starboard give not fire until he fall off, that
the Prize may receive our full Broad-side. Steady: Port a little; give fire Gunner:
His Fore-mast is by the board. This last Broad-side hath done great Execution. Cheerly
my Mates, the day will be ours; He is shot a Head; He bears up before the Wind to
stop his Leaks: Keep her thus; Well Steered. Port, Port hard; Bear up before the Wind, that we may give him our Starboard Broad-side. Gunner, Is
there great store of Case-shot and Langrel in our Guns? Yea, yea. Port, make ready to board him; Have your Lashers clear, and able men with them. Edge towards
him Guns when you give fire; Bring your Guns to bear amongst his Men with the Case-shot.
Well steered, we are close on boord. Give fire Starboard; Well done Gunner; They lie
Heads and Points aboard the Chase. Come, Aboard him bravely; Enter, Enter. Are you lached fast? Yea, Yea, We will have him before we go here-hence. Cut up the Decks; Ply your Hand-Granadoes
and Stink-Pots. He cries out Quarter; Quarter for our Lives, and we will yield up
Ship and Goods. Good Quarter is granted, Provided you will lay down all your Arms,
open the Hatches, hawl down all your Sails and furle them; loose the Lachings, we
will sheer off our Ship, and hoise out our Shallop. If you offer to make any Sail,
expect no Quarter for your Lives. Go with the Shallop, and send aboard the Captain,
Lieutenant, and Master and Mates, with as many more as the Shallop will carry. So
we will leave the Man of War to put his Prisoners down into the Hold, and secure. And so likewise I have shewn
you thus much of the Practick part of Navigation, in which you may perceive that I have wrought the Ship in all Essays, in Words and
proper Sea-Phrases; and if I was at Sea, I should perform it both in Word and Deed:
therefore I leave it all to your Judicious Censures. And let not Ignorance, the Arch-enemy of Arts deceive you, and cause you to think that I have writ what I cannot do; but that I
can as easily turn him in the Theorick, which way I list, as I can the Ship in the Practick. And so I will conclude with Ovid, when he sailed in the Straight Ionian,
Nothing but Waves we view in Sea where Ships do float,
And Dangers lie, huge Whales, and all Fish play:
Above our Heads, Heaven's Star-embroidered Coat,
Whose Vault contains two Eyes, for Night and Day;
Far from the Main, or any Marine Coast,
'Twixt Borean Blasts, and Billows, we are tost.
If Ovid in that straight Ionian Deep
Was tost so hard, much more are we on Seas
Of larger Bounds, where Staff and Compass keep
Their strict observance: Yet in this unease
Of Tackling Boards, we so the way make short,
That still our Course draws nearer to our Port.
Between the Stream and silver-spangled Skie
We rolling climbe, then hurling fall beneath;
Our way is Serpent-like, in Meads which lie,
That bows the Grass, but never makes a Path:
But fitter, like young Maids and Youths together,
Run here and there, all where, and none know whether.
THe ARTS, saith Arnobius, are not together with our Minds sent out of the Heavenly Places; but all are found out on Earth, and are in process
of time sought and fairly forged by a continual Meditation. Our poor and needy Lives
perceiving some casual Things to happen preposterously, while it doth imitate, attempt,
and try, while it doth slip, reform, and change, hath out of these, some Fiduous Apprehension
made by small Sciences of Art, the which afterwards by Study are brought to some perfection.
Yet the Practice of Art is not manifest but by Speculative Illustration; because by Speculation we know that we may the better know. And for this cause I chose a Speculative Part; And first of Geometry, that you may the better know the Practice.— To begin then.
I. A Point is that which hath no Parts.
A Point is supposed to be a Thing indivisible, or that cannot be divided into parts; yet it is the first of all Dimensions. It is
the Philosopher's Atome. Such a Nothing, as that it is the very Energie of All Things. In God it carrieth its Extremes from Eternity to Eternity; which proceeds from the least
imaginable thing, as the Point or Prick noted with the Letter A; and is but only the
Terms or Ends of Quantity.
[geometrical diagram]
II. A Line is a supposed Length, without Breadth or Thickness.
A Lines Extremes or Bounds are two Points, as you may see the Line a; b is made by moving of a Point from a to b. A Line is either straight or crooked; and in Geometry of three kinds of Magnitudes, viz. Length, Breadth, and Thickness. A Line is capable of Division in Length only, and may be divided equally in the Point C,
or unequally in D, and the like.
[geometrical diagram]
III. The Ends or Bounds of a Line are Points.
You are to understand, the Ends or Bounds of a finite Line is A, B, as before: but
in a Circular Line it is otherwise; for there the Point in its Motion returneth again to the Place where it first began, and so maketh the
Line infinite.
[geometrical diagram]
IV. A Right Line is the shortest of all Lines, drawn from any two of the said Points,
As you may see the Right Line AB straight, and equal between the Points A and B, without bowing, which are the
Bounds thereof.
V. A Superficies is a Longitude, having only Latitude.
Superficies is That which hath only length and breadth, whose Terms and Limits are two Lines. In the first kind of Magnitude the Motion of a Point produceth a Line: So in the second kind of Magnitude, the Motion of a Line produceth a Superficies. This is also capable of two dimensions, as the length AB and CD, and the breadth
AC and BD; and may be divided into any kind or number of Parts,
[geometrical diagram]
VI. The Extremes of a Superficies are Lines.
As the Ends of a Line are Points, so the Bounds or Extremes of a Superficies are Lines; as before, you may see the Ends of the Lines AB, and BD, and DC, and CA.
VII. A Plain Superficies lieth equally between his Lines.
So the Superficies ABCD is that which lieth direct and equally between his Lines. And whatsoever is said of a Right Line, the same is meant of a Plain Superficies.
VIII. An Angle is when two Lines are extended upon the same Superficies, so so that they touch one another in a Point, but not directly.
As you may see the two Lines AB and BC incline one towards the other, and touch one the other, in the Point B. In which Point, by reason of the bowing inclination of the said Lines, is made the Angle ABC. And here note, That an Angle is most commonly signed by three Letters, the middlemost whereof sheweth the Angular Point, as in this Figure, when we say Angle ABC, you are to understand the very Point at B.
[geometrical diagram]
IX. A Right Angle is that which is produced of a Right Line, falling upon a Right Line, and making two equal Angles on each side the Point where they touch each other.
As upon the Right Line CD suppose there doth stand another Right Line AB, in such sort that it maketh the Angles on either side thereof; namely, the Angle ABD on the one side: equal to the Angle ABC on the other side; then are either of the two Angles Right Angles; and the Right Line AB, which standeth erected upon the Right Line CD, without bowing or inclining to either part thereof, is a Perpendicular to the
Line CD.
X. An Obtuse Angle is thatwhich is greater than a Right Angle.
So the Angle CBE is an Obtuse Angle, because it is greater than the Angle ABC, which is a Right Angle; For it doth not only contain that Right Angle, but the Angle ABE also, and therefore is Obtuse.
[geometrical diagram]
XI. An Acute Angle is less than a Right Angle.
Therefore you may see the Angle EBD is an Acute Angle; for it is less than the Right Angle ABD, in which it is contained by the other Acute Angle ABE.
XII. A Limit or Term is the End of every Thing.
As a Point is the Limit or Term of a Line, because it is the End thereof; so a Line likewise is the Limit and Term of a Superficies, and a Superficies is the Limit and Term of a Body.
XIII. A Figure is that which is contained under one Limit or Term, or many.
As the Figure A is contained under one Limit or Term, which is the round Line; also the Figures B and C are contained under four Right Lines: likewise the Figure E is contained under three Right Lines, which are the Limits or Terms thereof; and the Figure F under five Right Lines: And so of all other Figures.
[geometrical diagram]
And here note, We call any plain Superficies, whose Sides are unequal (as the Figure F) a Plot, as of a Field, Wood, Park, Forest, and the like.
XIV. A Circle is a plain Figure contained under one Line, which is called a Circumference; unto which all Lines drawn from one Point within the Figure, and falling upon the Circumference thereof, are equal one to the other.
As the Figure AECF is a Circle contained under the Crooked Line AECD, which Line is called the Circumference. In the middle of this Figure is the Center or Point B, from which Point all Lines drawn from the Circumference are equal, as the Lines BA, BE, BD, BC; and this Point B is called the Center of the Circle.
XV. A Diameter of a Circle is a Right Line drawn by the Center thereof, and ending at the Circumference, on either side dividing the Circle into two equal Parts.
So the Line ABC in the former Figure, is the Diameter thereof, because it passeth from the Point A on the one side, and passeth also by the Point B, which is the Center of the Circle; and moreover, it divideth the Circle into two equal parts, namely, AEC being on one side of the Diameter, equal to AFC on the other. And this Observation was first made by Thales Milesius; For, saith he, if a Line drawn by the Center of any Circle do not divide it equally, all the Lines drawn from the Center of that Circle, from the Circumference, cannot be equal.
XVI. A Semicircle is a Figure contained under the Diameter, and that part of the Circumference cut off by the Diameter.
As in the former Circle, the Figure AFC is a Semicircle, because it is contained of the Right Line ABC which is the Diameter, and of the crooked Line AFC, being that part of the Circumference which is cut off by the Diameter: Also the part AEC is a Semicircle.
XVII. A Section or Portion of a Circle, is a Figure contained under a Right Line, and a part of the Circumference, greater or less than a Semicircle.
So the Figure ABC, which consisteth of the part of the Circumference ABC, and the Right Line AC, is a Section or Portion of a Circle, greater than a Semicircle.
[geometrical diagram]
Also the other Figure ACD, which is contained under the Right Line AC, and the parts of the Circumference ADC, is a Section of a Circle less than a Semicircle. And here note, That by a Section, Segment, Portion, or part of a Circle, is meant the same thing, and signifieth such part as is greater or lesser than a
Semicircle: So that a Semicircle cannot properly be called a Section, Segment, or part of a Circle.
XVIII. Right-lined Figures are such as are contained under Right Lines.
.
XIX. Three-sided Figures are such as are contained under three Right Lines.
.
XX. Four-sided Figures are such as are contained under four Right Lines.
.
XXI. Many-sided Figures are such a [...] have more Sides than four.
.
XXII. All Three-sided Figures are called Triangles.
XXIII. Of Four-sided Figures, A Quadrat or Square is that whose Sides are equal, and his Angles right, as the Figure A.
[geometrical diagram]
XXIV. A Long Square is that which hath right Angles, but unequal Sides, as the Figure B.
[geometrical diagram]
XXV. A Rhombus is a Figure Quadrangular, having equal Sides, but not equal or right Angles, as the Figure C.
[geometrical diagram]
XXVI. A Rhomboides is a Figure whose opposite Sides are equal, and whose opposite Angles are also equal: but it hath neither equal Sides, nor equal Angles, as the Figure D.
[geometrical diagram]
XXVII. All other Figures of Four Sides are called Trapezia's.
[geometrical diagram]
XXVIII. Such are all of Four Sides, in which is observed no equality of Sides or Angles, as the Figures L and M, which have neither equal Sides nor Angles, but are described by all Adventures, without the observation of any Order.
XXIX. Parallel or Aequi-distant Right Lines, are such which being in one and the same Superficies, and produced infinitely on both sides, do never in any part concur; as you may see
by the two Lines AB, CD.
[geometrical diagram]
XXX. A Solid Body is that which hath Length, and Breadth, and Thickness, as a Cube or Die; and the Limits and Extremes of it are Superficies, as the Figure I.
[geometrical diagram]
XXXI. Axis is the Diameter about which the Sphere or Globe is turned.
[geometrical diagram]
XXXII. The Poles of a Sphere are the Extremes or Ends of the Diameter, and are terminated in the Superficies of the Sphere.
[geometrical diagram]
XXXIII. A Sphere is defined by Euclid to be made, when the Diameter of a Semi-circle remaining fixed, the Semicircle is turned about, till it be returned to the Place whence it began to move at first.
.
Geometrical Theoremes.
I. ANY two Right Lines crossing one another, make the contrary or vertical Angles equal. Euclid. 15. 1.
II. If any Right Lines fall upon two parallel Right Lines, it maketh the outward Angles of the one, equal to the inward Angles of the other; and the two inward opposite Angles, on contrary sides of the falling Line, also equal. Euclid 29. 1.
III. If any Side of a Triangle be produced, the outward Angle is equal to the inward opposite Angles, and all the three Angles of any Triangle are equal to two Right Angles. Euclid 32. 1.
IV. In Aequi-angled Triangles all their Sides are proportioned, as well such as contain the equal Angles, as also the subtendent Sides.
V. If any four Quantities be proportional, the first multiplied in the fourth, produceth a Quantity equal to that which is made by multiplication of the second in the third.
VI. In all Right-angled Triangles, the Square of the Side subtending the Right Angle, is equal to both the Squares of the containing sides. Euclid 47.1.
VII. All Parallellograms are double to the Triangles that are described upon their Basis, their Altitudes being equal. Euclid 41.1.
VIII. All Triangles that have one and the same Base, and lie between two Parallel Lines, are equal one to the other. Euclid 37.1.
PROBLEM I. Ʋpon a Right Line given, how to erect another Right Line which shall be perpendicular to the Right Line given.
THE Right Line given is AB, upon which from the Point E it is required to erect the Perpendicular EH. Opening your Compass at any convenient distance, place one Foot in the assigned Point E, and with the other make the two Marks C and D, equal on each side the Point E; then opening your Compasses again to any other convenient distance wider than the former, place one Foot in C,
and with the other describe the Arch GG; also (the Compasses remaining at the same distance) place one Foot in the Point D, and with the other describe the Arch FF. Then from the Point where those two Arches intersect or cut each other (which is at H) draw the Right Line HE, which shall be Perpendicular to the given Right Line AB, which was the thing required to be done.
[geometrical diagram]
PROBL. II. How to erect a Perpendicular on the end of a Right Line given.
LEt AB, be a Line given, and let it be required to erect the Perpendicular AD. First upon the Line AB, with your Compasses opened to any small distance, make five small Divisions, beginning at A, noted with
1, 2, 3, 4, 5. Then take with your Compasses the distance from A to 4, and place one Foot in A, and with the other describe the
Arch e e: Then take the distance from A to 5, and placing one Foot of the Compasses in 3, with the other Foot describe the Arch h h, cutting the former Arch in the Point D: Lastly, from D draw the Line DA, which shall be perpendicular to the given Line AB.
[geometrical diagram]
This operation is grounded upon this Conclusion, viz. These three Numbers 3, 4, and 5, make a Right-angled Triangle, which is very necessary in many Mechanical Operations, and easie to be remembred.
PROBL. III. How to let fall a Perpendicular upon any Point assigned, upon a Right Line given.
THe Point given is C, from which Point it is required how to draw a Right Line which shall be perpendicular to the given Right Line AB.
[geometrical diagram]
First from the given Point C, to the Line AB, draw a Line by chance, as CE, which divide into two equal parts in the Point D. Then placing one Foot of the Compasses in the Point D, with that distance DC, describe the Semicircle CFE, cutting the given Line AB in the Point F. Lastly, If from the Point C you draw the Right Line CF, it shall be a Perpendicular to the given Line AB, which was required.
PROBL. IV. How to make an Angle equal to an Angle given.
LEt the Angle given be ACB, and let it be required to make another Angle equal thereunto.
First draw the Line EF at pleasure; then upon the given Angle at C (the Compasses opened to any distance) describe the Arch AB; and also upon the Point F, the Compasses unaltered, describe the Arch DE; Then take the distance AB, and set the same from E to D; Lastly draw the Line DF: So shall the Angle DFE be equal to the given Angle ACB.
[geometrical diagram]
PROBL. V. A Right Line being given, how to draw another Right Line which shall be parallel to the former, at any distance required.
THe Line given is AB, unto which it is required to draw another Right Line parallel thereunto, at the distance AC or DB. First open your Compasses to the distance AC or BD; then placing one Foot in A, with the other describe the
Arch C; also (at that distance place one Foot in B, and with the other describe the Arch D. Lastly, draw the Line CD, that it may only touch the Arches C and D: So shall the Line CD be parallel to the Line AB, and at the distance required.
PROBL. VI. To divide a Right Line into any number of equal Parts.
LEt AB be a Right Line given, and let it be required to divide the same into five equal Parts.
[geometrical diagram]
First, From the given Line A, draw the Line AC, making any Angle from the end of the given Line which is at the Point B. Then draw the Line BD equal to the Angle CAB. Then from the Points A and B, set off upon these two Lines any Number of equal parts, being less by one than the Parts into which the Line AB is to be divided, which in this Example must be 4. Then draw small Lines from 1 to 4, from 2 to 3 twice, and from 1 to 4, &c. which Lines crossing the given Line AB, shall divide it into five equal Parts, as was required.
PROBL. VII. A Right Line being given, how to draw another Right Line parallel thereunto, which shall also pass through a Point assigned.
LEt AB be a Line given, and let it be required to draw another Line paralle thereunto, which shall pass through the given Point C. First, Take with your Compasses the distance from A to C, and placeing one Foot thereof at B, with the other describe
the Arch DE; then take in your Compasses the whole Line AB, and place one Foot in C, and with the other describe the Arch FG, crossing the former Arch in the Point H: Then if you draw the Line CH, it shall be parallel to AB, the thing required.
[geometrical diagram]
PROBL. VIII. Having any three Points given which are not scituated in a Right Line, How to find the Center of a Circle which shall pass directly through the three Points given.
THe three Points given are A, B, and C; now it is required to finde the Center of a Circle whose Circumference shall pass through the three Points given.
[geometrical diagram]
First open your Compasses to any distance greater than half the distance between B and C; then place one Foot
in the Point B, and with the other describe the Arch FG; then the Compasses remaining at the same distance, place one Foot in the Point C, and with the other turn'd about make the marks F and G in the former Arch, and draw the Line FOG at length, if need be.
In like manner open your Compass at a distance greater than half AB; Place one Foot in the Point A, with the other describe the Arch HK: Then the Compasses remaining at the same distance, place one Foot in the Point[Page 31] C, and turning the other about, make the marks HK in the former Arch. Lastly, draw the Right Line HK, cutting the Line FG in O, so shall O be the Center, upon which you may describe a Circle at the distance of OA, and it shall pass directly through the three given Points ABC, which was required.
PROBL. IX. How to describe a Circle in a Triangle, that shall only touch the three Sides; and to find the Centre.
LAy down the Triangle ABC, the three Sides equal; then divide the Sides of the Triangle AB in two equal parts, as at E, and draw the Line CE; and likewise divide BC, and draw the Line AD; and where they cross one the other, as at O, that is the Center: Therefore put one Point of the Compasses in the Center O, and extend the other to either side, and describe the Circle GF, which will only touch the Sides A BC of the Triangle.
[geometrical diagram]
PROBL. X. How to lay down a Triangle in a Circle, and to find the Center of the Circle in the Triangle.
DRaw the three Sides of a Triangle AB C, it is no matter if they be equal or not; then put one Foot of your Compasses in the Point B, open the other to more than half the length of the greatest side, as to C; and
with that distance describe the Arch FHDG; and so removing the Compasses to C, cross the former Arch at F and D, and draw the Line DF. Again, the Compasses at the same distance, put one Foot in A, and describing a small Arch, cross the former Arch at H and G; and laying a Ruler over the Intersections of these two Arches at H and G, draw the Line GH; and where these two Lines cross one the other, as at K, that is the Center of the Triangular Points. From it extend the Compasses to either of the Points, and describe the Circle ABC, and the Triangle will be within the Circle.
PROBL. XI. Any three Right Lines being given, so that the two shortest together be longer than the third, To make
thereof a Triangle.
LEt it be required to make a Triangle of these three Lines A, B, and C, the two shortest whereof, viz. A and B together, are longer than the third C.
[geometrical diagram]
First draw the Line DE equal to the given Line B; then take with your Compasses the Line C, and setting one Foot in E, with the other describe the Arch FF: also take with your Compasses the given Line A, and placing one Foot in D, with the other describe the Arch GG, cutting the former Arch in the Point K. Lastly, from the Point K, if you draw the Lines KE and KD, you shall constitute the Triangle KDE, whose Sides shall be equal to the three given Sides ABC.
PROBL. XII. Having a Right Line given, How to make a Geometrical Square, whose Sides shall be equal to the Right Line given.
THe Line given is RI, and it is required to make a Geometrical Square whose Sides shall be equal to the Line RI, First draw the given Line RI, then (by the first and second Problem) upon the Point B raise the Perpendicular BC, making the Line BC equal to the given Line RI also: Then taking the said RI in your Compasses, place one Foot in C, with the other describe an Arch at D; The Compass at the same distance, set one Foot in A, and cross the former Arch at D; then draw the Lines D, C and DA, which shall conclude the Geometrical Square ABCD, which was required.
[geometrical diagram]
PROBL. XIII. Two Right Lines being given, How to find a third which shall be in proportion unto them.
LEt the given Lines be A and B; and it is required to find a third Line which shall be in proportion unto them.
[geometrical diagram]
First draw two Right Lines, making any Angle at pleasure, as the Lines LM and MN, making the Angle LMN: Then take the Line A in your Compasses, and set the length thereof from M to E; also take the Line B, and set the Length [Page 33] thereof from M to F, and also from M to H: Then draw the Right Line EH, and then from the Point F draw the Line FG parallel to EH. So shall MG be the third Proportional required: For Arithmetically
say,
As ME to MH: So is MF to MG 18.
[...]
PROBL. XIV. Three Right Lines being given, To find a fourth in proportion to them.
THe three Lines given are ABC, unto which it is required to find a fourth Proportional Line. This is to perform the Rule of Three. As in the last Problem, you must draw two Right Lines, making any Angle at pleasure, as the Angle EFG; then take the Line A in your Compass, and set it from F to I; then take the Line B in your Compasses, and set that from F to K; then take the third given Line in your Compasses, and set that from F to H, and from that Point H draw the Line H L, parallel to IK; So shall the Line FL be the third Proportional required.
[geometrical diagram]
Note, That these Lines are taken off a Scale, that is divided into 20 parts to an Inch: To do it Arithmetically say,
As FI is to FK: So is FH to FI.
[...]
Here note, That in performance of the last Problem, That the first and third Terms, namely the Lines A and C, must be set upon one and the same Line, as here upon the Line FE, and the second Line B must be set upon the other Line FG, upon which Line also the fourth Proportion will be found.
PROBL. XV. How to work the Rule of Proportion by a Scale of equal Parts, and such other Conclusions as are usually wrought in Lines and Numbers, as in Mr. Gunter's 10 Prob. 2 Chap.
The Scale of Inches is a Scale of equal Parts, and will perform (by protraction upon a Flat or Paper) such Conclusions as are usually wrought in Lines and Numbers, as in Mr. Gunter's 10 Prob. 2 Chap. Sector, may be seen, and in others that have writ in the same kind. This way Mr. Samuel Foster hath directed in the I Chap. of his Posthumus Fosteri.
An Example in Numbers like his Tenth Probl.
As 16 to 7: So is 8 to what?
Here because the second Term is less than the first, upon the Line AB, I set AC the first Term 16, and the second Term AD 7, both taken out of the Scale of equal parts: thence also the third Number 8 being taken, with it upon the Center C, I describe the Arke EF, and from A draw the Line AE, which may only touch the same Arke; then from D, I take DG, the least distance from the Line AE, and the same measured in the same Scale of equal parts, gives 3½, the fourth Term required.
[geometrical diagram]
But if the second Term shall be greater than the first, then the form of working must be changed, as in
the following Example.
EXAMPLE.
As 7 to 16: So 21 to what? — 48.
Upon the Line AB, I set the second Term 16, which is here supposed to be AD; then with the first Term 7 upon the Center D, I describe the Arke GH, and draw AG that may just touch it: Again, having taken 21 out of the same Scale, I set one Foot of that Extent upon the Line AB, removing it until it fall into such a place, as that the other Foot being turned
about, will justly touch the Line AG before-drawn; and where (upon such Conditions) it resteth, I make the Point C; then measuring AC upon your Scale, you shall find it to reach 48 Parts, which is the fourth Number required.
The form of Works (although not so Geometrical) is here given, because it is here more expedite than the other by drawing Parallel Lines; but in some Practice the other may be used. I have been the more large upon this,
because in the following [Page 35] Treatise I shall quote some more remarkable Places in Posthuma Fosteri: and the Solution of Proportions must be referred thither, the form of their Operations being the same with this.
In them therefore shall only be intimated what must be done in general, the particular
way of working being here directed.
PROBL. XVI. To divide a Right Line given, into two parts, which shall have such proportion one to the other as two
given Right Lines.
THe Line given is AE, and it is required to divide the same into two parts, which shall have
such proportion one to the other, as the Line C hath to the Line D.
First, From the Point A draw a Right Line at pleasure, making the Angle BAE; then take in your Compasses the Line C, and set it from A to F; and also take the Line D, and set it from F to B, and draw the Line BE: Then from the Point F draw the Line FG, parallel to BE, cutting the given Line AE in the Point G: So is the Line A B divided into two parts in the Point G, in proportion to the other, as the Line C is to the Line D.
[geometrical diagram]
Arithmetically, let the Line AE contain 40 Perches or Foot, and let the Line C be 20, and the Line D 30 Perches; and let it be required to divide the Line AE into two parts, being in proportion one to the other, as the Line C is to the Line D.
First, Add the Lines C and D together, their Sum is 50: Then say by the Rule of Proportion, If 50 (which is the Sum of the two given Terms) give 40, the whole Line AE; What shall 30 the greater given Term give? Multiply and divide, and you shall have in the Quotient 24 for the greater
part of the Line AE; which being taken from 40, there remains 16 for the other part AG: For
As AB is to AE: So is BF to EG.
[...]
PROBL. XVII. How to divide a Triangle into two parts, according to any proportion assigned, by a Line drawn from any Angle thereof; and to lay the lesser part unto any Side assigned.
LEt ABC be a Triangle given, and let it be required to divide the same by a Line drawn from the Angle A, into two parts, the one bearing proportion to the other, As the Line F to the Line G; And that the lesser part may be towards the Side AB.
By the last Problem divide the Base of the Triangle BC in the Point D, in proportion as the Line F is to the Line G (the lesser part being set from B to D.) Lastly, draw the Line AD, which shall divide the Triangle ABC in proportion as F to G.
As the Line F, is to the Line G:
So is the Triangle ADC, to the Triangle ABD.
PROBL. XVIII. The Base of the Triangle being known, To perform the foregoing Problem Arithmetically.
SUppose the Base of the Triangle BC be 45, and let the Proportion into which the Triangle ABC is to be divided, be as 2 to 4. First add the two proportional Terms together, 2 and 4, which makes 6; then say by the Rule of Proportion, If 6, the Sum of the Proportional Term, give 45 (the whole Base BC) What shall 4 the greater Term given? Multiply and divide, and the Quotient will give you 30, for the greater Segment of the Base DC, which being deducted from the whole Base 45, there will remain 15 for the lesser Segment BD.
As 2/4 is to 45: So is 4 DC 30.
[...]
PROBL. XIX. How to divide a Triangle (whose Area or Content is known) into two Parts, by a Line drawn from an Angle assigned, according to any Proportion required.
LEt the Triangle ABC contain 9 Acres, and let it be required to divide the same into two Parts, by a Line drawn from the Angle A, the one to contain 5 Acres, and the other 4 Acres. First, measure the whole length of the Base, which suppose 45; Then say, If 9 Acres the quantity of the whole Triangle, give 45 the whole Base, What parts of the Base shall 4 Acres give? Multiply and divide, the Quotient [Page 37] will be 20 for the lesser Segment of the Base BD; which being deducted from 45 the whole Base DC, then draw the Line AD, which shall divide the Triangle ABC according to the proportion required.
If 9 Acres give 45, What shall 4 Acres give?
[...]
PROBL. XX. How to divide a Triangle given into two parts, according to any Proportion assigned, by a Line drawn from a Point limited in any of the Sides thereof; and to lay the greater or lesser part towards
any Angle assigned.
THe Triangle given is ABC, and it is required from the Point M to draw a Line that shall divide the Triangle into two parts, being in proportion one to the other, as the Line N is to the Line O; and to lay the lesser part towards B.
[geometrical diagram]
First, from the limited Point M draw a Line to the opposite Angle at A; then divide the Base BC in proportion as O to N, which Point of Division will be at E; then draw ED parallel
to AM: Lastly, from D draw the Line DM, which will divide the Triangle into two parts, being in Proportion one to the other, as the Line O is to the Line N.
PROBL. XXI. To perform the foregoing Problem Arithmetically.
IT is required to divide the Triangle ABC, from the Point M, into two parts in proportion as 5 to 2.
First divide the Base BC according to the given Proportion; then because the lesser Part is to be laid
towards B, measure the distance from M to B, which suppose 32: Then say by the Rule of Proportion, If MB 32, give EB 16, what shall AR 28 (Perpendicular of the Triangle) give? Multiply and divide, the Quotient will be 14, at which distance draw a Parallel Line to BC, namely D; then from D draw the Line DM, which shall divide the Triangle according to the required Proportion.
PROBL. XXII. How to divide a Triangle (whose Area or Content is known) into two Parts, by a Line drawn from a Point limited, into any Side thereof, according to any number of Acres, Roods, and Perches.
IN the foregoing Triangle ABC, whose Area or Content is 5 Acres 1 Rood, let the limited Point be M in the Base thereof; and let it be required from the Point M to draw a Line which shall divide the Triangle into two parts between Johnson and Powell, so as Johnson may have 3 Acres 3 Roods thereof, and Powell may have 1 Acre and 2 Roods thereof.
First, Reduce the quantity of Powell's, being the lesser, into Perches (Observe, 160 square Poles contains 1 Acre, half an Acre contains 80 Perch, a quarter or one Rood 40 Perch.) which makes 240. Then considering on which side of the limited Point M this part is to be laid, as towards B, measuring the part of the Base from M to B 32 Perch, whereof take the half, which is 16, and thereby divide 240, the Parts belonging
to Powell, the Quotient will be 15, the length of the Perpendicular DH, at which Parallel-distance from the Base BC, cut the Side AB in D, from whence draw the Line DM, which shall cut off the Triangle DBM, containing 1 Acre 2 Roods, the part belonging to Powell: Then the Trapezia ADMC (which is the part belonging to Johnson) contains the residue, namely, 3 Acres 3 Roods.
[...]
PROBL. XXIII. How to divide a Triangle according to any Proportion given, by a Line drawn parallel to one of the Sides given.
The following Triangle ABC is given, and it is required to divide the same into two Parts, by a Line drawn parallel to the Side AC, which shall be in proportion one to the other, as the Line I is to the Line K.
First (by the 16th Problem) divide the Line BC in E, in proportion as I to K; then (by the 27th Problem following) find a mean Proportional between BE and BC, which let be BF, from which
Point F draw the Line FH, parallel to AC, which Line shall divide the Triangle into two parts, viz. the Trapezia AHFC, and the Triangle HFB, which are in proportion one to the other, as the Line I is to the Line K.
PROBL. XXIV. To perform the foregoing Problem Arithmetically.
LEt the Triangle be ABC, and let it be required to divide the same into two parts, which shall be
in proportion one to the other, as 4 to 5, by a Line drawn Parallel to one of the Sides.
First let the Base BC, containing 54, be divided according to the proportion given; so shall the lesser
Segment BE contain 24, and the greater EC 30; Then find out a mean Proportional between
BE 24, and the whole Base BC 54, by multiplying 54 by 24, whose Product will be 1296; the Square Root thereof
is 36, the mean Proportional sought, wch is BF. Now if BF 36 give BE 24, what shall AD 36? The Answer is HG 24, at which distance
draw a Parallel Line to the Base, to cut the Side AB in H, from whence draw the Line HF, Parallel to AC, which shall divide the Triangle as was required.
[geometrical diagram]
[...]
PROBL. XXV. To divide a Triangle of any known Quantity into two Parts, by a Line Parallel to one of the Sides, according to any Number of Acres, Roods, and Perches.
THe Triangle given is ABC, whose Quantity is 8 Acres, 0 Roods, and 16 Perches; and it is desired to divide the same (by a Line drawn up parallel to the Side AC) into two Parts, viz. 4 Acres, 2 Roods, 0 Perches; and 3 Acres, 2 Roods, and 16 Perches.
First, Reduce both Quantities into Perches (as it is hereafter taught) and they will be 720, and 576; then reduce both these
Numbers by abbreviation into the least proportional Term, viz. 5 and 4; and according to that proportion, divide the Base BC 54 of the given Triangle in E: then seek the mean Proportion between BE and BC, which Proportion is BF 36,
of which 36 take the half, and thereby divide 576, the lesser Quantity of Perches, the Quotient will be HG 32, at which Parallel-distance from the Base, cut off the Line AB in H, from whence draw the Line HF parallel to the Side AC, which shall divide the Triangle given, according as it was required.
PROBL. XXVI. From a Line given, To cut off any Parts required.
THe Line given is AB, from which it is required to cut off 3/7 Parts.
[geometrical diagram]
First, draw the Line A C, making any Angle, as CAB; then from A set off any 7 equal Parts, as 1, 2, 3, 4, 5, 6, 7; and from 7
draw the Line 7 B. Now because is to be cut off from the Line B, therefore from the Point 3, draw the Line 3 D, parallel to 7 B, cutting the Line AB in D; So shall AD be the 3/7 of the Line AB, and DB shall be 4/7 of the same Line.
As 7 is to AB: So is A 3 to AD.
PROBL. XXVII. To find, a Mean Proportional between two Lines given.
IN the following Figure, let the two Lines given be A and B, between which it is required to find a Mean Proportional. Let the two Lines A and B be joyned together in the Point E, making one Right Line as CD, which divided into two equal Parts in the Point G; upon which Point G, with the distance GC or GD, describe the Semicircle CFD: Then from the Point E, where the two Lines are joyned together, raise the Perpendicular EF: So shall the Line EF be a Mean Proportional between the two given Lines A and B. For,
As ED
is to EF:
So EF
to CF.
9
12
12
16
PROBL. XXVIII. How to finde two Lines, which together shall be equal in Power to any Line given; And in Power the one to the other, according to any Proportion assigned.
IN this Figure let CD be a Line given, to be divided in Power, as the Line A is to the Line B.
[geometrical diagram]
First, divide the Line CD in the Point E, in proportion as A to B (by the 16th Probl.) Then divide the Line CD into two equal Parts in the Point G, and on G, at the distance GD or GC, describe the [Page 41]Semicircle CFD, and upon the Point E raise the Perpendicular EF, cutting the Semicircle in F. Lastly, draw the Line CF and DF, which together in Power will be equal to the Power of the given Line CD; and yet in Power one to the other, as A to B.
PROBL. XXIX. How to divide a Line in Power according to any Proportion given.
FIrst, Divide the Line C D in the Point E, in proportion as A to B: Then divide the Line CD in two equal Parts in the Point G, and upon G as a Center, at the distance CD, describe the Semicircle CFD, and on E raise the Perpendicular of EF, cutting the Semicircle in F: Then draw the Line CF and DF, and produce the Line CF to H, till FH be equal to FD, and draw the Line D. H. Lastly, draw the Line FK, parallel to DH: Then shall the Line CD be divided in K; so that the Square of CK shall be to the Square of KD, as CE to ED, or as B to A.
[geometrical diagram]
PROBL. XXX. How to enlarge a Line in Power, according to any Proportion assigned.
IN the Diagram of the 28th Problem, let CE be a Line given, to be enlarged in Power as the Line B to the Line C.
First (by the 16th Problem) find a Line in proportion to the given Line CE, as B is to C, which will be CD; upon which Line describe the Semicircle CFD, and on the Point E erect the Perpendicular EF: Then draw the Line CF, which shall be in power to CE, as C to B.
PROBL. XXXI. To enlarge or diminish a Plot given, according to any Proportion required.
LEt ABCDE be a Plot given, to be diminished in Power as L to K.
Divide one of the Sides, as AB in Power as L to K, in such sort that the Power of AF may be to the Power of
AB, as L to K; then from the Angle A draw Lines to the Point C and D. That done, by F draw a Parallel to BC, to cut AC in G, as FG: again, from G draw a Parallel to DC, to cut AD in H. Lastly, from H draw a Parallel to DE, to cut AE in I: So shall the Plot AFGHI be like ABCDE, and in proportion to it, as the Line L to the Line K, which was required.
Also if the lesser Plot was given, and it was required to make it in proportion to it as K to L; then from
the Point A draw the Lines AC and AD at length; also increase AF and AI: That done, enlarge AF in Power as K
to L, which set from A to B; then by B draw a Parallel to FG, to cut AC in C, as B C: likewise from C draw a Parallel to GH, to cut AD in D: Lastly, a Parallel from D to HI, as DE, to cut AI being increased in E; so shall you include the Plot ABC DE, like AFGHI, and in proportion thereunto, as the Line K is to the Line L, which was required.
[geometrical diagram]
PROBL. XXXII. How to make a Triangle which shall contain any Number of Acres, Roods, and Perches, and whose Base shall be equal to any (possible) Number given.
LEt it be required to make a Triangle which shall contain 6 Acres, 2 Roods, 25 Perches, whose Base shall contain 50 Perches. You must first reduce your 6 Acres 2 Roods, and 25 Perches, all into Perches, after this manner.
First, Because 4 Roods makes 1 Acre, multiply your 6 Acres by 4. makes 24; to which add the 2 odd Roods, so have you 26 Roods in 6 Acres 2 Roods; then because 40 Perches makes 1 Rood, multiply your 26 by 40, which makes 1040, to which add the 25 Perches, and you shall have 1065, and so many Perches are contained in 6 Acres, 2 Roods, and 25 Perches.—Now to make a Triangle that shall contain 1065 Perches, and whose Base shall be 50 Perches, do thus; double the number of Perches given, namely 1065, and they make 2130; then because the Base of the Triangle must contain 50 Perches, divide 2130 by 50, the Quotient will be 42 ⅗ which will be the length of the Perpendicular of the Triangle. This done, from any Scale of equal Parts, lay down the Line BC equal to 50 Perches; then upon C raise the Perpendicular CE, equal to 42 ⅗ Perches, and draw the Line AE, parallel to BC; then from any Point in the Line AE, as from G, draw the Line BG, and GC, including the Triangle BGC, which shall contain 6 Acres, 2 Roods, 25 Perches, which was required.
PROBL. XXXIII. How to reduce a Trapezia into a Triangle, by a Line drawn from any Angle thereof.
THe Trapezia given is ABCD, and it is required to reduce the same into a Triangle.
[geometrical diagram]
First, extend the Line D C, and draw the Diagonal BC; then from the Point A draw the Line AF, parallel to CB, extending it till it cut the Side DC in the Point F. Lastly, from the Point B draw the Line BF, constituting the Triangle FBD, which shall be equal to the Trapezia ABDC.
And so I have concluded what I did intend of Geometrical Problems: Neither had I gone so far as I have, in regard Mr. William Leybourn hath ingeniously and very fully demonstrated them in his First Book of his Compleat Surveyor. But no Book (as I remember) now extant of Navigation, hath the foregoing Problems so large. Besides, I shall direct (in the following Treatise) the Mariner to Survey any Plantation or Parcel of Land very exactly and easily, by his Sea-Compass.
THE Mariners Magazine; OR, STURMY's Mathematical and Practical ARTS. The Second Book.
The ARGUMENT.
YOu're come to see a Sight, the World's the Stage;
Perhaps you'l say, 'Tis a Star-gazing Age.
Come out and see the Ʋse of Instrument,
Can Speculation yield you such Content?
That you can rest in Learning; But the Name
Of flying Pegasus, or swift Charles-Wain.
And would you learn to know how he doth move
About his Axis, set at work by Jove?
If you would learn the Practice, read, and then
I need not thus intreat you by my Pen,
To tread in Arts fair Steps, or gain the way;
Go on, make haste, Delinquent, do not stay.
Or will you scale Olympick Hills so high?
Be sure take fast hold on Astronomy;
Then in that fair-spread Canopy no Way
From thee is hid, no not Galaxia.
They that descend the Waters deep, do see
Our great God's Wonders there, and what they be.
They that contemplate on the Starry Sky,
Do see the Works that he hath fram'd so high.
Then learn the Worlds Division, and that Art
Which I shall shew you in this Second Part.
IN this Book is contained both a general and particular Description, Making, and Use of all the
most necessary Instruments belonging to the Art of Navigation; As the Mathematical Ruler, on which are these Scales following; viz. The Line of Chords, Points, Leagues, Longitude, Natural Sines, Tangents, Secants, at one End; at the other is Dialling Scales, viz. The Art of Dialling of all sorts, resolved by the Chords and Gnomon Line, and Scale of Six Hours; Scale of Inclination of Meridians, and two Scales inlarging Hours; Lines upon any reclining, inclining, or declining, Plain without a Center, called the greater and lesser Pole: On the other side is a Line of Artificial Signs, Tangents, and Numbers; A Meridian Line, according to Mercator's or Mr. Edward Wright's Projection; And Tables for the making of these Scales, with a Line of Longitude and Reduction, which are the Lines on the Mathematical [Page 46] Scale; also, A Portable most useful Travis-Scale, with a Table for to make it, with artificial Rhombs, Points, ½ Quarters, and Tangent-Rhombs, and the making of the Sinical Quadrant, and so ordered, that by the help of an Index, and Lines thereon, it shall answer most of the useful Questions in Astronomy and Navigation. Also the making the plain Sea-Chard, and the true Sea-Chard, and particular Chards for any Place; with the most useful and necessary Semicircle, that will protract any Angle, or run upon any Chard, without drawing Rhomb-lines to fill the Chard; that so, by help of this Instrument, the Chard may serve for many Voyages. Also the Making and Use of a Compleat Instrument, made in the manner and on the back-side of a Nocturnal, with 31 of the most useful and easiest Stars to be known in the North and South Hemisphere, of the first, second, and third Magnitude; which in a Moment, the Instrument being rectified, sheweth the Hour of the Night that any Star cometh to the Meridian, with his Declination N. and S. Also a Table of the Declination, Right Ascension, Latitude, and Longitude calculated from Tycho's Tables, rectified for the year 1671. On the other side a Nocturnal so ordered, that it shall give you the Hour of the Night by the North-Star, and the brightest Guard, and his bearing every Point of the Compass from the Pole, whereby you may take the true Declination; and also being so rectified, sheweth the Suns place in each Sign and Degree in the Ecliptick every day in the year. The Making and the Use of the Cross-staff, Back-staff, Quadrant; The Making and Use of the small Pocket-Instrument, on which is contained the most useful Lines, Scales, and Proportions, that in an Instant will shew the Diameter of any sort of Ordnance at the Bore, and the length and weight of the Gun, and Shot, and Powder, in Brass or Iron; and the Diameter and Names of each Piece, Diameter of the Shot to each Piece, and the weight of any Iron Shot, the Diameter being given in Inches, with the breadth and length of the Ladle; And how many Paces point-blank any Piece will shoot, and of Randoms for the sixth Point of the Quadrant, which may by this Instrument be answered near enough for so short a time, to give any reasonable Man an answer
to any useful Question in the Art of Gunnery. Also the Description of the Mariners Azimuth-Compass, so ordered that it shall measure all kind of Grounds whatsoever, whether Wood-land
or other; and for taking of Heights and Distances, whether accessible or inaccessible:
And by the help of the aforesaid Semicircle, to protract any Plot of a Field or Plantation whatsoever, as soon as any Instrument, as the Plain Table, the Theodolit or Circumferenter, with much delight and pleasure to the Ingenious Mariner, it agreeing so well with his Travisses at Sea. All which shall be shewn in the following Treatise in its due Place.
A DESCRIPTION OF INSTRUMENTS. CHAP. I. Of Instruments in general.
THe particular Description of the several Instruments that have from time to time been invented for Mathematical Practice, would make a Treatise of it self; and in this place is not so necessary to be insisted on every of the
Inventors in their Construction. To omit therefore the Description and Superfluity of unnecessary
Instruments, I shall immediately begin with the Description of those which are the Grounds and
Foundation of all the rest, and are now the only Instruments in esteem amongst Navigators and Mariners at Sea, which are chiefly these; viz. The Mathematical Ruler, the Plain Scale, the Sinical Quadrant, the Plain Sea-Chard, and the True Sea-Chard, the particular Chard, the Semicircle or Protractor, the Nocturnal, the Cross-staff, Back-staff, and Quadrant; the Gunter's Scale, and the Mariner's Azimuth-Compass. Now as I would not confine any Man to the Use of any particular Instrument for all Imployments; so I would advise any Man not to incumber himself with Multiplicity, since these
aforesaid are sufficient for all Occasions. These special Instruments have been largely described already by divers; As namely, by Mr. Blundevil, Mr. Wright, Mr. Gunter, and others: but not fitted with Tables for the making of them, or demonstrated so plain to the Capacities of Seamen, as they are here. Therefore in this place it will be very necessary to give a particular
Description of them, because that if any Man hath a desire to any particular Instrument, he may give the better direction for the making thereof, or making of it himself.
Forasmuch as there is a continual use both of Scales and Chords, which are on the Mathematical Scale, in drawing of Schemes in the Art of Navigation, and all other sorts in this Treatise; Therefore we will demonstrate the fundamental Diagram of the Mathematical Scale, that all Mariners may understand (that have not the knowledge already) the making of them, which is
a most commendable Vertue in an expert Mariner. I could wish that all Masters and Mates were able to make their own Instruments, that if they should be long at Sea, and by disaster break or lose their Instruments; or if any in the Ship discovers the Practice, he may be able to make more for himself
and others, without the help of the Artificer's Labour, and supply that defect by their own pains.
This Diagram plainly sheweth the making of the Scale of Degrees or Chords, and Points of the Mariner's Compass, in a Right Line B 8, being the Degrees, containing in all 90; and F 8 is the Scale for the Points of the Compass, being in all 8 Points for the ¼ part of the whole Circle.
Now for the Sines, Tangents, and Secants, you shall note, That the Semi-diameter AB must be divided into a Radical Number, for the more ease in Calculation; as into 100, or 1000, 10000, 100000; and that by
the Table of Natural Sines, [Page 48] Tangents, Secants, Chords, and Points, which I have fitted on purpose for this Work. You may take off so many Numbers as the Table directs you, as shall be shewn.
[geometrical diagram]
Here followeth a Table of 90 Degrees of the Quadrant. He that desires it larger, may make it to the Parts of a Degree. I have joyned the Chord proper to it, which is the Natural Sine of half the Arch doubled.
For Example, If you double the Natural Line of 6. 15. 25. 30 Deg. you shall produce the Chords of 12. 30. 45. 60 Degrees; thus 10453 is the Sine of 6 Degrees, being doubled, the Sum will be 20906 the Chord of 12 Degrees; and so of the rest, as in the Table following.
This done, Proportion the Radius of a Circle to what extent you please; make AB equal thereto, which must be divided into equal
Parts, as before-directed, by half thereof, and this Table, the Chord of any Arch proportionable to this Radius, may speedily be obtained. As for Example, Let there be required the Chord of Thirty Degrees, the Number in the Table is 518; or in proportion to this Scale of 100 equal Parts, AB is 52 almost; I take therefore 52 from the Scale of equal Parts, and set them from B towards 8 to h and o, and draw the Line h o, which is the Chord desired 30 Degrees: Thus may you find the Chord of any other Arch agreeable to this Radius. Or if your Radius be of a greater or lesser extent, if you make the Base of your Right Angle AB equal thereunto, You may in like manner find the Chord of any Arch, agreeable to any Radius given. Only remember, That if the Chord of the Arch desired exceed 60 Deg. AB which is divided into 100 equal parts, you must continue the Base AB in the division of such parts, as need shall require.
In this manner is made the Line of Chords in the Fundamental Diagram answerable to that Radius.
And in this manner you may find the Chord of the Rhomb, Points, halfs, and quarters, and the Sines, Tangents, and Secants of any Arch proportionable to any Radius, by help of these Tables following [which is an abbreviation of the Canon of Natural Sines, Tangents, and Secants] and proportioning the Base AB thereunto, which is the Scale of equal parts; as by Example may more plainly appear.
A Table for the Angles which every Rhomb maketh, with the Meridian, and the Chords of every Quarter and Point of the Compass.
Let there be required the Chord of the first Point of the Scale, 11 Deg. 15, in this Table, as I have fitted for every Point, Half, and Quarter, for ¼ of the Compass.
The Numbers: answering to 11 Deg. 15 Min. is 195. I take therefore with my Compasses 19, or reckon so many on the Scale of Equal parts, which is joyned with a Scale intended to be made; and so with a Square for that purpose, as shall be shewed, mark from F towards 8 the first Point 11 Deg. 15, where the Radius of the Circle is AB; and so of the rest.
The Scale of Longitude.
THis Scale is made also by the Table of Degrees and Chords, as before.
EXAMPLE.
It is required to know how many Miles make a Degree in the Parallel of 10 Deg. If you extend the Compasses from A, to the Complement of the Latitude 80 Deg. in the Line of Sines, and setting one Foot in F, turn that distance from F toward A, you will find it reach
59 Miles nearest, in the former Diagram.
Another EXAMPLE.
It is required in the Latitude of 60 Degrees to know the Miles answering to a Degree. In that Parallel extend the Compasses from A to the Complement of the Latitude 30, in the Line of Sines; and setting one Foot of the Compasses in F, turn that distance towards A, and you will find it reach 30 Miles, that makes a Degree in that Parallel; and so of the rest.
But if it be required how to make a Scale of Longitude in Miles answerable to the Radius of the same Scheme, for the Parallel of 10 Degrees, you will find in the Table, the Chord for 10 Degrees is 17.5 for the first Mile, and for 60 Degrees 1000, take 100 from A to B, as you was before-directed, and so do with the rest,
until you have made the whole Scale. Remember, that 60 Miles must begin where the first Degree of the Chords doth on the Scale, and so diminish towards the Pole 90 Degrees of the Scale, as reason will give you,
SINES.
NOte, That a Sine falls always within the Quadrant of a Circle, as CD, which is the Sine of the Arch BC 60 Degrees; and by the Table of Natural Sines, to every Degr. of the Quadrant which I have fitted for this purpose, whose Radius is 1000, you shall find the Sine of 60 deg. to be 86.6. I take therefore with my Compasses 86 from my Scale of Equal Parts, and set them from A towards 8 in the Line of Sines for 60 Degrees, where the Radius of the Circle is AB, and CE is the Complement thereof, or Sine of 30 Degrees of the Arch C 8, the Number in the Table answering 30 Degrees is 500; take therefore with your Compasses 50 equal Parts of A B, and lay it from A upon the Line of Sines for 30 towards 8; and so of the rest.
A Table of Natural Tangents to every Degree of the Quadrant.
De
Tan.
De
Tan.
De
Tan.
De
Tan.
De
Tangents.
1
17
19
344
37
753
55
1428
73
3270
2
34
20
363
38
781
56
1482
74
3487
3
52
21
383
39
809
57
1559
75
3732
4
69
22
404
40
839
58
1600
76
4010
5
87
23
424
41
869
59
1664
77
4331
6
105
24
445
42
900
60
1732
78
4704
7
122
25
456
43
932
61
1804
79
5144
8
140
26
487
44
965
62
1880
80
5671
9
158
27
509
45
1000
63
1962
81
6313
10
176
28
531
46
1035
64
2650
82
7115
11
194
29
554
47
1072
65
2144
83
8144
12
212
30
577
48
1110
66
2246
84
9514
13
230
31
600
49
1150
67
2355
85
11430
14
249
32
624
50
1191
68
2475
86
14300
15
267
33
649
51
1234
69
2601
87
19081
16
286
34
674
52
1279
70
2747
88
28336
17
305
35
700
53
1327
71
2904
89
57289
18
324
36
726
54
1376
72
3177
90
0000000
Infinite.
A Tangent Line is always falling without the Quadrant, and is drawn at the end of a Semidiameter at Right Angles, as B 6 in the Fundamental Diagram, which is the Tangent of the Arch BC 60 Degrees, as in the Table of Tangents you shall find it to be 1732 equal parts, which take with your Compasses from A, when you have continued the Line beyond B, take 173 parts, and that will reach from B to G, the Tangent of 60 Degrees in the Scale, and 8 H is the Complement Tangent 30 Degrees 577 parts; therefore take 57 parts, it will reach from B to the length of 30 Degrees; and so of the rest.
A SECANT.
A Table of Secants to every Degree of the Quadrant.
De
Sec.
De
Sec.
De
Sec.
De
Sec.
De
Secants.
1
1000
19
1057
38
1269
56
1788
74
3627
2
1000
20
1064
39
1286
57
1836
75
3863
3
1001
21
1071
40
1305
58
1887
76
4133
4
1002
22
1078
41
1325
59
1941
77
4445
5
1003
23
1086
42
1345
60
2000
78
4809
6
1005
24
1094
43
1367
61
2062
79
5240
7
1007
25
1103
44
1390
62
2130
80
5758
8
1009
26
1112
45
1414
63
2202
81
6392
9
1012
27
1122
46
1439
64
2281
82
7185
10
1015
28
1132
47
1466
65
2366
83
8205
11
1018
29
1143
48
1494
66
2458
84
9566
12
1022
30
1154
49
1524
67
2559
85
11473
13
1026
31
1166
50
1555
68
2669
86
14335
14
1030
32
1179
51
1589
69
2790
87
19107
15
1035
33
1192
52
1624
70
2923
88
28653
16
1040
34
1206
53
1661
71
3071
89
57298
17
1045
35
1220
54
1701
72
3236
90
0000000
18
1051
36
1228
55
1743
73
3420
Infinite.
37
1252
A Secant Line is drawn always from the Center of the Circle, until it cut the Tangent Line; as A G in the foregoing Diagram cuts the Tangent of the Arch BC 60 Degrees in G: so is AG the Secant of 60 Degrees, which in this Table of Secants is found 2000 equal parts; therefore take off such parts as are in proportion to
AB 200, it shall reach from A to G for the Secant of 60 Degrees, and AH is the Complement-Secant, or Secant of the Arch 8 C, 30 Degrees, which in this Table of Secants is found to be 1154; therefore take with your Compasses, or other Instruments, 115 equal parts, and it shall reach from A towards G for the Secant of 30 Degrees, as you may find by the Scale in the Diagram.
A Versed Sine is found by substracting his Complement-Sine out of the Radius. Example. For to know the Versed Sine of 60 Degrees, you must substract EC or AD, which is the Complement or Sine of 30 Degrees, viz. 500 out of the Radius 1000, or Sine of 90, AB, the remain will be DB 500, for the Versed Sine of the Arch BC 60 Degrees. In like manner E 8 will be found 134 for the Versed Sine of the Arch C 8, being 30 Degrees; and so work in like manner for any other Degree. The Word Versed is a sufficient Direction, to let them understand, that do not, That the Degrees of this Scale, or sort of reckoning, begins at B or F, and continues to 180 Degrees, the Diameter of the Circle, or the Line of Sines Reversed, by putting the two beginnings of Degrees together of the Quadrant or Seale, and so begin to count at one End; for 80 Degrees must be placed 10, for 70 Degrees 20 Deg. and so to 180; and of the first 90 or middle of the Scale, count the Sun's greatest Declination 23 Degrees 30 Min. towards both ends, that is, 47 Degrees asunder in that distance; by the side thereof must be placed the Reversed six Northern Signes, according to the Sun's Declination, and place in the Ecliptick at such Declination: And likewise 23 Degrees 30 Min. the space for dividing the Reversed Southern Signes toward 180; and are reckoned double, as occasion requireth.
Either of the Semidiameters AB or AF, the Sides of the Quadrant, you may take the equal divisions thereof, and make a Scale of Leagues or Miles, or Equal Parts, for the demonstration of all plain Triangles, which you cannot be without, having it upon the Ruler.
CHAP. II. A Description of what Instruments of Brass, Steel, Iron, and Wood, you must be provided with before you can make Instruments for Mathematical Uses.
BEfore we explain the other half of the Fundamental Diagram and Semicircle, it will be necessary for to give a Description of what Innstruments in Brass, Steel, Iron, or Wood, you must have by you in readiness, before you can make a Mathematical Instrument; That Men that are ingenuous may be provided in some measure with such, before they
go to Sea, in spending their spare-time on this Practice. In brief, they are these. First, For Instruments of Wood, you must be provided with several Scales of Equal Parts,Scales of equal parts of several lengths, which must be exactly and carefully divided, the length you intend
to make the Radius of the Instruments by it. First, divide this Line into 10 equal parts, and each 10 into 10 more; so is your Line divided into 100; and so you may continue it into 200, 300, 400, so much as you please,
as the Instrument you are making will require; which you may quickly see by the Table. You must be fitted with some pieces of Box (dry,Scales of Box or other Wood. clean from Knots, straight, and smooth planed) or other Wood, on which you may make what Scale you please. You must have by you a true Square of Brass and Wood,2 Squares. such as you may see in this Figure,2 Cramps. with a pair of Cramps made of Iron, with Screws to fasten the Scale of Equal
[geometrical diagram]
[Page 53] Parts, and the Scale to be made together, so as they may not slip, whereby may be made no mistake in Graduating.
Or for small Scales,Scales fastened with Nails, on Deal-Boards. Brass a Gauge. you may fasten the Scale of Equal Parts, and the Scale to be made by it, on a piece of Deal Board, with the Heads of Scuper Nails, so as they will not stir; but for greater Instruments, and Cross-staves, and Gauging-staves, you must do by them as in this Figure. You must have a Gauge made of Brass, with a good Steel Pin, for the drawing of straight Lines on your Scale, for the division of the Columnes for Graduation. You must have two or three Sorts and Sets of Steel Letters and Figures, and Figures for Ornament,Steel Letters and Figures. with a neat Hammer to use with them: And the Figures, and Letters, and Ornament-Figures, set in an Alphabet-Box, with written Letters and Figures before them, for the ready finding of them; with Characters of the Signes, and Planets, and Stars, in like manner.
The Instrument that you graduate with, the Edge must be very thin and sharp,Gauging Instruments. Brass Compasses, together with an Arch & 4 Points, and you may have several of them; or the end of a Pen-knife may do for a shift. You must have a Brass pair of Compasses to go with an Arch and Screws, to fasten at any distance; and four Steel Points to take in and out; two long Points for to reach a great distance. I have a pair by me will extend 3 Foot,; on a large Scale of Artificial Sines, Tangents, and Numbers, they are to be used. The other are short Points. One is to be made round for a Center-Point, that it may not go too far into the Wood; and the other pointed like a Dutch Knife, and the Shoulder fitted square as the other Points, to be fastned, and taken in and out at pleasure.To be taken in and out The use. A pair of Dividers. The Use of these Points is to draw Circles on round Instruments, as Nocturnals, and the like. You may have two pair of Dividers, the least 3 Inches and ½, and the biggest 7 ½ Inches long. I hold them best that go with a Bow at the Head, and to be set together by
a Screw in the midst. Be sure they be made of good Steel. These are to divide equal Parts, and any other equal Division. You must have for
great Instruments, as Bows, Quadrants, and the like, a pair of Beam-Compasses, for to sweep the Arches of them.Beam-compasses. Hand-vice. Files. You should have a Hand-Vice, so made as to screw into the edge of a Board for your use, and to take out again; with three or four sorts of small Files, for to file and make Pins, which you will have occasion for. These Brass and Iron Instruments or Scales you may now give direction to an ingenuous Smith (Thomas Moone) in Bristol (if you cannot have them before of these sorts) and he will fit you with them:Where they may be made or bought. Or you may have them ready made of Walter Hayes, at the Cross-Daggers in Moor-Fields, with many useful Instruments in Brass; Or of Andrew Wakely, Mathematician, at his Shop on Redriff-Wall, near the Cherry-Garden Stairs; Or in Bristol of Philip Staynred, Math.—And now I have shewn the Practitioner what Instruments he must be furnished with, I will return to the Explanation of the other half of
the Fundamental Diagram of the Mathematical Ruler. I had almost forgot a Receit for setting off the Graduation, when it is newly done on Box-Instruments, which is this. Take Charcoal,To set off the Instrument. and beat it to a fine Powder, and temper it with Linseed-Oyl; and let it be rubb'd on the Instrument newly made, and lie so on it for a time, untill it be pretty dry; and then with some
S [...]llet-Oyl rub the Instrument, and make it clean: So will you have the Graduation and Figures set off very neatly on Box Instruments, with Black.
CHAP. III. The Explanation of the other half of the former Semicircle; being a Description of the Fundamental Diagram, of the Dialling-Scale on the Mathematical Ruler.
THis annexed Diagram sheweth plainly the Description of the Dialling-Scales on the Mathematical Ruler; It being the most easie and exact Instrument used in that Art, as by the use will be manifest in the Seventh Book.
[geometrical diagram]
How to make the Diagram.
FIrst, Make a Semicircle by a less Radius, as ADB, and upon the midst of the Arch at D, with the distance DA describe the Quadrant-Arch, as AEB, which must be divided into six equal Parts, for the 6 Hours in the ¼ of the Sphere; which is sufficient to resolve the whole; and from each Point draw Lines to the Center at D; So will it cut the Line AB in 1, 2, 3, 4, 5, 6, for the Hour-Lines upon the said Scale for Dialling: and thus you see it is a Tangent Line, for which use it is more certainly done by this Table of Natural Tangents for three Hours, if you do but observe where the Right Line DE cuts the Tangent Line AB, which you see in the middle or Center of the Semicircle at R; therefore you must begin to make this Scale in the midst, and lay the distance of parts answering the Hours both ways from R towards [Page 56] B and A: As by Example, To Graduate 2Hours and 4 Hours, you see in the Table, the Numbers answering to 2 Hours and 4 Hours in the first Column to the left hand; is in the second 60 Minutes, or in the third 15 Degrees, and in the fourth Column the Tangent-parts 267; therefore if you take 267 such Parts whereof the Semidiameter RB is divided into 1000, as was shewed in the former Diagram, and put one Foot of the Compasses with that extent at R the midst or 3 Hours, and turn the other toward B, it will make the distance of 4 Hours; and turn that distance towards A, it will be 2 Hours of the Scale: And so do with the rest of the Hours, and distance of the Minutes.
A Table for the dividing of the Hours and Minutes upon the Dialling-Scale.
Ho.
M
D. M.
Tan. par.
3
10
2 30
43
20
5 00
87
30
7 30
131
40
10 00
176
50
12 30
221
2 4
60
15 00
267
10
17 30
315
20
20 00
363
30
22 30
414
40
25 00
466
50
27 30
520
1 5
60
30 00
577
10
32 30
637
20
35 00
700
30
37 30
767
40
40 00
839
50
42 30
916
0 6
60
43 00
1000
In like manner for the Scale of Inclination of Meridians, you must take out the Tangent-parts out of the Table of Tangents to every Degree, and graduate in the same manner as before, from the Center which is the midst of the Scale 45 Degrees, as is shewn plain in the Diagram.
For the Gnomen-Line, as others call it the Line of Latitude, Let BA be the Semidiameter; so on B describe the Quadrant ABC, whose Arch AC divide into 90 Degrees, from whence you may project the Line of Sines BC.
Now from each Degree of those Sines, draw Lines toward the Center of them at A, and note where they cut the Arch of the Quadrant BD: Then from B as a Center, take the distance of each of these Intersections, and lay them on the Line BD; so shall you have the Division of the Gnomon-Line, or Line of Latitude.
For the more ready making of this Scale, here is a Table of Latitudes calculated to the 90 Degrees of the Quadrant, and the way to calculate it your self. As for Example, To find the Latitude-parts for 30 Degrees of Latitude,
A Table of Latitudes for Dialling.
Deg.
Par.
Deg.
Par.
Deg.
Par.
Deg.
Par.
Deg.
Par.
Deg.
Par.
80
992
60
926
45
817
30
632
15
354
78
989
59
920
44
807
29
617
14
332
76
985
58
915
43
797
28
601
13
310
74
980
57
909
42
787
27
585
12
288
72
975
56
903
41
776
26
568
11
265
90
1000
70
969
55
896
40
765
25
551
10
242
89
69
965
54
890
39
753
24
533
9
219
88
68
962
53
883
38
741
23
515
8
195
87
67
958
52
875
37
729
22
496
7
171
86
66
954
51
868
36
717
21
477
6
147
85
998
65
950
50
860
35
704
20
458
5
123
84
64
945
49
852
34
690
19
438
4
98
83
63
941
48
844
33
676
18
419
3
74
82
62
936
47
835
32
662
17
397
2
49
81
61
931
46
826
31
648
16
376
1
25
First, Find the Sine thereof in the Natural Table of Sines, which will be found to be 50000; which sought for in the Table of Tangents, giveth an Arch of 26 Deg. 34 Min. Then the Proportion will hold,
As the Radius
100000
To the Secant 45 Deg.
141421
So is the Sine of 26 Deg. 34 Min.
44724
Ʋnto the Latitude-parts
63249
Which answers to the Radius 100000: But in my Table the Parts 632 answer to the Radius 1000, which will be sufficient for the Graduating the Line of Gnomons or Latitude.
But observe, To make 30 Degrees of Latitude on your Scale, you must take off [Page 57] 632 such Parts as the Line is divided into 100, or 1000, as you have been shewn in the former Diagram.
How to make the Line of Chords, you have been fully instructed already in the former Figure; which is only by dividing the Arch of the Quadrant AD into 90 equal parts; And from A as a Center, take the distance, and lay them down in a straight Line AD: So shall you have the Line of Chords or Sublemes. Or you may do it by the Table of Chords, as before-directed.
How to make the two Lines or Scales of Inlarging Hour-Lines upon any reclining Plain, without a Center, called by me the greater and the lesser Pole.
Invexed, you have a Table ready fitted for the making thereof.
First, You must make choice of the length of this Scale, that is in Proportion to the former Lines of the Scale.
A Table of Tangents for 5 Ho. to every 5 Min. of an Hour, for inlarging the Hour-Line Scale.
Hours.
Mi.
Deg Min
Tan. pa
Hours.
Mi
Deg. Min.
Tan. pa
5
1 15
21
5
46 15
1044
10
2 30
43
10
47 30
1091
15
3 45
65
15
48 45
1140
20
5 00
87
20
50 00
1191
25
6 15
100
25
51 15
1245
30
7 30
131
30
52 30
1303
35
8 45
153
35
53 45
1363
40
10 00
176
40
55 00
1428
45
11 15
198
45
56 15
1496
50
12 30
221
50
57 30
1569
55
13 45
244
55
58 45
1647
1
60
15 00
267
4
60
60 00
1732
5
16 15
291
5
61 15
1822
10
17 30
315
10
62 30
1920
15
18 45
339
15
63 45
2027
20
20 00
363
20
65 00
2044
25
21 15
388
25
66 15
2272
30
22 30
414
30
67 30
2414
35
23 45
440
35
68 45
2571
40
25 00
466
40
70 00
2747
45
26 15
493
45
71 15
2945
50
27 30
520
50
72 30
3171
55
28 45
548
55
73 45
3430
2
60
30 00
577
5
60
75 00
3732
5
31 15
606
10
32 30
637
15
33 45
668
20
35 00
700
25
36 15
733
30
37 30
767
35
38 45
802
40
40 00
839
45
41 15
876
50
42 30
916
55
43 45
957
3
60
45 00
1000
The first 3 Hours must be divided into 10 parts, and each of them into 10 more, which stand for 100,
or as you have been shewd for 1000. You must have two of these Lines of Equal parts, of two proportionable Lengths, for the greater and lesser Pole; And so take of the Tangent-parts answerable to every 5 Minutes of an Hour: As you see the first and second Columns of the Table are Hours and Minutes, the third Degrees, and the fourth Tangent-parts. So the Tangent of the first 2 Hours of the Scale or 30 Degrees, is 577 Parts; take of your two Scales 57 Parts; First of the largest Radius for 2 Hours on the greater Scale, and the like number of the smaller Radius, or Line of Equal Parts for 2 Hours of the lesser Pole-Scale. And so in the same manner you must work to finish the whole Scales of what Radius you please, by these Tables, as hath been directed.
The Use hereof is fully shewn in the Seventh Book, 29th and 30 Chap. of the Art of Dialling.
These Scales are sufficient to make any sort of Dials, in any Latitude (as is there shewn) with ease and exactness.
There are two Lines called by the Names of Style and Substyle-Scale; but is only for this Latitude, but may be found for any. But the Scales before-explained are most useful, and do the same, as you will find in the Seventh
Book and Twelfth Chapter of the Art of Dialling. And these are the Scales of one Side of the Ruler.
CHAP. IV. The Scales or Lines on the Back-side of the Mathematical Ruler, are these: A Line of Numbers, A Line of Artificial Tangents, A Line of Sines, A Meridian Line according to Mercator's or Mr. Wright's Projection; and the Scale of Equal Parts, by which the Numbers were taken off for the Graduating these Scales; and a Line of Longitude or Equinoctial, with a Scale of Reduction, as followeth.
I. HOw to divide the Line of Numbers is thus. You must prepare a Ruler of what length you please, and also a Scale of Equal Parts, divided into 100 or 1000: You must count them. But if you divide the Artificial Tangents and Sines with the Line of Number, you were best to divide the Line into 2000 Parts; so will you have 100 on the Line of Numbers. This Table is taken out of the Logarithms, by rejecting the Index or first Figure. It is best to omit the first Number, by reason they will take up so much room; and begin at 1 or 11, and take the Logarithm-part at 41 for the first 10th or Integer. But if you intend to make 100 on your Line of Numbers, first take 100, which is reckoned 1000, as you see in the foregoing Table, Parts of the Scale of Equal Parts, for the first 10 or middle of the Scale: Then suppose you were to make the first 2 or 20, take with your Instrument or Compass 301 equal Parts, and lay it from 1 to 2, and the same distance will reach from 10
in the middle to 20. In the like manner do with the rest; for 3 or 30 the equal parts
is 477, and for 4 or 40, the Log. parts is 602: So you may easily perceive how to do it, by what is written.
A Table for the Division of the Line of Artificial Numbers.
II. How to make the Line of Artificial Tangents on the Ruler.
THe Artificial Tangents are made in the same manner as before directed, beginning upon a Right Line of Numbers, omitting the first 30 Minutes, and beginning at 40 Minutes. The Tangent-parts are 106, taken off the former Scale, and applied as before-directed upward, will make 40 Minutes on your Scale: So the first and 89 Degree, the Tangent-part answering thereunto is 241; with them do in like manner, and so of the rest, until
you have finished the whole Line or Scale, as you may see in the Figure.
The following Table is so plain to be understood, that I need write no more, but, That the first Column to the left hand is Minutes, The second Tangent-parts answering to the Minutes and Degrees over each Column to 30 Degrees, and after to every 20 Minutes, as you may see in the Table.
III. A Table for the Division of the Line of Artificial Tangents to 45 Deg. and the Minutes fit to be set thereon.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Deg.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Tang. parts.
Minutes.
89
88
87
86
85
84
83
82
81
80
79
0
1
2
3
4
5
6
7
8
9
10
11
0
00
241
546
719
844
941
1021
1089
1147
1199
1246
1288
10
46
308
577
742
862
956
1033
1099
1156
1207
1253
1295
20
76
366
610
765
879
970
1045
1106
1165
1215
1260
1301
30
96
410
640
786
895
983
1056
1119
1174
1223
1267
1308
40
106
463
668
806
911
996
1067
1129
1183
1231
1274
1314
50
162
505
694
826
927
1009
1078
1138
1191
1238
1281
1321
Min.
70
12
13
14
15
16
17
18
19
20
21
22
23
0
1327
1363
1396
1428
1457
1485
1511
1536
1561
1584
1606
1627
10
1333
1369
1402
1433
1462
1489
1516
1541
1564
1587
1610
1631
20
1339
1374
1407
1438
1466
1494
1220
1545
1568
1591
1613
1634
30
1345
1380
1412
1442
1471
1498
1524
1549
1572
1595
1617
1638
40
1351
1385
1417
1447
1476
1503
1528
1553
1576
1599
1620
1641
50
1357
1391
1422
1452
1480
1507
1532
1557
1580
1602
1624
1645
Min
60
24
25
26
27
28
29
30
31
32
33
34
35
0
1648
1668
1688
1707
1725
174 [...]
1761
1738
1795
1812
1828
1845
10
1651
1671
1691
1710
1728
1746
20
1655
1675
1694
1713
1731
1749
1767
1784
1801
1818
1834
1850
30
1658
1678
1697
1716
1735
1752
40
1662
1681
1700
1719
1737
1755
1773
1790
1806
1823
1839
1855
50
1665
1684
1704
1722
1742
1758
Min
50
45
36
37
38
39
40
41
42
43
44
45
0
1861
1877
1892
1908
1923
1939
1954
1969
1984
2000
20
1866
1882
1898
1913
1928
1944
1959
1974
1989
40
1871
1887
1903
1918
1934
1949
1964
1979
1994
IV. How to make the Scale or Line of Artificial Sines to 90 Deg. and Minutes fit to be set thereon.
HOw to make this Line, was shewn by making the last; only that is sufficient to 45 Degrees, and this must be to 90 Degrees. And if your Line of Equal Parts be divided into 100, or as they be reckoned 1000, you may omit the last Figure of the Number: But if you number the Scale to 2000, as the Tables are made to, if you would set off the Sine of 30 Degrees, the Parts answering thereunto is 1698; therefore take off your Scale of Equal Parts with your Compasses 169, and it will reach from the beginning, to 30 Degrees on the Line of Sines.
So I hope you understand how to do the rest, it being made so plain and easie for
the meanest Capacity, by what hath been writ already.
V. How to make a Meridian Line according to the true Sea-Chard, or Mercator and Mr. Wright's Projection.
THis Line is made out of the Table of Meridian parts, called also the Division of the Meridian Line. To every 10 Minutes of Latitude, nearer we have no Chards or Plots made, which I have as yet seen; but they may be made by Mr. Wright's Tables to every Minute, if any Person will be so curious.
For the Graduating this Line in the Scale, you must note the Number answering to the first Degree is 200; therefore divide the Degrees of the Aequinoctial into 20 equal Parts, which stand for 200 of the Numbers of your Table. As by Example,
Suppose you would make the first 10 Degrees from the Aequator, towards either of the Poles, on the Scale; the Number answering 10 Degrees is 201, omitting the last Figure 0: Therefore you may take out of the Line of Longitude (which is Equal Parts, or the same Line by which you made all the rest) 201 or 20 Parts, and lay that distance for the beginning
of the first 10 Degrees; and for 20 Degrees 40, 8; and for 30 Degrees 62, 9; and so of the rest.
But if you are to make a particular Line, you must take the difference of the Degrees and Minutes, as shall be fully shewn in the Treatise of making a general and particular Sea-Chard, according to Truth, and Mr. Wright's Projection; but what hath been done already will serve for both, if you follow
direction.
There is demonstrated and shewn the making of Mercator's Scale, to measure Distance in any Parallel of Latitude in any true Sea-Chart.
VI. How to Calculate a Table, and by it how to take out the Numbers, and make a Scale of Reduction, to be used in Surveying of Land.
A Table for the Division of the Scale of Reduction.
Statute Acres.
10
272 25
11
255 00
12
189 06
13
161 10
14
138 91
15
121 00
16
106 35
16 ½
100 00
17
094 21
18
84 03
19
75 42
20
68 7
21
61 73
22
56 25
23
51 47
24
47 27
25
43 56
26
40 27
27
37 35
28
34 73
29
32 37
30
30 25
31
28 33
32
26 59
33
25 00
34
23 55
35
22 21
36
21 1
37
19 89
38
18 85
39
17 90
40
17 02
THis Scale consisteth of one part or Line of Numbers and Artificial Sines on the Ruler, for the more ready use thereof, as will be shewed: I shall first shew the Calculation
and Proportion used in making the Table, which is thus as followeth.
Example, For a Perch whose Measure is 21 Foot (which is the Irish Chain) this must be done by the back Rule of Three.
As 16½ squared, to 100 Acres:
So is 21 squared, to 61, 73 Acres.
So a piece of Ground being measured by the Statute-Chain of 16½ Foot to the Perch, should contain 100 Acres.
Then the same piece of Ground being measured by the Irish Chain of 21 Foot, will contain but 61 73/100 Acres, as you may see in the Table, which is near 61 Acres 2 Quarters 38.
By the Line of Artificial Numbers extend the Compasses from 16 ½ to 21, the same will reach twice repeated from 100 unto 61, 73 in the same
Line of Numbers.
To make this Line on the Scale, Take the Numbers off the Line of Numbers of the same Scale you make this Line upon.
I shall place the first 10 on the Scale, to this Number answers 272. 25″; therefore extend the Compasses from 100 to 272 ¼, or from 27 2/10, and lay one Foot of the Compasses at A, and the other will reach to B the distance to 16 ½. From 16 ½ you must lay
all your other Numbers. As suppose you would set down 14 on the Scale, the Statute Numbers answering thereunto is 138 and 91″. Extend the Compasses from 100 to 138 and 91″, and that distance will reach from 16 ½ at B, to 14 of the
Scale. The like if you would set off 20, the Numbers to that is 68. 7; and that distance will reach from 16 ½ at B, to 20: and so do with
the rest. Thus have I done with this Scale, being sufficient to resolve all manner of Mathematical Conclusions whatsoever. The Use follows in the succeeding Treatise.
[geometrical diagram]
CHAP. V. A Table for the Division of the Artificial Rhomb, or Points, Halfs, and Quarters on the Travis-Scale.
Points.
Nor. South.
Deg. Min.
Sine parts
Tang Rhomb.
Tang. qu.p
N. b. E. 1
2 48
688
689
S. b. E. 2
5 37
990
992
S. b. W. 3
8 26
1166
1
1177
1
N. b. W. 4
11 15
1290
7
1298
N. N. E.
14 3
1385
1398
S. S. E.
16 52
1462
1481
S. S. W.
19 41
1527
2
1553
2
N. N. W.
22 30
1582
6
1617
N. E. b. N.
25 18
1630
1673
S. E. b. S.
28 7
1673
1727
S. W. b. S
30 56
1710
3
1777
3
N. W. b. N.
33 45
1744
5
1824
N. E.
36 33
1774
1870
S. E.
39 22
1802
1914
N. W.
42 11
1827
4
1957
4
S. W.
45 00
1849
4
2000
N. E. b. E.
47 48
1869
S. E. b. E.
50 37
1890
S. W. b. W.
53 26
1904
5
N. W. b. W.
56 15
1919
E. N. E.
59 03
1933
E. S. E.
61 52
1945
W. S. W.
64 41
1956
6
W. N. W
67 30
1965
E. b. N.
70 18
1973
E. b. S.
73 7
1989
W. b. S.
75 56
1986
7
W. b. N.
78 45
1991
East & West.
81 33
1995
84 22
1997
87 11
1999
90 00
2000
THe Use of this Tabe is easily understood: The first Column is the Number of Points in one quarter of the Compass, and the second their Names in the whole; The third the Degrees answering to each quarter of a Point in the Quadrant; The fourth the Sines and Equal Parts answering thereunto; The fifth the Tangent-Rhombs; The sixth the Tangent-parts answering to each Quarter and Point to 45 Degrees ½, which is sufficient.
The making of this Scale is all one in a manner as you made the former; only the Line of Sines is there but once made, and here the Parts answering each Quarter are twice put down, or in two Lines marked with N. S. which stands for to shew the Line to be Northing, Southing; and E. W. signifies Easting and Westing. The first is the Sine, the second is the Complement that any Point or Quarter maketh an Angle with the Meridian. The Line marked with T. is the Tangent-Rhomb and Quarters, and the first Line is a Line of Numbers, which you have been already shewn to make.
The Traverse Scale.
One Example I will give the Learner, notwithstanding it is so easie; for some there are that will not understand, though
they see it often done; yet (to my knowledge) are Mates to good Ships.
EXAMPLE.
Suppose you was to set the first and seventh Rhomb or Artificial Point on the Scale, which is 11 Degrees 15 Minutes, the Equal Parts answering thereunto is 1290; therefore take 129 of your Scale of Equal Parts, and lay it from the beginning upwards, and you have by that distance the first and
seventh Rhomb of your Scale.
In like manner do for any other of the Points and Quarters by these Numbers, until you have finished the Scale; and when you have done, you have an Instrument the most easie, ready, and necessary that I know of, for the working of Travises, and correcting your dead Reckoning, which shall be shewn in the Part of Sailing by
the Plain Chard, in the Fourth Book. On the back side of this Scale you may set a Line of Chords and Equal Parts, and Points, for the ready protraction of Angles.
CHAP. VI. How to make a Quadrant which will resolve many Questions in Astronomy, by the help of an Index; and also very useful in Navigation.
AFter you have made choice of the Radius of your Quadrant CD, draw Parallel Circles thereunto, to hold the Degrees of the Quadrant and Columns, for the Figures, Points, and Quarters, as P 8. Then divide the Semidiameter or Side of the Quadrant CD and CM into 60 equal parts, and draw Parallels [Page 65] to each of the Divisions, as Sides, CD and CM. First divide them into 6, and then
each of them into 10 more, as you see in the Figure; at every 5 Parts make a Point, for the ready numbring of the Divisions.
fol.65.
Make an Index answerable to the Radius CD, with a Line of Sines on the first Side, and a Center-Ear to put over the Center-Brass-Pin of the Quadrant C, as occasion shall require: And on the other Edge make a Line of Equal Parts, equal to the 60 Divisions of the Side CD, with an Ear in like manner to remove at pleasure.
Make on the Edge a Tangent-Line; from it you must take the Sun's Declination, as you shall be fully shewn in the Use thereof.
This Quadrant will serve excellent well for a Protractor, with a long Index divided into 100 or 200 Equal Parts, with an Ear as the former, and a Needle put into a Stick, to put through the Center of the Index and Quadrant on any Point, in a Plain or Mercator's Chard, by which you may Protract any Rhomb without drawing Lines upon the said Chard; as likewise the Protractor or Semicircle which follows may do the same, being made in the same manner. On the back side of
the Quadrant you may put Mr. Gunter's or Mr. Samuel Foster's Quadrant, or any other as you shall think fit.
SECT. I. Having the Latitude of the Place of the Sun's Declination, It is required to find the Time of the Sun-Rising and Setting.
The Latitude 51 Deg. 30 Min. Northward, and the Declination 20 Degrees, the difference of Ascension will be found thus.
First, Lay the Center Ear at E of the Index, over the Brass-Pin in the Center at A of the Quadrant, and lay the Edge of the Index to EL, to the Latitude of the Place on the Arch DM; and take of the Tangent-Line on the Edge of the Quadrant 20 Degrees the Sun's Declination; and lay that distance from the Center at A towards D, at that distance run with your Eye along the Parallel-Lines, and mark where it toucheth the Edge of the Index; there follow that Parallel-Line to the Arch, and reckon the Degrees from B to that Parallel-Line will be 27 Deg. 14 Min. the difference of Ascension between the Sun's Rising and Setting, and hour of 6, according to the time of the Year.
The Degrees resolved into Hours and Minutes, is 1 Hour 49 Min. which is 4 of the Clock and about 11 Min, for the Sun Rising in the Morning, and 7 of the Clock 49 Min. his Setting in the Evening. In the same manner you must work for all Latitudes.
SECT. II. Having the Latitude of the Place, and the Distance of the Sun from the next Aequinoctial Point, To find the Amplitude.
So the Latitude being 51 Deg. 30 Min. and the place of the Sun in one Degree of Aquarius, that is 59 Degrees from the next Aequinoctial Point; therefore set the Ear at S of the Line of Sines of the Index on the Pin at A, and the Edge thereof to the Latitude, and reckon 59 Degrees the Sun's distance from the first Aequinoctial Point, from the Center to C along the Line of Sines of the Index; there note the Line that cuts the 59 Degrees following with your Eye, to the Degrees in the Arch, and reckon the Minutes of Degrees from M to the Edge of the Index, and you will find it about 33 Deg. 20 Min. the Amplitude required.
SECT. III. Having the distance of the Sun from the next Aequinoctial Point, To find his Declination.
The Sun being either in 29 Degrees of Taurus, or 1 Deg. of Aquarius, or 1 Deg. of Leo, or 29 Deg. of Scorpio, that is 59 Degrees from the next Aequinoctial Point, To find his Declination do thus: Put the Ear of the Line of Sines on the Pin and Edge of the Index; put to 23 Deg. 30 Min. in the Sun's greatest Declination, reckoned from M on the Arch; then count the Sun's distance 59 Deg. on the Deg. of Sines of the Index: From the Center put one Foot of your Compasses by the Degree, with the other take the nearest distance to the Line or Side CM; apply that distance in the Line of Sines of the Index, from S along, and the other Foot will reach to 20 Degrees, the Declination required when the Sun is in the aforesaid Degrees and Sines. In like manner you must do for any other Degrees of the Sun's Place.
SECT. IV. Having the Latitude of the Place, and the Declination of the Sun, To find the Sun's true Amplitude from the true East and West.
This is a most excellent ready way by this Quadrant, and as near the Truth as any Man can make any rational use of this Problem at Sea: It is thus. Suppose the Latitude to be 13 Degrees, and the Sun's Declination 20 Degrees Northward, the Sun's true Amplitude of Rising and Setting is required, from the true East and West.
Set the Ear of the Side of the Index on which is the Line of Sines on the Center, and Edge to the Latitude 13; then count from M 20 Degrees of Declination, and carry your Eye upon the Parallel-Line from that Degree of the Arch, and mark what Degree it cuts of the Index and Line of Sines, as in this Question it doth 20 Degrees 23 Minutes, and that is the true Amplitude required.
Secondly, Suppose you was about the Cape of Virginia, in Latitude 37 Degrees and 30 Min. and Declination 10 Deg. If you work as before-directed, you may find the true Amplitude to be 12 Deg. and about 36 Min. You may estimate the Min. but you cannot Steer by a whole Deg. when you have rectified your Compass by this; therefore this is sufficient for that Use, to shew you the difference between
the true Compass and the Steering Compass, if you observe his Rising and Setting by it.
Note, The Amplitude is the distance of Rising or Setting of the Sun or Stars from the true East and West Points upon the Horizon.
As for the foregoing Work,—In the Latitude of 13 Deg. the Sun or Star having North-Declination 20 Deg. therefore they will rise 20 Deg. 33 Min. to the Northward of the East, and set 20 Deg. 33 Min. to the Northward of the West. But if the Declination had been 20 Deg. South, then they would have risen 20 Deg. 33 Min. Southward of the East, and set 20 Deg. 33 Min. to the Southward of the West.
And so if you bring these Degrees and Minutes into Points and Quarters, and use the Variation-Compass upon the Instrument of the Moon in the First Book, you may readily rectifie the Compass you Steer by.
SECT. V. The Ʋse of the Quadrant and Variation-Compass in the First Book, on the Instrument of the Moon for shifting of Tydes.
This Instrument contains two Parts or Rundles, which are the two uppermost in the aforesaid Instrument made of Wood or Brass, moving one upon the other, as there you may see. The biggest of the two uppermost
Rundles represents the Compass you Steer the Ship by, which is subject to Variation: but the upper Compass doth represent the true Compass that never varieth, whereby you have a most necessary Instrument to rectifie the Compass, as Mr. Wakely hath made Tables to be used with it; but this will serve for use as near by the Quadrant.
Admit I am in the Latitude of 27 Deg. and Declination 20 Deg. Northward, and I observe the Sun's Rising and Setting to be due East and by North, and West by South Point of my Steering or Variation-Compass; the Variation in that Latitude is required.
The Sun having North-Declination, and in that Latitude of 27 Deg. if there be no Variation the Sun will rise (as you may presently find his Amplitude by the Quadrant and Index, 22 Deg. 34 Min. which is but 4 Min. not to be taken notice of,11 d. 15. to each Point. above) E. N. E. and sets W. N. W. But according to the foregoing Propositions, the Sun did rise at E. b. N. and set at W. b. N. Therefore it plainly appeareth that there is a full Point Variation.
Therefore on the Variation-Compass on the Instrument of the Moon, you must always bring the true Point of Rising and Setting on the upper Compass, to touch the false Point or Rising and Setting, found by Observation and Steering-Compass, on the [Page 68] middle Rundle, being set in this Position. You will find the E. b. N. to be the true E. N. E. and the W. and by N. to be the true W. N. W. and the N. b. E. to be the true N. and the S. b. E. to be the true S. and the S. E. ½ Point Southerly, to be the true S. E. b. S. ½ Southerly; and the South 3.1/4 East, to be the South ¼ West: And so you may do with ease in all other Observations, in like manner as you have been shewn, by Points, Halfs, and Quarters, which is on the two Trundles; and be sure nearer than ¼ of a Point I never did see any man Steer or sheape a Course.
SECT. VI. To know the Variation by the Quadrant.
You may do the same thing by the Quadrant, without the help of the Rectifier before spoken of, if you will remember, That this Quadrant hath eight Points, or ¼ of the whole Compass, by which you may orderly reckon the whole, and set the Index to the greatest difference either from the East, Southward, or Northward, or West. In like manner as in the foregoing Proposition, the true Amplitude of Rising and Setting was 2 Points, or 22 Deg. 34 Min. E. N. E. Set the Index to the Degrees and Points, reckoning the Deg. from D on the Arch of the Quadrant, toward E the Scale of Leagues of the Index; then reckon the Point and Degree taken by Observation, which is 11 Deg. and 15 Min. a just Point of the Compass: therefore it being but E. by N. short a Point of the true Amplitude, therefore the E. b. N. of the Steering-Compass, respecteth the true E. N. E. and the N. b. E. respecteth the true N. and so account all round the Compass a Point more than the Steering-Compass sheweth: And if you would know which way the Variation is, you see it is a Point more from the E. than your Compass sheweth Northerly.
But if the Steering or Azimuth-Compass, had shewn a Point more than the true Amplitude found by the Quadrant and true Point, the Variation had been Westerly.
But suppose the Amplitude found had been a Point Southerly, E. b. S. and the Sun's Rising and Setting had been a Point Northerly; by Observation of the Ship-Compass, you see there is two Points difference: therefore set the Index to two Points, from M the East or West side of the Quadrant, as in this Proposition you must reckon it, and you may see plainly that the East Point by the Steering-Compass is the true East-South-East Point; and the South Point is the true South-South-West Point; and the North is the true North-North-West Point; and so of all the rest: And the Variation is Southerly. So that you see how readily this Quadrant doth these things, when the Points of the Compass is imprinted in a Mans mind, which must be, and is in all Masters and Mates.
Suppose I would know by the Quadrant the true Point of the Compass, when Bootes Arcturus riseth and setteth: In the Latitude of 40 Deg. Bootes Arcturus Declination is 20 Deg. 58 Min.
Set the Side of the Index and Sine to the Latitude of 40, and count the Declination 21 Degrees almost, from M on the Arch; and run your Eye up the Parallel, and it will cut the Index about 27 Deg. 50 Min. which is reduced into Points and Quarters by allowing Gr. 15 Min. to a Point, his Rising will be almost E. N. E. ½ a Point Northerly, his Setting W. N. W. ½ Westerly. But if the Declination of a Star of the South side the Aequinoctial, the Rising had been E. S. E. ½ Southerly, and his Setting W. S. W. ½ Southerly.
In the like manner you may know the Rising and Setting of any Star in an instant, by this Quadrant and Index, which I hold to be as necessary an Instrument as Seamen can use, in respect of its plainness, and brevity, and portability, so made as you
see the Figure, the larger the better: And on it you may work all manner of Travisses to the distance of 60 Leagues or Miles which is on the side of the Index. It being so plain and easie, I need not write any thing thereof; but for the Learner's sake, take these few Rules following.
SECT. VII. To finde the Number of Miles answering to one Degree of Longitude, in each several Degree of Latitude.
In Sailing by the Compass, the Course sometimes holds upon a great Circle, sometime upon a Parallel to the Aequator, but most commonly upon a crooked Line, winding towards one of the Poles, which Lines are well known by the Name of Rhombs.
If the Course hold upon a great Circle, it is either North or South under some Meridian; or East or West under the Aequator.
Deg.
Min.
Miles.
00
00
60
18
12
57
25
15
54
31
48
51
36
52
48
41
25
45
45
34
42
49
28
39
53
08
36
56
38
33
60
00
30
63
01
27
66
25
24
69
30
21
72
32
18
75
31
15
78
28
12
81
23
09
84
15
06
87
08
03
In these Cases every Degree requires an allowance of 60 Miles, or 20 Leagues; every 60 Miles or 20 Leagues will make a Degree difference in the Sailing; therefore as was shewn in the first Diagram, and use of the Line of Sines, may be sufficient here, which is the Rule of Proportion.
But if the Course hold East or West, on any of the Parallels to the Aequator,
As the Radius is to 60 Miles, or 20 Leagues, the Measure of one Degree of the Aequator:
So is the Sine-Compl. of the Latitude, to the Measure of Miles or Leagues to one Degree in that Latitude.
But if you would know by the foregoing Quadrant the Miles answering to a Degree in each Parallel of Latitude, it is thus.
Set the Ear E on the Center-Pin, and reckon the Degrees of Latitude from D: to which set the Edge of the Index, and note the Parallel-Line that is at the Degree; carry your Eye on it to the Side CD, and from the Center to that Line you have the Number of Miles answering a Degree in that Latitude.
EXAMPLE.
In the Latitude of 18 degrees 12 min. set the Index 18 gr. 12 min. from D, and the Parallel-Line rising with that Degree, with your Eye or a Pin follow to the Edge, and you will find it to be 57 Miles, the Miles answering one Degree of Longitude and 51 Miles, in the Latitude of 31 gr. 48 min. as in the foregoing Table; and so work for any other Latitude in like manner.
But if the Course hold upon any of the Rhombs between the Parallel of the Aequator and the Meridian, we are to consider besides the Aequator of the World to which we Land, which must be always known.
First, The difference of Longitude, at least in general.
2. The difference of Latitude, and that in particular.
3. The Rhomb whereon the Course holds.
4. The distance upon the Rhomb, which is the distance we are here to consider, and is always somewhat greater than
the like distance upon a great Circle. The first follows in the next Proposition.
I. To find how many Leagues do answer to one Degree of Latitude, in every several Rhomb.
In this Table is the Degrees of every quarter Point, ½, and whole Point in the Quadrant; as the first quarter is 2 gr. 49 m. so the half Rhomb is 5 gr. 37 m. the third is 8 gr. 26 m. and the first Point from the Meridian is 11 gr. 15 m. and so you may plainly see the rest.
Rhombs.
Inclination to the Meridian.
Number of Leagues.
Gr. Min.
Leag par.
2 49
20 2
5 37
20 10
8 26
22 22
1
11 15
20 39
14 4
20 62
16 52
20 90
19 41
21 24
2
22 30
21 65
25 19
22 12
28 7
22 68
30 56
23 32
3
33 45
24 05
36 34
24 90
39 22
25 87
42 11
26 99
4
45 00
28 08
47 49
29 78
50 37
31 52
53 26
33 57
5
56 15
36 00
52 4
38 90
61 5
42 43
64 41
36 76
6
67 30
52 26
70 19
59 37
73 7
68 90
75 56
82 31
7
78 45
102 52
81 34
136 30
84 22
205 14
87 11
407 60
8
90 00
ad infinit.
As the Sine-Complement of the Rhomb from the Meridian, is to 20 Leagues or 60 Miles, the Measure of 1 Degree at the Meridian:
So is the Radius or Sine of 90, to the Leagues or Miles answering to one Degree upon the Rhomb.
Suppose by the Quadrant it were required to answer this Question,
Sailing N. N. E. from 40 Degrees of North-Latitude, How many Leagues shall the Ship run before it can come to 41? By reason this is the second Rhomb from the Meridian, and the Inclination thereof is 22 deg. 30 m.
Therefore set the side of the Index EL to the second Point from the Meridian N. N. E. 22 d. 30 m. and reckon from C 20 Leagues towards D, and with your Eye or a Pin follow the Parallel-Line to the Index, and you will find it cut 21 Leagues 65 parts (or better than ½ more) the number of Leagues you must Sail before you can reach 1 Degree.
You may do the same by the Travis-Scale thus. Extend the Compasses from 2 Points nearest the end of the Scale, and greatest Number of the Line of Numbers that is N. N. E. 2, and E. N. E. 6 Points, unto 20 Leagues on the Line of Numbers; remove the Compasses to 100 in the Line of Numbers, and the other Point of the Compass will reach to 21 Leagues 6/10 ½ or 65 parts, as before in the Line of Numbers.
This may be found also by a Line of Chords and Equal Parts, if you draw a Right Line, and take with your Compasses 20 parts, and lay it from one end on the Line; then take 60 deg. and sweep an Arch, and take 2 Points with your Compasses, and lay from the Meridian on that Arch from N. N. E. and draw the Secant or Rhomb-Line, at 20 Leagues draw a Perpendicular or Line at Right Angles there to the former, and measure the distance from the Center, to the Intersection of the Line drawn from 20, with the Rhomb-Line on the Scale of Leagues or Equal Parts, and you will find it the same as before. And so the Quadrant shews you all at one sight, if you understand without more words. By the Artificial Sine and Number, Extend your Compasses from the Sine of the Rhomb 67 deg. 30 to 20 in the Line of Numbers, the same Extent will reach from 8 Points, or 90 deg. or 100 in the Line of Numbers, to 21 Leagues 65 parts, as before.— This consider in general; I shall shew you more particularly in 12 Proofs (how of these four, any two being given, the other two may be found, both by Mercator's Chart, and all other ways, as is usual) when I come to treat more particularly of Navigation.
II. By one Latitude, Rhomb, and Distance, To find the difference of Latitudes.
Let the place given be C in the Latitude of 40 Degrees, that is in the Center of the Quadrant, the second Latitude unknown; The distance upon the Rhomb 21 Leagues 65 parts of a League; the Rhomb N. N. E. the second from the Meridian: Therefore set the Index to the Point, and count 21 Leagues 10 parts, and run your Eye up the Parallel-Line you there meet with, and reckon the Leagues from the Center C to that Line, and you will find it 20 Leagues; and such is the difference of Latitude required.
It is easie to be understood how to lay it down by the Plain Scale; therefore I shall forbear to write any more of that Way.
As the Radius, to the Co-sine of the Rhomb from the Meridian:
So the distance upon the Rhomb, to the difference of the Latitudes.
Extend the Compasses from the Sine of 90, to the Co-sine of the Rhomb 67 deg. 30 m. the same distance will reach from 21-65 Leagues in the Line of Numbers, to the difference of Latitude 20 Leagues. In like manner you must work for all such Propositions, let the Number be greater or less, by either Instrument.
The Travis-Scale is the same manner of Work, as the Artificial Sines, Tangents, and Numbers; For extend the Compasses from 8 Points, to 2 Points, the same distance will reach from 21: 65 in the Line of Numbers, to 20 the difference required.
III. By the Rhomb and both Latitudes, To find the Distance upon the Rhomb.
As suppose the one place given were C the Center of the Quadrant, in the Latitude 40 deg. the second place in the Latitude 41 deg. and the Course the second from the Meridian.
Set the Index to the Rhomb, and account 20 Leagues, which is 41 deg. the second Latitude, and carry your Eye on that Parallel that leads to the Index; and there it will cut the distance upon the Rhomb, which in this Question is 21 Leagues 65 parts.
Extend the Compasses from the Co-sine of the Rhomb from the Meridian, to the Radius or Sine of 90—
The same Extent will reach from 20 Leagues, the difference of Latitude, to 21: 65 in the Line of Numbers, the distance upon the Course required.
IV. By the distance and both Latitudes, To find the Rhomb.
Suppose the Place given was at C, in Latitude 40 deg. and the second Place a Degree or 20 Leagues further Northward, and the distance was 21-65 Leagues upon the Course.
From the Center C reckon 20 Leagues towards D, follow that Parallel, and set the Index, and count the distance until it touch the Parallel, and look in Arch of the Quadrant, and you will find the Rhomb 22 gr. 30 m. or N. N. E. 2 Points from the Meridian.— Or, Extend the Compasses from the distance upon the Rhomb 21: 65, to the distance of Latitudes 20 Leagues; The same Extent will reach from the Radius or Sine of 90, to the Sine-Complement of the Rhomb 67 deg. 30, which was required.
V. By the difference of Meridians, and Latitude of both Places, To find the Rhomb.
As if the Place given was C the Center of the Quadrant, 40 deg. and 20 Leagues was the difference of Latitude Northward, that is 41 deg. and the difference of Longitude 8 Leagues 45 parts of a League.
First, count from C the difference of Latitude 20 Leagues, on that Parallel count 8 Leagues 45 parts; to that put the Index; and in the Arch you will find the Course 22 gr. 30. from the Meridian.
Extend the Compasses from 20 Leagues to 8: 45, the same Extent will reach from 90 to the Tangent of the Rhomb 22 gr. 30 min. as before.
VI. By the Rhomb and both Latitudes, To find the difference of Longitude, or departure from the Meridian.
Let the Rhomb be 2 Points from the Meridian, the one Latitude given 40 deg. the other Latitude 41 deg. the difference 20 Leagues.
Set the Index to the Point and Rhomb 20 gr. from the Meridian, and count 20 Leagues the difference; on that Parallel reckon the Leagues between the Side and the Index, and you will find it in this Question 8 Leagues 45 parts, the Meridians-distance required.— Or, Extend the Compasses from the Tangent of the Rhomb 22 gr. 30, to Radius 90, the same Extent will reach from the difference of Latitude 20 Leagues, to the departure from the Meridian 8 Leagues 45 parts.
These six last Propositions depend one upon the other, as you may plainly see; which may be sufficient for the
Explanation of the Quadrant, by which may be understood much more.
CHAP. VII. How to make a most Ʋseful Protractor.
THis Instrument is always to be made in Brass or Copper, but best in Brass. On the Center C draw the Semicircle BO, and divide it into two 90, or 180 Degrees, as you may see the Figure; and let the sixteen Points of the Compass PP be set in the inward Circle, with the Quarter-Points. And let the
[Page 73]Index AE be two Diameters and ½ long, and so fitted as the Semidiameter of the Circle may be the distance from the Center, for the ready setting to any number of Degrees, or Points and Quarters the Edge thereof, and divide from the Center to the end of the Index into 100 equal Parts, which are accounted sometime Leagues, and reckoned by the Surveyors of Land Perches, or any other Denomination of Numbers: You may call it for protraction according as you have occasion to use it.
Let the Index be fastned to the Center with a Brass Rivet, and through the midst of the Rivet there must be a Hole drill'd; you must put the Pin or Needle spoken of in the last Chapter, upon any Point assigned, in any Chart or Protraction whatsoever. You may divide the Edges into equal parts, by which you may make a Meridian-Line on the blank Charts, according to Mercator's or Wright's Projection.
And now you have a necessary Instrum [...]nt, which will protract any Travis or piece of Land upon Paper, with as much speed as any Instrument I ever yet knew, and readier by much, the use whereof shall be shewn in this Treatise.
CHAP. VIII. The Projection of the Nocturnal.
IT consists, as you see, of three Parts. The greatest or handle-part hath on it two
Circles divided: On the first or outmost is the Ecliptick, divided into 12 equal Parts, in which is put the 12 Signes; and each of those 12 Parts is divided into 30 equal Parts or Degrees, in each Signe, numbered 10, 20, 30, as you see the Figure plainly sheweth. The inward Circle is the 12 Months of the Year, set in by a Table of the Sun's place every day of the Year, accounting the Degrees of the outward Circle, and the number of Days in each Month equally divided and set down 10, 20, 30.
Note, Where to begin to divide the Months and Days is thus. Observe the brightest Guarde, or by Calculation or the Globes find when he comes to the Meridian just at 12 a Clock at Night. In the following Tables the Right Ascension of the brightest Guard is 223 Degrees 31 Minutes; from it substract 12 Hours, or 180 Degrees, the Remainder is 43 deg. 31 min. the Right Ascension of the Sun the 26th day of April, in 16 deg. of Taurus, which must be uppermost next the Zenith in the middle.
The other Part or middle Rundle equally divideth the outward Circle into twice 12 Hours; and within that is a Circle equally divided into 32 Parts, or Points of the Mariner's Compass projected thereon. The upper part is an Index, the length is from the Center to the Foot of the Instrument; all three being fastned with a piece of Brass, so Rivetted that the Center is an Hole through which you may see the North-Star. You may make it in Brass, or good dry Box.
The Ʋse of the Nocturnal.
THe Use of the Nocturnal is easie and ready. Let the Tooth or Index of the middle Circle be set to the Day of the Month, and it will cut in the outward Circle the Sun's Place in the Ecliptique. Then hold the Instrument on high, a pretty distance from you, with the Foot AB right with the Horizon level: Then look through the Hole of the Center, and see the North-Star, turning the long Index or Pointer upwards or downwards, untill you see the brightest of the Guards over or under the Edge that comes straight to the Center. Then look on the Hour-Circle by the Edge of the Pointer, and it shews the Hour of the Night, and likewise the Point of the Compass the Guard beareth from the Pole; by the which you may have his Declination by the following Tables exactly.
The Hour of the Night may be also found by the Right Ascension of the Sun and Stars. Thus, When that you see any Stars in the South, whose Right Ascension is known, and also the Right Ascension of the Sun for that day, you shall substract the Sun's Ascension from the Star's; that which remaineth divide by 15, to bring it into [Page 74]Hours;Here place the North Nocturnal for the Pole Star and Guards for 15 Degrees makes an Hour, and 4 Minutes make a Degree; thereby you have the Hour of the Night. If the Sun's Ascension be more than the Star's, in such case you shall add 360 Degrees to the Ascension of the Stars, and then substract the Sun's Ascension, as before directed, you have the time also.
CHAP. IX. How to use the Pole-Star's Declination and Table, and thereby to get the Latitude.
THe Pole-Star, being so very well known to all Sea-men, is therefore made the most use of by them: Therefore know, That this Table is made for the year 1660. the true Declination being 2 deg. 30 min. but will serve for many years after. This Table is made contrary to the two former Tables; for whereas the North Point of the Nocturnal is the first Point you reckon from, and was on the former Nocturnal reckoned from South: so of this Nocturnal you must take the Point at the Tooth for North, and so reckon forward North and by East, and so on to East and South-East, South and West, to North again.
And likewise in the Table, you must begin in like manner at that part of the Table that lies directly under the Pole; which, as before-said, is properly called the North, and so proceed about the Pole, ascending from this lowest or North Point of the Meridian, as was said before, to the North-East, East, and South-East, so to the South or highest Point of ascending, being directly over the Pole: From the South they descend again by the West, and so return to the North again.
Observe this, That the brightest of the Guards is the first of the little Bear, which is the Star you are to observe, and is almost in opposition to the Pole-Star.
Note, That when the Guard-Star is under the Pole, then the Pole-Star is above the Pole; and when the Guard-Star is above the Pole, then the Pole-Star is under the Pole so many Degrees and Minutes as the Table shews you.
The Use of this Table and Instrument is this: Look with the Nocturnal, and see what Point the Guards bears from the Pole, as before-directed; and if you find the Guard is not on a full Point, stay a little longer until he is just, and then observe the height of the Pole-Star exactly as you can; then knowing by your Dead Reckoning within a Degree or two what Latitude you are in, look for the nearest to that Latitude in the top of the Table; and if you find the Point of the Compass which the Guard-Star is upon, in the first Column of the Table, and in that Line under the Column of your Latitude, you shall find the number of Degrees and Minutes the Pole-Star is either above or below the Pole, according to the direction of the last Column of the Table, which you must thus make use of; If the Star be any thing above the Pole, substract the Number in the Table from the height of the Star observed: but if the Star be under the Pole, then add the Number found in the Table to the height observed, by which you shall have the height of the Pole.
For Example. Estimate the Latitude to be near 40 Degrees, and observe the Pole-Star. Suppose you find the Altitude 40 Degrees, and the Guard-Star bears N. N. E. from the Pole; therefore look for N. N. E. in the first Column, and right under your estimated Latitude 40, in the same Line with N. N. E. you will find the Declination to be 1 Degree 30 Minutes; substract that from 40 Deg. the Altitude observed leaves the true Latitude 38 Degrees 30 Minutes.
d.
m.
40
00
01
30
38
30
But if the Guard-Star had been S. S. W. then the Pole-Star had been 1 Degree 33 Minutes under the Pole; which being added to the Altitude observed 40 Deg. the Latitude would have been exactly 41 Degrees 30 Minutes by the Star. So the Star's Altitude by observation being 55 Deg. the Guard bears from the Pole-Star S. E. b. S. the Declination against that Point is 2 Degrees 30 Minutes, added to 55. had been 57 Degrees 30 Minutes for the Latitude: but if your estimated Latitude had been near 50, and the Guard bear from the Pole North-West [Page 75] by North, the Declination against that Point is 2 deg. 30 min. substracted from 55 Degrees, the Altitude observed, there remains 52 deg. 30 min. the Latitude of the place by the Star.
A Table of the North-Star's Declination in these several Latitudes.
The True Point of the Compass.
0
20
30
40
50
60
70
D. M.
D. M.
D. M.
D. M.
D. M.
D. M.
D. M.
If the former of the Guards be ascending from the North or lower part of the Meridian.
North.
2 10
2 10
2 10
2 09
2 09
2 08
2 07
Above the Pole.
N. b. E.
1 53
1 53
1 53
1 52
1 52
1 51
1 49
N. N. E.
1 31
1 31
1 30
1 30
1 29
1 28
1 25
N. E. b. N.
1 06
1 05
1 04
1 03
1 02
1 01
0 58
N. E.
0 39
0 38
0 37
0 36
0 35
0 33
0 30
N. E. b. E.
0 10
0 09
0 08
0 07
0 06
0 04
0 01
E. N. E.
0 18
0 19
0 20
0 21
0 22
0 23
0 26
Ʋnder the Pole.
E. b. N.
0 49
0 50
0 50
0 51
0 52
0 53
0 56
East.
1 15
1 15
1 16
1 17
1 18
1 19
1 21
E. b. S.
1 38
1 39
1 39
1 40
1 41
1 42
1 44
E. S. E.
2 00
2 00
2 00
2 00
2 00
2 01
2 02
S. E. b. E.
2 15
2 15
2 15
2 15
2 16
2 16
2 16
S. E.
2 25
2 25
2 25
2 25
2 25
2 25
2 25
S. E. b. S.
2 30
2 30
2 30
2 30
2 30
2 30
2 30
S. S. E.
2 29
2 29
2 29
2 29
2 29
2 29
2 29
S. b. E.
2 22
2 22
2 22
2 22
2 22
2 22
2 22
If the former of the Guards be descending from the South or upper part of the Meridian.
South.
2 10
2 10
2 10
2 11
2 11
2 11
2 12
S. b. W.
1 58
1 53
1 54
1 53
1 55
1 55
1 57
S. S. W.
1 31
1 32
1 32
1 33
1 34
1 35
1 38
S. W. b. S.
1 07
1 07
1 08
1 10
1 11
1 13
1 13
S. W.
0 39
0 40
0 41
0 40
0 43
0 44
0 47
S. W. b. W.
0 10
0 11
0 12
0 13
0 14
0 16
0 19
W. S. W.
0 19
0 19
0 17
0 16
0 15
0 13
0 10
Above the Pole.
W. b. S.
0 48
0 47
0 46
0 45
0 44
0 43
0 42
West.
1 15
1 14
1 13
1 12
1 11
1 10
1 08
W. b. N.
1 39
1 39
1 38
1 37
1 36
1 35
1 33
W. N. W.
2 00
1 59
1 59
1 58
1 58
1 57
1 56
N. W. b. W.
2 15
2 15
2 14
2 14
2 14
2 13
2 12
N. W.
2 25
2 25
2 25
2 25
2 24
2 24
2 24
N. W. b. N.
2 30
2 30
2 30
2 30
2 30
2 30
2 30
N. N. W.
2 29
2 29
2 29
2 29
2 29
2 29
2 29
N. b. W.
2 22
2 22
2 22
2 22
2 22
2 21
2 21
I hope the young Seamen are pleased for Examples, it being made so plain to their Capacity, and as profitable for their Use as any
Rule whatsoever.
CHAP. X. How to make a most Ʋseful Instrument of the Stars, and by it to know most readily when any of 31 of the most notable Stars will come to the Meridian, what Hour of the Night, at any time of the Year, at the first sight.
THis Instrument consisteth of two parts, which is two Rundles; on the back side of the foregoing Nocturnal it may be very fitly made: On the matter or greater Rundle are three Circles divided; the outermost is the 12 Months of the Year, begun the 10th of March, the day the Sun enters into Aries; and the Days equally divided to the Number of Days in each Month. The second Circle representeth the 24 Hour Circle, divided equally into 24 Hours, ½ and ¼, beginning the 10th of March at 12 a Clock at Noon, the time the Sun comes to the Meridian, and the first Degree of Aries. The third and inward Circle is the Aequinoctial, divided into 360 Degrees; by them is accounted the Right Ascension of these 31 Stars in the Table following.
A Table of the Longitude, Latitude, Right Ascensions, and Declinations, of 31 of the most Notable Fixed Stars: Calculated from Tycho his Tables, rectified for the Year of our Lord, 1671.
Longitude.
Latitude.
Ascension.
Declination.
Nor. Sou.
The Whale's Tail
27 56 ♓
20 47 S
06 45
19 48
S
2
The Bright Star in the South Foot of Androm.
09 39 ♉
27 46 N
25 57
40 44
N
2
The Bright Star in the Right Side of Perseus.
27 17 ♉
30 05 N
44 16
48 36
N
2
The Bright Star of the 7 Stars or Pleiades.
24 24 ♉
04 0 N
52 00
23 03
N
3
The South Eye of the Bull Aldebaran.
05 12 ♊
05 31 S
64 17
15 48
N
1
The Bright Star in the left Foot of Orion Riges.
12 17 ♊
31 11 S
74 44
8 37
N
1
Orion's right Shoulder towards the East.
24 12 ♊
16 6 S
84 23
07 18
N
1
The glittering Star in the Mouth of the great Dog.
On the other Rundle or upper part, is placed all these aforesaid Stars; and any other you may set thereon, if you follow this Rule.
For Example, First set the Index to the 10th day of March of the upper Circle; on the under Circle, which will be at 12 a Clock at Noon, or 360 Degrees, stop it fast there, that it may not move, until you have placed the Stars on it as you intend to set thereon. As suppose you would set the Whale's Tail on the Rundle in his place; look in the foregoing Table, and you will find his Right Ascension 6 deg. 45 min. account that from the 10th of March, and on the Aequinoctial Circle, and lay a Ruler from the Center over the 6 deg. 45 min in the Matter and Aequator or inward Circle, and draw the Line from the Center to the outward Edge of the upper Circle, and thereon set the Name of the Star, next to that the Declination of the Star, and the Letter S or N. representing South or North Declination: on the inward Circle, set before each Star the Magnitude of the Star, whereby you may know the better, as the Figure following shews you all plain.
Take this Example more. Suppose you would set the Lion's Heart in his place; In the Tables I find his Right Ascension is 147 deg. 43 min. reckon that Number on the Aequinoctial Circle next the Rundle, and draw the Line as before-directed, 1 signifying the First Magnitude, secondly his Name, thirdly 13 deg. 33 N, for his North Declination.
The Instrument in this posture, you will find the Lion's Heart will come to the Meridian at almost 10 a Clock in the Evening the 10th day of March in any Latitude.
How to know the Hour of the Night any Star comes to the Meridian in any Latitude.
YOu have been in a manner shewed it before in the last Example. Set the Index or Hand of the upper Rundle to the Day of the Month, and right against the Star is the Hour of the Night, in the Matter the Star will be on the Meridian.
For Example, Suppose you would know the Hour of the Night the Bull's Eye comes to the Meridian the 20th Day of October; Set the Index to the Day, and right against the Bull's Eye is ¾ of an Hour past 1, the time in the Morning that Star will be on the Meridian South. And in the same manner you may see the Stars, and Hours they come to the Meridian that Night and Day. For note the upper half of the Circle, and 12 Hours is the Day-hours, and the lower and Handle-half is the Night-hours. You begin to reckon the Day-hours on the left side of the Instrument, and the Night-hours on the right side; so round with the Sun.
How to know what Stars are in Course at any time or Day of the Year.
THe Course and seasonable coming to the Meridian of the Stars, and what are fit to be observed, is shewn you at once, the Instrument in the former posture, if you look against each Star, you have the Hour of the Night and Day, being the whole 24 Hours. This is so plain, you need no further Precept.
How to know the Hour of the Night, by the Stars being on the Meridian.
SUppose it were required to know the Hour of the Night the 10th of December, the brightest of the 7 Stars being on the Meridian South: Set the Index to the day of the Month, and right against the brightest of the 7 Stars, is half an Hour past 9 at Night, which is the Hour required.
WHen the Mariners pass the Aequinoctial Line towards the South, so that they cannot see the North-Star, they make use of another Star, which is the Constellation called the Centaur; which Star, with three other notable Stars which are in the same Constellation, maketh the Figure of a Cross (betwixt his Legs) for which cause they call it the Crosiers. And it is holden for certain, That when the Star A (which of all four cometh nearest the South-Pole) is
[celestial diagram]
North and South by the Star B, then it is rightly scituated to take the Height by: And because this Star A, which is called the Cocks-foot, is 30 Degrees from the South-Pole, it cometh to pass, it being scituated as before-said, we take the Height thereof
(which is then the greatest that it can have) this Height will truly shew how far
we are distant from the Aequinoctial: For if the said Height be 30 Degrees, then we are under the very Aequinoctial: But if it be more than 30 deg. then are we by so much past the Aequinoctial, toward the South: And if it be less than 30 deg. so much as it wanteth, we are to the North of the Aequinoctial. And here it is to be noted, That when the Guards are to the North-East, then is the Star in the Crosiers fitly scituated for observation, because then they are in the Meridian.
CHAP. XII. How to make the Cross-Staff.
THe Mariner's Cross-Staff is that which by the Astronomers is called Radius Astronomicus, by which we observe the Celestial Lights above the Horizon. The Mathematicians have invented many kinds of Instruments, whereof the Cross-Staff and Quadrant are the most useful above all the rest. At Sea it is not every Mans Work to make and mark a Cross-Staff, and other Instruments, for want of Practice needful thereunto; yet notwithstanding it is fit and necessary
that a Master, his Mates, and Pilot, who are to have the Use of it, should at least know when it is well [...]ade.
For to mark well a Cross-Staff, you shall make a plain flat Board of good dry [Page 79]Wood, fifteen or sixteen Inches broad, and about four Foot or three Foot long: Paste it well with good Paper; draw along the one Side a Right Line, as in the next following Figure CAD; out of the Line C draw a Square Line upon AC, as CB, and upon the Center C draw the Arch AEB, being a Quadrant or fourth part of a Circle; divide that into two parts; the one half thereof, as AE, divide into 90 Equal Parts
or Degrees, thus; first into three Parts, and each of the same again into three; these Parts
each into two, and each of the last Parts into five: so the Arch AE shall then be divided into 90 Parts. Then take a right Ruler, lay the one end on the Point or Center C, the other upon each Point of the foresaid several Divisions, and draw small Lines out of C, through each of the foresaid Points or Degrees of the Quadrant, so long as they can stand upon the Board, as you may see it plain in the Figure. Then take with a pair of Compasses, just the half length of the Cross that you would mark the Staff after; prick it from the Point C towards B; as by Example, from C to F, and from D to G; joyn these two Points with a Line to one another; and even into such Parts as that Line is cut, and divided by the aforesaid Lines coming out of the Center of the Quadrant, must your Staff be marked, whether the Cross be long or short, as appeareth by the Lines HI and KL, which are drawn for Crosses: the half thereof is so long as CH, or CK, or CF. If the aforesaid Quadrant, for want of good Skill or Practice, be not well divided, or Lines not well drawn, the Staves being marked thereafter will also be faulty. Therefore they may be marked more exactly
by Points equally divided, in manner as followeth.
Prepare you a Staff, draw thereon a Right Line so long as your Staff, and take with a sharp pair of Compasses the half Length of the Cross after which you desire to mark your Staff: prick it so often along the aforesaid Line, as it can stand upon the same. Divide each of the Lengths of the half Cross into 1000 Equal Parts. Then prick upon the Staff you will mark from the Center-end, just half the Length of the Cross; and mark there a small thwart Stroke. Off from thence prick for each Degree so many of the same Parts as the Cross is divided in his half Length, like as is marked in the Table here annexed for every Degree. For the first Degree you shall mark off from 90 the aforesaid thwart Stroke 176, for the fourth Degree 724, for the 10 Degree 1918 of those Parts, and so of the rest. If you cannot divide the half Cross, by reason he is so little, into 1000, divide him into 100, and leave out the two
last Figures, and that shall satisfie your desire: For 30 Degrees take 73, and for 40 Degrees 114, and for 10 Degrees 19 Parts, and so of the rest.
CHAP. XIII. How to use the Cross-Staff.
SEt the end of the Cross-Staff to the outside of the Eye, so that the end of the Staff come to stand right with the Center of the Motion of the Sights. Then move the Cross so long off from you or towards you, holding it right up and down, and winking with
your other Eye, till that the upper end come upon the middle or Center of the Sun or Star, and the lower end right with the Horizon, the Cross then shall shew upon the side of the Staff belonging thereunto, the Degrees of the Altitude of the Sun or Star. Note, The Staff is marked with two Lines of Numbers, with 90 Degrees next the Eye, and diminishing from 90 to 80 and 70, 60 towards the outmost end: The Complement-Sine beginneth towards the Eye-end, and encreaseth contrarywise towards the outmost end, as from 10, to 20, 30. The first
Number sheweth you the Altitude, the second Number is the Sun's distance from the Zenith.
The Sun or Stars being high elevated from the Horizon, the Cross cometh nearer the Eye than when they are but a little elevated, and do stand neer the Horizon; thereby the eye mak [...]th (seeing now to the lower, and then again to the upper end of the Cross) greater [Page 81] motion in looking up and down, than when the Sun or Star doth stand low. And insomuch the Center of the Sight, by such looking up and down together with the end of the Staff, a Man seeth then smaller Angles then if it did remain stedfast, in regard whereof the Cross cometh nearer to the Eye than it should, and there is found too much Altitude. This being found by many, besides my self, by experience, they were therefore wont
to cut off a piece of the end of their Staff, or set the Crosses a Degree and ½ or two Degrees nearer the Eye; but it is not the right means for to amend the aforesaid Errors. The best means of all in my opinion is this; That upon each several Height which men will observe, they do try with two Crosses set upon the like Degrees, how the Staff must be set, that they may see the end of the same two Crosses right one with the other.
[geometrical diagram]
Having found that, and then taking off one of the Crosses, and setting the Staff again, in the same manner as before, all Errours will be so prevented, which by the lifting up or casting down of the Eye, might any manner of way happen.
EXAMPLE.
I desire to observe the Sun or any Star in the South: I make my Estimation, as neer as I can, how high that shall stand, or take the Height of them a little before they shall come to the South, which I take to be 50 Degrees. I set therefore the two [Page 82]Crosses each upon their 50 Degrees, and the end of the Staff in the hollow of the Eye-bone, on the outside of the Eye, and bow the Head forward or backward, or over the one side or the other, till I see the utmost end
of both the Crosses right one with the other, according as is shewn by these Lines AB and CD, as is apparent enough by the foregoing Figure; That the Sight-beams over the ends of the Crosses shall then agree with the Lines which might be drawn over the end of the Crosses, to the Point or Center at the end of the Staff, which doth agree with the Center of the Quadrant, or the beginning of the Equal Points upon which the Staff is marked. Keeping in memory such standing of the Staff, I take off the one Cross, and set the Staff again in the aforesaid manner to the Eye, and observe without any errour of the Eye.
In taking the Height of the Sun with the Cross-Staff, Men do use red or blew Glasses, for the saving and preserving the Eyes; yet it is notwithstanding a great let, and very troublesom for the Sight, especially if it be high: therefore the Quadrant and Back-Staff is much better, as will be shewed in the next Chapter.
Thus I have shewed you how to take an Observation by the Fore-Staff. The next thing that followeth in course will be to shew you how to work your Observation; which to do, take notice of these following Rules.
To Work your Observation.
IF the Sun hath North Declination, and be on the Meridian to the Southwards of you, then you must substract the Sun's Declination from your Meridian Altitude, and that Remainder is the Height of the Aequinoctial, or the Complement of the Latitude North. But if the Sun hath South Declination, you must add the Sun's Declination to your Meridian Altitude, and the Sum is the Height of the Aequator, or the Complement of the Latitude North. If the Sun hath North Declination, and be on the Meridian to the Northwards, then add the Sun's Declination to his Meridian Altitude, and the Sum is the Height of the Aequator, or the Complement of the Latitude South, if the said Sum doth not exceed 90 deg. but if it doth exceed 90 deg. you must substract 90 deg. from the said Sum, and the Remainder is your Latitude North.
If the Sun hath South Declination, and be to the Northwards at Noon, you must then substract the Sun's Declination from his Meridian Altitude, and the Remainder is the Complement of your Latitude South. When the Sun hath no Declination, then the Meridian-Altitude is the Complement of the Latitude. If the Sun be in the Zenith, and if at the same time the Sun hath no Declination, then you are under the Aequinoctial.
But if the Sun hath North Declination, and in the Zenith, then look how many Degrees and Minutes the Declination is, and that is the Latitude you be in North.
But if your Declination be South, then you are in South Latitude. If you observe the Sun or Star upon the Meridian beneath the Pole, then add your Meridian Altitude to the Complement of the Sun or Stars Declination, and the Sum is the Height of the Pole.
Rules for Observation in North Latitude.
SUppose I am at Sea, and I observe the Sun's Meridian Altitude to be 39 deg. 32 min. and the same time the Sun's Declination is 15 deg. 20 min. North; I demand the Latitude I am in.
Suppose I were in a Ship at Sea the 18th of April, Anno 1667. and by Observation I find the Sun's Meridian Altitude to be 62 deg. 15 min. The Latitude is required.
deg.
min.
The Meridian Altitude
62
15
The Declination North, subst.
14
18
The Complement of the Latitude
47
57
90
00
The Latitude I am in, required
42
03
Admit you were in a Ship at Sea the 5th of November, Anno 1679. and I find the Sun's Meridian Altitude to be 24 deg. 56 min. The Latitude is required I am in.
deg.
min.
The Meridian Altitude
24
56
The Declination South, add
18
37
The Complement of the Latitude
43
33
90
00
The Latitude I am in
46
27
Suppose a Ship at Sea the 28th of May, Anno 1666. and I find the Sun's Meridian Altitude by Observation 56 deg. 45 min. The Latitude is required I am in.
deg.
min.
The Meridian Altitude
56
45
The Declination North, Subst.
22
46
The Complement of the Latitude
33
59
90
00
The Latitude required I am in
56
01
Admit a Ship at Sea the 11th of June 1668. and find the Sun's Meridian Altitude by Observation 79 deg. 30 min. North, It is required the Latitude I am in.
deg.
min.
The Meridian Altitude
79
30
The Declination North
23
31 add.
103
01
90
00
The Latitude I am in
13
01 required.
Suppose I were at Sea the 14th of May 1693. and the Meridian Altitude of the Sun was 69 deg. 07 min. North, I demand the Latitude the Ship is in at that time.
SUppose I at Sea in a Ship the second of June, Anno 1666. and I find the Sun's Meridian Altitude by Observation to be 64 deg. 45 min. The Latitude the Ship is in, is required.
deg.
min.
The Meridian Altitude North
64
45
The Declination North, add
23
15
The Complement of the Latitude
88
00
90
00
The Latitude the Ship is in-
02
00 South.
Suppose a Ship at Sea the 28th of December, Anno 1695. and in Longitude 169 deg. East, and I find the Meridian Altitude by Observation to be 59 deg. 52 min. The Latitude the Ship is in, is required. The Declination in the Meridian of Bristol for the 28th of December, is 22 deg. 25 min. and the daily difference of Declination is at this time 8 min. Therefore if you look in the Table of Proportion following, you will find the Proportional Minutes to be about 4, which you must add to the Declination of the Meridian of Bristol, and the Sum will be the true Declination for the Longitude 169 deg. East, which is 22 deg. 29 min.
deg.
min.
The Meridian Altitude North
59
52
The Declination South, substr.
22
29
The Complement of the Latitude
37
23
90
00
The Latitude the Ship is in, which was
52
37 required.
Suppose I were at Sea in a Ship the 29th of June, 1679. and I find the Sun's Meridian Altitude to be 62 deg. 30 min. North, The Latitude is required.
deg.
min.
The Meridian Altitude North
62
23
The Declination North, add
22
26
The Complement of the Latitude
84
49
90
00
The Latitude the Ship is in
05
11 South.
Admit I am in a Ship at Sea the 20th of January 1667. the Sun's Declination 20 deg. 4 min. and the Sun's Meridian-Altitude 79 deg. 36 min. South, I require the Latitude the Ship is in.
Answer, 9 deg. 30 min. South.
Admit a Ship were at Sea, the Sun's Declination 13 deg. 53 min. South, and the Sun's Meridian Altitude 80 deg. 43 min. South, The Latitude is required.
deg.
min.
The Declination South
13
53
The Meridian Altitude
80
43 add.
94
36 the Sum.
Substr.
90
00
The Latitude the Ship is in
04
36 South.
If you observe the upper part of the Sun, you must substract 16 min. But to the contrary, if you observe the lower part of the Sun, you must add 16 min. for the Sun's Diameter, and the Sum will be the true Altitude of the Sun's Center.
SUppose I am at Sea, and observe the Brightest of the 7 Stars upon the Meridian, and find his Meridian Altitude to be 47 deg. 20 min. and the Latitude were required.
deg.
min.
The Declination of this Star is
23
03 North.
The Meridian Altitude
47
20
Substract the North Declination
23
03
The Complement of the Latitude
24
17
90
00
The Latitude I am in
65
43
Admit I were at Sea, and observe Hydra's Heart on the Meridian, his Altitude is 36 deg. 15 min. and his Declination is 7 deg. 15 min. South, The Latitude of the Place is demanded.
deg.
min.
The Meridian Altitude is
36
15
The Declination is South
07
15 add.
The Height of the Aequinoctial above
43
30 the Horizon.
90
00
The Latitude the Ship is in
46
30 required.
This you see is plain, and needs no further Precept but what is already said.
CHAP. XIV. A Description of the Back-Staff or Quadrant.
THe Back-Staff or Quadrant is a double Triangle, as this Figure following sheweth; whereof the Triangle ABC the Arch is equally divided into 60 deg. and the other Triangle is divided equally into 30 deg. ADF the Vanes are fitted neat. In proportion to him, the Use followeth.
The Ʋse of the Back-Staff or Quadrant.
SEt the Vane G to a certain number of Degrees, as the Altitude of the Sun requireth; and looking through the Vane F, to the upper Edge of the Slit of the Sight of the Horizon, if you see all Skie and no Water, then draw your Sight-Vane a little lower towards E: but if you see all Water and no Skie, then put your Eye-Vane up higher towards F; and when you have done so, observe again; and then if you see
the Shade lie upon the upper part of the Slit, on the Horizon-Vane, and you at the same time do see the Horizon through the Sight-Vane, then that is all you can do untill the Sun be risen higher; and tending the Sun until he be upon the Meridian, you will perceive he is descending, or as we commonly say he is fallen, you will
see nothing but Water; your Vanes fast in this posture, you have done observing the Sun upon the Meridian that day: Therefore reckon the Degrees from B to the upper side of the Vane G; to it add the number of Degrees from E to the Eye-Sight, and their Sum is the Distance of the Sun from the Zenith to the upper Edge of the Sun; to which Sum if you add 16 minutes, which is the Sun's Semidiameter, you will have the true distance of the Sun's Center from the Zenith or Complement of the Meridian Altitude. Note this, If you observe the upper Limb of the Sun by the upper [Page 86] part of the Shade, then it is the upper Limb that gives the Shade; but if you observe the lower part of your Shade, then it is the lower side of the Sun that gives the Shade: Therefore you must substract 16 min. from what your Back-Staff gives you, and the Sum or Difference gives you the right Distance of the Sun from the Zenith. You may have the Altitude of the Sun from your Quadrant, if you work thus; from C to G is 40 deg. for the Vane stands at 20 deg. from D to F is 16 deg. being added together makes 56, the Altitude or Height of the Sun above the Horizon, which you may use as you were shewn by the Fore-Staff: But in regard the English Navigators work their Observation by the Complement of the Sun's Altitude, when he is upon the Meridian, being so ready to be counted by their Quadrant; Therefore we will direct you in general, and after in particular Rules.
The Figure of the Quadrant
First, If the Sun hath North Declination, and you in North Latitude, and the Sun upon the Meridian, South of you; then if you add the Sun's Declination to his Zenith-Distance, that is the Complement of the Sun's Meridian Altitude, the Sum will be the Latitude you are in.
But if the Sun hath South Declination, you must substract the Complement of the Meridian Altitude, and the Remainder will be the Latitude the Ship is in.
If you be to the Southward of the Aequinoctial, and the Sun to the Northwards of the Aequinoctial, in such case you must add the Sum of the Declination to the Zenith-distance, and the Sum will be your Latitude South.
But if the Sun be to the Northwards of the Aequinoctial (that is, have North Declination) [Page 87] you must substract the Declination from the Zenith-distance, and the Remainder will be the Latitude South.
If you understand the fore-going Rules given of the Use of the Fore-Staff, you cannot mistake the Use of the Quadrant or Back Staff.
We will now come to Examples what are needful.
Observe these Rules for North Latitude.
ADmit we were in a Ship at Sea the fifth of May, Anno 1694. and by Observation I find the upper side of the Sun to be distant from the Zenith 37 deg. 36 min. the Sun being upon the Meridian, I require the Latitude the Ship is in.
deg.
min.
The Sun's Distance from the Zenith
37
36 his upper Edge.
The Sun's Semidiameter, add
00
16
The Center of the Sun from the Zenith
37
52
North Declination
19
02
The Latitude required, the Ship is in
56
54
Suppose a Ship at Sea the 29th of July, Anno 1682. and I find the Complement of the Sun's Meridian Altitude by observation to be 32 deg. 54 min. The Latitude of that Place the Ship is in, is required.
deg.
min.
The Complement of the Altitude is
32
54
The Sun's Semidiameter add to it
00
16
The Distance of the Sun's Center from Zenith
33
10
North Declination, add
16
07
The Latitude the Ship is in
49
17
Suppose a Ship were at Sea the 13th of Sept. 1683. and I find the Complement of the Sun's Meridian Altitude, or Distance from the Zenith, 45 deg. 42 min. I demand what Latitude the Ship is in.
deg.
min.
The Complement of the Altitude is
45
42
The Sun's Semidiameter add to it
00
16
The Distance of the Center of the Sun
45
58
The Declination South, substract
00
07
The Latitude the Ship is in
45
51
Admit a Ship were at Sea the fourth of December, Anno 1690. and the Complement of the Sun's Meridian Altitude that day were 49 deg. The Latitude the Ship is in, is required.
deg.
min.
The Complement of the Meridian Altitude
49
07
The Sun's Semidiameter—Add
00
16
The Center of the Sun distant from the Zenith
49
23
The Sun's Declination South, substract
23
30
The Latitude the Ship is in North
25
53
Suppose I were in a Ship at Sea the 23d of May, Anno 1695. and I am also in Longitude to the East of the Meridian of London 135 deg. and I find the Complement of the Meridian Altitude by Observation to be 13 deg. 12 min. The Latitude is required.
That is, by reason the Sun is to the Northward of my Zenith, and the Declination more than the true Distance from the Zenith or Complement of the Meridian Altitude; therefore substract 13 deg. 28 min. from 22 deg. 20 min. and the Remainder is the true Latitude 8 deg. 52 min. North.
EXAMPLE.
Let the Complement of the Sun's Altitude be ZS, the Altitude 76 deg. 32 min. in the North BS, the Declination North ES 22 deg. 14 min. if you add the Altitude SB 76 deg. 32 min. to the Declination SE 22 deg. 14 min. the Sum is BE 98 deg. 46 min. the Distance of the Aequinoctial from the Horizon in the North BZ 90 deg. being substracted from it, remaineth for ZE, the Distance of the Aequinoctial from the Zenith towards the South, 8 deg. 46 min. just BP the Latitude of the Place, and Altitude of the Pole above the Horizon.
d.
m.
BS
76
32
SE
22
14
BE
98
46
90
00
B
08
46
[geometrical diagram]
Let the Complement of the Sun's Altitude be ZS 13 deg. 28 min. the North Declination ES 22 deg. 14 min. being more than the Distance of the Sun from the Zenith, substract ZS 13 deg. 28 min. the Complement of the Altitude from SE the Declination, there remaineth 8 deg. 46 min. the Distance of the Aequinoctial from the Zenith ZE or Latitude of the Place BP, as before. I have been the more large on this, by reason I would
have Learners perfect in it, it being most useful Questions.
When that you Sail far Northward or Southward, that the Sun goeth not down, as they find that Sail about the North Cape, and to Spitsberghen, or Greenland; and that you would observe the Altitude by the Sun, also when he is in the North at the lowest,
First, There must be added to the Altitude of the Sun taken above the Horizon, the Complement of the Sun's Declination; that is, the Distance betwixt the Sun and the Pole, that Number sheweth the Altitude of the Pole.
Secondly, Or else the observed Altitude must be substracted from the Declination, that which remaineth is the depression or depth of the Aequinoctial under the Horizon in the North, just to the Altitude of the same in the South, the Complement thereof is the Altitude or Height of the Pole.
Thirdly, If you take the Complement of the Sun's Altitude, and substract from it the Complement of the S [...]n's Declination, there remaineth the Distance of the Pole from the Zenith, or the Altitude of the Aequinoctial in the South; the Complement thereof is the Altitude or Height of the Pole.
Fourthly, Or else if you add the Declination to the Complement of the Altitude, and you substract 90 Degrees out of that Number, there remaineth the Depth of the Aequinoctial in the North under the Horizon; that being substracted out of 90, there remaineth the Altitude of the Pole.
Directions for Observation in South Latitude.
ADmit a Ship at Sea the 7th of July, Anno 1695. and I am in Longitude 135 deg. East, and the Sun being upon the Meridian, I find the Complement of his Meridian Altitude by Observation to be 42 deg. 34 min. North; The Latitude is demanded, the Ship is in.
deg.
min.
The Complement of the Meridian Altitude
42
34
The Sun's Semidiameter, add
00
16
The Sun's Center distant from the Zenith
42
50
The Declination North, substract
21
16
The Latitude the Ship is in
21
34
Admit I were in a Ship the fifth of November, Anno 1687. and in Longitude 120 deg. West, and the Complement of the Sun's Meridian Altitude by Observation is 31 deg. 37 min. North; The Latitude is required, the Ship is in.
deg.
min.
The Complement of the Meridian Altitude
31
37
The Sun's Semidiameter, add
00
16
The Sun's Center distant from the Zenith
31
53
The Declination South
18
37
The Proportional Minutes
00
05 Added.
The Sun's Declination in the Meridian given
18
42
Which add to the Zenith-distance
31
53
The Latitude the Ship is in
50
35
Suppose I were in a Ship at Sea to the Southwards of the Aequinoctial, the third of January, Anno 1683. and I find the Sun upon the South part of the Meridian, and by Observation his Meridian Altitude is 75 deg. 38 min. The Latitude the Ship is in, is required.
In this Figure let C be the South, and P the North Pole, DE the Aequinoctial, AB the Horizon, Z the Zenith. Let AF be the Altitude of the Sun above the Horizon, in the North 58 deg. DF South Declination 8 deg. If you substract the Declination DF 8 deg. from FA the Altitude, there remains 50 deg. the Height of the Aequinoctial above the Horizon in the North; that being deducted out of 90 deg. there remaineth AP 40 deg. for the Depth of the North Pole under the Horizon, just to BC the Elevation or Altitude of the South-Pole above the Horizon in the South.
[geometrical diagram]
deg.
min.
Sun's Altitude
58
00
South Declination
08
00
Height of the Aequator
50
00
90
00
The Latitude is
40
00
The Ʋse of the Fore-Staff in South Latitude for the Sun and Stars.
SUppose I were at Sea in a Ship the second of June, Anno 1694, and I find the Sun's Meridian Altitude by Observation to be 59 deg. I demand the Latitude the Ship is in.
deg.
min.
The Meridian Altitude North
59
00
The Declination North, add
23
15
The Complement of the Latitude
82
15
90
00
The Latitude required
07
45
Admit I were at Sea in a Ship to the Southward of the Aequinoctial, the 12th of January, Anno 1682. and in Longitude 135 East; and I find by Observation the Meridian Altitude 63 deg. 34 min. North: There is required the Latitude the Ship is in.
The Declination for this Meridian, the Lands-end of England, is about 19 deg. 33 min. the daily difference in Declination at this time is 14 min. Therefore if you look in the Table of Proportion, you will find the Proportional Minutes to be 5, which you must add to the Declination of the former Meridian, and the Sum will be the true Declination for the Longitude of 135 deg. East, which is 19 deg. 38 min.
Admit a Ship were at Sea the third of August, Anno 1675. and I find the Sun's Meridian Altitude to be 59 deg. 36 min. North, The Latitude is required.
deg.
min.
The Meridian Altitude North
59
36
The Declination North, add
14
42
The Complement of the Latitude
74
18
90
00
The Latitude the Ship is in
15
42
Suppose a Ship at Sea, the Sun's Declination being 21 deg. 42 min. South, and the Sun's Meridian Altitude 74 deg. 23 min. South, The Latitude is required the Ship is in.
deg.
min.
The Complement of the Sun's Meridian Altitude
15
37
Substracted from the Sun's Declination
21
42
The Latitude the Ship is in
06
05
This being made so plain and easie to be understood, need no more Precedent: But observe
this, If you observe the upper Edge or part of the Sun, you must substract 16 minutes; if the lower part, add 16 minutes for the Semidiameter of the Sun, and the Sum sheweth the true Altitude of the Center of the Sun.
CHAP. XV. Directions for Observing the Stars.
SUppose I am at Sea in a Ship, and I observe the bright Star in the left Foot of Orion Rigel, upon the Meridian, and find his Altitude 44 deg. 32 min. his Declination is 8 deg. 37 min. North; The Latitude is required, the Ship is in.
deg.
min.
The Meridian Altitude
44
32
The Declination North
08
37 substract.
The Complement of the Latitude
35
55
90
00
The Latitude the Ship is in
54
05
Suppose I am at Sea, and I observe the South Balance of Libra, and by Observation of the Star upon the Meridian, I find his Altitude 39 deg. 27 min. and his Declination South 14 deg. 27 min. I require the Latitude I am in.
I have furnished the Practitioner with all useful and needful Examples, which I thought necessary for direction, which explains the following Tables, and shews the most easie and perfect way of Observation, and how to work them on
either side the Aequator. Others I confess have been larger, but none more plain: for he that cannot understand
these Rules and Directions, is not fit to be a Mate of any Ship or Vessel, nor fit to be ranked among the Ingenious Mariners.
CHAP. XVI. The Description and Ʋse of the most Ʋseful Quadrant for the taking Altitudes on Land or Sea, of the Sun or Stars, backwards or forwards, or any other Altitude of Hills, Trees, Steeples, or Castles, or any thing whatever.
THis Quadrant is made of well-seasoned and smooth dry Box Wood or Pear-tree. The Sides or Semidiameter of the Circle is about 19 or 20 Inches. C V and CH the Arch of the Quadrant is divided into 90 Degrees first, and each Degree into 6 Equal Parts, each Part being 10 Minutes, which is near enough for Sea or Land Observations, and numbred as you see from 10 to 90 deg. The two Sides next the Center, EF and GF, are divided each of them into 100 Equal Parts:
[geometrical diagram]
[Page 93] That which is next the Horizon GFH, are called the Parts of Right Shadow: the other Side EFV, is the Parts of contrary Shadow. In the Center at C there is a Brass-Pin, and on it hang the Thred and Plummet; and on the Side there is a Sight made of Brass at E. There is also an Horizon-Vane, let in upon the Center C, with two Laggs that the Brass-Pin comes upon in the middle of the Slit; and a Shade-Vane and Sight-Vane, for Back-Observation. The Use of the Quadrant is,
EXAMPLE.
Admit I am ashore upon any Land or Island, and would know the Sun's Meridian Altitude, and true Latitude of the Place. Take the Altitude thus; The String and Plummet being hanged on the Center C, turn the Brass-Pin to the Sun, and hold up the Center until the Shade of the Brass-Pin strikes on the Sight and Line of E, the Thred and Plummet playing easily by the Side: mark where it cuts the Arch of the Quadrant, as at F, that is the Sun's Altitude, and reckoned from H; and the Latitude is found by the same Rules as you have been given in the Use of the Fore-Staff. The best way to hold the Quadrant steady, is to skrew it with a Brass-Pin through at K, to a Staff set perpendicular, and then you may raise it by degrees, as the Sun rises.
PROPOSITION I. For Back-Observation at Sea.
Take the Handle of the Quadrant at H in your Hand, after the Vanes are set on, and fix the Shade-Vane; then hold your Quadrant as upright as you can; then bring your Sight-Vane to your Eye, and look through your Sight upon the Horizon-Vane. You must be sure to hold your Quadrant, so that the upper part of your Shad [...] Vane, may be upon the upper part of the Slit on your Horizon-Vane, and look through the Slit for the Horizon: But if you cannot see the Horizon, but all Skie and no Water, you must draw your Sight-Vane a little lower down towards H; but if, on the contrary, you do see all Water and no Skie, then slide your Sight-Vane a little higher towards V, and then make Observation again; and then if the upper
part of the Shade do lie upon the upper part of the Slit, and you see the Horizon at the same time, then it is well, and you must wait a little longer as your Judgment
thinks fit, till the Sun is upon the Meridian, and so do as you did before; and if the Sun be to the Westward of the Meridian, and falling, you will see all Water and no Skie, the Work is done for that time and day. Then look what Degrees the Shade-Vane is put at, which in the Figure is at 70 deg. which note. Look also what Degrees and Minutes do stand against your Sight, which substract from the former Degrees by the Shade-Vane, and the Remainder is the Sun's Meridian Altitude. As in the Figure, The Sight-Vane is at 25 deg. 30 min. which taken out of 70 deg. the Remainder is 44 deg. 30 min. the Sun's Altitude, or the distance of the upper part of the Sun from the Horizon; from which if you substract 16 min. which is the Sun's Semidiameter, the Remainder will be the Distance of the Sun's Center from the Horizon, or the true Meridian Altitude. And the way of working your Observation, is the very same as you have been given in the Use of the Fore-Staff.
PROP. II. Any Point being given, To find whether it be level with the Eye, or not.
Take the Quadrant and look through the Sight at E and Center-pin C, unto the Point given, or the Place you would know whether it be level, or not. If the Thred fall on CH the Horizontal Line, then is the Place level with the Eye: But if it should fall within, upon any of the Divisions, then it is higher; if without the Quadrant, then it is lower than the level of the Eye.
PROP. III. To find the Height of an House, Steeple, Tower, or Tree, from the Ground, at one Observation; and the length of the Ladder which will Scale it.
If you can approach the bottom or foot of the Thing whose Height you desire, the thing is easily performed by this Quadrant or Cross-Staff, holding up your Quadrant to the Place whose Height you would know, and looking through the Sight on the Side EC, going nearer or further from it, till the Thred cut 45 deg. or fall upon 100 Parts in the Quadrat: So shall the Height of the Thing above the level of your Eye, be equal to the Distance between the Place and your Eye.
If the Thred fall on 50 parts of a right Shadow, or 26 deg. ½ or Vanes on the Cross-Staff, set to the Number of Deg. the Height is but half the Distance.
If the Thred cut 25 Parts in the Quadrat, or 13 deg. 55 min. in the Arch of the Quadrant, it is but a quarter of the Distance: But if it fall on 75 Parts, or 36 deg. 53, it is three quarters of the Distance. The Rule is,
As 100, to the Parts on which the Thred falleth:
So is the Distance, to the Height required.
And on the contrary,
As the Parts cut by the Thred, are to 100:
So is the Height, to the Distance.
[geometrical diagram]
But when the Thred shall fall on the parts of the contrary Shadow, if it fall on 50 Parts, or 63 deg. 30 min. as it doth at C, the Height is double unto the Distance CD. If on 25, it is four times the Distance. If the Thred fall upon the contrary Shadow, this is the Rule,
As the Parts cut by the Thred, are unto 100:
So is the Distance, unto the Height.
On the contrary,
As 100, are unto the Parts cut by the Thred:
So is the Height, unto the Distance.
These are the Rules Mr. Gunter shews by the Quadrat, And what hath been said [Page 95] of Height and Distance, the same may be understood of Height and Shadow; but here follows more useful Rules than these before-going.
PROP. IV. The Distance being given, To find the Altitude.
Suppose EFGD were a Tower, or Steeple, or Tree, or House, whose Altitude you would know, and you cannot come so neer as to measure between your Station of 45 deg. and the Base of the Thing, by reason of some Wall or Moat; yet by the Proportion of the Line of Quadrature, you may help your self by going backwards.
Thus if you could not measure the Distance from B to D, then go backward from B to A, until the Thred cut the 26 deg. 30 min. of your Quadrant; and measure the Distance between B and A, as suppose it to be 32 Foot or Yards, equal to the Height DE 32; The whole Line DA being 64 Feet or Yards, which is double to the Height. By the Tables,
Suppose then the Angle made by the Thred on the Quadrant ADB, be equal to the Angle EAD 26: 30 min. and the Distance AD be 64 Yards, or 192 Foot, to find the Height DE, I say,
As the Radius 90 Degrees
100000
To the Tangent of the Angle EAD 26 deg. 30 min.
969773
So is AD 64 Yards
180618
To the Height required, DE 32 Yards
X, 50391
PROP. V. The Distance being given, To find the Distance from the Eye to the top of the Tower.
Let the Distance AD be 192 Feet, the Angle at the Eye A 26 deg. 30 min. and the Distance from the Eye or Hypotenusa AE is required.
As the Sine of the Angle AED 63 deg. 30 min.
995179
Is to the Radius 90 deg.
10
So is AD 192 Feet
1228330
To AE 214 5/10 Feet
233151
PROP. VI. Some part of the Distance being given, To find the Distance from the Eye or Hypotenuse.
Let the Part of the Distance given be AB 96 Feet, and it is required to find the Distance from the Eye or Hypothenuse EB, which is the Length or Hypothenuse to the Triangle DBE. First find the other Angles thus:
As the Sine of the Angle ABE 135 deg. 5 min. or 44 deg. 55 min.
984885
To AE 214 5/10 Feet
333142
So is the Sine of the Angle EAB 26 deg. 30 min.
964952
To the Distance from the Eye BE 135 5/10 Feet or Hypotenuse
1298094
313209
PROP. VII. Some part of the Distance being given, To find the Altitude.
Keep the Angle and Distance from the Eye found by the former Proposition.
As the Radius 90 deg.
10
Is to the Sine of the Angle EBD 44 deg. 55 min.
984885
So is EB 135 6/10
313225
To ED the Height required, 95 8/10 Feet
1298100
which is the same as before found, without sensible difference. By the same Rule you may find the nearest Observation in the Figure to the Tower.
PROP. VIII. To do the same thing by the Quadrant, and Scale of Equal Parts, another way.
Without Calculation, by your Quadrant or Scale of Equal Parts, you may be resolved of all the foresaid Propositions, by the help of a Line of Chords; you may lay it down and demonstrate it, as you see the Figure, by the same Scale of Equal Parts as you measured, the first Distance, will answer all the rest. This is so plain, it needs no other Precept.
Here is another way to find the Length of the Scaling-Ladder without Calculation, which in many Cases is the chief thing looked after; which cannot be so well done
by the Quadrat, as by observing the Angles of the Quadrant; and this is the best way I know.
Let your Station be any where at random, or as neer as you can come to the Foot of the Tower or Wall, for the Ditch, or Moat, or Cannon shot: As suppose at B, and observe there the Angle of the Height of the Thing, which let be any Degrees whatsoever, as here is 45 deg. I say, If you go so far backwards from this Station of B, toward H, till you make the thing appear just at half the aforesaid Angle, which is here 22 deg. 30 min. the half of 45 deg. That then this Distance from B to H is the true Length of the Sloap-side BE, without farther trouble; and a Ladder of that Length will Scale the said Moat or Wall, allowing only the Height of your Eye from the Ground.
PROP. IX. Part of the Distance being given, To find the Remainder of the Distance.
Let part of the Distance given be AB 96 Feet, and the Remainder of the Distance cannot be measured, by reason of danger of Shot, or Moat of Water, or some other Impediments; Therefore by the 7 Rule I found the Angle at B to be 44 deg. 55 min. So that the Angle BED 45 deg. 05 min. is the Complement thereof: Which known, I say,
As the Radius
10
To the Sine of the Angle BED 45 deg. 5 min.
985011
So is the Distance from the Eye BE 135 6/10 Feet
313225
To the Remainder of the Distance BD 96 Foot
298236
PROP. X. By the Height of the Sun, and the Length of the Shadow, To find the Height of any Tree, Tower, or Steeple.
This Conclusion may be tried by a little Quadrant or Pocket-Instrument, by which you may take the Sun's Altitude to a Degree, ½ or ¼, which is near enough for these Conclusions.
[geometrical diagram]
Suppose DE to be a Turret, Tree, or Steeple, whose Height is required to be found by the Shadow it makes on Level Ground, the Rule is thus, viz. Let the Height of the Sun be 37 deg. 00 min. and the Length of the Shadow 40 Foot, the Rule is,
As the Radius 90 deg.
10
To the Length of the Shadow 40 Foot
160206
So is the Tangent of the Sun's Height 37 deg.
987711
To the Height of the Thing desired
147917
which is found to be 30-14 Parts, which shews the Height to be a little above 30 Foot.
Here is another way to do the same without the help of a Quadrant and Sun's Altitude, viz. Set up a Staff of any Length, suppose 3 Feet in Length, as CB, and [Page 98] the Shadow which it makes from B to A is 4 Feet; Because the Shadow of the Tower from the Base thereof to B is 40 Feet, I say,
As the Shadow of the Staff, is to the Height of the Staff:
So is the Shadow of the Steeple, to the Height of the Steeple.
The Operation may be performed by Natural Numbers, or by Logarithms, thus, viz.
A Constant Kalendar; OR AN ALMANACK For Three Hundred Years. But more exactly serving
for Nineteen Years, BEING THE CIRCLE of the MOON, OR THE GOLDEN NUMBER. With New Exact
TABLES OF THE Suns Declination, Rectified by the best Hypothesis, until the LEAP-YEARS 1695. BY Capt. SAMUEL STURMY.
AN ALMANACK For XXXII. Years. According to the English and Foreign Accounts.
Anno Dom.
Prime.
[...]pact
Sund. Letter
Shrove Sunday.
Easter Sunday.
White Sunday.
Diff.
1664
12
12
CB
Feb. 21
10
May 29
1
1665
13
23
A
5
March 26
14
0
1666
14
04
G
25
April 15
June 03
0
1667
15
15
F
17
7
May 26
1
1668
16
26
ED
2
March 22
10
0
1669
17
7
C
21
April 11
30
0
1670
18
18
B
13
3
22
1
1671
19
29
A
March 05
23
June 11
5
1672
01
11
GF
Feb. 18
7
May 26
0
1673
2
22
E
9
March 30
18
1
1674
3
3
D
March 1
April 19
June 07
5
1675
4
14
C
Feb. 14
4
May 23
0
1676
5
25
BA
6
March 26
14
0
1677
6
6
G
25
April 15
June 03
1
1678
7
17
F
10
March 31
May 19
0
1679
8
28
E
March 2
April 20
June 08
5
1680
9
9
DC
Feb. 22
11
May 30
0
1681
10
20
B
13
3
22
1682
11
1
A
26
16
June 04
1683
12
12
G
18
8
May 27
1684
13
23
FE
10
March 30
18
1685
14
04
D
March 1
April 19
June 07
1686
15
15
C
Feb. 14
4
May 23
1687
16
26
B
6
March 27
15
1688
17
7
AG
26
April 15
June 03
1689
18
18
F
10
March 31
May 15
1690
19
20
E
March 2
April 20
June 08
1691
1
11
D
Feb. 22
April 12
May 31
1692
2
22
CB
7
March 27
May 15
1693
3
3
A
Feb. 26
April 16
June 4
1694
4
14
G
18
8
May 27
1695
5
25
F
Feb. 3
March 24
12
The Ʋse of the Almanack for 32 Years.
THis Table sheweth first the Date of the Years; secondly, the Prime, or Golden Number; thirdly, the Epact; fourthly, the Dominical Letter for these Years; and then in their Order the Chief Movable Feasts (viz.) Shrove-Sunday, Easter-day, and Whitsunday, upon which all the rest depend. The Foreign Account is commonly ten days before us; but their Movable Feasts fall sometimes at the same time with ours, sometimes 1, 2, 3, 4, or 5 Weeks before ours, as you see in the last Column of the Table.
Of the Terms.
THere are four times of the Year appointed for the Determining of Causes; these are called Terms. Two of these Terms (viz.) Hillary Term, and Michaelmas Term, are at a constant time of the Year: but Easter Term and Trinity Term are sooner or later, as those Feasts happen. Each of these Terms hath several Returns, and each Return hath four Days belonging to it. The first is the Day of Return or Essoin, for the Defendant in a Personal Action, or the Tenant in a Real Action. The second is the Day of Exception, for the Plaintiff or Defendant to lay an Exception. The third is the Day whereon the Sheriff must return the Writ. The fourth is the Day of Appearance in the Court. These four Days follow each other in order, except a Sunday or Holyday take up any of them, and then the Day following serves for both Occasions.
The Beginning and End of Hillary Term and Michaelmas Term, with all their Returns, you shall find in this following Kalendar, which are constant if no Sunday hinder them.
Easter Term begins Wednesday Fortnight or 17 Days after Easter, and ends the Munday after Holy-Thursday, or the Munday before VVhitsunday: It hath these five Returns.
Quind. Pasc. A Fornight after Easter.
Tres. Pasc. Three Weeks after Easter.
Mens. Pasc. A Month after Easter.
Quinq. Pasc. Five Weeks after Easter.
Crast. Ascen. The Day after Holy-Thursday.
Trinity Term begins the Friday after Trinity-Sunday, which is next Sunday to Whit-sunday, and hath these four Returns.
Crast. Trin. The Munday after Trinity-Sunday.
Oct. Trin. A Week after Trinity-Sunday.
Quind. Trin. A Fortnight after Trinity-Sunday.
Tres. Trin. Three Weeks after Trinity-Sunday.
The Exchequer opens four Days before Trinity-Term; but eight Days before the other Terms.
Lo! here a Trade surpasseth all the rest;
No Change annoys the Lawyer's Interest.
His Tongue buyes Lands, builds Houses, without Toil;
The Pen's his Plough, the Parchment is his Soil;
Him Storms disturb not, nor Militia Bands.
The Tree roots best, that in the Weather stands.
How to Rectifie the Tables of the Sun's Declination at any Time by Prostaphaereses.
Prostaphaereses of the Suns Declination.
Day Month.
Jan.
Feb.
Mar.
Apr.
May.
June.
July.
Aug.
Sept.
Octob.
Nov.
Dec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
Sec.
1
17
36
42
40
28
8
15
33
42
41
30
9
2
18
37
43
39
28
7
15
34
43
42
29
8
3
19
37
43
40
27
6
16
35
43
42
29
7
4
20
38
44
40
27
5
17
35
43
42
28
6
5
20
37
44
39
26
4
18
34
43
41
28
5
6
21
37
43
39
25
4
19
35
43
41
28
4
7
21
38
43
38
25
3
19
35
44
41
27
3
8
22
39
43
38
24
2
19
35
43
40
26
2
9
24
38
44
38
23
2
20
36
44
40
25
1
10
24
39
45
37
23
1
21
36
43
40
24
1
11
25
39
44
37
22
0
21
37
43
40
24
0
12
25
40
44
37
22
1
22
38
43
39
23
0
13
26
41
43
37
21
1
22
38
43
39
22
1
14
26
40
44
37
20
2
23
38
44
38
21
2
15
27
41
44
35
20
3
23
38
43
38
21
3
16
28
42
43
35
19
4
24
39
44
37
20
4
17
28
41
42
35
19
4
25
39
43
38
20
5
18
29
41
43
35
18
5
26
39
43
37
19
6
19
30
41
43
34
17
6
26
39
43
37
18
7
20
31
42
42
34
17
7
27
40
43
36
17
8
21
31
42
43
33
16
8
27
40
43
36
16
9
22
32
42
43
33
15
8
28
41
44
35
16
10
23
32
43
43
32
14
9
29
40
43
35
15
11
24
32
43
42
32
14
10
29
41
42
34
14
12
25
33
44
42
31
12
11
30
41
43
34
13
12
26
33
44
41
3 [...]
12
12
30
41
42
33
13
13
27
34
43
41
30
11
13
30
41
42
34
12
14
28
35
43
41
29
9
14
31
41
43
32
11
16
29
35
43
41
29
9
14
31
41
43
32
11
16
30
35
41
29
9
14
32
42
42
32
10
16
31
36
41
8
33
31
17
The Ʋse of the Table.
When the Declination increaseth, then add; and for decreasing, substract.IN this Kalendar, Printed for the Year 1665, 66, 67, and 1668. the Sun's Declination to be trusted sufficient; but for any Year after 1668. the Rule is thus.
For Example. I would know the Sun's Declination for the Year 1689. you must always substract 1668. from the Year given, which is here 1689. the Remainder is 21 Years; which being divided by 4, the Quotient is 5 Leap-Years, and 1 remains, which sheweth it is the first Year.
Now I desire to rectifie the Table for the first day of April, which in the Kalendar you have 8 deg. 36 min. and in this Table you have 40 Seconds;8 d. 39 m. Declination 1 April 1689. which multiplied by 5 Leap-years, give 200 Seconds, that is 3 m. 20 sec. to be added to 8 deg. 36 min. So have you 8 deg. 39 min. for the Sun's Declination in 1689.
To find the Sun's Declination upon every Day of the Year.
THe Sun's Year (that is, the time that the Sun goeth out of a certain Point of the Ecliptick, and returneth again to the same) is not of 365 days just; but about 5 Hours and 49 Minutes more (that is, little less than 6 Hours;) Wherefore after three Years, there is always added to the fourth four times 6 Hours, that is, a Day more in February, for to count the Year or the Revolution of the Sun in even Days; therefore that fourth Year is called Leap-year: Therefore when we describe the Sun's Declination in Tables, we always use to make four several Tables, for four such Years following one the other; and yet by reason of the foresaid difference, that four
Revolutions of the Sun do not justly make up one Day, but wants about 48 min. bringeth in process of time so great a difference in the Declination, that it is needful every twenty Years to renew such Tables.
How to find the Leap-years, it is thus: Divide the Year of our Lord above 1600. by 4; If the Division doth fall out even, without any over-plus, that Year then is a Leap-year of 366 Days: But if out of the Division there remain any Number, that Remainder sheweth how many Years that Year propounded, is after the Leap-year.
For EXAMPLE.
I desire to know what Year the Year 1666. is. Leaving 1600, I divide 66 by 4, and find there remains 2; for 16 times
4, or 4 times 16, is 64; that taken from 66, there remains 2; whereby I find the Year 1666. to be the second Year after the Leap-year. In the like manner you must work for any other Years: Only note this, If nothing remaineth upon the Division out of the Quotient, then it is a Leap-year if it be even.
As for EXAMPLE.
It is required to know what Year 1692 is. Leaving the 1600, divide the 92 by 4, and nothing remains upon the Division, but is even 23 in the Quotient; whereby I find that Year 1692. is a Leap-year.
For to know the same by the foregoing Tables, it is thus. Each Month hath 12 Columns; The first thereof shews the Days of the Month; The second Column, having the Dominical Letters, shews the Days of the Week; The third Column having two Rows of Figures, the first of them shews the Epact of the Moon, and the other the Hour of the Day, reckoning the said Hours always from Noon; the fourth Column shews the Chief Days of the Year, and the Terms and their Returns which are fixed and certain; and in the void places it shews the Rising and Setting of the Sun in this Latitude, and the Place of the Sun every 10 Day or Degree. These four Columns of themselves are fit for Mens ordinary use, and may be made with a little Art and
Pains to perform all the Conclusions which the yearly Almanacks shew and teach, as you shall see by the following Rules and Observations.
The fifth Column of the foregoing Tables shews the Sun's Declination for every Day of the Year, for all these Years in the first Column under-written, which are all Leap-years. The sixth Column shews the Daily Difference of the Sun's Declination. The seventh Column shews the first Year from the Leap-year: The eighth, the Daily Difference of the Sun's Declination in that Year. The ninth shews the second Year from the Leap-year; The tenth, the Difference; The eleventh, the third Year from the Leap-year; The twelfth, the Difference every Day of the Sun's Declination, as you see in the Tables. This Table following shews the Leap-years, First, Second, and Third Years, as they are plainly expressed in the Head of each Table.
For to find the Sun's Declination, Look for the Day of the Month in the left hand of the Table, and in the common Angle of meeting you will find the Declination which you seek after.
I. EXAMPLE.
I desire to know the Sun's Declination for the 22 of May, in the Year 1693. being the first Year after the Leap-year. In the Head of the Table I find the Month and Year; on the left hand of the Table I find the Day; and in the Common Angle or Line of Meeting, I find the Declination I look for to be North 22 deg. 13 min.
II. EXAMPLE.
Upon the 5th of November in the Leap-year 1692. I desire to know the Declination of the Sun. In the Head of the Table I find the Month and Day, and in the first Column to the left hand I find the Day of the Month, and in the Common Line of Meeting, under the Year, I find the Sun's Declination required, to be 18 deg. 37 m. South Declination; and his Difference in 24 Hours, 15 min.
The foregoing Tables of the Sun's Declination is rectified properly for the Meridian of the most famous and Metropolitan City of London. The Constant Kalendar I borrowed out of Ingenious Mr. Philips's Purchaser's Pattern, at the end of page 247. With some addition it is very useful with the foregoing Tables.
Of the Difference and Aequation of Declination in divers Places of the Earth.
A Table by which you may proportion the Sun's Declination to any other Meridian.
The Difference in Declination daily.
M
M
M
M
M
M
M
M
M
00
03
06
09
12
15
18
21
24
M
M
M
M
M
M
M
M
M
Degrees of Difference of Longitude either East or West.
Deg. 15
0
0
0
0
0
0
1
1
1
30
0
0
0
0
1
1
1
2
2
45
0
0
0
1
1
2
2
3
3
60
0
0
1
1
2
2
3
3
4
75
0
0
1
2
2
3
4
4
5
90
0
0
1
2
3
4
4
5
6
105
0
1
2
2
3
4
5
6
7
120
0
1
2
3
4
5
6
7
8
135
0
1
2
3
4
5
7
8
9
150
0
1
2
3
5
6
8
9
10
165
0
1
2
4
5
6
8
10
11
180
0
1
3
4
6
7
9
11
12
NOte this, They that are more Easterly from the Meridian of London, have the Declination less when the Sun declineth from the Line, and increaseth in Declination either Northward or Southward, as well between the 10th of March and the 12th of June, as between the 13th of September and the 12th of December; and more when the Sun returneth again towards the Line, whether it be North or South of the Line, as well between the 12th of December and the 10th of March, as between the 12th of June and the 13th of September.
On the contrary, They that are more Westerly from the Meridian of London, when the Declination increaseth North or South, have more Declination, and less when the Declination decreaseth; that is, when the Sun is going towards the Aequinoctial, either on the North or South side of the Line; the reason is, because the Sun cometh to the Meridian Eastward, to them that live there, always before it doth to us; and them that live more Westerly, have him later to their Meridian.
EXAMPLE I. Of those that are more Easterly, which increase in Declination.
On the 26th of March, the first Year after the Leap-year, I desire to know the Declination of the Sun at Noon at Bantam in the East-Indies. [...] I find by Globes, or the Plat of Mercator, that Bantam is to the Eastward of the Meridian of London about 110 Degrees; we do not esteem of a Degree or two, because it amounteth to nothing in this Practice. The Sun for his Course round the Heavens and Earth, which is 360 Degrees, hath need of 24 Hours; What time will 110 Degrees have? Facit 7 Hours, and something more not worth the noting; whereby the Sun comes to the Meridian 7 Hours sooner at Bantam, than it doth at London; That it is 12 a Clock at Noon at Bantam, when it is 4 of the Clock in the Morning with us at London. The Sun's Declination for the 26th of March, is 6 deg. 25 min.: The Difference of the Declination of the Day following, you find is 23 min. which it is increased; Therefore I say, If in 24 Hours the Declination increaseth 23 Minutes, How much then in 7 Hours? Facit almost 7 Minutes, that the Declination is less than it is at London. So that the Declination at Bantam that Day, is but 6 deg. 18 min. North: And on the contrary, when the Declination decreaseth, work, and you will have the Declination South, Eastward, or Westward.
EXAMPLE II. The Ʋse of the Table.
On the 17th of September in the same Year, I desire to know the Declination that day at Noon at Bantam. The Declination for the Meridian of London is that Day 1 deg. 52 min. and the Difference of the Declination of the Day following is 24 min. decreased; and, as was said before in the last E [...]mple, the difference of Longitude is 110 deg. Therefore I look in the Head of the foregoing Table, for the nearest Number to the Difference 24, and find it to fall just even on the Head of the last Column; then look on the left hand of the Table for the Difference of Longitude, and I find 105 deg. nearest, and in the Common Angle of Meeting I find 7, which is to be substracted from the Declination in the Meridian of London abovesaid, 1 deg. 52 min. and the Remainder will be the Declination for the Meridian or Longitude I am in, which is 1 deg. 45 min. South: But if the Declination decreaseth, as it doth here increase, then you must have added.
deg.
min.
In the Meridian of London the Declination
01
52
The Minutes Proportional substracted
00
07
The Declination for 110 deg. Longitude of Bantam, East
01
45
The Declination of 110 deg. West of the Meridian of London
A Ship coming on the seventh of November, in the third Year after the Leap year, into the great South-Sea, thwart of the Coast of Peru, in Longitude 76 deg. The Pilot desireth the Declination there at Noon in that Meridian.
deg.
min.
In the Meridian of London the Declination is
19
08 South.
The Minutes Proportional added
00
03
In the Longitude 76 deg. the Declination
19
11 West.
In the Longitude of 76 deg. East, the Declination is
19
05
Two Ships being in Company, they parted at the Lands-end of England: The one Sails Eastwards, and cometh upon his Reckoning upon the 28th of September 180 Degrees on the other side the Globe of the Earth (being the first Year after the Leap-year) and by the foregoing Tables finds the Sun's Declination 5 deg. 57 min. The other Ship Sails Westwards, and meeteth the first Ship at the aforesaid place, by his Reckoning not the 28th, but on the 27th of September, and findeth the Declination in these Tables for that Day; so that they differ in the Time one Day, and in Declination 24 min. the which proceedeth from this cause: The first having Sailed against the Rising of the Sun 180 Degrees, hath shortned his time 12 Hours; the other hath Sailed with the Sun 180 Degrees, hath lengthned his time 12 Hours, and thereby hath one Night less than the first. Seeing then in 24 Hours increaseth 24 Minutes, he that Sailed Eastward must reckon 12 Minutes Declination less, and he that Sailed Westward 12 Minutes more than the Table doth shew; and so both of them shall keep one manner of Declination, to wit, 6 deg. 9 min.
A Table of the Refractions of the Sun, Moon, and Stars, according to the Observation of thrice Noble Tycho Brahe.
Altitudes.
Sun.
Moon
Stars
Altitudes.
Sun.
Moon
min.
min.
min.
min.
min.
0
34
33
30
18
06
06
1
26
25
21
19
05
06
2
20
20
15
20
04
05
3
17
17
12
21
04
05
4
15
15
11
22
03
04
5
14
14
10
23
03
04
6
13
13
00
24
03
04
7
12
13
08
25
02
03
8
11
12
07
26
02
03
9
10
11
06
27
02
03
10
10
11
05
28
02
02
11
09
10
05
29
02
02
12
09
09
04
30
01
02
13
08
00
04
31
01
02
14
08
08
03
32
01
01
15
07
08
03
33
01
01
16
07
07
02
34
01
01
17
06
07
02
35
01
01
THe Refraction of the Sun, Moon and Stars, causeth them to appear higher above the Horizon than they are: Therefore the Refraction is alway [...] to be substracted from the A [...]ude observed, that the tr [...] [...]ltitude may be had.
As for Examp
The Sun's Meridian Altitude by Observation being 9 Degrees, I require the true Altitude.
deg.
mi.
Altitude by Observation
9
00
Refraction substract
0
10
The true Meridian Altitude
8
50
Of the Refraction of the Sun, A Dutch Ship being upon the Discovery of a North-East Passage to the East-India, was forced to Winter in Nova Zembla: the Mariners beheld the Sun 14 days sooner than he should by his Declination, and by Computation 5 Degrees under the Horizon; which is caused by the gross Vapours, and thickness of the Air neer the Horizon.
THis is the Chief and most useful Observation of any Almanack, and may as well be performed by this, as by any other. To this purpose, you must
by the general Kalendar at the beginning hereof, know the Dominical or Sunday Letter for the Year; then considering with your self, whether it be the beginning, midst, or end of the
Month (as you must do in any Almanack) find this Letter in the beginning, midst, or end of the Month, and reckoning from it to the Day of the Week, either Munday, or Tuesday, or whatsoever other Day it is, right against the Day of the Week you shall find what Day of the Month it is. Here is no difficulty in this; only when it is Leap-year you see there is two Sunday Letters, the first of these you may use only to the 24th of February, and the other all the Year after.
For Example. In the 1668. the Dominical Letter ED the first Sunday in January, is at the first E, which is at the fifth Day of the Month; the first Sunday in February is at the second Day of the Month; but the first Sunday in March is at the first D, which is at the first Day of the Month, and so all the Year after.
II. To know what Day of the Week any Notable Day will fall upon, in any Year.
First find the Dominical Letter in the former Table; then find your Letter in your Month next before the Day you desire, and so from thence count the Days of the Week, till you come to the Day desired. Thus if you would know what Day of the Week Lady-day, or the Annunciation of the Lady Mary falls upon this Year 1668. the Dominical Letter is D; this is three Days before the said Day, therefore that falls upon a Wednesday.
But now in the Year 1669. when the Dominical Letter is C, Lady-day will be upon the Thursday. This will be in a short time as ready to you, as if these Letters were painted out for you in Vermilion.
III. To find the Time of Sun Rising and Setting.
This is set down for most of the Days in the whole Year, for London; and may serve for all the East, South, and West Parts of England: And this is done after somewhat a briefer manner than is usual, making the Minutes which are placed in the midst, to serve both the Hours of Setting and Rising; which you must understand thus: The 7th of February you shall find these Figures, 4. 59. 8. that is, the Sun that Day sets at 4 h. 59 m. that is 59 m. after 4. and riseth at 59 m. 8 h. that is 59 m. before 8. or almost 7 a Clock. And so you must account them always, remembring, That
as the Minutes follow the first Figure, so they must be reckoned in Time after: as they stand before the last Figure, so they must be reckoned in Time before it.
And think it not preposterous that the time of the Sun's Setting is set down before the Rising; for the Sun's Setting is of most use, and the other serves in a manner for the filling up of the Column.
ho.
min.
If you double this time of Sun Setting
04
59
You have the Length of the Day
09
58
If you substract it from
12
00
You have the time of Rising, differing in shew from the Kalendar
07
01
But all one in effect; and this doubled, shews the Length of the Night
THis is set down in the Kalendar, about every tenth Day, to every tenth Degree; so that reckoning a Degree for each Day between, you shall have the Place of the Sun exact enough for most ordinary Uses. Thus the 10th of March the Sun enters into Aries; therefore the 15th Day, or five Days after, the Sun is in five Degrees of Aries.
V. To find the Day and Hour of the Change or New Moon, and thereby the Full and Quarters.
FIrst you must find the Moon's Epact for the present Year you are in: This Number is found out in the First Book, Page 12. and also in the Table before at the beginning of the Kalendar. The Change also may be found out by the Golden Number; yet that would stand so scattering and without form, that it is much handsomer and
readier to find out by this Epact, which runs for the most part in a Constant Order, only here and there skipping a
Day or a Number, which is marked with this ✶.
Having found out the Epact for this present Year, turn to the Month you desire, and there find out the said Number of the Epact in the third Column of the Months, and mark what Day of the Month it stands against; for that is the Day of the Change or New Moon. Likewise if you have respect unto the Dominical Letter, which is by it, you shall see what Day of the Week it is.
Now here in this Column there are two Rows of Figures; The first shews the Epact-Number, and the next the Time of the Day reckoned by the Hours from Noon, which are plain to understand till you come to 12 Hours after Noon, which is Midnight; but then the Numbers above 12, you must reckon to the Morning of the next Day.
So that these Hours after Noon,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23,
24,
are all one with these,
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
the next Day in the Morning.
Thus in the Year 1666. the Epact being 4, and the Dominical Letter G, you shall find this Epact-Number 4 against the 21 of July being Saturday; and the Figure o standing by it, shews that the New Moon is just at Noon.
Again, You shall find the Epact-Number against the 16th of November, being Friday; and the Figure of 2 standing by it, shews that it is about 2 Hours after Noon the Moon changeth. Now this is the true time of the New Moon, according to the Moon's mean Motion; which though it may differ half a day from the true Change, yet it seldom differs so much, and is better for the following Conclusion than the true time.
Having first found out the time of the New Moon, you may from thence reckon the Age of the Moon, and find the Quarters and Full Moon.
Thus the Moon's Age is
Days
Hours
Min.
At the First Quarter
7
09
11
At the Full Moon
14
18
22
At the Last Quarter
22
03
33
An Whole Moon
29
12
44
Or else observe the Dominical Letter that is against [...]e Epact, or Day of the New Moon; and where you find that Letter again, that is the First Quarter; for the Full Moon take two Weeks and one Day, which will fall upon the Letter next to it; for the Last Quarter take one Week more, which will fall upon the Letter of the Full Moon.[Page 121] Thus if the New Moon fall upon A, the First Quarter falls upon the next A, and the Full Moon on the next B a week after, and the Last Quarter on the next B. And thus you have this brief Kalendar or Constant Almanack for many Years; only for the more exactness in the Hour of the Moon's Change and Age, it is restrained to 19 Years: For though the Change of the Moon (for the most part) hapneth again upon the same Days, for several Revolutions of the Prime or Golden Number XIX; yet not upon the same Hour of the Day, but alters every Revolution 7 Hours, 27 Minutes, 30 Seconds, proceeding forward for the most part; but the Leap-years coming in with a Day more than ordinary, keeps this Motion so much backward, that in 300 Years it neither gains nor loseth a Day, only differeth in the Hour of the Day; yet for the more exactness, it will be better to renew this every 19 Years. All these things this brief Kalendar shews plainly, with little or no trouble more than in an yearly Almanack. I shall now proceed to some other Conclusions. I have been very large already, in the First Book, of Things concerning the Use of the Moon in other Conclusions; to which I refer you for any thing of the Tides, or the Southing of the Moon, or the Rising or Setting of the Moon, or what else is necessary in Navigation.
I thought to have entred my Figure of the Sea-Compass, for the Surveying of Land, which was promised in the Argument; As likewise the Gunner's Scale and Gauging Rod: But I refer you to the several Books in the following Treatise, where the Figure and the Use of it, is together for your satisfaction.
THE Mariners Magazine; OR, STURMY'S Mathematical and Practical ARTS. The Third Book.
CHAP. I. Of the Nature and Quality of Triangles.
THis Third Book is as it were a Key to these that follow, the Subject whereof is Trigonometry; therefore I hold it convenient before I come to the following Practice, to say something concerning Plain Triangles at least, although that Subject be handled by divers more able Mathematicians already, whose Works are extant, (viz.) Pitiscus, Snellius, the Lord Napier, Master Gunter, Master Norwood, Master Gellibrand: so that those who desire to make a farther scrutiny into Trigonometry, may peruse the forementioned Authors.
Before we come to shew how the Quantity of the Sides and Angles of any Triangle may be found by the Tables of Artificial Sines and Logarithms, and by the Lines of Artificial Sines, Tangents, and Numbers, take these following Theorems, as Necessaries thereunto.
[geometrical diagram]
I. A Triangle is a Figure consisting of three Sides, and three Angles, as in the Figure DBC.
II. Any two Sides of a Triangle, are called the Sides of the Angle comprehended by them; as the Sides CB and DB, are the Sides containing the Angle CBD.
III. The Measure of an Angle is the Quantity of an Arch of a Circle, described on the Angular Point, and cutting both the containing Sides of the same Angle; As in the Triangle following, the Arch CB is the Measure of the Angle at A, the Arch KD is the Measure of the Angle at E, and the Arch FG is the Measure of the Angle at H. Each of these Arches are described on the Angular Points A, H, E, and cut the containing Sides.
[geometrical diagram]
IV. A Degree is the 360 part of a Circle.
V. A Semicircle containeth 180 Degrees.
VI. A Quadrant containeth 90 Degrees.
VII. The Complement of an Angle less than a Quadrant, is so much as that Angle wanteth of 90 Degrees; as if the Angle AHE should contain 41 deg. the Complement thereof would be 49 deg. For if you take 41 from 90, there will remain 49 deg.
VIII. The Complement of an Angle to a Semicircle, is the Remainder thereof to 180 Degrees.
IX. An Angle is either Right, Acute, or Obtuse.
X. A Right Angle is that whose Measure is a Quadrant.
XI. An Acute Angle is less than a Right Angle.
XII. An Obtuse Angle is greater than a Quadrant.
XIII. A Triangle is either Right-Angled, or Obtuse-Angled.
XIV. A Right-Angled Triangle is that which hath one Right Angle; as the Triangle AHE is Right-Angled at E.
XV. In every Right-Angled Triangle, that Side which subtendeth or lieth opposite to the Right Angle, is called the Hypothenusa; and of the other two Sides, the one is called the Perpendicular, and the other the Base, at pleasure: But most commonly the shortest is called the Perpendicular, and the longer is called the Base. So in the former Triangle, the Side AH is the Hypothenusa, HE the Base, and AE the Perpendicular.
XVI. In every Right-Angled Triangle, if you have one of the Acute Angles given, the other is also given, it being the Complement thereof to 90 deg. As in the Triangle AHE, suppose there were given the Angle HAE 49 deg. then by consequence the Angle AHE must be 41 Degrees, which is the Complement of the other to 90 Degres.
XVII. The three Angles of any Right-Lined Triangle whatsoever, are equal to [Page 125] two Right Angles, or to 180 Degrees: So that if of any Right-Lined Triangle, you have any two of the Angles given, you have the third Angle also given, it being the Complement of the other to 180 Degrees.
[geometrical diagram]
So in this Triangle ABC, if there were given the Angle BAC 36 Degrees, and the Angle ACB 126 Degrees, I say by the consequence, there is also given the third Angle. For if you add the two given Angles together, and substract it from 180 Degrees, there will remain ABC 18 Degrees: The two given Angles being taken from 180, the Remainder is the Angle required.
In all plain Triangles whatsoever, the Sides are in proportion one to the other, as the Sines of the Angles opposite to those Sides. So in the Triangle ABC, the Sine of the Angle ACB is in such proportion to the Side AB, as the Sine of the Angle CAB is to the Side BC. And so of any other.
CHAP. II. Containing the Doctrine of the Dimensions of Right-Lined Triangles, whether Right-Angled or Oblique-Angled; and the several Cases therein resolved, both by Tables, and also by the Lines of Artificial Numbers, Sines, and Tangents.
I Come now to shew you how a Plain Triangle may be resolved; that is, by having any three of the six Parts of a Plain Triangle, to find a fourth by the Instruments before-mentioned.
In all the Cases following I have made use of but two Triangles for Examples; one Right-Angled, the other Oblique-Angled: But in either of them I have expressed all the Varieties that are necessary; so that
any three Parts being given in any of them, a fourth may be found at pleasure.
The Sides of any Plain Triangle may be measured by any Measure or Scale of Equal Parts; as an Inch divided into 10 Parts, or 20, 30 Parts; or likewise into Inches, Feet, Yards, Poles, Miles, or Leagues.
[geometrical diagram]
Draw a Line at pleasure, as AB;How to lay down an Angle by the Line of Chords. and from the Point. A let it be required to protract an Angle of 41 deg. 24 min. First extend the Compasses upon the Line of Chords, from the beginning thereof to 60 deg. always, and with this distance setting one Foot upon the Point A, with the other describe the pricked Arch BC: Then with your Compasses take 41 deg. 24 min. (which is the Quantity of the inquired Angle) out of the Line of Chords, from the beginning [Page 126] thereof to 41 deg. 24 min. Keeping the Compasses at this distance, if you set one Foot thereof upon B, the other will reach upon the
Arch to C. Lastly, draw the Line AC. So the Angle CAB shall contain 41 deg. 24 min.
To find the Degrees contained in an Angle.Suppose CAB were an Angle given, and that it were required to find the Quantity thereof. Open your Compasses, as before, to 60 deg. of your Chord; and placing one Foot in A, with the other describe the Arch BC. Then take in your Compasses the Distance CB, and measuring that Extent upon the Line of Chords, from the beginning thereof, you shall find it reach 41 deg. 24 min. which is the Quantity of the inquired Angle.
If any Angle given or required shall contain above 90 Degrees, you must then protract it at twice, by taking first the whole Line, and then the Remainder.
The several Cases of Right-Angled Triangles, may not only be applied to Navigation, but also in the taking of Heights, as is shewn elsewhere: And the Oblique-Angled Triangle, for the taking of Distances, taught in this following Treatise.
In the resolving of Plain Triangles there are several Cases, of which I will only insist on these, which have most relation to the Work in hand.
Of Right-Angled Plain Triangles.
CASE I. In a Right-Angled Plain Triangle, The Base and the Angle at the Base being given, To find the Perpendicular.
SUppose that the Line CA (in the following Figure) in the Right-Angled Triangle, were a Tree, Tower, or Steeple, and that you would know the Height thereof; you must observe with your Instrument the Angle CBA, and measure the Distance BA.
So have you in the Right-Angled Triangle ABC, the Base 405 Foot (Miles or Leagues the denomination might have been as well) and the Angle at the Base 32 deg. and it is required to find the Perpendicular AC.
Now because the Angle CBA is given, the Angle BCA is also given, it being the Complement of the other to 90 deg. and therefore the Angle BCA is 58 Degrees: Then to find the Perpendicular CA, the Proportion is,
As the Sine of the Angle BCA 58 deg. (which is)
9928420
Is to the Logarithm of the Side BA 405 Foot
2607455
So is the Sine of the Angle CBA 32 deg. (which is)
9724210
The Sum of the Second and Third added
12331665
The first Number substracted from the Sum
9928420
To the Logarithm of the Side CA
2403245
The nearest Absolute Number answering to this Logarithm 2403245, is 253 fere; and that is the Length of the Side CA in Miles or Leagues, or the Height of the Tree, Tower, or Steeple, which was required.
A GENERAL RULE.
IN all Proportions wrought by Sines and Logarithms, you must observe this for a General Rule, (viz.) To add the second and third Numbers together, and from the Sum of them to substract the first Number; so shall the Remainder answer your Question demanded, As by the former Work you may perceive, where the Logarithm of the Side BA 2607455 (which is the second Term) is added to the Sine of the Angle CBA 9724210 (which is the third Term) and from the Sum of them, namely from 12331665, is substracted 9928420, the Sine of the Angle BCA, [Page 127] which is the first Number, and there remaineth 2, 403245, which is the Logarithm of 253 almost, and that is the Length of the Side required.
[geometrical diagram]
To resolve the same Work by the Line of Sines and Numbers.
YOu may work these Proportions more easily by help of the Line of Sines,Place here the Line of Numbers, Sines, and Tangents.Tangents, and Numbers, on your Scale, the Proportion being as before.
Therefore if you set one Foot of your Compasses at 58 deg. in the Line of Sines, and extend the other Foot to 405 in the Line of Numbers, the same will reach from the Sine of 32 deg. to 253 in the Line of Numbers, which is the Length of the Side AC, which was required.
Or otherwise, Extend the Compasses from the Sine of 32 deg. to the Sine of 58 deg. in the Line of Sines; the same Extent will reach from 405 in the Line of Numbers, to 253, as before, the Work is much abbreviated, there being no need of Pen, Ink, nor Paper, or Tables; but only of your Compasses.
CASE II. The Base and the Angle at the Base being given, To find the Hypothenusa.
IN the same Triangle ABC, Let there be given (as before) the Base AB 405 Foot, Miles, Leagues, or Perches, and the Angle ABC 32 deg. and let it be required to find the Hypothenusa BC. Now because the Angle CBA is given, the other Angle BCA is also given; and the Proportion is,
As the Sine of the Angle BCA 58 deg.
9, 928420
To the Logarithm of the Side 405 Foot
2, 607455
So is the Sine of the Angle CAB 90 deg.
10, 000000
The Sum of the second and third Number
12, 667455 added.
To the Logarithm of the Side BC, which is
2, 679035
The Absolute Number answering to this Logarithm is 478; and so many Feet, Miles, Leagues, Perches is the Hypothenusa, according to the denomination of the Question; that is, whether it be Feet, Perches, Miles, or Leagues. By either of these the Work is the same way.
By the Line of Numbers and Sines.
AS was said before, the Work is altogether the same with the Tables; For the Proportion being,
Extend the Compasses from the Sine of 58 deg. to 405 in the Line of Numbers; the same Extent will reach from the Sine of 90 deg. to 478 in the Line of Numbers, and that is the Length of the Side BC. Or you may extend the Compasses from the Sine of 58 deg. to 90 deg. the same Extent will reach 405 to 478, as before.
CASE III. The Hypothenusal and Angle at the Base being given, To find the Perpendicular.
IN the same Triangle let there be given the Hypothenusal BC 478 Feet, Poles, Miles, Leagues, and the Angle at the Base CBA 32 deg. To find the Perpendicular CA.
The Angle CAB is a Right Angle, or 90 Degrees; Therefore the Proportion is,
As the Sine of the Angle CAB 90 deg
10, 000000
Is to the Logarithm of the Side BC 478
2, 679428
So is the Sine of the Angle CBA 32 deg.
9, 724210
To the Logarithm of the Side AC 253
12, 403638
The Number answering to this Logarithm is 253 fere; and that is the Length of the Side CA in Feet, Poles, Miles, or Leagues.
Here the Work is something abbreviated; for the Angle CAB being a Right Angle, and being the first Term, when the second and third Terms are added together, the first is easily substracted from it, by cancelling the Figure next your left hand, as you see in the Example; and so the rest of that Number is the Logarithm of the Number sought.
By the Line of Sines and Numbers.
EXtend the Compasses from the Sine of 90 Degrees, to 478; the same Extent will reach from the Sine of 32 Degrees, to 253.
Or, Extend the Compasses from the Sine of 90 Degrees, to the Sine of 32 Degrees; the same Extent will reach from 478, to 253; and that is the S [...]le CA.
CASE IV. The Hypothenusal and Angle at the Base being given, To find the Base.
LEt there be given in the Triangle the Hypothenusal BC, and the Angle at the Base CBA; and by consequence the Angle BCA the Complement of the other to 90 degrees: Then to find BA, the Proportion is,
The nearest Number answering to 2, 607848, is the Logarithm of 405: And so many Foot or Poles, or if the Question be Miles or Leagues, is the Base or Parallel of Longitude AB.
Now you see the former Figure is turned, and therefore very fitly may have other Denominations (or Names) So that in the Art of Navigation, it will not be unfit to call one of these Sides the Parallel-Side, as AB, or Side of Longitude, or Meridian Distance; the other the Perpendicular-Side, or the Side of Latitude, as CA; and the Hypothenusal, the Side of Distance CB, and the Arches to lay down from the Chords, as before-directed.
By the Line of Sines and Numbers.
THe Angle given, as before, Extend the Compasses from the Sine of 90 deg. unto 478. the same Extent will reach from the Sine of 58 deg. to 405 in the Line of Numbers.
Or, Extend the Compasses from the Sine of 90 deg. to the Sine of 58 deg. the same Extent will reach from 478 to 405, which is the Length of the Base turned up, or Parallel-Line of Longitude, as before said, AB.
CASE V. Let the Perpendicular be the Difference of Latitude 253 Leagues, and the Angle at C, S. W. b. W. 1 deg. 45 min. Westerly, or 58 deg. Let it be given to find the Hypothenusal or Distance upon the Rhomb.
IF the Perpendicular or Difference of Latitude 253 Leagues AC be given, and the Angle at ACB, S. W. b. W. 1 deg. 45 Westerly, or 58 deg. Then by consequence the Angle ABC, or Complement of the Rhomb is also given; taking the first out of 90 deg. then the Hypothenusal may be found thus.
As the Complement Sine of the Rhomb 32 deg. at B
9, 724210
Is to the Logarithm of the Difference of Latitude 253
12, 403121
So is the Sine of the Angle or Radius 90 deg.
10, 000000
To the Logarithm of the Hypothenusal, or Distance upon the Rhomb or Course sailed 478
2, 678911
Here because the Angle CAB is a Right Angle, or 90 Degrees the Radius, and comes in the third place, I therefore only put an Ʋnity before the second Term, and so substract the first Term, and the Remainder is 2, 678911; the Absolute Number answering thereunto is 478, the Side required.
By the Line of Numbers.
EXtend the Compasses from the Sine of 32 deg. to 253 deg. the same Distance will reach from the Sine of 90 deg. to 478, the Side required.
Or, The Distance between the Sine of 32 deg. and 90 deg. will be equal to the Distance between 253 and 478, and giveth the Side required.
CASE VI. The Hypothenusal or Distance Sailed, and the Perpendicular or Difference of Latitude given, To find the Rhomb or Angle ABC.
IN the foregoing Triangle, there is given the Hypothenusal or Distance sailed, CB 478 Leagu [...]s, and the Perpendicular or 253 Leagues difference of Latitude, and it is required to find the Angle ABC, and by it the Rhomb.
As the Logarithm of the Hypothenusal CB 478 Leagues
2, 679428
Is to the Right Angle or Radius 90 deg. CAB
10, 000000
So is the Logarithm of the Perpendicular 253 CA
2, 403121
To the Complement Sine of the Rhomb, or Sine of the Angle ABC 32 deg.
9, 723693
The nearest Number answering to 9, 723693, is the Sine of 32 deg. which deducted from 90 deg. there remains the Angle of the Rhomb 58 deg. or S. W. b. W. 1 deg. 45 VVesterly.
By the Line of Numbers.
EXtend the Compasses from 478, to the Sine of 90; the same Distance will reach from 253, to 32 deg.
Or, Extend the Compasses from 478, to 253; the same Extent will reach from the Sine of 90, to the Sine of 32 deg, which is the inquired Angle ABC, and the Complement of the Rhomb.
CASE VII. The Hypothenusal, and the Parallel of Longitude, and the Radius given, To find the Rhomb or Course Sailed.
IN the foregoing Triangle there is given the Hypothenusal or Distance Sailed, CB 478 Leagues, and the Right Angle CAB 90 deg. the Radius, and the Parallel of Longitude or Base 405 Leagues, to find the Course or Rhomb sailed, or the Angle ACB.
As the Hypothenusal or Distance sailed 478 CB
2, 679428
To the Right Angle CAB Radius or Sine of 90 deg.
10, 000000
So is the Parallel of Longitude, or Base AB 405 Leagues
12, 607455
To the Sine of the Angle of the Rhomb or Course sailed 58 or S. W. b. W. 1 deg. 45 Westerly
9, 928027
By the Lines of Sines and Numbers.
EXtend the Compasses from 478 in the Line of Numbers, to the Sine of 90 deg. the same Extent will reach from 405, to the Sine of 32 deg.
Or, Extend the Compasses from 478 to 405; the same Extent will reach from the Sine of 90, to the Sine of 32 deg. ACB, the Angle of the Rhomb or Course sailed, which was required.
CASE VIII. Having two Angles, and a Side opposite to one of them, To find the Side opposite to the other.
IN the Triangle QRS, is given the Angle QSR 25 deg. 30 min. and the Angle QRS 45 deg. 20 min. and the Side QS 305 Feet; And it is required to find the Side QR.
Here note, That in Oblique-Angled Plain Triangles, as well as in Right-Angled, the Sides are in proportion one to the other, as the Sines of the Angles opposite to those Sides: Therefore,
As the Sine of the Angle QRS 45 deg. 20 min.
9, 851997
Is to the Logarithm of the Side QS 305
2, 484299
So is the Sine of the Angle QSR 25 deg. 30 min.
9, 633984
The Sum of the second and third Terms
12, 118283
The first Term substracted
9, 851997
To the Logarithm of the Side QR
2, 266286
The nearest Absolute Number answering to this Logarithm is 185; and so many Feet is the Side QR.
By the Line of Sines and Numbers.
THe Line of Sines and Numbers will resolve the Triangle by the same manner of Work, as in the other before. For if you extend the Compasses from the Sine of 45 deg. 20 min. to 305 Foot, the same Distance will reach from 25 deg. 30 min. to 185 Foot, and so much is the Side QR.
Or, Extend the Compasses from the Sine af 45 deg. 20 min. to 25 deg. 30 min. the same Distance will reach from 305 to 185, the Length of the Side inquired.
[geometrical diagram]
In like manner if the Angle RQS, 109 deg. 10 min. and the Angle QRS 45 deg. 20 min. and the Side QS 305 Foot, had been given, and the Side RS required, the manner of Work had been the same: For,
As the Sine of the Angle QRS 45 deg. 20 min.
9, 851997
Is to the Logarithm of the Side QS 305
2, 484299
So is the Sine of RQS 109 deg. 10 min. (or 70 deg. 50 min.)
The Absolute Number answering to this Logarithm is 406, and so much is the Side RS.
In this Case, because the Angle RQS is more than 90 Degrees, you must therefore take the Complement thereof to 180 deg. so 109 deg. 10 min. being taken from 180 deg. there remains 70 deg. 50 min. whose Sine is the same with 109 deg. 10 min. And so you must work with all Angles above 90 Degrees; and so will the Complement to 180, as before-directed, effect the same thing.
By the Line of Numbers and Sines.
EXtend the Compasses from the Sine of 45 deg. 20 min. to 305 Feet, the same Distance will reach from 70 deg. 50 min. to 405.
Or, The Compasses extended from the Sine of 45 deg. 20 min. to 70 deg. 50 min. the same Extent will reach from 305, to 406 in the Line of Numbers, which is the Side RS required.
CASE IX. Two Sides, and an Angle opposite to one of them being given, To find the Angle opposite to the other.
IN the same Triangle, let there be given the Side QS 305, and QR 185 Feet, together with the Angle QSR 25 deg. 30 min. and let it be required to find the Angle QRS. The Proportion is,
As the Logarithm of the Side QR 185
2, 267172
Is to the Side of the Angle QSR 25 deg. 30 min.
9, 633984
So is the Logarithm of the Side QS 305
2, 484299
The Sum of the second and third Numbers
12, 118283
The first Number substracted from the Sum
2, 267172
To the Sine of the Angle QRS 45 deg. 20 min.
9, 851111
The nearest Degree answering to this Sine is 45 deg. 20 min. which is the Angle required QRS.
By the Line of Sines and Numbers.
EXtend the Compasses from 185, to 25 deg. 30 min. the same Distance will reach from 305, to 45 deg. 20 min. the Angle QRS.
Or, Extend the Compasses from 185, to 305; the same Extent will reach from 25 deg. 39 min. to 45 deg. 20 min. as before.
CASE X. Having two Sides, and the Angle contained between them given, To find either of the other Angles.
FOr the performance of this Problem, Suppose there were given the Side RS 406, and the Side RQ 185, and the Angle comprehended by them, namely the Angle at R, 45 deg. 20 min. and it were required to find either of the other Angles.
First, Take the Sum and Difference of the two Sides given; their Sum is 591, and their Difference is 221. Then knowing, that the three Angles of all Right-lined Triangles, are equal to two Right Angles, or 180 Degrees (by the 17th Theor. of Chap. 3.) Therefore the Angle SQR being 45 deg. 20 min. if you substract this Angle from 180 deg. the Remainder will be 134 deg. 40 min. which is the Sum of the two unknown Angles at Q and S; the half thereof is 67 deg. 20 min.
The Sum and Difference of the Sides being thus found, and also the Half-Sum of the two unknown Angles, The Proportion by which you must find the Angles severally is,
As the Logarithm of the Sum of the Sides 591
2771587
Is to the Logarithm of the Difference of the Sides 221
344392
So is the Tangent of the Half-Sum of the two unknown Angles 67 deg. 20 min.
10379213
The Sum of the second and third Number
12723605
The first Number substracted from the Sum
2771587
The Tangent of 41 deg. 50 min. (is this)
9952018
Which added to the Half-Sum, makes
109 deg. 10 min. Greater Angle.
The greater of the Angles required, Substract 41 deg. 50 min. from the Half-Sum, leaves the lesser Angle at S
25 deg. 30 min. Lesser Angle.
½ Sum6720Tang.415010910Lesser Angle2530
[geometrical diagram]
By the Line of Sines and Numbers.
EXtend the Compasses from the Sum of the Sides 591, to the Difference of the Sides 221; the same Extent upon the Line of Tangents, will reach from the Half-Sum, to the Tangent of the found Angle 41 deg. 50 min.
Or else extend the Compasses from the Difference 221, to the Tangent of the Half-Sum of the unknown Angles; the same Distance will reach from the Half-Sum 67 deg. 20 m. in the same Line, to the Tangent of 41 deg. 50 min. which added to, or substracted from the Half-Sum, as before is shewn, will give the Quantity of either of the two unknown Angles.
CASE XI. Two Sides and their Containing Angle given, To find the third Side.
THere is given RS 406 Paces, and RQ 185 Paces, and the Angle at R 45 deg. 20 min. which is by the 10 Case,
As the Sum of the Sides given RS + RQ 594 p.
2771587
Is in proportion to their Difference RS + RQ 221
2344392
So is the Tangent of the Half Sum of the 67 deg. 20 min.
10 [...]79213
Two opposite Angles Q and S unknown, 2, 3 Numb.
1 [...]723605
To the Tangent of the Angle
41 d.
50 m.
9952018
Which added to the Half Sum
67
20
Leaves the greater Angle at Q required
109
10
Whose Complement to 180 deg. is
70
50
Then say,
As the Sine of the Angle found 109, or 70 deg. 50 min.
9,975233
Is in proportion to his opposite Side RS 406 Paces
2,608526
So the Sine of the Angle given at R 45 deg. 20 min.
9,851997
To his opposite Side required QS 305 Paces
12,464523
The Logarithm of the Side required
2, 485290
By the Line of Sines and Numbers.
EXtend the Compasses from the Sine of 70 deg. 50 min. to the Logarithm-Side RS 406 Paces; the same Extent will reach from the Sine of 45 deg. 20 min. to the Side 305.
Or, Extend the Compasses from the Sine of 70 deg. 50 min. to the Sine of 45 deg. 20 min. the same Distance will reach from 406, to 305 Paces, which is the Length of the Side QS, which is required.
CASE XII. Three Sides of an Oblique Triangle being given, To find the Angles.
[geometrical diagram]
IN this Triangle SQR, Let the three Sides known,
The Side SR
406
The Side SQ
305
The Side QR
185
And it is required to find the three Angles.
To perform this, you must first let fall a Perpendicular from the Point Q, upon the Side SR, which you may do by setting one Foot of your Compasses in the Point Q, and open the other to the Point R, draw the Arch RE, and divide the Space ER into two equal parts; and so the Perpendicular will fall upon the Point B.
This substracted from the whole Line 406, leaves for the part within the Arch 261; the half thereof is 130 ½, which is the Place B where the Perpendicular will fall, reckoned from the Angle R; and by this Perpendicular you have divided the Triangle into two Right Angles, whose Sides are known: For RB being 103 ½, substracted from the whole Line SR 406, leaves for the remaining Part 275 ½. Now having those two Sides of these two Right-Angled Triangles, and the two first given Sides, 305 and 185, being the two Hypothenusals thereof, you may by the opposition of Sides to their Angles, as in the 6 Case; or by the Sides and Hypothenusal, as in the 7 Case, find the Angles.
By the Line of Sines and Numbers.
EXtend the Compasses from 406, to 490; the same Distance will reach from 120, to 145 SE Leagues, the Side required.
These are the most needful Cases in the Resolution of Plain Triangles, which might have been set forth with much Variety and Inlargement; but I rather strive
to shew the best and plainest way. The Practitioner being perfect in what hath been said before, we will proceed to our intended Discourse of NAVIGATION.
THE Compleat Sea-Artist; OR THE ART OF NAVIGATION. The Fourth Book. CHAP. I. Of Sailing by the Plain Chard, and the Ʋncertainties thereof; And of Navigation.
THE Art of Navigation, is a Knowledge by certain Rules for to Steer a Ship through the Sea, from the one Place to the other; and may not improperly be divided into two parts,
namely, the Common, and also the Great Navigation.
The Common Navigation requireth the Use of no Instruments but the Compass and Sounding-Lead, as chiefly consisting in Practice and Experience, in Knowledge of Lands and Points, how they lie in Distance and Course one from the other, and how they are known at
Sea, in knowledge of Depths and Shoulds and varieties of Grounds, the Course and Setting of Tides, upon what Point of the Compass the Moon maketh High-water in each several place, and the like; which must be reckoned partly by the Information
of skilful Pilots, but far better by a Man's own Practice and Experience.
The Great Navigation useth, besides the foresaid Common Practice, divers other Artificial Instruments and Rules, which they must take out of Astronomy and Cosmography. It is therefore needful, that every Pilot and Officer, that takes charge of any Ship or Vessel in the Practice of the Great Navigation, be first and chiefly well instructed in the principal Points of the foresaid Arts; that is, that he know the Order and understand the Division of the Sphere of the World, and the Motions of the Heavens, especially the Eighth, Fourth, and First; Together with the contriving or Making
and Use of Instruments, as I have shewn briefly in the Second Book. Know this, Without this Knowledge it is impossible to perform great Voyages (not before attempted) over the Sea. In regard such Knowledge may be attained to, by good Instruction, we have set forth the same in this Treatise, for the benefit of all such young [Page 138] Sea-faring Men, as are desirous to be Sea-Artists or Navigators, so clearly and plainly as the brevity of the same could suffer to be done.
The Defects and Imperfections of this Art are many; partly in the Skill or Theorick, partly in the Practick.
After a long Voyage, the Ship supposed to be near the Shore, the Commander or Master requires from their Mates an Account of their Judgement how the Land or Cape bears from them, the Course and Distance of it when they see it: He that comes nearest
the Shore, is supposed to have kept the best Reckoning. I have known some that have not been scarce able to number and make five Figures, have gone neerest the Shore than the best Artist in the Ship; but they have been wonderfully mistaken, to my knowledge, in other Voyages. I went a Voyage to Barbadoes in the Rainbow, and took our Reckoning from Lundy, in the Mouth of Severn; and in the Ship were 12 Practitioners and Keepers of Account; eleven of them kept it by the Plain Chart, and my self made use of Mercator and Mr. Wright's Projection. When we came in the Latitude (which was 400 Leagues from the Shore) every Man was ready to give his best Judgement of his Distance off the Shore: But they all fell wonderfully short of the truth; for he that should have had the
best Reckoning, was 300 Leagues short, and most of all the rest was 268 and 250 Leagues; and he that was accounted an excellent Artist aboard the Ship, was 240. But by the Reckoning kept by Mercator's Chart, which wanted but three Leagues short of the Island. In the same Ship, going from thence to Virginia, they also fell short, by the same way of Account by the Plain Chart, 90 Leagues the nearest; and those that were advised to keep it by Mercator, found it come but 4 or 5 Leagues short of the Cape of Virginia: But coming from thence home, they got their Credit mended; they came all within 30,
20, and 10 Leagues of the Shore.
So I say, If the Course and Distance had been first agreed upon from the Place they
were bound to, to be just the same, unto the Cape or Land they first descried; If men differ then, there is something in that, in respect of
the uncertainty of the Longitude: A bad Reckoning may prove better than a good.
But we find that there is near 180 Leagues difference Error, between the Meridian of Barbadoes and Lundy, and much more in the Distance; and in some Charts about 620 Leagues Errour, in the Distance between Cape Fortuna, the South Cape of Anian Fretum, to Cape Hondo by the River Depiscadores; and these Errors may be ascribed partly to the uncertainty of the Longitude, and partly unto the Plain Chart, and Sailing by it, which makes some Places nearer than they are, and other Places
far more distant than they are, and scituated much out of their true Course or Rhomb.
Secondly, Men many times commit great Errors in bad Steerage, and careless looking
to the Compass; for I have known many Seamen when their trike or turn have been out, and the Log hove, they have told the Master or his Mate, they have Steered ½ a Point a Weather the Course; besides, the Points of the Needle or Wyres being touched by the Load-stone, are subject to be drawn aside by the Guns in the Steerage, or any Iron neer it, and liable to Variation, and doth not shew the true North and South, which ought continually to be observed by a good Meridian, or as some call it an Azimuth-Compass, which is the proper Name. Such a one you have described, by which I Survey Land with, as is shewn in the following Treatise; so the Variation ought to be carefully allowed.
I found 11 degr. in a Field at S. George's and Bristol being four miles distant; and I made five Observations, as in p. 330. and differed ¼ of a degr. only.Besides, on Land there is great difference in the same Country and Places, as Diallists well know, by taking often the Declination of several Walls; as also Mr. Gunter's Observations at Limehouse, for the finding of the Variation, found it ½ a Degree more, and other Places of the same Ground less; and Moetius saith, he hath found a Degree or two difference. This difference at Land must needs shew the uncertainty we have at Sea. Besides, many times the Ship is carried away by unknown Currents, which when they be discovered by their Ripplings, as also some by reason of Trade-Winds, we set them in our Journal; as also if we meet with any Soundings, as there is in divers Places 100 Leagues off the Land or Islands, to my knowledge, I would advise all Learners to be careful to put down all such remarkable things as neer as he can, their Latitude and Longitude. So I believe did Moetius, to remember the Current that set between Brasilia and Angola, in the opposite Coasts of Africa, where he instanceth, [Page 139] That an able Master bound to St. Helen's, in 16 Degrees of South-Latitude, in the mid-way betwixt both Coasts, and being in the Parallel of Latitude thereof, steered East, was notwithstanding carried by the unknown Motion of an unknown Current 800 Miles Westward, and yet stemmed the Current with a fair Wind, and at last made the Coasts of Brasilia.
From the 10 of April to the 15 of July, the Current sets near North-West. From the 15 of July to the 12 of October, there is no Current perceivable. From the 12 of October to the 13 of January, it sets South-West; And from the 13 of January to the 12 of April, it seems to have no Motion perceivable. Again,
Currents is a means of great mistake in keeping of a Reckoning; for Captain Luke Fox in his North-West Discoveries, and the rest, complained fearfully of the fast Lands of Ice upon those Coasts, that so alters the Current, that in some Places they cannot make good their Course they steer upon, by three Points; especially in Davis his Streights, where steering East-by-South, they scarce could make good South-East-by-South, which is four Points of the Compass, and the Error at least 70 Leagues.
I have also perceived a good Current to set to the Eastward, E. S. E. about the Western Islands, and the Madera's, in several Voyages I have made to the West-Indies; but more especially I have observed it in my last Voyage to Barbadoes. I went out of England in Company with Captain Jeremy Blackman, in the Eagle bound to the East-Indies, and a Dutch Ship in his Company, and one of Plymouth for the Isle of May: So we kept company together as far as the Madera's, but intended never to see it that Voyage; for we reckoned our selves 25 Leagues, and some more, to the Westward of the Meridian of the Maderas: But being in the Latitude near about we had espied the Land; and being becalmed, drove with the Current by the Eastern end of the Island, betwixt Porto Sancto, and the Desarts or Rocks that lie off from that end. I compared Reckoning with most aboard each Ship that kept Account, and found some 30 Leagues to the Westward of the Island; and thereby in five Voyages made before that way, knew by Experience there is a Current sets strongly near about it E. S. E. Besides, several Ships of London and the West-Country have mist it, after much labour and trouble to find it. Snellius instanceth, That one of good repute, sailing out of Holland twice, mist it and came home. I shall not here trouble you with more Instances, nor
multiply needless Questions, nor strive to branch them out in their several Varieties;
but give you those which are most useful and necessary: And then if my time will permit,
I will shew you some Arts which will as much delight you to learn, and this as briefly as I can.
As for the first and most useful Questions in Navigation, is this;How we keep our Reckoning. By the knowledge of the Rhomb or Course you sailed upon, and the distance of Miles or Leagues that you sailed thereon, to know your difference of Latitude and Longitude (that is, how much you are Northerly or Southerly in respect of Latitude, or Easterly or Westerly in respect of Longitude.) This is the most ordinary manner of keeping of Account by most Masters and Mates, of the Ships Way, which is called the Dead Reckoning. And to keep this Account, first you see, That the knowledge of the Rhomb they sailed is always supposed to be had of the Log-board, supposing the Compass by which we steer, either doth or should shew the same exactly; and so you have the
Distances in Miles and Leagues, put down every half Watch upon the Log-board, with the Course sailed, and Winds By or Large: Therefore we will come to the first Question, and Resolve it by the Traverse-Table following, and also by the Traverse-Scale in the Fifth Chapter of the Second Book. I have shewed by the Sinical Quadrant already, in the Sixth Chapter of the Second Book: And we will resolve it also by the Artificial Sines and Tangents on the Ruler, and the Tables.
But know this, I never knew any Course steered at Sea, nearer than to half a Point; for there is no Halfs nor Quarters marked on the Compass.
The First Proposition. Questions of Sailing by the Plain, Ordinary Sea-Chart.
I. Sailing 57 Leagues upon the first Rhomb, How much shall I alter my Parallel of Latitude?
THe Angle that any Point makes with the Meridian, we call the Rhomb; but the Angle that it makes with any Parallel, is called the Complement of the Rhomb. Unto every Point of the Compass there answers 11 deg. 15 min. therefore the fifth Rhomb from the Meridian makes Angles therewith of 56 deg. 15 min. namely, S. W. b. W. S. E. b. E. N. W. b. W. N. E. b. E. whose Complement 33 deg. 45 min. is the Angle of the same Rhomb with every Parallel.
Now admit I sail from A to D, S. W. b. W. 57 Leagues, I demand the difference of Latitude EA.
[geometrical diagram]
First, by the following Traverse-Table, at the Head of the Table, over every Column, is put the Figure of Halfs, Quarters, and whole Rhombs; and in one of the Columns over head is N. S. and at the foot E. W. and so is numbred at the Head, from the left hand to the right. N. S. stands for Northing. Then the Rhombs are reckoned at the bottom, from the right hand back again; The Margent of the Tables shews the Leagues sailed; and over E. W. or under E. W. shews how much you have sailed East or West from the Meridian. N. S. shews North or South from the Latitude. As in this Example, The distance sailed is 57 Leagues on the fifth Rhomb; therefore under
3 Rhomb.
N S
W E
47 39
31 67
E W
N S
5 Rhomb.
Distance Sailed, in the Side, I enter with 57 Leagues, and in the Common Angle or Line of Meeting, I find 31. 67/100 over N. S. in the Foot; and in the next Column, over E. W. is 47. 39, as you see in the Table in the Side: So that the Difference of Latitude is 31 Leagues and 67/100 Parts of a League. And if it were required to find the Departure, you see it to be 47 Leagues and 39/100 Parts. This is very plain and easie, you need no farther Precept.
By the Traverse-Scale.
EXtend the Compasses in the Line of Numbers from 100 to 57, the same Distance will reach from 5 Points to 31, and about 7/10 in the Line of Numbers.
EXtend the Compasses from 100 in the Line of Numbers, to 57, as before; the same Distance will reach from the Sine-Complement of the Rhomb, to the Difference of Latitude, which is the same way as by the Traverse-Scale.
By the Tables of Artificial Sines and Numbers, by the Fourth Case of Plain Triangles.
As the Radius, which is the Sine of 90 deg. or Angle at E
1000000
Is to the Distance run 57 Leagues AD
175587
So is the Sine Complement of the Rhomb at D 33 deg. 45 min.
974473
To the Difference of Latitude required AE 31 Leag. 7/ [...]
150060
In like manner you may find the Difference of Latitude for any Distance run upon any Point of the Compass: But remember to add the second and third Numbers together, and from it to substract the first or uppermost.
II. Sailing 57 Leagues upon the first Rhomb, How far am I departed from the Meridian of the Place from whence I came?
By the Traverse-Table.
THis Question was answered in the last Example, and found over E. W. to be 47 Leagues and 39/100, as you may see in the small Table in the foregoing Side. In the like manner you may find the Difference of Latitude and departure from the Meridian, for any Distance run upon any Point of the Compass, which is the Use of that Table.
By the Traverse-Scale.
EXtend the Compasses from 100 in the Line of Numbers, to the Distance run 57 Leagues; so is the Sine of the Rhomb; that is, put one Point of the Compass on 5 Points, in the Line of East and West of the Scale, and the other will reach to the Departure from the Meridian 47 Leagues 39/100 Parts.
By the Tables of Sines and Numbers, by the Fourth Case of Plain Triangles.
As the Radius or Sine of 90. deg. at E
1000000
Is to the Distance run 57 Leagues AD
175587
So is the Sine of the Rhomb 56 deg. 15 min. A
991084
To the Departure from the Meridian to 47 39/100 ED
166671
By the Artificial Lines on the Ruler.
EXtend the Compasses from 90 deg. to 57; the same Distance will reach from 56 deg. 15 min. to 47 39/100 Leagues.
Or, Extend the Compasses from 90, to 56 deg. 15 min. the same Distance will reach from 57 Leagues, to 47 39/100, as before.
III. Sailing upon the fifth Rhomb, until I alter my Latitude 1 deg. 35 min. I demand how far I have Sailed?
AS sailing from A to C, S. W. b. W. till the Difference of Latitude be 31 Leagues 67/100, I demand the Distance run AC.
[geometrical diagram]
First, By the Traverse-Table, Look in the Foot of the Table for the fifth Rhomb, and over N. S. in that Column, look for 31 Leagues 67/100, and in the Common Angle of Meeting, to the left hand, under Distance Sailed, you will find Distance Sailed 57 Leagues AC required.
By the Line of Sines and Numbers.
EXtend the Compasses from the Complement-Sine 33 deg. 45, to 31 67/100 the Difference of Latitude; the same Extent will reach from 90 deg. to 57 Leagues.
Or, Extend the Compasses from 33 deg. 45 min. to 90; the same Distance will reach from 31 67/100 Leagues, to 57 Leagues, the Distance AC, as before.
Say by the second Case in Plain Triangles,
As the Sine-Complement of the Rhomb, 33 deg. 45
9,744739
Is to the Difference of Latitude 31 67/100 Leagues
3500648
So is the Sine of 90 deg. Radius
10000000
To the Distance run AC 57 67/100 Leagues
3755909
IV. Sailing upon the fifth Rhomb, until I have altered my Latitude 31 67/100, or 1 deg. 35 min. How much am I departed from my first Meridian?
AS sailing from A to C, S. W. b. W. till the Difference of Latitude AB be 31 67/100 Leagues, I require BC my departure from my Meridian.
By the Traverse-Table.
AS in the last Case, find 31 67/100 Leagues over the fifth Rhomb, in the Foot, and in the next Column to the left hand, over E. W. is 47 39/100 Leagues, the Departure required.
EXtend the Compasses from the Complement-Sine of the Rhomb, to 33 deg. 45, to 31 67/100 Leagues; the same Distance will reach from 56 deg. 15 min. the Sine of the Rhomb, to 47 39/100 Leagues, the Departure from the Meridian.
By the fourth Case of Plain Triangles.
As the Sine of 90 deg.
10000000
To the Difference of Latitude AB 31 67/100
2501059
So is the Tangent of the Rhomb 56 deg. 15
10175107
To the Departure from the Meridian 47 39/100 Leagues
2676166
In the like manner, by the Departure from the Meridian, you may find the Difference of Latitude.
V. Sailing upon some Rhomb between the South and the West 57 Leagues, and finding I have altered my Latitude 1 deg. 35 m. I demand upon what Point I have sailed.
SUppose I had sailed from A to C (being a Rhomb between the West and South) 57 Leagues, and then find the Difference of Latitude 31 67/100 Leagues, I demand the Angle BAC.
By the Traverse-Table.
NUmber 57 Leagues in the Column of Distance Sailed, and in that Line or Common Angle of Meeting, you must find the Difference of Latitude 31 67/100 Leagues, at the Foot of the Table in the fifth Rhomb, which was required.
By the Line of Sines and Numbers on the Scale.
EXtend the Compasses from the Distance run 57 Leagues, to the Sine of 90; the same Distance will reach from the Difference of Latitude, to the Sine-Complement of the Rhomb 33 deg. 45 min.
By the fifth Case of Plain Triangles.
OR, Open the Compasses from 57 Leagues the Distance, to 31 67/100 the Difference of Latitude; the same Distance will reach from the Sine of 90, to the Sine of 33 deg. 45 min. the Sine-Compl. Rhomb.
As the Distance on the Rhomb AC 57 Leagues
2755874
Is to the Difference of Latitude 31 7/10 Leagues AB
2501059
So is the Sine of 90 deg. B
10000000
To the Compl. Sine of the Rhomb at C 33 d. 45 m. the Sum
12501059
The first Number substract
2755874
The Sine of the Angle
9745185
The Sine-Complement of the Rhomb is C 33 deg. 45, substracted from 90 degrees, there remains the Angle of the Rhomb at A 56 deg. 15 min. which is five Points, namely, S. W. b. W. We neglect some part of a Minute, which is not to be regarded.
VI. Sailing upon some Rhomb between the South and the West 57 Leagues, and finding I have altered my Latitude 1 deg. 35 min. I demand my Departure from my first Meridian.
By the Traverse-Table.
NUmber 57 Leagues in the Column of Distance Sailed, and in that Line or Angle of Meeting find 31 67/100 Leagues, and in the Column to the left hand you will have 47 39/100 the Departure from the Meridian.
By the Sixth Case of Plain Triangles.
Distance run AC 57 Leagues
Sum 88 Leagues
2947923
Diff. of Lat. AB 31 67/100 Leagues
Remain 26 7/10 Leagues
2424881
5372804
Departure from the Meridian BC 47 ⅔ Leagues
2681402
This is thus done. To the Distance run, add the Difference of Latitude, and also substract it from the same, noting the Sum and Remainder; then add together the Logarithm of this Sum and Remain, and half that is the Logarithm of the Distance from the first Meridian.
By the Line of Numbers.
EXtend the Compasses from the Distance 57 Leagues, to 31 ⅔ the Difference of Latitude; the same Distance will reach from 88 the Sum, to the Departure, as before, 47 ⅔ Leagues.
Or, Extend the Compasses from 57, to the Sum 88 ⅔ Leagues; the same Distance will reach from 31 ⅔, to 47 ⅔, as before, which is the Departure required.
All things that have been done by the Artificial Sines and Numbers, are done by the Traverse-Scale, or Artificial Points, Halfs, and Quarters, and Tangent-Rhombs, with the Line of Numbers in the Traverse-Table; and this agreeing very well in Leagues and 100 Part of a League.
CHAP. II. What must be observed by all that keep Account of a Ship's Way at Sea; And to find the true Point of the Ship at any time, according to the Plain Chart.
I Might have further inlarged and multiplied Questions, but that I think these sufficient for any Use at present; and therefore I will be
brief, and come to the most material Business, (viz)
The whole Practice of the Art of Navigation, in keeping of a right Reckoning, consists chiefly of three Members or Branches.
First, Well experienced in Judgment, in estimating the Ship's Way in her Course upon every shift of Wind; allowing for Leeward-way, and Currents.
Secondly, In duly estimating the Course or Point of the Compass on which the Ship hath made her way good; allowing for Currents, and the Variation of the Compass.
Thirdly, The diligent taking all Opportunities of due observing the Latitude.
The Reckoning arising out of the two first Branches, we call our Dead Reckoning; and of these Branches there ought to be such an Harmony and Concent, that any [Page 145] two being given, as you see by the Work before-going, a third Conclusion may thence be raised with Truth.
As, Having the Course and Distance, to find the Latitude of the Ship's Place.
Or, By the Course and Difference of Latitude, to find the Distance:
Or, By the Difference of Latitude and Distance, to find the Course.
But in the midst of so many Uncertainties that daily occur in the Practice of Navigation, a joynt Consent in the th [...]ee Particulars, is hardly to be expected; and when an Error ariseth, the sole Remedy to be trusted
to, is the Observation of the Latitude, or the known Soundings when a Ship is near Land: and how to rectifie the Reckoning by the observed Latitude, we shall shew.
I would advise all Sea-men to yield unto Truth in this particular, That about 24 of the common English Sea-Leagues, are to be allowed to vary a Degree of Latitude, Sailing due North or South, under the Meridian; otherwise they put themselves to many Uncertainties in their Accounts.
First, In Sailing directly North or South, where there is no Current, finding their Reckoning to fall short of the observed Latitude, they take it to be an Errour in their Judgement, in concluding the Ship's Way by estimation or guess to be too little.
And secondly, If there be a Current that helps set them forward, that there is a neer agreement between the observed
and the Dead Latitude, they conclude there is no such Current.
Or lastly, If they stem the Current, they conclude it to be much swifter than in truth it is: And thus one Error commonly
begets another. But supposing a Conformity to the Truth, we shall prescribe four
Rules for correcting a Single Course.
THE LOG-BOARD.
SS 1666.
Hours.
Course.
Knots.
Half Knots.
Fathom.
By or large.
2
S. E. b. S.
7
½
2
L
4
S. S. W.
6
½
L
6
E. b. N.
9
L
8
N. b. E. ½ E.
8
L
10
N. N. W. ½ W.
7
½
L
12
W. N. W.
9
L
2
S. E. b. S.
8
½
2
L
4
S. S. W.
7
½
L
6
S. W. b. S.
6
L
8
S. W.
9
L
10
S. E.
8
½
3
B
12
S. E.
9
B
But first of all it is most necessary to shew how we do keep our Reckonings at Sea, by the Log-board, and also by our Journal-Book.
The first Column is for Time.
The second for the Ship's Course.
The third for the Knots.
The fourth for the Half Knots.
The fifth for the Fathoms.
The sixth is to put down the Sailing Large; that is, to make her Way good on the Point she Sails, signified by L; and Sailing By the Wind, signified by B; that is, to give allowance to your Course according to the Lee-way you have made (by taking in or having out more Sail, or by Currents or Variation) those several Distances
Our English or Italian Mile by which we reckon at Sea, contains 1000 Paces, and each Pace 5 Foot, and every Foot 12 Inches; the 120 part of that Mile is 41 ⅔ Feet, and so much is the space between the Knots upon the Log-line: so many [Page 146]Knots as the Ship runs in half a Minute, so many Miles she Saileth in an Hour; or so many Leagues and so many Miles she runneth in a Watch, which is four Hours, the time in which half the Company belonging to the Ship watcheth at once by turns.
EXAMPLE.
Nine Knots in half a Minute, is nine Miles in an Hour, which is nine Leagues and nine Miles in a Watch, which is 12 Leagues or 36 Miles in all. Every Noon, after the Master Mates having observed the Sun's Altitude, or every Day at Noon, they take the Reckoning from the Log-board, and double the Knots run, and then divide the Product, which is the number of Miles run, by three; the Quotient is the Leagues run since the former Noon. Or else add up the Knots, and multiply them by 2, and divide by 3, you have the same: But be sure it is all
upon Course. We throw the Log every two Hours, and we never express the Course nearer than ½ a Point of the Compass.
Mr. Norwood gives full satisfaction in his Seaman's Practice, by his own experience, That in our ordinary Practice at Sea, we cannot, if we will yield Truth the Conquest, allow less than 360000 of our English Feet to vary one Degree of Latitude upon the Earth, in sailing North or South under any Meridian. According to this Measure, there will be in a Degree 68 2/11 of Miles of our Statute-measure, each Mile 5280 Feet; and by the common Sea-measure, 5000 Feet to a Mile, there will be 72 Miles or 24 Leagues in a Degree, which we will take for truth.
Now if you would have shewn the Miles of a true Degree, allowing 60 to a Degree, the Miles must be enlarged proportionally, and the distance between every one of the Knots must be 50 Foot; as many of these as run out in half a Minute, so many Miles or Minutes the Ship saileth in an Hour; and for every Foot more, you most allow the 10 part of a Mile. And so, if you will work the old way by Leagues, you must reduce them by Arithmetick into a Degree, and 100 parts of a Degree; or Miles or Minutes may serve. For I have seen no Chart that the Meridian is divided into more parts than 6 times 10, which is 60 Minutes; or Mercator's Chart 20 times 3, which is 60: So the small Divisions on the Dutch Mercator's Charts, every 3 is a Mile or Minute, which is near enough for any use at Sea; and these Degrees are not above ½ an Inch upon the Aequator.
Sailing East or West berween any two Places, and using a Log-line that hath a Knot at every 7 Fathoms, and to reduce it into such Miles 60 to a Degree, each containing 6000 Feet, the Proportion in Number of these two is this, As 6 to 5; for 6 Knots of 7 Fathoms makes 5 of 8 2/6 Fathom, or 50 Feet. Admit a Man keeps a Reckoning of his Ship by a Log-line of 7 Fathoms, and by it find the distance of two Places 1524 Miles, or 508 Leagues, and would know the distance by a Log-line of 50 Feet to a Knot, or 6000 Feet to a Mile: Say then by the Rule of Proportion, As 6 is to 5: So is 1524 to 1720 Miles, whereof 60 Miles make a Degree or 20 Leagues.
Next we will work the Courses of the Log-board, and by it find the difference of Latitude, and departure from the first Meridian.
A Ship being in the Latitude of 47 deg. 30 min. North, and Longitude 00 degrees, the first Course of the Log-board is S. E. b. S. 16 Miles, and S. S. W. 13 Miles, E. b. N. 18 Miles, and N. b. E. ½ E. 16 Miles, N. N. W. ½ W. 15 Miles, and W. N. W. 18 Miles, and S. E. b. S. 18 Miles, and S. S. W. 15 Miles, 8. W. b. S. 12 Miles, and S. W. 18 Miles, S. E. 18 Miles by the Wind, the Wind at W. S. W. and E. S. E. The Ship made two Points Leard-way on the two last Courses.
I demand the Difference of Latitude, and departure from the Meridian the last 24 Hours, and the Latitude I am in.
There are several ways to work Traverses; but the most necessary and readiest is by the Traverse-Scale, and the following Table; the first is to the 10 part of a League, and the Table to the 100 part of a League or Mile. We shall work the former Traverse by the Tables following, and you at leisure may work it by the Trav [...]se-Scale, and find the neer agreement of both without any sensible Error.
You must put down the Courses made good upon each Point of the Compass, and the number of Miles or Leagues you find sailed on them by the Log-board; in such manner as I have done in this Table: then according to the Rhombs, look in the Table following for the Point, Half, or Quarter sailed, and the Distance in Miles or Leagues, in the right hand or left hand Column; and count the four Points and Quarters in the head of the Table, and the four next the East and West, from the left hand to the right hand, in the foot of the Table.
Put down in four Columns N. S. E. VV. and under put what answers each Point.
As for Example. The first Course sailed is three Points from the Meridian; namely, S. E. b. S. under that Column I count 16 Miles in the side, and find against it 13 30/100 Miles Southing, and 89/100 Miles Easting. I put it down in the Table in its place, 13. 30 under South, 8. 89 under East. In the like manner you must do by the rest. Likewise the last Course sailed S. E. but by reason of 2 Points Leeward-way, it is but E. S. E. that is, 6 from the South; therefore I reckon them in the Foot of the Table, and right against 18 I find 06. 89 Southward, and 16. 63 Eastward, which you may put down as I have done in the Table. In the like manner you must do if your Course were North or VVesting. This is so plain it needs no farther Precept.
Then add up the Sums in each Column, and substract the lesser out of the greater, the Remainer is the Difference of Latitude and Departure: As I find that the Ship hath gone but 38 96/100 Miles to the Southward, and the Latitude she now is in is 46 deg. 49 min. and the Eastward but 20 parts of 100 of a Mile: Therefore her Course is neer South she made good the last 24 Hours.
CHAP. III. A Formal and Exact Way of Setting down and Perfecting a Sea-Reckoning.The Rule of keeping a perfect Sea-Reckoning is best set down in particular after the
general true Sea-Chart, in Chap. 17. of Great Circle Sailing.
THis being the most necessary Rule in this Art of Navigation, How to keep an Exact Reckoning; Although the Course and Distance cannot be so truly and certainly known, as the Latitude may be; yet we must endeavour in these also to come as neer the truth as may be;
the rather, for that some Reckonings must necessarily depend wholly upon them. Therefore we come now to shew an Orderly
and Exact way of Framing and Keeping a Reckoning at Sea; for which purpose I have inserted this Table following, which sheweth how much a Ship is more Northerly or Southerly, and how much Easterly or VVesterly, by sailing upon any Point[Page 148] or Quarter-point of the Compass, any distance or number of Miles or Leagues proposed.
Mr. Norwood many Years since laid the ground of making this Table, after this Proportion, [As Radius is in Proportion to Distance run: So is the Sine-Complement of the Rhomb, to the Distance of North or South: And so is the Sine of the Rhomb to the Distance of East or West] as you may see by the first and second Case of Plain Triangles.
Therefore for every Point and Quarter-point from the Meridian, there are four Columns: In the first thereof is set down the number of Leagues or Miles run or sailed upon that Point or Quarter-point of the Compass; The second sheweth how much you have altered the Latitude, that is, how much you are more Southerly or Northerly, by running so far upon that Point or Quarter-point; The third sheweth how much you are more Easterly or Westerly, by running or sailing that Course and Distance, as you have been before directed.
Note this, The Numbers set in the first Column from 1 to 10, are also to be understood from 10 to 100, or from 100 to 1000: and
the Figure of the fourth place answers to the Figure in the first.
Course
Distance.
Southing.
Westing.
South
100
995
98
½ Po. W.
70
697
68
3
29
3
Leagues.
173
1721
168
As, Suppose a Ship sails away South ½ a Point Westerly 173 Leagues or Miles; we set down this Number thus. Look into thefirst Column for the ½ Point, or ½ the first Rhomb from the Meridian against 10 is sometimes made use of, and understood to be 100. I find in the second
Column against it 995 (or you may have the same Number at 100 towards the Foot of the Table, omitting the last Figure) and then in the third Column you may see 98; also against 70, or 7, there is 697, and in the second 68; and in
the third against three in the first Column is 29, in the second is 2 and 9/100, which is almost 3/10; therefore I put down 3.
These summed up as in the Table, shews that the Ship sailing upon the first half Point from the Meridian, as namely, S. ½ W. is to the Southwards of the Place she departed 172 1/10 Leagues or Miles, and to the Westward 16 Leagues and 8/10. If you desire more Exactness, you may use all the Places for the greatest
Number, which is 100, (viz.)
A Traverse-Table for every Point, Half-Point, and Quarter-Point of the Compass, to the 100 part of a League or Mile; which gives the Difference of Lat. and Departure from the Meridian.
A Traverse-Table for every Point, Half-Point, and Quarter-Point of the Compass, to the 100 part of a League or Mile; which gives the Difference of Lat. and Departure from the Meridian.
A Traverse-Table for every Point, Half-Point, and Quarter-Point of the Compass, to the 100 part of a League or Mile; which gives the Difference of Lat, and Departure from the Meridian.
A Traverse-Table for every Point, Half-Point, and Quarter-Point of the Compass, to the 100 part of a League or Mile; which gives the Difference of Lat. and Departure from the Meridian.
Beforegoing, if you take all the Numbers in the Table, they will stand as here appeareth, where the Southerly Distance is 172 16/100 Leagues, and the Westerly is 16 95/100 Leagues.
But I hold it more convenient to omit the last Figure to the right hand, and so take the Tenths, as in the second Example; and then in all things it will agree with the Traverse-Scale, on which if you extend the Compasses from 100 to 73, the same Distance will reach from the first ½ Point next the Line of Numbers, to 172 1/10; and from the half Point of the Westing, to 16 9/10 Leagues, as before.
As also, If you extend the Compasses from 172 1/10 the Difference of Latitude, or from 72 1/10 to 6 8/10, which stands for 16 8/10 on this or the like occasion,
and apply this Distance from 4 Points on the Tangent-Line of the Scale, and the other Point of the Compasses will reach to ½ Point, which is from the South Westerly, as before.
Now for the Point and ½ Points reckoned at the Bottom, it is thus.
Admit a Ship fails 57 Leagues or Miles North-West and by West, or the 5th Rhomb from the Meridian; I would know how much I am to the VVestward, and how much to the Southward.
Distance
3 Rhomb.
3
N S
E W
57
474
317
Sailed
E W
N S
5 Rhomb.
5
Therefore look in the bottom of the Table for the 5th Rhomb, and in the Side for 57 Leagues or Miles; and in the Line of Meeting over the fifth Rhomb you have 47.39 or 47 4/10 for the Westing, and 31.67 or 31 7/10 almost for the Northing.
Now had you been to find the Northing and Westing of the third Rhomb from the Meridian, as N. W. b. N. to 57 Leagues distance, the Northing would be 47 4/10, and the Westing 31 7/10, as you see signified by the Letters N. S. and E. W. at the head of the Table, and North N. S. under E. W. at the foot of the Table. This is so plain, it needs no further Precept.
Or by the Traverse-Scale, Extend the Compasses in the Line of Numbers from 10 or 100, to 57 Leagues; the same Distance will reach from 3 Points in the next Line, with 5 Points of the Easting and Westing, to 47.4 Leagues or Miles; that Distance will reach from 5 Points in the Line of N. and S. to 31.7 Leagues, as before.
And the Compasses extended from 47 4/10, to 31 1/10 on the Line of Numbers; the same Distance will reach from 4 Points in the Tangent-Line, to 5 Points from the Meridian, or 3 Points if the Case so required, as if it had been N. W. b. N. The like do in all such Questions.
Likewise by the Traverse-Scale, Let the Course be given N. W. b. W. and Departure 47 4/10, To find the Distance and Difference of Latitude.
Extend the Compasses from 5 Points in the Line of E. W. of the Scale, to the Departure; the same Distance will reach from 10 or 100 in the Line of Numbers, to 57 the Distance; And also from 5 Points in the Line of N. S. to 31 7/10 the Difference of Latitude. I make this plain by the Scale, by reason the Compasses and the Scale, are more portable than the Book and Table.
A larger Example I will give you of the Tables and Traverse-Scale together, whereby you may perceive, That the Artificial Numbers, Points, and Quarters agree in all things with the Table; nay, I hold the Scale the best of the two, for the ready allowing for Variation, and for Currents, which is done by removing the Compasses from one Point or Distance to another. Now let the Question be this,
Suppose a Ship sail from the Island of Lundy, in Latitude 51 deg. 22 min. North, and Longitude 25 deg. 52 min. to the Island of Barbadoes, in Latitude 13 deg. 10 min. North, and Longitude 332 deg. 57 min. By the Plain-Chart, Difference of Latitude is 764 Leagues, and Longitude 1059 Leagues, and I sail these several Courses, (viz.) S. S. W. ½ W. from A to B 400 Leagues, S. W. b. S. ½ W. 125 Leagues, and S. W. 180 Leagues, and S. W. b. W. ½ Westerly 190 Leagues, W. S. W. 146 Leagues, and W. b. S. 159 Leagues, and South 8 Leagues 7 7/10: All these Courses and Distances I set down as followeth. In the first Column is expressed the Days of the Month, and Distance sailing upon each Course; The second, the Day of the VVeek; The third, the Course sailed; The fourth, the Distance from the Meridian; The fifth, the Place and Point of each Course by Letters; The sixth, the Distance sailed; The seventh, eighth, ninth, and tenth, the Northing, Southing, Easting, and VVesting, which is the Difference of Latitude and Departure from the Meridian in Leagues 1/10 Parts; The eleventh Column is the Latitude; The twelfth, the Longitude; The thirteenth, the Variation of the Compass.
Da. Month.
Da. Week.
Course sailed.
Distance from the Meridian.
The Places.
Dist. sailed.
Northing.
Southing.
Easting.
Westing.
Latitud. D [...]gr. Min.
Longit. Degr. Min.
Variation.
Apr.
21
f
S. W. ¾ W.
S.W. 54 d. 12 m.
From A to K.
1306 Leagu.
764 Leagues
704 Leagu.
1058 Leag.
1058 Leag.
51 22
25 52
Easterly 5 m.
13 10
332 57
May
2
2
S. S. W. ½ W.
S. W. 2 Po. ½
From A to B.
200 200
Current sets E. S. E.
176 4/10 176 4
94 3/10 94 3
33 44
16 26
Cur. sets by estim. E. S. E.
6
G
S. W. b. S. ½ W.
S. W. 3 Po. ½
From B to C.
100 20 5
773 155 39
634 127 32
28 28
12 28
Variat. 00 m.
10
d
S. W. S. W. ¼ W.
S. W. 4 Points. S W. 3 ¾
From C to D.
100 80 12
707 566 87
707 566 80
22 06
5 42
Variat. 00 m.
15
b
S. W. by W. ½ W.
S. W. 5 Po. ½
From D to E.
100 90
471 424
882 794
17 38
35 19
Westerly 2 degr.
19
f
W. S. W.
S. W. 6 Points.
From E to F.
100 40 6
383 153 23
924 370 55
15 50
350 24
Westerly 4 degr.
24
d
W. by S.
S. W. 7 Points.
From F to G.
100 50 9
195 98 38
981 490 92
13 11
342 46
Westerly 5 degr.
The Course made good.
S. W. 48 d. 34 m.
1212 3/10
763 8/10
862
13 11
342 46
This done, add up the South Column, which Sum is 763 8/10 Leagues; which reduced into Degrees, [...] by dividing by 20 and multiplying the odd Leagues under 20 by 3, and adding the Minutes in the Tenths, you will find the Difference of Latitude in Degrees to be 38 deg. 11 min. which substracted from 51 deg. 22 min. there remains 13 deg. 11 min. the Latitude of Barbadoes.
deg.min.512238111311Latitude of Barbadoes.Add up the Sums of the West Column, which is 862 Leagues; that converted into Degrees, is 43 deg. 6 min. Substract that from the Longitude of the Island of Lundy; if you cannot, add to it 360 deg. So 25 deg. 52 min. added to 360 deg. makes 385 deg. 52 min. Then the Difference of Longitude substracted from it, 43 deg. 6 m. there remains 342 deg. 46 min. the Longitude the Ship is in.
You must note, The Degrees are such that 60 Miles or min. makes a Degree of Longitude or Latitude, or of a Great Circle.
Note, The day we set sail, we put down the day of the Month and VVeek, the direct Course to the Port we are bound to, and the Place marked with two Letters, as [Page 155] in this Table A for Lundy and K for Barbadoes; and also under Distance, the number of Leagues upon a straight Course; and under Northing and Southing, the Difference of Latitude in Leagues and Tenth Parts; and under Latitude, the Latitude of the two Places; and under Longitude, the Longitude of the two Places, and also the Variation of the Compass from whence we set out first: which you may see all plain in the head of the Table, in the Common Angle of Meeting with the 21 of April.
And remember, You have the Latitude and Longitude given you; therefore by it you must find the direct Rhomb and Distance, as you have been shewed by the second and sixth Case of Plain Triangles.
Now if you would set down this Reckoning on the Plain Chart severally, you must extend your Compasses from one of the Parallels of Longitude, to the Latitude you are in; as also take off so many Leagues of the Meridian Line, as your Departure hath been, reckoning 5 Degrees for 100 Leagues, and every Degree for 20 Leagues.
As for Example.
Suppose we would set down the first Distance of South and West, Extend your Compasses from the Parallel of 40 deg. to your Latitude you are in 33 deg. 44 min. And also extend another pair of Compasses on the Aequinoctial, if there is one divided; if not, on the Meridian, which is all one; and take off 16 deg. 26 min. by one of the Parallels of the Meridian: or take off 188 Leagues 6/10, which is 9 deg. 26 min. the Difference to the Westward from your first Meridian; and so let the Compasses of the Difference of Latitude run upon the Parallel of 40 deg. and the other Compasses with 188 6/10 or 9 deg. 26 min. on one of the Parallels of North and South, until they meet in the Point B: (And so add the Meridian-difference of the second Place to the first; and the Difference of Latitude of the second Place, substract from the first,But add the difference of Latitude as you sail to the Northward by reason you are going from the North Pole toward the South or Aequator.) As for your Degrees of Longitude, you must know where you begin the first Meridian; and as you go to the Westward substract the Difference of Degrees of Meridians, and as you sail to the Eastward add the Difference of Degrees, and you have the Longitude in Degrees where you are.
So that this may suffice for a President, to lay down on your Draught or Blank Chart the Point of the Place of the Ship, by the Meridian-distance and Difference of Latitude; and as you have been directed, so are the Points C, D, E, F, G set down, So that you need not pester the Chart with Rhomb-lines, as formerly; but take the Difference between the Latitude and Meridian-distance off the Line of Numbers, and apply that Distance to the Tangent-line of Rhombs on the Traverse-Scale, and that will presently shew you the Point or Rhomb between any two Places assigned.
The drawing of the Plain Sea-Chart, and the way of sailing thereby, is the most easie and plainest of all others: And
though it be fit to use only in Places neer the Aequinoctial, or in short Voyages, yet it will serve for a good Introduction to the other kinds of Sailing. Therefore
we shall not lose our labour; for in all kinds of Sailing the same Work must be observed
with some caution.
First make the Square ASTB, of what length and breadth you please, and divide each Side into as many equal Parts as your occasion requires; and then: draw straight Lines through these Parts, crossing one the other at Right Angles, so making many little Geometrical Squares, each of which may contain one Degree: but I have made this, by reason of its largeness; to contain 10 Degrees. Note, That the Degrees of the Meridian at the Aequinoctial are all of equal distance to the Poles, which is a gross Error, which shall be shewn in the following Discourse. So that you may make the Meridian-Line in your Chart 25 deg. 52 min. to the Westward of the Meridian of Lundy: Or you may divide the two Sides into Degrees as far as you think fit, and every Degree into 60 Parts, which is the old way; and I know most Mariners will not be directed a new way of dividing the Degrees each of them into 10 Parts; so each Part will contain about 2 Leagues; and that division of double Leagues is near enough for the Mariners use. You may suppose each of these Parts to be subdivided into 10; so every Degree will contain 100 Parts, which is a very ready way if you keep your Account by Arithmetick, by Decimals or 10 Parts. This is so plain, it needs no further Precept; therefore we will proceed to the use of it.
Now your several Courses and Traverse-Points are laid down on your Chart, from [Page 157]Lundy at A, to B the first, second to C, third to D, fourth to E, fifth to F, the sixth
to to G, which is the Point the Ship is in when you cast up this Reckoning.
764
00
763
08
000
02
1058
862
196 Leagues short of Barbadoes at G.
Now to know how far you are short of the Island by the Plain Chart, substract the Sum of the South Column 763 8/10 from the Difference of Latitude 764 Leagues, and you will find you are but 2/10 parts of a League to the Northward of your given Latitude, which is not to be regarded; and also substract the Sum of the West Column, from the Difference of Longitude, and the Remainer is 197 Leagues, which you are short at G of the Barbadoes; and being in the Latitude of 13 deg. 11 min. the Island bears off you due West at K: So that you should sail 197 Leagues on that Point West, before you should be arrived at your Port by the Plain Chart.
But by the true Sea-Chart you are arrived at G, which is the Island of Barbadoes:The Errors of the Plain Sea-Chart. For the true Meridian-distance is but 865 Leagues betw [...]n Lundy and Barbado [...]s, and the Plain Chart makes it 1059 Leagues; and the true Course from Lundy to Barbadoes is but 48 deg. 34 min. which is S. W. a little above a quarter of a Point Westerly; and the Plain Chart makes it 54 deg. 12 min. S. W. which is above ¾ of a Point; And the true Distance is but 1152 Leagues, and by the Plain Chart it is 1366 Leagues. By this you may plainly perceive, that no Island, nor Cape, or Head-Land, can be truly laid down in the Plain Chart in its true scituation, but near the Aequinoctial only, and near about the same may be used without sensible Error, because there only
the Meridians and Parallels are equal; but on this side or beyond the Aequinoctial, there is Error committed proportionally to the Difference of the Meridian and Parallel; that is, The true Difference of Longitude found out by the Plain Chart, hath the same proportion to the true Difference of Longitude, that the Parallel hath to the Meridian. But most Mariners will not be drawn from this plain easie way of Sailing, notwithstanding the have
it plainly demonstrated to them by us: But those that take the true way of Sailing,
find the Credit and Benefit of it, to the shame of those that are so obstinate, conceited,
and grounded in Ignorance.
B [...]t in the following Discourse I will use my endeavour to make things so plain, that if the Ingenious Mariner will but spend half an hours time at the setting forth of his Voyage, to find by direction his true Course and Distance, and Meridian-distance, and put it at the Head of his Journal, as you see in the Table, he shall use his Plain Sailing, all the rest of his Voyage; and he shall have direction how to use it by the Chart made according to the Globe. But something more of this way, according to my Promise.
CHAP. IV. How to Correct the Account, when the Dead Latitude differs from the Observed Latitude.
WE are come now to make good what was promised in the second Chapter, to prescribe four Precepts for correcting a Single Course.
I shall be brief, in regard Mr. Collins, in pag. 22. of his Mariner's Scale new Plained, hath imitated Moetius a Hollander, a Latin Author, in these Examples; but good Rules, the oftner writ, the more they get.
The First EXAMPLE.
IF a Ship sail under the Meridian, if the Difference of Latitude be less by Estimation, than it is by Observation, the Ship's Place must be corrected and enlarged under the Meridian; and the Error is to be imputed either to the Judgment in estimating the Distance run, in making it too little; or if the said Distance be estimated by a sound experienced Judgment, it is to be supposed you stem some
Current.
Admit a Ship sail from A, in the Latitude of 36 deg. directly South, 70 Leagues, or 3 deg. 30 min. and by Estimation is at B, but by Observation he is in Latitude 32 deg. The Reckoning rectified, the Ship's Place is in the Point C; but if the Difference of Latitude be more by Estimation, than it is by Observation, the Judgment may err, in supposing
the Distance run to be too much. In this Case, the Distance is to be shortned, and the Correction must be made according to the Latitude observed under the Meridian.
Admit a Ship sail South from A, in the Latitude 36 deg. untill she have altered her Latitude 3 deg. 30 min. by Estimation being at B, in Latitude 32 deg. 30 min. and if the observed Latitude be 33 deg. 00 min. the Ship's Place corrected is at C, and not at B.
RULE II.
SUpposing no Current, If the Dead Latitude differ from the Observed Latitude, the Error is in misjudging the Distance run, which is to be made longer or shorter, as the Case requires.
Admit a Ship sail from A, S. S. E. ¾ Easterly 70 Leagues, and is by Estimation at P in the Latitude of 33 deg. but if the observed Latitude be 32 deg. 30 min. admit at B, then a Line drawn through B, parallel to NA, crosseth the Line of the Ship's Course at Q, which is the Corrected Point where the Ship is: So that the Distance is inlarged 10 Leagues 4/10, the whole Distance AQ is 82 Leagues 4/10.
[geometrical diagram]
The same manner, If the Ship had sailed 94 Leagues on the same Course, and by Estimation were at the Point R, in the Parallel of 32 deg. and by Observation the Latitude were found to be 32 deg. 30 min. In this Case the Ship's Distance is to be shortned, by drawing the foresaid Line BQ parallel to NA; and it will cross the Line of the Ships Course at Q, the Corrected Point where the Ship is.
By the Traverse-Scale.
EXtend the Compasses from 100, to 94; the same Distance will reach from 2 ¾ Points, to 82 4/10 Leagues in the Line of Numbers.
RULE III.
SUppose there is some Current, and you can depend upon the Observed Difference of Latitude, and Log-distance, as both true; then the Error may be imputed to the Rhomb, which alters by reason of the supposed Current.
Especially when you sail in Rhombs near the East and West; for then if the Dead Latitude differ from the Observed Latitude, the Error is to be imputed either wholly to the Rhomb, or partly to the Rhomb, partly to the Distance.
If wholly to the Rhomb, then retain the observed Difference of Latitude, and Distance by Observation, and thereby find the Departure from the Meridian, by drawing a new Rhomb-line.
But if your Judgment would allot the Error partly to the Rhomb, partly to the [Page 159]Distance, keep the observed Difference of Latitude: And for the Departure from the Meridian, let it be the same as was by the Dead Reckoning.
Suppose a Ship sail East by South ½ a Point Southerly 72 Leagues, from the Latitude of 36 deg. from A to M, and by Dead Reckoning should be in the Latitude of 35 deg. If the Observed Latitude be 35 deg. 20 min. which is at S; In this Case, if the Error be wholly imputed to the Distance, the Line SX being drawn parallel to NA, would cut off or shorten the Distance as much as the Measure MX, which is 26 Leagues; which because it seems absurd and improbable, is not to be admitted of: Wherefore
imputing the Error to the Rhomb only, place one Foot of the Extent AM in S, and with the other cross the Line NA at L; and so is AL the Departure from the Meridian required; whereby the Rhomb-line, if it were drawn, will be ordered to pass through F the Cross.
By the Traverse-Scale.
IF you extend the Compasses from 100, to 72 the Distance; the same Extent will reach from the Difference of Latitude by Observation, to the true Rhomb, which is almost East by South: and if you apply that Distance to one Point on the Line N. S. of the Scale, the other will reach to the Departure required 70 6/10 Leagues.— Which is far better than the other way.
The Fourth PRECEPT, CASE, or EXAMPLE.
IF a Ship sail East or West, and the Dead and Observed Latitude doth agree, the Reckoning cannot be corrected; but if they differ, the Error will be partly in the Rhomb, and partly in the Distance: In such a Case keep the Meridian-distance, and the Difference of Latitude is the Distance you are gone to the Northward or Southward of the East and West.
By the Traverse-Scale.
EXtend the Compasses from 100, to the Distance sailed; the same Extent will reach from the Difference of Latitude by Observation, to the true Course: So that you may in a moment do all these Questions and Cases by the Traverse-Scale, and Line of Numbers and Artificial Points and Quarters thereon. If you have but the perfect Use of it, I know there is no Instrument whatsoever more ready to resolve any useful Question, and correct your Reckoning.
Lastly, If by frequent Observation you find the Ship is still carried from the East or West, either Northward or Southward, you may conclude some Current to be the cause thereof: Keep the Distance by Dead Reckoning and Observation, and the Difference is the Distance from the Parallel.
We will not multiply too many Examples, but rather advise the Ingenious to make use of such as his need shall require; for
understanding what hath been said, will be advantageous to the Practitioner.
CHAP. V. How to allow for known Currents, in Estimating the Ship's Course and Distance.
THis Subject hath been largely handled by Mr. Norwood, at the end of his Sea-mans Practice; and by Mr. Philips, in his Advancement of Navigation, page 54, to 64. As also how to find them out by comparing the Reckoning homeward with the Reckoning outward, which was kept betwixt two Places: Therefore [Page 160] I shall be brief, and demonstrate by Scale and Compass, what they have done by Tables.
First, This is easie to be understood, If you sail against a Current, if it be swifter than the Ship's way, you fall a Stern; but if it be slower, you get on head so much as is the Difference between the Way of the Ship, and the Race of the Current.
EXAMPLE.
mil.South8Current3Goes a head58If a Ship sail 8 Miles South in an Hour, by Log or Estimation, against a Current that sets North 3 Miles in an Hour, that substracted from 8, leaves 5 Mile an Hour the Ship goes a head South: But if the Ship's way were 3 Mile an Hour South, against a Current that sets 8 Mile an Hour North, the Ship would fall 5 Miles an Hour a Stern.
mil.Current8Ships Way3Falls a stern5Admit a Ship runs East 4 Miles an Hour, and the Current runs also 3 Miles an Hour, What is the true Motion of the Ship? Answer, 7 Miles an Hour a Head.
Admit a Ship cross a Current that sets North East-by-North 4 Miles an Hour; the Ship sails in a Watch, or 4 Hours, 9 Leagues East-by North, and in two Watches more she sails 13 Leagues E. N. E. by the Compass.
Now it is required what Course and Distance the Ship hath made good from the first place of setting out from A.
[geometrical diagram]
First draw the Right Line AL, then with the Chord of 60 Deg. describe the Quadrant on it; to be sure take 90 deg. off the Line of Chords, and lay it from N to O; then draw the North Line AP; then set off the Ship's first Course one Point from the East from N to G, and draw the Line AG, and from A to B lay off the first Distance 9 Leagues: Then prick off the Course of the Current, being 5 Points from N to F, and draw the Line AF, being the Course N. E. b. N. of the Current. And because the Current in 4 Hours sets 5 ⅓ Leagues forward in its own Race, therefore draw the Line BC, parallel to AF, that is, take the nearest Distance from B to AF, and sweep a small Arch, and from B to the upper Edge of the Arch, draw the Line BC thereon, put from B to C 5 ⅓ the Currents Motion, and draw the Line AC, which shews the Course the Ship hath made good the first Watch.
Now for the second Course, draw CH parallel to the Line AL, and with the Radius or Chord of 60 deg. upon C as a Center, draw the Arch HZ, whereon prick 22 deg. 30 min. or 2 Points for E. N. E. for the Ship's second Course from the East; and draw CZ, whereon prick down the Distance sailed 13 Leagues from C to D; then draw DW parallel to AF, as you did BC; then because the Current sets 10 ⅔ in two Watches, therefore prick down 10 ⅔ Leagues from D to W, and draw the Line AW; which being measured upon the same Scale of an Inch divided into 10 parts, shews the Ship's direct Distance is 35 6/10 Leagues; whereas if there had been no Current, the direct Distance had been AR 22 2/10 Leagues: Then measure the Arch NE, and you will find it 35 deg. which is a little above 3 Points from the East. So the Point the Ship hath made good is North-East-by-East a little Northerly; whereas if there had been no Current, the Course had been NS, that is, East and by North ¾ of a Point Northerly, and had been at R, but now the Ship is at W, therefore distant from it equal to RW 15 6/10 Leagues. The prick'd Lines are the Courses and Arches without a Current.
This is a good way to work these Questions: If you have no Compasses, draw on a Slat or Quadrant to work Traverses by; if you have, that way is the soonest done by them after the same manner. Some
will expect, that knows me, some other sort of Questions, (besides these most useful beforegoing:) For them, and their leisure-time, I have
inserted these six Questions following.
QUESTION I. A Ship Sails 40 Leagues more than her Difference of Latitude, and is departed from the Meridian 80 Leagues, I demand her Difference of Latitude.
[geometrical diagram]
MAke a Right-Angled Triangle, so that the Base FG be equal to her Difference 40 Leagues, and the Perpendicular GH equal to her Departure 80 Leagues: Then continue the Base FG, and find the Center point E unto H and F, so it will be E, and G 60 Leagues for the Difference of Latitude sought.
Arithmetically.
Square GH 80, you have 6400, [...] which divide by GF 40, the Quotient is 160; from whence substract GF 40, there remains 120; the half is 60, for the Difference of Latitude sought.
QUEST. II. A Ship Sails 20 Leagues more than her Difference of Latitude, and but 10 Leagues more than her Departure from the Meridian, I demand her Distance Sailed.
[geometrical diagram][geometrical diagram]
IN the Triangle ABC, you have EB 20 Leagues more than the Difference of Latitude AC; and AD, 10 Leagues more than the Departure from the Meridian BC.
First, with the double of either Number, which here I take, the double of EB 20, wch is 40 Leagu. and lay from F unto G; then I take the other Number AD 10, and add it thereunto, as GH. Now on the midst of FH, as at K, making it the
Center, I describe the Semicircle HIF: Then on G erect the Perpendicular which cuts the Yrch in I; then measuring [Page 162] GI, it will be equal to DE 20 Leagues, which added to the two former Numbers 20 and 10, you have in all 50 Leagues for the Distance sailed, required.2010200200The Square 400
Anabically: 2 AD:X:EB = DEq 400, whose V q is 20, [...] (20 the Root.
QUEST. III. Two Ships Sail from one Port; The first Ship Sails directly South, the second Ship Sails W. S. W. more than the first by 35 Leagues, and then were asunder 76 Leagues; The Question is, How many Leagues each Ship Sailed.
FIrst draw the Meridian-line AB, and from A draw a W. S. W. Course as AC continued, and from C lay down the 35 Leagues unto D. Now draw the Chord-line of 6 Points, as BC; then take 76 Leagues, and lay it from D to cut the Chord-line in E. Lastly, from E you must draw a Parallel Meridian, which will cut the Rhomb-line in F; so measuring EF, you shall have 45 68/100 Leagues, that the first Ship sailed directly South: So the second Ship sailed 35 Leagues more, therefore must Sail in all 80 68/100 Leagues, which is the Distance required.
[geometrical diagram]
By the Artificial Tables of Sines and Numbers.
As the Side ED
76 Leagues co: ar.
811919
To the Sine of the Angle ECD 56 deg. 15 min
991985
So is the Side CD
35 Leagues
154407
To the Sine of the Angle CED 22 deg. 31 min.
958311
which substract from 56 deg: 15 min. you have the Angle at D 33 deg. 44 min.
Then,
As the Sine of the Angle at F 67 deg. 30 min. co. ar.
003438
Is to his opposite Side
ED 76 Leagues add
188081
So the Sine of the Angle at D 33 deg. 45 min. add
974455
To his opposite Side
FE 45 [...]8/100 Leagues
165974
So the South Ship Sailed 45 68/100 Leagues; and the other W. S. W. 80 68/100 Leagues.
QUEST. IV. Two Ships Sailed from one Port: The first Sails S. S. W. a certain Distance; then altering her Course, she Sails due West 92 Leagues: The second Ship Sailing 120 Leagues, meets with the first Ship. I demand the second Ship's Course and Rhomb, and how many Leagues the first Ship Sailed S. S. W.
DRaw the first Ship's Rhomb from A unto E, being S. S. W. then lay her Distance sailed West 92 Leagues from A unto C, and from C draw a S. S. W. Course, as CD continued: Next take 120 Leagues, and lay it from A, so that it shall cut the continued Line in D: so drawing AD, you shall have the second Ship's Rhomb, near W. S. W. Lastly, measuring CD equal to AB, you shall find it to be 49 ½ Leagues that the first Ship sailed S. S. W.
[geometrical diagram]
For the Course,
As the Side AD
120 Leagues,
co: ar.
792082
Is to the Sine of the Angle at B 67 deg. 30 min.
996562
So is the Sine of the Side BD 92 Leagues
196379
To the Sine of the Angle BAD 45 deg. 6 min.
985023
Unto which add the Angle FAB 22 deg. 30 min. you have the second Ship's Rhomb 67 deg. 36 min. being near W. S. W. whose Complement is the Angle ADB 22 deg. 24 min.
QUEST. V. Two Ships Sail from one Port 7 Points asunder: The one Sails in the S. W. Quadrant, and departs from the Meridian 57 Leagues; and the other Sailed in the S. E. Quadrant, and was departed from the Meridian but 25 Leagues, and then are both fallen into one Latitude; I demand the Rhomb or Courses of each Ship.
FIrst draw an East and West Line continued; and making choice of a Point at D, upon D erect a Perpendicular, which will be a Meridian-line, as DA continued. Now from D lay down the West Ship's Departure DB 57 Leagues; also the East Ships Departure 25 Leagues DC: so their whole Distance will be CB 82 Leagues. Now upon the Point at B, or else as here at C, draw an Angle of the Complement of 7 Points, or one Point, which is W. b. N. as CF the prickt Line; but if their Courses had been more than 8 Points, then you must lay it to the Southward of the West Line.
[geometrical diagram]
Now from the midst between B and C, at E, draw another Meridian-line, until it cut the former Rhomb-line CF in the Point G: So taking the Distance from the Point G unto C, lay the same from G until it cut the Meridian-line in the Point A, which is the Place and Port you Sailed from.1 Course S. W. ½ W. 2 Course S. S. E. ½ E. Lastly, From A you shall draw their Rhombs or Courses, as AB, which is 4 ½ from H to N from the South, Westwards; and the Eastward Ships Course is AC 2 ½ Point from P to N, from the South, Eastward.
As the Sum of their Departures CB 82 Leagues
191381
To the Difference of their Departure SB 52 Leagues
150515
So is the Sine of the Sum of their Courses CAB 78 deg. 45 min.
999080
To the Sine of the Difference of their Courses, Sum
1149595
SAB 22 deg. 30 min. the Sum
958214
Now 22 deg. 30 min. added to 78 deg. 45 min. the half is 50 deg. 37 min. ½; that is 4 Points ½ or S. W. ½ W. for the one Ships Course Sailed from A to B: and 22 deg. 30 min. substracted from 78 deg. 45 min. the half is 28 deg. 07 min. ½; that is, 2 Points and ½ S. S. E. ½ a Point Easterly, for the other Ship's Course.
QUEST. VI. From the Port at A I Sail S. S. W. unto B, and from B I Sail N. W. b. W. unto C, and from C I Sailed unto my first Port at A, E. b. N. Now having Sailed in all 120 Leagues, I would know how many Leagues I have Sailed upon each Point.
FIrst draw AB a S. S. VV. Course, at any convenient distance; then from B draw a N. VV b. VV. Course, and from A draw the opposite Course of E. b. N. which is VVest by South, which will cut BC in C; so continue the Sides of the Triangle AB unto E, and AC unto F. Then lay BC from B unto D, and AC from D unto E. Then take
120 Leagues, and lay the same from A unto F: Next draw the Line EF, and from D and B draw Parallels thereunto, which will cut AF in G and H. Lastly, measuring AH, you shall have 33
⅔ Leagues that you have Sailed S. S. VV. And measuring HG, you shall have 39 Leagues 6/10 parts that you have Sailed N. VV. b. VV. Also measuring GF, you shall have 46 ¾ Leagues near, that you have Sailed E. b. N. which makes in all near 120 Leagues.
[geometrical diagram]
Arithmetically, By the Table of Natural Sines in the Sea-mans Kalendar.
First, Add up all the Sines of the Angles together, Which is
deg.
min.
45
00
7071
56
15
8314
78
45
9790
25175
Then by the Rule of Three,
As 25194, to 120 Leagues: So
45
00
To the Distances sailed
S. S. W. 33 68/106 AB.
56
15
N. W. b. W. 39 60/100 Leag.
78
45
E. b. N. 46 72/100 CA.
I might have added several other Questions of this nature, but I hold these sufficient; for those that understand how these
are done, may do any of the like nature: But the way of demonstrating and laying of
them down, as you see in the Figures, I never saw before of any other Mans Work. Therefore now we will come to the true
way of Sailing, and Use of the true Sea-Chart.
CHAP. VII. The Disagreement betwixt the Ordinary Sea-Chart, and the Globe; And the Agreement betwixt the Globe and the True Sea-Chart, made after Mercator's Way, or Mr. Edward Wright's Projection.
THe Meridians in the ordinary Sea-Chart are Right Lines, all parallel one to another, and consequently do never meet; yet they cut the Aequinoctial, and all Circles of Latitude, or Parallels thereunto, at Right Angles, as in the Terrestrial Globe: But herein it differeth from the Globe, for that here all the Parallels to the Aequinoctial being lesser Circles, are made equal to the Aequinoctial it self, being a great Circle; and consequently, the Degrees of those Parallels, or lesser Circles, are equal to the Degrees of the Aequinoctial, or any other great Circle, which is meerly false, and contrary to the nature of the Globe, as shall be plainly demonstrated.
The Meridians in the Terrestrial Globe do all meet in the Poles of the World, cutting the Aequinoctial, and consequently all Circles of Latitude, or Parallels to the Aequator, at right Spherical Angles: so that all such Parallels do grow lesser toward either Pole, decreasing from the Aequinoctial Line.
As for Example, 360 Degrees, or the whole Circle in the Parallel of 60 Degrees, is but 180 Degrees of the Aequinoctial; and so of the rest: Whereas in the ordinary Chart, that Parallel and all others are made equal one to another, and to the Aequinoctial Circle, as we have said before.
The Meridians in a Map of Mercator or Mr. Wright's Projection, are Right Lines, all Parallel one to another, and cross the Aequinoctial, and all Circles of Latitude, at Right Angles, as in the ordinary Sea-Chart: But in this, though the Circles of Latitude are all equal to the Aequinoctial, and one to another, both wholly, and in their Parts and Degrees; yet they keep the same proportion one to the other, and to the Meridian it self, by reason of the inlarging thereof, as the same Parallels in the Globe do. Wherein it differeth from the ordinary Sea-Chart, for in that the Degrees of great and lesser Circles of Latitude are equal; and in this, though the Degrees of the Circles of Latitude are equal, yet are the Degrees of the Meridian unequal, being inlarged from the Aequinoctial towards either Pole, to retain the same proportion as they do in the Globe it self; for as two Degrees of the Parallel of 60 Degrees, is but one Degree of the Aequinoctial, or any Great Circle upon the Globe, so here two Degrees of the Aequinoctial, or of any Circle of Latitude, is but equal to one Degree of the Meridian, betwixt the Parallel of 59 ½ and 60 ½ and so forth of the rest.
Now for the making of this Table of Latitudes, or Meridional Parts, it is by an addition of Secants; for the Parallels of Latitude are less than the Aequator or Meridian,Radius 10000, to Secant 20000. in such proportion as the Radius is to the Secant of the Parallel.
For Example. The Parallel of 60 Degrees is less than the Aequator; and consequently, each Degree of this Parallel of 60 Degrees is less than a Degree of the Aequator or Meridian, in such proportion as 100000 Radius, hath to 200000 the Secant of 60 Degrees.
Now how Mr. Gunter and Mr. Norwood's Tables are made, which are true Meridional Parts, is by the help of Mr. Edward Wright's Tables of Latitude. Mr. Gunter's is an Abridgment, consisting of the Quotient of every sixth Number, divided by 6, and two Figures cut off.
As for Example. In the Tables of Latitude for 40 Degrees, the Number is [...]
deg.
parts.
(43:
712
That divided by 6, the Quotient is 43 deg. 712 parts of the Aequator, to make 40 Degrees of the Meridian. And Mr. Norwood's Tables of Meridional parts, is an Abridgment of Mr. Wright's Table of Latitudes; namely, [Page 167] every sixth Number cutting off four Figures to the right hand, as for 40 Degrees: as before the Number is 2622/7559 in regard it wants but a little of [...]000 cut off, we make the Meridional parts 2623, as you will find by his Table. So this Table sheweth how many Parts every Degree, and every Tenth part of a Degree of Latitude in this Chart, is from the Aequinoctial; namely, of such Parts, as a Degree of the Aequator containeth 60. And this which I here exhibit, and call a Table of Meridional Parts, is also an Abridgment of the Table of Latitudes of Mr. Wright's, namely, the first Numbers, omitting always the three last Figures.
As for Example▪ All the Numbers are for 40 deg. 26. 227. 559; omit the three last, and divide the rest by 3, and in the Quotient is 8742, the Meridional parts for 40 Degrees; and so of the rest: So that this Table sheweth how many Parts every Degree, and every Tenth part of a League, and every Tenth Minute of Latitude in this Chart. is from the Aequinoctial to the Poles; namely, of such Parts as a Degree of the Aequinoctial contains 20 Leagues. This is large enough for our Uses at Sea, and as ready, being in Leagues, by cutting off the last Figure, which is a Tenth: For I could never see any Draught or Plat made according to Mr. Wright's Projection, excepting his own in his Book, that is divided into more Parts than 6; for all the Mercator's or Dutch Charts as I have [...]een, are divided into 6 times 10, which is 60 Minutes: But he that desires a larger Table, may make use of Mr. Wright's Tables of Latitudes.
The Use of this Table shall partly appear in the Problems following, and may be illustrated after this manner.
PROBLEM I. How to find by the following Tables what Meridional Parts are contained in any Difference of Latitude.
YOu must take the Meridional Parts answering to each Latitude, and substract the lesser from the greater; so the Remainer is the Number of Meridional parts contained in the Difference of Latitude proposed.
As, Let one Latitude be 51 deg. 20 min.
12002
Meridional Parts.
And the other Latitude be 13 deg. 10 min.
2657
9345
The Meridional Parts contained in the Difference of Latitude are 934 5/10 Leag.
The Degrees are over the Parts, and the Minutes are on each side under the Degrees; and in the Common Angle of Meeting or Line with the Minutes, is the Meridional Parts you desire.
PROBL. II. The Latitudes of two Places being given, and Difference of Longitude of both Places, To find the Rhomb and Distance.
TO the intent the Application may be the more evident, our Examples shall be of two Places before-expressed on the Plain Chart.Longitude.d.m.Lundy25523600038552Barbad.33257Differ.525520104018 ⅓1058
Suppose the Latitude of the Island of Lundy in the Mouth of Seavern, to be at A, 51 deg. 22 min. and the Latitude of Barbadoes 13 deg. 10 min. at B, and the Difference of Longitude 52 deg. 55 min. CD, that the Barbadoes is to the VVestward of the Island of Lundy; The Course and Distance from the one Place to the other is demanded.51 d.22 m.13103812 (42037604764
First you may demonstrate the Question by the Scale. Draw the Right Line AC for the Meridian; and in regard the Difference of Latitude is 38 deg. 12 min. convert them into Leagues, by multiplying them by 20, the Number that goes to a Degree,[Page 168]
[geometrical diagram]
and the odd Minutes divide by 3, and the Difference in Leagues will be 764; which lay from A to B,As was shewed in the last Example. for the Common Difference of Latitude. Then take the Difference of the two Latitudes inlarged, 934 5/10 Leagues, and lay from A to C; then draw the two Parallel Lines, as BE and CD. Then 52 deg. 55 min. the Difference of Longitude, converted into Leagues, as before-directed, is 1058 [...]/3, which lay from C to D, and draw the Line AD, which is the true Course from A to D, and the Distance according to the True Chart inlarged: Therefore AE is the true Rhomb-line and Distance found out, produced by the former Work. And as D is the true Point by Mercator's Chart of Barbadoes, so is E the true Point of the same Place of Barbadoes by the Plain Chart; and AE the true Distance, BAE the true Course. And as CD is the true Longitude by the Globe, so is BE the true Meridian-distance between Lundy and Barbadoes.Then you must put down the Course and Distance and Meridian distance in the Head of your Journal. You may work afterwards by the Rules of the Plain Chart; and you need not work Mercator's way any more, without you have a Mercator's True Chart; and to work by that, you shall be directed in the following Discourse. Therefore to work by the Plain Rules all the Voyage after, measure AE, and you will have 1157 Leagues for the Distance; and for the Course take GH, and apply it to the Points on the Scale, and you will find the true Course S. VV. a little more than a Quarter VVesterly, which is all one Course with the True Sea-Chart; but the Distance inlarged is 1408 Leagues AD. Now by the Plain Sea-Chart the Course is BAF, S. W. above ¾ of a Point VVesterly; and the Distance is AF 1306 Leagues: So that the Plain Chart sheweth the Distance more than it is by 149 Leagues, and the Course more VVesterly by half a Point, and the Meridian-distance too much by 194 Leagues, which is a gross Error; and in such Distances grosly are these Men mistaken, that use a Plain Chart.
ADmit a Ship is at A, in Latitude 51 deg. 22 min. North, as is Lundy, and sails, or is to sail to E, in Latitude 13 deg. 10 min. according to the Plain Chart corrected, which is Barbadoes; or by Mercator's Chart, Barbadoes is in the Point D, and the Difference of Longitude is 1058 Leagues, which is 52 deg. 55 min.. First find the Difference of Latitude inlarged, as is before-directed in the first Problem, and found to be 934 5/10 Leagues.
Now you have given AB the Difference of Latitude 38 deg. 12 min. inlarged from B to C, and CD the Difference of Longitude 52 deg. 55 min. whereby the Angles and Hypothenusal shall be found by the Fourth and Fifth Case of Plain Triangles.
But because in this kind of Projection, the Degrees of Longitude and Latitude are not equal (except in Places near the Aequinoctial) the Degrees of Latitude at every Parallel, exceeding the Degrees of Longitude, in such proportion as the Aequinoctial exceeds that Parallel; therefore these Differences of Longitude and Latitude must be expressed by some one common measure; and for that purpose serves the foregoing
Table, which sheweth how many Equal Parts are from the Aequinoctial, in every Degree of Latitude, to the Poles; namely, of such Equal Parts as a Degree of Longitude contains 20 Leagues.
Wherefore, as before-directed, multiplying 52 deg. by 20, and dividing the odd Minutes, being 55, by 3, it will be 18 ⅓ Leagues; added to the former Sum, makes 1058 ⅓ Leagues, for the Meridional parts contained in the Difference of Longitude. Also by the last Problem, I find the Meridional parts contained in the Difference of Latitude to be 934 5/10 Leagues: So that AC is 934 5/10 Parts, and CD 1058 ⅓ of such Parts.
Therefore, By the Second Case of Plain Triangles.
As the Difference of Latitude inlarged AC is 934 5/10 parts
297057
Is in proportion to the Radius 90 deg.
10
So is the Difference of Longitude in such Parts CD 1058 ⅓
1302448
To the Tangent of the Rhomb at A 48 deg. 33 min.
1005391
Extend the Compasses from 934 5/10 Leagues the inlarged Latitude, to 1058 ⅓ Leagues; the same Distance will reach from the Radius to the Tangent of the Course 48 deg. 33 min. which is the Course from Lundy to Barbadoes, S. W. a little above a quarter of a Point Westerly.
By the Fifth Case of Plain Triangles.
As the Sine-Complement of the Rhomb at D 41 deg. 27 min.
982083
To the Difference of Latitude AB 764 Leagues
288309
So is the Radius o 90 deg.
10
To the Distance AE 1154 2/10 Leagues
306226
Extend the Compasses from the Complement-Sine of the Rhomb 41 deg. 27 min. to the Sine of 90 deg. the same Extent will reach from the true Difference of Latitude 764 Leagues, to the Distance AE 1157 Leagues, which is required.
PROBL. III. The Latitude of two Places, and their Distance given; To find the true Course and Point, or Place you are in, by Mercator's Chart.
ADmit I sail from the Island of Lundy, in the Latitude 51 deg. 22 min. in the Southwest Quarter of the Compass, 1154 2/10 Leagues; and then find my self in the Latitude of 13 deg. 10 min. I would know what Point of the Compass I have sailed upon, and my Difference of Longitude to the Westward.
The Difference of Latitude AB is 38 deg. 12 min. which reduced into Leagues is 764 Leagues.
As the Distance sailed 1154 2/10 Leagues AE
306226
Is in proportion to the Radius 90 deg.
10
So is the true Difference of Latitude 764 Leagues AB
288309
To the Sine-Complement of the Rhomb 41 deg. 27 min. at D
982083
that is, S. W. ¼ W. or Southwest 3 deg. 33 min. Westerly, the Course that the Ship hath sailed upon.
Extend the Compasses from 1154 Leagues the Distance, to the Sine of 90; the same Distance will reach from the Difference of Latitude 764 Leagues, to 41 deg. 27 min. the Co-sine of the Rhomb: The Sine is 48 deg. 33 min. that is, 4 Points and above a Quarter from the South Westward from the Meridian.
Secondly, For the Difference of Longitude.
Find by the First Problem the Difference of Latitude inlarged, as is there directed, 934 5/10 Leagues: Then it is,
As the Radius 90 deg.
10
To the Difference of Latitude in Parts 934 5/10 AC
297057 Inlarged.
So is the Tangent of the Rhomb 48 deg. 33 min. A
1005395
To the Difference of Longitude in Parts 1058 Leagues
302452
Extend the Compasses from the Sine of 90 deg. to the Difference of Latitude inlarged 934 5/10 Leagues; the same Extent will reach from the Tangent of the Course 48 deg. 33 min. to 1058 Leagues: which laid off from C to D, shall be the Point or Place in Mercator's Chart where the Ship is.
Or, 1058 Leagues ⅓ converted into Degrees, by dividing by 20, the Quotient is 52 deg. 55 min. the Difference of Longitude required.
PROBL. IV. Sailing 1154 Leagues upon the 4 ¼ Rhomb from the Meridian, or 48 deg. 33 min. from the South Westerly, I demand the Departure from the Meridian.
As the Radius 90 deg.
10
To the Distance sailed 1154 Leagues AE
306226
So is the Sine of the Rhomb 48 deg. 33 min. at A
987479
To the Departure from the first Meridian 865 Leagues
293705
Extend the Compasses from the Sine of 90 deg. to 48 deg. 33 min. the same Extent will reach from the Distance sailed 1154, to the Meridian Departure 865 Leagues,[Page 172] BE is the true Meridian Distance, which you may set in the Head of your Journal, to substract your Daily Distance from your first Meridian,
PROBL. V. Both Latitudes and the Meridian Distance of two Places being given, To find the Difference of Longitude, and Course and Distance on the True Sea-Chart.
THis is a most useful Problem, when the Mariner hath cast up his Traverse: Suppose a Ship sail upon the S. W. Quarter of the Compass, from Latitude 51 deg. 22 min. unto Latitude 13 deg. 10 min. and the Departure from the first Meridian to the Westward 865 Leagues.
You must find first the Difference of Latitude inlarged, as is before-directed in the first Problem 934 5/10.
As the true Difference of Latitude AB 764 Leagues
288309
Is to the Meridian-distance or Departure BE 865 Leagues
293701
So is the Difference of Latitude inlarged AC 934 5/10 Leagues
297057
To the Difference of Longitude in Leagues 1058 CD
590758
302449
By the Line of Numbers.
EXtend the Compasses from AB the true Difference of Latitude 764 Leagues, to BE 865 Leagues Meridian-distance; the same Extent will reach from AC. 934 5/10 Leagues the Difference of Latitude inlarged, to the Difference of Longitude 1058 Leagues; which laid off upon the Parallel-Line from C to D, is the Point and Place of the Ship in Mr. Wright's or Mercator's Chart.
As the true Difference of Latitude 764 Leagues AB
288309
Is to the Meridian-distance 865 Leagues BE
1293701
So is the Radius 90 deg.
10,
To the Tangent of the Course 48 deg. 33 min. at A
1005392
By the Artificial Lines on the Scale.
EXtend the Compasses from AB 764 Leagues, to BE 865; the same Distance will reach from 90 deg. to the Tangent of 48 deg. 33 min. that is, 4 Points and above a Quarter from the South Westward, that is, S. W. ¼ Westerly, the Course the Ship hath kept.
As the Sine of the Course at A, 48 deg. 33 min.
987479
Is to the Radius 90 deg.
10
So is the Departure from the Meridian 865 Leagues
1293701
To the Distance sailed AE 1154 2/10 Leagues
306222
By the Scale.
EXtend the Compasses from the Sine of 48 deg. 33 min. at A, to the Sine of 90 deg. the same Extent will reach from 865 Leagues BE, to 1154 Leagues AE, the Distance sailed.
PROBL. VI. By the Difference of Longitude, and one Latitude, and the Course, To find the other Latitude and Distance run.
SUppose I sail from Lundy, in Latitude 51 deg. 22 min. North Latitude, S. W. 3 deg. 33 min. Westerly, until my Difference of Longitude be 52 deg. 55 min. that [Page 173] is, from C to D, which is the Place of the Ship in Mercator's Chart; I demand how much I have laid the Pole, and how far I am from Lundy?
As the Tangent of the Rhomb 48 deg. 33 min.
1005395
To the Difference of Longitude CD 1058 Leagues
302448
So is the Radius
10
To the Difference of Latitude in Leagues 934 5/10. AC
297053
By the Artificial Lines on the Scale.
EXtend the Compasses from 48 deg. 33 min. to 1058 in the Line of Numbers; the same Extent will reach from 90 deg. to 934 5/10 Leagues.
Or, Extend the Compasses from 48 deg. 33 min. to 90 deg. the same Distance will reach from 1058 Leagues, to AC 934 5/10 Leagues, as before.
Now the Meridional parts answering the Latitude of 51 deg. 20 min. is 12002;1200293452657 from it substract 934 5/10 here found, and there remains 2657, which Number I look for in the Table, and find it under 13 deg. and in the Line of 10 min. which is the Latitude of the second Place where the Ship is; and the Difference of Latitude is 38 deg. 12 min.
The Distance may be found as before, in the second and fifth Problems.
PROBL. VII. By the Course and Distance, and one Latitude, To find the other Latitude, and Difference of Longitude.
SUppose I sail S. W. 3 deg. 33 min. Westerly, 1157 Leagues, and by observation find my self in the Latitude of 13 deg. 10 min. I require the Latitude of the Place from whence I came, and the Difference of Longitude between the two Places.
For the Difference of Latitude,
As the Radius B 90 deg.
1000000
To the Distance sailed 1154 Leagues AE
306222
So is the Sine-Complement of the Course E, 41 deg. 27 min.
982083
To the D fference of Latitude 764 Leagues
288315
Extend the Compasses from the Sine of 90 deg. to 1154 Leagues; the same will reach from the Sine of 41 deg. 27 min. to 764 Leagues, which converted into Degrees, is 38 deg. 12 min. the Difference of Latitude; which added to 13 deg. 12 min. the Latitude of the last Place, the Total is 51 deg. 22 min. the Latitude of the first Place required.
The Difference of Longitude is found as before in the third Problem, saying,
As the Radius, To the Difference of Latitude inlarged 934 5/10:
So is the Tangent of 48 d. 33 m. To the Difference of Longitude in Leagues 1058, which is 52 deg. 55 min.
Now to convert the Difference of Longitude found in any Latitude into Leagues, do it after this Example.
Suppose two Places in one Parallel of Latitude, as in the Parallel of 51 deg. 22 min. whose Difference of Longitude is 52 deg. 55 min. I require the Distance of those two Places.
As the Radius
10
Is in proportion to the Compl. Sine of the Latitude 51 d. 22 m.
You must understand, That the Leagues of Longitude in any Parallel of Latitude, are in proportion to the Distance in Leagues, as the Aequinoctial is to that Parallel; or, as the Semidiameter of the one, is to the Semidiameter of the other, as was said in the Seventh Chapter.
CHAP. VIII. How to divide a Particular Sea-Chart, according to Mercator and Mr. Wright's Projection.
[...]IF it be a Particular Chart you would make, you must first consider the two L [...]titudes you would make the Chart for; and out of the foregoing Table of Meridional parts, take the Numbers answering to each Latitude, and substract the lesser out of the greater, and the Remain is the Numbers which you must take for the extreme Parallel of Latitude.
Deg. Min.
Parts. Merid.
Differ.
Parts. Equal
30
13401
1975
20
13342
59
1921
10
13284
58
1863
55 00
13226
58
1805
50
13168
58
1747
40
13110
58
1689
30
13052
58
1631
20
12995
57
1573
10
12938
57
1516
54 00
12882
57
1459
50
12825
57
1402
40
12768
57
1345
30
12712
56
1288
20
12656
56
1232
10
12600
56
1176
53 00
12545
55
1120
50
12490
55
1065
40
12435
55
1010
30
12380
55
955
20
12325
55
900
10
12271
54
846
52 00
12217
54
792
50
12163
54
738
40
12109
54
684
30
12055
54
630
20
12002
53
576
10
11949
53
523
51 00
11896
53
470
50
11843
53
417
40
11790
53
364
30
11737
53
311
20
11685
52
258
10
11633
52
206
50 00
11581
52
154
50
11539
52
102
40
11479
52
51
30
11426
51
As for Example, I would make a Blank Mercator's Chart from the Latitude of 49 deg. 30 min. to 55 deg. 30 min. and for 10 Degrees of Longitude. [...] Look in the Table of Meridional Parts, and for the Latitude of 49 deg. 30 min. you will find the Number answering thereunto is 11426, and the Numbers for the Latitude of 55 deg. 30 min. is 13401; the least substracted from the greater, the Remainer is 197:5 Equal Parts for the length of the Meridian-Line.Divide the Difference 197:5 by 20, the Quotient will shew you the number of Degrees of the Aequinoctial and min. that makes the length of the Meridian Line; as for 6 deg. from A to D, will be 9 d. 55 m. of the Aequator. Therefore first draw the Line AB, DC for the Meridian-Line, and cross it with two Perpendiculars, as BC and AD: Then divide one of the Parallels of Latitude into 10 Equal parts, as AD, and subdivide each of those Degrees into 20 equal parts or Leagues that makes a Degree of Longitude and Latitude at the Aequator; and suppose each of these 20 Parts to be divided into 10 parts more, so will a Degree be divided into 200 parts: Then take with your Compasses 1975 Equal parts out of the Line AD, and lay from A to B, and from D to C, for the extreme Parallels of Latitude; and through each Degree of Longitude marked with 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, draw Meridian-lines parallel to the first Meridian: Then out of the Table of Meridional parts collect the Numbers answering to every 10 Minutes of Latitude, as in the first Column of this Table annexed,You may divide every 10 Minutes into two Equal parts; so is every Degree divided into 5 min. as AB & DC into 10 m. the second Column is the Number answering the Minutes of Latitude in the Table of Meridional parts, which substracted the lesser from the greater, the Remain is the Difference, as in the third Column 51 deg. for the Difference of the two lowermost Numbers: Then add the Numbers together in the fourth Column in this manner; 51 for the first 10 minutes, and 52 added to it, makes 102 for 50 Minutes; and 52 added to 102, makes 154 for the Number of Equal parts you must take out of the Line AD for 30 min. from A to 50 Deg. of Latitude,From D. and lay it on both sides of the Chart, and draw the Parallels of 50 Degrees of Latitude; and so do of the rest, as you see in this Table. And for 51 Degrees the Number is 470; take 470 and lay it upwards from A to 51 Degrees on both sides, and draw the Parallels of 51 Degrees of Latitude; and so do with all the rest.
CHAP. IX. The Projection of the Meridian-Line by Geometry, and how to make a Scale of Leagues for to measure Distances in any Latitude.
THE Projection is the ground-work of Mr. Wright's Table of Latitudes, in his Book called, The Correction of Errors in Navigation, where he sheweth how to make it, and hath also made a Table by the continual addition of the Secants of every Minute, which shews how much you are to lengthen the Degrees of Latitude in your Chart, that so there may be a true proportion between the [Page 176]Degrees of Longitude and Latitude in all Places. Which Table I have abridged, and made it more plain and easie, by reducing it into Leagues and Tenth parts, as hath been shewed before. We will here shew you how to do the same by Geometry, and also how to make a Meridian-line answerable to any Line of Longitudes, and a Scale of 100 Leagues to measure any Distance in any Latitude.
First, Make the Quadrant ABC, of what largeness you please, and divide the Limb thereof into 90 Degrees, and number them from B towards C; Then divide the Side of the Quadrant into 5 Equal parts, which are five Degrees of the Aequinoctial. Then divide the first Degree from the Center, as AD, into 6 Equal parts, and through them draw Parallel-lines to AC: You may divide each of the other four Degrees from D to B into 20 Equal parts, which are 20 Leagues, which makes a Degree of Longitude at the Aequator; and so you may number them as you see, from 10 to 100: So the whole Line AB will be your Radius, and the length of 110 Leagues, or five Degrees and a half of Longitude of your Chart. And because the Degrees of Longitude are to be of one length in all Latitudes, therefore the Degrees of Latitudes must encrease, as the Secants of the Latitudes increase. Therefore if you would know how long one Degree of Latitude must be in the Latitude of 50 Degrees, lay a Ruler on 50 Degrees, and on the Center A, and draw the Line AH. Now the Radius being AD, the length of one Degree of the Aequator, this Line A h, or h K,
[geometrical diagram]
being both of one length, is the Secant of 50 Degrees to that Radius, and must be the length of one Degree of Latitude in a Chart from 50 Degrees to 51 Degrees, as you may presently try by the former Chart; and so the Line AC which is the Secant of 20 Degrees, is the length of one Degree of the Meridian-line in the Latitude of 20 Degrees; and so for any other Latitude. The six Lines divided in the first Degree AD, are 10 Minutes apiece; and so you have the Secant of every 10 Minutes of Latitude, and their length in every Latitude, for a particular Chart, and for a general Chart, which hath in it North and South Latitude.
You may divide the Quadrant's Side AB into 10 Equal parts, and subdivide them into 10 more; so will D ♓ be 10 Degrees of the Aequator, and e ♑ will be [Page 177] the length and Secant of 10 Degrees in the Latitude of 20 Degrees, and L ♈ the length and Secant of 10 Degrees in the Latitude of 60 Degrees, which is twice the length of one Degree of the Aequator: So that you may presently try the truth of this Projection, how it agrees with the Globe: Whereas one Degree of Latitude in the Globe, is equal to two Degrees of Longitude in that Latitude of 60 Degrees; so here AL the Secant of 60 Degrees, is twice the length of AD the measure of one Degree of Longitude in the Blank Chart; and L ♈ is twice the length of 10 Degrees D ♓: So every Degree is two of the Aequator in the Latitude of 60 Degrees of a general Chart; and by the Globe, in the Latitude of 75 deg. 30 min. one Degree of Latitude is equal to four Degrees of Longitude. So in the Quadrant, AR is four times the length of A D: and so the Proportion will hold in any other
Latitude.
A Scale of Leagues from the Latitude of 25 Degrees, to the Latitude of 57 Degrees 00 Minutes.
How to make the Scale of Leagues.
THE Quadrant being drawn, as before-directed, take 110 Leagues and lay from A to B, and draw the Line MNB at Right Angles thereunto: and if it be for a particular Chart, as that before-going, draw Lines from the Center through every particular Latitude; as you see in the Quadrant I have done, to make a Scale for the blank Chart before-going, from the Latitude of 49 deg. 30 min. to 55 deg. 30 min. So that if you would know the length of 110 Leagues in the Latitude of 50 Degrees, lay a Ruler upon 50 deg. in the Arch of the Quadrant and the Center, and draw the Line A ♌, and that is the length of 110 Leagues in that Latitude. So that if you draw Parallel-lines to MN, through every 10 Leagues in the Side AB, you will have the length of every 10 Leagues in every Latitude, as you may plainly see in the Quadrant: and so you may do for every League, as you see the little Checkers betwixt the Latitude of 20 deg. and 30 deg. for 10 Leagues between Latitude of 50 deg. and 56.
Suppose you would know the length of 40 Leagues in the Latitude of 50 deg. Extend the Compasses from A to K, and that Distance is 40 Leagues in that Latitude: And in like manner work by the rest in any other Latitude.
If you would make this into a Scale, as the Figure YM in N; First in the Quadrant extend your Compasses from the Center A, to the Intersections of the Lines drawn through every Degree MNB, and lay them down upon the Side of the extreme Latitude of your Chart, as A, O, P, Q, Y, M, with the small Arches, as you see I have done from M to NY, and that is the length of the Meridian-line of [Page 178] your Scale or Degree of Latitude MY; therefore draw the Parallel-lines YN and M m for the extreme Parallels of your Scale: Then extend your Compasses from Y in the Quadrant, to each of the Intersections of these small Arches that are drawn from the Intersections on the Tangent-line MN; and from 25 deg. uppermost, lay that Extent downward for the Parallel of 55 deg. of Latitude, as the Line above the lowermost; and so lay down all your Latitudes by these small Arches, in like manner; and so neatly divide the Side of your Scale MY of Deg. of Latitude: Then draw Parallel-lines to all these Degrees, as you see: Then extend the Compasses from the Center of the Quadrant to M, for the length of the lowermost Line of your Scale M m for 110 Leagues. Then extend the Compasses from the Center of the Quadrant to N, which is the length of 110 Leagues in the Latitude of 25 deg. and it is the Distance of the uppermost Line YN of your Scale; and draw N m the outside of your Scale 120 Leagues: So take every 10 Leagues from the Center A, in the Line AM, in Latitude 56 Degrees, and divide the lowermost Line of the Scale; and the like do in the Latitude of 25, for to divide the uppermost Line of the Scale; and draw Lines through each of them, which will divide all the rest of the Parallel-lines in each Latitude into 10 Leagues apiece, and number them as you see I have done; and divide the first 10 Leagues by the Meridian-line of the Scale, into 10 Equal parts below and above, and draw Lines through each of the Divisions: So have you neatly divided your Scale, and every Degree of Latitude thereof, into Leagues, to 100 and 10 odd Leagues; which will measure any Distance in a Chart, made according to the Degrees of Latitude and Longitude in the foregoing Chart.
For to know the Rhomb between any two Places, shall be shewed in the Use of the general Sea-Chart following, by a Protracting Quadrant, and also how to find the Place of any Ship in Mercator's Chart, and to lay down any Traverses.
CHAP. X. The Way of Sailing by a Great Circle.
WE will now shew you the way of Sailing by the Arch of a Great Circle, which is the most true way of Sailing of all others, if a Man is sure to have Winds, that he may, keep neer the Arch that goes over any two Places propounded. But as there is a great deal of uncertainty
in having a constant Wind by the Arch, so likewise the Trade-winds many times lye wide of this Arch many Leagues; besides many dangers of Rocks, and Sands, and great Currents, and danger of Pirates; which by keeping near the Arch may lead Men into many inconveniences; which may be a greater trouble to a Man,
more than by sailing a few Leagues the more, for his best advantage and more security; besides the trouble in that way,
of keeping of Accounts, which Men that watch every 4 Hours, cannot allow so much time every Noon, not will be perswaded to do it once in three or four Days, in regard Mercator's Char [...] comes neer the very truth. Let the Wind, and Sea, and Currents, or Pirates, drive a Ship ever so far wide of the Arch, yet it is all line by that True Chart. If you keep a true Account of the Ship's way, allowing for Variation of the Compass, and Setting of Currents, and having of the Sea, you may at any time have the [...]ue Point where the Ship is, and how to shape a Course the most ready and convenient way to the Port you are bound to: Yet, I say, these Men that are perfect in this way of sailing,
may see by their Mercator's Chart the dangers that lie between any two Places, and shun them; and likewise make many
Voyages, where the Winds may favour them, sailing by the Arch, and no danger of Rocks or Sands to trouble them, which will prove a great advantage when your Course lies neer East and West; for sailing upon these two Points, Men trust altogether upon their Dead Reckoning (by the two former ways) but by this way you may hol [...] your self,Plain Sailing, Mercator's. by altering your Latitude many Degrees, by which you may often recti [...]ie your Account.
For Example, Admit you were to sail from Avero on the Coast of Portugal, to the [Page 179]Bay on the back side of Aquamacke neer Virginia,This Arch of a Great Circle over two Places in Latitude 40 d. 00 m. and Difference of Longitude 70 deg. is the Line PW in the Diagram of the 14. Chap. of part of the Globe in Plano. which lye both neer the Latitude of 40 Degrees; and suppose the Difference of Longitude between these two Places be 70 Degrees: The Distance of the two Places East and West is 53 deg. ½ and something more; but the Distance in the Arch of a Great Circle is but 52 deg. and a little more, that is, 1 deg. and about ½ less, which is but a little benefit to this, which is the chiefest, That
in sailing between two such Places by the Arch of a Great Circle, you will in the first half of the way raise the Pole 5 deg. 41 min. and then in the other half depress the Pole as much; so that in the whole Voyage you will alter your Latitude 11 deg. 22 min. by several Courses; by which you may rectifie your Dead Reckoning, which you cannot do in sailing upon a Parallel of East and West; by which you see it is the best way of sailing, as well as the nearest, especially
in such occasions, if the Wind favour you.
Now concerning this way of sailing, Mr. Edward Wright our Noble Countryman did first lay down a way, in his Book of Correction of Errors in Navigation, pag. 63, 64, and 65, by Geometry, which Captain Santanstal did comment upon in his Book called the Navigator, which was only of a Parallel Course; for any other way he said but little or nothing.
Mr. Norwood in his Book of Trigonometry hath added many Problems of sailing by the Arch of a Great Circle; for those that will or can, may by his pains find out all things in this way of sailing:
But as they are difficult, and the way unknown to most Sea-men how to calculate; so they are tedious to those of the best skill: Therefore I commend
Mr. Philips his Tables, in his Book called The Advancement of Navigation, and likewise his Plain Figures in his Book of the Geometrical Sea-man. I shall likewise by Geometry, and by Calculation, give you some satisfaction. Either way shall be done with speed, and as exactly as
need be required.
The true Distance between two Places in the Arch of a Great Circle contained betwixt them, is thus to be found out.
If the two Places have no Latitude (being both under the Aequator, and one of them also no Longitude, the Longitude of the other being less or not more than 180 Degrees, the Longitude is the Distance.
But if the Longitude be greater than 180 Degrees, substract it out of 360 Degr. the Remainer is the Distance.
If two places be in one Meridian, and have the same Longitude both, and but one hath Latitude North or South, the Latitude is the Distance.
But if both Places agreeing in Longitude, have Latitude also of like denomination (that is, both Latitudes Northerly, or both Southerly) then substract the lesser out of the greater, the Remain is the Distance.Latitude.Landse.50:15Ribedev.43:30Distance7:452014015Leagues155Mult. by3Miles465
But if both Places in one Meridian, have one Northerly Latitude, and the other Southerly Latitude, add them together, for the Sum is the Distance in Degrees.
CHAP. XI. How to find the true Distance of Places, one of them having no Latitude: The other having Latitude and Difference of Longitude less than 180 Degrees, To find • 1 Their Distance in a Great Circle. , • 2 The Direct Position of the First Place from the Second. ,
and • 3 And the Second Place from the First.
The First Scituation.
FIrst, If any two Places being proposed, the one under the Aequinoctial, the other may be in any other Latitude given, either North or South, and the Difference of Longitude of the Places being known; you may find the three things before spoken of in any Question, by the following Directions. We call the [Page 180]Angle that the Rhomb leading from one Place to another, makes with the Meridians, the Position of these Places: But in regard the Arch of a Great Circle, drawn between two Places, is the most neer distance from the one Place to the other;
therefore the Angles which that Arch makes with the Meridian of those Places, we call the Angles of Direct Position: or direct way of two Places one from the other.
To draw a Great Circle from the Amazones over Lundy, put the Difference of Longitude on the West Side from AE to G; and draw the prickt Line NS, N for the North, and S for the South Pole: and through G draw the Azimuth Circle NGS, and draw the Parrallel of Latitude CZC; and through Z draw the Arch AEZRVQ, for the Circle which passeth from AE Amazones to Z Lundy.Now in the following Diagram, let A be the Entrance of the great River of Amazones, under the Aequator; AQ the Arch of the Aequator, or Difference of Longitude; and let C represent the Island of Lundy, lying in Latitude 51 deg. 22 min. Northerly, and CQ the Meridian thereof: and suppose the Difference of Longitude AQ to be 41 deg. 22 min.
[geometrical diagram]
How to do these Questions Geographically.
FIrst, With an Arch of 60 Degrees describe the outward Meridian AEECQIF.
Secondly, Draw AEQ the Aequinoctial Line. Thirdly, Take 51 deg. 22 min. of the Line of Chords, and lay it from Q to C; and draw the Line CD through the Center, and the Line EF at Right Angles thereunto. Fourthly, Take off your Scale of ½ Tangents, counting from 90,-41 deg. 22 min. and lay from Q to A, for A represents the River of Amazones. Now draw the Circle CAD through A, the Horizon thereof is EF; then measure FK, and apply it to the Line of ½ Tangents, as before directed: and you will find the Angle of Direct Position to be 48 deg. 25 min. Take that Number out of your Line of ½ Tangents, from 1 Degree forwards towards 90, and lay it from H to L for the Pole, and draw a Line from L through A, it will cut the Line in I; so measuring CI on the Line of Chords, it will be 61 deg. 57 min. for the Distance, which is 1237 ⅓ Leagues and 3712 Miles.
By the Tables.
THen in this Triangle CAQ, Right-Angled at Q, there is required CA the nearest Distance of the two Places in the Arch of a Great Circle; and the Angle ACQ of Direct Position from the Island Lundy to the Amazones: and the Angle[Page 181] CQO being the Complement of the Angle of the Direct Position of the Island of Lundy.
For the nearest Distance CA,
As Radius, is to Co-sine of Difference of Longitude 41 d. 22 m.
987534
So is Co-sine of the Latitude or Difference 51 deg. 22 min.
979699
To the Co-sine of the Distance 61 deg. 57 min.
967233
which 61 deg. 57 min. converted into Leagues, is 1237 ⅓ as before, the nearest Distance between those two Places.
For the Angle of Direct Position from the Amazones toward Lundy, NAER,
As the Radius, to the Sine of the Differ. of Longitude 41 d. 22 m.
982011
So is the Co-tangent of Difference of Latitude 51 deg. 22 min.
990267
To the Co-tangent of the Angle of Position 27 d. 50 m. NAER
972278
For the Angle of Position ACQ,Measure NR on the half Tangents, and the Angle of Position, and the Distance is all one as before.
As Radius 90, is to Co-sine of Differ. of Lat. 51 d. 22 m. QC
989273
So is Co-tangent of Differ. of Longitude 41 deg. 22 m. AQ
1005522
To the Co-tang. of the Angle of Direct Position 48 d. 25 m. ACQ
994795
The same Proportions will hold by the Artificial Lines on the Scale.
And thus you see, he which will sail the nearest way from the Amazones to the Lizard, shall at first shape his Course 27 deg. 50 min. from the Meridian to the Eastward; that is, N. N. E. almost ½ a Point Easterly. Now if the Wind should serve that you might sail this Course, it is to be understood, that in this kind of sailing he is not to continue this
Course long; but to shift it, and incline more and more to the Eastward, as often as occasion requires: which how it may be done, shall be shewed in the following
Discourse.
PROBL. II. How to find the Great Circle's greatest Latitude N. or S. or Obliquity.
NOte, Without the knowledge of the true Quantity of the Obliquity or Latitude of that Great Circle which will pass directly over the Places propounded, there can be no compleat Demonstration, much less Arithmetical Calculation of things pertaining thereunto; therefore it is needfull that the true Quantity
of each Great Circle's Obliquity be diligently found to exact certainty: which to do, in some Cases is very easie,
and in some again more difficult. Therefore I will propound Rules for the several Scituations following, except those that are scituate under the Aequator, or under the same Meridian.
If one Place be under the Aequator and hath no Latitude, and the other hath any Quantity of Latitude, and the Difference of Longitude being less than 90 deg. as before 41 deg. 22 min. it is easily found, thus:
The greatest Obliquity in the foregoing Diagram is HRV,
As the Sine of the found Distance 61 deg. 57 min.
994573
Is to the Sine of the Latitude 51 deg. 22 min.
1989273
So is the Radius (added to the last Number) To the Sine of the greatest Obliquity 62 d. 16 m.
994700
So 62 deg. 16 min. is the greatest Obliquity or Latitude from the Aequator, of that Great Circle extended over those two Places.
But if the Difference of Longitude be 90 deg. as AEH, and one of the Places have no Latitude, and the other have any Quantity of Latitude; then it is evident to reason, as in the foregoing Diagram may appear, that the second Place is scituate in the very Point of the greatest Obliquity, which is never above 90 Degrees, as HN; and the other Place is in the very Point of Intersection of the said Great Circle with the Aequator: For note, That every Great Circle that passeth over any two Places propounded, cuts the Aequator in two opposite Points 180 deg. from each other, as the Ecliptick Line doth in the two Points of Aries and Libra; and the greatest Obliquity of that Circle is 23 deg. 30 min. the Sun's greatest Declination, and never any more.
Now if one Place have no Latitude 00 deg. 00 min. and the other have any Quantity of Latitude, the Difference of Longitude being more than 90 deg. to find the Obliquity of the Great Circle passing over those Places.
As admit one Place Latitude 00 deg. 00 min. and the other 51 deg. 22 min. Difference of Longitude 138 deg. 38 min. Distance betwixt them is near 107 deg. Therefore take the Distance 107 deg. out of 180, and there remains 73 deg. Then,
As the Sine of the Remainer 73 deg.
998251
Is to the Sine of the Latitude 51 deg. 22 min.
989273
So is Radius
10
To the Sine of the greatest Obliquity 54 deg. 25 min.
991022
So that 54 deg. 25 min. is the greatest Obliquity of the Great Circle extended over these two Places. And so you may work for any Questions of this nature.
The second Scituation.
SEcondly, There may be two Places scituated in divers Parallels of Latitude, betwixt the Artick and Antartick Poles, that may have one Degree and Minute of Latitude, yet may have several Degrees of Longitude.
Admit there be two Places both in the Latitude of 51 deg. 22 min. and their Difference of Longitude be 52 deg. 55 min.
1. To find the nearest Distance of those two Places.
2. The Direct Position of the one Place from the other.
How to do these Questions Geographically.
TAke off the Line of Chords the Latitude of the Place 51 deg. 22 min. and lay from AE to X, and from Q to C; and take of the ½ Tangents the same Latitude, and lay from K to O; and through these three Points draw the Parallel of Latitude XOC; the Difference of Longitude laid from Q to L, draw the Meridian Circle NLS, the second Place is at R, and first at C the Meridian-circle cuts at R: Therefore draw the Circle from C through R to B, and measure HI on the ½ Tangents, and you will find it 68 deg. 46 min. for the Angle of Direct Position HCI. Now from the ½ Tangents take 68 deg. 46 min. and lay it from the Center K to E, and from E draw through the Point of Intersection at R the prickt Line ERF; and because it cuts the Line in F, therefore measure CF on the Line of Chords, and you will find it 32 deg. 18 min. for the true Great Circles Distance, which is 646 Leagues, or 1938 Miles.
In the Seventh Problem of sailing by Mercator's Chart, yo may see there was required the Distance of these two Places measured in the Parallel, and found to be 660 5/10: but here is required the nearest Distance in the Arch of a Great Circle: Work thus by the Tables.
For the Distance,
As the Radius, Is to the Sine Comple of the Lat. 38 d. 38 m. RN
979541
So is the Sine of the Differ of Longitude 26 d. 27 m. RFN
964876
To the Sine of half the Distance 16 deg. 09 min. RF
944417
Which doubled is 32 deg. 18 min. and this converted into Leagues and Miles, as before, is 646 Leagues, and 1938 Miles, the nearest Distance, and less than the Distance measured in a Parallel by Miles 42.
To find the Direct Position,
As Radius 90, Is to the Sine-Compl. of the Lat. 58 d. 38 m. RN
989273
So is the Tangent of ½ the Differ. of Longitude 260 d. 27 m. RNF
969678
To the Co-tangent of the Angle of Position 68 d. 46 m. NRF
958951
Which sheweth, that if you will go the nearest way from C to R, you must not go West, though both be under one Parallel; but must first shape your Course from C from North 68 deg. 46 min Westerly, that is almost W. N. W. and so by little and little inclining to W. b. N. and then W. and W. b. S. and almost W. S. W. as before.
How to find the Obliquity.
TWo Places having Latitude both the same, as 51 deg. 22 min. and towards the same Pole, whether North or South, and Difference of Longitude 52 deg. 55 min. or any number of Degrees under 90: If above 90, take it out of 180, and work with the Remainer the same manner
of way.
PROBL. IV.
As Radius 90, To Co-tangent of the Latitudes 51 d. 22 m. RN
990267
So is the Co-sine of ½ the Differ of Longitude 26 d. 27 m. RNF
995197
To the Compl. Tan. of the greatest Obliquity 54 d. 25 m. NF
So that the greatest Obliquity is 54 deg. 25 min. And the same Proportion will hold for any Question of this nature.
We might proceed to frame many Questions touching those two Places; but these being the most material, I leave the rest to
your own Practice, to use as much brevity as I may. I might have shewn you the Side and Angles; but in regard they are Spherical I omit it, and shall demonstrate them at last in Plano: But you may follow these Rules, if you cannot apprehend the Diagram; but some may desire the Triangle, therefore I lay it down.
[geometrical diagram]
In this Triangle CRN, let the two Places be C and R, and let N be the North Pole; then CN or RN either of them are 38 deg. 38 min. the Complement of the Latitude and the Angle CNR is the Difference of Longitude: There is required CFR the nearest Distance, and the Direct Position of the one to the other, NCR or NRC; for in this Case those two Angles are equal: And seeing NC and NR are equal, therefore let fall the Perpendicular NF, the Triangle NCR is divided into two Right-angled Triangles CNF and RNF, which are every way equal.
The Third Scituation. One Place having North Latitude, and the other Place having South Latitude, of different Quantities, and the Difference of their Longitudes less than 90. As I omit one Place having North Latitude, as Lundy, 51 deg. 22 min. the other South Latitude, as the Rio de la Plata, 35 deg. 00 min. Difference of Longitude betwixt them 45 deg. 55 min. I demand the Distance, the Angle of Position, and the greatest Latitude or Obliquity of the Great Circle that passeth over these two Places.
AFter you have described the outward Meridian NESA, take from the Line of Chords the Latitude 51 deg. 22 min. and lay it from E to P, and draw the Line PCO and HCM at Right Angles to P, take off the half Tangent Line the Difference
[geometrical diagram]
[Page 185] of Longitude 45 deg. 55 min. and lay it from E to F, and draw the Meridian-Circle NFS, whereon lay the Latitude of Rio dela Plata 35 deg. from F to R, by taking 35 out of the Line of Chords, and laying it from E to 35, and the ½ Tangent of FE from C to Pole, and draw the prick'd Line Pole 35, which cuts the Circle NFS in R, the Rio dela Plata: and through R draw the Circle PRO, and measure MN on the half Tangents, you will find the Angle of Position to be RPE 36 deg. 2 min. Then take the half Tangent of 36 deg. 2 min. and lay from C to K, and draw the prick'd Line from K through R, and it will cut the Line at T; therefore measure TP on the Line of Chords, and that is the measure of RP 95 deg. 18 min. for the Distance, or 1906 Leagues, or 5718 Miles: The greatest Latitude or Obliquity is from AE to L; and VLW is the Parallel of 68 deg. 21 min. the greatest Obliquity required,
PROBL. V.
Then by the Tables,
As Radius, To the Co-sine of Differ. of Longitude 45 d. 55 m.
984242
So is the Co-tangent of the greater Latitude 51 deg. 22 min.
990267
To the Tangent of the first Arch 29 deg. 5 min.
974509
The less Latitude 35 deg. and 90 deg. makes 125 deg. Take the first Arch 29 deg. 5 min. therefrom, and there remains 95 deg. 55 min. Take this out of 180 deg. and there remains 84 deg. 5 min. the second Arch: Then
As Co-sine of the first Arch 29 deg. 5 min.
994005
Is to the Co-sine of the second Arch 84 deg. 15 min.
901318
So is the Sine of the greater Latitude 51 deg. 22 min.
989273
Out of
180d
00′
Take
84
42
To the Co-sine of 84 d. 42 m.
890591 Sum
And there remains
95
18
The true Distance 95 d. 18 m.
896586
which was required.
Now to find the Obliquity, Take both their Latitudes as if they were North, or both South, and the Complement of the Difference of Longitude to 180 deg. which here is 134 deg. 05 min. half that is 67 deg. 2 min. 30″: both the Latitudes added together make 86 deg. 22 min. half that is 43 deg. 11 min. it being too little, I added about 1 deg. 20 min. to the half, to find the mean and true Latitude 44 deg. 31 min. by which I find the Obliquity, as I proved by this Operation.
As Radius, To Co-tangent of the Latitude 44 deg. 31 min.
1000732
So is Co-sine of half the Difference of Longitude 67 deg. 2 min.
959158
To the Co-tangent of the Obliquity 68 deg. 21 min.
959890
Now to find whether 68 deg. 21 min. be indeed the true Obliquity, make these Proofs of it.
As Radius, To Co-tangent of Obliquity 68 deg. 21 min.
959890
Take from it the Tangent of the Latitude 51 deg. 22 min.
990267
There remains the Co-sine of Differ. of Longitude 60 deg. 14 min.
969623
Again,
As Radius, To Co-tangent of Obliquity 68 deg. 21 min.
1959890
Take out the Co tangent of the other Latitude 35 deg. 00 min.
1015477
There remains Co-sine of Differ. of Longitude 73 deg. 51 min.
944413
Now both the Longitudes found,6014735113405 73 deg, 51 min. and 60 deg. 14 min. added [Page 186] together, makes just 134 deg. 05 min. the Difference of Longitude at first propounded betwixt those two Places; which proves, That the greatest Obliquity of the Great Circle that passeth directly over these two Places, the Island Lundy and Rio dela Plata, so scituate, is 68 deg. 21 min.
Now if it so happen that both the Latitudes be of the same Quantity, as one Place North Latitude 11 deg. 30 min. and the other Place South Latitude 11 deg. 30 m. and the Difference of Longitude betwixt the two Longitudes 55 deg. 48 min. To find the true Great Circles Distance betwixt such Places, first divide the Difference of Longitude into two equal parts, and then take one Latitude and half the Difference of Longitude, and find the Distance belonging to one Latitude, which doubled, yields the whole Distance betwixt the Places propounded: As Longitude 55 deg. 48 min. halfed is 27 deg. 54 min. and Latitude 11 deg. 54 min. Then work thus.
As Radius, To Co-sine of Differ. of Longitude 27 d. 54 m.
994633
So is the Co-sine of the Latitude 11 deg. 30 min.
999119
To the Co-sine of 30 deg. half the Distance
993752
which doubled is 60 deg. the whole Distance betwixt one Place South Latitude 11 deg. 30 min. and and another North Latitude 11 deg. 30 min. having 55 deg. 48 min. Difference of Longitude. And so work for any two Places so scituate.
Geographical Questions. Two Ships being at Sea, their Difference of Longitude was 53 deg. Now upon a day they observed the Sun being between them; the North Ship found the Sun's Meridian Height 33 deg. and the South Ship 77 deg. the middle Latitude between them was 15 deg. North of the Aequinoctial Line: I demand the Angle of Position, and Distance from the North Ship to the South?
Meridian Altitude.33 d. Compl. 57 deg.77 d. Compl. 13 deg.Sum 70½ Sum 35Mid. Lat. 15Nor. Ship 50 lat.South Ship 20 lat.FIrst add the two Meridian Altitudes Complement together, 33 deg. and 77 deg. Complement 57 and 13, the Sum is 70, the half Sum is 35 deg. the middle Latitude 15; add the middle Latitude and half Sum together, it makes 50 deg. the
[geometrical diagram]
[Page 187]North Ships Latitude; and substract the middle Latitude from the half Sum, and the Remain is 120 deg. the Latitude of the South Ship. The North Ships Latitude is laid from Q to N 50 deg. the Difference of Longitude QF 53 deg. Through F describe the Great Circle Meridian PFB, on which lay down the South Ships Latitude 20 deg. as FC, and so draw the Great Circle NCD through the Intersection of the prick'd Line IH, with the Meridian at C; for that is the Latitude of the second Ship: So the Angle of Position is NCQ, whose measure is CG on the half Tangents 48 deg. 58 min. from the South Westwards; and the Distance is NH 48 deg. 22 min. that is, 1683 ⅔ Leagues, or 5051 Miles, the nearest Distance of the two Ships, which was required. How to do it by the Tables, you have been shewed in the last Example.
The Fourth Scituation.
PROBL. VII. The Latitude of two Places being given, together with the Difference of Longitude, To find, • First, The nearest Distance in the Arch of a Great Circle. , • Secondly, The Direct Position from the first Place to the second. And, , • Thirdly, From the second Place to the first. And, ,
and • Fourthly, The Circles greatest Obliquity that passeth over those two Places.
ADmit L be the Latitude of Lundy 51 deg. 22 min. and Longitude 25 deg. 52 min. and B is the Latitude of Barbadoes 13 deg. 10 min. and Longitude 332 deg. 57 m. and Difference of Longitude 52 deg. 55 min.
[geometrical diagram]
Lay down first the Latitude 51 deg. 22 min. from Q to L; secondly, the Difference of Longitude QF 52 deg. 55 min. and draw the Meridian-circle P through F to S; then lay down the Latitude of Barbadoes from Q to 13, and take of the half [Page 188]Tangent Line QF 57 deg. 55 min. and lay from C to K, and draw the prick'd Line, and he will cut the Meridian PFS in B, the Latitude of Barbadoes, 13 deg. 10 min. through B draw the Circle LBN, so the Angle of Position is BLQ, whose measure is RO 67 deg. 51 min. that is, W. S. W. 21 min. Westerly; which taken off the Line of half Tangents of your Scale, and laid from C to Pole, and draw the Line from Pole through B, and it will cut the Limb in G: Therefore measure GL on the Line of Chords, you have 57 deg. for the Distance, or 1140 Leagues, or 3420 Miles.
To find the Distance in Questions of this Nature,
1 As the Radius, Is to the Co-sine of the Diff. of Longit. 52 d. 55 m.
978030
So is the Co-tangent of the greater Latitude 51 deg. 22 min.
990267
To the Tangent of the first Arch 25 deg. 44 min.
968297
Take 25 deg. 44 min. from 76 deg. 50 min. the Complement of 13 deg. 10 min. the less Latitude, and the Remain is 51 deg. 6 min. the second Arch.
2 As the Sine of the first Arch 25 d. 44 m.
995464
To find the Distance.
Is to the Co-sine of the second Arch 51 d. 06 m.
979793
So is the Sine of the greater Latitude 51 d. 22 m.
989273
1969066
To the Co-sine of the Distance 57 d. 00 m.
973602
1306 P.1157 M.1140 G.Leag. 17166
which is the nearest Distance in the Arch of a Great Circle, by 17 Leagues less than Mercator's Chart by the Rhomb, and less by 166 Leagues than by the Rhomb on the Plain Chart; which confirms this to be the nearest of all ways of Sailing betwixt any two Places.
To find the Angle of Position,
3 As the Sine of the Distance 57 deg. 00 min.
992359
Is to the Sine of the Difference of Longitude 52 deg. 55 min.
990187
So is the Co-sine of the Latitude 13 deg. 10 min.
998843
Add the two last: Substract the first Numbers
1989030
There remains the Sine of the Angle of Direct Position
996681
Which is 67 deg. 51 min. from the South part of the Meridian Westward, as namely, W. S. W, 20 m. Westerly.
Now to know the Distance and Angle of Position, you must put Barbadoes at B, on the West side of the Circle 13 deg. from AE, and draw the Parallel of Latitude 51 deg. 22 min. and lay off the Difference of Longitude from AE to F, and draw the Meridian-circle PFS, and it will cut the Parallel of Latitude in L; therefore from B draw the Circle from L to K: And if you follow your former Directions, BD will be the Measure of
BL the Distance found, as before, 57 deg. 00 min. and LBP the Angle of Position, whose Measure is RG 36 deg. 26 min. and the Great Circles greatest Obliquity is CO 54 deg. 40 min. For,
4 As the Sine of the second Arch 51 deg. 6 min.
989111
Is to the Sine of the first Arch 25 deg. 44 min.
963767
So is the Tangent of Difference of Longitude 52 deg. 55 min.
1012157
1975924
To the Tangent of the Direct Position 36 deg. 26 min.
986813
From B towards L, which is 3 Points, 2 deg. 41 min. from the Meridian, namely, N. E. b. N. 2 deg. 41 min. Easterly, you must sail first from B towards L; but alter your Course, still increasing toward the Eastward, as shall be shewed.
5 As Radius, To Co-sine of less Latitude 13 deg. 10 min.
998843
So is the Sine of the Angle of Position 36 deg. 26 min.
977370
To the Co-sine of the greatest Obliquity 54 deg. 40 min.
976213
These are the Scituations of all Places upon the Terrestrial Globe; so that there cannot be any two Questions, but, in respect of each other, they will be found in one of these four kinds; except
they fall in one Meridian, or on the Aequator: and these Directions you have in the Tenth Chapter: Therefore if you will seriously observe these short Directions already given, and
as follows, you shall never have your Expectation deceived.
CHAP. XII. How to describe the Globe in Plano, by the Mathematical Scale.
THese, and all other Questions of this nature, concerning the Resolution of any Spherical Triangle, may very easily be performed by the Globe: But because the Globe is a chargeable Instrument, and so every one cannot have it, therefore several Men, have for several Uses, invented
several ways to Project the Globe upon a Plain, as Mr. Gunter hath them in his Book of the Sector. The fittest for this purpose will be that of Gemma Frisius, which is most used in the Great Maps of the World, the Projection whereof is as followeth.
First, By the Chord of 60 deg. describe the Circle AENES, and by the Chord of 90 deg. divide it into four parts, as AEE a Cross Diameter for the Aequator, and NS for the great Meridian: Then by taking off every 10 deg. of the Chord, you may divide each Quadrant into 90 deg. and number them as in the Figure: Then if you take off your Line of ½ Tangents in your Scale every 10 deg. and 5 deg. and lay them from the Center C on the four Sides of the Quadrants, as you see the Figure, and number them as they are in the Figure; so shall you divide the Diameters into his parts AEE the Aequinoctial, NS the Meridian, which are half Tangents. You may do it also without [Page 190] the Scale, by your Ruler, if you stop one end of your Ruler at N, and turn the other end about to the several Degrees in the lower Semicircle ESAE: And also if you keep one end of your Ruler fixed in the Point AE, and lay the other end about to the several Degrees in the Semicircle NES; so have you the Meridian-diam ter divided into half Tangents likewise.
Now you have divided the Diameters, they must guide you in the drawing of the Meridians from Pole N to Pole S, which are perfect Circles; as likewise are the Parallels of Latitude.How to find the Centers of the Meridian-circles. You may find the Centers in the Diameter AE, if you extend the Compasses from the first Degree on the half Tangents, to the Secants of every 10 Degrees, and with that Distance put one Point at 10 deg. in the Semidiameter AEC, and in EC will the other Point be the Center of the Meridian of the first 10 deg. from AE: and do the like from E in the same manner, for any other Degree. To draw the Meridian of 50 deg. Longitude, take the Secant of 50 deg. off the Scale, and one Point will stand in the Semidiameter AEC, at 50 deg. and the other will stand in the Center at East, and likewise at West for 50 deg. on the other side: And so do for the rest; and so you may find the Center of any Circle whatsoever, upon the Cross Semidiameter belonging to it, which you must continue beyond the Great Circle,How to find the Centers of the Parallels of Latitude. where the Center will be in many Questions. For the Parallels of Latitude, it is thus: Take the Complement-number of Degrees off your Line of Tangents, put one Point in the Degree of Latitude, the other will stand in the Center.
For Example. If you will draw the Parallel of Latitude for 60 deg. take off the Tangent-line of your Scale 30 deg. the Compl. of 60 Latitude, and the other will fall upon V the Center of the Parallel of 60 deg. in the Semidiameter NS continued beyond the Circle.
So, Take the Tangent of 40 deg. and it will draw the Parallel of 50 deg. whose Center is at ♓: and so do in drawing all Parallels of Latitude. You may draw them also by making several Trials, until your three Points be in a Circle, and also draw the Parallels of Latitude: with the same Distance find their Centers; but if you can, by the Scale is the surest way.
The Four Scituations that are in the Globe.
The First Scituation.
AE is a Point of Intersection for the Mouth of the River of Amazones, Z Lundy, CRV the Obliquity 62 deg. 16 min. E the other Point of Intersection with the Aequator, NRV the measure of the Angle of Position. which applied to the Aequator from AE inwards, shews you 27 deg 50 min. from Amazones to Lundy. Now if you will know the Distance in such Questions, measure it in the Meridian that agrees with the Angle of Position; as namely, for this Distance AEZ, you must measure from N in the Meridian-line of 27 deg. 50 min. and you will find it 61 deg. 57 min. And so do for to measure any other Distance.
The Second Scituation.
I is the first Places Latitude, ♈ is the Difference of Longitude 52 deg. 55 min. ♈ is likewise the second Places Latitude; and ♍ H is the measure of the Angle of Position, which measured in the Semidiameter AEC, will be found 68 deg. 46 min. In that Meridian measure l r the Distance, and you will find it from N towards C, to reach 32 deg. 18 min. Remember to measure the Distances from the Poles in the same Meridian, of the Number of Degrees of the Angle of Position: The greatest Obliquity of that Circle N r l is at W 54 deg. 25 min. Intersection of the Aequator at Y, W is a Meridian of greatest Obliquity.
The Third Scituation.
L is the first Places Latitude 51 deg. 22 min. North; E p is the Difference of Longitude 45 deg. 55 min. R is the second Places Latitude, or Rio de Plata, ▵ the greatest Olliquity 68 deg. 22 min. m n the Measure of the Angle of Direct Position: applied to EC will be found 36 deg. 02 min. in that Meridian: from the Pole measure the Distance LR, and you will find it 95 deg. 18 min. P the Intersection of the Great Circle, passing over the two Places in the Aequator.
The Fourth Scituation.
The first Latitude is at L Lundy 51 deg. 22 min. Difference of Longitude counted from E 52 deg. 55 min. that Meridian will cut the Latitude of Barbadoes 13 deg. 10 min. at b: M ♓ the Measure of the Angle of Direct Position 67 deg. 51 min. and b L measured in that Meridian is 57 deg. 00 min. the Distance. Now to know the Angle of Position from Barbadoes, being Westward from Lundy, set it on the West side of the Figure, as at B; and likewise if the first Place be to the Eastward, put his Latitude to the East side of the Meridian.
Now to know the Angle of Position from Barbadoes, and Distance, and Obliquity, B □ O is the Arch of the Great Circle that passeth over these two Places; □ is Lundy, q h is the Measure of the Angle of Position 36 deg. 26 min. B □ measured in that [Page 192]Line, you will find the Distance 57 deg. O is the greatest Obliquity 54 deg. 40 min. NO ☉ S a Meridian of the greatest Obliquity: d is the Intersection with the Aequator.
CHAP. XIII. By Arithmetick how to Calculate exactly for any Degrees and Minutes of Obliquity; What Degree and Minute of Latitude the Great Circle shall pass through for any Degree and Minute of Longitude, from the Point of Obliquity, or of its Intersection with the Aequinoctial.
NOte these Rules well, for they serve for all Questions of this nature, what Difference of Longitude soever any Point or Place hath from the Meridian of its next Obliquity, which is ever 90 deg. or less; the Complement thereof to 90 deg. is the Difference of Longitude of that Point or Place, from the Meridian of that Great Circle's next Intersection with the Aequator.
The first sort of RULES are these.
By the Latitude of two Places, the Difference of Longitude betwixt them, and the Obliquity of the Great Circle passing directly over both Places given; To find the Difference of Longitude of each Place from the Meridian of the greatest Obliquity.
Let this be our Example. Lundy in North Latitude 51 deg. 22 min. and Barbadoes in North Latitude 13 deg. 10 min. and Difference of Longitude 52 deg. 55 min. and the greatest Obliquity 54 deg. 40 min. Work first with the less Latitude, to find the Difference of Longitude from the Meridian of Obliquity of both Places.
RULE I.
Thus, As Radius, and Co-tangent of Obliquity 54 deg. 40 min.
1, 985059
Take from it the Co-tangent of less Latitude 13 deg. 10 min.
1, 063090
There remains the Co-sine of Differ. of Longitude 80 d. 27 m.
921969
The Difference of Longitude of the second Place from the Meridian of the greatest Obliquity. And by reason the Difference of Longitude from the Obliquity is more than the Difference of Longitude betwixt the two Places, therefore substract the Difference of Longitude 52 deg. 55 min. from 80 deg. 27 min. and there remains the Difference of the first Place from the Meridian of Obliquity 27 deg. 32 min. and the first Place is betwixt the second Place and the Meridian of Obliquity. But if the Difference of Longitude from Obliquity had been less than the Difference of Longitude betwixt the two Places, then substract the Longitude from the Difference of Longitude betwixt the two Places, and the Remain had been the Difference of Longitude from the Meridian of Obliquity, to the first Place.
The Use of these Rules are,
1. To find where to place the Meridian of the greatest Obliquity betwixt any two Places, in a blank Chart, or Mercator's Chart or Plat, to trace out the Voyage, as we find that it is 80 deg. 27 min. the Difference of Longitude of the second Place from the Meridian of Obliquity, and 27 deg. 32 min. from the first.
2. You see you find the Difference of Longitude betwixt the Obliquity and any Latitude propounded, by the last Rule.
3. By the Obliquity of the Great Circle, and the Difference of Longitude from the Obliquity, to find the true Latitude.
As Radius, To Co-tangent of Obliquity 54 deg. 40 min.
1985059
Take from it Co-sine of Differ. of Longitude 80 deg. 27 min.
921086
There remain Co-tangent of the Latitude 13 deg. 10 min.
1063073
And so likewise if you work with 27 deg. 32 min. by the same Rule, you will find the other Latitude 51 deg. 22 min. and by it find from any Place through what Latitude the Great Circle passeth every 10 deg. or 5 deg. or more or less Quantity of Longitude from the Obliquity; and thereby find the Latitude, to make marks in every Meridian; and so trace to out the Great Circle Arch, in your Mercator, or Mr. Wright's Plat or Blank Chart.
As thus, for 2 deg. 28 min. of Longitude more, added to 27 deg. 32 min. makes 30 deg. 00 min.
RULE III.
As Radius, To Co-tangent of Obliquity 54 deg. 40 min.
1985059
Take from it the Co-sine of Differ. of Long. from Obliq. 30 d. 00
993753
There remains the Co-tangent of the Latitude 50 deg. 41 min.
991306
And by the same Rule I made this
Differ. of Longit from Obliq. Lundy
Latitude from Lundy.
Deg. Min.
Deg. Min.
27 32
51 22
30 00
50 41
35 00
49 07
40 00
47 13
45 00
44 56
50 00
42 12
55 00
38 58
60 00
35 12
65 00
30 48
70 00
25 46
75 00
20 03
80 27
13 10
Barbadoes.
Table, of an Arch of a Great Circle, extended from Latitude 51 deg. 22 min. to Latitude 13 deg. 10 min. Difference of Longitude 52 deg. 55 min. setting the Point of the greatest Obliquity upon a Meridian-line, that so it might be the better protracted on Mercator's Chart: This is, for every 15 deg. Difference of Longitude, these are the Latitudes the Great Circle will pass through; so that you see there is 52 deg. 55 min. added by 5 to the Difference of Longitude of the first Place from the Meridian of Obliquity, which makes 80 deg. 27 min. the Difference of Obliquity of the second Place, which was the Difference of Longitude from Obliquity at first found. In like manner work for any other Place.
CHAP. XIV. How by the Scale of Tangents to make a Part of the Globe in Plano, whereby you may trace out the Latitudes to every Degree of Longitude; or every 5 or 10 Degrees, as neer as you will desire, without Calculation.
BY the Line of Tangents on the side of your Mathematical Scale, you may make the following Projection, which was made by Mr. Philips in his Geometrical Seaman, pag. 5. by Tables and Geometry: But here you may save that labour, if you have a Scale with a Line of Tangents on it.
First, Consider of what length your Tangent or Side of your Quadrant must be, and accordingly set off your Radius from A towards D, as I have done, by taking off 77 deg. of the Tangent-line of my Scale, and set it from A the Pole to D, for 13 deg,It serves for the same numb. of deg. in South Latitude. on the North side of the Aequator, or 13 deg. of North Latitude, which is the Complement[Page 194] of 77 to 90 deg. Then make the other side of the same length, and draw the Quadrant ADE, the Radius is always a Tangent of 45 deg. Then with your Compasses take off the Line of Tangents the several Degrees, and draw the Arches or Parallels of Latitude, as you see I have done in the Figure. Thirdly, divide the Limb of the Arch DE into 90 deg. and through every 5 or 10 deg. draw Lines of Longitude, or Meridian-lines. The Arches of Latitude must be numbred as in the Figure; but the Lines of Longitude you may number from either side, as occasion requires.
You may, if you will, when occasion requires, divide a Circle into four Quadrants, and draw the Lines of Longitude from the Center; and you may make this as large or as little as you will, by the Tables of Natural Tangents in the Second Book, as you have been there shewed how to lengthen or shorten your
Radius: You may number the whole Circle of Longitude into 360 deg.
The Blank Quadrant, being thus made, will serve for many Examples; especially if you make it upon a Slat Stone, that you may wipe the Arch, that is lightly drawn by a Slat Pen betwixt any two Places, off at pleasure.
You may set down therein the two Places you are to sail between, according to their
Latitudes and Longitudes; and then only by your Ruler draw a streight Line from the one Place to the other, which will represent the Great Circle which passeth between the two Places, and will cross those Degrees of Longitude and Latitude, which you must sail by exactly. You may do it by the Difference of Longitude only, if you will, as shall be shewed in this Example, for proof thereof.
Of a Voyage from Lundy, in Latitude 51 deg. 22 min. and Barbadoes, in Longitude 332 deg. 27 min. Difference 52 deg. 55 min. and Latitude 13 deg. 10 min. To find by what Longitude and Latitudes the Arch of a Great Circle drawn between those two Places doth pass.
First, Let the Line AD represent the Meridian of the Island of Lundy,You may measure the Distances thus. Take BL the Distance of the two Places, and put it upon the Meridian BZ of Barbadoes the second Place, & you will find the Parallels of Latitude to be 57 d. the true Distance, as before sound. marked out by L for its Latitude 51 deg. 22 min. and the Longitude thereof 25 deg. 52 min. at D, which is set down according to its Longitude and Latitude. Then from D in the Limb or Arch of the Quadrant, count the Difference of Longitude 52 deg. 55 min. and this is the Meridian of the Island of Barbadoes, on which you must mark out the Latitude 13 deg. 10 min. at B; lay a Ruler from the first Latitude L to the second at B, and draw the streight Line LB, which representeth the Arch of a Great Circle between the two Places; and if you guide your Eye along in this Line, you may readily and truly perceive by what Longitudes and Latitudes you should sail: For where this Line crosseth the Arches of Latitude and the Lines of Longitude, that shews the true Longitude and Latitude of the Arch, according to your desire.
Differ. of Longit. substract
Differ of Longit. from Obliqu.
Latitude
Differ of Longit. added.
D. M.
D. M.
D. M.
D. M.
52 55
27 32
51 22
00 00
2 28
30 00
50 41
02 28
5 00
35 00
49 07
07 28
5 00
40 00
47 13
12 28
5 00
45 00
44 56
17 28
5 00
50 00
42 12
22 28
5 00
55 00
38 58
27 28
5 00
60 00
35 12
32 28
5 00
15 00
30 48
37 28
5 00
70 00
2 [...] 46
42 28
5 00
75 00
10 30
47 28
5 27
80 27
13 10
52 55
52 55
Barbad.
Now the truth hereof will more evidently appear, if you compare the Latitudes and Longitudes which this Line intersecteth, with this Table, as before, Calculated by me for every [...]deg. of Longitude. You may see by the Figure, that the Line BL in the Points a, b, c, d, e, F, g, h, i, k, L, doth cross the Parallel of Latitude, as you see in the fourth Column of this Table, at the same number of Degrees from the first Meridian, as you see in the fifth Calu [...]n. The first Column is the Number of Degrees of Longitude, the second is the Difference substracted, the third is the Degrees of Longitude from the Meridian of the greatest Obliquity, the fourth and fifth are, as before, Latitude and Longitude added, for the Difference of Longitude from Lundy. For Example: I would know what Degree of Latitude 47 deg. 28 min. of Longitude from the first Meridian doth cross; and I see by the Figure it is at [...]0 deg. of Longitude, which the Table sheweth is 20 deg. [...]0 m. of Latitude, and so of the rest, 75 deg. from Meridian of Obliquity, and Longitude 338 deg. 24 min.
And so in like manner you may lay down upon the former Quadrant any two Places, howsoever scituated, by their Longitude and Latitude, of Difference of Longitude, in the manner as you have been shewed in the last Example.
RULE IV. By the Latitude, and Difference of Longitude from the Obliquity, to find the true Great Circle's Distance.
As Radius 90, To Co-sine of the Latitude 51 deg. 22 min.
979541
So is the Sine of the Difference of Longitude 27 deg. 32 min.
966489
To the Sine of the Distance 16 deg. 47 min.
946030
How by the Latitude well observed, and the Rhomb discreetly rectified, to find
the Latitude, Rhomb, and Longitude, and Distance, having two of them known, by an
Arithmetical Rule, and by them to prick the same down in a Blank Chart or Mercator's
Plat.
So that 16 deg. 47 min. is the Great Circle's Distance from the Point of the greatest Obliquity of that Great Circle.
RULE V. By the Obliquity of the Great Circle, to find the true Latitude to any Quantity of a Great Circle's Distance, from the Point of his greatest Obliquity.
As Radius, To Sine of greatest Obliquity 54 deg. 40 min.
991158
So is Co-sine of the Distance from Obliquity 16 deg. 47 min.
998109
To the Sine of the true Latitude 51 deg. 22 min.
989267
Again,
As Radius, To Sine of greatest Obliquity 54 deg. 40 min.
991158
So is Co-sine of Distance from Obliquity 73 deg. 47 min.
944602
To Sine of true Latitude of Barbadoes 13 deg. 10 min.
935760
RULE VI. By the Great Circle's Distance from the Point of Obliquity, and the Latitude given; To find the Difference of Longitude betwixt the Place and the Meridian of greatest Obliquity.
As Radius, To the Sine of Distance from Obliquity 16 d. 47 m.
946052
Take from it the Co-sine of the Latitude 51 deg. 22 min.
979541
Remaine the Sine of Difference of Longitude 27 deg. 33 min.
966511
By these four last Rules you may be confirmed of the truth of the former Work, of [...]acing of the Great Circle by Longitude, Latitude, and Distance, from the Point of its greatest Obliquity.
This is worth your Observation, That the Complement to 90 Degrees of the Great Circle's Distance of a Place from the Point of the Great Circle's Obliquity, is always the Great Circle's Distance from the Point wherein that Circle intersecteth the Aquinoctial.
Whereby to find what Rhomb you are to sail upon, that you may keep in or neer the Arch of a Great Circle, extended from one Place to another.
RULE VII. By the Difference of Longitude from the Obliquity, and Latitude given; To find the Great Circle's Distance from the Point and Meridian of greatest Obliquity.
As Radius, To the Sine of Difference of Longitude from Obliquity 80 deg. 27 min.
999393
So is Co-sine of the Latitude 13 deg. 10 min.
998843
To the Sine of the Great Circle's Distance from Obliq. 73 d. 47 m.
998236
So that if you take out 16 deg. 47 min. in the Distances of Obliquity before found out of 73 deg. 47 min. the Remain will be the true Distance of the two Places 57 deg. as was found in the second Rule of the fourth Scituation. But if the Point of greatest Obliquity had been betwixt the two Places, you must have added them together, and the same
had been the Distance of the two Places.
RULE VIII. By the Great Circle's Distance from the Obliquity, and the Latitude given; To find the Rhomb.
As Radius 90 d. and Tang. of Great Circles Distance 16 d. 47 m.
947943
Take out the Co-tangent of the Latitude 51 deg. 22 min.
990267
There remains the Co-sine of the Rhomb 67 deg. 50 min.
957676
Having therefore in the four Scituations, before-going, been directed how to find the Distance betwixt the two Places, by the Arch of a Great Circle, and the greatest Obliquity of any Circle, you may by the first Rule of Chap. 13. find the Difference of Longitude of each Place, from the Point of the Great Circle's greatest Obliquity; and then by the fourth and seventh Rules of Chap. 15. find the Great Circle's Distance to the Obliquity, and by the eighth Rule find the Rhomb to be sailed on, from either Place towards the other.
RULE IX. To find how far a Man should sail upon a Rhomb, before he change his Course a Point Half a Point, or a Quarter of a Point.
You may try this by your Protracting Quadrant, on your Blank Plat or Chart, made according to Mr Wright's or Mercator's Projection; where the Voyage is truly and carefully traced out as before.
Or you may Arithmetically try every Point, or Quarter of a Point, or Half Point, as you see cause, by these Rules.
Having found by the eighth Rule the Rhomb was 67 deg. 50 min. add to it a Quarter of a Point, or 2 deg. 29 min. it makes 70 deg. 19 min. W. N. W. ¼ W. and Latitude 49 deg. 07 min. and Difference of Latitude 2 deg. 15 min. and it is required the Distance.
As Radius 90 deg. To Sine of Difference of Latitude 2 d. 15 m.
859394
Take out the Co-sine of the Rhomb 70 deg. 19 min.
952739
There remains the Sine of the Distance 06 deg. 42 min.
906655
Therefore 06 deg. 42 min. is 134 Leagues, may be sailed W.N.W. ¼ Westerly, from the Latitude 51 deg. 22 min. to Latitude 49 deg. 07 min.
RULE X. By the Great Circle's Distance, and the Difference of Latitude given; To find the Rhomb.
As Radius, To Sine of Difference of Latitude 2 deg. 15 min.
859394
Take out the Sine of the Distance 6 deg. 42 min.
906655
There remains the Co-sine of the Rhomb 70 deg. 19 min.
952739
RULE XI. By the Rhomb, and Distance upon it given, To find the Difference of Latitude.
As Radius, To Co-sine of the Rhomb 70 deg. 19 min.
952739
So is Sine of the Distance 06 deg. 42 min.
906655
To the Sine of Difference of. Latitude 2 deg. 15 min.
859394
Which 2 deg. 15 min. taken from the first Latitude 51 deg. 22 min. there remains the Latitude 49 deg. 07 min. which are good Proofs:
RULE XII. By the Obliquity of the Great Circle, and the Latitude given; To find the Difference of Longitude from the Meridian of Obliquity.
As Radius 90, To Co-tangent of Obliquity 54 deg. 40 min.
985059
Take out the Co-tangent of the Latitude 49 deg. 07 min.
993737
There remains Co-sine of Difference of Longitude 35 deg. 00 m
991322
35 deg. 00 min. is the Difference of Longitude from the Meridian of Obliquity. You you may also try it by the Difference of Longitude in the fifth Column, 7 deg. 28 min. in the foregoing Table, added to 27 deg. 32 min. makes 35 deg. as before.
RULE XIII. By the Latitude, and Difference of Longitude from the Obliquity given; To find the Great Circles Distance from the Meridian of Obliquity.
As Radius, To Co-sine of the Latitude 49 deg. 07 min.
981592
So is the Sine of the Difference of Longitude 35 deg. 00 min.
975859
To the Sine of the Distance 22 deg. 05 min.
957451
Which is the Distance from the Meridian of the greatest Obliquity: Then you may proceed to the Rhomb next to be failed on.
Now if you add 2 deg. 49 min. to the former Rhomb 70 deg. 19 min. it makes 73 deg. 8 min. that is W. S. W. half a Point Westerly, and the Difference of Latitude 4 deg. 11 min. which you may find by the foregoing Tables, or thus by the ninth Rule.
As Radius, Is to Co-sine of Difference of Latitude 4 deg. 11 m.
886301
Take out the Co-sine of the Rhomb 73 deg. 08 min.
946261
There remains the Sine of the Distance 14 deg. 34 min.
940040
Which 14 deg. 34 min. converted into Leagues is 291 Leagues ⅓ the Distance you may sail W. S. W. ½ a Point Westerly, from Latitude 49 deg. 07 min. to Latitude 44 deg. 56 min. and the Difference of Longitude is 10 deg. as you may see by the former Table, by the Latitude 44 deg. 56 min. is 17 deg. 28 min. and the Difference of Longitude from Obliquity is 45 deg. 00 min. and the Longitudes according to my Globes made by Hondius 8 deg. 24 min. and so work until you have calculated the Distance for every ¼ Point, or rather half Point, because ¼ of a Point cannot be well steered upon. It is no matter if you do not exactly keep the Arch of a Great Circle, for the Reasons before given: but as neer to it as you may conveniently, as Mr.
Norwood hath sufficiently answered in his ninth Problem of Sailing by a Great Circle.
But let me advise you to get into your Latitude short of the Place you are bound to, for fear of mistakes in your Reckoning, and so over-shoot your Port to a greater disgrace, than the credit of Great Circle Sailing will bring you; and then to know what Distance you are to sail in that Parallel of Latitude.
RULE XV. By the Latitude, and Difference of Longitude given: To find the Distance upon a Course of East or West.
As Radius, Is to Co-sine of the Latitude 13 deg. 10 min.
998843
So is the Sine of Difference of Longitude 5 deg. 00 min.
894029
To the Sine of the true Distance 4 deg. 52 min. which is 97 Leagues ⅓.
892872
After you have made some progress in your Voyage, you may make use of these most excellent Rules, whereby a Mariner may make his Conclusion most certain.
RULE XVI. By the Difference of Latitude, and Rhomb sailed on; To find the Distance. You have it in the Fourteenth Rule before-going.
.
RULE XVII. By the Latitude, and Distance sailed upon an East or West Course, To find the Difference of Longitude; as 20 Leagues sailed in Latitude 51 deg. 22 min.
As Radius, To Sine of the Distance 1 deg. 00 min.
1824185
Take away Co-sine of the Latitude 51 deg. 22 min.
979541
Remains the Sine of Difference of Longitude 1 deg. 36 min.
844644
Therefore sailing 20 Leagues East or West in the Parallel of 51 deg. 22 min. the Difference of Longitude made is 1 deg. 36 min. which is a good Rule when you are in the Latitude of the Place you are bound to. In the first Rule you have how for to find the Difference of Longitude from Obliquity, Chap. 13. and likewise in Rule 12. where the Difference of Longitude is 35 deg. substract it from 27 deg. 32 min. and the Remain will be the Difference of Longitude of two Places, one in Latitude 51 d. 22 min. the other 49 deg. 07 min. as you have been directed before.
So that what hath been written will satisfie any ingenious Spirit, to make use of
these Rules in these four Scituations; and these four will answer any thing required, in all sorts of Great Circle sailing. I shall now make the blank Mercator Plat; and trace out the Arch of a Great Circle; and likewise shew how by Latitude and Longitude to find the Place of any Ship in Mercator's Chart.
CHAP. XVI. How to make the most true Sea-Chart, and the Ʋse thereof in Mercator's and Great Circle Sailing, called a General Chart.
FOr the manner of the Division, Let the Aequator be drawn and divided, and crossed with Parallel-Meridians, as before directed; only one Degree of Longitude in the Particular Chart before-going, is 10 deg. of the Aequator of this General Chart. You have been directed how to make the Meridian-line off the Scale which is for a General Chart; and the same Rule makes this. Look in the Table of Meridional parts, and you will find the Difference between the Aequator and 40 deg. of Latitude in the Meridian-line to be 874, 2 which is 874 Leagues and 2/10; that divided by 20 is 43 deg. 42 min. 2/10 of the Aequator, therefore take out of a Scale of Equal parts, answerable to each Degree of the Aequator, divided into 20, which you must reckon 200 Parts, take 874 Parts, and that Distance will reach from the Aequator to 40 deg. of Latitude. Always remember in a General Chart to omit the last Figure in the Table, which is Tenths, 2/10 in this.
And for 50 deg. take 1158 such Equal parts, which is 57 deg. 54 min. will reach from the Aequator at Ae, to 50 deg. of Latitude in the Meridian-line: And so do for any other Degree or Minute of Latitude, until you have made the Chart, as I have done the Figure following.
How to make the Meridian-line by the former Geometrical Projection.You may divide the Meridian-line by the Projection in the Quadrant, making the side thereof answerable to 5 Degrees of the Aequator of the Chart, as in this.
Suppose you would know the Distance betwixt 40 deg. and 50 deg. of Latitude, in the Meridian-line of a Chart.—Take the middle of 10 deg. which is 45 deg. out of that Latitude in the Quadrant, and it will reach from 40 to 50 deg. of Latitude in the Chart, which you may soon try. And so work for any other Latitude.
A Scale of Leagues.You have also there a Scale of Leagues for every Parallel of Latitude, to measure any Distances in the Chart; for every Degree is 200 Leagues in this Chart, as you may soon apprehend, without more words, by the former Directions.
The Protracting Quadrant.The Protracting Quadrant (you may see the Figure following) shews you all at one sight, without more words, how to make it by dividing
it into eight Points, and each Point into four Quarters, and an Arch within into 90 deg. and a Libal or Index to be rivetted to the Center, and a Hole drilled through the Rivet, to put a Pin through the Center of the Quadrant upon any Place assigned, and let him square by the Parallel-Meridians and Parallels of Latitude; so laying the Index over the second Place, the Limb of the Quadrant will shew you the Point of the Compass, and what Angle it makes with the Meridian, or bearing of the first Place from the second, as we have shewn by divers Arithmetical Rules, for your more certain and exact direction, how to keep your Reckoning upon your Mercator's Chart, or Blank; and to know first and afterwards what Rhomb you are to sail upon, keeping in or neer the Arch of a Great Circle; and to know what Longitude and Latitude you are in, after some progress made in your Voyage.
You shall have here also the way how you shall trace out the Arch of a Great Circle betwixt the Places in a Blank or Mercator's Plat, and how to prick down upon your Chart any Distances of Longitude and Latitude.
Before in the third Rule of Chap. 13. you have the way how to calculate for any Degree and Minute of Obliquity, and any number of Degrees of Longitude; what Degrees and Minutes of Latitude the Great Circle shall pass through, by the same Rules I have calculated, for every 10 Degrees of Longitude, reckoning from the greatest [Page][Page]
A Generall Sea Chart According to Mercator.
The Index
Fasten this in with a Rivet to the Centor of the Quadrant A that it may turne upon
it with a hole through the Rivet.
A Scale of equall parts for the deviding the Meridian Line by the Table of Meridionall
parts.
[Page 201]Obliquity, to the Intersection of the Great Circle with the Aequator, viz.
Greatest Obliquity G 54 deg. 40 min.
For Deg. Lat.
10
20
30
40
50
60
70
80:27
For 90 deg. Longit.
54 d. 15 m.
52:58
50:41
47:13
42:12
35:12
25:46
13:10
For 00 deg. Latitude.
In this following Mercator's Plat, G standeth at 54 deg. 40 min. of North Latitude, and AE just 90 deg. of Longitude Westward, and AEN 90 deg. Eastward from G, being the two opposite Points in the Aequator, 180 deg. from each Point of Intersection AEAE.
In every Meridian betwixt G and AE, on both sides a Meridian drawn at every 10 deg. of Longitude in the Chart, make a mark at Latitude found in the Table, by the Meridian-line of the Chart; and having so marked every Meridian betwixt G and AE, then by these marks ye may draw Arches from one to another: but it will suffice to draw Right Lines from Mark to Mark, as from G the greatest Obliquity of the Great Circle, to the next Meridian on each side, and likewise to the next, until you come down to the Aequator at AE. on both sides; so have you pourtrayed on this Mercator Plat a Great Circle Arch from Lundy to Barbadoes, one being in North Latitude 51 deg. 22 min. at L, the other at B in North Latitude 13 deg. 10 min. with Difference of Longitude 52 deg. 55 min. from C unto L, to C under B.
Note, To make a perfect Circle: the Latitudes of the Arch are the same on the South side of the Aequator, as you have found them on the North side. You might have marked out only so much of the Great Circle from the first Latitude, as you see I have done from the side of the Meridian-line, and Latitude of Lundy 51 deg. 22 min. at l, to Latitude 13 deg. 10 min. at B, by the Difference of Longitude in the last Column of the Table in Chap. 13. and Difference you will find of Longitude is 52 deg. 55 min. by the former Direction: The Figure makes all plain in the Chart or Blank.
Two Places in one Latitude, as in the second Scituation Latitude 51 deg. 22 min. as in the following Chart, one at H, and the other Place at L in the same Latitude, and Difference of Longitude 52 deg. 55 min. the neerest Distance is not upon a Parallel directly from H to L, but from H to sail from 51 deg. 22 min. by G the greatest Obliquity, in Latitude 54 deg. 40 min. W. N. W. almost, then W. b. N. and W. b. S: W. S. W. the other half from G to L, which is the neerest Distance by 42 Miles; for the Distance by the Parallel is 660 5/10 Leagues, but by the Arch of a Great Circle is but 646 Leagues: And one would not think but the Parallel were the neerest, to look in the Plat: but he that knows the Globe, conceives that by the Arch that goes neerer the Poles, cross the Meridians, to be the neerer; therefore the Arch must be the neerest way. And sailing into several Latitudes, you have the benefit to correct your Dead Reckoning, which you cannot so well do by keeping a Parallel of East and West. These Directions may be sufficient for any Questions you will have any way in Great Circle Sailing.
But he that will take the pains, may find great delight in this sort of Practice:
Yet I must conclude, That although it is the neerest way, it is not the convenientest
way for Seamen, for several Reasons best known to them that keep an Account of the Ships way, which I could lay down here; but in regard it is needless, I leave every one to his
mind, and shall shew you the way how I did keep my Account at Sea, by the Plain Chart and Mercator's Chart; and how to measure Distances in Mercator's Chart, in any Parallel also: which, if you have a better way, publish it, that others may gain benefit by
it; for you will not hurt me any way; but rather I desire, that all the Nauigators in England did exceed me (for His Majestie's sake, whose Subjects we are) and hope that the
Neighbour-Nations will once know, That the English Mariners are not less known in Art, than by their Courage, which the Dutch know by dear-bought Experience.
CHAP. XVII. How to keep a Sea-Journal, that so every Sea-man, Navigator, and Mariner, may not be ashamed to shew their Account to any Artist, and by it benefit themselves and others.
I Would not have any ingenious Sea-Artist, that hath a long time kept Account of a Ship's way, and hath been Commander or Mate many years, to think I prescribe him Rules, and to perswade him out of his beaten Path (No, I think that a hard matter.) But
we prescribe Rules for those that are but new Learners, that so they may have a perfect Method and Way
of keeping Account of a Ships way at Sea; that if the Master should perceive an Ingenious Practicioner aboard, and by examining his Journal find him able, might at his return home give him encouragement, by speaking in his
behalf to other Men to make him a Mate; and that is the way to encourage Artists: But I confess the greatest Dunces have commonly the best Imployments, and many abler
men before the Mast: which is great pity, that the deserving Men had not their right. But what shall I
say? There is such an aversment in Fate. Therefore I shall proceed to our Journal. I conceive it will be fit to have a Book in Folio, that a sheet of Paper makes but two Leafs, and to keep the left side of your Book void, that you may write all the Passages of the Voyage; that is to say, when you set Sail, with what Wind, and what Ships are in company with you, and how far you keep company; what Storms, and how the Wind was: and likewise put down the time that you come by any misfortune, of cracking
or breaking a Mast or Yard, or if any Men should die; and also what Damage you receive by any Storm, and the like Occurrences, as you shall think requisite; and what Currents and Variation you meet with. But before all this, put down the Title of the Voyage, over the left-hand Page, in these or such like Words, viz.
A JOURNAL of our Intended VOYAGE by God's Assistance from Kingrode-Port Bristol in Latitude 51 deg. 30 min. to the Island of Madara in Latitude 32 deg. 10 min. and from thence to Barbadoes in Latitude 13 deg. 10 min.
The right side of your Book throughout may be divided into 13 Columns, by Lines, as you may see in the following Example.
In the first must be expressed the day of the Month, in the second the Letter of the Week-day that Year; put it once in the top of the Page: In the third Column the Months; make him large enough to put down the Latitudes you make by Observation of the Sun or Stars, and Currents, and how they set: In the fourth, the Course steered by the Compass: In the fifth, the Variation of the Compass, if there be any; or else the Variation by Currents, if there be any. Set down the Angle of the Rhomb, it made with the Meridian in the sixth Column; and in the seventh, the Distance sailed in Leagues or Miles: In the eighth, ninth, tenth, and eleventh Columns, set down the Northing, Southing, Easting, and Westing: In the twelfth, the Latitude by Dead Reckoning; and in the thirteenth Column, the Difference of Longitude from the first Meridian, according to Mercator's Chart, or the Arch of a Great Circle, or a Polar Chart or Globe.
A Journal of our Intended Voyage, by God's Assistance, in the Good Ship the Eliz. of B. S. S. Commander, from Kingrode in Latitude 51 d. 30 m. to Madara in Latitude 32 d. 30 m. and from thence to Barbadoes, in Latitude 13 d. 10 m.
March
25
a
Set sail out of Kingroad, in Company with the John bound to Cales, and Ann [...] bound to Virginia; the Wind at E. N. E. thick rainy Weather.
The Journal of our Intended Voyage, by God's Assistance, in the C. (of B.) S. S. Commander, from Lundy, in Latitude 51 d. 20 m. to the Island of Madara, in Latitude 32 d. 30 m. Angle of Position, or Course S. S. W. 1 d. W. Distance 411 Leagues, Meridian Distance 167 Leagues, Difference of Longitude 11
d. 16 m. from Madara to Barbadoes, in Latitude 13 d. 10 m. Angle of Position or Course S. W. 61 d. 14 m. Distance 798 Leagues, Meridian Distance 698 Leagues, Difference of Longitude 41 d. 40 m.
51 deg. 20 min.
Course
5 East
Degr. from the Merid. SW 25d. SW 61 d. 14 m.
Distan.
Diff. La
Diff. La.
M. dep.
M. dep.
51 d. 20′
diff. Lo.
32 deg. 30 min.
SSW
Variation.
411
377
377
167
167
32 d. 30′
11d. 16′
13 deg 10 min.
SWbW half W
798
387 lea.
387 lea.
698
698
13 d. 10′
41 d. 40
Latitude by Observation.
Course by Compass.
Variation of Compass.
Degrees from the Meridian.
Dist. sailed.
North.
South.
East.
West.
Lat. By dead R.
Diff. of Longit.
Degr. Min.
Points.
Degrees.
Degrees.
Leag. 10
Lea. 100
Lea. 100
Lea. 100
Lea. 100
D. M.
D. M.
26
a
Set sa [...]l March 25.
S b W half W
5 deg 30 min. E
SW 22 d. 30 m.
48
44 35
18 37
49 07
27
b
S b W half W
5 deg. 30 min. E
SW 22 d. 30 m.
49
45 27
18 75
46 51
28
c
44 deg. 31 min.
S b W half W
5 deg. 30 min. E
SW 22 d. 30 m.
51
47 12
19 52
44 30
28
Add up the Numbers, the sum is 148
136 74
56 64
04 14
29
d
SSW
2 deg. 45 min. E
SW 25 d. 15 m.
43
38 87
18 38
42 34
30
e
40 deg. 27 min.
SSW
1 degree East
SW 22 d. 30 m
46
42 50
17 60
40 26
31
f
38 deg. 30 min.
SSW
0 degr. East
SW 22 d. 30 m.
39
36 03
14 92
38 38
Add up the Numbers, the sum is 128
117 40
50 90
Correction by Observation 2 9
2 70
01 10
8
The sum corrected is 130 9
120 10
52 00
38 30
07 43
Difference of Latitude, Depart. from first Merid. 278
256 84
108 64
1
G
April, a Current sets E b S.
SW b S
11 d. 30 m. Cur.
SW 22 d. 30 m.
43
43 42
17 99
36 17
2
a
SW b S
11 d. 30 m. Cur.
SW 22 d. 30 m.
45
41 57
17 22
34 12
3
d
32 deg. 30 min.
SW b S
11 d. 30 m. Cur.
SW 22 d. 30 m.
42
38 80
16 07
32 19
Set to th'Eastward by the Current in 3 days 22 Leagues and almost a half E b S by
estimat.
Add up the Numbers, the sum is 134
123 79
51 28
Correction by Observation 3 9
3 66
1 50
11
The sum corrected is 131
120 13
49 78
32 30
10 46
4
c
Madara Island bears West distant. 8.58
8 58
Difference of Latitude, Depart. from first Merid. 418
376 97
167 00
32 30
11 16
24
v
Set sail April 23. from Madara.
SW b S
5 d. 30 m. East by Current.
SW 28 d. 30 m.
36
31 75
16 97
30 55
25
c
SW b S
2 d. 45 m. East
SW 30 d. 45 m
46
39 46
23 65
28 57
26
d
27 deg. 43 min.
SW b S
SW 33 d. 45 m.
30
24 94
16 67
27 42
27
e
25 deg. 54 min.
SW
SW 45 degr.
51
36 06
36 06
25 55
28
f
23 deg. 47 min.
SW
SW 45 degr.
60
42 43
42 43
23 48
29
G
22 deg. 40 min.
SW
SW 45 degr.
28
19 80
19 80
22 47
Numbers added, the sum is 151
194 44
155 58
Correction by Observation 3
2 30
1 90
7
Sum corrected is 154
196 74
157 48
22 40
29
Difference of Latitude, Depart. from first Merid. 572
573 71
324 48
20 47
30
a
SW b W
00 degr.
SW 56 d. 15 m.
45
25 00
37 42
21 25
1
b
May
SW b W
00 degr.
SW 56 d. 15 m.
47
26 11
39 08
20 07
2
c
18 deg. 57 min.
SW b W
00 degr.
SW 56 d. 15 m.
43
23 89
35 75
18 56
3
d
SW b W
00 degr.
SW 56 d. 15 m.
44
24 44
36 58
17 43
4
e
16 deg. 50 min.
SW b W
00 degr.
SW 56 d. 15 m.
39
21 67
32 43
16 37
Add up the Numbers, the sum is 218
121 11
181 26
Correction by Observation 7 8
4 33
6 50
13
Sum corrected is 210 2
116 78
174 76
16 50
4
Difference of Latitude, Depart. from first Merid. 782 2
690 49
499 24
31 06
5
e
16 deg. 14 min.
WSW half W
5 degr. West
SW 67 d. 30 m.
31
11 86
26 64
16 15
6
G
14 deg. 03 min.
WSW half W
5 d. 30 m. West
SW 67 d. 30 m.
46
17 60
42 50
15 22
7
a
WSW half W
5 d. 30 m. West
SW 67 d. 30 m.
33
12 63
30 49
14 44
8
b
14 deg. 03 min.
WSW half W
5 d. 30 m. West
SW 67 d. 30 m.
30
11 48
27 72
14 09
Add up the Numbers, the sum is 140
53 57
127 35
Correction by Observation 5 2
2 00
4 80
6
Sum corrected is 145 2
55 57
132 15
14 03
8
Difference of Latitude, Depart. from first Merid. 927 4
746 6
631 39
38 50
9
c
13 deg. 48 min.
W b S half W
5 degr. half
SW 84 d. 20 m.
55
5 39
54 73
13 47
10
d
W b S half W
2 degr. 50 min.
SW 87 d. 11 m.
60
2 94
59 92
13 39
11
e
13 deg. 10 min.
W b S half W
00 deg. 00 min.
SW 78 d. 30 m:
51
9 95
50 02
13 09
Ship is in Lat. Barbadoes 69 lea. Ea.
Add up the Numbers, the sum is 166
18 28
164 67
11
Difference of Latitude, Depart. from first Merid. 1092
We will frame a Reckoning between the three Places before-mentioned, from Lundy to Madara, from thence to Barbadoes, whose Distance in their Rhombs, and Difference of Latitude, and Meridian-distance, I have put over in the head of the left-hand page, as you may see, answers to the words under. And in truth, I have found these Distances very near the truth; In two Voyages I differ but two Leagues, and that I was short. I worked it first out of a Mercator-Chart, and in Plain Sailing took the Product of that Work for my Distance, and Meridian-distance, and Course, as you have been already shewn in the first Question in Mercator-Sailing.
You see by the left-hand Page that we set sail the 25th day; but we entred it not in the right-hand Page until the 26th day at Noon: for it is to be understood,Not that I do affirm the Variation to be Easterly for I know it to be 1 d. 30 m. West; but being neer the truth or not, it serves to exemplifie the Rule, that being
the end for which this Example is made. That since her setting sail March 25. to Noon of the 26th day, the Ship steers away and makes her Way good on the S. b. W. ½ W. Point of the Compass; but the Variation being 5 deg. ½ or half a Point to the Eastward, as you see in the fifth Column, therefore the Point she hath made good upon is only S. W. 22 deg. 30 min. as is expressed in the sixth Column: Upon this Rhomb she sails 48 Leagues, as in the seventh Column appears: And answerable thereunto I find in the Traverse-Table before-going, the Southing to be 44 35/1 [...]0 Leagues, or by the Traverse-Scale 44 4/10. Leagues; and the Westing 18 37/100 Leagues by the Traverse-Scale 18 4/10 Leagues, as here in the ninth and eleventh Column appears by the Figures plainly set down. The Figures to the left hand signifie Leagues in this Journal, or Miles; and the two Figures to the right hand signifie the 100 part of a League: The Southing being 44 35/100 Leagues, which is 2 deg. 13 min. nearest; if that be substracted from the Latitude from whence you came, Lundy 51 deg. 20 min. it makes the Latitude the Ship is in at Noon to be 49 deg. 07 min. as appears in the twelfth Column. In the same manner, the second entrance, being the 27th of March, sheweth, that from the 26th day at Noon, to the 27th day at Noon, she made her way
good upon the S. b. W. ½ W. Point of the Compass; but the Variation being 5 deg ½ Easterly, therefore the Angle of the Rhomb which the true Meridian was from the South to the Westward S. W. 22 deg. 30 min. and sailing 49 Leagues, the Southing is 45 27/100 Leagues, and the Westing 18 75/100 Leagues: So the Latitude is now 46 deg. 51 min. So the third Entrance is the 28th day, the Course and Variation the same as before, and the Distance 51 Leagues; the Southing 47 12/100 Leagues, the Westing 19 52/100 Leagues: So the Latitude now is 44 deg. 30 min. You must understand the like manner of working of all the rest. What hath been said
of a Reckoning may suffice; but it is of very good use to set down the Longitude in the last Column, and a Rule how to convert the Easting and Westing, that is, the Leagues or Miles in the East and West Column, into Degrees and Minutes of Longitude. I will give you this General Rule, that you may do it neer enough, without any sensible Error, on your Mercator Chart, or Polar Chart or Globe, provided these Rhombs differ not much one from another; by which Rule I found the Longitude for every Sum in the Journal. Say then,
As the Difference of Latitude,
To the Departure from the Meridian:
So is the Difference of Latitude in Meridional parts,
To the Difference of Longitude in Leagues or Miles.
The Difference of Latitude in the South Column summ'd up (as you must do as often as you have any notable Difference betwixt your observed Latitude and Dead Latitude) is 136 74/100 Leagues; omit the last Figure to the right hand 4/100, and then it will be 136 7/10.
The Departure from the Meridian in the West Column is 56. 64; omit the last Figure, it is 566: So you put them down.But if you sail all on one Course, the Rule is in the third Probl, of Mercator Sailing.
The Meridional parts for the Latitude 51 deg. 22 min. is
12002
The Meridional parts for the Latitude 44 deg. 30 min. is
As Radius 90 deg. To Tang. Rhomb: So difference of Lat. in Merid. parts, To difference of Longitude in Leagues or Miles.
As the Sum of the South Column, or Difference of Latitude 1367
313576
Is to the Sum of the West, or Departure from the Meridian 566
275281
So is the Difference of Latitude in Meridional parts 2043
331026
606307
To the Difference of Longitude in Leagues 84 6/10
292731
Which reduced into Degrees is 4 deg. 14 min. for the 28th of March. So still you must remember to take the Sum of Difference of Latitude and Departure from the first Meridian. There are several other Rules you may see laid down before, for a Parallel-Course of East and West, and other Rules to find the Longitude; as occasion requires, you may make use of them: But this Rule saves you trouble, and comes neer enough in sailing several Courses.
8 min. is2 7/10 Lea. neer1201 to 520so 27:) 273640104014040But let us proceed with our Journal. I observed the Meridian Altitude of the Sun the third day at Noon, that is from 30 at Noon to 31. I find my Latitude by observation 38 deg. 30 min. which, by Dead Reckoning it, is but 38 deg. 38 min. so the Difference is 8 min. Southerly; but being assured of a good Observation, I correct the Dead Reckoning thereby, by this Rule of Proportion, saying,
As the Sum of the North Column corrected is 1201
307954
To the Sum of the East Column corrected 520
271600
So is the aforesaid Increasing Southerly 27
143136
414736
To the Increasing Westerly 1 17/100 Leagues
106782
[...]
Which is 1 League 1/10, and something more, not to be taken notice of. This Rule of Proportion Mr. Norwood hath laid down in page 111. of his Seaman's Practice, in the Description of his Journal in Miles, from Barmoodoes or Summer Islands to the Lizard; which method I do in many things follow, but not all: But this Rule that I propose is by the Traverse-Scale, which I hold best, which is thus.
By the Traverse-Scale.
Extend the Compasses from the Point made good in the last summing up, to the number of Leagues or Miles Difference of Latitude by Observation, and by Dead Reckoning, in the Line of Numbers; the same Distance will reach from some Points from the East and West, to the Difference of East or West.
As for Example.
Extend the Compasses from 2 Points and a little more (which was the Sum of the Course made good the 31 of March) unto 2 7/10 Leagues, which is 8 min. or thereabouts in the Line of Numbers; the same Extent will reach from 6 Points to 1 1/10 Leagues and something more in the Line of Numbers, and that is the increasing Westerly. You may also with the same Extent correct the Distance, if you put one Foot at W or 100 in the Line of Numbers, the other will reach to the Distance 2 9/10 Leagues corrected by Observation, as you see I have done in the Journal. So you see, That understanding perfectly the Use of the Traverse-Scale, you may do the same, and more readily, as Mr. Norwood doth with his Table, to every Degree and Minute of the Quadrant, without sensible Error.
Now this Difference being found, I add therefore and put down in the South Column the Difference 2 70/100 Leagues, and the West Column 1 16/100 Leagues, and under Distance 2 9/10 Leagues: Now the same corrected is by observation 130 9/10 Leagues, Distance 120 10/100 Leag. Southing and Westing 52 Leag. 8 min. substracted from the Dead Latitude, make 38 deg. 30 min. the true corrected Latitude according to observation: [Page 205] Then I sum up the first Sums of the 28 of March, and this Sum corrected 31 of March together, and you have the Distance 278 Leagues,This is because I find my self to the Northward, that is, less to the Southward by
11 m. than by Dead Reckoning, & therefore less to the Westward 1 50/100 Leag. or so much to the Eastward, by reason my Course is in the S. W. Quarter, it
must be corrected in the contrary Quarter.Difference of Latitude 256 84/100 Leagues, and Departure 108 64/100 Leagues, and by the Rules before-given 256 8/10 Leagues Southing, and 108 6/10 Leagues Westing; and with the Difference in Meridional Leagues 364 7/10 Leag. I find the Difference of Longitude in Leagues 154 [...]/10 Leagues, converted into deg. and min. is 7 deg. 43 min.
In like manner, upon the third of April I should be in Latitude 32 deg. 19 min. but by very good observation, I find the Ship in the Latitude 32 deg. 30 min. that is, not so much Southerly by 11 minutes: therefore to correct it by Observation, I put under Distance 3 9/10 Leagues, and in the South Column 3 66/100 Leagues, and in the East 1 50/100 Leagues, and under Dead Latitude 11 min. I substract the corrected Difference of Distance out of the Sum over it, and likewise the corrected Difference in the North Column out of the Sum in the South, and likewise the East out of the West Column, and add the 11 min. to the Dead Latitude, and then you have the Sum corrected; but if there be any Current, you may set it down, and allow for it, and note it down, as is that Example following the first of April to the third, and by your Traverse-Scale presently find how much the Current hath set you to the Eastward.
But if your Course be neer the East and West,In sailing East and West, you have a Rule in Probl. 7. of sailing by Mercator or Mr. Wright's Chart. it is sufficient to correct it in Latitude only, as in the Example of the 12th and 13th of May; for in that Case the Longitude cannot be corrected but from some further ground. Now to set down this Reckoning upon the Plain Chart, or common Sea-Chart, it is needless and unnecessary: The better way is to set down every one of the Sums as they are corrected by Observation, in the same manner as you are directed in
the latter end of the third Chapter of this Book; and so by the total Sums of the Difference of Latitude and Departure from the first Meridian, or Latitude and Meridian-distance, you may set it down on your Draught or Chart as often as you please with ease.
Now to set off every Sum corrected in Degrees of Latitude and Leagues of Longitude, you have a Scale of Leagues or Miles for that very purpose, and Directions how to do it, in the ninth Chapter of this Book: But if you are desirous to set down your Reckoning in a Mercator or Mr. Wright's Chart, on in the Polar Chart, you have in the 12th and 13th, or last two Columns of your Journal, the substance and principal scope of your Reckoning set down as often as you sum up or correct your Reckoning: namely, your Latitude and Longitude; which whensoever you have a desire to set down in the foresaid Chart, or any other graduated Chart, with Degrees of Longitude and Latitude, you may readily do it.
As for Example. Suppose I would set down the Plat of the aforesaid Journal from the 25th of March to the 13th of May, I find the Latitude against the 25th of March 51 deg. 20 min. and the Latitude of the Barbadoes 13 deg. 10 min. and the Difference of Longitude 52 deg. 35 min. Therefore in the Latitude of 13 deg. 10 min. I draw or point out an occult Parallel, and reckon 52 deg. 35 min. from the Island Lundy towards the West: I draw by that Longitude an occult Meridian;I hope this way will find good acceptation with the ingenious Mariner or Artist. the Intersection of this Meridian with the foresaid Parallel is the Point representing Barbadoes, or the Place of the Ship; and the like is to be understood of any of the other: And so I put down in the General Chart of Mercator the 8 Points of the Ship's Place, 1 a, 2 b, 3 c, 4 d, 5 e, 6 f, 7 g, 8 h, as there you may see. This form of
keeping a Reckoning is the most fit and agreeable of all others as I have seen or heard of, to all sorts
of Charts, Maps, or the Globe it self, and to all kinds and ways of Sailing whatsoever.
CHAP. XVIII. A Description of the following Table of the Latitude and Longitude of Places, and the way how to find both.
THE ancient Geographers, from Ptolomy's time downward, reckoned the Longitude of Places from the Meridian, which passeth through the Cabo Verde Islands; and others have the beginning at the Canary Islands; and Jodocus Hondius beginneth at the Isle Pico one of the Azores; and Mr. Emery Mullineux doth account the Longitude from the Westermost parts of St. Michael's, another Island of the Azores: who, albeit they differ greatly in respect of the beginning of each of their several
Longitudes, they come all to a neer agreement for their Difference of Longitude from any particular Meridian or Place: And for the exact setling of Latitudes, we have many certain helps; but the Longitude of Meridians hath still wearied the most able Masters of Geography. By Latitude and Longitude the Geographers strive to represent the Parts of the Earth, that they may keep Symmetry and Harmony
with the Whole.
Now the Longitude of any Place is defined to be an Arch or Portion of the Aequinoctial, intercepted between the Meridian of any Place assigned; as the Meridian that passeth through the Lizard, the most Southern Land of England, or any other Place from whence the Longitude of Places is wont to be determined. Many have endeavoured to set down divers ways
how to find by observation the Difference of Longitude of Places; but the most certain way of all for this purpose, is confessed by all
Learned Writers to be by the Eclipses of the Moon: But now these Eclipses happen but seldom, and are yet more seldom and in very few places observed by the
skilful Artist in this Science; so that (some there are) but very few Longitudes of Places designed out by these means.
If you would know how to find out the Longitude of any Place by the Eclipse of the Moon, you must first get some Ephemerides, as the Practick Tables, or Mr. Vincent Wing's Directions in his Harmonicon Coeleste, pag. 150. or any other Learned Mathematicians Calculation, and see what hour such an Eclipse of the Moon shall happen at that Place for which the said Tables were made; then afterwards you must observe the same Eclipse in that Place whose Longitude you desire to know. Now if the time of the Eclipse agree with that other for which the Tables were made, then you may conclude, that both Places have the same Longitude, and are scituated under the same Meridian. But if the number of the Hours be more than the Place you are in is scituate Eastward, you must therefore substract the less Number out of the greater, and the Remainer must be converted into Degrees and Minutes.
A TABLE OF THE LONGITƲDE and LATITƲDE Of the most Notable Places, That is, HARBOURS, HEAD-LANDS, and ISLANDS OF THE WORLD.
Newly Corrected, and Composed after a new manner, by beginning the said Longitude at the Meridian of the most Southern Port of England, the Lizard.
By Capt. SAMUEL STURMY Math.
The Sea-Coast of Newfound-Land, and New-England.
The Places Names.
North Latitude
West Longit.
D.
M.
D.
M.
Cape Honblanto
52
11
50
14
Belile
51
02
48
44
Cape Bonavista
49
19
47
42
Trinity Bay
48
54
49
04
Bacalao Island
48
40
46
55
Consumption Bay
48
21
47
49
Cape St. Francis
48
01
47
27
Cape Daspaire
47
36
46
03
Cape de Raca
46
27
46
30
Bay Bulls
47
28
47
11
St. John's Harbour
47
47
47
21
Plasantia Bay
47
32
47
41
Cape St. Larinso
47
10
48
59
Island St. Paly
47
36
50
18
Cape Raya
48
05
52
49
Cape Deganica
54
01
53
21
New-Eng. Cape S. Charles
52
48
52
23
Cape Brittan
46
01
52
57
Cape Salila
43
46
55
22
Cape Codde
42
21
61
32
Boston
42
39
64
36
Plymouth
42
07
62
35
Nantucket
41
08
60
17
Martins Tinyard
41
17
61
12
The Sea-Coasts on the main Continent in America, or West-India.
The Places Names.
Latitud.
Longit.
D.
M.
D.
M.
Elizabeth Island
41
02
62
04
Bloik Island
40
55
62
36
Long Island
40
45
63
16
Cape May
39
55
64
45
Virginia. Cape Charles
37
47
65
26
Cape Henry
37
01
65
38
Cape Hatcrass
35
49
65
46
Cape Fara
34
06
69
56
Cape de Catocha
21
23
80
37
Cape de Camaron
16
05
76
19
Cape de Gratias
15
31
70
56
Cattergaine
10
24
65
06
Bay Tonto
12
10
62
36
Cape St. Roman
11
55
60
36
Cape Dacodara
11
08
56
38
Cape Trag, or 3 Points
11
17
55
41
Cape Brama
09
21
54
16
Cape Dasbassas
08
20
53
11
Suramo
06
09
50
56
Suranam
05
58
49
52
The West-India Islands.
The Places Names.
North Latitud.
West Longit.
D.
M.
D.
M.
Defonsseaca Island
12
23
48
30
La Burmuda
32
25
56
00
Behama
27
57
73
06
Teavis
27
27
71
04
Sigvatro
26
18
68
45
Guatro
25
47
68
00
Guamina
25
15
67
53
Tiango
24
33
66
30
Guanahimo
23
50
66
39
Mayagnana
23
05
66
51
Caycoss
22
05
64
31
Amiana
21
40
64
38
Inagua
21
19
67
03
Yamatta
22
32
67
49
Soamia
24
20
68
50
Javaqua
25
10
71
30
Yamia
24
22
70
10
St. John
18
30
60
42
Santa Cruce
17
42
59
18
Anguilla
18
48
57
00
St. Martin
18
35
56
47
St. Bartolama
18
15
56
33
Barbada
17
18
55
39
Antego
16
32
54
52
Dassijada
16
00
54
36
Marigallatita
15
41
55
26
Dominica
15
00
55
05
Mattalina
14
20
54
44
St. Lucia
13
30
54
43
Barbadus
13
10
52
58
Tobago
11
12
53
06
Point Degallaia
10
45
53
31
Gianada
12
10
54
32
St. Vincent
12
50
54
28
Guardadupa
16
00
55
31
Monsariat
16
20
55
41
Maves
17
00
56
27
St. Cristova
17
30
56
45
Island Devas
15
57
57
28
Island Blanco
12
20
56
52
Margaita
11
28
56
37
Turtuga
11
30
57
40
Island Derickilla
12
19
58
01
Boca
12
19
58
53
Island Deavos
12
29
59
22
Bonoga
12
32
60
54
Quissa
12
25
60
39
Moagos
12
20
61
55
East end of Hispaniola
18
47
62
28
Middle of Hispaniola
18
30
64
58
West end of Hispaniola
18
25
68
26
East end of Jamaica
18
00
71
58
Jamaica Harbour
18
15
72
57
West end of Jamaica
18
38
74
57
The East end of Cuba
22
00
75
56
Caimanis
19
41
77
41
Grand Caiman
19
21
78
45
Santavilla
17
28
77
50
Mosquito
14
50
76
04
Guanabo
16
33
81
19
Guanabimo
16
10
83
04
Cozumal
19
25
84
56
Lasalleerauas
22
00
87
58
The Island Delas
23
30
91
58
Abraio
25
50
94
00
Labarmaia
22
55
93
16
Island Dearanas
22
36
93
14
Triango
21
23
93
05
Zarka
20
50
93
00
The Island of Proudanco
13
27
81
16
St. Andrea
12
42
80
57
The Sea-coasts of Brazilia.
The Places Names.
South Latitud.
West Longit.
D.
M.
D.
M.
The River Amazones
00
00
41
30
The Island of St. Paul
00
55
14
36
The Island of Ascension
07
48
20
06
Cape Blanco
02
25
22
29
Island Rocas
03
42
17
16
Island Farnando
03
40
15
16
Abratho
05
00
17
56
Cape St. Raphall
06
10
19
36
Cape St. Augustin
08
25
18
28
River St. Mignall
09
30
19
01
The River Roall
11
21
20
41
River Gianda
14
49
22
06
Cape de Abeotho
17
52
21
42
St. Harbara
18
11
21
06
Island Ascension
17
19
17
01
Trinidada
19
50
14
24
St. Maria Dagasta
19
38
12
14
Island de Martin
19
00
08
03
Island de Pidos
21
52
05
51
Cape St. Toma
21
47
23
38
Cape Frio
22
52
24
43
Cape St. Maria
35
00
37
11
River de Platta
35
50
45
52
Port St. Juliano
50
00
52
30
The Streights of Magellane
53
30
56
30
Cape de Sancto Spirito
52
20
58
30
Cape Victoria, West end of the Streights
52
30
65
40
Lima Cape
12
00
80
30
Cape Guya, Cape Blancoo
06
10
85
30
Cape St. Frainsco
01
30
80
30
Cape St. Frances
North.
West
Point de bon matre
07
30
80
00
Nombre de Dios. W. Sea
10
00
77
30
Nova Albion, or New-England, in the South Sea, the back side of it
46
00
162
30
Cape de Fortuna (Aniar. fra.
55
30
170
E 00
Insulae Salamonis
So. Lat.
W. Long.
Nombre de Jesus
05
50
169
30
Tapan Insule
36
N. 00
153
E 00
Cape de Buena Desco
01
S. 00
155
00
The East India Islands.
The Places Names.
South Latitud.
East Longit.
D.
M.
D.
M.
Hipon Island
06
45
125
20
Bantam, East-India
06
15
115
34
Jamba Islands
01
49
122
25
South end of Sumatra
05
52
125
48
Middle end of Sumatra
01
30
120
49
North end of Sumatra
05
28
116
35
Gomaspala
05
40
116
29
Niobar
07
00
115
04
Island Desombro
08
00
114
44
Island Rusta
09
50
114
33
Quarinibar
11
10
114
44
Ghitra Andomaio
12
00
114
41
Island Dandemajo
13
00
114
39
Island Decocoss
14
30
115
12
Celloan
07
50
98
39
Doda Safia
09
40
93
02
Andaio
11
30
90
51
Garine
10
50
90
56
Moique
09
05
91
29
Cuballa
08
53
91
07
Island de Prosoll
10
23
90
55
Island de Zocha
11
12
90
45
Chorebaman
12
32
87
55
Sucatra
12
18
74
01
Abdelcari
12
12
71
44
Apoluria
09
S. 20
90
50
Adu
05
39
88
50
Degomo
02
40
87
35
Piedros Blanco
06
10
76
55
Diego gratiosa
08
30
78
15
Set Hermanas
03
02
70
25
Domes Caicuhas
03
21
65
24
Island Quellallo
03
40
64
20
De Almiranta
03
57
63
28
Agnalaga
09
00
66
15
Asdore Has
09
05
65
02
John de Nava
09
00
63
26
Cosmobodo
09
40
61
52
Donatall
08
20
59
57
Aignos
09
30
58
18
John de Comoro
09
00
57
20
Pemba
05
09
53
30
Zanziba
06
26
53
35
Mansia
07
50
53
08
John Demiz
10
48
54
24
Comoro
11
20
55
48
Mohalla
12
11
56
25
Foanna
12
09
57
03
Mayatta
12
40
57
55
St. Christopher
14
30
56
03
John de Nova
17
20
55
29
Baslas de India
22
10
55
22
North end of St. Laurence
25
37
60
54
St. Apohima
20
50
65
54
Domscascahas
20
50
66
54
Moroslass
20
10
68
44
Dosgarias
15
20
70
43
St. Branda
17
13
74
44
Englands Forest
20
50
71
14
Diego Roize
20
05
74
54
John de Lisbone
25
24
68
32
Romoras
28
19
81
21
The Sea-coast on the Main Continent in the East India.
The Places Names.
North Latitud.
East Longit.
D.
M.
D.
M.
Malacca
01
41
116
14
Queda
06
47
117
44
River de Care
10
45
118
54
River Bongale
22
09
121
33
Aicopoir
20
19
112
39
Samnabron
18
30
108
52
Arme Gon
14
35
100
27
Naga Patam
11
21
99
59
Cape Comorin
07
50
97
39
Cochin
09
40
97
29
Callant
10
48
97
27
Mongalar
12
40
97
19
Dodall
17
01
98
55
Goa
14
40
97
01
Chaul
18
10
98
51
Calecut in East India
11
30
92
58
Macao in the K. of Pegu
19
30
112
49
Domon
19
54
99
01
Surrat
21
00
99
36
Dio
20
48
96
57
River Decinda
24
55
95
39
Gudar
24
50
89
28
Cape Muchoaridan
25
32
82
39
Cape Russallgat
22
07
84
39
Cape de Ponto
18
19
79
09
Dofar
17
00
75
04
Cape de Matriaia
15
33
72
39
Adon
13
08
66
56
Cape Guardafuy
11
40
71
24
Cape de Baslos
04
30
65
19
Magadox
02
30
59
24
Molinda
02
S. 42
52
11
Tanga
05
20
52
01
Cape Faslto
08
02
52
24
Dagnada
15
17
53
26
Cape Corintes
23
30
48
51
Cape St. Marin
25
40
46
59
River St. Lussea
28
25
46
09
Bay Doliagoa
33
18
43
59
The Sea-coast from Cape Bone Esprance to Guiney.
The Places Names.
South Latitud.
East Longit.
D.
M.
D.
M.
Island Desistian
36
57
11
44
Island Degiaiatica
37
56
14
04
Cape Agullas
36
20
33
54
Cape Bonee sprance
35
50
32
54
Cape Sacos
29
40
30
14
Ascention Island
07
48
05
24
St. Elana
16
03
10
08
St. Elana Nova
16
03
19
48
Bassas
17
45
27
35
Cape Lado
10
00
29
23
Cape Padron
06
00
29
04
Cape Lopas
01
00
25
21
Anabona Island
01
22
22
56
Island St. Mathaos
01
40
07
45
Island St. Toma
00
N. 10
23
34
Island Chocos
00
40
23
50
River Gaboan
00
10
27
16
River de Angai
01
00
27
30
Island de Principas
01
50
25
14
Island Defarnanda
03
10
26
06
River Boilin
02
42
27
29
River Decainaronas
04
00
27
09
The Sea-coast from Samsons River to the River of Gambo, Coast Guiney and Barbary.
Where the Table is begun, on the Coast of Terra Nova.
By multiplying the Hours by 15, and dividing the Minutes of Hours, if there be any, by 4, so will the number of Degrees arise; and if there remain any Minutes after the Division, they must be multiplied again by 15, and so will the number of Minutes of Degrees arise, by which these Places are distant from each other, which Distance is called the Difference of Longitude of that Place for which the Tables were calculated, if the other Place be Eastward of the first; but if it be more Westward, it is to be substracted from the Longitude of the other.
And this is the way we have endeavoured to settle the Longitude, with as much neerness to the truth as possible we could. I have not only made use
of my own Calculation of the Difference of Meridians of Places, as I have often used at Barbadoes and Virginia, or any other Place, from the Meridian of the Lizard; but I have also obtained them from the best Geographical Charts that are yet discovered, and the latest Tables made; and so by consulting with the able and skilful Mariners, that have used the East and West India; by the first we have informed our selves for the setling ehe Longitude of Places in the East India, with the best approved Authors: as in page 161 of Harmonicon Coeleste we find the Difference of Meridians betwixt Calicut in East-India and London to be 5 hours and 50 min. which being converted into Degrees and Minutes as before directed, is 37 deg. 30 min. the Difference of the Meridian of London and Calicut; and the Difference of the Meridian of London and the Lizard,[Page 218] 5 deg. 24 min. added to it, gives the Difference of the Meridian of Calicut and the Lizard, it makes 92 deg. 54 min. the Difference of Longitude to the Eastward of the Meridian of the Lizard.
And Macao in the Kingdom of Pegu, whose Difference of Meridians with the City of London is 7 ho. 9 min. which is 107 deg. 15 min. the former Difference added makes Macao to the Eastward of the Lizard 112 deg. 39 min. The Difference betwixt the Tables before-going, and the Eclipses, in the Difference of Meridians of Calicut and London, is very small, the Tables 4 min. more; and the Difference between the Eclipses and the Tables is 22 min. more. Then the Observation of Macao and London, being so small, it may very well be born withal: And we have setled the Longitude of the West India, according to long and approved Experience of Voyages of my self and others, from the Lizard to Barbadoes, and to Cape Henry and Charles the Capes of Virginia.
The Latitude of a Place is the Distance of the Zenith, or the Vertical Point thereof from the Aequator, or the Height of the Pole elevated above the Horizon. You have been shewed several ways already, for the finding the Poles Elevation above the Horizon. but this Rule will not be impertinent to this Place, being not named before, which is by the Stars thus.
You must observe some Fixed Star in the Heavens, which is neer the Pole, and that never sets in that Region: Thus, you must observe the least and also the greatest Altitude of the said Star, when he doth come to the Meridian under the Pole, and also above the Pole; which done, you must add the least Altitude to the greatest, and so the half of the deg. and min. thus numbred together, will be the Elevation of the Role, or Latitude of the Place.
An Example whereof may be this. The first Star of the three in the Tail of the Great Bear, in his least Altitude, observed at Bristol, is about 10 deg. 59 min. and the greatest Altitude of the same, when he is above the Pole, is found to be neerest 91 deg. 59 min. both which Numbers being added together, do make 102 deg. 58 min. the half of that same is 51 deg. 29 min. the true Latitude or Elevation of the Pole.
You may take notice, I begin the Longitude at the Meridian of the most Southern Parts of England at the Lizard, and increases on each side of that Meridian, from 1 deg. to 180 deg. both Eastward and Westward,; therefore you must note, That by these Tables all Places, that lie to the Eastward of the Meridian of the Lizard, are called East Longitude; and all Places on the West side of the Meridian of the Lizard, is called West Longitude.
Therefore a Ship being in East Longitude, sailing to the Eastward, she increaseth her Longitude; but sailing to the Westward, it decreaseth. And likewise if a Ship be to the Westward of the Lizard, that is, in West Longitude, and saileth to the Westward, the Longitude increaseth; but sailing to the Eastward, the Longitude decreaseth.
You must note, the Sun riseth to the Eastward, therefore all the Stars, and are carried West; and that all Places that are to the Eastward of the Meridian of the Lizard, the Sun comes to their Meridian first, according the time it is to the Eastward of the Meridian of the Lizard: As you may note what was before directed, That every 15 deg. is an Hour, and 4 min. a Degree: Therefore in the former Example of Calicut, whose Difference of Meridians is 5 ho. 50 min. that is to say, the Sun is on the Meridian in the East India at Calicut at 10 min. past 6 a Clock in the morning here at the Lizard, that is, 5 ho. 50 m. sooner than he comes to the Meridian of the Lizard, to make here 12 a Clock at Noon. And so on the contrary, lesser to the West by every 15 deg. As for Example.
The Difference of Longitude betwixt the Meridian of the Lizard and Barbadoes is 52 deg. 58 min. that converted into Time is 3 hours 42 min. the time the Sun comes to the Meridian of the Lizard, before it comes to the Meridian of Barbadoes; that is to say, it is our 3 a Clock 42 min. past at the Lizard in the afternoon, before it is 12 at Noon in the Barbadoes.
You may take notice, I took my first Latitude and Longitude from the Northern parts of Newfound-land, to the Westward at Cape Homblanto, neer Bell-ile, and so have coasted all round the Bay of Mexico, and taken the West India Islands in the way, [Page 219] and so round the Coast by Brazil, and through the Straights of Magellane to Nova Albion, where Sir Francis Drake was on the back side of New-England, in the South Sea; and from thence to the East India, first the Islands, and then the Main Land, and back by Cape Bon Esprance, and round the Coast of Guinney and Barbary down from Tangire, and upon the Christian Shore to Giblitore and Tocke in the Canary Islands and Westward Islands, and so along the Coast from Cales to Callis, and from the Lizard to Newcastle, and from thence along the Coast of Scotland to Skey Island, and along the Coast from Calis to the Scaw, and along the Coast from the Lizard to the Isle of Man, and round the Coast of Ireland to the Sea-coast of Iseland, and so from the Scaw round the Sound, by the Nase of Norway to Archangel, and about by the Sea-coast of Greenland by the North-west Discovery, to the Coast of New-found Land, where first I began; whereby you may see I have traced a Path, or coasted round to
the most Chief Harbours, Head-lands, and Islands in the World, by the Tables. And so I shall conclude with these Verses in Mr. Philips's Preface.
STƲRMY's MATHEMATICAL AND Practical Arts. The Fifth BOOK. SHEWING A new Way and ART of SƲRVEYING of LAND by the MARINERS AZIMƲTH or AMPLITƲDE-COMPASS; By which you may SƲRVEY and PLOTT with ease and Delight, all manner of Grounds, either small Inclosures, Champions, Plains, Wood Lands, or any other Uneven Grounds.
AND ALSO, How to take the Plott of a whole Town; and a most Excellent way to be satisfied whether his Plott will Close, before he begins to Protract the same. ALSO, The ART of GAGEING all sorts of Vessels, as Cube-Vessels; or to Measure Square Vessels, as Cylinder
Vessels, and Pipes, Hogsheads, and Barrels: and to Measure Vessels that are part out;
And also to Measure Brewers Tuns, or Oval Tuns, or Mash-Fats, or Cone-Vessels, or
Brewers Coppers, or any other Vessels. AND LIKEWISE, How to Measure exactly all kind of Plain Superficies, as Walls, Timber Work, Roofs
of Houses, Tyling, Board, and Glass, and Wainscot, Pavement, and the like; As also
Timber and Stone; And of Measuring of SHIPS. The ART of GƲNNERY, On a new invented Scale, which resolves most Questions in a moment in that ART in a most Excellent Compendious Form, never by any set forth in the like manner before
in the ART of GƲNNERY: With divers Excellent Conclusions, all resolved, both Arithmetical and Geometrical and Instrumental and by Tables; Being framed both with, and without the help of Arithmetick; As also divers sorts of Artificial FIRE-WORKS, both for Recreation, and Sea and Land Service. By Capt.SAMƲEL STƲRMY.
LONDON, Printed by William Godbid, Anno Dom. M.DC.LXIX.
The ART of Surveying of Land By the SEA-COMPASS: The DESCRIPTION of the COMPASS, and STAFF, and CHAIN. The Fifth Book.
CHAP. I.
I Have been all this while a shewing the Mariner, How to describe and make his own Instruments, and the use thereof in Navigation; I am also willing to shew him the great use there may be made of his Sea-Compass, commonly called the Azimuth, or Amplitude-Compass, which all ingenious Mariners carry to Sea.
This Compass requires but little description, it being so well known to all Sea-men; for it is
the same in a manner as they Steer the Ship by: But it is called by the name of a
Meridian Compass. The Chart within the Box is divided as you see in this Figure, each quarter into 90 Degrees,
beginning at North and South, numbred East and Westward; on the Glass there is a
Brass Circle, and Diameter, that goes over the Center of the Compass-Chart, the Brass Circle is about 7 ½ Inches Diameter, and about 6/10 of an inch broad; The
outward Circle is divided into 360 Degrees by 90 Degrees in each Quarter, as you see
the former was; the Figure makes all plain to the meanest capacity, and numbred as
you there see from 15 Degree each way from each opposite Point; The inward Circle
is the Hours answering each 15 Degrees and Quarters of the Horizon, and they are numbred as you see in the Figure.
There is a Circle the like divide the high of the Needle and Box-Chart, with lines drawn up the Box at 90 Degrees every way, that the Degrees of the upper
Circle, and lower Circle, and Chart, may agree.
In the Diameter FGHK there is a right Line drawn in the midst, as GH, and at each
end is two slits of an inch and an half long, each of them, as FG and HK; which are
cut right in the middle: by which in taking any Angle, you must be sure to set the
North Point right under the slit and line of the one; and the South Point under the
slit and Line of the other: and so must you always, when you take the Angle of any
[Page 2] two stations from one place to another, You must be sure to keep the two slits in
the Brass Diameter over the North and South Point of the Chart; and turn the Index that is riveted to the Center at C to the Object, and look through the sights that
stand upon it; when you find you see it plainly, and have made a good observation
of the Angle, look what Degrees the edge of the Index cuts, and upon what Quarter of the Compass: and that number of Degrees is the Angle of the two places from the Meridian.
The sights that stand perpendicular on the Index are 1 inch and ¾ long; the further sight hath a wire that goeth through the midst
thereof, by which we cut the Object: that sight next unto you hath only a slit.
Through which you must see the Wire and Object you look at in one and the same Line,
when you make any Observation.
Betwixt the two sights is a right line drawn through the midst, and at the further
sight is fastened a perpendicular of Brass with a right line through the midst as
BD: this perpendicular is fastned with two small Brass screws at M: O to the further
sight with a wire; and at D is a hole where is fastned a Silk thred twisted and screwed
through a small hole in the Eye-sight at S, and fastned with a small wooden pin.
The Use of the Azimuth-Compass.And this is for to take the Sun's Azimuth at any time of the day, by turning the Eye-sight to the Sun; and the slits over the
North and South point of the Chart (as before directed) you may set the Index to what Degree you please; and when the shadow made by the Thred DS, comes upon the
Line in the midst of the Index on the Line SA, and on the perpendicular Line RD; then on that instant take the Sun's
Altitude by a Quadrant or Staff, and note it down: and likewise the Degrees cut by the Index at the perpendicular end, and that is the Sun's Magnetical Azimuth at that time. When you have done,The Amplitude Compass. you may unscrew the perpendicular from the sight; and then you have the Compass ready to take an Amplitude of the Sun's Rising or Setting: but more of that in the
following Treatise, when we shall touch upon Astronomy
When you make use of the Compass for Surveying of Land, you have a Brass socket screwed fast to the bottom of the
inward Box that holds the Chard; In that socket you put the head of your three legged Surveying-Staff with a small screw on the side to fasten it to the head, that it may not stir when
the Compass is set North and South as before directed; then you may turn the Index and sights to what object you please, and be sure of your Angle from the Meridian
if your Chard be good, and the Needle well touched and placed. Those that make them should have
a special care of that; and that the points of the Needle be fastned and cemented
together with a little Tin,How the Need [...]s of the Compass must be set. so that they do not stir abroad, as I have seen many Charts carelesly made, doth; It might be to the shame of them that make them; and likewise
the Wires put off one side 5 Degr. more or less, as if in all places there were still
½ a Point-variation, which is a lazy trick as well as faulty in most places. I would
advise all Ingenious Mariners to make a constant practice of taking observation of the Sun's Altitude or Azimuth; and Steer a Course, and make allowance accordingly, as hath been shewed elsewhere,
with the Wires of the Needle put exactly under the Meridian, as this Compass before-going the Points are; and then in all things this Instrument will come to the Truth, as well as a Needle of greater charge, and Plain Table and their appurtenances of 3 l. price: or the Theodolite, and Circumferenter and Veracter. And yet I cannot but highly commend these Instruments as very useful for Land-men which have Money enough. Neither dare I reject as useless,Of the Devices of Instruments. either the Topographical Instrument and Cross-Staff of Mr. Diggs, the Familiar Staff of Mr. John Blagrave, the Geodetical Staff, and Topographical Glass of Mr. Arthur Hopton, the Sector Cross-Staff, and the Pandoron of Mr. George Atwel, or any other witty Invention which hath been devised for the Exact Plotting, and
Speedy Measuration of all manner of Superficies, as Land, and the like. But in regard the Authors have in their own Works to their exceeding
Commendation described the Making and Use of the said Instruments, I shall say no more.
And for the Mariners Compass in a manner to do the same things for the Surveying of Land, or Plantations, or the like, I hope will be well taken and accepted of all Ingenious Mariners, for whose sake I take these pains.
Let the Glass over the Chard be as clear as possibly you can get him.
The Figure of the Staff is plain, it needs no further description: It is to be had at any Instrument-Makers.The Staff.
[depiction of a three-legged surveying-staff]
Of Chains, the several sorts thereof.
Of Chains there are several sorts, as namely Foot-Chains, each Link containing a Foot or 12 Inches; so the whole Pole or Perch will contain 16 ½ Links or Feet according to the Statute Pole.
The Chains now used and in most esteem among Surveyors are Three. The First I will name is Mr. Rathborn's, which had every Perch divided into 100 Links; and that of Mr. Gunter's which had 4 Perches or Poles divided into 100 Links: so that each Link of Mr. Gunter's Chain is as long as Four of Mr. Rathborn's. And this year Mr. Wing hath described a Chain of 20 Links in a Perch for the more ready use thereof in his Art of Surveying; Therefore when we have taken the Angles, and Plotted a piece of Ground, we will shew how to know the contents thereof in Acres, Roods, and Perch by the two last Chains.
SECT. I. Mr. Gunter's Chain.
MAster Gunter's Chain is a Chain most used amongst the Surveyors of this Age, and is always made to contain 4 Poles, and each Pole 25 Links, and each Link 7 Inches 92/100 of an Inch, and each Pole according to the Statute contain 16 ½ Feet, the whole Chain is 100 Links in the Four Pole or 66 Feet. In measuring with this Chain you are to take notice of only Chains and Links; saying, such a line measured by the Chain contains 64 Chains 45 Links, or thus distinguished 64. 45. and this is all you take notice of in Surveying of Land.
Now for the ready counting of the Links; at every Perch let there be two Curtain-Rings fastned, and one Ring at every 5 Links: so you may readily count the Rings at either end. If the Ingenious Mariner wants a Chain; he may mark a six Thred-line or small Belch as before directed with Red Cloth marks
and White for distinction, or bitts of Leather as we mark our Dipsey line; and be sure to stretch him well first; or if you can, let it be a Top-gallant Brace half worn; then measure them exactly: and mark him as before directed, and you may
measure any place of Land or Plantation, or any distance, as well in dry weather, as with a Chain, without sensible Errour,
SECT. II. Cautions to be used, and to be observed in the use of any Chain.
VVHen you have occasion to measure large distances, or otherwise, you may by chance
mistake or miss a Chain or two in keeping your account, which will breed a considerable Errour; and also
in measuring of distances, in going along by a Hedge side you can hardly keep your
Instrument-Chain & mark in a right line; and therefore the distance will be more than in reality
it is. For avoiding these mistakes you ought to provide ten small sticks, which let
him that leadeth the Chain, carry in his Hand before; and at the end of eve [...]y one of those Chains, stick one of these Sticks or Arrows into the [Page 4] Ground, which let him that followeth take up; so going on until the whole number
of Sticks be spent, and then you may conclude you have measured Ten Chains without further trouble: and these Ten Chains if the distance be large, you call a Change, and so you may denominate every large distance by Changes, Chains and Links in a piece of Paper you keep the account by. If the distance be far, you must set
up a Cloth upon a Stick for a mark betwixt your Instrument and the further mark, and see through your Instrument both the marks in one; then you may be sure to go straight with the Chain.
SECT. III. How to reduce any Number of Chains and Links into Feet and Yards.
IN taking of Heights and Distances hereafter taught, it is necessary in the Practice
of my Geometrical Conclusions to give your measure, in such cases, in Feet and Yards by reducing of your Chains and Links thus.
Multiply your Numbers of Chains and Links, as one whole Number by 66, cutting off the Product the two last Figures towards the
right Hand; so shall the Figures to the left Hand be Feet, and the Figures cut off
shall be 100 parts of a Foot.
5:32
Examples.
8:06
66
66
3192
4836
3192
4836
351,12
531,96
[...]
Let it be required to know how many Feet are contained in 5 Chains, 32 Links. Set down the Chains and Links with a Comma (:) thus and these Multiplyed by 66, the sum will be 351 Feet and 12/100 parts of
a Foot as thus you see it stand 351:12. This is the Rule by Mr. Gunter's Chain.
If you divide the Feet by 3, the Quotient will be 117 Yards. Now if you have less than 10 Links as 6, you must always remember to put (0) to supply before the 6, and Multiply the
number as you see in the last Example.
SECT. IV. How to cast up the Content of any piece of Land in Acres, Roods and Perches by Mr. Gunter's Chain.
BY a Statute made the 33 of EDVVARD the I. an Acre of Ground ought to contain 160 square Perches, and every Rood of Land 40 square Perches, and every Perch contains 16 ½ Foot; and 4 Perches, Poles, or Luggs in breadth, and 40 in length makes an Acre: which multiplyed together is 160, Half an Acre is 80, a Quarter 40 square Perches.
[geometrical diagram]
Suppose the Figure ABCD were a square piece of Ground as the Marsh of Bristol, and were 15 Chain 16 Links every way: Then to find how many Acres, Roods and Perches are in it, do thus. Square the sides, that is, Multiply one in the other, and cut
off the 5 last Figures to the right Hand, and that before is Acres: what remains Multiply by 4 (for 4 Roods makes an Acre) and cut off 5 Figures as before, and the comma: is Roods; and that which [Page 5] remains, multiply by 40 the number of Perch in a Rood, and cut off 5 Figures to the right hand of the Product; and in like manner you have the odd Perches. This Example will make all clear and plain.
So you will find 15 Chains, 16 Links Multiplyed together, as before directed, will produce 22, 98256: the 5 last Figures cut off to the left hand, remains before the Comma 22, which is 22 Acres, and the 5 Figures multiplyed by 4, the Product is 393024:
15:16
15:16
90 96
1516
7580
1516
5 Figures cut off on the left hand, the Comma is 3 Roods, and the 5 last Figures multiplyed by 40, cut off 5 Figures, and the rest will be 37 Perch.
Acres 22⌊98256
4
Roods 3⌊93024
40
Perch 37 ⌊20960
By the Line of Numbers.
Extend the Compasses on the line of Numbers on the Scale of Scales from □ which is at 160 unto the side of the Square AB 15 Ch: 16 [...]in. which is 60 Perch and above ½, the same distance will reach from the same 60 ½ Perch, to 22 Acres, 3 Roods, and 37 Perch, and the 20960 part of a Perch.
SECT. V. How to Measure a Long Square Piece of Ground by a Chain of 20 Links to a Perch, according to Mr. Wing.
MAster Wing in his Art of Surveying, in the 113 Page hath described a Chain of 20 Links in a Perch, which is somewhat more ready, if you will reckon the Land in Perches for small parcels of Land.
Suppose a piece of Land be in length 36 Perches and 16 Links, and in the breadth 3 Perches 2 Links; By this Chain I desire to know the Contents thereof, having 20 Links in a Perch, I desire to perform the operation in a Decimal way; Count by half the number of Links, and then the Sums will stand thus; and cutting off 2 Figures, and you have for the Contents of the piece of Ground 114 Perches 2/100 parts. But I would advise the Practitioner in greater parcels of Land, to follow Mr. Gunter's Chain, the Surveyors all generally making use of him; therefore, for further Use of Mr. Wing's Chain, I refer you to the Page of his Book of the Art of Surveying.
368
31
368
1104
114:08
SECT. VI. To Measure a Long Square piece of Ground.
LEt the long piece of Ground be ABCD whose length AB is 11 Chains 25 Links, or 45 Perch; and his breadth AC 8 Chains, or 32 Perches.
[geometrical diagram]
Multiply one by the other, as before in the last Example, directed by Mr. Gunter's Chain; and you will find the Contents of the Ground to be 9 Acres, no Roods; but 0 Perch.
[...]Extend the Compasses from the □ Center, which is at 160. Perch unto the length AB 45, and the same distance will reach from the bredth AC 32 Perch, to 9 Acres.
SECT. VII. To Measure a Triangular Piece of Ground.
SUppose the Base of a Triangular piece of Ground AB whose measure is 11 Chains 25 Links, or 45 Perch, and the perpendicular CD 32 Perch, or 8 Chains; Take the half thereof, and Multiply one in the other, will produce the Contents of the whole Triangle to be 4 Acres, 2 Roods, 0 Perch.
[geometrical diagram]
By the Line of Numbers.
[...]Extend the Compasses from the ▵ Center, which is at 320 — unto the Base AB — 45, the same extent will reach from the perpendicular CD 32 unto the quantity of Acres, which is 4 A. 5/10 as before.
SECT. VIII. To Measure a piece of Ground of Four unequal sides called a Trapezia.
LEt the Ground given be A, B, C, D; after you have taken the Angles with your Compass, and noted them down in a piece of Paper or Field-Book, (as shall be shewn in the following Discourse) You must Protract or lay down the
Figure as I do this, by a Scale of equal parts of 10, or 15, or 20, or 25, or 30 parts divided
[geometrical diagram]
[Page 7] into an Inch; I have laid down all the following by 20 parts or Perch in an Inch: then draw the Diagonal Line AC, and with your Compasses take the distance AC, and apply it to your Scale of 20 Perch to an Inch; and you will find it 60 Perch, or 15 [...]hain; and then if you let fall the perpendiculars BF and DE, and measure them in the like manner, you will find by your Scale BF 20 Perch or 5 Chain, and DE 24 Perch, or 6 Chain. [...]
In respect the Base is common to both the Triangles: You may therefore add the two perpendiculars together 20, and 24, the sum will be 44, the half thereof is 22 Perch. This Number being multiplyed by the whole length of the Common Base AC, 60 Perch, giveth 1320 Perch, that divided by 160, gives the Contents of the Trapezia or piece of Ground to be 8 Acres, 1 Rood, 0 Perch. You might have multiplyed half the Base AC 30 by the sum of the two perpendiculars 44, [...] and it gives you the same as before.
By the Line of Numbers.
Extend the Compasses from the Trine ▵ Center at 320 to half the length of the Diagonal AC— 30
The same will reach from the Sum of the perpendicular 44, to the quantity of Acres, which is 8 25/100 Acres.
After this manner you may measure a piece of Ground of 5-6-7-8, or any number of Sides, by bringing it into Triangles and Trapezias, as shall be shewn.
SECT. IX. To Measure a Piece of Ground which is a perfect Circle.
THe proportion of the Circumference of any Circle, to its Diameter, [...] is as 7 to 22. Example. In this Circle ABCD let the Diameter thereof be 56 Perches, Feet, or Inches, which multiplyed in it self giveth 4136.Acres as 5:1:24 Per. This Number multiplyed by 11, gives 45496, which being divided by 14, [...] the Quotient will be 3249 10/14, that is the Area of the Circle.
[geometrical diagram]
How many Poles and Feet, or square Inches, is in any Circle whatsoever, you may know better by these Rules; First, If you know the Diameter, and would find the Circumference, say, as 7 to 22, so the Diameter 56 to the Circumference 176; Or if you know the Circumference, and would find the Diameter, say, as 22 to 7, so is the Circumference 176, to the Diameter 56.
The Diameter and Circumference being thus known, the Rule to find the Content is this.
The Diameter being 56 Perch, and the Compass 176, the half of both these multiplyed together, and divided by 160, you have the quantity of Acres 15, Roods 1, Perch 24, which is the Contents of that Circle.
By the Line of Numbers.
Extend the Compasses from O Center at (203 7/10 unto the Diameter AC 56, the same distance will reach again from 56 to the quantity of Acres 15 4/10.
I confess though the ordinary proportion of 7 to 22, is somewhat too much; yet it
is but about 1 in 3000, which will breed no great difference in these Questions.
IN the Oval ABCD, let the length be given 40 Perch, [...] or Feet, or Inches; and DB 30 of the same measure: Then to find the Quantity in Perch, Feet or Inches; if you work by the same Denomination of Feet and Inches, as I do of Perches.
[geometrical diagram]
Multiply AC by DC 30, the Product Multiply by 491, and from that Product cut off 5 Figures as before directed, and as in the Margin, the Contents will be 5 Acres, 3 Roods, 22 Perch, 72/100 parts of a Perch.
By the Line of Numbers.
Extend the Compasses from the 0 Center in the Line at 203 7/10 to the length of the Oval AC 40, the same distance will reach from the breadth DB 30 unto the Contents in Acres, 5 Acres, 9/19 fere.
SECT. XI. To Measure a piece of Ground lying in form of a Sector of a Circle.
[...]LEt the Sector be ABC, whose sides is AB, or AC 48 Perch, and the Arch thereof BC 30 Perch: Then to find the Contents in Acres, Multiply AB 48, by BC 30, the Product divide by 320, the Quotient is 4 Acres, and 160, the Remain divide by 180 the ¼ of 320, and the Quotient is the Roods; if any thing remain, it is Perch; So the Contents of this piece of Ground is 4 Acres, 2 quar. 0 Perch.
[geometrical diagram]
By the Line of Numbers.
Extend the Compasses from the ▵ Center at 320 unto the side, or Semidiameter AB or AC 48, the same distance will reach from BC 30, to the quantity of Acres 4 and 5/10.
SECT. XII. To Measure a piece of Ground that is a Segment, or part of a Circle.
LEt the Segment be ABC; AB 60 Perch, DC 18 Perch: [...]Multiply the Chord of the Segment 60 by the perpendicular Height, and the Product divide by 225 the Gage-Number, and the Quotient will be the Acres, the Remain divide by 1/10 and it is found 4 Acres, 3 Roods, 8 Perch, the quantity of Ground in the Segment,
[geometrical diagram]
By the Line of Numbers.
Extend the Compasses always from 225 unto the Chord of the Segment AB 60: the same distance will reach always from the perpendicular Height, unto the quantity of Acres 4 and 8/10 fere.
SECT. XIII. Having a Plot of Ground with the Content in Acres, To find how many Perch of that Scale was contained in one Inch, whereby it was Plotted.
LEt the long Square piece of Ground, be containing 9 Acres; First, Measure the Plott by a known Scale, which suppose it be of 10 Perch in an Inch; so measuring AB, you find it 22 5/10 Perch; also the breadth 16 Perch.
[geometrical diagram]
By the Line of Numbers.
If you work by the Rule of the last Square Plott, you will find the Contents 2 25/100 Acres. Now on the Line of Numbers, take with your Compasses the distance between 2 25/100; and 9 Acres, the half distance thereof will reach from 10 on the Line of Numbers unto 20: so I conclude the Ground was cast up by a Scale of 20 Perch in an Inch. [...]
SECT. XIV. A Piece of Ground being measured by the Statute-Perch of 16 ½ Feet, To know how many Acres it is, it being measured by a Perch of 21 Foot, which is the Irish Perch.
[...] The Square of a 11 is 121. and of 14:196 196:121:1440 [...]EXample. The last Piece of Square Ground being found 9 Acres by the Statute-Perch of 16 ½ Foot; and you would know how many it is by the Irish Perch of 21 Foot.
By the Line of Numbers.
Extend the Compasses from the Irish-Perch of 21 Foot, to the English Statute-Perch of 16 ½ Foot, the same distance will reach from 9 Acres turned twice over unto 5 6/10 Acre, fere: Or in the Scale of Reduction extend the Compasses from 16 ½ Foot to 21, the same distance will reach from 9 to 5/10 in the Line of Numbers. And so of any other measure. Now by reducing the 9 Acres into Perches, it makes 1440 Perch; and because the greater measure is to be reduced into the lesser, Multiply the given Quantity 1440 by 121 the Square of 11, which 11 was found, thus. 16 ½ being a Fraction, it reduced into halfs, makes 33 divided by 3 is 11; So the Irish-Perch 21 Foot in halfs is 42 divided by 3 is 14, those two Numbers squared a 11 is 121, the Square of 14 is 196, the Product of two Numbers 1440 multiplyed by 121, the Product is 174240: that divide by 196, the Square of 14, and the Quotient is 888 192/196 Perch, reduced into Acres, is 5 Acres, 5 Roods, 8 Perch 192/196, almost 9 Perch according to the Irish-measure.
Suppose you had been to Reduce Irish-measure into Statute-measure; then multiply 1440 by 196, and the Product would have been 282 240: that divided by 121; and the Product had been 2332 ½ Perches fere, which makes Acres 14, Roods 2, Perch 12 ½ fere, Statute-measure.
By the Line of Numbers.
Extend the Compasses in the Scale of Reduction from the Number of Feet in Customary measure, as I do from 21 Foot to the Perch of Irish-measure: the same will reach from 9 Acres Irish in the Line of Numbers to 14 Acres, 2 Quarters, and 12 Perch Statute-measure.
Or if you extend the Compasses from 21 in the Line of Numbers to 16 and ½, the extent turned twice over from 9, will fall upon 14 and ½ Acres, and a little more.
So that if you remember in all sorts of measure to reduce your Fraction into the same Denomination, and seek out the least proportionable terms: by Dividing by 3 if half Foot, and squaring these terms as before directed, you have a Rule that serves for all sorts of Customary or Irish measure whatsoever.
CHAP. II. How to take the Plott of a Field at one Station taken in the middle thereof by the Azimuth-Compass.
BEfore you go into the Field, you must Rule a piece of Paper in 8 Columns as you see the Figure following in this Chapter, makes all plain, without any more Description, which is called a Field-Book. Secondly, when you come into the Field, first place Marks at the several Angles of the Field, as at ABCDEF, in the following Figure; then make choice of some convenient place about the middle thereof at 0, to fix
your Compass, if you can see all the Marks; and be sure the Brass-Diameter and Slits before descriebd, be set directly over the Meridian or North and South Line of the Chard, and there fixed.
This done, direct your Sights to your first Marks at A, Marking what Degree the Index cutteth, which let be 36 Degr. 45 Min. you may estimate the Minutes; This you must note down in your Field-Book in 1 and 2 Column thereof, as you see in the Book it is plain set down; then measure the distance from 0 the place of the Compass to A your first Mark, which let contain 8 Chains, 10 Links, which must be placed in th 3 and 4 Column of your Field-Book, as you see in the Figure of the Book.
Then direct your Sights to B the second Mark, and note the Degrees cut by the Inde [...] which let be South Easterly 80 Degrees 45 Minutes, and the distance 8 Chains 75 Lin. You must put down in the Field-Book, as before; First, the Letter B; Secondly, the Inclination to the Meridian cut by the Index South Easterly 80 Degr. 45 Min. in the third Column; then 8 Chains 75 Links in the fourth, as you may see in the Columns in the Book, all plain; then direct your Sights to C your third Mark, and note the Degrees cut by the Index, which let be S. E. 16 degr. 45 min. and the distance OC 10 Chains 45 Links, put the same down in the Field-Book likewise, as before directed; then direct your Sights to D, and note the Degrees cut by the Index, which let be S. W. 32 degr. 00 min. the distance OD, 8 Chains 53 Links, Note it down in the Book, as before.
Then direct your Sights to E, the Index cutting 72 degr. 45 min. North Westerly; and the distance OE 8 Chains 15 Links: They must be noted in the Book as the rest are.
Lastly, direct your Sights to F your last Mark, the Edge of the Index cutting in the upper Brass Circle N. W. 18 degr. 00 min. the distance OF 9 Chains 55 Links; then will the Observation stand in the Field-Book as in the following Table or Figure; then by a line of Chords, or by the Protractor you may presently Protract the exact Figure of the Field upon Paper thus, By a line of Chords; take 60 degr. and draw the Circle. Secondly, draw the Meridian line of North and South, and parallel-line of East and West. Thirdly, your first Observation and degr. cut by the Index, was 36 degr. 45 min. N. E. Therefore take 36 degr. 45 min. off the line of Chords, and lay from the Meridian line M to N. Fourthly, the distance OA was found 8 Chains 10 Links; take with your Compasses off any Scale of equal parts, 8 Chains and 1/10 that stands for 8 Chains 10 Links; lay this distance from O to A, and draw the prick'd line OA.
Then secondly, take out of the line of Chords the second Angle S. E. 80 degr. 45 min. and lay from the South towards the East on the blind-Arch, and through it draw the line OB; then take off the same Scale of equal parts 8 Chains 75 Links, that is, 8 parts, and 75 parts of 1 divided into 100, and lay it from O to B, and prick the line from O to B, and draw the line AB, which measure with your Compasses, and apply it to the Scale of equal parts as before, and you will find the side AB 8 Chains 75 Lin. The like do by all the other Angles and distances in the same manner as you have been shewed in the two first Angles. The Figure makes it so plain, it need no further precept; and you may put down the Number in the side, as I have done.
Now by the Protractor in the Second Book described.
Turn the Protractor after you have laid down all the Angles in the NE. and S. E. quarterly, and lay the Center-Pin of the Protractor on the Point O, as before; and the Diameter on the Meridian-line [...]You may lay the Diameter-Edge thereof on the North and South line, and through the Center, put a Pin on the Center of the Plott at O, and note the degr. and distances in the Field-Book, as before: the first was North Easterly 36 deg. 45 min. put the Edge of the Index to 36 degr. 45 min. in the Arch of the Protractor; and by the Edge account in the Scale thereof 8 Chains 10 Links; and by the side thereof draw the line AO, prick'd as before; and so do by the rest of the Angles and sides in like manner, and you may presently draw a Plott of Ground you have measured. This is very plain, and may be understood by the meanest capacity in this Art. The Observations Marked in the Field-Book stand as in the following Table.
The manner how the Field-Book must be Ruled.
Mark.
D. M.
Ch.
Lin.
A
N. E.
36 45
08
10
B
S. E.
80 45
8
75
C
S. E.
16 45
10
45
D
S. W.
32 00
8
53
E
N. W.
72 45
8
15
F
N. W.
18 00
9
55
You are to note, that every Degree in the uppermost Brass-Circle is supposed to be divided into 60 parts, which is called Minutes, which cannot be expressed in regard of the smalness of the Instrument or Circle of Brass on the Glass of the Compass; and therefore the odd minutes must only be estimated: so must the odd Links taken off your Scale of equal parts, and it will breed no sensible Errour.
CHAP. III. How to take the Plott of a Field at one Station taken in any Angle thereof, by the Sea-Compass.
IN the Figure of the Field following, place your Compass at M, set your Brass-Diameter right over the North and South points or Line; then first direct your Sights to A, which let be supposed to be 22 degr. 15 min. N. E. which Note in your Field-Book, as before; then measuring the distance with your Chain MA which let be 8 Chains 46 Links, which place in your Field-Book according to former directions.
Secondly, Direct your Sights to B, the degrees cut off by the Index, suppose 42 degr. 45 min. and suppose the distance measured to be MB 10 Chains 21 Links, and put them down also in the Field-Book.
Thirdly, Direct your Sights to C, the degr. cut is 66 degr. 30 min. and the distance MC 1 [...]Chains 64 Links, put these in your Field-Book also, as before; and in the same manner you must deal with the other marks DM, and EM, and MF, and MG; and so having them all in the Field-Book, they will stand as followeth.
CHAP. IV. How to Measure any Piece of Ground be it never so Irregular; And how to reduce the Sides into Triangles or Trapezias, and to cast up the Content thereof in Acres and Perches.
SUppose you were to Measure a Piece of Ground, or Wood, or Marsh, or any place whatsoever, by your Compass and a Line marked as the 4 Pole-Chain before described in the first Chapter; and if you cannot see all the Angles by reason of the bigness thereof, then you must measure round about by the sides thereof, as in this Figure following; and the Observations made in the Field are set down in the Field-Book following, so plain, that it need no further precept.
Suppose you made your first Observation at A in the Field in the following Figure, (the Compass being rectifyed as before directed) you direct your Sights along the hedge to the Mark in the corner at B, and the Index cutts 54 degr. from the South Westwards, and the distance is 5 Chain 12 Links, which set down in your Field-Book thus, A B bears S W 54 degr. 00 min. distance 5 Chain 12 Links; Then make your second Station at B, and direct your Sights to C, the Index cuts N W 45 deg. 00 min. distance 2 Chain 89 Links, which note down in your Field-Book, as you did before in the second place; and so do by all the rest. From C to D N W 76 degr. distance 3 Chains 35 Links, from D to E N E 31 degr. distance 4 Chains 55 Links, from E to F N E 56 degr. distance 2 Chains 57 Links, from F to G NE 21 degr. distance 2 Chains 24 Links, from G to H S E 51 degr. 00 min. 2 Chains 95 Links, from H to K S E 34 degr. 3 Chains 25 Links, from K to A SW 4 degr. 2 Chains 95 Links; Thus you see all the Observations plainly set down in the Field-Book, you may proceed to Protracting your places of Observation and Marks in the Field, and your degrees and length of Lines orderly placed in your Field-Book; We proceed two ways to examine the truth: Thus the Rule is.How to know if the Sides be rightly measured before you go out of the Field, and Protract the Platform thereof.
First,
As the Radius or Sine of 90 degr. is to the length of the side of the Field in Chains and Links, or Perches and 100 parts; so is the Sine of the degree cut by the Index to the length of the Parallel of East and West in Chains and Links, or Perches and 100 parts.
Therefore by your Scale extend the Compasses from the Sine of 90 to the length of the side of the Field in the Line of Numbers, the same distance will reach from the degrees cut by the Index to the length in the Parallel of East or West.
Secondly,
As the Radius or Sine of 90 degrees to the length of the side of the Field in Chains and Links: or Perch and 100 parts; so is the Complement Sine of the degrees cut by the Index to the length of the Meridian of North or South in Chains, or Links, or Perch, or 100 parts.
Wherefore Extend the Compasses from the Sine of 90 degrees to the length of the side of the Field in the line of Numbers; the same distance will reach from the Sine Complement of degrees cut by the Index to the length of North or South in the Meridian.
So that you see the 4 last Columns in the Field-Book are noted North and South, East and West.
Now to know by the Chains and Links, the first Observation from A to B, is S W 54 degr. and the distance A B is 5 Chains 12 Links; therefore by the last Rule extend the Compasses from 90 degr. to 5 Chains 12 Links in the Line of Numbers, that distance[Page 15] will reach from 54 degr. cut by the Index to 4 Chains 14 Links in the line of Numbers, which is the distance in the Parallel of West, and also the same extent will reach from the Complement of 54 degr. which is 36 degr. to 2 chains 97 links in the line of Numbers, which is the distance in the Meridian South, and put it in the South Column of your Field-Book, as you did 4 chains 14 links in the Column of West; and so you may do with the rest of the Observations.
But the most sure way and least Errour, is, to convert your chain and links into Perches and 100 parts of a Perch, and then you can Protract in Perch and 100 parts the better.
Thus if you Multiply the number of chains found in the side by 4, by reason 4 Perches are in every chain; and if there be above 25 links in the place of links, divide by 25, and the Quotient will shew the odd Perch to be added; and what remains is links: that Multiply by 4 likewise, the Product will be 100 parts of a Perch.
As for Example.
The first side AB his distance is 5 chains 12 links, multiplyed by 4, makes 20 Perch, [...] 48/100 parts, which put in the next Column to it; Now if you extend the Compasses from 90 degr. to 20 Perch 48/100 parts, the same distance will reach from the Sine of 54 degr. cut by the Index to 16 Perch 56 parts, which is in the West Column, and the same extent will reach from the Complement 54 degr. which is 36 degr. to 11 Perch — 88 parts, [...] which I put in the South Column; and by the same Rule I work in like manner by the rest of the Observations. The second side BC is 2 chains 89 links, reduced as before, makes 11 Perch 56/100 parts; and so work by them, as before directed, you shall have all your Numbers stand as in the following Figure of the Field-Book.
Houses name.
Angle with Merid.
Cha. Lin.
Pol. 100 pts.
North.
South.
East.
West.
AB
S W 54 00
5 12
20 48
P. pts.
11 88
16 56
1
BC
N W 45 00
2 89
11 56
8 16
8 16
2
GD
N W 76 00
3 35
13 40
3 32
13 00
3
DE
N E 31 00
4 55
18 20
15 72
09 40
4
EF
N E 56 00
2 67
10 68
6 00
8 88
5
FG
N E 21 00
2 24
8 96
8 40
3 20
6
GH
S E 51 00
2 95
11 80
7 32
9 20
7
HK
S E 34 00
3 25
13 00
10 72
7 28
8
KA
S W 4 00
2 95
11 80
11 68
0 24
9
The Sum
41 60
41 60
37 96
37 96
Under the line are the distances in the East Column; and likewise the Figures on the inside the Meridian, are the points of North, as 1, 2, 3, 4, 5, 6 Column; and the Figures on the outside are 7, 8 to A, and are taken out of the South Column, and working as directed in the two first sides, you may find all the rest. The Figure makes all plain to the meanest Capacity; and so you will have the true Plott of your Ground, or Park, or Wood-land, or Plantation, or place whatsoever, drawn on Paper or Parchment. Now if there be any Houses by the hedge-side, made a mark in your Field-Book in that Angle, and how many Chains or Perch from the place you observe, and by it insert it into your Plott.
How to draw the Plott by the Protractor.There is a House in the first side and Angle AB, SW 54 degr. about 2 Chains 42 Links, or 9 Perch 21 parts, and reckon 1 Chain 24 Links, or about 5 Perch; therefore put it down in your Book, with the Man's Name that ows the House, as I have done, (John Cooke) or the like; and if any House, Church, or Castle, be in the middle, take the Angle thereof from any Point, and measure the distance, and note it in your Book, and enter it into your Plott, as I have done this House. By these rules you may compleatly take a whole Parish, Plantation, or Island: Now if you draw a Plott by the Protractor described in the Book of Instruments, You must rule your Paper or Parchment with an obscure plummet Merid. Lines, and Parallel Lines about 1 inch and ½ asunder; and put the Pin, the Center and Rivet upon any Point, and turn the side of the Protractor on the Meridians, and look in the Field-Book for the Angle, and put the Edge of the Index to the degrees, and count the Perch on the Index-side, there make a Mark with your Pin for the second place, and draw a Line from that place by the Edge of the Ruler to the Centers for the side of the Hedge or Field.
As for Example.
Suppose you were to draw the side AB in the Plott with your Protractor, lay the Protractor on the out-side of the Meridian-Line, and the Diameter-Edge thereof to the Meridian-Line; then in the opposite Degree and Quarter as in this Example is NE: I put the Foot of the Index to 54 degrees, and from the Center, the Edge points SW 54 degr. Number the Chains, or Links, or 20 Perch 42 parts, and from that Number to the Center draw a Line by the Edge thereof, and you have the Side AB; by this Rule you may gain all the rest. There is no Man that understands any thing in these Arts, but knows readily how to Plott a Field by the Rule before-going without more Directio [...]s, for they will be all the same.
CHAP. V. How to find the just Quantity or Content of any Piece of Ground in any Form.
VVE have shewed in the fore-going Chap. how to Measure the Geometrical Square, the Parallelogram, the Triangle, Trapezium, the Circle, and the like. Now we will shew you how to cast up the Contents thereof more fully. Suppose the fore-going Figure A, B, C, D, E, F, G, H, K, be a Plott drawn or Protracted by a Scale of 10 Perch in an Inch, and the exact Contents thereof is required. Now because it is an Angular Plott, neither in the form of a Square, Parallelogram, Trapezium, nor Triangle; therefore all such Plotts must be reduced into some of these forms: which to effect, I reduce the main Body of the Field into the Trapezium ACEK, and the residue of it into 5 Triangles, as ABC, and CDE, and EFK, FHK, and FGH.
Now to know the just Quantity of Acres, Roods, and Perches the Field contains, I first measure the Trapezium ACEK, I measure with my Compasses the length of the perpendicular CO, and apply it to my Scale of 10 Perch in an Inch, and find it 14 Perch 88/100 parts; and likewise the perpendicular KP, and find it 10 Perches, 50 parts, which I add together, and they make 25 Perches 38 parts, which I Multiply in half the Base AE 16:43, and the Product is 416:99:34: Therefore if you cut off 4 Figures to the right-hand, you will have the Contents of the Trapezium 416 Perch, and cut off 2 Figures of the 4, and before the Comma is 99 parts of 100 of a Perch and 34/1 [...]0 remains, which is not to be taken notice of.
In like manner for the Triangle ABC, I multiply half the perpendicular: 4∷75 by the whole Base 25 Perch 26 parts, or the whole perpendicular by half the Base, as before; it is all one, and the Product is the Content of the Triangle ABC 119 Perch 98 parts: and so likewise for the Triangle CDE, multiply half the base 9 Perch 52 by 12 Perch, and 20 DY the perpendicular, and the Product is 116 Perch 14: is the Contents of that Triangle. Likewise in the Triangle EFK, the length of the perpendicular FR is 7 Perch: 00 and the half length of the base EK is 14 Perch 70, multiplyed as one whole Number, the Product is 102 Perch 90 parts 00: the Contents of that Triangle EFK: the 4 Triangle FHK perpendicular: 6 P. 10: H q base FK 11 P. 50 the Contents is 70 Perch: 15: the first Triangle FGH the perpendicular GS is 7 P. 92 and the base FH, the half thereof is 6 P. 50: multiplyed together, and the Product is 51 P. 48 parts for the Triangle FGH.
Lastly, I add the several sums together, and they give the Content of the whole Figure in Perches and 100 parts.
Which if you will reduce into Acres, Roods and Perches,Reduce Perches into Acres, Roods, and Perches. you must note, that 16 Foot and ½ Square is a Perch, and 160 of these Perches makes an Acre; therefore Divide 877 Perch by 160, the Quotient shews the Acres to be 5; that which remains above 40 divide by 40, and the Quotient will be Roods 1: and that which remains will be 37 Perch. This is a General Rule for reducing of Perches; so that the whole Plott of Ground yieldeth the Content of the said Field 5 Acres, 1 Rood, 37 Perch and 64/109 of a Perch.
This is the way to cast up the Content of any Irregular Field, by reducing it into Trapeziums and Triangles, and adding their several Products into one Sum, which ought heedfully to be regarded, it being one of the most material works belonging
to the Practice of a Surveyor; for unless he be perfect herein, he can never perform any Work of that nature aright.
I have been brief and plain in shewing the Art of Surveying by the Sea Compass; I might have been longer, but to avoid Prolixity, I think what is right is sufficient:
If any desire a larger Discourse, he may make use of Mr. Leybourn's Compleat Surveyor, or Mr. Wing's Art of Surveying, and others that have writ at large of the Use of the Plain Table, which is the most easy and useful Instrument in the Art of Surveying of small Inclosures. But the Compass fitted as before, will with a little labour do any thing as exact as the Plain Table, Theodolite, Circumferentor, Protractor, or any other Instrument; especially large places, as Woods; Parks, Forrests, or Plantations or the like: and what Direction I have given in Use of the Sea-Compass will serve for an Introduction to all other sorts of Instruments for Surveying of Land.
CHAP. V. How to take the Height of any Island, or Mountain in the Sea by an Example made by the Author of the Height of Tenariff.
MAny Learned Men have writt of the Great Incredible Height of several Mountainous Hills and Islands in the World. For taking of the Height of Islands in the Sea, none have greater opportunities than Sea-men. By them may all Men be informed of the truth of such like things; and also of several
good Stars that be to the Southward, as the Crofiers and Cannobas in the Stern of the Ship, and any other, did they but observe any such Stars from a known Latitude, and take their Meridian-Altitude and time of Night; or if they cannot, or will not Calculate, and find by it the Stars Altitude, and Longitude, and Declination: yet if they bring it home, and give it to some able Artist, he will do it, and all
Men shall have the benefit of it, and by it, it would be a great help to Navigators.
It is reported of Aristotle, Mela, Pliny, and Solinus of the Invincible Height of Athos a Hill in Macedonia, and of Caucasus; and of Cassius in Syria, and many other places: and among the rest one of the most miraculous things which
they have observed of the Mountain Athos, is, that it being a Hill and Mountain situated in Macedonia, it casts a shadow into the Market-place in Myrrhina, a Town in the Island Lemnos, distant from Athos 86 Miles to the Eastward. It is no marvel it cast so large a shadow, seeing by Experience of the shadow of
a Mans Body, we find it extraordinary long at Sun-Rising, as well as at Sun-Setting. They report it is Higher than the Region of the Air. Julius Scaliger writes from other Mens Relations, that the [Page 19] Island Pico of Tenariff riseth in height 15 Leagues, or 60 Miles: Most Writers agree that it is the highest
Mountain in the World, not excepting the Mountain Slotus; it self, which I question whether any mortal Man ever see Slotus, besides the Monk of Oxford, who by his skill in Magick conveyed himself into the utmost parts of the World, and
took a view of all places about the Pole: yet that this Island cannot be so high, shall appear by the following Demonstration
and Observation made by Me; all Sea-men that have used to Sail to the Canaries know, that the Snow is not off it above 2 Months in a Year, that is, June and July.
Patricius not content with the former Measure of 60 Mile high, reaches to 70 Mile high; Now
that any Snow is generated 60 or 70 Mile above the plain superficies of the Earth and Water, is more then ever they can perswade any Men that understand
these things, seeing that the highest Vapours never arise by Ptolomy 41 Miles, and by Erat [...]sthenes's Measure 48 Miles above the Earth; that is, There is never no Rain, Dew, Hail, Snow,
or Wind, but still a clear serenity.
I have been and passed by Tenariff several times my self, in the Year 1652 I was there in the Castle-Frigot of London Cap. John Wall Mr. and Loaded our Ship at Garrachica right under the Pico; and since bound to the West-Indies in the Year 1656, in the Society of Topsam, a Ship I had Command of, was put by Westerly Winds to the Eastward, that we had sight of the Pico of Tenariff, it bore off us South; about Noon I was resolved to make Observations of the height thereof, to try Conclusions
with my Quadrant of 20 Inches Semidiameter, described in Chap. 16 of the second Book; and by the Rules of Quadrature I made these Observations following. On the 5th. of May 1656 I observed, and found the Sun's apparent Meridian-Altitude 81 degr. 44 min. his Declination 19 degr. 08 min. the Latitude 27 degr. 20 min. the Latitude of Pico I found formerly to be 28 degr. 20 min. difference of Latitude 60 min. or miles, which in the following Figure I make one half of my Horizontal Base AD; then at the same time observing the Height or Altitude of Pico, I found it 24 degr. 14. min. Therefore according to the Sphericality of this Terrestial Globe consisting of the Earth and Sea, I demonstrated the following Figure. The Section of the Arch A E F S represents the superficies of the Horizon of Tenariff; or a part of an Arch of a great Circle, the Meridian A D, the difference of Latitude 60 mile, D C Pico the perpendicular, E B a second Observation, N the North part of the Horizon, S the South part,See Fig. 98 in the following Schemes. ☜ C the Port of Garrachica.
Now to find the perpendicular Height thereof, you have the Rules at large set down in the 16th. Chap. of the Description and Use of the Quadrate and Quadrant; and by the first Rule of the Quadrant I found the perpendicular Height D Pico 27 min. or mile, if the Sea were a flat plain as a Table-board, as the Right-Line A B D S represents.
But having 3 days of Fair Weather, in sight of Pico I made a second Observation, and ran to it with my Ship until I made an Angle with the Pico of 45 degr. 00 min. as Pico E C: or Pico BD; and had the apparent Meridian-Altitude of the Sun 81 degr. 29 min. the Declination 19 degr. 22 min. Latitude the Ship is in 27 degr. 53 min. difference of Latitude 27 min. or miles, being equal to the Height, as B D to the perpendicular D C Pico, the Angle of 45 degr. being the most sure as can be made by any Instrument which confirms the first, and the Height of the Pico of Tenariff to be 27 min. or miles High, if the Sea were flat as a Board.
But touching the Hypothesis, that the Earth and Sea makes a Round Body, It is generally agreed upon by all the Philosophers, Astronomers, Geographers, and Navigators Antient and Modern; and therefore the distance of a degree 60 min. reckoned in the Heavens by Observations of the Sun or Stars is more than 60 miles upon the superficies of the Earth and Sea,, as appears by several Experiments made by able Artists: but especially by the Labour and Industry of our own Countrey-Man Mr. Richard Norwood, as you may see in his second Chap. of his Book the Sea-mans Practice, made by him betwixt York and London. He makes it evident that 1 degree of a Great Circle[Page 20] on the Earth, is near 367200 Feet, which in our Statute-Poles of 16 and ½ Feet to the Pole is 22254 Poles, and about half, and these reduced into Furlongs at 40 Poles to a Furlong, makes 554 Furlongs and 14 Perch; and lastly, these reduced into English-miles of 8 Furlongs to a mile, makes 69 miles and 4 Furlongs 14 Poles, that is 69 and ½ miles and 14 Poles to 1 degree upon the superficies of the Earth and Sea. And seeing a Degree is the 360 part of any Circle, equally divided in the Circumference; Therefore if we can find how many Feet, Perches, Furlongs, or Pieces, are in a Degree of known measure: then can we presently resolve how many of the same known measure are in the Circumference of any Circle so divided on the Earth and Sea;How to know the Circumference of the Earth. for if there is 367200 Feet in one degree of a Great Circle upon the superficies of the Earth and Sea, therefore it is evident, that if you multiply 367200 by 360 degr. the Product is 132192000 Feet, which reduced into Poles, is 8011636, and these reduced into Furlongs, are 200290 Furlongs, and 36 Poles; And lastly, these reduced into miles, are 25036 English miles, 36 perch for the Circumference of the Earth and Sea.
How to find the Diameter & distance to the Center of the Earth and Sea.And now if you desire the Diameter and Semidiameter of the Earth, as it is proved by Archimedes, That the proportion of the Circumference of a Circle is to the Diameter thereof almost as 22 to 7; therefore by the Rule of Proportion, Multiply the Circumference of the Earth; namely, 132192000 by 7, and divide the Product 925344000 by 22, the Quotient is 42061091, which is the Diameter of the Earth in Feet: and the half thereof, namely, 21030545 is the Semidiameter of the same, or distance of the superficies of the Earth and Water, to the Center, being 21 millions of feet, and a little more; [...] and these reduced into miles, as we did the Circumference, shews the Diameter of the Earth to be 7966 miles, and somewhat more: and the distance to the Center or Semidiameter 3983 miles; and thus is found the Circumference, Diameter, and Semidiameter of the Earth and Sea, and also the quantity of a degree of the same measure in English measures of Feet, Perch, Roods, and miles. Therefore if you do still retain a degr. in the Heavens to be 60 minutes, you may find how many Feet is in a mile on the Earth and Water, if you divide 367200 feet by 60, the Quotient will be 6120 feet; which doubled, and divided by 33, and half-feet to a perch, the Quotient is 370 perch, and 30 foot remains: divide 370 by 40 perch to a furlong, and the Quotient is 9 Roods or furlongs, and 10 perch or poles remains, divided by 8 Roods to a mile, the Quotient is 1, and 1 remains; so that a minute in the Heavens by this Rule and measure upon the superficies of the Earth and Water, contains 1 mile, 1 rood, 10 perch, and 30 foot; therefore my degr. 60 min. of Latitude at my first Observation, is found by these Rules to be 69 and ½ miles, 14 perch my distance upon the Arch of a Great Circle from the Latitude of Pico: therefore working by the Rules given of Quadrature in the 16th. Chapter of the 2 Book, the true Height of Pico will be found to be 31 29/108 miles,The true height and distance from the Eye. and the distance from the Eye to the Top of the Pico A, P, will be found by the Rules in the 16th. Chap. 76 22/100 miles.
And working the second Observation by the same Rules, your difference of Latitude 27 minutes B D will be found to be 31 miles, 2 furlongs, 14 perch, 18 foot, which is 31 ¼ miles and a little more; which is almost the same Height found by the first Observation:
and the distance from the Eye to the Top of the Pico is 44 24/100 miles.
By these several Rules you may find the Height of any Mountainous Island at Sea, or High-Land on the Main-Land, if you can come bring it North or South of you, and make any Observation of the Sun or Stars; or if you will but trust to a Log-Line marked after the former Experiment, that a mile doth contain 6120 feet, or 1020 fathoms; and so 3 miles or a League contains 18360 feet, or 3060 fathoms: then if you intend to keep a Log by ½ minute-Glass: and because half a minute is 1/120 part of an hour, divide 6120 by 120, the Quotient is 51 feet. Therefore so many 51 feet or knots she runs out in ½ a minute, so many miles she sails an hour. By this Rule you may keep your Reckoning exactly; for I had Experience in sailing North and South by a Log-Line marked after the rate of 6000 in a mile, that is by the same Rule 50 feet or 8 fathoms, and 2 feet to every knot, that I have run or sailed almost 22 Leagues to raise or depress the Pole 1 degr. on a Great Circle; and if any have impartially taken the [Page 21] same notice and care, (or will) they shall find the like. But many that follow the
old Rule 300000 feet to a degree, and 5000 feet to one mile, and 60 mile to a degree, or the 120 part of an hour 41 ⅔ feet, or 7 fathoms to the knots upon the Log: when they Sail North or South, and find the Log to fall too short of their Observation, imputes it to ill Steerage, Sometimes to the Variation of the Compass, or some Errour in their Plotts, or some Current, or other accident, but will not believe the truth a great many without they had Angels
should tell them so, a great many have so much Ignorance and Obstinacy. But for confutation
of some of the Antient Authors, especially Cleomedes, whom I take to be a Man which did never see, nor observed Tenariff Pico; He affirms, that there is no Hill found to be above 15 furlings in Height; and of Mr. Hughes, He saith, that if Mercury himself should affirm a Hill to be above 4 miles in Height, he will not believe him; neither will I believe them that are of that
Opinion, be they what they will, without they could prove the foregoing Observation
not good, and produce better of their own made by the Pico of Tenariff; and so much for the Altitude of Hills at Sea.
CHAP. VI. How to find the Distance of a Fort, or Walls of a City, or Castle, that you dare not approach for fear of Gun-Shot; Or the Breadth of a River or Water, that you cannot pass, or Measure over it, made by 2 Stations, with the Quantity of the Angle at each Station.
SUppose from some private place as at A, you espy a Castle, Fort, Tree, or place whatsoever, that you dare not approach for fear of Gun-Shot, Marsh-Grounds, or a River betwixt you, or some other Impediments, that you cannot make your second Station in any open place, but are forc'd to make it in some other secure Place at B; therefore
plant your Instrument or Compass at A, and direct the Sights to C and B, take the Quantity of the Angle C A B 46 degr. 00 min. and go to B, and take the Quantity of the Angle A B C 79 degr. 0 min. then measure the distance of the 2 Stations A and B 350 fathoms.
Then by a Plain Scale, or by the Line of Sines on the Scale of Scales, you may presently resolve the distance, as I do by the Tables.See Figure 99 in the following Schemes. ☜ 46 d.00 m.79001250018001250550
SUppose you were to take the Breadth of a River, as I have at Crocken-Pill, which runs betwixt Glocester-shire and Somerset-shire, and found the breadth of the Water upon a Spring-Tide 40 Perch or a Furlong; you must do it thus. Being on the Riverside as in the former Figure at E, there set your Compass; Observe some mark on the other side of the Water, as at D; then set a mark at E, and go square-wise either to the right-hand, or to the left from these 2 marks, so far, until you spie the mark D on the other side the Water doth justly make an Angle of 45 degr. with the mark E; and this will be when you come to F; then measure carefully F E, the distance of the 2 Stations, and that shall be equal to the breadth of the River: so that if FE be 10:20:30:40:50: or 100 Poles, or Yards, or Feet, the breadth is the same. The like may be done by any other Angle, as if you go to G, and make an Angle of 26 degr. 30 min. in D; then is the distance GE twice the breadth; but ever if you can get an Angle of 45 degr. for that is the best and readiest Angle to find out such a distance; therefore if you can, use no other.
And the like way of Working you may do at Sea, if you gain the Sight of any Cape, Head-Land, or Island, set it by your Compass when you see it, without altering your Course, make an Angle of 45 degr. And by your Plain-Scale if you have kept a good account of your Way by the same Rules as before, you shall
have the true distance of your Ship from the first Place, or Cape, or Head-Land, or Island whatsoever: Or you may get the Slope-side D F or D G if you measure it with your Compasses, and apply it to the same Scale of equal parts by which you put down the distance E F or E G. Thus you may find the distance from the Ship, to any Cape; These are made so plain by the Rules before-going, that it need no further precept.
SECT. II. Being upon the Top of a Hill, Tower, Steeple, or a Ships Top-Mast-Head, there observing the Angle of distance from you, To find the true distance thereof.
YOu may do this by your Quadrant; thus. Let the height of the Hill, Tower, Steeple, or Ships Top-Mast-Head be 40 yards, or any other measure: and from it you see an House, Tree, or Place whatsoever, and you desire the distance from you. You have been shewed already to find the height of a Tree, Tower, Hill, or Steeple; by this Rule we will shew you how to stand upon them, and take the distance from any thing else, viz.
Let the height of the Tower, or Mast, or Hill be 40 yards, and let the Angle of distance taken with your Quadrant be 80 degrees, being 10 degrees under the Line of Level; this is the Rule for all such Questions.
As the Tangent Compl. 80 degr. which is 10 d.
924631
is to the height 40 Yards.
260206
So is the Radius
10
to the distance from the Top of the Tower or Hill 226 9/10
SECT. III. By the way of your Ship, and any 2 Angles of Position, to find the Distance of any Island, Cape, or Head-Land from you.
YOu have been shewed how to do it with a right-Angle of 45 degr. already; but with a little more trouble, you shall learn to do it by any 2 Angles whatsoever.
As for Example.
Suppose you were Sailing full South from A towards B, and from A should espy Land at C bearing 2 Points from you to the Westward, as S S W, or S W 22 deg. 30 min. and Sailing still upon your Course until you come to B, you observe the Place bears from you just 4 Points, or S W 45 degr. which is the double of the Angle observed at A. If in this manner you double any Angle; that is, let your first Angle be what it will, you must Sail until you have doubled that Number; then you may assure your self that the distance you have Sailed between A and B, is justly equal to the distance between B and C, B being the second Place where you made your last Observation, and
C being the Place observed. So that if A B be 12 miles, B C is likewise 12 miles; and this you may do without further trouble or Calculation, and may lay it down by
your Plain-Scale, as I have done this following Figure.
In all such Questions remember that the Angles at the second place of Observation, shall be either just the double, if you go nearer
to the Place, or else just the half if you go further off than the Angle at the first place.See Figure 100 in the following Schemes. ☜ Therefore the first Angle you may take at Random, no matter what it is, so you be careful to observe when you
be just upon the double, or the half; so that by Calculation you may resolve it almost
with as little trouble as a Right-Angle, which is made plain thus.1800450135022301573018000d. 2230 In the Triangle ABC the acute Angle being the outward at B, being 45 degr. the obtuse or inward-Angle being the Complement thereof to 180 degr. must be 135 degr. and the Angle at A being 22 degrees 30 min. being added to this, makes 157 degr. 30 min. which Substracted from 180 degr. there must needs rest for the Angle at C 22 degr. 30 min. Now this Angle at C being equal to the Angle at A 22 degr. ½; therefore the side A B opposite to the one Angle, must needs be equal to the side B C opposite to the other Angle, as you see by this Case.And by these Rules you may find all the opposite sides.
CHAP. VII. How to take the Distance of divers places one from the other, remote from you, according their true Situation in Plano, and to rotract (as it were) a Mapp thereof by the Compass and Pplain-Scale.
THe Problem serveth chiefly to describe upon Paper or Parchment all the most Eminent and Remarkable
places in a Country, Town, or City, whereby a Mapp thereof may be exactly made by help of a Table of Observations following, as with a little Practice you may soon perceive.
Upon some high Piece of Ground make choice of 2 Stations as A and P, from whence you may plainly discern all the Principal Places which you
intend to describe in your Mapp; then at A Plant or set your Compass fixed, and turn the Index about to P; and let A and P bear one of the other North and South, as you see marked with the Letters N and S: and then direct your Sights to the several Marks from A to B, C D E F G H I K L M observing what degr. the Index cutteth. As suppose your Instrument fixed at A, and the Sights directed to B, the Index cutteth N E 83 degr. 50 min. and likewise the Index directed to C, cuts 82 degr. 5 min. and so in like manner take the rest of the Angles, as you see them in the Table following, which were noted down by you in a Paper-Book when they were taken.
The Stationary Distance 730 Perch, or 2 miles 90 Perch.
Places
Angles
deg.
min.
A B
N E
83:
50
A C
S E
82:
05
A D
S E
64:
50
A E
S E
56:
20
A F
S E
45:
26
A G
S E
41:
30
A H
S E
24:
40
A I
S E
09:
00
A K
S W
11:
00
A L
S W
16:
00
A M
S W
23:
00
Next measure the Stationary distance A P, which was found 730 Perch, which you must Note down likewise in your Book; then plant your Compass, and fix him at P, that the Chard may stand North and South on the Stationary-Line P A, then turn the Index to your first Mark K, the Index cuts N W 24 degrees; Likewise turn the sights to L, and mark the Inclination to the Meridian, and put it down N W 17 degr. and so do by all the rest of the former Marks or Points; and Note them down as you see in this Table P K: P L: P M: P I: P D: P B: P C: P E: P G: P F: P H: and where the Lines Intersect each other, drawn from the two Places A and P, there must you describe
the several Places, to which you made Observation, where you may Write the Name of
the Places.See Figure 110 ☞
Lastly, If you would know the Distance of any of the Places thus described,See Figure 101 ☜ one from another, you have no more to do, but open your Compasses to the two Places on the Paper; and then apply it to the same Scale, by which you laid down the stationary Distance AP, which in this Figure was laid down by a Scale of 20 Perch to an Inch: the like is to be understood of Fathoms, Yards, or Feet; and so applyed, it will without farther trouble effect your desire.
And you may Protract it by help of your Line of Chords, and Line of equal parts, as this you see is done; or by the help of your Protractor, as before directed: and if there is any other Notable Castle, or Tower, or Place, lying in a right-Line with your Observation upon any Hill, you must remember always in taking of Inaccessible Heights and Distances; as also in Plotting Unpassable Distances, by reason of Water, that you take these two stationary distances as far asunder as may be. And if at any time you require the Altitude of a Church, Castle, or Tree, standing upon a Hill, you must perform it at two Operations; first by taking the Altitude of the Church, or Castle, or Tree together as one Altitude; and secondly by taking the Altitude of the Hill alone; then by substracting the height of the Hill from the whole height, the remainder shall be the height of the Castle, or the like.
And here Note also, That in the taking of all manner of Altitudes, whether accessible or inaccessible, you must always add the height of your Instrument from the Ground to the height found, the total is the true height. And thus much briefly touching this Matter.
The ART of Gaging of Vessels. CHAP. VIII. The Ʋse of the Line of Numbers, and the Lines on the Gaging Rod or Staff, and the Rules in Arithmetick in Gaging of all sorts of Vessels, (viz.) to Gage a Cube-Vessel, to Measure any Square-Vessel, and a Cylinder-Vessel; Also, Barrels, Pipes, or Hogsheads; to Measure a Vessel part out, to Measure a Brewers-Tun, or a Mash-Fat, to Measure a Cone-Vessel, to Measure a Rising or Convex Crown; and also a Convex or Falling Crown in a Brewers-Copper; also a Brewers Oval Tun.
PROBL. I. The true Content of a Solid Measure being known, To find the Gage-Point of the same Measure.
THe Gage-Point of a Solid Measure is the Diameter of a Circle whose superficial Content is equal to the solid Content of the same measure; so the solid Content of a Wine-Gallon according to Winchester measure, being found to be 231 Cube-Inches: if you conceive a Circle to contain so many Inches, you shall find the Diameter thereof to be 17:15 by this Rule. Example. A Wine-Vessel at London is said being the 66 Inches in length, and 38 Inches in the Diameter, would contain 324 Gallons. If so, by the Line of Numbers we may divide the space between 324 and 66 into two equal parts, the middle will fall about 146, and that distance will reach from the Diameter 38 unto 17:15 the Gage-point for a Gallon of Wine or Oyl after London measure: the like reason holdeth for the like measure in all places. Thus likewise you may discover the Gage point for Ale-measure, an Ale-Gallon, as hath been of late discovered containing 282 Cubique-Inches; for as 1 is to 1:273, so is 282 to 356, 3 whose square-root is 18:95 the Gage-point for Ale-measure, because of Wast and Soil exceeding that of Wine above two Inches: or you may find it as before by the Content 256, 3 and the length 66, and the Diameter 38, as before. There are several other Rules to find it, but these may satisfy to
save Prolixity, Mr. Phillips, and others, have found and proved by Example. That there is 288 ¾ Cubique Inches in an Ale-Gallon, which I believe is the Truth: But that which is received by Authority, are these
sorts of measures, the Wine-measure is 231 Cubique Inches, and for Ale 282 Cubique-Inches or Beer; and for Drie things, as Corn 272 Inches. These Rules are undeceivable with Authority.
Therefore take notice you must be very careful in all your measures of all sorts of Vessels, their length, breadth, and dephth, as also of the Head and Bong; for all small Errours in them may increase too much in the Content: for the mistake of a quarter of an Inch in a large Vessel, may make you misreckon a Gallon in the Content; therefore how to be careful is best known to the Practicioner more than I can declare
by many words.
PROBL. II. The Description of the Gaging-Rod, or Staff.
THe most useful Gaging-Rod is 48 Inches or 50 in length, upon one square there is 2 Lines, a Line of Numbers, and a Line of 48 Inches, every Inch divided into 10 parts for the ready measuring of any Vessels, length, breadth, or depth.
But for the measuring of Great-Vessels, there is two Staffs divided into Inches and 10 parts, made to slide.
On the second side is two Lines, the first to Gage by the Head, and the second by the Bong, which added together multiplyed in the length, will give the Contents; As by Example in the following Problem, and Use of a Table of Wine measure.
And the third square is two Diagonal Lines, for the Gage of Wine the first; and for Ale, the second: which shews the Contents to the 1/10 part of a Gallon according to 282 Cubique-Inches in a Beer or Ale-Gallon, the Use in Probl. 7.
On the fourth side is a Line of Segments, or 63 Gallons divided into 1000 parts, as you may have the Use by the following Table in Probl. 8. The making of this Staff is best known to the Instrument-Maker, by reason it must be exactly done; and you may have them of Mr. Philip Standridge in Bristol, and by Mr. Hays, and John Brown in London, Mathematical Instrument-Makers.
PROBL. III. The Description of Symbols of words for Brevity in Arithmetick.
VVHere these following Characters, are placed, you are to Work by these Rules; and that will resolve your Question.
+ Plus or Addition, which is as much as to say add.
− Minus or Substraction, then you must substract.
× In or Multiplication, now you are to multiply.
/2 To Divide by 2 or any other Number under the line.
= Equal to the thing desired.
q Square the Number given.
2 − q Twice squared, when 2 stands before the Letters.
C Cube the Numbers.
Z Sum and Z q Square of the Sum.
√ q To Extract the square Root.
Z The Sums of the Squares.
X Difference, and X. difference of the squares.
X q Squares of the difference.
AE The Rectangular or Plain of them, which is the Product of 2 Numbers multiplyed.
SUppose we have a Cubical Vessel to measure, whose sides let be ABCDEF, which let be every way 24 Inches, and I desire to know how many Gallons of Wine or Ale the same will hold.See Figure 102 in the following Scheme. ☞
For Beer or Ale by the Line of Numbers.
Extend the Compasses always from the Gage-Point (which for Ale is 16 8/10) unto the side of the Cube 24 Inches, the same extent will reach from the same 24, turning twice over unto 49 Gallons, and better.
For Wine.
Extend the Compasses always from the Gage-Point, which for Wine is 15 2/10 unto the side of the Cube 24 Inches: the same extent will reach from the same 24, turning twice over unto 59 85/200 Gallons, which is almost 60 Gallons of Wine.
The Arithmetical way.
AB,/282 C = Gall. of Ale 49. AB,/231 C = Gallons of Wine 59 [...]5/100.
PROBL. V. How to Measure any Square Vessel.
SUppose we have a Square Vessel to Measure, whose side AB let be 72 Inches, and breadth AC 32,See Figure 103 ☞ and the depth CD 8 Inches.
By the Line of Numbers for Ale.
You must first find a mean proportion between the length AB 72, and the breadth AC 32, by multiplying it together, and taking the Square Root thereof, or taking the middle way between 72, and 32 on the Line of Numbers, and you will find it 48 for the mean.
Now Extending the Compasses from the Gage-point 16 8/10 to the mean Number 48 Inches, the same extent will reach from the depth CD 8 Inches, turning twice over unto 65 36/200 Gallons.
For Wine.
To find how many Wine-Gallons it is, Work by the Gage-point 15 2/20 as you did in the last Rule, and you will find near 79 8/10 Gallons; or you may find a mean proportion between the breadth AC 32, and the depth CD 8: which will be 16 Inches, and so Work according to the former Rule.
How to Work the same without the Gage-Point.
Example for all. Extend the Compasses from the Ale-Gallon 282 unto the length AB 72: the same distance will reach from the breadth AC 32 unto 8: 17/104 Gallons: for an Inch depth, so for 8 Inches you may presently find it to be 65 36/100 Gallons.
For Wine-Gallons.
Take the Numbers 231 the Gage-point, which by the former work you shall find 9 975/1009 Gallons for 1 Inch depth.
The Mathematical way.
ABX: ACX CD / 282 = Ale-Gallons 8 17/100
ABX AC XCD / 231 = Wine-Gallons 9 975/1000
The Browers Coolers are measured all one as this Vessel is.
SUppose the Diameter of the Head AB be 24 Inches, and the length thereof AC be 30 Inches, To find the contents in Ale-Gallons.See Figure 104 ☜ Extend the Compasses always from the Gage-point, which for Ale is 18 95/100 Inches unto the diameter 24 Inches; the same distance will reach from the length 30 Inches turned twice over unto 48 13/190 Gallons.
For Wine.
Extend the Compasses from the Gage-point 17 15/100 unto the diameter 24; the same distance will reach from 30 turned twice to 58 7/10 Gallons.
The Arithmetical way.
For Ale. ABqXAC / 359 = Ale-Gallons (viz.) 48 13/190
For Wine. ABqXAC / 294 = Wine-Gallons, (viz.) 58 75/100
PROBL. VII. How to Measure a Globe-Vessel.
SUppose the diameter or height of the Globe be AB 24 Inches: Then to know the Contents in Ale or Wine, it is thus.See Figure 105 ☜
For Ale.
Extend the Compasses from the Gage-point, which is 23 21/100 unto the diameter AB 24 Inches; the same distance will reach from the same 24 turned twice over unto 25 2/3 Gallons of Ale.
For Wine.
Extend the Compasses from the Gage-point 21 unto the diameter 24 turned twice over, as before, you shall have 31 ⅓ Gallons.
The Arithmetical way. [...]
For Ale AB.C / 540 = Gallons, (viz.) 25 60/100
For Wine. AB.C / 440 = Gallons, (viz.) 31 40/190
PROBL. VIII. How to Measure a Barrel, Pipe, Butt, Punching, Hogshead, or small Cask.
SUppose you have a Cask to measure, whose length is AB 27 Inches, and depth at the Bong CD 23 Inches, and breadth at the Head EF 20 Inches.
You are to find a mean-diameter between the Head and the Bong by these Rules.
Take the difference between 23 and 20, which is 3: which being multiplyed always by 7, the Product here is 21, and divided by 10, the Quotient will be 2 1/10 which added to the lesser diameter 20, you have 22 1/19 for the mean-diameter.
Another way to find the mean diameter is thus. A Vessel having 20 Inches diameter at the Head, 23 Inches [...] the Bong, I would know the mean-diameter: 20 and 23 makes 43, the half is 21: 50 the lesser taken out of the greater, the difference is 3, which reduced into 10 is 300; then divide by 45, the Product is 6/10 added to 21 50/100 makes 22 1/10 Inches the mean-diameter required.
Extend the Compasses from the Gage-point always 18 95/100 unto the Mean-diameter 22 1/10, the same will reach from the length 27 Inches turned twice over, to 36 7/10 Gallons.
For Wine.
Extend the Compasses always from the Gage-point 15 15/200 unto 22 1/10, the same will reach from 27 to 44 and 8/10 Gallons, as before.
The Arithmetical way.
M stands for Mean, D. for Diameter, MD for mean-Diameter.For Ale. CD 2 q +: EFqXAB / 1077 = Gallons (viz.) 36 55/10 [...].
For Wine. CD 2 q +: EFqXAB / 880 = Gallons 44 78/100.
See Figure 106 ☞There is another way to Work this Vessel or Question, by the Mean-diameter which was before found to be 22 1/10 Inches; and that is after the Cylinder-Vessel, which may be resolved by the Line of Numbers, as before, and by
Arithmetick thus.
These sort of Cask Gaged 5 several ways.For Ale. MDqXAB / 359 = Gallons, (viz) 36 73/100
To know what it shall hold in all.Take the measure with your Rod from the Bong hole at C to the lower part of the Head at F, as the Line FC, which in the Example is near 25 4/10 Inches: so if you would know how much Ale the [...]ask will hold, you shall find the Bong Hole to cut in the Diagonal Line 36 7/10 Gall.Extend the Compasses from 231 to 282, the same extent will reach from the Content in Wine-Gallons 44 8/10 to 36 7/10 the Contents in all Gallons. And for Wine it will cut 44 ¾ Gall, the Contents required.
A Table for the Gaging of Vessels.
Head.
Bong.
Head.
Bong.
D The Diameter in Inches.
G pts.
G pts.
G pts.
G pts.
01
0,001
0.002
31
1,089
2,178
02
0,004
0.009
32
1,160
2.321
03
0,010
0.020
33
1,234
2.468
04
0,018
0 036
34
1,310
2.620
05
0,028
0.056
35
1,388
2.776
06
0,041
0.081
36
1,469
2.938
07
0,056
0.111
37
1,551
3.102
08
0,072
0.145
38
1,636
3.272
09
0,092
0.183
39
1,724
3.448
10
0,113
0.226
40
1,813
3.625
11
0,137
0.274
41
1,904
3.809
12
0,163
0.326
42
2,000
4.000
13
0,192
0.383
43
2,096
4.191
14
0,222
0.444
44
2,194
4.388
15
0,255
0.510
45
2,296
4.588
16
0,290
0.580
46
2,398
4.796
[...]7
0,328
0.557
47
2,504
5.007
18
0,367
0.734
48
2,611
5.222
19
0,409
0.818
49
2,721
5.442
20
0,453
0.906
50
2,833
5.665
21
0,500
1.000
51
2,948
5.895
22
0,548
1.097
52
3,065
6.129
23
0,600
1.199
53
3,184
6.367
24
0,653
1.305
54
3,305
6.609
25
0,708
1.416
55
3,428
6.856
26
0,766
1.532
56
3,554
7.108
27
0,826
1.692
57
3,682
7.364
28
0,888
1.777
58
3,813
7.625
29
0,953
1.906
59
3,945
7.890
30
1,020
2.040
60
4,080
8.160
By the same Rule you may know what an Ale-Cask will hold in VVine.
The Use of the two Lines upon the Rod marked Head [...]nd Bong; and of this Table for Wine-measure.
The Use of this Table is the Root of the usual Making and Use of the lines on the Rod or Staff only. In the Table you have the perfect Number, but you must number upon the Staff, for 10 account 100, and every small Division is 10; and you must estimate the parts of these small Divisions: then is the Work all one as with this Table, (viz.)
You must measure the Diameter first at the Head, and find the Number in the Table, or Staff belonging to it; then measure the Diameter at the Bong, and likewise in the Table, or on the Staff, find the Number belonging to that; then add those two together, and multiply the Sum thereof by the Inches of the Vessels length, measured from Head to Head in the Inside.
The Table and Staff shews for 20 Inches at the Head.
0,453
For 20 Inches at the Bong.
1,199
These 2 added together, make
1652
27
Which being Multiplyed by
11564
27, the length,
3304
Makes
44604
According to this Operation, it should be 44 Gallons 604/1000 parts, which difference is of no Moment in these Conclusions.
The Table of Segments.
Gals
Parts.
Gal.
Parts.
Gal.
Parts.
63
10000
42
6288
21
3712
½
9705
6223
½
3647
62
9530
41
6158
20
3582
½
9390
6094
½
3517
61
9280
40
6040
19
3452
½
9170
5976
½
3387
60
9065
39
5913
18
3321
½
8962
5850
½
3255
59
8862
38
5787
17
3189
½
8765
5724
½
3123
58
8661
37
5662
16
3056
½
8580
5600
½
2986
57
8491
36
5535
15
2918
½
8404
5476
½
2847
56
8319
35
5415
14
2775
½
8236
5354
½
2703
55
81 [...]4
34
5294
13
2630
½
8072
5234
½
2556
54
7990
33
5174
12
2481
½
7909
5115
½
2405
53
7829
32
5057
11
2328
½
7758
5000
½
2250
52
7672
31
4943
10
2171
½
7595
4885
½
2091
51
7519
30
4826
9
2010
½
7444
4766
50
1928
50
7370
29
4706
8
1846
½
7297
4646
50
1764
40
7225
28
4585
7
1681
½
7153
4542
50
1509
48
7082
27
4462
6
1420
½
7012
4400
50
1329
47
6944
26
4338
5
1235
½
6877
4276
50
1138
46
6811
25
4213
4
1038
½
6679
4150
50
1935
45
6679
24
4087
3
1035
½
6613
4024
50
830
44
6548
23
3960
2
720
½
6483
3906
50
602
43
6418
22
3842
1
470
½
6353
3777
50
295
PROBL. VIII. By the Line of Segments on the Rod or Staff, and also by a Table, How to find the Quantity of Liquor in a Cask that is part full.
SUppose you would know the Quantity of Liquor in a Cask whose depth at the Bong is 23 Inches, as before, and let the Liquor be in height 16 Inches, and the whole Cask to hold 44 85/190 Gallons.See Figure 107 ☜
By the Line of Numbers on the Staff, the proportion will be, as the whole depth 23 Inches is to the depth in Liquor 16 Inches, so is 1000 to 692 parts.
Which being sought for in the Segment-line on the Staff, you shall have in the Line by it 46 15/1 [...]0 Gallons.
Now if you extend the Compasses from 63 to 46 75/100 Gallons; the same distance will reach once from 44 85/100, the Contents of the whole Cask to 33 6/10, that is, 33 6/10 Gall. of Wine in the Cask.33 − 6/10
Then by this Rule always as 63 Gallons is to 46 75/100 Gallons, so is 44 85/100 Gallons to 33 60/100 Gallons of Wine in the Cask.This table is made to ½ or 50/100 parts of a Gallon: by same Rules you may make him Quarts or Pints.
By the same Rule Work for Beer or Ale.
To Work this Arithmetically is somewhat tedious; wherefore I have here Calculated a Table whereby you may perform it very easy by help of the Rule of Three.
Example.The top or Bong is uppermost in the Column to the left hand 63 Gal. the bottom is ½ to the right hand Colum.
In the last Vessel whose depth at the Bong is 23 Inches, and depth in Liquor 16 Inches
The first Rule of Proportion.
As 23 is to 16, so is 10000 to 69 21 parts, which sought for in the nearest Number in the table of Segments you shall have against it 46 75/100 Gallons nearest.
As the whole Radius 63 Gallons is to 46 75/100, so is the Gage of your Vessel 44 85/100 Gallons to 33 54/100 Gallons near, as before.23 − 7:10000 to 30493043 answers 15:76/10063; 15.76 so 44 85/290 = 11 22/100 Gall.
After this manner of Working you shall have for 7 Inches depth of Liquor 11 22/120 gallons.
And so by these Rules you may work for any other Cask.
PROBL. IX. How to Measure a Brewers Tun, or a Mash-Fat.
LEt the Tun be ACDE, whose Diameter in the bottom let be ED 98 Inches, and the Diameter at the top AC let be 90 Inches, add both the Diameters together, you have 188 Inches; then take the half thereof, and it is 94 Inches, this is the Mean-diameter FG; then get the height of the Tun, which let be AB 40 Inches. Now to know how many Barrels of Ale or Beer it will hold according to 36 Gallons to the Barrel, you shall Work thus.
By the Line of Numbers.
Extend the Compasses always from 113 7/10, which is the Gage-point for a Barrel unto the Mean-diameter 94, the same distance will reach from the height 40 Inches turned twice over unto 27 ½ of a Barrel.
The Arithmetical way by the Mean-diameter.
FGqXAB / 359 = Gallons 984 5/10.
Which being divided by 36, you have 27 Barrels 121 ½ Gallons.
Or thus for Barrels.
FGqXAB / 12924 = Barrels 27 35/ [...]90 [...], near as before.
This Arithmetical way by the Mean-diameter is not absolute true, yet near enough for Brewers Tuns, by reason there is difference of Diameters between the bottom and the top; yet it is seldom above 7 or 8 Inches: But to have an Exact way which also serveth for Coopers or any, take this way for Working this Tun for an Example.
Divide 985 1/10 Gallons by 36 Gallons in a Barrel, and the Quotient 27 Barrels, and 13 Gallons remains: so the one will hold 27 Barrels 13 Gallons 115/1000 parts.
Or thus for Barrels.
EDq + AC q, + EDXAC. ZXAB / 38772 = 27 Barrels 378/1000.
PROBL. X. How to Measure a Cone-Vessel, such as is a Spire of a Steeple, or the like, by having the Height and the Diameter at the Base.
SUppose the Diameter at the Base AB be 98 Inches, and the height DC 490 Inches.
Then by the Line of Numbers for Barrels of Ale or Beer.
Extend the Compasses from the Gage-point 169 9/10 unto the Diameter AB 98, the same will reach from the height of the Cone DC 490,See Figure 109 ☜ turned twice over unto 121 4/10 Barrels ferè.
This is the best Proportion to Work for great Cones to have it in Barrels, but small Cones have it in Gallons.
Then thus Work.
Extend the Compasses from the Gage-point 32 82/100 unto the Diameter of the Base 98, the same will reach from the height of the Cone 490 twice turned unto 4369 547/1009 Gall.
The Arithmetical way. For Gallons.
ABq × DC / 1077 = 4369 547/1000 Gallons.
Which being divided by 36, you have 121 34/100 Barrels.
Or thus for Barrels.
ABq, XDC / 38772 = 121 347/1000 Barrels.
An Example.
The Brewers Tun before measured may be measured, after this manner by Cones, by this Example in this Figure I have proportioned the same Tun in this Cone, as you may prove thus by, the Rule of Proportion to find the Diameter on the top EF 90, it was before.
Work thus.
As CD 490 is to AB 98, so is CG 450 to EF 90 Inches; and so back again, to find the height of the greater Cone, say, as the difference of the Diameters 8 Inches is to the height of the Tun 40 Inches: so is the Diameter of the bottom AB 98 Inches to the greater height DC 490 Inches, from whence substract 40, there remains the height of the lesser Cone GC 450 Inches.
Now Working as before, for the Contents of each Cone.
The greater Cone will be found to be
4369 Gall.
547 Parts.
And the lesser Cone to contain
3384 Gall.
432 Parts.
Which substracted from the greater Cone, there remains 985 Gallons 115/1000 Parts.
985 Gallons
115/1000
For the Brewers Tun, as before found in the 9 PROBL. which is 27 Barrels 13 Gallons. 1/10
PROBL. XI. How to Measure a Segment or portion of a Globe or Sphere, which serves for a Convex Signet or Rising, or Falling Crown in a Brewers Copper.
ADmit you have the Diameter of the Crown AB 80 Inches, and the height thereof CD 6 Inches.
A Convex Rising Crown.
See Figure 110 ☞The Falling Crown is nothing but this Figure, the upper part turned down.
Note that a Crown is seldom less then 2 Inches, nor above 12 Inches; for in Bristol in all their Crowns belonging to the Brewers Coppers, the least that was found, was 1 6/1 [...]Inch, and the greatest height or depth 11 2/10 Inches.
By the Line of Numbers.
Extend the Compasses from the Gage-point 18 95/100 unto the Diameter AB 80, the same distance will reach from half the height CD 6, which is 3 being turned twice unto 53 Gallons ½ ferè.
The Arithmetical way for Ale or Beer-Gallons.
ABq. X ½ CD / 359 = 53 48/100 Gallons.
PROBL. XII. How to Reduce Ale-measure into Wine; And likewise to Reduce Wine-Gallons into Ale.
For Example.
THere is a Vessel that holds 60 Gallons of Ale; the Question is how many Gallons of Wine it will hold.
The Proportion of the backward Rule of 3.
As 282 Ale is to 231 Wine ∷ so 60 Ale to
73; 19/77 Wine-Gallons.
Or thus. As 94 is to 77 ∷ so is 60 Ale to
The reason is thus, 231 Ale-Gallons is 282 Wine-Gallons, or 77 Ale-Gallons is 94 Wine-Gallons.
Or, as 282 to 231, so is 116. 4 to 95. 30, and as 231:282, so is 95:30 to 116 [...]/10, or extend the Compasses from the Ale-gallon 282 to the Wine-gallon 231, the same distance will reach from 60 to 73 2/10 Gallons, or from 77 to 94, or from 94 to 77.
PROBL. XIII. How to Measure a Brewers Oval Tun.
LEt the length in the bottom be AB 120 Inches, and the breadth EF 90; let the length at the top be CD 112 Inches, and the breadth 84; also the depth 40 Inches CA.
See Figure 111 ☞A Brewers Oval Tun.
Now to Work this, you must find a Mean-proportion between the length in the bottom 120, and the breadth 90 Inches.
The Arithmetical way.
AB × EF ♈ q = 103 − 92. That is, Multiply the two Numbers together, and of the Product thereof extract the Square Root▪ so shall you have the Mean-proportional Number.
By this Rule you will find a Mean-proportion between the 120 at the bottom, and the breadth 90, to be 103 and 92.
And likewise for the top between 112 and 84, will be 97; then as before, you shall find the Sum to be 200 − 92; the half thereof is the Mean-diameter 100; [...]6/10 [...]Inches; so shall you Work all one, as you did in the Round Tun.
How to get the Mean-diameter by the Line of Numbers.
Let the Numbers given be 120, and 90. Extend the Compasses from 90 to 120: Divide that in half, the same distance will reach from 90 to 103 92/100 almost 104 the Mean Number required; and so likewise between the Number 112 and 84, you will find it 97; then as before, you shall find the Sum 200:92 and ½ 100 46/100 Inches.
The Arithmetical way, as before, is thus.
M. Diam: XCA / 12924 = 31 3/36 Barrels of Ale or Beer.
And in Gall. M. Diam. XAC = 1119 219/1000 Gallons of Beer or Ale.
By the Line of Numbers for Barrels.
Extend the Compasses always from the Gage-Point 113 7/10 to the Mean-diameter 100 46/100: the same will reach from the height 40 turned twice over, to the Quantity of Barrels of Ale or Beer 31 2/10.
PROBL. XIV. How to Gage a Vessel by Oughtred's Gage-Rule.
THis is an Instrument by taking the length in Inches and 10 parts, and is as Exact as any way Instrumental extant; both the Diameters at the Head and Bong, with a Line called Oughtred's Gage-Line.
The Use is thus.
Take the Diameter at the Bong with those Divisions before said from that end where the Divisions begin to be numbred, and set that down twice: and on the Diameter of the inside the Head in this manner,063063051177 and then add them together, as here you see the length in Inches. Suppose to be (30. 82) then say, as 1, is to 1:77, so is 30:82 to 54 55/100 of a
Gallon, being a little more then ½ a Gallon, or 54 Gallons ½ the Content of a High-Country Hogshead; and so you may do by any other great or small sort of Cask.
CHAP. IX. Wherein is shewed both Arithmetically and Instrumentally How to Measure exactly all kind of plain Superficies, as Walls, Timber-work, Roofs of Houses, Tyling, Board, Glass, Wainscot, Pavement, and the like; as also Timber and Stone.
PROBL. I.
FOrasmuch as it is very requisite for a Compleat Artist to know how to Measure all manner of Buildings, as Walls, Timber-work, Tyling, and such like; I shall in the following Example make Illustration thereof.
Note this, that Walls and Tyling are measured by the Rod of 18 Feet, Wainscot by the Yard or Feet, and Board and Glass by the Foot only. Therefore measuring any of these things, consideration must be had to the just Form and Figure thereof: Then by the following Rules you will soon have the Area Content thereof.
As for Example.
Suppose there be a Wall in the Form of the Figure, and it is required to know how many Perch, or Rods, Yards, and Feet are contained therein.
The Arithmetical way for Perch.
ABXAD, or thus, ABXAD.ZX2/16 ½ = 76 Perch.
Extend the Compasses always from 16 ½ to the length 66, the same Extent will reach from the height 19 Foot unto the true Contents of the Wall 76 Perch.
[geometrical diagram]
Then to bring it into Rods, and Feet, and Yards, Work as before. Multiply 66 by 19, the Product is 1254 Feet; which Divide by 324 (because there is so many Square Feet in a Rod,) and the Quotient is 3 Rods, and 282 remains, which divide by 9 (for so many Feet is contained in a Yard,) and the Quotient is 31 Yards, and 3 remains which is Feet; so this Wall being 66 Foot long, 19 Foot high, there contains 3 Rods ¾ Yard, 4 Feet (for as 81 Feet or 9 Yards is a quarter of a Rod.
But suppose ABC be a Govel Dormant Pike, such you must measure them as Triangles, to bring it into Feet. Multiply 16 the Perpendicular by half the Base AC, the Product is 160, the Contents in Feet double it: divide by 33 half Feet, the Quotient is 9 Perch, 11 ½ Feet remains. Work by the Line of Numbers as in the last Rule, and it will be 9 Perch 7/10, (or divide 160 by the Quotient 17 Yards and 7 Feet, remains the true Contents of the Dormant Pike, that is 1 Rod, 8 Yards, 7 Feet.
[geometrical diagram]
But in Measuring of Chimneys which require more Workmanship then other ordinary Walls, they are usually accounted at double measure. First measure them as single-measure. Take the length of the brast Wall EF, and the 2 side Angles DE, and FC, which Multiplyed into the height CB the Product of that Multiplication doubled, yieldeth the Content.
According to the Customary measure allowed for Chimneys that stand in a Govel or Side-wall, (but if the Chimney stand by himself, the Back is to be measured with the rest of the Chimney; but the Back standing against a Govel or Wall is accounted part of the Wall, and must not be measured with the Chimney.
[geometrical diagram]
Admit this Figure LIK. GH. AB. DC be a Chimney to be measured, and according to double measure, the Content is required.
First measure the Base CF the Brast-wall DC, and FC the side Angles, which together makes 24 Feet; next take the height of the Square CB 18 15/100 Feet, which Multiplyed together, the Product is 435:60 Feet for the Content of the Figure ABDC. Then for the □ and Brast-Wall GH, and side Angles is 15 Feet, height n H 6:26 Feet: X as before, makes 93:90 Feet, for the Content of the Square GH m:n.
In like manner of Working, you will have the Contents of the Square IK:RV:92:16; likewise the Chimney-S [...]aft in compass is 9 Feet, and 8 Foot high. × together as before, is 72 Feet for the Contents: add these 4 Sums together, the Sum is 693:66 Feet doubled is 1387 Feet [...] 32/100 Feet the Content of the Chimney according to Customary measure.
Feet. Parts.
The Squares. ABDC:
435:60
The Squares. GHmn:
93:90
The Squares. IKRV:
92:16
The Shaft IL:
72:00
The Sum
693:66
doubled.
693:66
The Total Sum.
1387:32
Which reduced into Perches as before, is 5 Perch 26 Foot 07/100 Parts; or into Rods, is 4 Rods 10 Yards 1 Foot 12/100 according to these Measures: But it is fit the Master-Workman should Measure it, and should have so much Arithmetick, as to Multiply and Divide, or else he cannot be a Compleat Workman in every part.
Note that after the same order Slate-work and Tyling are measured either by Perch or Rods of 18 Foot square. Note that Roofs of Houses, and Timber-work, Partition-Floors, and the like, are reckoned by the Square of 10 Foot, but Worked by the same Rule, as have been already delivered in this Problem; therefore it needs no other precept.
PROBL. II. How to Measure Boards, Glass, Pavement, Wainscot, and the like.
IN the last Problem we have shewed that Boards, Glass, Pavement and Wainscot, and the like, they are commonly accounted by the Foot or Yards; Therefore to make this plain, we shall instance only upon Boards which are cut out in long squares commonly.
How to Measure them.
Take the length and breadth in Inches and Parts, Multiply one by the other, the Product will shew the Content in Inches; (that divide by 144 the Numbers of Inches in one Foot, the Quotient will tell you the Number of Feet, and the remainder is Inches.
For Example.
Su [...]h Rules are made by Walter Hays in More-Fields, and in Brictly by P. Staynard.Admit I have a Board that is 7 Foot long, and 18 Inches broad; Multiply 84 Inch. which is in 7 Feet by 18 Inches, the Product is 1512; which divide by 144 the Number of Inches in a Foot of flat-measure, and the Quotient shews 10 feet, and 72 remains, which is ¼ 144; therefore the Board contains 10 foot ½; but many times the Board falls out to be broader at one end, then it is at the other, add together the breadth at each end; then take the ½ for the true breadth.
And Work as before; But commonly Artificers have a Useful Line put upon their Rules for their ready Measuring of Board and Timber-measure; but this is the Exactest way, though that is near; and what have been said of Board-measure, only the same is to be understood is the way of Measuring not only Boards and Glass; but likewise all manner of Wainscot, Pavement, Floors, and such like; they depend upon one and the same Geometrical Ground, though they be reckoned by different measures, as you see by the Perch, Rod, Square, Yard or foot according to the Custom of the Place, therefore needs no further Example.
Extend the Compasses always from 12 Inches unto the breadth 18 Inches, the same extent will reach from the length 7 foot, unto the Number of Square foot in the Board, which is 10 5/10 foot.
AB 18 Inch × AD 7 foot/ = 10 4/10 feet.
PROBL. III. The Mensurations of Solid Bodies of Timber and Stone, and first of Squared-Timber.
VVHatsoever hath length, breadth and thickness, is called a Solid-body; as Timber, and Stone, and the like, which are usually measured by the foot: and therefore you are to observe that a foot of Timber or Stone is accounted a foot square every way in the Form of a Die; whereby it plainly appears that a foot of Timber is 12 times more than a foot of Board, which it 144 Inches; but a foot of Timber must be 1728 Inches.
To find the Square-Root by the Tables.For Timber that is squared you may find the Contents thereof on this wise; First find a Mean betwixt the two Sides at the End. Admit the height at the End be AC 16 Inches, the breadth thereof AB 25 Inches the half Sum.
By the Tables
120412
Add together
139794
Sum.
260206
The half Sum.
130103
is the Square.
Root and Mean-proportion between 16 and 25 which is 20.
Divide, and take the middle between 16 and 25, and you will find the Mean 20 as before; Then to know how many foot of Timber is in a Square of 16 Inches in height, 25 Inches broad, and 14 foot long.
Extend the Compasses always from 12 Inches unto the Mean-proportion or side of the Square 20 Inches; the same will reach from the length 14 foot turned twice over to 38 9/10 foot of Timber.
The Arithmetical way.
Thus. AB × AC. × AD / 144 = to the Contents in feet. 38 128/144
Or thus. AB × ACAD / 1728 = reduced into inches 168 in 14 foot 38 1536/1728 feet, as before found.
Yet it is common with the Carpenters to add the broad and narrow side together,A common Erro [...] amongst most Carpenters. and to take the ha [...] thereof; the true Square that way is very erroneous, especially when the difference between the side is much.
[geometrical diagram]
In the former Example one side is 25, the other 16, the Sum 41, the half 20 50/100 inches; that is, half an inch too much, as was proved by the former Rules; that is just 20 for the Mean or true Square: so that by taking ½ the 2 sides 20 ½, it makes the piece of Timber 40 foot 4/10, when indeed it is but 38 9/10 feet, which is 1 foot and half too much.
Now if a Piece of Timber that is tapering, the Common Rule is to take the Mean betwixt both ends; and so to Work as in the last Form, but it is not absolute true.
For Example.
Admit a Piece of Timber were Square at one end 25 inch. and at the other 15 inches, and 14 foot long.
This is the absolute Arithmetical way.
ZAB, DE + AE. XAD / 432 = Contents 39:7/ [...].
PROBL. IV. How to find how many Inches in length will make one Foot of Timber, being alike in the Squares.
SUppose you have a Piece of Timber that is four-square 16 inches every way, and you would know how many inches in length will make one foot of Timber.
By the Line of Numbers.
Extend the Compasses always from 12 inches to the side of the Square, which in this Question is 16 inches, the same turned twice over from 12 inches, will reach to 6 ¾ inch. in length for one foot of Timber.
1728/AC16q = Cf 6 192/256 or 6 75/100 Inches as before.
Having the side of a Square piece of Timber, at the end and the length in feet, to find how many feet is contained therein.
Admit the side of the Square at the end be AC 16 Inches, and the length thereof 14 foot, Then
By the Line of Numbers.
Extend the Compasses always from 12 Inches unto the side of the Square AC 16 inches; the same distance will reach from the length 14 foot turned twice over unto 24 [...] [...]/ [...]feet in the Piece of Timber.
The Arithmetical way.
AC 16 q × BD 14 foot./144 = The Content in Feet 24 12 [...]/1 [...]
PROBL. V. How to Measure Round-Timber five several ways.
ADmit you were to Measure a Piece of Round-Timber, as a Tree whose Diameter or thickness at the end is 20 inches; I desire to know how many inches in length will make one foot of Timber.
By the Line of Numbers.
Extend the Compasses from the Diameter AB 20 inches unto the constant Number 13, the same distance will reach from the same 13 turned twice over unto 5 ½ inches for one foot, as AD.
The Arithmetical way. 2220/ABq = AD 5 50/100 inches for one foot.
Having the Diameter of a Piece of Timber, as admit it to be 20 inches, and the length suppose 15 foot; To find the Contents in feet.
By the Line of Numbers.
Extend the Compasses always from 13 54/100 to the Diameter AB 20 inches, the same distance will reach from 15 the length turned twice over unto the Contents 32 7/10 feet in the tree.
Square the Diameter AB 20, and it is 400; treble it by 3, and it is 1200, multiply it by the length 15, and the Product is 18000, that divide by 550, and the contents is 32 717/1000 feet of timber in a round piece or tree, which is 32 foot, and about ¾ quar.
Here is likewise another brief Rule, Arithmetically thus.
Square the Diameter AB 20, and it will be 400; Multiply that by 11, and it is 4400, divide it by 14, and the Quotient is 314, and 4 remains, which multiplyed by the length 15, the Product is 4714; that divide by 144, and the Quotient is 32 7/10 as before.
Or else you may find the Contents of the Circle by this Rule, as 7 is to 22, so is the Diameter to the Circumference; or multiply half the Diameter by half the Circumference, and the Product is the Content of the Circle; that Multiply by the length, and divide by 144, gives the Content of the Timber or Tree in feet or parts.
Now the common way used by Artificers, is to measure round a Piece of Timber or Tree, and to take the one fourth part for the Square, which is very erroneous and false.
For Example.
The measure of the Compass or Circumference by the Rule before-going is 62 9/10 inch. of the round piece of timber or tree the ¼ thereof is 15 70/100 inches, which they take to be the Square; which Multiplyed into it self, produceth 243:66 for the Area of the Base; which Multiplyed by the length 15 foot, the Product is 365490, the Contents in feet and parts; that divided by 144, the Quotient is 25 371/1000 that is differing from the Truth no less then 7. 356/1000, that is,
7 foot and about a quarter too much: the Buyer hath then his due; but I conceive they agree in the Price to stand to that measure, by reason of the wast in Chips before it is brought into Squares; but the best way will be to measure the tree right, and afterwards allow for the Wast; or else in time the Error will be taken
for Truth, and Truth will be accounted Error, as it is by too many this day.
How to Measure a Round piece of Tapering Timber.
Admit the Diameter of the Great End of a piece of tapering timber be AB 20 Inches, and the Lesser End CD 16 Inches, and the length EF 15 foot. To find the Contents, add both the Diameters 20 and 16, the Sum is 36, the half is 18 for the Mean.
Then, Extend the Compasses always from 13 54/100 to the Mean-diameter 18, the same will reach from the length 15 foot turned twice over unto 26 51/190 foot. Or by Arithmetick, Square 18 the Mean-diameter, and it makes 324; that treble by 3, the Product is 972; that Multiply by the length 16, the Product is 14586, that divide by 550, and the Quotient is 26 51/100 foot, the Contents of the taper-piece of timber is 26 foot and half.
PROBL. VI. How to Measure a Pyramedal piece of Timber.
This piece of timber is measured by this Rule,A right Lined sharp Piece of Timber is called a Pyramide(viz.)
ADmit you have a piece of timber to measure, whose length at the Base is 25 inches AB, and breadth AC 16 inches: and the length of the piece DE 15 foot.
[geometrical diagram]
By the Line of Numbers.
First, by the Line of Numbers find a Mean-proportion between 25 and 16 by dividing it into 2 parts, and the middle will fall upon 20 inches, the Mean-proportion required.
Then extend the Compasses always from 20 78/100 unto the Mean-diameter 20 Inch. the same distance will reach from the length 15 foot turned twice over unto 13 88/100 foot of timber.
The Arithmetical way.
AB × AC × DE / 432 = 13 foot 888/1000 parts of timber in the Piece.
AB × ACX ⅓ DE / 144 = 13 foot 888/1000 parts, that is 13 foot and above three quarters.
PROBL. VII. How to Measure a Conical piece of Timber.
ADmit you had a Cone Piece of timber whose Base or Diameter at the End AB is 28 inches, and the length thereof CD 15 foot, it is required to know how many feet of timber is in the Piece.
But the Base thereof Round it is a Cone.
[geometrical diagram]
Extend the Compasses always from 23 43/100 unto the Diameter AB 28, the same distance will reach from the length 15 foot turned twice over unto 21 4/19.
The Arithmetical way.
ABq × CD / 550 = 21 38/100 foot of timber in the Cone Piece.
And by the former Rule you may Measure any part of a Cone or Pyramide-piece. Admit you were to cut a Piece of 5 foot at the greater End, and you find the Diameter EF 18:95 Inch. First, Mean-diameter 18:95 and 28 Inch. added is 46; 95 the half is 23:48 the Mean: then extend the Compasses from 13 54/100 unto the Mean-diameter 23 48/100; the same distance twice repeated from the length 5 foot, will reach to 15 [...]7/10 foot in the ⅓ of the Cone at the great End; And likewise to Measure EFHG the Diameter HG is 9. 45 added to 18 95 EF, the Sum is 28:4.
The ½ is 14; 20 Inches the Mean-proportion the length 5 foot; by the former Rule you will find in that Piece of timber 5 50/100 foot; and to Measure the little Cone GH 9 45/100 inches diameter and 5 foot long; Work as to Measure the whole Cone, and you will find it 81/1 [...]0 parts of a foot.
foot.
parts.
15
07
5
50
0
81
21
38
And so you have truly Measured the Pieces, as you may find by adding them up, and they make 21 foot 38/199 parts, as you found in the whole Cone at first; and so by finding the Area of the Circle and [...]part, you may find the Segment of any Cone or Pyramide that is Square in the sides by the Area thereof; by the same Rules you Measure Stone. It is needless to make more Examples in this thing.
CHAP. X. For the Burden of a Ship, or her Tunnage, Take these Rules following.
SECT. I.
SUppose you were to Gage a Ship that the length of her Keel is 45 foot, the breadth of the Beam 17 foot, the depth of her Hould 9 foot always to find the Tunnage.
Multiply the breadth by the length, and with the Product Multiply the depth in Hold, and divide by 100, and the Quotient will shew you the Tunnage to be in this Example 68 85/100 tun.
Or extend the Compasses always from 100 to 17 the breadth, the same distance will reach from 45 the length, to 7 65/100.
Then extend the Compasses from 1 unto 7 65/100, and the same distance will reach from the depth in Hold 9 foot to the tunnage 68 85/100 tun of King's tunnage.
But for Merchants Ships who give no allowance for Ordnance, Masts, Sails, Cables, and Anchors, which is all a Burden, and no tunnage
You must siork thus for the tunnage.
SECT. II.
45 × 17 × 9/95 = 72 45/100 Tun Burden.
Or, extend from the Gage-point 95 always to the length of the Keel 45, the same will reach from the breadth 17 of the Beam to a 4 Number, as to 8 1/1 [...]; then extend from 1 to 8 1/10 the same distance will reach from the depth in Hold 9 foot to the Burden 72 45/100 tun.
SECT. III. Having the Proportion of any one Ship in Burden, with the length of her Keel-Timbers; To Build another of any Burden according to that.
PROPORTION.
ADmit I have a Ship of 80 Tun, the length of her Keel is 46 foot. Now I am to Build a Ship whose Keel must be 65 foot; I desire to know how many Tun she must be.
Extend the Compasses from 46 foot unto 65 foot, the same extent will reach from the Burden 80 Tun, being turned 3 times over unto 225 7/19 tuns.
ADmit you had a Ship of 226 tuns, and the length of her Keel is 65 foot; Now I would Build a Ship of twice the Burden, that is 452 tuns; Now I would desire to know the length of her Keel.
Extend the Compasses from 226 unto 452, the ⅓ of that distance will reach from the length of her Keel 65 foot unto the length of the greater Ship's Keel 81 9/19 foot ferè.
The Arithmetical way.
65. C. × 452/226 = 81 3/1 [...]0 foot ferè.
I could have insisted upon more Examples, but it is to no purpose; by reason the Carpenters have these Rules in Practice most of them; and for Gaging and Measuring Ships, the breadth and sharpness of her bottom is to be considered; and to abate something of 95 the Divizer, or add something to it according to Judgement and Reason; and so likewise to 100, to find
the tunnage.
CHAP. XI. The Application of the Line of Numbers in Common Affairs, as in Reduction of Weight and Measure of Cheese, Butter, and the like.
I Have added this Chapter, not for that I think it absolutely necessary; but only because I would
have the absolute applicableness of the Rule to any thing be hinted at, that it may
be known; that any thing may be measured by Rule, as well as by Weight, so far as there is Proportion considering that, and any thing else; the Application
of which I leave to the Industrious Practitioner, only here I give a hint.
What have been said of other things in Reduction, is general in any other, as from 12 to 10 either Shillings or Inches to tenths, as of a Shilling, or tenths of a Foot, or Pence or Farthings, Ounces or Chauldrons, Hundreds, either weight or tale. The Rule is thus, (viz.) In either one shilling, or foot, hundred, or the like. If 100 is 12 d. what shall 66 be? facit 8 pence or 8 Inches; that is, Extend the Compasses from 100 to 12, the same will reach from 66 to 8; and so of all other.
If 100 be 112 P. what shall 50 be? facit 66 Pound; If 100 be 8 Pints, what shall 25 be? facit 2 Pints; If 100 be 48 farthings, what shall 30 be? facit 14.4, that is 3 d. 2 f. ½ near.
If 100 be 36 bushels, what shall 24 be? facit 8 bushels ½ and better.
If 100 be 60 min. what shall 50 be? facit 30 min. or ½ an hour.
If 100 be 120, what shall 80 be? facit 96. The like is for any Line of Reduction. Now if you would know how many there must be in any greater Number then one; then
say, By the Line of Numbers thus: If 48 farthings be one shilling, how many shillings is 144 farthings? facit 3 shillings. For the extent from 48 to 1, will reach from 144 to 3; And again, if a Mark and a half be one Pound, how many Pound is 12 Mark? the extent from 1:50 to 1 shall reach from 12 to 8; which reason must help you to
call it 8 Pound. Again, if 3 Nobles be one Pound, what is 312 Nobles? facit 104 l. the extent from 3 N. to 1 P. will reach from 312 to 104: Further, if a Chauldron of Coles cost 36 shill. what shall ½ a Chal. cost.? facit 18. But more to the matter; If 36 bushels cost 30 s. what shall 5 bush. cost? facit 4 s. 2 d. If one week be 7 days, how many days in 39 weeks? as 1 to 7, so 39 to 273 days in 39 weeks; As 8 furlongs make 1 mile, how much is 60 furlongs? facit 7 ½ miles, for the extent from 1 to 8, gives from 60 to 7:50, and the like of all other.
CHAP. XII. The most Excellent Gunners Scale, Which resolves the Chief Principles of the whole Art of Gunnery, in a very brief and Compendious form, never by any set forth in the like nature before;
with divers Excellent Conclusions both Arithmetical, and Geometrical, and Instrumental; and by Tables being framed both with, and without the help of Arithmetick. As also divers Artificial Fire-Works, both for Recreation, and for Sea and Land-Service.
SECT. I. The Qualifications every Gunner ought to have, and the Properties, Duty, and Office of a Gunner.
HE ought to have skill in Arithmetick, to work any Conclusion by the single and double Rule of 3, to abstract both the Square and Cube Roots, and to be perfect in the Art of Decimal Arithmetick, and to be skilful in Geometry; to the end he may be able through his knowledge in these Arts, to measure heights, depthts, breadths and lengths; and to draw the Plot of any Piece of Ground, to make Artificial Fire-Works which are used in the time of War: A Gunner that hath a Charge ought to have in readiness all necessary things for his Artillery:
As Wheels, Axle-trees, Ladles, Rammers, Sheep-skins to make Spunges, Gun-powder, Shot, Tampions, Chain-Shot, Cross-bar-shot, Canvas, or Strong Paper to make Cartredges, Fire-works, Artificial Torches, Dark Lanthorns; again, to Mount and Dismount Guns, Hand-spikes, Coyns, Budge-Barrels to carry Powder, and Baskets to carry Shot to your Piece. When leisure will permit, he is to choose good Match-cords, to Arm his Linstocks in readiness to light, for to give Fire, and also a pair of Caleper Compasses to measure the Diameters of Shot, or the Muzzle, or Base-ring, or the like; and also a small Brass pair of Scales and Weights, a Ruler divided into Inches, and 8 Parts in every Inch, for the ready measuring of Cartredges, how to fill them.
A Gunner should never be without such a Scale as this as I have here described, and to know the Use thereof perfectly; and thereby
be ready to give a reasonable answer to any Man of any Question belonging to any sort
of Ordnance used in England in a moment, as this Scale will do, as shall be shewn: He should always carry a pair of Compasses with him to measure the Diameter or Bore of any Piece; and also the length of the Cylinder within, the better to fit her with a Shot, and proportion a Charge.
A Gunner ought to know the Names, Length, Weight and Fortification of every Piece about the Chamber (that is as far as the Piece is Laden with Powder;) and be able to tell readily how much Powder is a due Charge for any Piece, what Shot is fit, how many Matrosses must attend the same, how many Horses or Oxen will draw the said Piece, or Men, if occasion be; He must be careful in making Choice of a sober honest Man, for the
Yeoman of the Powder; and he must not beat up the Head of his Powder-Barrels with an Iron tool, but with a Wooden Mallet, which can never Fire the same: A Gunner ought to trie his Piece, to know whether it be true bored or not, to proportion his Charge according to the thinnest side of the Metal, and accordingly take his Observation
at the Britch of the Piece, just over, where by his Art he finds the middle of the Bore within the Piece is; by which means a good Shot may be made out of a bad Piece.
Before he makes a Shot, he is to consider, that if the Piece lie point blank, or under Metal, he ought to put in a sufficient Wadd after the Shot, to keep it close to the Powder; for if it should not be close, but some distance between the Powder and Shot, the Piece will break in the vacant place; but in case you mount your Piece, put no Wadd after the Shot.
And one chief thing is to know very well how to Disport his Piece, be it either true bored, or not true bored, which he may try first.
When a fit Man is entertained, the Mr. Gunner (whom he serves,) should bring him to his Pieces, and give him the Denominations of his Piece, and parts thereof; which when he hath learned, which is the base-ring, and trunnion-ring, the mussel-ring, and the like, (you may see their names all plain in the Fig. of the Gun without more words;) and likewise the Crows, Handspikes, the Coyn, and the like; and how far in the Bore is called the Chamber of the Piece: These things, with the Gunner's care well understood, he may give them further Directions, (viz.) But it is great pity, that the Gunners at Sea did not exercise the Sea-men in this knowledge, as the Corporal doth in Mustering of them with their Musquets; for want of the like knowledge, the greatest part of common Sea-men, are as dull and ignorant, when they be required to stand by a great Gun in time of Fight; and therefore it would be much for the Credit and Honour of our English Nation, to train up their Sea-men in this knowledge especially; but it is taken notice of, that if any man have any
Art above another, he is afraid to let another see him do any thing, or understand from
him such knowledge, for fear he will be in a short time as able as himself; which
many do attain unto in a short time to be as able as himself without their help;
therefore it is more for their Credit to teach them what they know.
SECT. II. Who were the Inventors of Gun-powder, and some Principles of Philosophy fit to be known.
SOme Italians have writ that Archimides the Philosopher was the first Inventor of Guns and Gun-powder: or whether this be truth or not, Learned Men are of divers minds; Munster, and Gilbert Cognot have written, that Guns were devised first in the year 1370 by a Monk, whom Munster calls Bertholdus; sithen our Country man Dr. Dee in his Mathematical Preface, and Discourse of Menader saith, that an English-man was first Inventor of Gun-powder in another Country, and they first made use of it from him; also our English Chronicles do report, that in the year 1380 a Monk did accidentally let fall a spark of Fire upon Brimstone and Saltpeter beaten to Powder in a Morter covered with a Slat-stone, he seeing this mixture blow off the Stone from the Morter, did thereupon devise a
kind of Powder, and taught the Venetians how to use the same in Pipes of Iron against the Ganvates.
Every Simple Body is either Bright and Light, or else Gross and Dark, and Ponderous,
and according to the variety and difference, it is always naturally carryed towards
some one or other part; the World hath height as upwards, or depth as downwards; and the depth dependeth upon the Influence of the height.
All pure and rare bodies ascend, as the Fire more than the Air; but the thick and
gross bodies descend, as the Earth more than the Water.
Nothing worketh naturally, but in that which is contrary to it, and more feeble; the
form working, is aided by the Qualities; and the matter suffering, which suffereth
by the Quantity.
Nature is extremely curious, as well of her perfection, as her conservation; and then
when all things conspire, as well the Action that cometh from the Agent, as the Passion
from the Patient, hath proportion.
Accident hath its variety from the Subject, and goeth not from one thing unto another.
Every Corporal thing reposeth in its natural place: Motion may be made any where within
the Orb of the Moon. Nature admitteth no Empress.
A body rarifying it self, the place thereof increaseth as the body increaseth, the
resistance of the moved proportion to the Mover, furthereth the motion; the longer
the Chace of a Piece, the louder the Report; also the force of the stroke dependeth on the swiftness of
the Course.
SECT. III. The Description and Use of the Gunners Scale, upon which is all sorts of Ordnance, from the Canon, to the Base of their Weight, Lading, Shot, and all other things appertaining to them.
THis Scale is made according to the Diameter of our English Ordnance, but 8 inch. long, the Diameter of a Canon-Royal; and it may be made of Silver, Brass, or Box, or any other f [...]e grained Wood, that will not warp. Upon one side I have set the Names of all sorts
of Ordnance, and in the Angle of meeting with the Names, is the diameter of the bore; and betwixt that and the next less diameter, is first the common length of such Pieces; and upon the step of breadth, is how many Paces these Pieces shoot point blank, and right in the Angle of meeting, betwixt the two diameters with the Angle of meeting with the Names, is first the weight of the Gun, the breadth of the Ladle; and thirdly, the length; fourthly, the weight of the Charge in Powder; fifthly, the diameter of the Shot; sixthly, the weight of the Shot; seventhly, a Line of Inches; eighthly, each Inch divided into 10 parts, and likewise into 8 parts, which are parts and half quarters, which the Line of Diameters of the bore comes from. The degrees in the divisions, and on the thickness and length thereof, there is a Line of Numbers, by which you work all the most useful Questions in Gunnery, as you will find in the following page.
The Use of this side is thus.
Suppose you come to a Piece of Ordnance, and it is desired to know what Piece it is; take the Scale, and put it into the bore of the Piece, mark the step of a Diameter that fits it, and the Angle of Diameter goes down into the Line of Inches, and parts, and that diameter goes into the side in the Angle of meeting, and tells you the Name of the Piece: Betwixt the next less Diameter, right under, you have as before, the common weight of the Piece, the breadth and length of the Ladle, weight of Powder, diameter of Shot, and weight.
As for Example.
Admit I came to a Gun, and found by the former directions, that her diameter of the bore is 4 ¼ Inches. And in the Angle of meeting in the side, I find her Name is Demiculvering, lower then ordinary; at the end thereof I find 9 or 10 foot the usual length, and betwixt the next less diameter and the step is 174 the paces the Piece carries the Bullet in a level-line, point blank, right against weight in the next less Diameter, which is 4 Inches, is the usual weight 2000 l. breadth of the Ladle 8, and length 12 Inches, the weight of the Powder 6 ¼ or 4 ounces; and next the diameter of the Shot 4 Inches; and next, the weight 9 l. So that you see the next less diameter is the diameter of the Shot, as well as of a less Piece of Ordnance. This I have made plain to the meanest capacity: Here they are set down in this Table following.
The Explanation of the Scale may serve likewise for the Table; only take notice, that under Inches and Parts, is to be understood the first; to the left hand is Inches, and the other is so, many 8 parts of an Inch.
As for Example.
Admit you enter the Table with a Saker of the lowest sort, the height of the bore is 3 4/8 Inches, 8 foot long, the weight 1400, breadth of the Ladle 6 4/8, length 9 6/8 Inch. weight of the Powder 3 pound 6 ounces, diameter of the Shot 3 2/8, weight of the Shot 4 pound 12 ounces, and the paces the Piece carries, by Alex. Bianco's Tables is 150 of 5 foot to the Piece.
Observe that the Ladle is but 3 diameters of the Shot in length, and 3/5 parts of the Circumference from the Canon, to the whole Culvering, I allow the Charge of Powder to be about two diameters of the Piece: from the Culvering to the Minion; the Charge to fill two diameters and a half; all from the Minion to the Base three diameters of Powder.
The names of the Pieces of Ordnance.
Diameter of the Bore.
Length of the Gun.
Weight of the Gun in pounds.
Breadth of the Ladle.
Length of the Ladle.
Weight of the Powder.
Diameter of the Shot.
The weight of the Shot.
He shoots point blank.
The
Inches. Parts.
Feet. Inches.
Pounds
Inches. Parts.
Inches. Parts.
Pounds. Ounces.
Inches. Parts.
Pounds. Ounces.
Paces.
⌊8
⌊8
⌊8
⌊8
⌊8
A Base.
1:2
4:6
200
2:0
4:0
0:8
1:1
0:5
60
A Rabanet.
1:4
5:6
300
2:4
4:1
0:12
1:3
0:8
70
Fauconets.
2:2
6:0
400
4:0
7:4
1:4
2:2
1:5
90
Faucons.
2:6
7:0
750
4:4
8:2
2:4
2:5
2:8
130
Ordinary Minion.
3:0
7:0
750/800
5:0
8:4
2:8
2:7
3:4
120
Minion of the largest size.
3:2
8:0
1000
5:0
9:0
3:4
3:0
3:12
125
Saker the lowest sort.
3:4
8:0
1400
6:4
9:6
3:6
3:2
4:12
150
Ordinary Sakers.
3:6
9/9:0
1500
6:6
10:4
4:0
3:4
6:0
160
Sakers of the oldest sort.
4:0
10:0
1800
7:2
11:0
5:0
3:6
7:5
163
Lowest Demiculvering.
4:2
10:0
2000
8:0
12:0
6:4
4:0
9:0
174
Ordinary Demiculvering.
4:4
10/10:0
2700
8:0
12:6
7:4
4:2
10:11
175
Elder sort of Demiculvering.
4:6
12/13:0
3000
8:4
13:4
8:8
4:4
12:11
178
Culverings of the best size.
5:0
12/10:0
4000
9:0
14:2
10:0
4:6
15:0
180
Ordinary Culvering.
5:2
12/13:
4500
9:4
16:0
11:6
5:0
17:5
181
Culvering of the largest size.
5:4
12/10:0
4800
10:0
16:0
11:8
5:2
20:0
183
Lowest Demicanon.
6:2
11:0
5400
11:4
20:0
14:0
6:0
30:0
156
Ordinary Demicanon.
6:4
12:0
5600
12:0
22:0
17:8
6:⅙
32:0
162
Demicanon of great size.
6:6
12:0
6000
12:0
22:6
18:0
6:5
36:0
180
Canon Royal, or of
8:0
12:
8000
14:6
24:0
32:8
7:4
58:0
185
The Description of the other side of my Gunner's Scale.
Upon the other side is a Scale of 8 Inches divided into four quarters, and betwixt each quarter above it is three Columns; the Inches shews the height of all sorts of Iron shots from 2 ounces to 72 pound; and of Lead from 3 ounces to 806 pound ½, and of Stone from 1 ounce to 26 12/16 pounds; each distinguished from other by their names, written in the first Inch, the Table is in the sixth Section, and the weights and measures, accommodated into our English Averdupoiz weight of 16 ounces to the pound, and to our Foot of Assize of 12 Inches to the Foot. The Line of Inches being likewise divided into 10 parts, the whole into 80, may serve for 800; for Protraction as follows: There is also the Gunners Quadrant divided into 90 degr. in the outmost Limb, and in the second Limb within, is divided into the 12 points of the Gunner's Quadrant, and [Page]
[geometrical diagram with representation of a cannon]
[Page][Page 49]each point 4 parts; and in the third Limb is a Geometrical division of right and contrary shadows, for the ready taking of heights and distances; but there is also a Geometrical Quadrate, with each side divided into 10 parts, which stands for 100, and each 10 parts divided into 10 more, the Use thereof in taking of heights and distances is in the 16 Chap. of the Second Book of the Description of Instruments: But the Use for to level, or else to mount or Imbase any piece of Ordnance, is in the 34 Sect. of this Book. To the side thereof is fitted a piece of Brass of the same breadth as the Scale in thickness, with two holes within an Inch of each End, and two Screws fitted to serve the four holes, as you may see in the Figure to the side of the Scale, that if you would level or mount any piece of Ordnance, Screw the plate to the end of the side B, with both Screws, and put the plate in the bottom of the metal as far as he will go, and put the tomping in upon him to keep the plate fast, and then level or mount your Piece, as in 33 Section directed.
But if you will Imbase any piece of Ordnance to any place or point assigned, you must screw the plate to the end A, and let the side with the Line of Numbers be next the muzzle, and stop him with the tomping, as before; then Imbase your Piece, or put him under the Line of Level as you will, to what degree you please; and when you have done, screw the plate to the side A B, with a screw at one end, and a screw at the other, (there is also over the weight of the Shot a division of the right Ranges, and likewise a proportion of Randoms of any piece of Ordnance, upon any mounture from degree to degree; and likewise you may put the division of Inches in the 38 Section, for the number of Inches and parts from 5 foot to 14 foot long, requireth to mount her to any degree of mounture with great facility and ease. There is also Triangle-wise a plain Scale, that goes along down by the degrees of diameters, or steps, the Line is a Line of Chords, with the Gnomon-line, and a Line of six hours of the same Radius, and a Line of Rhumbs, with the Line of Sines; and this is for the making any sort of Dial in any Latitude by the following directions, and also for the Plotting any Triangle, or resolving any Question in Navigation, or Astronomy. You must remember, there is a Brass Pin in the Center at C for to hang the Plummet and String, with the Lope upon.
Thus I hope I have fitted all ingenious Gunners with a Scale so useful, that I will leave it to them to give me commendation for my labour and
pains. If I might advise Gunners of all sorts, that are able to have one of these Scales of Brass or Wood, to carry about him, to resolve any Question presently for his own credit, and it
is very portable and fit for his Pocket; but it is best to have a case of Leather
or Cloth to keep it clean; and you may carry a pair of Compasses with him, and by him you may resolve most of all the Questions in this Noble Art of Gunnery.
On the side of the Quadrant betwixt the Equinoctial, and the Radius, or Suns greatest Declination is a division to every 10 minutes of the Suns Amplitude Rising and Setting answerable to the Ecliptick Line, and the Declination on the other side the Figure, makes all plain to any Instrument-maker; without further precept.
SECT. IV. The Ʋse of the Line of Numbers on the Scale, for the help of such as cannot Extract the Cube and Square-Root.
How by knowing the weight of one Bullet, to find the weight of another Bullet, the height being given.
A Bullet of Iron of 6 Inches height, weigheth 30 l. what will the like Bullet of 7 Inches in Diameter or height weigh; always take these Rules.
Extend the Compasses from 6 Inches to 7 Inches Diameter, the same distance will reach from 30 l. weight, turned 3 times over unto 47 l. 10 ounces, the weight of a Shot 7 Inch. high.
That is Cube 6 makes 216, and Cube 7 makes 343; then by the Rule of Proportion Multiply 343 by 30, the Product is 10240, divide by 216, the Quotient is 47 pound, 138/216 or 63/100, which is 10 ounces ¼ as before; there is something less then 7 Cube Inches in one pound of Iron.
By the Tables of Logarithms.
The Logarithm of 6 is
07781512
The Logarithm of 7 is
08450980
Substract the uppermost Number out of the lower, the diff. increasing.
0669468
3
The last Number Multiply by 3, and the triple of this difference is
2008404
Added to the Logarithm of 30 l. with no ounces, (which is 3000)
34771212
Gives the Logarithm of the weight 47 64/100
36779616
Now to know how many ounces 64/199 is, work thus by the Rule of Proportion.
If 100 gives 64, what will 16 ounces give? Answer, 10 ounces 24/100; so the Shot of 7 Inches diameter weighs 47 l. 10 ounces 24/100 or 47 64/100 pound, the like way of work is with all such Questions.
SECT. V. Admit the weight of an Iron Bullet being 30 pound, the Diameter was 6 Inches, the weight being 47 64/100 what may the Diameter be?
16:100 10 24/100100001001000241024FIrst I will shew you how to turn 10 ounces and 24/100 into 100 parts of a pound; always say, If 16 give 100, what shall 10 give 24/100 (64 as you may see the work
in the Margent, where the weight is known, and the Diameter required.
Always Divide the weight 30 l. and 47 l. 64/100 into 3 equal parts, and that distance will reach from 6 Inches Diameter, to 7 Inches the Diameter required on the Line of Numbers.
By the Tables the Logarithm of 30 is
34771212
The Logarithm of 47 64/100 l. is
36779616
[...]
Uppermost Substracted from it, leaves the difference increasing,
2008404
The difference divided by 3, or the third part of this difference.
0669468
added to 6 Inch. Diameter, the Logarithm
7781512
Gives the Logarithm of 7. Inch. Diameter required
08450980
This is the most easy, ready, and certain way of Arithmetick; and so work for all such Questions, if three Numbers be given, to find a fourth in a Triplicated Proportion.
SECT. VI. The Geometrical finding the Diameter for the weight of any Shot assigned.
MR. Gunter in his first Book, Section 4, hath shewed the Making of the Line of Solids on his Sector: but this Rule shews the proportion of the Diameters in weight: having a Shot of one pound 2 pounds or 3 pounds weight of the Metal or Stone assigned; if it be of a pound, divide the Diameter into 4 equal parts, and 5 such parts will make a Diam. for a Shot of the said Metal or Stone that shall weigh just two l.
And divide the Diameter of a Shot that weighs just 2 l. into 7 equal parts, and 8 such parts will make a Diam. for a Shot of 3 l. weight; And divide the Liamet. of a Shot of 3 l. weight into 10 equal parts, and 11 such parts will make a Shot for 4 l. weight.
And divide the diam. for a Shot of 4 l. weight into 13 parts, 14 such parts will make a diam. of a Shot for 5 l. weight.
And divide the diam. of a Shot of 5 l. weight into 16 equal parts, 17 such parts will make a diam. for a shot that will weigh 6 l. and so dividing each next diam. into 3 equal parts more then the next lesser was Divided into, and it will with one part added from
a diamet. of a shot that will weigh just 1 l. more; and so you may proceed infinitely, increasing or decreasing, by taking one
part less, as is appointed to be Divided, for one l. less, and the next into 8 l. less, to abate 1 for the Remainder, infinitely decreasing it.
[geometrical diagram]
A second Geometrical way.
First you must have exactly the diamet. of a shot that weigheth one pound, and then describe a Circle, whose diamet. shall be just equal thereunto; and Divide it into 4 Quadrants,
[geometrical diagram]
[Page 52] with 2 diamet. cutting each other in the Centor orthogonally. Then take the Chord of the whole Quadrant 90 degr. BC in your Compasses, and lay it from the Centor of the first Shot one pound D to 2,The Diameter of a Shot of 1 pound is 1 Inch 94/100 parts. and so A 2 will be the Diameter of a Shot of 2 pound; Then extend the Compasses from 2 to the Chord C, and lay that distance from D to 3, so will A 3 be the Diameter of a Shot of 3 l. And so likewise extend the Compasses from 3 to C, it will reach from D to 4, and from 4 to C, and it reaches from D to
5, and from 5 to C, lay it still always from D to 6; and so continuing till you have
proceeded as far as you will: You shall find that if AB were the Diameter of one pound, A 2 is the Diameter of 2 pound, and A 3 is the Diam. of 3 l. and A 4 the Diam. of 4 l. and A 5 the Diam. of 5 l. A 6 the Diam. of 6 l. and lastly, A 8 is the Diam. of 8 l. and so you may proceed in like manner infinitely.
Likewise having the Diameter of a Shot of any weight, the double of the Diam. is the Diam. of a Shot which weighs 8 times as much. Thus the double of A 1, which is A 8, makes the Diameter of a Shot of 8 pound; and so the double of A 2, which is the Diameter of a Shot of 2 l. makes A 16, the Diameter of a Shot of 16 pounds, that is 8 times 2 pounds; and so the double of A 3 makes the Diameter of a Shot of 24 pounds, and the double of A 4 makes the Diameter of a Shot of 32 pounds, four times 8 being 32; and so you may proceed as you please, and find the bigness
of any Shot.
A third way.
This you may do also, having the Diameter of a Shot of one pound, double that diam. it will make a diam. of 8 pound; and treble the diameter of one pound, will make a diameter of a Shot of 27 pound, and quadruple or 4 times the same, will make a diam. of a Shot of 64 pounds, and 5 diameters will make a Ball of 125 l. and 6 diameters of a Shot of one l. will make a diameter of a Shot that will weigh 216 l.
Now it is convenient to shew how to find the Mean-divisions between these extremes; as for the diameter of a Shot of 2 l. 3 l. 4 l. 5 l. 6 l. 7 l. or what more you will; so as by such progression you may proceed from pound to pound, until you come to the last term of 216 pound; nevertheless the same manner of working will proceed infinitely. Lay the forementioned
6 diameters upon one and the same right Line; you must at the end of them draw another Right-line orthogonally, and set therein the diamet. of 2 such Shot given as at C, and from thence draw another Right-line parallel to the first, as GH, and then draw a Quadrant as A B, and from the Centre G draw right Lines through all the divisions of the diam. marked upon the right Line AF which are all equal, so shall you have 6 divisions to be divided; the first being divided already, and is the diam. of a Shot of 1 l. but the second division is to be in the Circumference or Quadrant divided into 7 parts equally, because it containeth the second diameter unto 8, for adding 1 to 7 it makes 8; the third division is into 19 equal parts, which being added to 8, makes 27; the fourth shall be divided into 37 equal parts; which together with 27, makes 64; the fifth shall be divided into 61 equal parts, which added to 64, makes 125; and lastly, the sixth place must
be divided into 91 equal parts, unto which adding 125, you shall make a diameter of a Shot of 216 pound justly.
Now forasmuch as these divisions are difficult to make well, within so small a Quadrant: you may therefore describe a greater, as the Quadrant LM, and there the divisions are more distinct, and larger than in the lesser they can be; Further, you may note,
that Fire-balls, Granadoes, and other Globous Artifices, must have the same proportion to their Grandures from their Ball of one pound, which may be exactly considered; and so by this Method you may make Balls of Lead, Brass, Stone, and Granadoes, Fire-balls, and all other Spherical Fire-works, of what weight you will, having one of one pound first, to lead you accordingly.
SECT. VII. To find what proportion is between Bullets of Iron, Lead, and Stone, by knowing the weight of one Shot of Iron; to find the weight of any other Shot of Lead, Brass, or Stone of the like Diameter.
THe proportion between Lead and Iron, is as 2 to 3, so that a shot of 2 pound of Iron, is of like diameter or height as 3 l. of Lead.
As for Example.
A shot of 6 Inches diameter weighs 30 pound, to find the weight of a shot of Lead of the same diameter.
By the Rule of Proportion.
First, if 2 gives 30, what will 3 give? multiply and divide, and the Quotient is 45, [...] the weight of a shot of Lead.
By the Tables, the Logarithm of 2 is
03010300
The Logarithm of 30 is
14771212
The Logarithm of 3 is
04771212
Add the 2 lowermost, the sum is
19542424
Substract the upper Num. the Remain is the
Log. of 45 l. the weight of the shot in Lead of the same diam.
16532124
Extend the Compasses from 2 to 30, the same distance shall reach from 3 to 45; (In like manner work by the rest following. [...])
The porportion between Iron and Stone, is as 3 to 8; so that a shot of 30 pound of Stone, is as big as the like shot of 80 l. of Iron; and 11 l. ¼ of Stone, is of the same diameter 6 Inches, as a shot of 30 l. of Iron and 45 l. of Lead; the proportion between Lead and Stone, is as 4 to 1; so that one shot of Lead of 40 l. is of the height as a Stone shot of 10 l.
The proportion between Lead and Brass, is as 24 to 19,
The proportion between Iron and Brass, is as 16 to 18.
By these Rules aforegoing you may Calculate with ease, if Iron shot be wanting, and the other to be had, what height and weight either shot of Lead, Brass, or Stone, ought to be of to fit any Pieces of Ordnance; and by the same Rules here is a Table faithfully Calculated; and doth shew the weight of any shot of Lead, Iron, and Stone, from 2 Inches diam. to 8 Inches, and Quarters of Inches; the proper Stone for this purpose is Marble, Pibble, Blew hand Stone; (there may be a little difference of weight in some sort of Stone: but these do neer agree in weight; you must remember in loading your Piece with a Shot of stone, you must not have so much Powder as you do with Iron-shot, but abate according to proportion, as is between Stone and Iron.
Inches.
Quart.
Iron. Poun. Ounc.
Lead. Poun. Oun.
Stone. Poun. Oun.
2
1 1
1 10 ½
0 7
2
1
1 9
2 6
0 9
2
2
2 2
3 3
0 12
2
3
2 14
4 5
1 1
3
3 12
5 10
1 7
3
1
4 12
7 2
1 13
3
2
6 0
8 15
2 4
3
3
7 5
11 .00
2 12
4
8 15
13 07
3 6
4
1
10 10 ½
16 0
4 0
4
2
12 10½
18 15
4 12
4
3
14 14
22 5
5 9
5
17 05
26 2
6 8
5
1
20 1
30 2
7 8
5
2
23 2
34 11
8 11
5
3
26 6
39 9
9 14
6
30 00
45 00
11 04
6
1
34 00
51 00
12 12
6
2
38 00
57 00
14 04
6
3
42 00
62 00
15 12
7
48 00
72 00
18 00
7
1
53 00
79 08
20 00
7
2
58 00
87 00
22 12
7
3
64 00
96 00
24 00
8
71
106 8
26 10
The use of the Table, to find the weight of any Shot of Iron, Lead, or Stone from 2 to 8 Inches Diameter.
This Table is exactly Calculated, and the use thereof is very easy; we will make it plain by
two Examples; I would know of Shot of 6 Inches, their weight in Iron, Lead, and Stone: the first Column is Inches, the second Quarters of Inch. the third Poun. and Ounc. of Iron, fourth Pounds and Ounces of Lead, fifth Poun. and Ounces of Stone.
Enter the Table with 6 Inches diam. in the first Column, and in that Line you shall have 30 poun. of Iron, 45 pound of Lead, 11 pound 4 ounces of Stone, the weight of 6 Inches diam. And likewise, for 4 Inches ¾ diam. the weight of an Iron Shot is 14 pound, 14 ounc. of Lead 22 pound 5 ounces, of Stone 5 pound 9 ounces; and so of the rest.
SECT. VIII. How by knowing the weight of one Piece of Ordnance, to find the weight of another Piece being of that very shape of the same Metal, or any other Metal.
FIrst, with a pair of Crallapers take the greatest thickness of your Piece, as at the Base-Ring; and also the Piece, whose weight you know not.
Example.
Admit a Brass Saker of 1900 weight, hath his greatest thickness 11 ½ Inches; Now I find the diam. of the other Brass Piece, whose weight I know not, to be 8 ¾: then always by these Rules:
If the greatest diam. and weight is given, to find less weight, or else the contrary.
As the Logarithm greatest diameter, 11 50/100
306069
The Logarithm of the least, 8 75/100
294200
The Difference increasing.
11869
3
× 3 or the Triple of this Difference Substract
35607
From the Logarithm of the weight given 1900
327835
Rest the Logarithm of 837, the weight required,
292228
Or extend the Compasses from 11 ½ to 8 75/100 Inches diam. the same distance will reach from the weight given 1900 pound turned 3 times over to 837 pound.
The Arithmetical way.
C 8 ¾ × 1900/C 11 ½ = 837 l. weight almost in Brass.
But if the Piece had been Iron whose weight you sought, you must always do as before with the Brass, and find the difference of their Metals by the last Problem, which is 16 to 18, then
say by the Tables,
As the Logarithm of Brass, 18,
125527
is to the Logarithm of weight in Brass 835
292272
So is the Logarithm of proportion of Iron 16
120412
The Sum
412684
to the Logarithm of the weight in Iron 744
287157
Or extend the Compasses from 18 to 837, the same distance will reach from 16 to 744 l. weight in Iron.
Arithmetical way.
X 837 by 16/18 = 744 l. of Iron almost.
SECT. IX. How to make a Shot of Lead and Stone, the Stone being put in the Mould in which the Leaden Shot should afterwards be cast, to be of the like Diameter and Weight as an Iron Shot is of.
Inches.
Quart.
Lead. Poun. Ou.
Stone. Poun. Oun.
Both together. Poun. Oun.
1
0 1 ⅔
0 0 ⅓
0 2
1
2
0 6¼
0 1 ¾
0 8
2
0 14
0 4
1 2
2
2
0 12
0 8
2 4
3
3 2
0 10
3 12
3
2
5 0
1 0
5 0
4
7 7
1 8
8 15
4
2
10 8
2 2
12 10
5
14 7
2 14
17 5
5
2
19 4
3 12
23 0
6
25 0
5 0
30 0
6
2
32 0
6 0
38 0
7
40 0
8 0
48 0
7
2
48 0
10 0
58 0
8
59 0
12 0
71 0
It is found by experience, that if you take 5 parts Lead, and one part Stone, it will come very near the matter, wanting not above 3 Ounces, which is nothing,
respecting the difference you shall find in Pibble Stones. Here you have a Table how much Lead, and how much Stone must be together, to make the equal of Iron Shot, from 1 Inch, and to every half in the first and second Column to 8 Inch. Diameter; the third Column is how much Lead, the fourth how much Stone, the fifth how much weight both together.
SECT. X. How by knowing what quantity of Powder will load one Piece of Ordnance; to know how much will load any other Piece whatsoever.
ADmit you have a Saker of three Inches three quarters at the bore diam. and it requires 4 pound of Powder; what will a Demi-Canon of 6 ½ Inch. require? Work by these Rules always.
As the Logarithm of 3 75/190 diam.
257403
The Logarithm of 6 50/1 [...]0 Inch. diam.
281291
the difference increasing,
23888
(3
The triple of the difference added
71664
The Logarithm of 4 l. of Powder, 0 ounces.
160206
to the Logarithm of 20 8/10 or 20 84/199 l. of Pow.
231870
[...]So that the Demi-Canon must have 20 pound 13 ounces for her Charge of Powder; reduce the Fraction as before in the Margin into ounces.
By the Scale, extend the Compasses from 3 75/100 to 6 50/100 Inches diam. the same distance turned three times over from 4, will reach to 20 84/1 [...]0 pound weight, as before.
The Arithmetical way.
C 6 50/100 ×4/3 75/100 C = 20 l. 13 ounces of Powder for to load a Demi-Canon.
You are likewise to understand that the Demi-Canon should be fortified so well as the Saker by this Rule.
The diameter of the Saker is 3 75/100 Inches
257403
The Demi-Canon diam. is 6 50/100 Inches
281291
the difference increasing,
23888
(3
The triple of the difference by (3)
71664
added to the Logar. of 1600 weight of Saker
320412
gives the Logar. of 8332 the demi-Canon,
392076
Also by the Scale, and Arithmetick Rules, as in the foregoing Rules you will find the weight of the Demi-Canon 8332 pound, proportionable according to the Saker; but suppose the Demi-Canon to be no more than 6000 weight, then you must use these Rules.
The supposed weight of the Demi-canon 6000
377815
add the weight of the Powder well fortified, is 20 84/100
331889
The sum is
709704
Substract the weight of the Gun well fortified 8332
392074
leaves the weight of the Powder 15 pound,
317630
Fifteen pound being a sufficient Charge for that Piece: or extend the Compasses from 6000 to 8332, the same distance will reach from 20 84/100 to 15 l. of Powder, as before.
Thus you are always to take care of over-loading your Piece, which error many run into, when they call a Piece a Demi-canon, they presently load her with so much as is allowed for such a Piece so named, seldom examining whether the Piece have Metal enough for such a Charge; by which mistake they endanger their own lives,
and others which stand near. Now, for easy plain Rules, I say you never had before
laid down in this manner, to resolve these things; for if you compare these Rules
with Nath. Nye, Master Gunner of the City of Worcester, or any other Art of Gunnery, you will find a great deal of difficulty in Cubing and Extracting the Cube Root, and with reducing and Fractions (which here you may do five Questions, for one that
way, and more true and near, therefore I compare them to his Rules.
How to make the true dispert of any true bored Piece of Ordnance.
Now we have found how to proportion Shot and Powder to any Piece of Ordnance true bored; before we Load and Fire,By the same Rules you may find the Diameter of a Shot with a String. let us find the true Dispert to direct the Shot to the assigned mark.
Girt the Piece about the Base Ring round at the Britch with a Thred, and also the Muzzle Ring at the Mouth, and divide them two measures into 22 equal parts, which you may presently
do, by applying it to a Scale, that hath an inch divided into 10 parts, and Divide the parts by 7, and Substract the greater out of
the lesser, and take half the difference, is the true Dispert.
As for Example.
Suppose when I have measured the length of each String, and Divided, it into 22 equal
parts, I find that 7 parts of the longer String is 11 inches, and 7 parts of the shorter is 9 inches; I Substract 9 out of 11, and the remain is 2, the half is 1, which is the true Dispert.
Another way to Dispert any Piece.
If you have a pair of Callipers, as in the general Figure ACB, as you take the diameter of a Shot, and apply it to a Scale Divided into 8 or 10 parts, to know the Contents thereof; so with the Callipers take the greatest thickness or diam. of the Base Ring, and by your Scale see how much that is; as admit that the length of the Line a:b:c:d, where the diam. of the Base Ring, then take the diam. of the Muzzle Ring; as admit it be a, b, as you may try by the Figure of the Gun in the general Figure; then Divide the difference b d into 2 equal parts, and one of them is the Dispert, put it upon the Muzzle of the
Gun as CB, and stick it fast on the top of the Muzzle Ring with a little Pitch or Wax, and from the Base Ring at A in the Figure to the top of the Dispert at B, take aim to the Mark you would
shoot to, and that is the way to hit; but it Callipers be wanting, take a Stick that is straight and flat, and 2 Strings with two Musket Bullets at the end, and two Loops made at the other end, the Stick being something more than
the diam. at the Base Ring, and put the Stick upon the top of the Ring at the Muzzle, as you see the Fig. HK on the Gun, and put the Strings so nearer and farther, until they only touch the side of the
matter of the Muzzle Ring, and mark the Loops on the Stick, and put the Stick on the Base Ring, and do in like manner, and mark the Sticks; and the Work will be the same, as it
were taken by the Callipers; and the difference of the two Notches on the Stick will be ab the Base Ring, and ab the Notches of the diam. of the Muzzle Ring, and half the difference bc or cd is the Dispert, as before, if the Piece be true bored.
A fourth way to Dispert a Piece of Ordnance.
If the Piece be not Chamber-bored, take the Priming Iron, and put it down in the Touch-hole, until it rest upon the Metal in the bottom of
the bore, there make a mark with the Base Ring; likewise apply the Priming Iron to the bottom of the Metal at the mouth, and so much higher as the mark is which
you made at the Base Ring, than the Muzzle Ring, the difference is the true Dispert.
SECT. XI. How to know whether your Piece be Chamber-bored.
FIrst you may Dispert your Piece the three first ways, and when they agree in one, take that for the true Dispert;
then with your Priming Iron take the Dispert this last way; which done, compare it with the other Dispert first
found, and what it wants, is the just difference of the Chamber from the Bore of the Piece.
Admit the Dispert truly found by the two first ways be three Inches, as by this last way is but two Inches, it shews that the Chamber differs from the true Bore on each side one Inch; so that if the Bore of the Piece be five Inches high, the Chamber being one Inch on each side lower, is, but three Inches high: the like Observation we would always have you to make, that you may not afterwards
be deceived in making Cartredges of Canvas or Paper to load the same.
SECT. XII. How to know what Diameter every Shot must be of to fit any Piece of Ordnance, or to choose Shot for Ordnance.
TAke the Diameter of the Bore of the Piece, and Divide into 20 equal parts, and one of those parts is sufficient vent for any
Piece, the rest of the 19 parts must be the height of the Shot; but now adays most Gunners allow the Shot to be just one quarter of an Inch lower than the Bore; which Rule makes the Shot too big for a Cano [...], and too little for a Faulcon; but if the mouth of the Piece be grown wider, then the rest of the Cylinder within by often shooting; to fit Shot to such a Piece, you must trie with several Rammers-heads, until you find the Diameter of the Bore in that place where the Shot useth to lie in the Piece; and a Shot of one twentieth part lower than that Piece is sufficient; therefore let Gunners remember to trie the Piece, as directed.
SECT. XIII. How to find what Flaws, Cracks, and Honey-combs are in Pieces of Ordnance.
THere is one good way, as soon as you have discharged a Piece of Ordnance, cover the mouth of the Piece close, and stop the Touch-hole at the instant time; if there be any unknown Cracks or Flaws which go through the Metal, a visible Smoak will come through those Cracks and Flaws; if not, the Gun is not cracked.
There is a way to reflect the Sun-beams when he shineth, with a Looking-glass or Steel
in at the hallow Cylinder of the Piece; for by this means a bright and clear light will be within, and by that light you
will see every Flaw, Crack, or Honey-comb.
But this way you may see at any time; take a Stick something longer than the Piece, cleave the end of the said Stick, for to hold an end of a Candle, light the Candle,
and put it into the cleft end of the Stick, and put it into the Piece; by this light observe by degrees whether from the one end to the other there be any
of the foresaid Flaws, Cracks or Honey-combs in the Piece.
This is a usual way likewise, if in striking a Piece upon several places of the Metal with a Hammer of Iron, you shall at any stroak hear
a hoarse sound; then without doubt there is Honey-combs: but if in so striking the Piece, you shall at every stroak hear a clear sound, then may you be sure your Piece is clear of any Honey-combs, Cracks, or Flaws.
SECT. XIV. How to find whether a Piece of Ordnance be true bored, or not.
FIrst, there must be provided a Staff, and two Rammer heads upon the Staff, and on
the Rammers heads there must be two right Lines drawn upon them; that is, Divide the
two Rammer heads that are the just height, and fit the bore into two equal parts opposite
to each other, and draw Lines thereon; the like do by the Staff, that the Lines on
the Rammer heads may stand alike, one at one end, and at the other end, as you see
in the general Figure LM.
And let the Staff come through one of the Rammer heads about 9 Inches longer than the Cylinder of the Gun; then lay a flat Stick on the Muzzle-Ring, and hold the side of the Quadrant on the Scale to the Stick, and it will by the String and Plummet find the middle, or upper and
lower place of the Metal; or by hanging a Plumb-Line and Quadrant before the concave, and the Stick on the top; then after you have found the Point,
and upper and lower place of the Metal, put the Rammer head L into the Gun, and let one hold him hard, and right with the Line or Mark on the upper part of the
Gun, and lower part with the Line on the Rammer head on the Staff above and below, whilst
you put in a Priming Iron in at the Touch-hole, and strike hard the Rammer head, make
a Mark; then pull him out, and apply the Line on the Rammer head to the Mark on the
upper and lower edge of the Muzzle of the Gun, and you may presently see how much the Mark is from the right Line of the Rammer
head, to the right hand, or to the left; that is, if the Mark is just on the right
Line, the bore is in the midst: but if you find it a quarter of an Inch on the right or left hand, so much lyeth the bore either to the right or left; and
in Shooting, the Piece must be ordred accordingly.
But now to know whether it is thicker upwards or downwards, or how the bore is; the
way to know this, find the Diameter of the Piece the Touch-hole, as is already taught in 10 Chap. bend a Wire a little at the very
end, that it may catch at the Metal when it is drawn out; after the Wire is fitted
thus, first put it into the Touch-hole till it touch the bottom of the Metal in the
Chamber; then holding it in that place, make a mark upon the Wire, just even with
the said Touch-hole; afterwards draw up the same Wire, untill it catch at the Metal
at the top of the Chamber; at that instant make a mark upon the Wire just even with
the Touch-hole: the difference betwixt the two marks, is the just wideness of the
Chamber, and the distance between the first mark, and the end of the Wire, having
half the Diameter of the Chamber of the Piece Substracted from it, will leave the half of the Diameter of the Piece, if the Piece be true bored; but if this number be more then half the Diameter, then the bore lyeth too far from the Touch-hole, and the upper part of the Metal
is thickest; but if less, the under part hath most Metal.
One Example will make it very plain.
Suppose that the Metal at the Britch be represented by ABCD, and the Metal at the
Muzzle by efgh, and the bore of the Piece I, whose Centre is l, or the bore K, whose Centre is m: (and I find the Diam. of the Piece to be 21 Inc. at the Touch-hole, the half thereof is 10 ½ inch. Then I find by a Wire the Diam. of the bore to be 5 Inch. but the bottom of the Metal is 8 ½ Inch. half the Diam. of the bore being 2 ½ Inches to a 10 ½ makes 13 to the bottom of the Metal; but if you add to 8½ half the diam. of the bore 2 ½ it is 11, which is half an Inch more than 10½, that shews the Centre of the bore to be at R, and the thinnest of
the Metal is undermost, and there he is like to break first; besides, it shews that
you must add half an Inch to your Dispert of a true bored Piece, to make a Dispert for the Piece to shoot well: but if you had found by the direction before given, that the ½ an
Inch had been less, as 10 only, and the greatest part of the Metal had been under; and
therefore you must cut the Dispert ½ an Inch shorter then a Dispert made for such a true bored Piece; and likewise if you find by the Rammer head, and prick with a Wire at the Touch-hole
½ an Inch difference to the right or left hand, as l or m, that side which is the thinnest, you must put the Dispert cut ½ an Inch shorter, the three Figures makes all plain as it is written, as you may see by the
direction of inches.
SECT. XV. Of Iron Ordnance what quantity of Powder to allow for their Loading.
YOu must first Calculate a Charge of Powder for the said Iron Piece, as if it had been a Brass Piece, and in case you want the weight of the said Iron Piece, you must find it as you were taught in Chap. 7; and when you have found it as is
taught in Chap. 9, how much Powder will Load the same if it were of Brass, then just 3 quarters so much is sufficient to Load an Iron Piece.
As for Example.
A Brass Saker of 1500 weight requires 4 l. what will an Iron Demi-Culvering of 2800 weight require? Work as in the 9 Chap. and you shall find 6 l. 6/10 or 6 l. 14 1/2 ounces, so well fortified as the Saker, will serve a Brass Demi-Culvering for a Charge.
The which we will likewise examine by the Rule in the 7 Chapter.
The Brass Saker's diam. is 3 35/100 inch. Logarithm
257403
The diam. of the Demi-Culvering Brass 4 50/100 inches
265321
The difference increasing.
7918 (3
The triple of the difference.
23754
The weight of the Saker added to it 1500
317609
gives the Logar. of the weight of of the Demi-C. Brass 2592 l.
341363
Or by the Scale, extend the Compasses from 3 75/100 to 4 50/109 the same distance turned 3 times from 1500, will reach
2592 l. as before.
The Arithmetical way.
4 50/100 C × 1500/3 ¾ C = is equal to the weight 2592 pound, which is the weight such a Demi-Culvering should be of that burneth 6 l. 14 ½ ounces of Powder.
To find what a Demiculvering of Brass of 28 hundred will require, Work thus.
The Logarithm of 2592
341363
The Logarithm of 2800
344715
The difference increasing.
3352
The one third of the difference,
1117
The weight in Powder 6 90/100 l. added
283884
gives the weight 7 l. 8 ounces
285001
Or extend the Compasses from 2592 to 2800, the same distance will reach from 6 9/10 to 7 ½ l. as before.
The Arithmetical way.
2800 × 6 9/10/2592 = 7 pound 8 ounces, as before.
Of which number you must take 3 Quarters for a Charge for the said Demi-Culvering, ¾ thereof, being 5 pound 10 ounces, will be a sufficient Charge for such a Piece; and also whatsoever you find on the Scale, and in the Table in the third Chapter for Brass Pieces, take three quarters thereof for the Charge of your Iron Piece, if they be near that weight.
SECT. XVI. To know what quantity of Powder should be allowed to a Piece of Ordnance not truly bored.
ADmit the diameter of the Metal of the Piece at the Touch-hole be 16 inches, and the diameter at the bore is 5 ¼ inches, the weight of the Piece 4850, as you may see by Chap. 7, such a Piece you may find in the ninth Chap. requires 11 l. for her due Charge, being near two Diameters of her bore in Powder; But by my Instrument in the general Figure, with the two Rammers heads at the two ends LM (at the Rammer
end L that was in the Gun at the Touch-hole, I find by the prick at S on the Rammer, the soule or bore to be
1 inch out of his place, or 1 inch from the middle of the Metal; then I conclude, that the thinnest part of the Metal
is 4 inches 3/2 parts, and the thickest side 6 and ⅜ parts; by which it appears, that one side
is just 2 inches thicker than the other side, as you may see plainly by this Figure; the Line AB divided
is the diameter or greatest thickness at the Touch-hole, every Division signifies an inch from the inward Circle to the outward Circle, is the thickness of the Metal; the
inward Circle signifies the bore of the Piece, which you may see is just an inch from the true bore or Centre of the outmost Circle; therefore you must work as if
the Piece were fortified no more than only so much as the thinnest part of the Metal is, which
here doth appear to be 4 inches ¾ parts, the ½ of the diameter of the bore is 2 ⅝ inches added, makes 7 from A to D, the Centre of the bore being the thinnest part of the
Metal, the whole diameter being 14, which is the true diameter, by reason the thinnest side of the Metal is but 4 ⅜ inches thick.
And by this you must proportion your Charge by the former being 16 inches, if the bore had been placed at C in the true Centre, then evermore by these Rules.
[...]
The Logarithm of greatest diameter 16 is
220412
The Logarithm of the less diameter 14: 0 l.
214612
The difference decreasing,
5800
3
The triple of the difference Substracted
17400,
from the Logarithm of 11 l. Powder 0 ounces,
204139
Leaves the Logarithm of the Powder 7 4/1 [...]pound
186739
So that 7 pound 4/19 or 6 ounces, is a sufficient Charge for such a false bored Piece; or extend the Compasses from 16 to 14, the same distance 3 times repeated from 11, will reach 7 4/19 pound, as before.
The Arithmetical way.
C 14 is [...] 2744 × 11/C 16 is / 4096 = 7 pound 6 ounces, as before.
SECT. XVII. How Moulds, Forms, and Cartredges, are to be made for any sort of Ordnance.
CArtredges are usually made of Canvas and Paper-Royal; first take the height of the
bore of the Piece of your Scale a little less 1/20 part of an inch of the diameter for the Vent, and three diam. of the Chamber of the Piece in breadth, cut the Paper and the Canvas, and for the Canon in height to the whole Culvering, is allowed about 2 diam. of the Piece, from the Culvering to the Minicn, the Charge the length of two diamet. and ½ all from the Minion to the Base 3 diameters of Powder, and make them at first about 4 diameters long, and according to the directions here given, mark them, or put a pound of Powder into each Cartredge, and measure how full it fils by your Scale for each Gun in your Ship, or Army; and by that Rule you may know how to make a Table, to make a Scale, to mark the Cartredge
for the full loading, or diminishing of your Powder, according to the goodness or badness of the Powder; and to the exraordinary over-heating of the Piece, having resolved for what sort of Ordnance are to serve you, and accordingly to have a form of Wood turned to the height of
the Cartredge, which is the 21 part of 22, the diameter of the bore, and ½ an inch longer than the Cartredge is to be, before you paste your Paper on the form, first
Tallow him, so will the Canvas and Paper slip off without starting or tearing; if
you will make for tapered bore Guns, your Forms must be accordingly tapered, if you make Cartredges of Canvas, allow one
inch for the Seams; but of Paper ¾ of an inch more than 3 diameters for the pasting. If once about the former, having a bottom fitted upon the end of
the former, and Cartredge; you must paste the bottom close, and hard round about,
then let them be well dryed; and then mark every one with Black or Red Lead, or Ink,
how high they ought to be filled; which if you have no Ladles, Scales, nor Weights,
these diameters of the Bullets make a reasonable Charge for the Canon 2 ¼ for a Culvering 3, and for the Saker 3 ½ for the lesser Pieces 3 ¾ of the diameter of the Bullet, and let some want of their weight against time they are over hot,
or else you may endanger your self, and others.
SECT. XVIII. How to make Ladles, Rammers; or Spunges for all sorts of Ordnance.
EVery Mr. Gunner doth, or should know how to Trace, Cut out, and also make up and finish all Ladles,
Spunges, and Rammers, and direct others how to make, and finish the same ready for
use.
You have in the Table in the third Chapter the length and breadth of the Ladle, answerable
to each Gun in inches and parts, and you must allow ½ a diameter more to inclose the head of the Staff within the Plate; the Button, or head of the
Ladle-staff must be the height of the Shot almost; for Spunges, their bottoms and heads are to be made of soft Wood, as Asp,
Birch, Willow, to be one diameter ¾ in length, and ¾, or a very little less of the height of the Shot, and covered with Sheep-skins, Wool, and nailed with Cooper's Nails, that together
they may fill the concave of the Piece.
Let the Bottom or Head of the Rammers be of good hard Wood, and the height, as before
one, and the length ⅓ of the diameter of the Shot at one end next the Staff, it must be so turned, that a Ferril of Brass may be put
thereon, to save the Head from cleaving, when you Ramme home the Shot, the Buttons must be bored ⅔ for the Staff to be put in and fastned with a Pin through,
and his length a Foot more than the concave of the Gun.
To make a Ladle for a Chamber-bored Piece, your Compasses opened to just the diam. of your Chamber within ⅛ part of an inch, Divide the measure into two equal parts, then set your Compasses to one of them, and by that distance draw a Circle on a Slat or Paper, the diam. of that Circle is ¼ shorter than the diam. of the Chamber, and take ⅗ parts of the Circle for the breadth of the Plate of the Ladle; and for
Cannons, the length ought to be twice and ⅔ parts to hold at two times, the just quantity
of Powder.
SECT. XIX. How the Carriage of a Piece should be made.
MEasure the length of the Cylinder of the bore, and once, and half that length should be the length of the Carriage, and in depth 4 diameters of the bore of the Piece at the fore-end, in the middle 3 and ½, and at the end next the ground 2 and ½, the
thickness the diameter of the Shot; the Wheels should be one half the length of the Piece in height; the Saker and Minion must exceed the former by 1/12 part, the Faulcon and Faulconet by one sixth part. Sea-Carriages are made less, as the Block-maker that makes them hath Rules for.
SECT. XX. To know whether the Trunnions of any Piece of Ordnance are placed right.
MEasure the length of the Cylinder of the bore from the Muzzle to the Britch, Divide the length by 7, and Multiply the
Quotient by 3, and the Product will shew you how many Inches the Trunnions must stand from the lowest part of the concavity of the Piece, and you must know that the Trunnions ought to be placed, so that ⅔ of the Piece may be seen above, or in that place where the Trunnions are set.
SECT. XXI. How much Rope will make Britchings and Tackles for any Piece.
IN Ships that carry Guns, the most experienced Gunners take this Rule. Look how many foot your Piece is in length, four times so much is the length of the Tackle, and their Britchings twice the length; and if the Ropes be suspected not to be good, they nail down Quoyners to the Fore-Trucks of heavy Guns, that he may not have any play; and if Britchings, and Tackles, and Quoyners should give way in foul weather, presently dismount her, that is the surest way.
The Rammers and Spunges are made of four Strand-Ropes best, and served close, and sewed with Yarn, that they may be stiff to Ramme home
the Shot and Wadd.
SECT. XXII. What Powder is allowed for Proof, and what for Action of each Piece.
FOr the biggest sort of Pieces and Canons; for Proof ⅘, and for Service ½ her weight of her Iron Shot, for the Culvering the weight of the Shot almost for Proof, and for Action ⅔, for the Saker and Fauloon ⅘, and for lesser Pieces the whole weight of the Shot, until they grow hot; and for Proof, the lesser Pieces give them 1 and ⅓ of the weight of the Bullet in Powder.
SECT. XXIII. The difference between the Common Legitimate Pieces, and the Bastard Pieces, and Extraordinary Pieces.
GƲnners call them Legitimate Pieces, as have due length of their Chase, according to the height of their bores; Bastard
Pieces are such as have shorter Chases, than the Proportion of their bore doth require;
and Extraordinary, are such Pieces as have longer Chases, than the proportion of their bore alloweth; and these are
called Bastard Canons, Culverings; and so likewise of Saker and Faulcon, which by your Scale, and the Rule thereon, you may presently find them.
SECT. XXIV. How Powder is made, and several ways to know whether Powder be decayed or no, by moisture or Age, in part, or in whole.
POwder was always made of Salt-peter, Brimstone, and Char-cole; but in these latter
times experience hath still mended the goodness or strength of it more, than it was
in former times by much: but briefly thus; the best sort that is made at this present
time is made of six parts Salt-peter, Brimstone, Char-cole, one part.
the Musquet or Pistol Powder is now commonly made of Salt-peter five parts, one part of Brimstone, and one of
Cole; Canon-Powder of Salt-peter four times as much as of Brimstone, and as of Cole. The reason why
Pistol-Powder being the strongest of 6 - 1 - 1 is not so good for the Canon as 4 - 1 - 1 the weakest, although you take but so much of the Pistol Powder as you find by an Engine to be of like strength with another quantity of Canon-Powder.
The reason why Canon-Powder is best for Ordnance, is, because it taketh up a grearer room in the Cylinder of the Piece, than Pistol-Powder; for in taking up much room it hath the greater length or fortification of Metal about
it in the Piece.
Suppose a Saker require four l. of great Powder for her loading, and I would know how much Pistol Powder is equal in strength to four l. of Canon-Powder, trying by an Engine made on purpose to try Powder, I find 3 l. of Pistol-Powder; therefore you easily concieve, that 3 pounds have but 3 quarters of the Metal of
the Piece to keep it from breaking, when 4 pound had a quarter more Metal, than the other had.
Nath. Nye Mr. Gunner found by an experiment made by him at Deriton the 17 of March Anno 1644, he loading a Saker-bore Piece of Iron,47 Pound of Metal. and the thickness of the Metal about the Chamber was 2 inch. and load her with 4 pound of weak Canon-Powder, which filled the Cylinder of the bore 9 inch. just, which 9 inch. in length, and two inch. in thickness is 225 inches of Metal about the Powder, which was 6 ounces more than the Piece should have had in proportion to Pistol-Powder: He fired, and the Piece went off safe; and he saith, he loaded her again with one pound and ¼ of fine Powder almost, which filled the bore but 2 inches and ¾, and had to its Fortification but 6, 8 ¾ inches, which in weight is 15 l. and when the Gun was discharged, it broke into divers pieces, as there is witness enough in that Town.
The harder the Corns of Powder are in feeling, by so much the better it is. Secondly,How to know good Powder.Gun-powder of a fair Azure or French Russet colour is very good, and it may be judged to have
all its Receipts well wrought, and the proportion of Peter well refined. Thirdly,
Lay 3 or 4 Corns of Gun-powder upon a white piece of Paper, the one three fingers distant from the other, and fire
one, if the Powder is good, they will all fire at once, and leave nothing but a white chalky colour
in the place where they were burned, neither will the Paper be touched; but if there
remains a grosness of Brimstone and Salt-peter, it is not good. Fourthly, If you lay
good Powder on the palm of your Hand, and set it on Fire, it will not burn you. Fifthly, To know
the best amongst many sorts of Powder, make a little heap of every sort, and then setting those heaps one from the other,
mark well when you put Fire into them, which of the heaps did take Fire the soonest;
for that Powder that will soonest be on Fire, smoak least, leave least sign behind it, is the best
sort of Gun-powder.
SECT. XXV. How to make an excellent good Match to give Fire to any Ordnance.
TAke Cords made of Hemp that is not very fine, or of Tow which is better, (although
it will consume sooner,) and twist it until you have made the Srands as big as a mans
little Finger; this done, boyl the said Cords in strong Lye ashes, and a little Salt-peter,
until all the Lye be wasted, and then make it up, and take the feces or remainer into your hand, and with the other draw the Match through twice or thrice,
then drie it, and keep it for special uses, for Vaults, Mines, and moist weather,
and it is very fit for your use any where.
SECT. XXVI. How to make Powder it shall not wast with time, and preserve that as is good to keep it from decaying.
THat time shall not wast it, take what quantity of Powder you will, and mix it with Brandy, and make it up in Balls, and drie them well in
the Sun, or in a warm place, and keep them in Earthen Pots well glazed until you have
cause to use them; this Powder will not wast with age, nor decay.
To preserve Powder that is good, all Gunners have, or should have that reason to keep their Powder and Store in as good a drie place as is to be had in Fort or Ship convenient, and every Fortnight
or at most three weeks turn all the Barrels, and Cartredges Barrelled up for readiness,
turn them upside down, so will the Peter be Dispert into every place and part alike;
for if it should stand long, the Peter will descend downwards always as it lies,
and if it is not well shaked and moved, it will want of its strength at top very much,
and one Pound at bottom with long standing, will be stronger than three at the top;
keep all Cartredges which are filled for the Piece against he is hot in Barrels by themselves, that you may know them by a mark when
need requires.
SECT. XXVII. To renew and make good again any sort of Gun-powder that hath lost its strength by long lying, or moisture, or any other means.
FIrst moisten the said Gun-powder with Vinegar or fair water, beat it well in a Mortar, and then sift it through a
fine Sieve, or a Search; with every pound of Gun-powder mingle one ounce of Salt-peter that hath been mealed; and when you have so done, beat and moisten
this mixture again, until you see by breaking, or cutting with a Knife that there
is no sign of Salt-peter or Brimstone in it; moreover, corn this Powder when it is incorporated with the Peter as it ought to be; then prepare a Sieve with
a bottom of thick Parchment made full of round holes; then moisten the Powder which shall be corned with Water, put the same, and also a little Boul into the Sieve;
and when you have so done, sift the Powder so as the Boul rolling up and down in the Sieve may break the clods of Powder, and make it by running through the holes to corn; and if it will not go through,
you must beat it again until it will.
SECT. XXVIII. To make Powder of divers Colours, and first to make White Powder.
TAke of Salt-peter 12 parts, of Brimstone two parts, and of Camphir one part, beat,
and sift, and incorporate all these things together; and after you have so done, beat
them again, and so oft until you are sure they are well incorporated, then moisten
it with Aqua Vitae; and when you have thus done, corn the Powder, as you are taught before.
To make Red Powder.
Take the same things, and work them as before directed for White Powder, and as that was moistned with Aqua Vitae, now you must moisten this with Vineagar, being Red as Blood, which will make the
Powder likewise so in moistening of it, and then corn it, as is before taught.
To make any Coloured Powder.
Boil the Vineagar in such transparent Colours as you would have the Powder to be of, as if Blew with Blew Bice, of Green with a little Verdigreace, and the
like: always take care that the Colour be not too thick, but very thin, or else it
will weaken the Powder that you do make.
SECT. XXIX. Of several sorts of Salt-peter, and a way how to make a sort of Salt-peter very excellent, with ease, and less cost than any way.
ARtificial Salt-peter is a mixture of many substances gotten with Fire and Water out
of drie Dirt and Earth, as out of Vaults and Tombs, and also Charnel-Houses; the best
of all is of Beast-dung converted into Earth, in Stables, or Dunghills of a long time
not used; and when it is to be made, it is made with a great deal of Charge. Another
excellent sort of Salt peter is made on Flower that is called Plaster that groweth
on Walls four parts, of Unslak'd Lime one part, and so boiled over the Fire with Water,
which is to no purpose to make relation how for to make full direction will fill my
live-sed sheets too fast; but this one way, which is the most easie and least cost,
I will write the Receipt thereof, which is this.
Take quick Lime, and pour warm Water upon it, and let it stand six days, stirring
it twice or thrice a day; and take the clear of this Water, set it in the Sun until
it be wasted, and the Salt-peter will remain in the bottom. To refine Salt-peter,
and make it fit for use, there is several ways, but this by Fire I shall only write
thereof. Do thus: take an Iron Pot or Skellet, and fill it with Peter, set it on the
Fire, and cover it close with an Iron Cover on the top, or with a Stone; when the
Salt-peter is melted, take Brimstone most finely beaten, and cast some thereon; kindle
it, and let it burn until all the upper part be burned, which when effected, will
leave the Salt-peter close like to a piece of Marble, for the Brimstone will burn
up the gross victiousness of the Salt-peter; It is to no purpose to give a further
relation of this, by reason every Gunner may have his Peter ready made refined and in Meal at the Powder-mens, or Chandlers;
or if he is constrained to make Peter or Powder, he may have several Books which give a full and large description of the making thereof,
as Nath. Nye Tarta glia, or Norton's; but for what is useful for a Gunner in particular, is sufficiently spoken already; therefore let it suffice now, having
shewed sufficiently how to make Powder, and trie the strength of Powder; to know what Shot and Powder is meet for every Piece, to find whether the Pieee be true bored or not, to load a Piece with discretion, if not true bored to make the Dispert; and also to know the difference
betwixt Iron and Brass Pieces. I shall come to touch how to make a good Shot either of Point-blank, or at Random, with as much ease and plainness as ever was
taught by any before.
SECT. XXX. How to Load and Fire a Piece of Ordnance like an Artist.
BEfore we shoot at a Mark, it is best to Load our Piece, in which, first observe the Wind, and be sure to lay your Budge-Barrel, or Cartredge
of Powder to Wind-ward of your Piece, and place your Linstock to Lee-ward, clear the Touch-hole, and Spunge her well, and
strike the Spunge on the Muzzle to shake off the foulness two or three blows.
Then let him stand on the right side of the Gun, and hold the Barrel, so that his assistant may thrust in the Ladle; being full, give
it a shog, then strike off the heaped Powder, he being on the right side likewise, with his Body clear of the Muzzle, put the Ladle
home to the Chamber stedily holding your Thumb upon the upper part of the Ladle-staff,
then turn the Staff until your Thumb be quite under it, and give a shake or two to
clear the Powder out of the Ladle; as you hale him out, keep him up that you may bring no Powder out with the Ladle; then with the Rammer put the Powder home gently, and after put in a good Wad, and thrust it home to the Powder, and give it two or three stroaks, to gather the loose Powder together, and it will fire the better; be sure your assistant have his Thumb on the
Touch-hole all this while; then put in the Shot with the Rammer home, and after him another Wad, and then with the Rammer give two
or three strokes more to settle it home, that there may be no vacuity between the
first Wad Bullet, and last Wadde; your Budge-Barrel and your self standing to Wind-ward
always, and your Piece by the Dispert directed to the Mark, Prime her, and let the Powder come from the Touch-hole to th [...] Base-ring, your Leg standing forward, and fire the Powder on the Base-ring, and draw back your Hand, and you have fired like the best of Gunners; but if you had given fire upon the Touch-hole, the Powder there would have endangered to have blowed the Cole and Linstock and allon of your
hand; therefore you must have a care of a great Touch-hole.
SECT. XXXI. The difference of Shooting by the Metal, and by a Dispert by a Right Range, and at Random, by the Figures following.
SHooting by the MetalShooting by the Metal. is the Figure AB, that is, admit you raised the Muzzle-ring, and the Base-ring, and
the Mark, and your Eye in a Right-line, if you put the Scale into the Muzzle with
the Plummet hanging to it, you shall find it differ 4 or 5 or 6 degr. according to the length or Mark of the Piece, and in regard of the several differences of the length and marks, or Diameter of her Base and Muzzle-ring, no certain proportion can be generally assigned; yet
for most Pieces it hath been well observed, that the Piece directed by her Metal, will shoot about twice as far as when the Mark is level and
set by a Dispert & Quadrant and the Sight-line parallel to the Horizon; so that admit
a Piece were laid by the Metal of Base and Muzzle-ring, and that it differed from a right
Level 6 degr. as you found by your Quadrant in the Scale, you fired at the Mark, the Gun so layed, and measuring the distance you find it 412 Paces, which is as much beyond
the Mark, as it is to it, which is the difference of Shooting by the Dispertof Shooting by a Dispert. or Axis of the bore in right bored Pieces following, this is chlled Point-blank; for if you acknowledge the higher the Muzzle
of a Piece is elevated, the farther the Shot is carryed in a Right-line. There can be no Point-blank
directly known, nor Rules to know them, without you take a Piece and make 11 Shots out of her, and so by it proportion a Table, as here is one following.
You must have leave, if possible, to Shoot so many Shots in a Piece at such elevations with the like goodness and quantity of Powder, as will make you such a Table of the Proportions of Right Ranges, called Point-blanks,
before you can have any guess certainly of Proportion for other Guns; but to make a good Shot at a Mark, first be sure your Guns Trunnions be placed right, the Carriage well made, the [Page 69] Platform clean swept, and that the ground be Level, and that the Carriage-Wheels
be one as high as the other, and whether the Axle-tree be placed just across the Carriage,
or not; or whether the Piece be true bored or not, if it be a true bored Piece, set your Dispert on the Muzzle-ring, just over then Centre of the Muzzle of the Piece, which may be done by holding a Line right before the Muzzle, and two Sticks or Notches
made in Sticks put on the Muzzle-ring, and by the Line you may easily find the middle
where to put the Dispert; and likewise if the Line touch the upper side & lower side
of the Metal alike, you may be sure the Gun lies Level, or by the Scale and Quadrant on it and Plumb-line, if it hang in the
Mouth of the piece, and no degrees of Altitude, but by the Long Side-line thereof, be sure the Piece is Level, and will carry the Bullet Horizontally in his violent Course; therefore
by your Crows and Standers by, or Matrosses set about the Piece to the Mark, as at D, that with your Eye two foot from the Base-ring you may see the Mark on the upper part of the Base-ring, the top
of the Dispert, and the Mark or Turret you Shoot at in a Right-line, as CD, and this
you may call Point-blank, and all Shooting in this Form, and no other.
A Table of Point-blanks.
d [...]g.
randoms
0
192
1
209
2
227
3
244
4
261
5
278
6
285
7
302
8
320
9
337
10
354
20
454
30
693
40
855
50
1000
On the upper part of the Base-Ring, the top of the Dispert, and the Mark or Turret
you Shoot at in a Right-line, as CD; and this you may call Point-blank, and all Shooting
in this form, and no other.To Shoot in a Right-Range, that is, as far as the Bullet doth go in a Right-line at
any Degree of Elevation.By the Rule of Proportion.Paces.248239519d.10:354254900200230703Sum485003285245484
But for Shooting in a Right-line called the Right Range of a Bullet out of any Piece for making of Batteries, or Shooting at Random at any advantage, you may make use
of this Table, until by your own experience, you have made out of a Gun by Shooting first Level, and afterward from degree to degree to 10 degrees mounture, or more in a Right-line.
The use of the Table of Right-Ranges, or Point-blanks before-going, it is found by
experience, that the Piece assigned at six degrees of mounture Shoot 200 Paces in a right or insensible crooked Line; I desire to know
how far the same Piece will Shoot in a straight Line, being mounted to 10 degrees? say by the Table if 285 the Number against 6 degrees giveth 200 paces, what will 354 the Number against 10 degrees give? I answer, 248 4/10 paces.
The Logarithm of 6 degr. is 285.
245484
Which done backwards, is 6 degr. against 285 in the last Table.
The Logarithm of the Paces known 200
230103
The Paces against 10 degr. is 354 is
254900
The Sum
485030
give the Logarithm of 248 4/10 Paces
239519
Or extend the Compasses from 285 to 200, the same distance will reach from 354 to 248 4/10 Paces as before
200 × 354/285 = 248 4/10 Paces, as before.
And the Figure is from E the top of the Castle to F the side of the Tower at 10 degr. mounture, carries the Bullet violently 248 Paces.
SECT. XXXII. How to Order and Direct a Piece, and amend an ill Shot that was made, either by the Metal, Level, Right-line, or Advantage,
or Mount.
AFter you have made one Shot, and find the Piece carry just over the Mark, then do all as hath been before taught again; and when
your Piece lies directly against the Mark, observe how much the last stroke of the Shot is above
the Mark, so much longer make your Dispert, that the top of it may be just seen from
the Britch of the Piece in a direct Line with the stroke of the Shot; and being so fitted, Level your Piece with this new Dispert to the assigned Mark, give Fire, and without doubt it will
strike the same; if the first Shot had struck under the Mark, then bring the Piece in all points, as before; mark how much of the Dispert is over the Shot, and cut
it just so short, as being at the Britch, you may discern the top of it, the Mark
of the Base-Ring, and the stroke of the Shot in a Right-line, when you percieve it
is of such a length, Level the Piece to the intended Mark as at the first, Prime, and give Fire.
If the first Shot had struck on the right Hand of the Mark, to mend it, you must Level
the Piece as formerly, you standing behind the Britch of the Piece, observe the stroke of the Shot over the Dispert, that part of the Base-ring which
you at that instant looked over in a Right-line towards the Dispert, and the stroke
of the Shot, set up a Pin with a little soft Wax on the Base-ring, so this Pin will
be in a Right-line with the Dispert and stroke, then Level your Piece to the Mark intended by this Pin, and the first Dispert, and without question you
will make a fair Shot; for when you Level by the Metal of the Base-ring where the
Pin is placed and the Mark; the Piece standing at that direction, look over the top
of the Dispert from the Mark in the Base-ring, and you shall find the Piece to lie
just so much to the left, as the former Shot struck to the right, from the intended
Mark, which should in all likelihood now strike the Mark.
But if the Shot be both wide and too low, then you must use both directions, as before
taught to make the next Shot; first regulate the Dispert by cutting it shorter, according
as the Mark of the Shot is lower than the intended Mark, then by the last Rule mend
the Shooting wide; these things done with care and diligence, you may be sure will
mend a bad Shot.
SECT. XXXIII. Of Shooting upon the Advantage or Random, at a Mark, beyond the Right-line of the
Pieces reach, or Right-range of the Shot, and of the Dead-range for every Degree.
IN the two last Sections we have shewed for the Right-range; now we come to shew for
the Dead-range, which consisteth of the right and crooked Range together in one, and
then called the Dead-range, which is the whole distance from the Platform upon which
the Piece intended or assigned is discharged unto the first fall or graze of the Bullet
on the Level-Line, or on the ground called the Horizontal Plain, (by reason the different
length of the Piece, and strength of the Powder increaseth or diminisheth the course
or fury of the Shot, and therefore more difficult to be found out, but only by Experience,
or Diagrams, Tables, or Scales made by Experiment; now it vis ery diffiault, and a
thing uncertain also to arrive herein unto exactness, without some Experience of
the Piece; and therefere every one that will learn to Shoot at Random, must draw his
Piece in a Level Ground, where first Shooting Level, he must observe that distance
in Feet and Paces; then Mount his Piece to one degree, and mark where that shall Graze, and thus find the distance of every degree from the Level to the 10 degree, and by these distances make a Table, to which [Page 71] annex the degrees against the distances, by which Table, and the Rule of Proportion, or Logarithms,
or Line of Numbers, find how far another Piece will convey her Shot from degree to degree; but in case you cannot have the Liberty, nor Powder to do the aforesaid, you shall have a Table here that was made out of a Saker eight Foot long (by Nath. Nye, as in the 40th. Chap. of his Book,) where he saith, he Loaded her with three Pound
of Powder, the Shot at one degree of mounture was carryed 375 Yards, or 225 Paces; the next Shot was at five degrees Random, and at that mounture the Shot was conveyed 416 Paces; and the next tryal
was at seven degrees mounture, and the Random produced 505 Paces; the last tryal was at 10 degr. mounture, which sent the Shot 630 Paces of five Fort to a Pace.
Whilst he made these Shots, he loaded the Piece himself with loose Powder, exactly weighed, and weighed the Wad also, and beat down the said Wad with four strokes
so near as he could by the same strength, as he did the time before; also he let the
Piece cool betwixt each Shot of it self, staying above half an hour betwixt each Shot,
he put no Wad after the Bullet, because the Piece was mounted, he tryed the strength of his Powder, and noted it down, to compare with other Powder, to know its strength by; and that is the way all Gunners must take, that intend to make good Shots at Random. All Mr Gunners should be able to draw an exact Description of the said Garrison, and every object
as lyeth near his Works by the Rules of the 7th. Chap. of the Art of Surveying by the Sea-Compass in this 5th Book; so that he may know what is within reach of his Guns, by which means he shall not be troubled to take Distances, but be ready at all times
to know his Distances by his Maps: then after he hath made one Shot, he may make another
Shot to any Distance he pleaseth.
Example.
Suppose I know the Distance by my Map where the first Shot grazed to be 704 Paces,
as you may see by the Figure out of the lowermost Gun of the Castle from S to the graze at A, the mounture of the Piece being 4 degr. how much must I mount the Piece to convey her Shot 900 Paces, as you may see by the Figure B the Gun, to C the Shots graze, or place required.
You must proportion these Distances of Random, to those in the Table following; saying,
if 704 Paces require 370 Paces, as is in the Table at 4 degrees of Random, what number will be found against the degree in the Table; I must Mount the Piece unto 900, and work by these Rules.Extend the Compasses from 704 to 370, the same will reach from 900 to 473.
370 × 900/704 = 473
The Logarithm of the Shot made 704 is
284757
The Logarithm in the Table against 4 degr. is 370)
256820
The Logarithm of 900 Paces Random is
295424
The Sum is
552244
gives the Logar. of 473 Paces,50546144 which I look for in − 267487 the Table of Randoms, but find no such number there,
but the next less is 461, and the next greatest is 550 against 7 degrees, the difference between these two Numbers, is 44 and 461 is 12 less than 473, and
12 is almost the one fourth part of 44,50547332 and therefore it shews that the Piece must be mounted at 6 degr. 15 min. or one Quarter to reach the Distance of 900 Paces as from B or b to C.
Here is a Table of Randoms that was made out of the fore-named Saker of 8 Foot long, and loaded with 3 Pound of Powder, and every Gunner is advised, if possible, to get Powder, to make one by his own experience, and always to keep some of the same Powder to try his Proportions by; the Rule in the 24 Section by any other Powder he shall have occasion to use, for this is 1 of the material things belonging to
a Gunner, without which knowledg hee an never make a good Shot; for at the time of a Leagure
he must expect often to change his Powder, as sometimes you shall have 9 Pound of one sort as good as 15 of another sort, as
by Instrument and Shooting you may have experience.
SECT. XXXIV. How to make an effectual Shot out of a Piece of Ordnance at Random.
EVery one that hath Charge of a Gun, must at one time or other get leave of his Commanders to make two or three Shots
at least out of the same Piece, and measure the Distance from the Platform to the
first graze of the Shot, you must apply it to this Table by the last Rules of Proportion
in the last Section, and find what deg. you shall need Mount the Gun to for any other Shot at any other time, when you shall have occasion; when you
have Loaden your Piece, as you are directed in Sect. 32, take the Distance to the Mark in the XVI Chap. of
the second Book of the Description and Use of the Quadrant I have shewn you; and also observe how many degrees the Platform is higher or lower than your Mark by your Quadrant on the back-side of your Scale; after you have done that, then Calculate by the last Rules what degree the Gun must be Mounted to, to reach the Mark, if the said Work be under the Platform, Substract
the Difference found by your Quadrant, out of the degree of the Random; but if the said Mark be higher than the Platform, Add the degr. of that Altitude to the degree of the Random, and at these corrected degrees Mount your Piece.
How to Mount your Piece by your Scalc and Quadrant thereon.
The use of the Quadrant on back-side of the Scale in the third Chap. for the Mounting of the Piece.To the side of the Scale or Quadrant is a piece of Brass fitted of the same breadth, with two Screws, and holes fitted
to screw the Brass Plate two Inches of the former length, without the Edge or Side of the Line of Numbers, for to take
any Angle that is under the Line of Level; for if you put the Brass into the Mouth
of the Piece, the Line of Numbers being next unto it, and put in the Tompkin into the Mouth likewise
to stop it fast in the middle of the Metal at the bottom, and then the standers by
raise the Britch with Crows to what degree you please; and so likewise if the Mark or degr. assigned be above the Line of Level, if the Scale will not stand fast by the degree of the Diameter that fits the Bore, putting of it just into the Mouth of the Piece, then screw the Brass Plate to a hole made on purpose for the other side, and turn
the degrees of the Diameter to the Bore, and fasten it with the Tompkin in the middle of the Mouth, as before;
and so this Instrument will be most useful for all things as belong to a Gunner, with less trouble and Charge, than any other that ever was made by any other Men,
and far more useful.
Then the Instrument being in the Mouth of the Piece, as before directed, mark diligently until the Plumb-Line, which proceeds from the
Centre of the Quadrant, cut these assigned degrees and Parts of degrees that you are to Mount the Gun by, in the Arch which is Divided into 90 degrees in the outward Circle thereof; your Gun so Loaded and fitted, as beforesaid, make your Shot, for without question, you will
make a good Shot, and strike or came near the Mark.
Suppose you make tryal of your Gun as is spoken of in the last Section 32, you find that at 4 degrees of Random upon a Level-Ground the Shot is conveyed 704 Paces, if you be called out
in hast upon Service against a City, or other Fort, and being appointed to play your
Gun towards it, you also find it to be beyond the reach of the Right-range of your Shot,
and the Distance being 560 Paces; and also that the place is lower than where you
can Plant your Gun by one degree and ⅙ or 10 min. then to know the degree of Mounture, you may work as by the last Rule, if 704 gives 370 against four degrees, what will 560 give? the Distance to the Mark, it will give you the Number 295; look
for this Number in the last Table 295, or the nearest Number to it, and against that
degree and Part of a degree, which must also be found by Substracting the nearest less Number out of the nearest
great Number, the greatest Number in the Table is 323, the nearest least is 274,
the Difference is 49, the difference betwixt 323, and 395, is 28, the half of 49 is
24 ½, which shews that degree is 2 and a little above ½ of Random; but because the Mark is lower than the Platform,
Substract one degree ⅙ or 10 min. out of 2 degr. 32 min. and the Remain is 1 degr. 22 min. the true height the Piece must be elevated, to reach the Mark; but if the Shot graze to the right or left,
you are to mend it by direction in Sect. 31, but ever by the Example or Direction
there.
Suppose the Shot graze over the Mark 20 Paces, Substract this 20 out of 560 the Distance,
and Mount the next Shot according as if the Mark were but 540 Paces distant, if 20
Paces too short, make the next Shot as 580 Paces, that is the degree that is found by that Proportion to reach so far.
SECT. XXXV. How to find the Right-Line, or Right-Range of any Shot discharged out of any Piece, for every elevation by one Right or Dead-Range given for the Piece assigned.
FIrst, you must have the help of this Table of Dead-Ranges,You may make use of this Table by the same Rules in the 32 Sect. and Table of dead-ranges, which was made by Experience of Mr. Norton, and the Table of Point-blanks, or Right-Ranges, was the same Shots in a Right-line
at every degree of Mounture, as this Table is the Dead-graze of the Bullet at the same Shot with
the same Gun, for he made 200 Shots for tryal.
Now although it be a thing very difficult, and likewise uncertain to arrive herein
to exactness, without some Experiments made with the assigned Piece and Powder; yet to come to a necessary nearness at first, far surer than by uncertain guessing
by this Table, or by my Scale, and the Rules therein directed.
As for Example.
Admit you were to seek the Right-Range of a Bullet, that the Piece was fired at 30 degrees Mounture, and the Dead-Range of the Bullet was known to be 2200 Paces.
Or look in this Table of Dead-Ranges against 30 degrees is 2150)
333243
And the Dead-Range given or known for 30 degrees to be 2800 Paces.
334242
The Number against 30 degrees in the first Table of Point-blanks 693
284073
The Sum
618315
Gives the Logarithm of 710 Paces for the Right-Range of
285072
The Bullet carryed violently in a Right-Line at 30 degr. Mounture.
But admit the Level Right-Range is given, and the Right-Range of 30 degrees Mounture be sought.
Work by these Rules.
The Logarithm of Point-blank 0 degr. is 192
228330
The Num. against 30 d. is 693 in the Table Point-blank
284073
The Level Right-Range of this Piece is 197 Paces
229446
The Sum
513519
Gives the Log. of 711 Paces for the Right-ranges required 285189
And as the Numbers are in the Logarithm, so you may do by the Line of Numbers.
693 × 197/192 = 711 Paces the Range in a Right-Line.
SECT. XXXVI. To know how much of the Horizontal-Line is contained directly under the Right-Line of any Shot called the Right-Range made out of any Piece at any Elevation assigned.
BE it propounded to find what part of the Horizontal-Line lyeth directly under the
Right-Range of the Piece assigned at 30 degr. Elevation, the Right-Range for 30 deg. Mounture, by the last Rule is found to be 711 Paces, Work with the Complement of
the Pieces Mounture 60 degr. thus always.
As the Radius 90 degr.
1000000
is to the number of Paces in the Right-Range 711 Paces
285186
So is the Compl. Sign of 30 degr. Mounture, which is 60 degr.
993969
to the Logarithm of the Horizontal-Base 619 Paces
279155
Now you find that 619 Paces lies under the Right-Range of the Shot, you may presently
find how much of the Horizon is contained under the Crooked-Range of the Shot, if
you Substract 619 the Horizontal Distance out of 2200, the Randoms at the first graze
of the Bullet from the Piece, the Remainer is the Horizontal Distance 1581 Paces, which lies under the way of the
Shot, as it goeth helically between the Right-Range, and the natural or perpendicular
motion, or before it make the first graze; the like in all other questions.
SECT. XXXVII. Of the violent, crooked, and natural Motion or Course of a Shot discharged out of any Piece of Ordnance assigned.
BY the third and fourth Proposition of the second Book of Tartaglia, his Nova Scientia▪ shews that every body equally heavy, as a Shot in the end of the violent motion thereof,
being Discharged out of a Piece of Ordnance, so it be not right up, or right down, the Crooked-Range, shall join with the Right-Range,
and to the natural Course and Motion betwixt them both.
[geometrical diagram]
As for Example.
The Right-Range being all the Right-Line AB which is properly called his Violent Motion,
and BC will be the mixt or Crooked-Range, and CD the Natural Motion, wherein from
A to B is the furthest part of the Violent Motion, and from C to d the end of the Natural Motion.
And in the seventh Proposition of the same Book, he proveth that every Shot equally
heavy, great or little, equally elevated above the Horizon, or equally Oblique or
Levelly directed, are among themselves like and proportional in their Distances, as
the Figure following sheweth, as A:E:F is like and proportional in the Right and Crooked-Ranges
unto H:I, and in their Distance or Dead-Ranges AF unto A:I.
And in his fourth and sixth Propositions of the same Book, he proveth that every Shot
made upon the Level hath the mixt or Crooked-Range thereof, equal to the Arch of a
Quadrant 90 degr. and if it be made upon any Elevation above the Level, that then it will make the
Crooked-Range, to be more than the Quadrant.
And if that be made Imbased under the Level, that then the Crooked-Range thereof will
be an Arch less than a Quadrant.
And lastly, in his ninth Proposition of the same Book, he undertakes to prove if one
Piece be Shot off twice, the one Level, and the other at the best of the Random at 42½
degr. Mounture, that the Right-Range of the length is but ¼ of the Right-Range of the best
of the Randoms, and that the Dead-Range of the Level is but 1/20 of the Dead-Range
of the best Random, whereto he that desires a further Demonstration of these Propositions
in his said second Book of Nova Scientia.
[geometrical diagram]
SECT. XXXVIII. How to make a Gunner's Rule, being an Instrument which will serve to elevate a Piece of Ordnance with more facility than the Gunner's Quadrant.
[geometrical diagram]
BEcause the Quadrant on the back-side of the Scale cannot be conveniently used at all times, especially when the Wind blows hard, and
being near the Enemies Guns, the Plumb-Line is so long, or too long before he stands still; to remedy this, the
Gunner's Rule was invented; the Figure thereof you may here see; this Ruler must be 10,
or 12, or 14 Inch. long according as the Gun will require, it must have a long slit down the middle thereof like an Eye-vane of
a Quadrant or Back-staff, the Head thereof make Circular according to your Gun, as you see in the Figure; the Instrument is described standing upon his Britch of
a Piece of Ordnance; in the middle of the small narrow slit you must place a Lute-string, a well twisted
Thrid, or Silk-string may serve; this Bead must be set to such an Inch and Parts, as you find is agreeable to such a degr. as you must Mount your Gun unto, and on one side the slit you must place a Division of Inches, and every Inch into 10 Parts Divided, and thus it will serve for all sorts of Guns; but if it be for a particular Gun, then on the other side you may place the degr. and min. when you shall find by the length of your Gun, how many Inch. and Parts goes to make one degree; but to use it with all sorts of Ordnance, let it only be Divided into Inches and Parts, the Bead stands at Inch. four in the Rule.
SECT. XXXIX. How to Divide the Gunner's Rule into degr. by help of a Table, sitting it for any Piece from 5 foot long to 14 soot long; and by the help of this Table, any Piece may be Elevated to any degr. without the help of a Quadrant, Ruler, or any other Geometrical Instrument whatsoever.
TO fit the Ruler for one Gun only, here is the Rule for the Deviation of the degr. Note, this Table hath 11 Columns, the first shews the length of the Piece in Feet and half Feet, the other 10 in the Head is 10 degr. and under is Inches and the 100 part of an Inch, from 1 degr. to 10 degr. and so you may take them out of the Table, and set them on the Rulers opposite side.
The use of the Table
If your Gun be 8 Foot long one Inch 68/109 makes one degree; your Gun 12 Foot long 5 Inches 9/100 makes 2 degrees.
Or you may set your Bead to 5 6/100 Inch, to Mount him 2 degrees.
SECT. XL. How to give Level to a Piece of Ordnance, with the Gunner's Rule at any Degree of Random.
YOur Piece being Loaded in all points, as is before taught, and you have brought the Piece in a Right-line with the Mark, the Dispert being placed upon the Muzzle-Ring; in
like manner place your Ruler upon the Base-ring, and let one standing by hold it,
for the Foot of it fitted round to the Gun, you may be sure to put it right, and you may estimate on its perpendicular near enough;
now having before the Distance to the Mark you intend to Shoot at; and admit you have
found it to be 461 Paces, and the first Shot you made for Practice out of that Gun, conveyed her Shot at two degrees of Mounture 274 Paces, then according to the Rules in the 32 Section, and the Tables
of Random, there I find 461 against 6 degr. which I must Mount the Gun to reach 461 Paces.
Then to find by this Table how many Inches, and hundred parts of one Inch 6 degr. will require; look in the Table above, and find on the left Hand in the first Column
the length of the Piece 13 Foot just, under 6 degr. in the Common-Angle, you shall find 16 44/100 Inches, and to that height I set the Bead on the Lute-string, to 16 44/100 Inch, or 16 4/10; for every Inch is Divided into 10 Parts, and every Part is supposed to be Divided into 10 more;
then cause the Piece to be Mounted higher or lower, until you bring the Bead, the top of the Dispert,
and the Mark all in one Line, stop the Piece in that position with a Coyn, Prime, and give Fire.
If you will Shoot by the Metal of the Piece, Substract the height of the Dispert out of the Inches found by the Table, and the Remainer, Mount your Piece unto; if the Dispert be 3 Inches ¼ long, Substracted from 16:44 found in the Table, leaves 12 19/100, or 122/50 of
an Inch, you must set the height of the Bead to Shoot the same Distance, by the Muzzle-Ring
without the Dispert.
SECT. XLI. How by the Table to give Level to a Piece of Ordnance, without the Gunner's Rule.
IF you have not a Quadrant, nor a Ruler, and would make a Shot at 4 degrees of Elevation, look in the Table, and find the length of the Piece, which suppose to be 9 Foot and half, right againg 9 ½ in the Angle under 4 degr. you shall have 7 Inches to be the length of any Stick, cut, and set it upon the Base-Ring, and bring the
top of the said Stick, the top of the Dispert, and the Mark in a Right-line with your
Eye, and Prime, and give Fire, and you will make as good a Shot, as if you had the
Ruler, and Bead, or Quadrant; if you will have no Dispert, take the Dispert, and Measure it upon the foresaid Stick
at the Base-Ring, and from it cut off its lenghth just, and the Remainer you may use
upon the Base-Ring, and it shall mount the Gun to 4 degrees, as before; and bring the top of the Stick, the Muzzle-Ring, and the Mark in a Right-line,
and you may be sure to make a good Shot; if the Dispert were 3 Inches, that cut from 7 Inches, the Remain is 4 Inches, for the length of the Stick to be set on the Base-Ring, for to Level the Piece without a Dispert.
SECT. XLII. How to make a Shot at the Enemies Lights in a dark Night.
UPon such occasion, to Shoot at a Light seen in the Night, Dispert your Piece with a lighted and flaming Wax-Candle, or with a lighted piece of Match, that with
your Eye you may bring the Base-ring, the fired Match on the Muzzle-Ring, and the
Enemies Light in a Right-line, (or mark) then give Fire, and you will make a good
Shot.
SECT. XLIII. How to make a perfect Shot at a company of Horse-men, or Foot-men passing by the
place where Ordnance doth lie upon a Level-Ground; and also to make a good Shot at a Ship Sailing upon a River.
TAke a Piece that will reach the way or Mark in a Right-Line that the Horse or Foot are to pass
by, then your Gun Loaded so with Powder as it may presently take Fire, and Shot fit for that use; and seeing a Tree, Bush,
or Hillock, or some turning cross way for his Mark, and when the Enemies come near
to that way in a Right-Line with his Gun, give Fire: and at Shooting at a Ship in a River, he must put his Piece to some evident Mark on the other side the River, and when the Head of the Ship shall
begin to be betwix the Piece and the Mark, and then give Fire.
SECT. XLIV. How to cause the same quantity bosh of Powder and Shot, discharged out of the same Piece, to carry close, or more scattering.
IN the Book of Mr. John Bate of Extravagants, he saith, take the quantity of a Pease of Opium, and charge it among the Case-Shot, and it will make the said Shot flie closer together,
than otherwise it would; for Opium is of congealing and fixative nature; this a Sea-man found by experience.
SECT. XLV. How a Shot which sticketh fast within the Concavity of a Piece, that it cannot be driven home unto the Powder, may be Shot out, without hurt to the Gunner, or hurt to the Piece.
VVHen any Piece of Ordnance is Charged with such a Shot as will not be driven home
unto the Powder, then the Gunner to save his Piece from breaking, must so Imbase the
Mouth thereof, or put him under the Line of Level, that fair Water for two or three
days being put into the Touch-hole at several times, may run into a Vessel set under
the Mouth of the Gun, to save all the Salt-Peter that was in the Powder: and then let the Gunner clear the Touch-hole, and put in as much Powder as possible he can, and Prime, and
give Pire, and it will serve to drive out the Shot.
But when a Shot hath lain long in a Piece, until he is grown rusty, and so stick fast;
put strong Vinegar in the Mouth of the Piece, and with the Rammer strike the Shot
until it doth move, then put in Vinegar until it run clear through the Powder and
Shot, Prime, as before, and give Fire with good Powder; and if it do not run through
after it hath stood 3 days, clear the Touch-hole, Prime, and give Fire.
SECT. XLVI. A Piece of Ordnance at the same Elevation, and towards the self-same place, with the like quantity of
Powder and Shot, discharged several times, what difference there is in their Ranges.
THere hath been a Piece discharged in the space of an Hour seaven times, with the like quantity of Powder,
Shot, and Mounture, and their Ranges have been fonud as followeth; the first Shot
was conveyed 418 Paces, the second 438, the third 442, the fourth 434, the fifth 427,
the sixth 412, the seaventh 395; so that the greatest difference from the first Shot
was 24 Paces; this every Gunner must take notice of, if he intends to Shoot well at Random; the reason of these things
is this, the first Shot of Powder found, the Chamber of the Gun moist, and the Air quiet, the second Shot, the Chamber was dryed, and the Gun in a good temper, and the Air moved towards the Mark with the first Bullet, and having
less assistance than the first, went beyond, and made the best Shot; and every Shot
after, will come shorter and shorter, as the Gun grows hotter and hotter; the reason is, by how much hotter the Piece is, by so much the more attractive is the concavity of the Piece made; and because the Shot is driven forth or expelled with no other thing, than
by the Air's exaltation, or Wind, caused through the Salt-Peter; and therefore the
oftner the Piece is Fired, and the more heat, the more attractive, which suppleth and retaineth continually
more of that Wind, which should serve to expel the Bullet; and therefore the virtue
expulsive in that Piece, doth more and more decrease, and the Shot flyeth not with that swiftness, as it did
before in the 2 first Shots, which dryed and brought the Gun into his best temper; but the third and fourth Shot is but little difference from
the first, but the rest will differ every Shot.
Nicholas Tartaglia doth report, that many Shots being made at a Battery by a Piece, it chanced by some occasion, the Piece rose up in such sort that the Piece touched the ground with its Mouth; a little Dog running by, did smell into the Mouth
of the Piece, and after a little time, was drawn almost to the further end of the concavity: they
pulled him out almost dead; this was done by the virtue Attractive.
SECT. XLVII. How to Weigh Ships sunk, or Ordnance under Water: or to know what empty Cask will carry any sort of Ordnance over a River.
NIcholas Tartaglia hath well collected from the Learned Archimedes, and hath calculated the Proportion of Stone, and other Mettals, what they will weigh
in the Water, and in the Air.
All Men know by reason, that whatsoever is heavyer than so much Water, as the body
of the Metal thrusteth out of the place, or Vessel, will sink; and being lighter than
so much Water, will swim.
Tartaglia saith, that ordinary Free stone weighing 93 l. in the Air, will weigh but in the Water, which is near as 2 is to 1, between the
Free-stone, and the Water. And that Marble-stone that weigheth 7 l. in the Air, will weigh but 5 l. in the Water, which is near as 7 to 2, between the Marble and the Water.
And Iron and Tin, that in the Air weigheth 19 l. will weigh 16 l. in the Water; so it is as 19 to 3.
Brass weighing in the Air 65 l. will in the Water weigh 55 l. and so Brass is to Water, as 65 to 10.
And Lead weighing in the Air 30 l. will weigh in the Water but 27 l. so Lead is to Water, as 30 to 3.
And lastly, Gold in the Air being 17 l. in the Water,Bristol, Septem. 12. 1667 by experience of a Spanish Ship called the Victory of Majork, sunk in Hungrode at the Pill; her Burthen about 300 Tuns; we weighed her with 4 empty Lighters of 30 Tuns a piece, by lashing the Lighters at Low-water, Head and Stern; and at High-water
the Ship we heav'd A shore by a Grabble, and did rise as the Water did flow; and the
Low-water or two afterward, the Ship was free, and swimmed; the Ship and Water was
Estimated to weigh 1000 Tuns, that the four Lighers weighed; she had no Goods in, but turned over as she
was, having done to Carein. it will weigh but 16 l. so Gold is to Water, as 17 to 1. He also layeth down Rules to weigh Ships, or Guns,
or any thing else in the Water, that hath not lain too long, and docked it self in
Oaze: for if the thing sunk be upon Sands or Rocks, it will weigh the better. He describes
Vessels Loadeu to be brought to the place where the thing is sunk, and a Globe of
Glass put in a Frame of Wood, and a place in the bottom of the Glass to put his Head
into the concave, he may both see and breath, and by a Windless Rope, and weight to
sink it, he may first let down the weight, and after have himself down in that Frame,
that is in a form of an Hour-Glass, to the bottom of the Sea, and do the work, and
sling the [...]hip, and Guns, and when he will come up to the top again, to un-wind the Rope, and
the Frame will be guided upright, and he and it will come to the top of the water
very safe, and fasten this Rope brought from the Ships, and un-lead the Vessels, then
will they Buoy up the Ship sunk, or any thing from the ground.
Or by a Float-Stage & Windless, Capston, and Trunk of Leather made so thick, that
no Water can come in, and with a pair of Glass Eyes fastned, that no Water can go
through, fitted to the Case of Leather within, and two Bladders blown at the brim
or too of the Water, made fast to the Case of Leather to swim, the Mouth of the Case,The best weighing of Ships, is where the water ebbs and flows much. while the Man goes down with Ropes fit to sling it, and makes them fast at liberty;
then hale him up after a time fit, and by your Vessels, as before, and Float-Stage,
Heave and Buoy up the thing sunk.
Know this, that 5 Tun of Cask will swim a Canon of 8 or 9000 weight, 4 Tun a Demi-Canon, 3 Tun a Culvering, and 2 Tun a Saker, with all things belonging thereunto, as Planks and Ropes.
SECT. XLVIII. How many Horses, Oxen, or Men will serve to draw a Piece of Ordnance.
FOr every hundred weight of Metal, one Man; so a Piece of 8000 Pound weight requires
80 Men, and as many more Men as the Carriage may be in hand weights; for every 00
weight of Metal, one Horse, then 16 Horses will draw a Cun of 8000 weight; but in
the Winter 24; also 17 Yoke of Oxen is thought sufficient to draw a Piece of 8000
weight, but in the Winter, they had need be one third part more.
SECT. XLIX. How Gunners may take a Plott of their Garrison, and every object near it.
HE may by the Compass and Ruler directed in the seventh Chapter of Surveying of Land, take the Plott of his Garrison, and every noted place or way within Gun-shot, and draw it into a Map, and have it in a readiness, and need not be troubled
to make Distances every time he hath an occasion to make a Shot; but by his Scale
and Map, may know if his Gun will reach any place thereabouts; and by the fourth Chap. of the General use of the
Canon of Logarithms in Mr. Gunter's Works, learn the Practice of Fortifications, there it is put down for him at large.
OF ARTIFICIAL FIRE-WORKS, FOR Recreation, AND SEA and LAND-SERVICE. CHAP. XIII.
SECT. I. A Description of the Mortar-Piece, and how to make one of Wood, and Past-Board (for a need,) Brass and Iron ones being
wanting.
THe same Metal that makes the best sort of Brass-Ordnance, they make Mortar-Pieces with, and by these Measures; if the Diam. or Bore be 9 Inches, let the Mortar be one Foot and half in length, and let the Chamber in which you Load your Piece with Powder be 3 Inches Diam. and 4 and a half deep; the thickest of the Metal above the Touch-hole 3 Inches, and the upper part thereof 1 Inch ½.
To make the Mortar-Piece of Wood and Past-Board.
Provide a Wooden-Ruler of such bigness as you desire to make the Diameter of the Morter, then grease your Ruler well, that the stuff may slip off that is put about him, which
is Past-Boards and Canvas, and very well plyed with hot Glue; and after let it dry
a little while on the Rowler, and another while off from the Rowler; and when this
kind of Trunk is very dry, put it on the Ruler, and set it in a Lathe, and cut off
both ends of the Trunk with a Chizel very even, then turn a Foot thereto with a shoulder
to put the Trunk upon, and in the middle thereof make the Chamber for your Powder; if the Piece be 8 Inches in the Mouth, let the thickness of the Past-Board-Trunk be two Inches thick, and 18 Inches long, the Britch or Foot 10, the Shoulder 2 Inches long, and 2 high, that when the Trunk is put on this Shoulder, and joyned with the
[Page 84] Wood, it may be just even with the same; the Bore into which you put your Powder must be two Inches high, and three deep, Plated with Copper, Lattin, (if it be possible)
as also all the rest of the Wood that goeth into the Trunk; when you have put the
Trunk into the Britch of Wood, nail it round about the Shoulder, by making holes for
the Nails, and then driving in the Nails upon that Wood, that you made to receive
the Past-boards or Trunk; then cover both Wood and Trunk with good Belch-Cord and
Glue again, and let it be well dryed, it will last a long time; and with such you
may Shoot Ballouns into the Air for Recreation.
SECT. II. How to fit and prepare Granadoes for the Mortar-Piece.
THe Shot of great Mortar-Pieces are most commonly one tenth part lower than the Bore,
because of Cording them, to sling into the Mouth of the Piece; and for fear of secret
Cracks, which cannot be easily espyed, they are coated with Pitch, so that being fitted
and prepared, they do but just fit the Bore.
How to make Fuses.
Every Ball hath an hole left to put in a Fuse, or piece of Wood, just like a Faucet
for a Spiggot; this hole must be just one quarter of the Diameter of the wooden Fuse, which Fuse must be in length three quarters of the height of
the Granadoe; make it taper, and then filled with composition, and driven gently into
the Powder that is in the Ball, leaving a little of it without: the Composition of
this Fuse is made thus; take one Pound of Powder, four Ounces of Salt-Peter, and one
of Brimstone, first beaten to Powder, and sifted in a Sieve severally, these Ingredients
being mixt together, your Composition is fit for use.
SECT. III. How to make Granadoes of Canvas for the Mortar.
THe operation of these Granadoes made of Canvas is quite contrary to these already set down: these are only Fit to
Fire a Town, they are not of so violent execution, as the precedent, yet altogether
as costly in the making; for the making of them, fit a piece of Canvas upon a round
Ball of Wood or Form, so big as you would have your Granadoe, then take this Composition
following; four Pound of Salt-peter, two Pound of Gun powder dust, and two Pound of
Brimstone; all these incorporated, and moistned with Oyl of Salt-peter; fill your
Case with this Compound, and cover it with Cords, and pierce the Sack full of holes,
and in every hole put an Iron Barrel, Charged like a Pistol; these must be driven
into the Sack unto the head, then let there be an hole about an Inch deep, which shall
serve to Prime it with Powder-dust, moisten it with Oyl of Petrol; be sure your Barrels
have great Touch-holes, that the rust through time may not choak them, and they will
be ready for service many years.
SECT. IV. How Granadoes are to be Charged in a Mortar, and Fired.
YOu must take great care in the Loading or Charging the Mortar, thus; first, weigh
the Powder to a Drachm that you put into the Chamber, and after it put a good close
Wadd of Hay, or a Tampion of Wood, then cut a Turf off the Ground that may just fill
the bottom of the hole or bore of the Mortar next the Wadd; your Granadoe being prepared, with a coat of Pitch and Cord, as before taught, sling it into the
Mouth of the Mortar; observe to have the Fuse of the Granadoe in the middle of the Bore, then go to the Britch, thrust up a Wire in the Touch-hole
to make sure, then [Page 85] Prime with the best drie Powder you have; for (believe me) this is a ticklish sort
of Shooting; without care, your Life, and Mortar-Piece is now at stake; but we will
give you very sure directions how to give Fire.
Provide small Fuses, such as we taught you to make before for the Shells, but less,
about a quarter of an Inch bore, three quarters of an Inch thickness, and eight Inches
long, fill these with good Powder-dust, moisten it with Oyl of Salt-Peter but a little,
and put it in with an Iron Rammer, try whether you like the time, they continue burning;
if too slow, abate Oyl of Peter; if too fast, add more to it.
Thus being prepared, the use is, (viz.) thrust the Pick of your Linstock in at one end of the Fuse you mean to give Fire
withal, bid one of your assistants come to one side of the Mouth of the Piece, and
give Fire to your Fuse, wherewith Fire the Fuse in the Mortar, and then with great
speed give Fire to the Touch-hole; these Fuses are very certain to give Fire, but
Match doth ofttimes fail.
SECT. V. How to make Hand-Granadoes to be hove by Hand.
THere is good use made of Hand-Granadoes in assaults, and Boarding of Ships, and there
be two sorts of them made; the first is shewed already,The Sdells are made of Glass, or nelld Clay, or Paper. the second is made by Sea-Gunners upon a Mould made with Twine, and covered over
with Cartredge-Paper, and Musquet-Bullets cut in two, put with Past and bits of Paper
thick on the out-side; after you have doubled the Shells, Paste on some at a time,
and let it drie, and then some more, until he is quite full; then dip him in scalding
Rozin, or Pitch, and hang him up, and he is for your use; but you must have the innermost
end of the Twine. which must be left out at the small hole for the Fuse; and before
you Pitch it, you are to wind it out, and stop the hole, and then Pitch it.
To Load them, fill these small Shells with Gun-Powder, then make a Fuse of one Pound
of Gun. Powder, six Ounces of Salt-Peter, and one of Char-cole; or if you will have
them of less durance, you may take the Composition made for the Fuses before spoken
of for great Granadoes, knock the Fuse up to the head within one quarter of an Inch,
which is only to find it by in the night; stop well the rest of the holes if any Chinks
are open, with soft Wax; then your first Shells must be coated with Pitch and Hurds,
lest it should break with the fall; and be sure when you have Fired the Fuse, suddenly
to cast it out of your hand, and it will do good execution.
SECT. VI. How to make Fiery-Arrows or Darts like Death Arrow-Heads.
MAke your Head of Iron, sharp and bearded, to stick fast; and to it have an Arrow
or long shaft of Wood, and about the middle of that Head make fast a Linnen Bag in
form of an Egg, leaving open at the end a hole, that it may be filled with the Composition
following; take one Pound of Peter, half a Pound of Gun-powder, and as much Brimstone
in Powder; all these Ingredients being mixt well, and mingled with Oyl of Petriol;
with this fill the Bag round about the Arrow-head, then let all be bound about with
Wire; and for Priming of these, dip Cotton-week into Gun-Powder wet with Water; and
well dryed again before it is used, and let the Arrows or Shafts be so put into the
Head, that when they be stuck in a House or Ship-side, or any where else, the Man
which endeavours to pull them out, may be deceived, and pull only the Shaft, and leave
the Head to burn the House, or Ship, or Mens Cloaths, or any thing else; if you throw
or shoot it well, it will Fire whatsoever combustible stuff or matter shall be near
it, as Sails, Timber, Pitch, Tarred places; and this will much assault Enemies in
storming a Work, or Boarding a Ship.
FIre-Pots, and Balls to throw out of Mens Hands, or with a Bascula, may be made of Potters-Clay, with Ears baked, and to it hang lighted Matches, and
throwing them, if it lighteth on a hard thing, it breaks, and the Matches Fire the
Powder, and the half Bullets of Musquets contrived upon them, as before, disperses,
and doth much mischief; their mixture is of Powder, Peter, Sulphur, and Sal-armoniack
of each one Pound, and 4 Ounces of Camphire pounded, and Searced, and mixed well together
with hot Pitch, Linseed-Oyl, or Oyl of Peter; prove it first by burning it, if it
be too slow, add more Powder, and if it be too quick, more Oyl, or Rozin, and then
it is for your use.
SECT. VIII. How to make Powder-Chests.
YOu must make them with 2 Boards to be nailed together, like the ridge of a House,
and one longer and broader to the bottom thereof; between the three Boards put a Cartredge,
then make it up like a Sea-Chest, and fill it with Pibble-stones, Nails, Stubs of
old Iron; then nail the Cover on, and the end to the Decks, in such a place as you
may Fire the Powder underneath through a hole made to put a Pistol in.
SECT. IX. How to make Artificial Fire-Works for Recreation and Delight.
VVE shall not describe the Moulds in particular, being needless; for such Men as are
inquisitive into these things, let them buy Mr. Babrington, or Bate, or Malthus, or Norton's Fire-Works; here we will lay down such Rules, as shall be as soon conceived without
Figures, only a Rowler or Mould for to make the Paper upon; and that may serve for
all the rest, they being made in the same manner.
To mak good experienced Rockets our way, do thus; get a Form or Rowler to be turned
in a Lathe, what thickness you please, and intend to make your Rockets, and let his
length be 8 times the Diameter; if it be ¾ of an Inch in thickness, the length will be 6 Inches, put so many Rowls
of Paper on this Form, until it is ½ an Inch thick, or make it ¼ Inch the whole then
Paste the upper side to the rest; then you must contract the Paper together an Inch
from the Mouth, thus: dip an Inch of the Case in Water, the Formor in him, and with
Twine, about ¾ of an Inch from the end gather it in; but let a Formor, or a thing
near the bigness be put into his Mouth, while you draw it in with the Twine and choak
it; you must remember to leave a small hole to put in a Wire through the Composition
half way the Rocket, as big as a Bodkin; then take out the Formor, and dry them,
and they are for use at any time; the Figure following makes all plain; A is the Mouth
of the Rocket, B so far the Bodkin must be thrust up the middle; you must be provided
with a smaller Bodkin, which when your Rocket is filled with the Composition, and
tyed to the Rod, you must thrust this Wire Bodkin in at the Mouth, straight up to
the midst of the Rocket, having a care not to thrust it more upon one side than the
other.
SECT. X. To make the Composition for Rockets of any size.
[diagram]
THe Reader may make use of these Rules, not upon trust out of Authors, but found by
Practice and Experience; and first for Rockets of 1 Ounce; you must use only Canon-Powder-dust,
being beaten in a Mortar, and finely Searsed; this makes him rise very swift, making
a great noise, but carries no Tail. These of more Operations are made by putting
one Ounce of Char-cole-dust to 8 Ounces of Powder; this Composition will hold for
Rockets of one, two, and three Ounces; but for these of four, take three Ounces of
Charcole-dust, to one Pound of Canon-Powder-dust, continuing that Rule, until you
come to Rockets of 10 Ounces; and also for Rockets of a Pound, take one Pound of Powder-dust,
and four Ounces of Char-cole-dust, and these are big enough for any Recreation or
Delight.
To fill the Rockets with this Composition.
Hold the Mouth downwards where it was Choaked, and with a knife put in so much as
you can of the Receipt provided for that size at one time, then with a Rammer fitted
to the Case, and with a Mallet give three or four indifferent knocks, then put in
more Composition until it be full, every time knocking the Rammer, as before, until
the Composition come within one Diameter of the bore of the top; then put down a piece of Paste-board, and knock it in hard,
prick three or four little holes therein; then put fine Pistol-Powder in almost to
the top, and upon that another cap of Paper, upon which put a Piece of Leather, that
it may be tyed on the top of the Rocket, and fast Glued on; then get a straight Twigg,
and bind it upon the Rocket with good Twine; it must be no heavier, than being put
upon your Finger an Inch and a half from the Mouth of the same, that it may just ballance
the Rocket, then it is prepared for use.
To give Fire to one or two Rockets.
Set your Rockets Mouth upon the Edge of any piece of Timber, that stands so high from
the ground, that the stick may stand perpendicular from it downwards or upon a side
of a Wale or Carriage-wheel, or any dry place whatsoever; then lay a train of Powder
that may come under the Mouth thereof; give Fire thereto, and you have done; but to
Fire more Rockets than one, that as one descendeth, the other may ascend by degrees:
make this Composition following; of Roch-Peter eight Ounces, Quick-Brimstone four
Ounces, and fine Powder-dust two Ounces, which lay in a Line, from one Rocket to another,
they being placed ten Inches, or a Foot one from another; give Fire to this Composition,
and you have your desire, if you did prick the Rocket with the Wire, as directed;
you shall see how gallant one will mount the Air, when the other is spent.
SECT. XI. How to make flying Serpents and Rockets that will run upon a Line, and return again.
FOr this you must provide a small Rowling-Pin about one quarter of an Inch in thickness;
upon which Roul seaven or eight thickness of Paper; fill them four Inches with Powder-dust,
sometimes putting between the filling a little of the Composition for Rockets of
ten Ounces, and at the end of four Inches choak him; fill two Inches more with Pistol-Powder,
then choak the end up, and at the other end put in a little of the mixture for Stars,
which follows, and choak between them and the Composition, and it is fit for use;
but divers of those with the Starry end downwards upon the head of a Rocket and Powder-train
to blow them out, when the Rocket is spent, they will first appear like so many Stars;
when the Stars are spent, taking hold of the Powder-dust, they will run riggling to
and fro like Serpents; and when that Composition is spent, they will end with every
one a Report, which will give great content to the beholder.
I did omit to speak of Runners in its proper place in the last Section, for that is
the Composition, which you must make them of, very carefully whether they be, double
or single, or those that carry Dragons, Men, or Ships, or other Shapes in motion,
least they shame their Master; the Line must therefore be fine, even, and strong,
and being rubbed over with soft Sope to make it slippery, and not easily to take Fire;
Those that turn Wheels, may have a further addition of Roch-Peter in their Receipts
to add pleasure and life to the beholders; You must have a piece of Cane as long
as the Rocket, and bind to the Rockets, and so that ones Head may be to the others
Vent, that when one hath carryed the Cane on the Line to the end thereof, the other
may Fire, and bring him back again to the Tower or place where it was fired; these
Figures are made with strong Paper or Parchment, and with Lattin, and Wire, and Twine,
until they be brought into these Shapes, and then painted like Ships, or Dragons,
or like the thing it carries with it.
SECT. XII. How to make Fire-Wheels, or as some call them Girondles.
FIrst be provided with Spinning-Wheels or the like, made easy to run round upon its
Axis, Horizontal, or Vertically; and put Flags on the top of the middle, to set out the
Wheel; bind Rockets to the Wheel, and Crackers betwixt each Rocket, with the Mouth
of one towards the Tail of the other; thus continued, until you have fitted the Wheel
quite round; which done, cover them with Paper pasted over, and coloured handsomly
to set it out, that one taking Fire, they may not Fire all, and daub Soap upon them
quite round, leaving the Mouth of one of them open, to give Fire thereto; the first
Rocket being burned, will set Fire to the rest one after another, keeping the Wheel
in a contional motion, until they be all spent; you may bind Fire-Lances to these
Wheels, either upright, or near over athwart, which will make to appear diversity
of Fire-Circles; you must take care to place your Wheels called Girondles at convenient distance from other Fire-Works, least they should make a confusion,
and spoil all the Work.
SECT. XIII. How to make divers Compositions for Starrs.
FOr Starrs of a Blew colour mixed with Red, the Composition is of Powder-mealed eight
Ounces, Salt-Peter four, Quick Brimstone twelve Ounces, Meal all these very fine,
and mix them together with two Ounces of Aqua Vitae, and half an Ounce of Oyl of Spike; which let it be very dry before you use it.
Another Composition which will make White and beautiful Fire; take Powder eight, Salt-Peter
24, quick Brimstone two, Camphire one Ounce, Meal these Ingredients, and Incorporate
them; make the Camphire with dipping your Pestil into a little Oyl of Almonds, and
it will Meal presently, and keep it close from the Air, or else it will become of
no use.
Another White-Fire which lasteth long, take Powder four, Salt-Peter 16, Brimstone
eight, Camyhire one, Oyl of Peter two Ounces, Meal these that are to be Mealed, and
mix them according to the former directions.
SECT. XIV. How to make and use the Starrs.
TAke little square pieces of Brown Paper, which fill with either of the foresaid Compositions
which you like best, fold it down, rowling it til you make it round, about the bigness
of a Nut or bigger, according to the size of your Rocket that you intend them for,
Prime them with drawing through them Cotton-Week, and they are prepared to make fast
to the Wheels: you may also make them thus; you must have a Rowler which must be as
big as an ordinary Arrow, which shall be to Rowl a length of Paper about, and Paste
it round, and dry it well, fill it with a Thimble, and thrust it down with a Rowler,
and then cut it in short Pieces about half an Inch long; then you must have in readiness
either hot Glue, or Size mingled with red Lead, dip therein one end of your short
Pieces, least they take Fire at both ends together; besides, it will not so easily
blow out; these being thus done, set them to dry until you use them, and in the top
of the Rocket, whereas in the 10 Section you were to fill it with Pistol-Powder, now
you must put none, but a very little, and that is to blow one of the bits of Starrs
out, which must stand in the room of the Powder, and on the top of that another Tire,
with strewing a little Powder and dust; and in like manner another, to a third or
fourth, putting a little small corned Powder between them, until you come unto the
top of the Rocket-case, there put a Paper over the Head of it, and tie it close about
the top, that none of the Powder come from between the Stars; the Cotton-Week is such
as the Chaundlers use doubled 6 or 7 times, dipped in Salt-Peter Water, or Aqua Vita, wherein some Camphire hath been dissolved; or for want of either, in fair Water,
cut it in divers pieces, Rowled in Mealed Powder dryed in the Sun, and it is done.
SECT. XV. How to represent divers sorts of Figures in the Air with Rockets.
VVHen you have divers Rockets to make for a great Fire-Work, let one be with a Report,
another with Starrs, another with Golden Hair or Rain, one with Silver Hair or Rain,
which it seems to be when you are right under; and upon the Head of another Rocket
place the Serpents, and they will make most delightful sport.
SECT. XVI. How to make Silver and Golden Rain, and how to use them.
YOu must provide store of Goose-Quills, which being provided, you must cut them off
so far as they are hollow; the Composition to fill them is, two Ounces of Cole-dust,
and one Pound of Powder well mixed; having filled many of these Quills, you shall
place them in the same place as I told you to put the Powder and Stars, putting a
little Pistol-Powder to blow them out, as you did the Stars, and fill the top of the
Case as full of them as you can, with the open end downwards; so soon as the Rocket
is spent, there will appear a Golden Shower, or Rain; or with the Composition for
White-Stars filled in the Quills, will make a Shower of Silver Rain.
SECT. XVII. How to make Fire-Lances.
MAke them thus; first, you must make Cartredges, or Cases just like the Cases for
Rockets, only those for a need may be made with Past-Board, and Glued, as they are
formed round, but the former is better; let them be filled with the drie Composition
for Stars in the 13th Section; Prime them with wet Gun-Powder, the lower end of the
Case is stopped with a piece of Wood, to the end they may be nailed and stirred when
and where they shall be used, the Wood being about three Fingers breadth long out
of the Case or Cartredge, or as long as you will.
SECT. XVIII. The manner how to make Balloons for the Mortar-Piece.
YOu must have a Formor or Ruler twice the length of the Diameter and of the bigness,
as you will have the inside of your Balloon, and upon that Formor put so many Past-Boards,
as you shall think sufficient for strength, then Paste or Glue them well together,
and choak him at the end with a String, leaving a small hole for a Port-Fire, which
must be made just like a Rocket but no holes in it as the Rocket hath, and of such
length as is fit: now to fill the Balloons, place all your Serpents within it together,
with Stars, Rockets, and Crackers, leave very little room within the Case, or Cartredge;
and being filled, put in as much Powder-dust as you can, that it may run every where
through the Chinks between the Serpents, Rockets, and Stars, that they may all Fire,
and that the said Powder-dust may break the Balloon; these things thus done, choak
up the other end close, and Charge it in the Mortar, as we have taught you to do the
Canvas Granadoe in the fourth Section, and you may shoot it when you please, and you
will make most excellent delight to the Spectators, and credit to your self; for this
is part of the way of Mr. Malthus's Fire-Works, which were the best that ever I practised.
SECT. XIX. A most precious Unguent for any Burning.
DIvers Men in the Practice of Fire-Works one time or other chance to be burned, or
blown in the Face by Powder; here you have Mr. Malthus's Salves, which is known by often Experience to be very good, and will fully cure
you.
The SALVES.
Take fresh Hogs-Grease, or Lard, as much as you please, and boyl, and take off the
Scum, until there arise no more Scum; then set the Lard three or four nights abroad,
after which it must be washed in running Water to take away the Saltish nature, and
to make it White; then melt it, and keep it for your use.
Otherwise,
The White of an Egg, and fresh Butter being mingled together, and well beaten into
an Oyntment, is excellent good.
SECT. XX. Another Salve most Excellent.
TAke a Stone of Quick Lime, and let it be dissolved in clear Water, and when the water
is settled, pour it out gently from the Lime through a Linnen cloath, then put as
much Sallet Oyl, as you have Water together, and beat it all to an Oyl; you may keep
it for such uses, and you have a most Sovereign Cure for all manner of Burning whatsoever.
THE MARINER'S MAGAZINE; OR, STƲRMY'S Mathematical and Practical ARTS. The Sixth Book. Wherein is Contained, The Definitions
of the Circles of the Sphere; With the manner how to Resolve all the most necessary Propositions thereunto belonging,
by a Line of Chords and Signs; As also, by Chords and Tangents, and half Tangents; and likewise by Calculation of the Tables of Artificial Signs and Tangents, and all usual Astronomical Propositions appertaining to the first Motions, being all of extraordinary Use; whereof
few of them have yet been Treated so plain in the English Tongue.
‘The Heavens declare the Glory of God, and the Firmament sheweth his handy-work, Psal. 19.’‘When I consider the Heavens the work of thy Fingers, the Moon, and the Stars which
thou hast ordained, what is Man, that thou art mindful of him? or the Son of Man,
that thou visitest him? Psal. 8.’
STƲRMY'S MATHEMATICAL and PRACTICAL ARTS The Sixth Book. Wherein is contained a Definition of the Circles of the Sphere, with the manner how to Resolve all the most necessary Propositions thereunto belonging, by a Line of Chords and Signs, or by Chords and Tangents; as also by Calculation by Tables.
CHAP. I.
IN the former Books are contained such Problems Geometrically, as are most necessary
for every professed ingenious Artist to understand and Practice; Now to the end the
Practitioner may Elevate his Thoughts to the contemplation of those Glorious Bodies,
the Sun, Moon, and Stars; I shall here in this place give a brief Survey of the first
Rudiments of Astronomy, for the help of young Practitioners and Mariners, for whom chiefly I take these pains;
I shall give a brief and succinct Explanation of the several Circles of the Sphere, better than we could in the foregoing RHYTHMES to be understood, and then shew how to resolve the most usual and common Problems
thereof; and after the Art of Dialling Geometrically and Instrumentally, and by Calculation, as promised.
1. This is to be understood, that Astronomers do imagine 10 principal Points, and
10 Circles to be in the hollow inside of the first Moveable Sphere, which are commonly
drawn upon any Globe or Sphere, besides divers other Circles which are not delineated,
but only apprehended in the Fancy.
The Points are the two Poles of the World, the two Poles of the Zodiack, the two Equinoctial Points, the two Solstitial Points, and the Zenith and Nadir.
2. The Poles of the World are two Points, which are directly opposite to one another, about which
the whole frame of Heaven moveth from the East into the West, whereof that which is seen on the North-side the Equinoctial, is called the Arctick-Pole.
3. And the other directly opposite to the former is called the Antrctick, or South-Pole, and can be seen only on the South-side of the Equator; a right Line imagined to be drawn from the one Pole to the other, is called the Axis or Axle-tree of the World.
4. All other Lines drawn through the Centre of the Sphere, and limited on each side of the surface of the Sphere, is a Diameter, but not an
Axis, unless the Sphere move round about it.
5. The Poles of the Zodiack are two Points directly opposite to each other, distant from the North and South Pole 23 degr. 31 min. and are Diametrically opposite, on which the Heavens move, from the West into the East.
6. The Equinoctial Points are in the beginning of Aries and Libra, to which Points the Sun cometh the 11 of March, and 13 of September, and makes the Days and Nights of equal Length in all places in the World.
7. The Point of the Summer Solstice, is the beginning of Cancer, which the Sun cometh unto about the 12 of June, and longest day, to the Inhabitants on the North part of the World, and the shortest day to the Inhabitants of the South.
8. And the Point of the Winter Solstice, is the beginning of Capricorn, to which the Sun cometh the 11 of December, and maketh the shortest Days of the Year to the Inhabitants of the North Hemisphere, and the longest to the Inhabitants on the South-side the Equinoctial.
9. The Vertical Point, or Zenith, is the Point directly over-head, and is the Centre or Pole of the Horizon, 90 degr. every way from the Horizon.
10. The Nadir is the opposite Point under our Feet.
Circles of the Sphere.
The Ten Circles are as followeth; The Equinoctial, or likewise called the Equator, which is the chief Circle in the Sphere, dividing the Heavens in the middle between the two Poles; the two Points of Aries and Libra, cut this Circle in opposite Points, and make the Days and Nights of equal length
over all the World.
2. The Meridian is a great Circle passing through the Poles of the World, and the Zenith of the Place; the Sun when he comes to this Meridian, it is Noon; the number of Meridians is as many as can be imagined Vertical Points from the West to the East, whereof the Cosmographers have described, 180.
3. The Horizon is distinguished by the names of Rational, or Sensible; the first is a great Circle
every where Equidistant from the Zenith, and divides the superior or upper Hemisphere, from the lower, and by chance are distinguished by the names of Right, Oblique, and
Parallel-Horizon.
A Right-Horizon have the Inhabitants under the Equator, who have the Horizon passing through the Poles of the World, and cuts the Equator at Right-Angles.
An Oblique-Horizon is such an one as cuts the Equinoctial Oblique, or aslope, or hath any degr. of Latitude from the Equator.
A Parallel-Horizon is one that hath the Poles for the Zenith and Nadir, and the Equinoctial for the Horizon.
The Sensible-Horizon is a Circle that divideth the part of the Heavens, which we see, from the part we
see not, called a Finitor.
4. The Zodiack is a great Circle, that divides the Equator into two equal parts; the Points of Intersection are Aries and Libra, the one half declining toward the North, the other to the South 23 degr. 31 min. his ordinary breadth is 12 degrees; but later Writers make it 14 or 16 degrees by reason of the Wandring of Mars and Venus.
In the middle thereof is a Line called the Ecliptick, from which the Latitude of the Planets are numbred both Northward and Southward;
the Circumference of this Circle containeth 360 degr. which is divided into 12 equal Parts called Signs, every one representing some living
Creature, either in Shape or Property, as you read in the Denominations; and also
every Sign containeth 30 degr. and every degree containeth 60 min. and every min. 60 seconds, and every second 60 thirds.
The Names and Characters of the 12 Signs.
♈
♉
♊
♋
♌
♍
Aries.
Taurus.
Gemini.
Cancer.
Leo.
Virgo.
♎
♏
♐
♑
♒
♓
Libra.
Scorpio.
Sagitarius.
Capricornus.
Aquarius.
Pisces.
5. The Six uppermost are the Northern, and Six undermost the Southern Signs.
Of the Colures.
6. These are two great Circles, or two Meridians passing through the Poles of the
World, crossing one the other at right Angles, and dividing the Equinoctial and the Zodiack into four equal parts, making thereby the four Seasons of the Year.
7. The Solstitial Colure is as before, a great Circle drawn through the Poles of the World, the Poles of the
Zodiack, and the Solstitial Points of Cancer and Capricorn, shewing the beginning of Summer and Winter.
8. The Equinoctial Colure, is a Circle passing by the Poles of the World through both the Equinoctial Points of Aries and Libra, shewing the beginning of the Spring and Autumn, when Days and Nights are equal.
9. The T [...]opick of Cancer is a lesser Circle of the Sphere, equally distant from the Equinoctial to the Northward 23 degr. 31 min. 30 seconds, wherein when the Sun is, he is entring Cancer, and making his greatest Northern Declination.
10. The Tropick of Capricorn is also a lesser Circle, equally distant from the Equinoctial Southward 23 deg. 31 min. 30 seconds, to which when the Sun cometh, which is the 10th of December, maketh his greatest Southern Declination.
11. Of the two Pole Circles.
These are two lesser Circles, distant so much from the Poles of the World, as the
Tropick of Cancer and Capricornus is from the Equinoctial 23 degr. 30 min. which are the Pole Points of the Zodiack, which moving round the Poles of the World, describe by their motion the said two
Circles; that about the North-Pole is the Arctick Circle, and that about the South the Antarctick Circle.
12. The first Six are called great Circles, and the other Four lesser Circles; by
the Centre of a Circle is meant a Point or Prick in the middle of a Circle, from whence
all Lines drawn to the Circumference are equal, and are known by the names of Radius.
13. That is said to be a great Circle, which hath the same Centre as the Sphere, and
Divides it into two equal halfs, called Hemispheres; and that is a lesser Circle, which hath a different Centre from the Centre of the
Sphere, and Divides the Sphere into two unequal Portions or Segments.
14. Of other Circles imagined but not described in a material Sphere or Globe.
Such are the Azimuths, Almicanters, Parallels of Latitude and Declination.
Azimuth or Vertical Circles pass through the Zenith, and Intersect the Horizon with right Angles; wherein the distance of the Sun or Stars from any part of the
Meridian are accounted, which are called Azimuth, and the East and West is called the Prime Vertical Azimuth.
15. The Sun or any Star having Elevation or Depression above or below the Horizon, are then properly said to have Azimuths; but if they be in the Horizon, either rising or setting, the Arch of the Horizon between the Centre of the Sun or Star, and the true Points of East and West, is called Amplitude.
16. Circles of Altitude called Almicanters, are Circles Parallel to the Horizon, and Intersect the Vertical Circles with right Angles, which are greatest in the Horizon, and meet together in the zenith of the place, in which Circles the Altitude of the Sun, Moon, or Stars above the
Horizon are accounted, which is the Arch of an Azimuth, contained betwixt the Almicanters, which passeth through the Centre of the Sun, Moon, or Stars, and the Horizon.
17. Parallels of Declinations are lesser Circles, all Parallel to the Equinoctial, and may be imagined to pass through every degree and part of the Meridian, and are described upon the Poles of the World; in like manner, the Declination of
the Sun or-Star is measured by the Arch of the Meridian between the Sun and Star, and the Equinoctial.
18. Parallels of Latitude in the Heavens, are lesser Circles described upon the Poles of the zodiack or Ecliptick, and serve to define the Latitude of a Star, which is the Arch of a Circle contained
betwixt the Centre of any Star or Planet, and the Ecliptick Line, making right Angles therewith and counted toward the North or South Poles of the Ecliptick, the Sun never passeth from under the Ecliptick-Line, is said therefore to have no Latitude.
19. Longitude of the Sun or Stars is measured by the Arch of the Ecliptick, comprehended between the Point of Aries, and a supposed great Circle passing from the Poles of the Ecliptick and the Sun or Stars Centre, and accounted according to the order and succession
of the Signs.
20. Longitude on the Earth, is an Arch of the Equinoctial contained between any assigned Meridian where it begins, and the Meridian of any other place, but accounted Eastward from the first place, as the Right-Ascention;
but in my Tables it is counted East and West from the Meridian of the Lands-end terminating at 180 degrees.
21. Right-Ascention is an Arch of the Equinoctial accounted from the beginning of Aries, which cometh to the Meridian with the Sun, Moon, or Stars, or any portion of the Ecliptick; and by it there are Tables made for the Sun, Moon, and Stars to know the time of
their coming to the Meridian, as by the help of the hour of the Star, the true time of the Night.
22. Oblique Ascention, is an Arch of the Equinoctial between the beginning of Aries and that part of the Equinoctial, that riseth with the Centre of the Sun or Star, and any portion of the Ecliptick in any Oblique-Sphere.
23. Ascentional-difference is the Arch of Difference between the Right-Ascention, and the Oblique-Ascention, and thereby is measured the time of the Sun or Stars before, and after Six.
24. Elevation of the Pole is the Height thereof above the Horizon, which is equal to the distance between the zenith, and the Equinoctial, whose Complement is equal to the distance of the zenith from the North or South Pole, or the Elevation of the Equator above the Horizon; these Circles I have inserted, to the end they may be perfectly known; for without
them, the Reader cannot well understand the following Problems of the Sphere that are depending thereon.
CHAP. II. The Projection of the Sphere in Plano, represented by the Analemma, and the Points and Circles before described.
THe Sphere may be Projected in Plano in straight Lines, as in the Analemma, if the Semi-diameters of the Circles given, be Divided in such sort as the Line of Signs in the Fundamental
Diagram of the Scale.
This Scheme is fitted for the Latitude of Bristol 51 degr. 28 min. and represents the Points and Circles of the Sphere before described.
Take with your Compasses the Chord of 60 degr. and upon the Centre C describe the Circle HZON (2.) Draw the Diameter HCO which represents the Horizon; and at Right-Angles thereunto, cross it with another Diameter ZCN.
Then with the Latitude of the place, prick off 51 degr. 28 min. from O to N, and from H to S; and of the same Line of Chords, take the Complement
of the Latitude 38 degr. 32 min. and prick off from HAE, and from O to Q, and draw NSC and AECQ.
Then take the Suns Declination 23 degr. 31 min. and prick off from AE to G and T, and with the like Chord do the same from N to Y
and g, for the Polar Circle; and the like do from Q to D and P, and from S to X and ♄; and
through these Points draw Parallels to the Equator Y g, and TSD, and G h: P, and X ♄.
And through the Centre draw the Ecliptick-Line TGP; and draw RS Parallel to the Horizon HCO, which is the Parallel of Altitude of the Hour of Six; and at any other distance,
draw Parallels of Altitude E I f.
(1.) Thus are the Points before defined, represented in this Diagram; N is the North-Pole-Arctick, S the South-Pole-Antarctick; g the North, X the South-Pole of the Ecliptick; C both the Equinoctial-Points of Aries and Libra.
T The Point of the Summer-Solstice— P the Point of the Winter-Solstice, Z the Zenith over our Heads, N the Nadir-points under our Feet.
(2.) The greater Circles are HCO the Horizon, ZCN the Axis thereof, or Prime Vertical Azimuth of East and West; HZON the great Meridian, and also the Colure of the Summer and Winter-Solstice,— AECQ the Equinoctial; T C P the Ecliptick; SCN the Axis of the World, the Hour-Circle of 6; and lso it represents the Colure of the Equinoxes.
(3.) The lesser Circles are there represented, T D the Tropick of Cancer; GP the Tropick of Capricorn; Y g the Arctick Circle, about the Pole North; X ♄ the Antarctick-Circle, or South-Pole.
(4.) Other Circles not described upon Globes, are there represented; E f represents a Parallel of Altitude called an Almicanter; the Prickt Arches Z ☉, and S I being Ellipses represent the Azimuths, or Vertical-Circles.
The Projection of the Sphere in Plano, by straight Lines or Signs.
The Prickt Arches from the Poles, represent the Meridian or Hour-Circles, which are also Ellipses; the Drawing thereof will be troublesom, and for that reason is not mentioned; and
how to shun them in the resolution of any Proposition of the Sphere, by Chords shall be shewn in the several Questions following; But the Sphere may be Projected in Plano by Circular-Lines, as in the general Astrolobe of Gemma Trisius, by help of the Tangent, and ½ Tangents in the Fundamental Diagram of the Scale, and
by the Directions in the 4 Book 12 Chapter beforegoing, and will resolve the same
things; the directions shall be one and the same, in both, in Letters, and represents
the same.
Any Line drawn Parallel to the Equinoctial AEQ, as pq, TD. Yg: doth represent Parallels of Declination.
And any Line drawn Parallel to the Ecliptick TP, represents a Parallel of Latitude
of the Stars and Planets in the Heavens.
(5.) Divers Arches relating to the motion of the Sun, and seen upon the Globes, and found by Calculation, are in the Convex-Sphere, represented in Right-Lines, and in the Concave-Sphere by Circular-Lines.
(1.) The Suns Amplitude, or Coast of Rising and Setting, from the East and West in
the Analemma, is represented by CL in North Signs, and by CF in South Signs.
(2.) His Ascentional-difference, or time of Rising and Setting from Six in Summer
by SL, in Winter by FH.
(3.) His Altitude at Six in Summer by RC, and his Depression at Six in Winter, by
Cb.
(4.) His Azimuth at the hour of Six in Summer, by RS, or CI, equal to hb in Winter.
(5.) His Vertical-Altitude, or Altitude of East and West, by MC in Summer, and his
Depression therein in Winter by CN.
(6.) His hour from Six being East and West, in Summer by MS, in Winter by Nh.
(7.) His Azimuth from the East and West upon any Altitude, is represented in the Parallel
of Altitude by the Convex-Sphere, where it Intersects the Parallel of Declination by I ☉; but in the Concave-Sphere may be measured on the Horizon HO, as CV, or CI, measured on the Line of half Tangents.
(8.) The Hour of the Day from Six, to any Altitude, is always represented in the said
Point of Intersection, in the Parallel of Declination, hereby q ☉, or in the Concave-Sphere by S ☉; and all these Arches thus represented in Right-Lines,
are the Signs of those Arches to the Radius of that Parallel in which they happen,
being accounted in the midst of that Parallel.
How to measure the Quantities of those respective Arches by a Line of Chords and Signs,
and by Half-Tangents; and consequently thereby to resolve the most useful Cases of
Spherical Triangles; as also by Calculation, is what I intend shall be the subject
of the Pages, viz. and the Art of Dialling by a Gnomical Scale.
The former Sphere or Scheme doth represent the Triangles commonly used in Calculation.
Thus the Right-Angled Triangle CK ☉, Right-Angled at K; supposing the Sun at ☉, is
made of CK, his Right Ascention, ☉ K his Declination; — KC ☉ the Angle of the Ecliptick,
and the Equinoctial being the Suns greatest Declination 23 deg. 31′ C ☉ K, the Angle of the Suns Meridian and Ecliptick.
In the Right-Angled-Triangle LON, Right-Angled at O; supposing the Sun at LON is the
Elevation of the Pole, NL the Complement of the Suns Declination, LO the Suns Azimuth
from the North.
LNO the Hour from Midnight, or Complement of the Ascentional-Difference, NLO the Angle
of Position, that is, of the Suns Meridian with the Horizon; and of the like parts,
or their Complements, is made the Triangle CML.
In the Right-Angled-Triangle CIS, Right-Angled at I; supposing the Sun at S, there
is given CS his Declination, IS his Altitude at the hour of Six, CI the Suns Azimuth
from the East and West at the hour of Six, ICS the Angle of the Poles Elevation, CSI
the Angle of the Suns Position.
In the Right-Angled-Triangle COM, suppose the Sun at M; dM the Suns Declination, Cd his hour from Six, CM the Altitude being East or West, dCM the Latitude, dMC the Angle of the Suns Position.
In the Oblique-Angled Triangle Z ☉ N, if the Sun be at ☉. ZN is the Complement of
the Latitude, and N ☉ the Complement of the Suns Declination, or distance from the
Pole. Z ☉ the Complement of the Suns Altitude, or height; ZN ☉ the Angle of the Hour
from Noon; NZ ☉ the Suns Azimuth from the North-part of the Meridian; Z ☉ N the Angle
of the Suns Position.
And thus we have shewed how the former Diagram or Analemma represents the Spherical Triangles used in Calculation; whereby, of the Six parts in each Triangle, if any three are
given, the rest may be found by Calculation from the Proportions, and that either
by Addition and Substraction, by the Artificial Signs and Tangents; and what is resolved
by either of these sorts of Tables, we will resolve with the first Tables, and with
Scale and Compasses, that you may see the near agreement betwixt them.
THis Proposition is propounded, in the first place, because many others depend upon
it; According to the Hypothesis of Ticho, it is to be suggested, and there is ascribed to the Sun a Triple Motion; first,
a Motion upon his own Centre, whereby he finisheth one Revolution in 26 Days.
2. A daily motion from the East into the West, whereby he causeth the Day and Night.
3. An opposite Motion from the West into the East, called the Annual Motion, whereby
he runs once round in a Year through the whole Ecliptick, moving near a degree in a Day; and thereby causeth the several Seasons of the Year.
The two later Motions are fancied out unto us, by a Man turning a Crane-Wheel, or Grind-stone 365 times round, while a Worm struggling against, and contrary to that Motion, creeps
once round the contrary way, but Obliquely and a slope; that is, from the further
side of the Wheel, towards the hithermost; and by this Motion the Sun is supposed
to describe the Ecliptick-Line, and continually to insist in this Course; the other
Planets, except the Moon, moving round him, or following after him, like Birds flying
in the Air, being subject to his Motions, and many of their own besides; many of which
Motions are removed by the Copernican supposition of the Earths Motion, which is a subject of much controversy among the
Learned; However, be it either the one, or the other, the Propositions hereafter
Resolved, vary not, by reason thereof: And so the Sun being supposed not to vary from
under the Ecliptick in respect of Latitude; the Proposition and Quaere, in effect
is what Longitude he hath therein, from the nearest Equinoctial-Point, which may be
found within a Degree for his Course in each day, from his Entrance into each Sign,
from that day of the Month.
The Day of the Month the Sun enters into each Sign.
♈ Aries, March 10.
♉ Taurus, April 9.
♊ Gemini, May 11.
♋ Cancer, June 12.
♌ Leo, July 13.
♍ Virgo, August 13.
♎ Libra, September 13.
♏ Scorpio, October 13.
♐ Sagitarius, November 11.
♑ Capricornus, Decem. 11.
♒ Aquarius, January 9.
♓ Pisces, February 8.
1. If the number of the Day of the Month given, exceed the number of that Day, in
which the Sun enters into any Sign; Substract the lesser Number from the greater,
and the Remainer is the Degree of the Sign, that the Sun possesseth.
ON the 12 of May I would find the Suns true place by the former Rules; The Sun enters Gemini May 11; which Substract from 12, the Remainer is 1, which shews the Sun to be in 1 deg. of Gemini the third Sign; that is, 61 degr. from the next Equinoctial-Point.
2. Example.
Let it be required to know the Suns Place the 4th of November; on the 11 day of that Month the Sun enters Sagitarius, and the 13th day of September he enters Libra: betwixt the 13 of September, and the 4 of November is 52 days, and consequently 52 degr. from the Equinoctial-Point Libra; then 30 taken from 52, there remains 22 degr. the Suns place in Scorpio, which is the thing required.
But here is a nearer Rule yet than this, to find the Suns place exactly, and that
is by Mr. Vincent Wing's Hypothesis, and Tables in Astronomia Britanica, how to Calculate his true place from Earth, the Rule is,
First, enter the Table of the Middle-Motion of the Sun, and write out the Epocha next going before the time given, under which set the Motion distinctly belonging
to the Years, Months, and Days, and Hours, and Minutes, if any be; (only in the Bissextile or Leap-Years,) after February a Day is to be added to the number of Days given; then adding them all together,
the sum will be the Middle-Motion of the Sun for the time given.
As for Example.
Suppose the time given be the 12 of May at Noon 1667, at which time the Suns place is required.
Time given.
Longitude ☉
Apog. ☉
S
D
M
S
S
D
M
S
The Epocha 1661 years Including 6 May. Days 12.
9
20
24
49
3
6
45
5
11
29
33
07
0
0
6
10
3
28
16
39
20
0
11
49
40
2
The Suns Mean-motion, or Longitude.
2
00
04
15
3
6
51
37
2. Substract the Apog. of the Sun from his Mean-Longitude, and the Remain will be his Mean Anomaly.
Example.
S
D
M
S
The Suns Mean-Longitude is
2
00
04
15
The Apogeum Substracted
3
06
51
37
The Suns Anomaly.
10
23
12
38
With the Suns Anomaly enter the Table of his Equation with the Sign on the Head and the deg. descending on the left hand, if the Number thereof be under 6 Signs; but if it be
more than 6 Signs, enter with the Sign in the bottom, and the degr. ascending on the right hand, and in the common-Angle you have the Equation answering
thereunto; only you must, if need require, remember to take the Proportional part.
Example.
S
D
M
S
In the Table of Equation answering to 23 degr. is
0
01
11
51
The Suns Mean-Longitude, add
2
00
04
15
The Suns true Longitude.
2
01
16
06
Therefore the true place of the Sun is in 1 degr. 16 min. 6 seconds of Gemini.
Another Example.
In the Year 1583 March 14 at Noon, in the Meridian of Ʋraneburg in Denmark, thrice Noble Ticho-Brahe, most excellently observed the Suns true place in 3 deg. 17 min. 40 seconds of ♈. The time at London was 1583 March 13 day 23 h. 8 m.
The Convex-Sphere, which resolves all the most useful Problems in Astronomy, by the Direction of 13 Problems following.The Concave-Sphere, which resolves 13 Problems, viz. and by them may be resolved, most of the useful Problems in Astronomy.
A Table of the Suns Mean Motion.The Apoche, or Radius.
Year.
Longit. ☉
Apog. ☉
S
D
M
S
S
D
M
S
Ch. 1
9
07
59
51
2
08
20
03
1581
9
19
48
55
3
05
22
55
1601
9
19
57
54
3
50
43
28
1621
9
20
06
52
3
06
04
00
1641
9
20
15
51
3
06
24
33
1661
9
20
24
49
3
06
45
05
1681
9
20
33
48
3
07
05
38
☉ Mean Motion in Years above 20.
Year.
Longit. ☉
Apog. ☉
S
D
M
S
S
D
M
S
20
0
00
08
59
0
00
20
33
40
0
00
17
57
0
00
41
05
60
0
00
26
56
0
11
01
38
80
0
00
35
54
0
11
22
10
100
0
00
44
53
0
11
42
43
200
0
02
29
45
0
03
25
26
300
0
02
14
38
0
05
08
08
400
0
02
59
31
0
06
50
51
500
0
03
44
23
0
08
33
34
600
0
04
29
16
0
10
16
17
700
0
05
14
09
0
11
59
00
800
0
05
59
01
0
13
41
42
900
0
06
43
54
0
15
24
25
1000
0
07
28
47
0
17
07
08
2000
0
14
57
34
1
04
14
15
3000
0
22
26
21
1
21
21
22
4000
0
29
55
08
2
08
28
30
5000
1
07
33
55
2
25
35
37
6000
1
14
52
42
3
12
42
44
☉ Mean Motion in years under 20
Years.
Longit. ☉
Apog. ☉
S
D
M
S
M
S
1
11
29
45
40
1
02
2
11
29
31
20
2
04
3
11
29
16
59
3
05
L 4
00
00
01
48
4
07
5
11
29
47
28
5
08
6
11
29
33
07
6
10
7
11
29
10
47
7
12
L 8
00
00
03
35
8
13
9
11
29
49
15
9
14
10
11
29
34
55
10
16
11
11
29
20
35
11
18
L 12
00
00
05
23
12
20
13
11
29
51
03
13
21
14
11
29
36
43
14
23
15
11
29
22
23
15
25
L 16
00
00
07
11
16
26
17
11
29
52
51
17
28
18
11
29
38
31
18
29
19
11
29
24
10
19
31
L 20
00
00
08
59
20
33
☉ Mean Motion in Months.
Longit. ☉
Apog. ☉
S
D
M
S
M
S
Janu.
00
00
00
00
00
00
Febr.
01
00
33
18
00
05
Marc.
01
28
09
11
00
10
April
02
28
42
30
00
15
May
03
28
16
39
00
20
June
04
28
49
58
00
25
July
05
28
24
07
00
31
Aug.
06
28
57
25
00
36
Sept.
07
29
30
44
00
42
Octo.
08
29
04
54
00
47
Nov.
09
29
38
12
00
53
Dec.
10
29
12
22
00
59
The Calculation. Apog. ☉
Long. ☉
S
D
M
S
S
D
M
S
The ☉ Apocha.
☞ 1581.
9
19
48
55
3
5
22
55
Years added 2
11
29
31
20
2
4
March.
1
28
9
11
10
Days 13
12
48
48
2
Hours 23
56
40
0
Minutes 8
20
0
The Suns Mean Motion.
0
1
15
14
3
5
25
11
Apogeum Substract.
3
5
25
11
The Anomaly of ☉
8
25
50
03
The Equator added to 115′ 14′
2
2
51
The Suns true place, with Observation.
♈
3
18
5
Agreeing.
(3) Example the time given the 10 of April 1665 at Noon; and admit by the former Rules we have found the Suns Mean Motion 29
degr. 0 min. 30′ his Apogeum 3 s. 6 d. 49 m. 29 s. his Anomally 9 s. 22 d. 11′ 1″; first find a Proportional part, the Equat. answering to 22 s. 1 d. 52′ 22″
The Equator answering to 23 d.
1 51 29
their difference
59
Then I say, if 1 deg. or 60 min. give 53 seconds, what shall 11 of the Anomaly give? by the Rule of proportion, [...]
Multiply 53 by 11, the Product is 583, which Divide by 60, the Quotient will be 9″
45/60; and because the Equation decreases I Substract it from the Equa. answering
22 degr. which is 1 d. 52″ - 12″ for the true Equation desired, which according to the Title, being added
to the Suns Mean-Longitude, giveth the true place of the Sun required.
Example.
S
D
′
″
The Suns Mean-Longitude.
0
29
00
30
The Equation added.
1
52
12
The Suns true Longitude.
1
00
52
42
D. ″ ″
Therefore the Suns true place is in 0 52:42 of Taurus; these Examples are sufficient for Direction, to find the Suns true place at any
time.
PROBL. II. The Suns Distance from the next Equinoctial-point; and his greatest Declination being given, to find the Declination of any Point required.
VVIth your Compasses take the Chord of 60 degr. upon the Centre C, describe the Circle HZON, and draw the Diameter HCO, which represents
the Horizon, and at Right-Angles, or perpendicular thereunto, draw ZCN, the Vertical
Azimuth of East and West, and take the Latitude of the Place, as in this Example,
is 51 d. 28 m. and prick it from O to N, and from H to S, and draw the Axis or Meridian of the Hour of Six NCS; then prick from Z to AE, and from N to Q 51 degr. 28 min, and draw the Equinoctial-Line AECQ, then the Suns place being given, take 23 deg. 31 m. and prick from AE to ♋, and from Q to P, and draw the prickt Line ♋ CP; then take
the Suns Distance from the next Equin.-Point, which in this Example shall be 61 deg. 18 m. out of the Line of Signs, and prick it from C to ☉, and through ☉ ♁ draw a Parallel-Line
to the Equinoctial, as TD, and it shall be a Parallel of Declination, and where it
cuts the outward Meridian, as at T; apply the Distance AET to the Line of Chords,
and you have the Declination 20 degr. 30 min. which was required; Or you may take the nearest Distance from ☉ to the Equator, and
apply it to the Line of Signs, and that will give you the Declination 20 degr. 30 min. as before; and if through ☉ ♁ you draw a Line Parallel to the Horizon HO, as ef, it is a Parallel of the Suns Altitude, and so have you the Sphere Orthographically
in Right-Lines in the Convex-Sphere; and if you follow the directions of the use of Tangents, and half Tangents in the
12 Chap. of the fourth Book of the Description of the Globe in Plano, you have the Sphere projected in Plain and Circular Lines, and fitted for the use
of divers Questions; the Direction in both Spheres by the Letters signify the same
thing; but observe what you are directed by Signs in the Convex-Sphere, is likewise to be done by ½ Tangents in the Concave-Sphere.
By the Tables in the Right-Angled Triangle CK O; we have given, first the Hypothenase C ☉ 61 degrees 18′; secondly, the Angle KC ☉ 23 degr. 31′, hence to find K ☉, the Rule is, as the Radius is in proportion 10
to the Sign of the Suns greatest Decl. 23 d. 31′ KC ☉
960099
So is the Sign of the Suns distance from the next Equinoctial-Point 61 degrees 18 min. C ☉
994307
to the Sign of his Declination required 20 degrees, 30′ K ♁
954406
Or extend the Compasses in the Line of Artificial Signs from 90 degr. to 23 degr. 30 min. the same extent will give the distance from the Suns Place, to his Declination.
The Sun being either in 1 deg. 18 min. of Gemini, or 29 deg. 42 min. of Capricorn, or 1 degr. 18′ of Sagitarius, or 28 degr. 42 min. of Cancer; that is, 61 degr. 18 min. from the next Equinoctial-Point; the Declination will be found to be 20 degrees 30 minutes.
PROBL. III. Having the Suns greatest Declination, and his Distance from the next Equinoctial-Point; to find his Right-Ascention.
IN the Foregoing Scheme, having drawn the Parallel of the Suns Declination TD, passing
through the Place at ✶ the extent S ♁, is the Sign of the Suns Right-Ascention from
the next nearest Equinoctial-Point, to the Radius of the Parallel TD; and therefore place the extent ST from C to X, and upon X as
a Centre with the extent S ♁, describe the Arch at k, a Ruler laid from the Centre just touching the extremity of that Arch, finds the
Point N in the Limb of the Meridian or Quadrant, and the Arch ON, applyed to the Line
of Chords, is 59 degr. 09 min. and so much is the Suns Right-Ascension in the first quarter of the Ecliptick.
In the Triangle CK ♁ we have given as before, (1.) the Angle of the Suns greatest
Declination KC ♁ 23 degr. 31 min. (2.) the Longitude of the Sun from the next Equinoctial-Point Aries C ♁ 61 degr. 18 min. hence to find the Suns Rght-Asscention, the Rule is,
As the Radius
10
to the Tangent of the distance 61 degr. 18 min. C ♁
1026162
So is the Co-Sign of the Suns greatest Decl. 23 deg. 31′ KC ♁
996234
to the Tangent of the Right-Ascention CK 59 deg. 9 m.
1022396
Or in the Concave-Sphere; if you draw the Meridian from N through ♁ to S, whose Centre will be found upon the
Equator, it will cut the Equinoctial in K; measure the distance GK on the Line of
half-Tangents, and you have 59 d. 09′, as before.
Or extend the Compasses from 90 d. to 66 d. 29, the same distance will reach from 61 deg. 18 m. to 59 deg. 9 min. which is the Suns Right-Ascention in 61 deg. 18 ♊.
But this you are to observe, that if the Right-Ascention sought, be in the second
Quadrant ♋ ♌ ♍, then you are to take the Complement of the Arch found to 180 deg. if in the third Quadrant ♎ ♏ ♐, adde 180 deg. to the Arch found; but in the last Quadrant, Substract the Arch found from the whole
Circle 360 degr. and you shall have the Right-Ascention desired.
Example 2.
The Sun in 28 degr. 42 min. of ♋, that is, 61 degr. 18 min. from the Equinoctial Point ♎; the Rule is as before. As the Radius is to the Co-Sign of the greatest Declination, so is the Tangent of the Suns distance
from the next Equinoctial-Point 61 degr. 18 min. to the Tangent of 59 degr. 09 as before, which taken from 180, is 130 deg. 51 min. which is the Suns Right-Ascention in 28 degr. 42 min. of Cancer.
Example 3.
The Sun in 1 degr. 18 min. of ♐, that is, 61 degr. 18 min. from the next Equinoctial-Point ♎, the work is the same as before; therefore to the
Arch found, I add 180 degr. a Semi-Circle; so 59 deg. 09 min. and 180, makes 239 deg. 09 min. the Right-Ascention of the Sun sought in 1 deg. 18 min. of Sagitarius ♐.
Example 4.
The Sun in 28 degr. 42 min. of Capricorn, 61 deg. 18 min. from the next Equinoctial Point ♈, the operation is the same with the former Example;
wherefore Substract the Arch found 59 deg. 09 min. from the whole Circle 360 deg. and there will remain 200 deg. 51 min. which is the Suns Right-Ascention in 28 deg. 42 min. of ♑ Capricorn.
PROBL. IV. The Elevation of the Pole, and Declination of the Sun being given; to find the Ascentional-Difference.
THis is represented in the Figure by SL in the Parallel of Declination, and it is
therefore to be reduced into the common Radius, therefore take the Radius of the Parallel ST, and prick it from C to X, as before; then take the extent SL,
and setting one Foot upon X, with the other draw the part of an Arch at a, lay a Ruler from C, that it may just touch the outside thereof, and it cuts the Circle
in d, and take the Chord or Extent Hd; and you will find it 28 deg. 0 min. which being converted into Time, is an Hour 52 min. and so much doth the Sun Rise before, and Set after Six in Summer; but so much doth he Rise after, and Set before Six in Winter, when he hath the same Declination South.
In the Right-Angled Spherical Triangle SLC are known. 1. SCL the Complement of the
Poles Elevation 38 deg. 32 min. 2. The Suns Declination 20 deg. 30 min. hence to find the Ascentional Difference SL.
As the Radius 90 deg. is in Proportion
10
to the Tangent of the Latit. 51 deg. 28 min. SCL
1009887
So is the Tangent of the Suns Declination 20 deg. 30′ SC
957273
to the Sign of SL the Ascentional-difference 28 d. 00 m.
967160
Extend the Compasses on the Artificial-Lines of Signs and Tangents, and you will find it, as before; or
if you take the distance NS, and prick it from S to K, and lay the Ruler from Cover
L, it will cut the Arch of the Meridian in d; then Measure the distance Nd on the Line of Chords, and it will be 28 deg. 00 min. as before found, that is one Hour 52 min.
PROBL. V. The Suns Right-Ascention, and his Ascentional-difference being given; to find his Oblique-Ascention, and Descention.
TO perform this, you must observe these 2 following Rules. 1. If the Suns Declination
be North, you must Substract the Ascentional-difference from the Right-Ascention,
and the Remain will be the Oblique-Ascention; but if you add them together, the sum
will be the Oblique Descention. 2. If his Declination be South, add the Ascentional-difference, and the Right Ascention together, the sum will be
the Oblique-Ascention; but if you make Substraction, the Remainer will be the Oblique-Descention.
Admit the Sun is in the 1 deg. 18 min. of Gemini by the second Problem, his Right-Ascention is 59 deg. 09 min. and his Ascentional-difference by the 4 Problem, is 28 deg. 0 min. therefore according to the first Rule, because his Declination is North, the difference thereof 31 deg. 09 min. is the Suns Oblique-Ascention, and the sum of them 87 deg. 09 min. his Oblique-Descention.
PROBL. VI. To find the time of Sun-Rising, and Setting, with the length of the Day and Night.
YOu must find the Ascentoinal-difference by the 4 Problem, which converted into Time,
allowing 4 min. of an hour for every degr. and 4″ seconds for every min. and the sum of Hours and Minutes, is his difference of Rising and Setting before
or after the hour of Six, which was found before to be 28 deg. or 1 hour 52 min.
Therefore when the Sun is in Northern Signs, add the sum to Six, and the Total is
the Semi-diurnal Arch, as in this Example, is 7 hours 5′2, or time of Sun-setting, and Substract it from Six, and the Remain is 4 h. 8′. m the time of Sun-Rising; double 7 ho. 5′2 m, it is 15 ho. 4′4, the length of the Day; Substract it from 24 ho. 00 m. and the Remain is 8 ho. 16 m. the length of the Night the 12 of May, in Latitude 51 deg. 28 min. at Bristol.
But if the Sun is in Southern Signs, make Substraction, as in this Example; the Sun
having 20 deg. 30 min. South Declination, or in 1 deg. 18 min. ♐; Substract 1 ho. 5′2 from 6, the Remain is 4 ho. 08 m. for the time of Sun-Setting, double it, and it is 8 ho. 16 m. the length of the day; add 1 ho. 52 m. to 6, the sum is 7 ho. 52 m. is the time of Sun-Rising; double it, it is 15 ho. 44 m. the length of the Night, in Latitude 51 deg. 28 m. in 1 deg. 18 m. of Sagitarius.
PROBL. VII. The Elevation of the Pole, and Declination of the Sun being given; to find his Amplitude.
MEasure the extent CL with the Compasses in the Line of Signs, and it will reach to 34 degr. 40 min. and so much doth the Sun Rise and Set to the Northward of the East and West in the
Latitude of Bristol, when his Declination is 20 deg. 30 min. North; but he Riseth and Setteth so much to the Southward of the East and West, When
his Declination is so much South.
Now on the Concave-Sphere, the extent CL on the Horizon, applyed to the Line of half-Tangents,
is 34 deg. 40 min. the Amplitude, as before.
If the Suns Parallel of Declination doth not meet with the Horizontal-Line HO, as
in Regions far North, the Sun doth not Rise nor Set.
In the Right. Angled Spherical-Traingle LOC of the 4 Problem, having the Angle LCO,
the Complement of the Latitude 38 degr. 32 min. and LO the Suns Declination 20 degr. 30 min. in 1 d. 18 m. ♊ his Amplitude by Calculation may be found thus.
The Artificial Lines by this Rule answers the same.
As the Co-Sign of the Latitude 51 deg. 28 min. LCO
979446
is to the Radius 90 degr.
10
So is the Sign of the Declination 20 degr. 30′ LO
954432
to the Sign of the Amplitude CL 34 deg. 40 min.
974986
This Rule is the common Rule Mariners make use of for the finding of the Variation
of the Compass at Sea, by comparing the Coast, or bearing of the Sun, observed by an Amplitude or
Azimuth-Compass at the Suns Rising or Setting, and by his bearing, found by these
Rules beforegoing, the difference sheweth the Variation.
As for Example.
Admit you observed the Suns Amplitude of Rising or Setting by your Compass in the first Chap. and fifth Book of the Art of Surveying described.
And by the Compass found, the Magnetical Amplitude
45 d. 55′ compl. 44 d. 05′ N.
By the Rules beforegoing find the true Amplitude,
34 d. 40 compl. 55 d. 20′ N.
Substract the less out of the greater, the difference
11 d. 15 m. Variation.
And by reason the Magnetical Amplitude is more than the true Amplitude; therefore
the Variation is 11 degr. 15 min. which is one Point Westerly; and if you are bound to a place that bear North of you,
you must Sail upon the North by West Point; or if you bare West, you must Sail W and
by S; and if South, the Course must be South and by East; or if you bear East, then
the Course must be East and by North, to make good a North, or West, or South, or
East Course; and so of all the rest of the Points you must allow in like manner.
2. But admit the Magnetical Amplitude observed by the Compass, were but 23 deg. 25 min. and the true Amplitude by the former Rules found to be 34 degr. 40 min. the upper Substract from the lower, the difference is 11 degr. 15 min. and by reason the Magnetical Amplitude is less than the true Amplitude, and the difference
11 degr. 15 m. which is one Point Variation Easterly; and so the North Point is the N by E Point,
and NE is the NE by E, and E is E by S, and South is S by W, and W is W by N Point
of your Sailing Compass, when you have such a Variation, and the Complement of the Amplitude is the Suns Azimuth
from the North or South part of the Meridian, according as your Declination is. And
this is sufficient for an Example to find the Variation of the Compass in any place or time.
As likewise by his Oblique-Ascension, and Ascensional difference; or by the Suns being
East and West by the following Rules, or by the Suns Azimuth at the hour of Six; as
likewise his Azimuth at any other time or place observed, as shall be shewn for the
help and benefit of young Mariners.
PROBL. VIII. Having the Latitude of the Place, and the Suns Declination, to find the time when
the Sun cometh to be due East and West.
IN the Parallel of Declination, the hour from Six is represented by SM; with that
extent upon the Point X draw the Arch b the Ruler laid from C to the outward edge of the said Arch, cuts the Circle at (e,) the distance O e applyed to the Line of Chords sheweth 17 deg. 20 min. it converted into Time is 1 h. 9 m. 20 sec. and so much after Six in the Morning, and before Six in the Afternoon, will the Sun
be due East and West; by the Concave-Sphere, if you lay a Ruler from A over M, it
cuts the Limb in (e,) measure N e on the Line of Chords, and it is the same 17 deg. 20 min. the Rule by Artificial Signs and Tangents holds as by Calculation. viz.
Suppose the Latitude 51 degrees 28 min. and Declination North 20 degrees 30 min. therefore in the Right-Angled Spherical Triangle ZNM are given (1) Z N the Complement
of the Latitude 38 degr. 30 min. (2) N M the Complement of the Sum Declination 69 degr. 30 min. Then I say.
As the Radius 90 is in proportion
10
To the Co-Tangent of the Declination 69 degr. 30 min. NM
957273
so is the Tangent of ZN compl. of Latitude 38 degr. 32 min.
990112
is to the Co-Sign of ZNM 17 deg. 20 min. which Reduced is 1 h. 9 m. of time, as before.
947385
Which 1 h. 9 m. added to 6 h. is 7 h. 9 m. the moment in the Morning, the Sun will be due East; and if you Substract 1 h. 9 m. 20 sec. from 6 h. 00 m. and the Remain will be 4 h. 50 m. 40 sec. the moment in the Afternoon the Sun will be due West.
PROBL. IX. The Elevation of the Pole, and the Declination of the Sun being given; to find the Suns Altitude when he is due East and West.
MEasure the extent CM on the Vertical-Circle, and apply it to the Line of Signs, and
it will reach to 26 degr. 36 min. or the same distance taken of the Concave-Sphere, and applyed to the Line of ½ Tangents,
shews the same number, and so much is his Altitude sought in Summer; but when he hath
the like Declination South, then so much is his Depression under the Horizon in Winter,
when he is East and West; if the Suns Parallel of Declination TM doth not meet with
the prime Vertical Circle CZ, the Sun cometh not to the East and West, as it happeneth
many times in small Latitudes, or Countreys betwixt the Tropicks.
In the former Diagram, the Suns Altitude when he is due East and West, is shewed by
the Arch CM, wherefore in the Triangle CVM we have given, (1) the Suns Declination
VM 20 degr. 30 min. (2) the Angle of the Poles Elevation MCV 51 deg. 28 min. to find his Altitude CM; I say,
This Rule will hold by the Artificial Lines, of Signs and Tangents.
As the Sign of the Angle of Latit. 51 d. 28 m. UCM
PROBL. X. The Elevation of the Pole, and Declination of the Sun being given, to find the Suns Altitude at the Hour of Six.
TAke the nearest distance from S to the Horizon CL, and apply it to the Line of Signs,
sheweth the Altitude to be 15 degr. 54 min. or the same taken of the Concave-Sphere, and measured on the Line of ½ Tangents,
sheweth the same; and so much is his Depression under the Horizon at Six, when he
hath South-Declination 20 degr. 30 min.
In the Concave-Sphere, you may see all the Triangles plain, and we have known in this
Triangle ZSN, (1.) The Complement of the Latitude ZN 38 degr. 32 m. (2.) the Complement of the Suns Declination NS 69 degr. 30 min. to find the Hypotenase ZS; therefore I say,
As the Radius 90 degr. is in Proportion
10
To the Co-Sign of 69 degr. 30 min. NS
954432
so is the Co-Sign of 38 degr. 32 min. ZN
989334
To the Sign of the Altitude 74 degr. 6 min. S 00
943766
Whose Sign is 15 degr. 54 min. is the Suns Altitude at the hour of Six, when he is 1 degr. 18 m. of ♐ in Latitude 51 deg, 28 m. Extend the Compasses from 90 degr. to 20 d. 30′, the same extent will reach from the Latitude 51 deg. 28 min. to 15 deg. 54 m. as before.
PROBL. XI. Having the Latitude of the Place, and the Declination of the Sun given; to find the
Suns Azimuth at the Hour of Six.
THis is represented in the Convex-Sphere by VZ in the Parallel of Altitude of the
Sun VSB; Prick VB from C to W, and with the distance VS draw the Arch upon W at h, and lay the Ruler just touching the said Arch, cuts the Circle in Y; the distance
HY measured on the Chords, sheweth the Azimuth, or the distance G 00, on the Concave-Sphere, applyed only to the Line of ½ Tangents,
shews the Azimuth to be 13 deg. 07 min. and so much is the Sun to the Northwards of the East and West of the hour of Six.
In the Right-Angled Spherical-Triangle ZNS of the general Diagram, we have known first,
ZN, the Complement of the Latitude 38 degr. 32 min. (2.) NS the Complement of the Suns Declination 69 degr. 30 min. to find the Azimuth of the hour of Six, represented by the Angle NZS.
I say,
As the Radius 90 is in proportion
10
to the Compl. Sign of the Latitude 38 deg. 32′ ZN
979446
So is the Co-Tangent of NS 69 deg. 30 min.
957273
to the Co-Tangent of the Azimuth NZS 76 d. 53′
936719
Or extend the Compasses from the Co-Sign of the Latitude to the Radius; the same extent will reach from the Tangent of the Declination, to the Azimuth 76 deg. 53 min. as before; the Suns Azimuth from the North part of the Meridian in the Latitude of 51 degr. 28 min. and Declination 20 degr. 30 North, (13 degr. 07 min is from the West.)
PROBL. XII. Having the Latitude of the Place of the Suns Declination, and his distance from the
Meridian being given, to find the Suns Altitude at any Time assigned.
BY this Case may be found the Suns Altitude on all hours, and the distance of Places,
in the Arch of a great Gircle; for the Suns Altitude on all hours thereby is meant,
that if the hour of the Day, the Declination and Latitude be given, the Suns Altitude
proper to the hour, or his Depression may be found.
Take the Chord of 60 degr. and describe the Arch HTPOD, draw the Horizontal-Line HCO, and from O to P prick
of the Chord of the Latitude 51 degr. 28 min. and from P to T and D set of the Complement of the Suns greatest Declination, 66
degr. 29 min. and draw the Parallel of Declination TD, and the Axis CSP, or the Meridian of the hour of Six; then draw the Radius TC, which is the Ecliptick-Line, and take off the Line of Signs, and prick down,
15
1
from 6 before it, and after it.
30
degr. 2
45
for the 3
60
Hours of 4
75
5
Then take the nearest distance from 15 degr. to CS, the Meridian of the hour of Six, or Axis, and prick it from S to 5 and 7; and likewise take the nearest distance fro 30 to
CS, and lay it from S to 8 and 4; and in like manner do with the rest, then will the
nearest distance from 4 5 6 7 8 9 10 11 to the Horizontal-Line HCO be the Signs of
their respected Altitude;
And so much is the Suns Depression under the Horizon at the hours of 8, 7, and 6 in
Winter; as you may soon trie by the same Division on the Parallel of the Suns greatest
Declination Southward DST
The Summer Alt. for the h. of
7
are
27:23
8
36:42
9
45:42
10
53:45
11
59:42
And the Winter Altitude for
the Hours of
9
are
5 d. 13
10
10:28
11
13:48
And so much is the Sun deprest under the Horizon in Summer at the hours of 3, 2, 1,
from Mid-night, as you may soon find it by the nearest distance from 9, 10, 11, in
the Line LD to the Line LO; if you apply that distance to the Line of Signs, you may
draw the Parallel of Altitude through each hour, as the Example is through 9 ho. e f, and measure He, or Of on the Line of Chords, and it is 45 degr. 42 min. the Suns Altitude at 9 a clock the 11 of June.
2. But admit the Latitude were 51 deg. 28 min. and his Declination 00 deg. 00 m. and suppose, for Example, seek his distance from the Meridian is 2 ho. 44 min. or 41 deg. 0′0 and the Sun upon the Equinoctial the 11 of March, and 13 of September.
Upon the Equinoctial at K is the Sun represented the 11 of March, or 13 of September, and distance from the Meridian 2 ho. 44, or 41 degr. 00 min. If in the Convex-Sphere you lay S ♁, from C to K the Right-Ascension 59 deg. 9 min. and through K you draw a Parallel to the Horizon, as dM, it will cut the Meridian in d, and measure the distance Hd on the Line of Chords, and it is 28 degr. 03 min. the Suns Altitude required, or the nearest distance from K to the Horizon-Line HCO
applyed to the Line of Signs, would have shewed the same. In the Concave-Sphere, if
you take the distance CK, and draw a small Arch at r; and take the nearest distance to the Azimuth-Line of East and West ZC from K, and
with that distance turn the other Foot over, and cross the Arch at (r,) and through K and r draw a Circle of Altitude, it will cut the Limb in E and F measure HE, or OF on the
Line of Chords, sheweth the Altitude 28 degr. 03 min. as before.
In the Concave-Sphere, you may see the Triangle ZAEK; and we have given first AEZ
the distance from the Zenith to the Equator, equal to the Latitude of the place, 51
degr. 28 min. Secondly, AEK the Suns distance from the Meridian 41 deg. to find the Suns Altitude from K to the Horizon.
I say, as the Radius 90 deg.
10
is to the Co-Sign of the Suns distance from the Merid. 42 d. 0′ AEK
987777
So is the Co-Sign of the distance from the Equ. to the Zen. 51 d. 28 m. AEZ
979446
to the Co Sign of ZK 61 degr. 53 min.
967223
Whose Complement is the Altitude Kp 28 deg. 03 min. which was required; you may in your practice draw a particular Figure for the Question,
and shew every Triangle by it self, by the Line of Chords, and half Tangents.
Secondly, when the Sun is in North Signs, ♈ ♉ ♊ ♋ ♌ ♍.
Let it be required to find the Suns Altitude at 10 a clock and 2 m. past before Noon, when the Sun is in entrance of Gemini, in Latitude 51 degr. 28 min.
First, By the Convex-Sphere, the nearest distance taken from the Suns place, to the
Horizon HC, applyed to the Line of Signs, sheweth 51 degr. 12. min. the Suns Altitude required; Or to find his distance from the Meridian, take ST,
and prick it from C to X; then with the distance S ♁ on X as a Centre, draw the touch
of an Arch at K, a Ruler laid over the Centre, over the outward edge of the Arch,
cuts the Arch of the outward Meridian in (n;) then measure On on the Line of Chords, and it is 60 deg. 2 min. S ♁, whose Complement is 29 deg. 58 min. the Suns distance from the Meridian OT; Wherefore in the Triangle NZ ♁, we have
known, (1,) ZN the Complement of the Latitude 38 deg. 32 min. (2,) N ♁, the Complement of the Suns Declination 69 degr. 30 min. (3,) the comprehended Angle ZN ♁, the distance of the Sun from the Meridian 29 deg. 58 min. to find Z ♁, and hereby the Suns Altitude 92 ♁, I say,
As the Radius 90
10
is to the Co Tangent of the Latitude 51 deg. 28 min. NZ
990112
So is the Co-Sign of the Angle from the Meridian 29 deg. 58 m. ZN ♁
993767
to the Tangent of the fourth Ark 34 deg. 36 min. N q
983879
From the Complement of the Suns Declination N ✶ 69 deg. 30 min. Substract N q 34 deg. 36 min. there remains 34 deg. 54 min.
As the Co-Sign of the fourth Ark. 34 deg. 36 min. N q
991547
is to the Compl. Sign of the Latitude 38 deg. 32 min. ZN
989334
So is the Co-Sign of the fifth Ark 34 deg. 54 min. q ♁
991389
1980723
to the Co-Sign of Z ♁ 38 deg. 48 min.
989176
Whose Sign is 51 deg. 12 min. p ♁ the Suns Altitude above the Horizon at 10 a clock, and 02 m past, and 1 a clock 58 min. past in the Afternoon, when he is in 1 deg. 18 m. of ♊ in Latitude of 51 deg. 28 min. North.
Now if you follow the Rules before-going, you may find the Suns Altitude by the Line
of Chords, and half-Tangents by this Figure to be the same.
Suppose the Sun in the Southern Signs ♎ ♏ ♐ ♑ ♒ ♓ in the opposite Point to the former,
having South-Declination 20 degr. 30 min. and be also distant from the Meridian 29 d. 58 m. take the Declination 20 deg. 30 min. and prick it from AE to B, and Q to R in both the Spheres, and draw the straight
Line in the Convex-Sphere BMR, and take from the Suns place in the opposite Sign in
the Parallel of Declination, his distance from the Meridian YR, and prick it on the
other side of the Parallel from B to ☉; and the nearest distance to the Horizon-Line
HC, applyed to the Line of Signs, shews the Altitude to be 13 d. 23 m. and in the Concave-Sphere take of the Line of half-Tangents, the Declination 20 degr. 30 min. and lay it from the Centre C to M, and the Axis CS continued; take the Complement of the Declination of the Line of Secants, and
place it from C on the continued Line or Axis, and that will be the Centre of the Parallel of Declination; or if you take the like
Complement 69 degr. 30 min. of the Line of Tangents, and put it from M on the Axis, it will be the Centre of the Parallel of Declination; therefore draw it BMR, and
it will cut the Meridian in ♁, the place where the Sun is.
Now, to find the Suns Altitude or Ark (Z ☉) or Z ♁; therefore to find how much it
is; you must find the Pole of the Circle N ♁ Z, which is done after this manner.
Lay a Ruler from Z to h, and it will cut the Circle in ♃; then take 90 deg. and prick it from ♃ to ♂, then lay a Ruler over from Z to ♂, and it shall cut the
Horizon in ♀, which Point ♀ is the Pole of the Circle Z ♄ n.
To measure the Ark Z ♁, you must lay a Ruler upon ♀ and ♁, which will cut the outward
Circle in the Point X, so shall XZ measured on the Line of Chords, give you the quantity
of d. contained in the Arch XZ, which will be 76 d. 37 equal to the Complement of the Suns Altitude. I have been the larger in this
precept, that it may be a Rule of Direction, to shew how the Ark of any great Circle
of the Sphere; the sides of all Spherical Triangles being such, may be measured whatsoever,
by his operation in the Concave-Sphere.
Observe the Figure we have given in the Oblique-Angled Triangle ZN ♁. 1. NZ as before,
the complement of the Latitude 38 deg. 32 min. 2. N ♁ 110 deg. 30 min. the same distance from the North-Pole. 3. the Angle ZN ♁ 29 deg. 58′ to find the Altitude ♄ ♁.
As the Radius 90
10
is to the Tangent of NZ 38 deg. 32 min.
990112
So is the Co-Sign of the Angle, ZN ♁ 29 deg. 58′
993767
to the Tangent of Nq 34 deg. 36 min. 4 Ark, as before,
983879
From the Ark N ♁ 110 deg. 30 min. Substract the Arch Nq 34 deg. 36 min. and there remains 75 deg. 54 min.
As the Co-Sign of Nq 34 deg. 36 min.
991547
is to the Co-Sign of ZN 38 deg. 32 min.
989334
So is the Co-Sign of the fifth Ark q ♁ 75 deg. 54 min.
938670
1928004
to the Co-Sign of Z ♁ 76 deg. 37 min.
936457
Now the Complement of Z ♁, is ♁ ♄ 13 deg. 23 min. which is the Suns Altitude required.
PROBL. XIII. The Suns Altitude, and his Distance from the Meridian, and Declination being given;
to find his Azimuth.
DEmonstrate the Question by the Line of Signs and Chords, with 60 degr. draw the Semi-Circle, draw the Horizon-Line HO, draw the Parallel of Altitude ef, and of ☉ B Parallel to the Horizon; then draw the Axis CN, and the Equinoctial CAE at Right-Angles or 90 degrees from N, as NAE; then draw the Parallel of Declination TL and (BR) by the Line of
Chords, and Signs; then take the extent IE, and set from C to P, and upon P with the
extent I ♁, draw the Arch (00) a Ruler laid from C, just touching that Arch, cuts
the Limb ({oil}) the Arch H {oil} measured on the Line of Chords, is 41 degr. 42 min. and so much is the Sun to the Southward of the East and West, the Complement is 48
deg. 18 min. and so much is the Suns Azimuth from the South part of the Meridian Westward. Now
by the Concave-Sphere, if you find the Centre of the Aximuth-Circle on the Horizontal-Line
at ☉, and draw the Circle from Z through ♁ N; measure the distance C ☿ on the Line
of half-Tangents, and it will be the Suns Azimuth from the East and West, as before,
41 deg. 42 min. (Complement 48 deg. 18 min. from the South.)
[geometrical diagram]
In the Oblique Spherical-Triangle Z N ♁, we have known. 1. Z ♁ the Complement of
the Suns Altitude 38 deg. 48 min. 2. The Angle ZN ♁ 29 deg. 58 m. the Suns distance from the Meridian. 3. The Complement of the Suns Declination N
♁ 69 deg. 30 m.
Now I work thus;
As the Complement-sign of the Altitude 38 degr. 48 min. Z♁
979699
is to the Sign of the Angle from the Merid. 29 deg. 58 m. ZN ♁
969859
So is the Compl. Sign of the Declination 69 deg. 30 N ♁
997158
1967011
to the Sign of the Suns Aximuth NZ♁ 48 deg. 18 min.
987312
Now admit the Altitude were 13 degr. 23 min. his distance from the Meridian 29 deg. 58′,Set VY from C to ☍, and with V ♁ draw the Arch at ♂. and his Declination South 20 deg. 30′.
By the foregoing Rules, you will find the Arch H ♑, measured on the Line of Chords
61 deg. 15 min. the Suns Azimuth from the East and West, whose Complement is 28 deg. 45 min. the Azimuth from the South part of the Meridian.
Or on the Concave-Sphere draw the Azimuth Circle Z ♁ ♄ N, and measure on the Horizontal-Line
C ♄, applyed to the Line of half-Tangents, is 61 deg. 15′ as before, and the Azimuth 28 deg. 45 min. H ♄ from the South part of the Meridian; and 151 deg. 15 min. his Azimuth from the North part of the Meridian O ♄.
Observe the general Diagram of the Concave-Sphere, you have in the Oblique-Angled
Spherical Triangle RN ♁. 1. Z ♁ the Complement of the Suns Altitude Z ♁ 76 deg. 37′. 2. The Angle ZN ♁ 29 deg. 58 min. his distance from the Meridian. 3. N ♁ 110 deg. 30′, the Complement of the Suns Declination. The Rule is to find the Azimuth,
As the Complement-Sign of the Altitude Z ♁ 76 deg. 37
998804
is to the Sign of the Angle from the Merid. ZN ♁ 29 d. 58 m.
969853
So is the Complement-Sign of the Declination N ♁ 69 d. 30 m.
997158
1967011
to the Sign of the Suns Azimuth NZ ♁ 28 degr. 45 min.
968207
PROBL. XIV. The Poles Elevation, with the Suns Altitude and Declination given; to find the Suns
Azimuth.
FIrst, draw a Diagram by the Rules beforegoing in the Convex-Sphere, that is, the
Elevation of the Pole ON 51 deg. 28 m. the Suns parallel of Altitude ♑ V 13 d. 23 m. and the parallel of the Suns Declination 20 degr. 30 min. BR; then take ♑ V, the parallel of Altitude, and prick [...] from C to 00; then take the distance V ♁, and on ☍ as a Centre, draw the Arch at
( [...]) a Ruler laid over C, and the outside of (♂) shall cut the Arch in (n,) then measure the distance (HW,) and the Line of Chords 41 deg. 15 min. as before.
Or draw the Concave-Sphere, and the Axis or Elevation of the Pole ON; and likewise
prick from H and O on both sides the Parallel of the Suns Altitude ♑ V 13 deg. 23 min. and take the same number of the Line of half-Tangents, and prick it on CZ, the Prime
Vertical Line of East and West from C upwards to (v,) and draw the parallel of Altitude, ♑ v, whose Centre will be upon the Vertical CZ continued or found, by taking the Complement
of the Altitude 76 deg. 37 min. of the Line of Secants, and prick it from C on the Vertical-Line continued, and that
is the Centre; draw ½ as (♑ v) thence draw the Parallel of Declination 20 deg. 30 min. by the former direction from AE to B, and from Q to R, and take 20 deg. 30 min. of the Line of half-Tangents, and prick it from C on the Axis to (m,) and draw B ♁ M R the Parallel, whose Centre is found upon the Axis continued, as
before, by the Complement Secant of the Declination 69 deg. 30 min.
Then where the Parallel of Altitude and Declination cross each other, which is at
♁, there is the Sun at that time; therefore draw the Azimuth-Circle, as before (12
Probl. directed,) from Z through ♁ to N, and it will cut the Horizon in ♄; then measure
C ♄, and it is the Suns Azimuth from the East and West, which applyed to the Line
of half-Tangents, shews 61 deg. 15 min. as before, whose Complement is 28 deg. 45 min. the Azimuth from the South; in like manner measure all Azimuths from the Prime Vertical
on the Horizon.
By Calculation; First, consider the Declination of the Sun, whether it be towards
the North or South, so have you his distance from the Poles; then add this distance,
the Complement of his Altitude, and the Complement of your Latitude all three together,
and from half the sum Substract the distance from the Pole or Complement of his Declination,
and note the difference.
Look well and observe the general Diagram in the Oblique-angled Triangle.
Z ♁ N the Complement of the Suns Altitude is Z ♁ 76 deg 37 min. 2. The Complement of the Suns Declination is N ♁ 69 deg. 30 min. and the Complement of the Latitude ZN 38 deg. 32 min. which known, you may frame your operation, thus.
Decl. South 20 d. 30 m.
The distance from the Pole 110 deg. 30 min.
Suns Altit. 13 d. 23 m.
Complement, or Sign is
76:37, Z ♁ 998804
Latit. North 51 d. 28 m.
Complement, or Sign of
38:32, ZN 979446
37 deg. 18 m.
(4) Sign, or sum is
225:39, 1978250
all
The Sign of 67 deg. 11′, or half sum of (3) 112:49
996461
So is the Sign of 2 deg. 19 min. or the difference 2:19
860662
Add the Radius (2)
2000000
From this Sum Substract 1978250 the fourth Sign
2857123
Take the half with the Radius (it is the 7th. Sign)
1878873
The half doth give the Mean-proportional Sign 14 degr. 22 min.
939439
And the double of 14 degr. 22 min. is 28 degr. 44 min. the Azimuth of the Sun from the South part of the Meridian; and it taken from 180
degr. 00 min. leaves 151 degr. 15 min. the Suns Azimuth from the North part of the Meridian, as before.
An Example for North Declination answerable to the first in (13 Probl.)
By the former Rules, you may find the Azimuth in each Sphere by Instrument. The Suns
Declination North 20 degr. 30 min. his Altitude 51 deg. 12 min. the Latitude 51 degr. 28 min.
Observe the Diagram in the Oblique-Triangle Z ♁ N the Complement of the Suns Altitude
Z ♁ 38 deg. 48 min. the Complement of the Suns Declination, N ♁ 69 deg. 30 min. the Complement of the Poles Elevation ZN 38 deg. 32 min. which known, the operation may be thus framed.
Suns Decl. North 20 d. 30 m. Compl.
69 d. 30′
As the Radius
Altitude 51:12 Compl.
38:48:979699
is to the Co-Sign of Alt.
Latitude 51:28 Compl.
38:32:979446
so is the Co-Sign of Lat.
The Sum
146:50:1959145
to a fourth Sign 22 d. 59′
The 1/1 Sum.
73:25:998154
as 4 S. is to the S. of ½ sum
The difference
3:55:883445
so is the Sign of the differ.
Add the Base 69 deg. 30 N ♁ (2)
2000000
3881599
The half of it
1922454
to a 7th. Sign 9 d. 39 m.
So is the Co-Sign 24 deg. 09 min.
961227
The double of it is 48 deg. 18 min. the Suns Azimuth from the South part of the Meridian, as before found, or 131 deg. 42 min. from the North part of the Meridian.
I have set down these two Examples thus particularly, to shew the agreement with the
former two; this note, that generally in all Spherical-Triangles where three sides
are known, and an Angle required, make that side which is opposite to the Angle required,
to be the Base, and gather the Sum, the half Sum, and the difference, as before.
Having these means to find the Suns Azimuth, we may compare it with the Magnetical
Azimuth, and so find the variation.
So the Magnetical Azimuth being more than the true Azimuth by 11 deg. 15 min. which is one Point of the Compass; therefore it shews the Variation to be one Point,
or 11 deg. 15 min. Westerly.
And suppose the Course by the Compass be East 8 Points from the North or South, or
90 degrees; and let the Variation be 11 deg. 15 min. to the Westward.
I demand the true Rumb;
Mr. Borough observed in 1580 d. 11 deg. 15 min. Variation Easterly in Line, House, Fields.
The Magnetical Rumb
90 d. 00 m.
Substract the Variation Westerly
11 d. 15
there remains the true Rumb NE
78 d. 45
Mr. Gunter 1662 found 6 deg. 15′
So that if the Variation be Westerly, you may concieve by looking upon the North-Point;
by the Variation one Point, that it being Westerly, it is always accounted to the
left hand, so the North-Point, is one Point to the right hand of his true place; and
you must Sail N by W, to make good a North way, and W by S to be a good West, and
S by E to be a direct South, and E by N to make good an East Course, which will make
an Angle with the Meridian of 78 deg. 45 min.
2 Example.
But suppose the Magnetical Azimuth by the Needle had been
37 d. 03′
And the Suns Azimuth found, as before, to be
48-18
Substract the lesser out of the greater, the difference NE
11 d. 15
And in regard the Magnetical Azimuth is less than the true Azimuth by 11 deg. 15 min. therefore the difference and variation is Easterly one Point, which is 11 deg. 15 min. and consequently all the Points, stand 11 deg. 15 min. or one Point to the left hand out of their true Places; and therefore, to make good
a North Course, you must Sail by your Compass N by E; and an East Course, Sail E by
South, and South Sail S by W; and to make good a West Course, Sail W by N; and so
it is to be understood of all other Courses or Points; for in this Example, the true
Course makes an Angle with the Meridian of 48 deg. 18 min.
The Year 1666 at Bristol, in Rownam Meadows, My self and Mr. Phillip Stainard, and some other friends Masters of Ships, took with us a Quadrant described in the 16 Chapter of the Second Book of 20 Inches Semi-diameter, and one Needle, and one Azimuth Compass, described in the First of
the Fifth Book, the Needle about 9 Inches long, the Chard 8 Inches; and in the Afternoon we made these Observations following.
Add the Radius to the Sum; Take the half is the Sign of half the Ark.
As the fourth Sign 26 degr. 27 min.
964893
is to the Sign of the half Sum 75 d. 21′
998564
So is the Sign of the difference 8 deg. 51′
918709 add
1917273 sum
to the seventh Sign 19 deg. 31 min. (add Rad.)
1952380
the half is the Mean proportional Sign of
35:19976190
☉ Obser. Altitude.
Magnetical Azimuth.
Suns true Azimuth.
Variation VVesterly.
Gr. M.
Gr. M.
Gr. M.
Gr. M.
44 20
72 00
70 38
01 22
39 30
80 00
78 24
01 36
31 50
90 00
88 26
1 34
37 42
95 00
93 36
1 24
23 20
103 00
101 23
1 23
Which doubled, is 70 deg. 38 min. the Suns Azimuth from the South part of the Meridian, or 54 degr. 41 min. the Complement of 35 deg. 19 min. doubled, is 109 deg. 22 min. the Suns Azimuth from the North part of the Meridian; and so of the rest, as they
are set down in this Table, viz. from the South part of the Meridian.
PROBL. XV. To find the Altitude of the Sun by the Shadow of a Gnomon set Perpendicular to the Horizon by Scale and Compasses; as also by Calculation.
VVIth your Compasses on a piece of Board, describe the Circle ABCD, place it Horizontal
with a Gnomon in the Centre O, cross it with 2 Diameters; then turn the Board, until the shadow
be upon one of the Diameters, at the end of the shadow make a Mark, as here at E;
lay down also the length of the Gnomon-Pin or Wire from the Centre on the other Diameter from O to F, draw a right-Line
from E to F, as EFH; then with the Chord of 60 deg. sweep the Arch GH upon E as a Centre; apply the distonce GH the Arch to your Line
of Chords, and that will give you the Altitude of the Sun required, as in this Example
will be 52 deg. 53 min.
So the Pin or Gnomon OF being 37 parts, and th [...] shadow OE 28, such equal parts, the Altitude will be found to be 52 degr. 53; or [...]e Gnomon being 28, and the shadow 37 parts, the Altitude will be IK 37 d. 07 m. or the shadow being 83, the Gnomon or Staff 100, the Tangent of the Angle will be 50 deg. 18 min. 20 the Altitude of the upper edge of the Sun or Angle HEG; from which, taking the
Semi-diameter of the Sun 16 m. 27″, there remains 50 d, 1′ 59″ the true Altitude of the Centre of the Sun.
After this manner, if you observe the greatest Meridian-Altitude of the Sun the 11
of June, and 10 of December, you shall by the difference of them find the distance of the Tropicks; the greatest
Declination of the Sun, and Elevation of the Equator, and Latitude of the Place.
As for Example.
At London the greatest Meridian-Altitude of the Sun is 61 deg. 59″ 30″, and the least 14 deg. 56′ 30″.
The Suns greatest Meridian-Altitude taken June 11, is
61 d. 59:30″
The Suns least Meridian-Altitude taken December 10
14:56:30
The distance of the Tropicks, take the half of
47:03:00
And it is the Suns greatest Declination, Substracted from the Alt.
23:31:30
Leaves the Elevation of the Equator,
38:28:0
Whose Complement is the Latitude of the Place,
51:32:0
PROBL. XVI. Having the Latitude of the Place, the Suns Declination, and the Suns Altitude; to
find the Hour of the Day.
BY the Line of Chords and Signs, by the Convex-Sphere, set the extent ST, from C to
X, and upon X as a Centre, with the extent S ♁, draw the Arch K, a Ruler laid from
C just touching the Arch, finds the Point (n,) the Arch (on) measured on the Chords, sheweth 61 deg. 10 min. the Suns distance from the hour of Six, viz. 4 ho. and almost one min.
The Rule is by the Tables; Add the Complement of the Suns Altitude, and the Complement
of the Suns Declination, or Distance of the Sun from the Pole, and the Complement
of your Latitude, all three together, and from half the sum; Substract the Complement
of the Altitude, and note the difference.
Thus in our Latitude of Bristol 51 deg. 28 min. the Declination of the Sun, being 20 deg. 30 min. Northward, and the Altitude 51 deg. 12′. I find the Sun to be 29 deg. 50 min. from the Meridian, as by this Example.
Altitude of the Sun 51 d. 12, the Compl. 38 d. 48, As the Radius 90
10
Declination North 20:30 the distance from the Pole 63 30. to the Sign of the Suns
distance from the Pole.
997158
Latitude North 51:28 the Compl. is 38:32 so is the Sign Compl. of Lat.
979414
The sum of all three 146:50 to a fourth Sign
976572
The half Sum — 73:25 as the 4 Sign is to the Sign of half sum
998154
The difference— 34:37 so is the Sign of the difference
The Mean Proportional between this seventh Sign, and the Sign of 90; that is, add
the Radius to the seventh Sign, and take the half, and it will be the Sign of the
Complement of half the Hour from the Meridian, which in this Example is found to be
14 deg. 55 min. the double of that is 29 deg. 50 min. which converted into Hours, doth give almost two Hours, it wants but 40 Seconds.
PROBL. XVII. Having the Azimuth of the Sun, the Altitude of the Sun, and the Declination; to
find the Hour of the Day.
THus the Declination being 20 deg. 30′, the Altitude 51 deg. 12 min. the Azimuth from the South, found by the 14 Problem, to be 48 degr. 18 min. I might find the time to be 29 deg. 58′, that is, almost 2 Hours wanting 8 Seconds; so the difference is 32 Seconds;
that is, by reason of the several Operations, which is near enough for the Mariners
use.
As the Co-Sign of the Declination 69 deg. 30 min.
997158
is to the Sign of the Azimuth 48 deg. 18 min.
987311
So is the Co-Sign of the Altitude 38 deg. 48 min.
979699
to the Sign of the Hour 29 deg. 58 min.
1967010
969852
PROBL. XVIII. How to find the Right-Ascension of a Star, and the Declination of a Star; having
the Longitude and Latitude of that Star given.
PRoject the Sphere Geometrically; that is, draw the great Meridian with a Chord of
60 deg. you may draw the Horizontal-Line HO, and Vertical ZN; then set the Latitude 51 deg. 28′ from O to N, and draw ZAE, and from H to S, and from N to Q▪ and draw the Equinoctial-Line
AECQ, then take the distance of the Pole of the Ecliptick, from the Pole of the World
23 deg. 31 min. and prick it from N to P, from AE to ♑, from S to A, and from Q to ♋; then draw the
Ecliptick-Line ♑, C, ♋, on which you must put the Longitude or Distance of the Star,
from the next Equinoctial-Point, as in this Example; The Star in the Mouth of the
great Dog Sirius, his Longitude is found by the following Rules to be 9 deg. 32 min. of Cancer; and his Latitude is 39 deg. 30 min. South, a Star of the first Magnitude; take 9 deg. 32 min. out of 90 deg. the Remain is 80 deg. 28 min. the distance of the Star from the next Equinoctial Point C, prick that from C to
K on the Ecliptick, and draw the Circle of Longitude P, KS, then prick the Latitude
39 deg. 30′ from ♑ to D, and from ♋ to F by the Chords; then take the same number of the
half-Tangents, and prick it from C to M, and draw the Circle of Latitude of the Star
parallel to the Ecliptick, as DMF, and where this Parallel cuts the Circle of Longitude,
as at ✶ that is, the place of the Star; then draw the Meridian-Circle from the North-Pole
through ✶ the Intersection to the South-Pole, and it cuts the Equinoctial in R; measure
CR on the Line of half-Tangents, and it gives the Right-Ascension Complement to 180
deg. which is 82 deg. 21 min. the Complement is 97 deg. 39 min. the Right-Ascension desired.
Now to find the Declination of the Star; lay a Ruler over P and the Ecliptick, at
K, and it will cut the Arch in L; take a Quadrant 90 deg. and prick it from L to P; lay a Ruler over p and P, and it will cut the Ecliptick in O; lay a Ruler over O, and ✶, and it cuts
the Lamb in e; measure Qe on the Line of Chords, and it is 16 deg. 14 m. the Stars Declination required, By the Concave-Sphere; the Convex-Sphere, will not so conveniently shew the true Scituation and Place of the Stars as this;
and therefore it is omitted.
The Stars have little or no alteration in their Latitude; but in their Longitude they
move forward about 1 degr. 25 min. in a hundred years, which is 85′.
By Noble Ticho, his Tables of Longitude and Latitude of the Stars, rectified by himself, to the beginning
of the year 1601.
The Latitude of the most bright Star Sirius in the Mouth of the great Dog, is 39 deg. 30 min. and his Longitude is 8 degr. 35 min. 30″ of ♋; I desire the Stars true Longitude, or to be rectified for this present
year 1667.
You must work by the Rule of Proportion, thus, if 100 give 85 m. what shall 66 the difference in years betwixt 1601, and 1667 give? Multiply, and
Divide, and the Quotient will be 56 m. 6″ added to the Longitude found in the Tables of the Stars in the second Book on
the back-side the Nocturnal, which 8 deg. 35 min. 30″ makes 9 d. 31′ 36″ of Cancer, the Longitude of the Star Sirius this year 1667; and so work to find the Longitude of any other Star in any other
year, past, or to come.
Take 9 deg. 32′ the Longitude of the Star, out of 90 deg. there remains 80 degr. 28′, his distance from the next Equinoctial Point; which being known, the First Rule
is,
As the Radius 90
10
is to the Sign of the Stars Longitude from the next Equ. Point 80 d. 28
CK 999396
So is the Co-Tangent of the Stars Latitude 39 deg. 30′ A ✶
1008389
to the Tangent of the fourth Ark 50 deg. 06 min.
1007785
Compare this fourth Ark with the Arch of Distance betwixt the Poles of the Ecliptick,
and the Poles of the World 23 deg. 31 min. if the Longitude and Latitude of the Star be alike, as in North Signs ♈ ♉ ♊ ♋ ♌ ♍,
and the Latitude is on the North-side the Ecliptick; or if the Longitude be among
the Southern Signs, as ♎ ♏ ♐ ♑ ♒ ♓, and the Latitude Southward; then shall the difference
between the fourth Ark found, and the distance of the Poles 23 deg. 31′ be your fifth Ark.
But if the Longitude and Latitude shall be unlike, as it is in this Example; as the
Longitude in a Northern Sign, and the Latitude South; or the Longitude in a Southern
Sign, and the Latitude North; then Add this fourth Ark found, to the distance of both
Poles 23 deg. 31 min. the sum of both shall be the fifth Ark.
Then the Rule is,
As the Sign of the fourth Ark 50 deg. 06′
988488
is to the Sign of the Fifth Ark 73 deg. 37′
998199
So is the Tan. of the Stars Long. from the next Equin. Point 80 deg. 28 m.
4077484
to the Stars Right-Ascension from the next Equin. Point 82 deg. 21 m.
2075683
82 d. 21 m. Substracted from 90 d. or 180 leaves 7 d. 39′ which added to 90 d. the sum is 79 d. 39′ the Right-Ascen. of Sirius required.
1087195
Then the Rule to find the Declination, is,
As the Co-Sign of the fourth Ark 50 deg. 06′
980716
is to the Co-Sign of the fifth Ark 73 deg. 37′
945034
So is the Sign of the Stars Latitude 39 deg. 30′
980351
1925385
to the Sign of the Stars Decl. required 16 d. 14′
944669
You have the proof of the Work by the foregoing Geometrical Rules; or you may take
this by Calculation, if there be no former errour, the Proportion will hold.
As the Co-Sign of the Latitude 39 deg. 30 min.
988740
is to the Co-Sign of the Right-Ascen. from the next Equ. Point 82 d. 21′
912424
So is the Co-Sign of the Declination 16 deg. 14 min.
998233
1910657
to the Compl. Sign of Longit. from the next Equinoctial 80 d. 28 m. or Sign of the Longitude 9 deg. 32 min. as was at first given.
921917
By these Rules and directions work, to find the Right-Ascension, and Declination of
any Star, which you desire to know.
PROBL. XIX. Having the Declination, and Right-Ascension of a Star; to find the Longitude and
Latitude thereof.
IN the former Diagram of the 18th Problem, you have the Right-Ascension of the Glistering-Star
in the great Dog's Mouth called Sirius; CR 82 d. 21 m. take it off the Line of half-Tangents, and prick it from C to R; and you have also
the Declination drawn Bye; then draw the Meridian-Circle from N, cutting the Point R, the Stars Right-Ascension
from the next Equinoctial Point, and Parallel of Declination in ✶ to S the South-Pole;
then take the distance of the Poles of the World, and the Poles of the Ecliptick 23
d. 31′, and prick from N to P, and from AE to ♑, and from S to A, and from Q to ♋, and
draw ♑ C ♋ the Ecliptick-Line; then draw the Circle of Longitude through P, through
the Intersection of the Parallel of Declination, and Meridian, which is the body of
the Star to A, and it will cut the Ecliptick in K; measure CK on the Line of half-Tangents,
and you have the Longitude of the Star from the next Equinoctial-Point 80 deg. 28 min.
And to find the Latitude; if you lay a Ruler over P and K, it will cut the Limb in
♄, prick 90 deg. from ♄ to ♃, and lay a Ruler over P and ♃, and it will cut the Ecliptick in the
Point ♂; lay a Ruler over ♂ and ✶, and it will cut the Limb in F; apply the distance
♋ F, to the Line of Chords, and it will be 39 deg. 30′, the Latitude required.
In the Triangle ZRC, by Calculation, we have 1. the Angle ZCR, the distance of the
Poles 23 degr. 31 min. 2. the side CR 82 degr. 21′, the Right-Ascension from the next Equinoctial-Point; then reason must guide
you, as by these Rules, to find the Longiture and Latitude of a Star.
As the Radius 90 deg.
10
is to the Tang. of the Angle ZCR 23 d. 31 m. the Poles distance
963864
So is the Sign of Right-Ascension CR 82 d. 21′ from the next Point
999611
to the Tangent of ZR 23 deg. 20 min.
963475
Which 23 degr. 20′ add to the South-Declination 16 degr. 14 min. makes 39 deg. 34′, ZR; but if the Declination had been Northerly, you must have Substracted the
Arch found out of it, and the Remain had been ZR; but if it had been more than the
Declination, Substract it out of the Arch found, and the difference had been ZR; so
with reason wave the Rule, as occasion requireth.
Then to find the Angle CZR, and the side CZ, the Rule is,
As the Sign of the fourth Ark ZR 23 deg. 20′
959778
is to the Sign of the Angle of distance of the Poles ZCR 23 d. 31′
960099
So is the Sign of Right-Ascen. from the next Equ. Point CR 82 d. 21 m.
999611
1959710
to the Sign of CZR 86 deg. 49 min. the fifth Ark
999932
Then;
As the Sign of ZCR 23 deg. 31 min.
960099
is to the Co-Sign of RC 23 deg. 20 min.
959771
So is the Sign of CRZ 9 degr. Radius
10
to the Sign of CZ 83 degr. 03 min.
999679
Which Angle CZR 86 deg. 49′ is equal to the Angle ✶ ZK.
Then to find the Latitude of the Star,
As the Sign of ✶ KZ Radius 90 degr.
10
is to the Sign of 39 deg. 34′ the 4 Ark and Declination R ✶
980412
So is the Sign of ✶ ZK 86 deg. 49 min.
999932
to the Sign of the Latitude of the Star desired ✶ K 39 deg. 30 m.
980344
And lastly, to find the Arch ZK, and by it consequently the Longitude CK.
Therefore, if you Substract 2 degr. 35 min. out of the Arch CZ 83 deg. 03 min. the Remain is 80 degr. 28 min. the true Longitude of the Glistering Star Sirius in the Mouth of the great Dog, from the next Equinoctial-Point at the time given.
But if the Meridian-Circle had cut the Ecliptick at K, and the Circle of Longitude
at Z; then in such cases, add the Arch found ZK to CZ, and the sum had been the true
Longitude from the next Equinoctial-Point C; and so work with reason, to find the
Longitude and Latitude of all other Stars, as by reason you did Substract ZK from
CK; therefore the Sign was above a Quadrant, if you Substract his distance from the
Vernal-Equinox 80 deg. 28 min. from 90 d. the Remain is 9 deg. 32′ of Cancer, the Sign and deg. the Star is in, as before.
PROBL. XX. Having the Meridian-Altitude of an unknown Star, and the distance thereof from a
known Star; to find the Longitude and Latitude of the unknown Star.
IN the 16th Chapter of the second Book of Harmonicon Coeleste, Mr. Vincent Wing hath this Example, and Observation, made by the Phoenix of Astronomy Ticho-Braghe in the year 1577, which we will borrow for an Example; it being a useful Rule for
all Ingenious Navigators, for by it they may find the Longitude and Latitude, and
consequently, by the foregoing Rules, the Right-Ascension, and Declination of those
good Stars for their use, that are in the South Hemisphere, viz. as they have been named by the Portugals; the South-Triangle, which Constellation hath 5 Stars, one of the Eastermost corner,
which comes last to the Meridian of the second Magnitude. The Crane, in which there is 13 Stars on the left Wing, and another on the right side the back
of the second Magnitude. The Phoenix, 15 Stars, the Water-Serpent hath 15 Stars. The Dorado, or Gilt-head-Fish, situate in the very Pole of the Ecliptick; and in that Constellation
is 4 Stars. The Chamelion, with the Flie, in which is 13 Stars; The Bird of Paradise, in which is 12 Stars; the Peacock, in which is 15 Stars; one in the head of the second Magnitude; the Naked Indian, in which is 12 Stars; and also the Bird Taican, or Brasilian Pye, in which Constellation is 7 Stars, two of them of the third Magnitude: Also two useful
Stars for Navigators; in one Constellation, which are Noah's Dove, which containeth 11 Stars, of which there are 2 in the back of it, of the second
Magnitude, which they call the Good Messengers, or Bringers of good News, and those in the right-Wing are consecrated to the appeased
Deity; and those in the left to the retiring of the Waters, in the time of the Deluge;
and they come to the Meridian about half an hour before the great Dog; and by the
Globes are about 21 degr. 30′ distant, from the nearest in the back; but I would have the Sea-men take him
exact, as likewise a good Constellation called the Crane Grus, or the Flamengo, as the Spaniards call it; this Astensine consisteth of 13 Stars, and hath 3 Stars of the second Magnitude,
that in the head is called the Phaenicopter Eye, and the other are on his Back, and the other in his left Wing; These Stars I would
desire those Mariners that Sail to the East or West-Indies; to take the Meridian-Altitude
thereof, and their distance from any known Stars, and by it you shall have all the
rest; for many times I have been Sailing between the Tropicks, and for 12 days together
have had no Meridian-Altitude of the Sun, by reason of close and cloudy weather,
which is bad for those that are bound to small Islands, and Cape-Lands; therefore
to the Southward, as well as to the Northward; these Stars will stand them in great
stead, and serve their turn, as well as the Sun, to find the Latitude thereof.
The Rule is thus; About the end of the Year 1577, Ticho observed the distance of rhe little Star in the breast of Pegasus from the bright Star of the Vulture, to be exactly 45 degrees 31′, and by the Meridian-Altitude thereof, he found the Declination thus.
The Angle sought, which Bi-sected, gives the Sign of 22 d. 27′ 17″ 958200
Which doubled, is 44 degr. 54′ 34″, is the Angle LFO, which is equal to the Arch DE, the difference of their
Right-Ascension, which I add to the Right-Ascension of the bright Star of the Vulture,
292 degr. 35′, and the Sum is 337 degr. 29′ 34″, is the Right-Ascention of the little Star in the breast of Pegasus.
Then having the Declination of this Star 22 degr. 20′, and the Right-Ascension 337 degr. 29′ 34″, the Longitude of the said Star by the last Problem (19) will be found to
be 18 degr. 36 ♓, and the Latitude thereof 29 degr. 24 min. North.
Now to draw the Diagram by Chords, and half-Tangents Geometrically, with the Chord
of 60 degr. draw the Circle; then draw ♑, ♋, the Ecliptick 23 degr. 31′ ♑, S, and by C draw the Equator; then by the Parallel of Declination, and Right-Ascension
of the Vulture, will find LO; therefore if you put the difference of Ascension from
♈ to E 22 degr. 30′ 20″, from the nearest Equinoctial-Point, and draw the Meridian-Circle FES, and
it will cut the Parallel of Declination at O; draw through O, as PON the Circle of
Longitude, and measure ♈ X on the Line of half-Tangents, and it is 11 degr. 24 min. from the nearest Vernal Equinox, Substracted from 30 degr. leaves 18 degr. 36′ of ♓ for the Stars Longitude; and as before directed, you may find the Latitude
♑ ♄ to be 29 degr. 24′.
PROBL. XXI. To find the Parallax of Altitude of the Sun, Moon, or Stars.
THe true Altitude of the Sun, Moon, or Stars, ought to be observed in the Centre of
the Earth, (if possible) whereto the Tables are conformed; but because we dwell upon
the Superficies of the Earth 4000 Miles nearest, or 3983 English Miles from the Centre
of the Earth; therefore, the Planets seem lower to us, than indeed they be; and therefore
to find the true place of the ☉ ☽ ✶, you must draw a Right-Line from the Centre of
the Earth, through the Centre of the Sun, Moon, or Stars; but the apparent visible
place is determined by a Line drawn from the Eye, through the Centre of the Star;
therefore the Parallax of a Star is an Arch of a great Circle, passing by the Zenith,
and the true place of the Star, the Arch of the same Circle intercepted between the
true and apparent place.
A Figure or Scheme shewing what the Parallax, or diversity of Aspects is.
In this Figure, C denotes the Centre of the Earth.
D the Place or Superficies of the Earth, from whence the ☉ ☽ or ✶ is seen.
☽ ☉ and ♂, their Place in their Orbs.
C ♂ I, C ☉ M, C ☽ N. the Lines of their true Place.
D ♂ G, D ☉ K, D ☽ M, the Lines of their visible or apparent Places.
Hence the Angle made by the Intersection of the said two Lines through the Body of
the Planet, is the Angle of Parallax, that is to say, in ♂ the Angle C ♂ D, which
is equal to the Angle I ♂ G, in the ☉ the Angle of Parallax, is the Angle C O D; and
lastly in the Moon it is the Angle C ☽ D, or N ☽ M, which is all one.
By this it is manifest, the nearer a Star is to the Horizon and Centre of the Earth,
the greater is the Parallax; and hence it is, that the Orbit of the Moon being nearest
to the Earth, her Parallax is greatest, and most perceptible, because the Semi-Diameter
of the Earth bears a sensible proportion to the Semi-Diameter of the Moons Orbit,
though it be very little, or nothing at all in comparison of the Orbs of ♄ ♃, and
the fixed Stars, which is caused by the Interval and vast distance which is between
them; but this last Problem hath no relation to the common use of Mariners, therefore
I shall not insist any further, but refer the Reader to Harmonicon Coeleste, where there is a full discourse, and Rules relating thereunto.
But these Astronomical Propositions as I know to be useful for Sea-men, I have here
inserted; for the first and second will find the Suns Place and Declination, together
with the 15 Probl. will find the Latitude.
The third, fourth fifth, and sixth will find the Suns Rising and Setting, as the 7,
8, 9, 10, 11, 13, 14, to find the Variation of the Compass, and the 12 to find the
Suns Altitude at any time assigned, and the rest being very useful Rules of the Stars,
by which you may have the Hour of the Day, and Night; for having the Latitude of the
Place, with the Declination and Altitude of the Sun, or any Star, they may find the
Hour of the Sun or Star from the Meridian by the 16 and 17 Problem; then comparing
the Right-Ascension of the Sun, with the Right-Ascension of the Star, they may have
the Hour of the Night in all these Propositions; I have been as plain and as brief,
as the several Resolutions thereof would permit me; and I do wish the Practitioner
as much delight in the Practice, as I have had in the Composing of it.
The Seventh Book. THE Mariners Magazine, OR, STURMY'S Mathematical and Practical ARTS.
THE ART OF DIALLING BY THE Gnomical Scale, AS ALSO BY CALCULATION. SHEWING The Making
of all sorts of DIALS, both within Doors and without, upon any Walls, Cielings, or
Floors, be they never so irregular, wheresoever the Direct or Reflected Beams of the
Sun may come, for any Latitude. AND How to find the true Hour of the Night, by the Moon and Stars: And how to Colour,
Guild, and Paint DIALS; And how to fasten the Gnomon in Stone or Wood. Never before made so plain to the meanest Capacity. By Capt. SAMUEL STURMY.
To my much Honoured Friend Isaac Morgan Esq; Collector of his Majesties Customs in
the Port of Bristol.
SIR,
NOt to inform your Judgment in any thing concerning the Subject Matter of these my
poor Labours (your Wisdom and approved Knowledge in all Learning being so general,
that I can add nothing unto it) but to inform the World how much I honour you and
your Vertues, and by how many Obligations I stand engaged to you for the many signal
Favours you have vouchsafed me since the time I came first into this Port, I dedicate
this Part of my Book, as the proper Part of the fruits of my spare time, in twenty
seven Dials, unto you presented as an unworthy New-years-gift; and that Dial-piece being the Subject of the whole Art of Dialling, I will name the Dials, that I may charge you to Patronage no more than you had; viz. Eight Verticals and Decliners, Eight Recliners and Incliners, and Eight Decliners, Recliners, and Incliners, a Globe with two Pole-Dials, and one Shadow-Dial, made on a Piece of Freez-stone, as is seen in the Frontispiece the Gnomon or Stile fastned by me, and likewise Painted and Guilded, which is well [Page] known by you and many others: And being desired by some Friends that saw my way,
and this Piece of Dialling, to Print it, by their importunity, according to the best of my judgment, I have
so done; and if any way profitably, then according to mine own desire. As it is,
I have made bold to make choice of you for the Patronage thereof, that it may gain
the more Credit by your Protection; And if any shall be offended at this Work, my
Device shall be a Dial, with this Motto, ASPICIO UT ASPICIAR; only to all Favourers of ART I am direct, erect, plain, as I am, Sir, to you, and desire to be,
SIR,
Your humble and affectionate Servant to be commanded, SAMUEL STURMY.
2. A Gnomon Scale or Line, as BD, the use thereof you may see in Chap. 7. and so forward in Degrees.
3. A Scale or Line of six Hours, for drawing the Hour-lines in any Dial, divided into
every 10 Minutes; the use you may read from Chap. 4. forward, as BEA.
4. A Scale of Inclination of Meridians, divided in Degrees as the Diam. BA; the use thereof is in Chap. 12. and so forward.
5. Upon the other side are two Lines or Scales, for the inlarging the Hour-Lines on
any Plain: The greater Pole is marked with +; the lesser is marked with −, called
the lesser Pole for distinction sake; and these Lines are divided into every 10 Minutes,
but may be by the Table into every 5 Minutes.
6. These Scales or Lines are on the Mathematical Scale, with the rest that are described
Book 2. called The Scale of Scales.
THE ARGUMENT.
REeader, read this; for I dare this defend,
Thy posting Life on Dials doth depend.
Consider thou, how quicks the Hour's gone;
Alive to day, to morrow Life is done.
Then use thy time, and always bear in mind,
Time's Forehead hairy is, but bald behind.
Here's that which will decline to thee, and show
How quick Time runs, how fast thy Life doth go.
Yet be ingenious, learn the Practick Part,
And so attain to Practice of this Art:
Whereby you shall be able for to trace
Out such a path, where Sol shall run his Race;
And make the greater Cosmus to appear,
According to each Season of the Year.
CHAP. I. The Preface of the kinds of Dials.
ALthough Gnomoniques pertain to Astronomy, yet I think it not amiss for the ease of the Reader, to place these in a distinct
Book by themselves.
Sun-Dials may be reduced to two sorts. Some shew the Hour by the Altitude of the Sun, as Quadrants,
Rings, Cylinders; and for the making thereof, you must know the Suns Altitude for
every day, or at least every tenth day of the year, and for every hour of those days.
The other sort shew the hour by the shadow of a Gnomon or Stile parallel to the Axis of the World; and of that I treat chiefly in this Book. These be all Projections
of the Sphere, upon a Plane which lies parallel to some Horizon or other in the World. And if upon such a Plane the Meridians only be projected, they shall suffice to shew the Hour, without projecting the other
Circles, as the Ecliptique, the Aequator with his Parallels of Declination, the Horizon with his Almicanters and Azimuths, which are sometimes drawn upon Dials more for Ornament than for Necessity.
FOr the better understanding of the Reasons of Dials, these Theorems would be known.
I. That every Plane whereupon any Dial is drawn, is part of the Plane of a Great Circle of the Heaven, which Circle is an
Horizon to some Country or other; That the Center of the Dial, representeth the Center of the Earth and World; and the Gnomon which casteth the Shade, representeth the Axis, and ought to point directly to the two Poles.
II. That these Dial Planes are not Mathematically in the very Planes of Great Circles; for then they
should have their Centers in the Center of the Earth, from which they are removed
almost 4000 miles; and yet we may say they lye in the Planes of Circles parallel to
the first Horizon, because the Semidiameter of the Earth beareth so small proportion to the Suns Distance,
that the whole Earth may be taken for one Point or Center, without any perceivable
Error.
III. That as all Great Circles of the Sphere, so every Dial Plane hath his Axis, which is a straight Line passing through the Center of the Plane, and making Right
Angles with it; and at the end of the Axis be the two Poles of the Plane, whereof that above our Horizon is called the Pole Zenich, and the other the Pole Nadir of the Dial.
IV. That every Plane hath two Faces or Sides: and look what respect or situation the
North Pole of the World hath to the one side, the same hath the South Pole to the other; and these two Sides will receive 24 Hours always: so that what one
Side wanteth, the other Side shall have; and the one is described in all things as
the other.
V. That as Horizons, so Dial Planes are with respect to the Aequator divided into first, Parallel or Aequinoctial; secondly, Right; thirdly, Oblique Planes.
VI. A Parallel or Polar Plane maketh no Angles with the Aequator, but lies in the Plane of it, or parallel to it; that is, hath the Gnomon erected on the Plane at Right Angles, as the Axis of the World is upon the Plane of the Aequator: because the Axis and Poles of the Dial are here all one with the Axis and Poles of the World, and the Hour-lines here meet all at the Center, making equal Angles, and dividing the Dial Circle into 24 equal parts, as the Meridians do the Aequator, in whose Plane the Dial lies.
VII. A Right Horizon or Dial Plane cutteth the Aequator at Right Angles, and so cutteth through the Poles of the World, that it hath the Gnomon parallel to the Plane, and so the Hour-lines parallel one to another; because their
Planes, though infinitely extended, will never cut the Axis of the World: yet have those Dials a Center, though not for the meeting of the Hour-lines, viz. through which the Axis of the Dial Circle passeth, cutting the Plane at Right Angles, and cutting also (neer enough
for the projecting of a Dial) the Circle of the World.
VIII. An Oblique Horizon or Dial Plane cutteth the Aequator at Oblique Angles; that is, hath for their Gnomon the side of a Triangle, whose Angles vary according to the more or less Obliquity
of the said Horizon: and the Gnomon shall always make an Angle with the Plane, of so many Degrees as the Axis of the World maketh with the Plane, or as either of the Poles of the World is elevated above the Plane.
IX. Every Oblique Horizon is divided by the Meridians or Hour-circles of the Sphere into 24 unequal parts; which parts are always lesser,
as they are neerer to the Meridian of that Horizon or Plane; and greater, as they are farther off: and on both sides of the Meridian
of the Plane, the Hour-circles which are equally distant in time, are also equally
distant in space. Whence it is, that the divisions of one Quadrant of your Dial Plane
being known, the divisions of the whole Circle are likewise known.
X. The Hour-lines in an Oblique Dial, are the Sections of the Planes of the Hour-circles
of the Sphere, with the Dial Plane: and because the Planes of Great Circles do always
cut one another in Halves by Diameters, which are straight Lines passing
[geometrical diagram]
through the common Center; therefore Lines drawn from the Center of the Dial, to
the Intersections of the Hour-circles with the Great Circles of the Plane, shall be
those very Sections, and the very Hour-lines of the Dial.
XI. Every Dial Plane being an Horizon to some place in the Earth (as was said Theorem I.) hath his proper Meridian, which is the Meridian cutting through the Poles of
the Plane, and making Right Angles with the Plane. If the Poles of the Dial Plane
lie in the Meridian of the Place, then is the Meridian of the Plane all one with the
Meridian of the Place, and the Gnomon or Style shall stand erected upon the Noon-line,
or Line of 12 a Clock, as in all direct Dials. But if the Plane decline, then shall
the Substyle Line, or Line which the Gnomon standeth upon, which is the Meridian of
the Plane, vary from the Line which is the Meridian of the Place; and this Variation
shall be East, if the Declination be West of the Plane: And contrarily, because the
Visual Lines, by which the Sphere is projected on Dial Planes, do, like the Beams
of a Burning-glass, intersect or cross one another in a certain Point of the Gnomon
(to be assigned at pleasure, and called Nodus) and so do all place and depaint themselves on the Dial Plane, beyond the Nodus, the contrary way.
XII. Dials are most aptly denominated from that part of the Sphere where their Poles lie, though some Authors have chosen to denominate them from the Circles in which
their Planes lie; as the Dial Plane which lieth in the Aequinoctial, or Parallel to
it, is called by many an Aequinoctial Plane; but I concur with those who would rather
call it a Polar Plane, because the Poles thereof are in the Poles of the World.
CHAP. III. How to make the Polar Dial, and how to place it.
THe Plane of the Polar Dial lieth in the Aequinoctial, where the 12 chief Meridians
or Hour-circles divide both the Aequinoctial and this Plane into 24 Hours or equal
parts; the Gnomon stands upon the Center at Right Angles with the Plane.
First draw the Horizontal Line AB, and cross the same at Right Angles with the Line
CD; now on the Center at G, with the Chord of 60 Degrees, or with the Tangent of three
hours, you may describe the Circle ACBD, and about it make the Square EFHI; then take
out of the Hour-line one Hour, and lay it from each corner, as EFHI both ways: also
do the like with two hours as you see done, and from the Center at G draw Lines to
those Hour-points: so shall you have the Hour-lines in the Aequinoctial Dial; CD being
the Meridian or 12 a clock Line, and AB the East and West Line, serving for 6 in the
morning at B, and 6 in the afternoon at A, and so number the rest of the Hours in
order: You need draw no more hours than from 4 in the morning unto 8 at night, for
this Latitude of Bristol, being neer 51 d. 30 min.
For the Gnomon or Stile, you must have a straight Pin or Wyre set upright in the Center,
of such length as you see convenient; but if you will have it of such a length as
may neither be too short nor too long, then take this Rule.
How by Calculation to find the length of the Stile, and Semidiameters of the Parallels
of Declination.
IF it were required to proportion the Stile to the Plane, suppose the Semidiameter
of the greatest Parallel upon the Plane were but 6 Inches, and the Parallels should
be the 5 d. of Declination, the Rule is general.
As the Tangent of 45 deg.
1000000
Is to the Tangent-parallel of Declination 5 deg.
894195
So is the Semidiameter of the Plane 6 inches OA
277813
To the length of the Stile 53 parts
172010
which shews that the length of the Stile must be 53/100 parts of an Inch divided
into 100 parts.
How to find by the length of the Stile, the Semidiameter of the Parallel Circles
of Declination.
SUppose the length of the Stile above the Plane to be 10 inches, and you were to find
the Semidiameter of the Tropick, whose Declination is known to be 23 deg. 30 min. the Rule is for this and any other Declination,
As the Tangent of 45 deg.
1000000
Is to the length of the Stile 10 inches
100000
So is the Co-tangent of Declination 23 deg. 30 min.
which shews the Semidiameter of the Tropick to be 23 inches: So if the Declination
be 20 d. the Semidiameter will be 27 inches 47/100; if 15 d. then 3732/100; if 10 d. then 56 71/100; if 5 d. then 114 305/1000; and so of any other height of the Stile: as admit it were 53/100
parts of an inch high, then the Semidiameter of 23 deg. 30 min. would be 1 21/100, and for 20 deg. it will be 1 57/100, and for 15 deg. 1 96/100; if 10, then 2 98/100; if 5 deg. then 6 inches; if Stile be 13/100 and 75/100, the Semidiameter 23 deg. 30 min. is 37/100 parts, as you may see the Figure makes all plain; and so of any other.
[geometrical diagram]
Of all Dials this is the plainest; for it is no more but divide a whole Circle into
24 equal parts: and this is the very ground to all the rest.
With this Dial, the Hour-lines being equally divided into 24 equal parts, on the inner
circle you may make a Mariners Compass, with the 32 Points drawn upon it, to know
in all Latitudes whether the Moon being upon such a Point maketh High-water; or upon
what Point the Moon must be, when at those Places set together it maketh High-water
or Full-Sea.
For to know upon what Point the Moon is, may be done two manner of ways; by setting
it by the Compass, or by reckoning according to the age of the Moon, and the Hour
of the day. The setting according to any Point, may not be done with a common flat
Compass as the Mariners steer by (as many, wanting better reason, think they may,
to their great mistake) by reason it doth only divide the Horizon into equal Points,
and sheweth in what Vertical Circle or Azimuth the Sun or Moon stands: But this must
be done with a Compass, which being elevated according to the Superficies of the Aequinoctial,
divideth the Aequinoctial so likewise into equal parts, as the common flat Mariners
Compass doth divide the Horizon. Such an Aequinoctial Compass, with a Dial in, as
abovesaid, is of fashion as hereafter followeth pourtrayed. Whereof the Wheel ABC
sheweth the Superficies of the Aequinoctial, the Wyre ED the Axle-tree of the World.
The foresaid Wheel must be all alike marked on both sides, as well under as above,
with the 32 Points of the Compass, and with twice 12 Hours: and right against the
East and West at Land M, must so hang upon two Pins, as upon an Axletree, that it
may be turned [Page 7]
[geometrical diagram]
up and down, and the Wyre at the under end at D, by the Quadrant FDG, may be set
unto any height of the Pole. If then you set such a Compass with the under bottom
level, the Line HK North and South, viz. H to the North, and K to the South, and the under end of the Wyre right against such
a Degree of the Quadrant FG, as the height of the Pole that you find your self in,
then shall the Wheel ABC stand equal with the Superficies of the true Aequinoctial,
and the Wyre ED with the Axletree of the World; and the setting by such a one, and
a common Compass, giveth great Difference. And the neerer the Aequinoctial, the greater;
as may be understood by the Examples following.
EXAMPLE I.
IN the height of 50 deg. or thereabouts, the Sun being in the beginning of Cancer, at his greatest Declination to the North, by a common Compass cometh not before half
an hour after seven of the Clock to the East, and at half an hour after four to the
West; that is, he goeth from the East to the South and round to the West in nine hours;
but from the West through the North, until again in the East, in 15 hours.
EXAMPLE II.
IN the height of 30 deg. he cometh a little before half an hour past nine of the clock to the East, and a
little after half an hour past two of the Clock in the West, and so goeth in less
than five hours and a half from the East through the South to the West; but from the
West through the North, until again in the East, he goeth more than 18 hours. Thirdly,
Being under the Line, and the Sun having no Declination, he ariseth in the morning
right in the East, and so rising higher and higher, continueth East until that he
goeth over our heads through the Zenith into the West; and so continueth West, still
going down West, until he cometh again to the Horizon: and so according to a flat
Compass he is the one half of the day East, and the other West, without coming upon
any other Point. It is not so with this Aequinoctial Compass. The Sun and Moon go
always a like time on every one Point of the Compass, to wit, from the East to the
South 6 hours, from the South to the West 6 hours, from the West through the North
to the East in twice 6 hours.
This Dial will serve for all Latitudes, if you put the end of the Wyre at D, to the
height of the Pole or Latitude of the place as beforesaid; so the shadow of the other
end at E will fall upon the Hours and true Points of the Compass, all the time the
Sun is to the North of the Aequinoctial; but when the Sun is to South of the Aequinoctial,
you must look for the Hours and Points of the Compass upon the under side of the Dial.
CHAP. IV. How to make the South Aequinoctial Dial, or Polar Plane.
THe Aequinoctial Dial we call that which hath his Poles in the Aequinoctial Circle,
of which there be three kinds.
1. The Direct or South Aequinoctial Dial, which faceth the Meridian directly, not
looking from him to the one side more than to the other, having his Poles in the Intersections
of the Aequinoctial and Meridian.
2. The East or West Aequin [...]tial Dials, which may also be called Aequinoctial Horizontal Dials, for an Horizontal
Di [...]l declaring just 90 Degrees from the South or North, becomes an Aequ [...]noc [...]ial Dial, as well as Horizontal, because there is his Polar height, upon the Intersection
o [...] the Horizon with the Aequinoctial: and though this Dial be of kin to both, yet his
Gnomon shews that he should be sorted rather with the Aequinoctial Dials, than with
the Horizontal. These two sorts are regular, having the Poles in the four notablest
Points of the Aequator. The third is somewhat irregular, but may be brought to Rule.
How to make the first of these, draw the Horizontal Line AB, and about the midst at
C let fall the Perpendicular CD, which is the Meridian or 12 a clock Line. Let CD
be equal to a Chord of 60 Degrees, or the Tangent of three hours, and through D draw
the Line FE, parallel to AB; make also DE and CB equal to D, so have you a true Square
CDEB. Now take one hour with your Compasses off your Scale, and lay the same both
ways from E towards B and D, as E 1. Do the like with two hours, and draw the pricked
Tangent-lines from C to these Marks.
Next, Let the length or height of the Gnomon or Stile be GH, equal to C [...], or 3 hours; so drawing a Line through GH, parallel to the Horizon, you shall find
it cut the former Lines drawn to the Center C, in the Points l, m, n, o, p: through
which Points, if you draw Parallel-lines to the 12 a clock Line CD, you shall have
all the afternoon hours as far as V: and the morning hours must be drawn in like manner
and distance, to the left hand or West side, beginning from 7 in the morning unto
12, as in the Figure following.
Note, that the height or length of the Stile is always 3 hours from the Meridian,
as you see HG, which you may make with Copper or Brass Plate, or Iron, in form as
you see shadowed, whose breadth on the top is here HR, which may be made more or less
as you please.
This Dial will serve in any Latitude, if the Plane be placed parallel to the hour
of 6, so that the Plane be even with the Pole of the World.
How to calculate the Height of the Stile, and the Points of Hour-distance from the
Meridian.
SUppose the length of the Horizontal Line AB or FE be 12 Inches, and that it were
required to put on all the Hours from 7 in the morning to 5 in the evening; here
we have 5 hours and 6 inches on either side of the Meridian, [...]herefore I allow 15 Degrees for an hour. The Rule to find the height of the Stile
is,
As the Tangent-compl. of the given Hour 15 deg.
1057194
Is to half the Horizon or Distance from the Meridian 6 inches
277815
So is the Tangent of 45 Degrees
1000000
To the [...]eight of the [...] inches and parts
220621
And likewise the distance of the Hour points of 9 and 3 from the Meridian will be
1 61/100, or 1 inch and 61 parts of 100.
How to find the length of the Tangent between the Substile and the Hour-Points.
HAving found the length of the Stile in our Example to be 1 inch 61 parts of 100,
then in this Example, as we find the first Hour, so find the rest.
As the Tangent of 45 deg.
1000000
Is to the Tangent of the Hour from the Meridian 15 deg.
942805
So is the height of the Stile 1 61/100 inches
220621
To the length of the Tangent-line between the Meridian or Substiler 43/100 inch
163426
Hours
An. Po.
Tang.
deg.
mi.
In.
par.
12
0
0
0
0
11
1
15
0
0
43
10
2
30
0
0
93
9
3
45
0
1
61
8
4
60
0
2
79
7
5
75
0
6
0
6
6
90
0
Infinit.
and the Hour-point of 1 and of 11 a clock. And so of the rest, take them off a Scale
of an Inch divided into 100 parts, and prick them from C and D both ways to BA and
EF, and draw the Hour-lines parallel to the Meridian; and so do with the rest, until
it be finished, as you may see by the Table.
CHAP. V. How to make the East Aequinoctial Dial, or the West Lat. 51 d. 30 m.
THis Plane is a right Horizon of those People who dwell under the Aequator, distant
from us 90 deg. of Longitude; as the South Aequinoctial Plane of the last Chapter was the Horizon of those who dwell under the Aequator in the same Longitude with
us: Therefore these Dials are in all Points alike, only the Substiler Line, which
in the South Aequinoctial Dial is at 12, is but 6 in the morning for our Country,
because of the difference of Longitude.
To pourtraict this on a Wall or Plane, first draw the Horizontal Line AB; then upon
the Center C describe the Semicircle ADB, whereon lay the Latitude of the place 51
d. 30 m. from A unto D; so drawing GD continued, you shall have the Hour of 6: then
with your Compasses take off your Scale 15 deg. of the Line of Chords, and turning
them off 6 times, divide the Arch DF into 6 equal parts, and [Page 10] draw prick'd or blind Lines to those Divisions, which would be all one as if you
had done it thus. CD being equal to the Chord of 60, or Tangent of 3 hours, you shall
make the Quadrant or true Square equal to the side thereof CDEF, and from the corner
at E, you shall lay down both ways towards D and F the hours of 1 and 2, from whence
draw Lines to the Center C: Next make choice of the length or height of your Pin or
Stile, which you must lay down from C to G on the 6 hour Line, drawing from the Point
G a Line perpendicular to the Line of 6, or parallel to the side CF, as GH, which
cuts the former Lines in the Points IKLMH; through which Points drawing Lines parallel
to the hour of 6, you shall have the morning hours from 6 to 11, and the hours before
6, from 4 in the morning, are equal as from 6 to 7 and 8.
[geometrical diagram]
How to make the West Aequinoctial Dial.
THe West Aequinoctial Dial erect, serving for the afternoon, is drawn by the same
Rules contrariwise like the East in all points, only it shews but the afternoon hours,
as the East shews the forenoon hours: When you have drawn on paper the East Dial,
and set it by guess in its scituation, go on the West side of it, and you may see
through the paper the picture by reflection of the West Dial; and so will the picture
of the backside of the West shew you the true picture of the East Dial.
The way to calculate the height of the
Hours.
Ang. Po.
Tang.
deg.
min.
In.
par.
5
7
15
0
2
68
4
8
30
0
5
77
3
9
45
0
10
0
10
2
60
0
17
32
11
1
75
0
37
32
12
90
0
Infinit.
Stile, and the distance of the Hour-lines from the hour of 6, is the same as in the
last Chapter of the Polar Plane: For suppose the length of the Stile to be 10 inches,
then the length of the Tangent-line belonging to the first hour will be 2 inches and
68 parts of 100, as you see in this Table for the rest of the hours, which taken off
a Scale of equal parts, and prick'd from the Aequinoctial from C towards F, and likewise
upon the parallel DE: so you will make a Dial all one as by the former way, which
is good proof, if you draw the Hour-lines through these two Points; and so of the
rest.
WHat an Oblique Dial is, and why it hath been so called, hath been shewed Chap. 2.
They be
Regular.
Irregular.
The Regular lie in some notable Circle of the Sphere; as first the Vertical Dial,
whose Plane lieth in the Horizon, for which cause many call it the Horizontal Dial.
Secondly, the South and North Horizontal Dials, whose Plane lieth in the East Azimuth,
and is commonly called the South or North erect direct Dial. As for the East and West
Dials, they belong to another place, as was said Chap. 5.
The Irregular are such as lie Oblique to the Horizon, as Reclining or Inclining Dials;
or else lie Oblique to the Meridians, as Decliners; or else Oblique to both, as Recliners
or Incliners declining, which are esteemed the hardest of all, because of their double
irregularity.
The Declination of a Plane is the Azimuthal Distance of his Poles from the Meridian
of the place East or West.
The Reclination is the distance of his Poles from the Zenith and Nadir of your place.
Inclination is the neerest distance of the Poles of the Plane from your Horizon; and
whatsoever the reclination of the upper face of a Plane is, the inclination of the
lower face is the Complement thereof.
CHAP. VII. How to make the Vertical Horizontal Dial.
DRaw first the Horizontal-line AB, which is the Hour-line of 6;Take off the Scale of Hours the hasts and quarters, and prick from D as you did the Hours, and draw the quarters in the Dial as you were directed for
the Hours. then take the Latitude of 51 d. 30 m. from the Gnomon-line, and lay it down both ways from the Center C, as to A and B:
Next take the whole Hour-line of 6, fixing one leg of your Compasses on A, describe
a little Arch towards D; do the like from the Point B, crossing the Arch at D; so
draw the Line AD and BD. Now upon these Lines you must transport the 6 hours from
D unto A, and also from D unto B, as you see by the Figures 1, 2, 3, 4, 5, from whence
drawing Lines [...]rom the Center C, you shall have the Hours as you see numbred from 4 in the mo [...]ning until 8 in the afternoon, which sufficeth for this Latitude 51 d. 30 m.
For the height of the Stile, take off your Line of Chords with your Compasses the
Latitude of the Place 51 d. 30 m. and lay from K to E, from 12 to neer 4, and so drawing CE, you have the height of
the Stile, which may be made in Brass or Copper Plate, as you see shadowed in the
Dial following.
Thus by Calculation,
As the Sine of 90 d. Is to the Sine of the Latitude 51 d. 30 m.
989354
So is the Tangent of the Ho. 15 d.
942805
To the Tangent of the Hour-line from the Meridian 11 d. 50 m.
Line of Chords, and prick from the Meridian 12 from K on each side, the Degree or
Tangent of each hour; And by the same Rule you may find the Quarters, and that you
may prick off in like manner, which is a way how to make an Horizontal Dial, as before,
Latitude 51 deg. 30 min.
CHAP. VIII. A South and North Erect Direct or Horizontal Dial, and how to make it.
THis belongs to an upright Wall looking full North or South, and the Plane of it lies
in the East Azimuth.
First draw the Horizontal-line AB, which serveth for 6 in the morning at A, and 6
in the afternoon at B; then from the Center lay down from C the Gnomon of the Latitudes
Complement 33. 30 both ways, as to A and B: Now with the whole Line of 6 hours from
A describe an Arch towards D, and with the same distance from B cross the same Arch,
and draw the two Lines AD and BD, whereon from D you must transport the hours, as
you see by the Figures 1, 2, 3, 4, 5. So drawing Lines through those parts from the
Center C, you shall have the hours from 6 a clock in the morning to six a clock in
the afternoon.
90Lat.5130Compl.3830With the Chord of 60 on the Center C describe the Semicircle ADB from which Line of
Chords take the Complement of the Latitude 38, 30, and lay down from the Meridian
at E unto F; so drawing CF, you shall have the height of the Stile [Page 13] above the Plane. This, if it be for a large Dial, as against a Wall, is best to be
made of a Rod of Iron; for small Dials a Brass Plate is best, and your Dial is done.
[geometrical diagram]
This Dial shews the Hours from 6 in the morning to 6 at night: The other hours before
and after 6, as far as four and eight, belong to the North face of this Dial. Because
the Almicantars may oft obscure the Intersections of the Hour-circles, you may avoid
that if you reduce this Dial to a Vertical Dial, for the South Horizontal Dial, being
the Vertical Dial of those People who live 90 degrees Southward from us, that is,
in 38 d. 30 m. of South Latitude.
Secondly, For the North face, imagine you had for the Gnomon a Wire thrust aslope
through the center of the Plane from the Southside Northward, and you will presently
conceive, that in the North Dial the Horizontal or 6 a clock Line will be lowest,
and that the Stile or Gnomon will turn upwards towards the North Pole, as much as
it turned downwards on the other side; and that all the Hours save 6 in the morning,
and 6, 7, 8 at night, may be left out in our Latitude, because the Sun shineth no
longer upon it; and those Hour-distances you may find and set off from 6 a clock Line,
as you did the Hours of like distance in the South face. Note in a South erect direct,
or a South erect declining Dial, the Stile always points downwards; but if it be a
North erect declining Dial, the Stile points upwards.
CHAP. IX. How to make a South inclining 23 deg. in the Latitude of 51 deg. 30 min.
SUppose that the Plane be so inclining, that the face thereof be towards the South,
and the North part be elevated 23 deg. above the Horizon, and that the South part be dipped as much under the Horizon; then
to find the height of the Stile above the Plane,d.m.Lat.5130Incl.23H. Stile.2830 you must substract the Inclination 23 deg. from the Latitude of the Place, which is here 51 deg. 30 min. so the Remainer being 28 deg. 30 min. shall be the height of the Stile. Now for drawing the Hour-lines, you shall do no
otherwise than you have done before in making the Horizontal Dial according to the
Stiles height 28. 30, as you may perceive in the Dial following.
Note, That if the Inclination of the Plane be more than the Latitude, then you must
substract the Latitude from it, so there shall remain the height of the Stile above
rhe Plane.
But if the inclination be South, that so the upward face of the Plane looks Northward,
then you are to add the Inclination to the Latitude of the Place; and if it exceed
90 degrees, you must then substract it from 180 degrees, so shall you have the Poles
height above the Plane towards the South part of the Dial. The Figure of this Dial
followeth.
CHAP. X. How to observe the Declination of any Declining Plane.
ALL perpendicular Planes, as Walls, lie in the Planes of one of the Azimuths; which
Planes cut always both Zenith and Nadir, and the Center of the Earth, as in the Figure
Z is Zenith and Nadir ESWN. Horizon EW is the Base or Ground-line, or any Horizontal
Line, drawn upon a Wall or Plane, looking full South or North: his Poles are at S
and N in the Meridian; wherefore he declineth not, but lieth in the East Azimuth EW.
[geometrical diagram]
AB is a Wall or Plane declining East by the Arch SP, to which AB or WE are equal:
for so much as the Wall bendeth from the East Azimuth, so much doth his Pole at P
decline or bend from the Meridian.
Now to find how much any Plane declineth, and so in what Azimuth he lies, one good
way is this. When the Sun begins to enlighten the Wall, or when he leaves it, then
is the Sun in the same Azimuth with the Wall: therefore take at that instant his Altitude,
and thereby get his Azimuth, according to Chap. 14. of the Sixth Book, so you shall have the Declination of the Wall.
Another way, if you have not time until the Sun cometh unto the Azimuth of the Wall,
or the Vertical of it, which cutteth the Pole thereof, then get the Suns Azimuth
as before when you can, and at the same time observe by the sight of your Circumferenter
the Suns Horizontal Distance from the Pole of the Plane; and by comparing of those
together, you may easily gather the Declination of the Wall: As in Example.
I observed the Sun to be gone West from the Pole of the Plane 72 deg. and by the Altitude of the Sun then taken, I found his Azimuth 62 deg. Here I reason thus: The Sun is gone from the Pole Vertical of the Wall 72 deg. and from the Meridian 62 deg. therefore the Meridian lies between the Pole of the Plane, and the Sun: And because
☉ P is 72, and ☉ S 62, therefort SP the Declination of the Plane is 10 deg. the Difference of 72 and 62; and the Declination is East. for the Sun is neerer to
the Meridian, than to the Vertical of the Plane.
And thus if you draw a rude Scheme of your Case, you may soon reason out the Declination,
better than do it blindfold, by the Rules commonly given.
And by those two last ways you may take the Declination not only of upright Planes,
but of Recliners also.
How to take the Declination of any Wall or Plane, without the help of a Needle or
Loadstone.
FOr Example. Suppose SNDE represent a Wall, or the face of the Plane whereon I am
to make a Dial, and I desire to know the Declination thereof from the Meridian Eastward
or Westward.
[geometrical diagram]
If you have no Instrument, take a plain Board, having one planed or straight side
or edge, which Board let be represented by DEVQ: apply the straight edge of the Board
ED to the side of the Wall or Plane, as in this Figure, and in the middle of the Board
at C, I set one foot of the Compasses, and the other opened to 60 deg. of my Line of Chords, I describe the Circle ZBHA. In the Center C, I erect or place
a Stile or Wire, as CO perpendicular to the Horizon, placing the Board as neer Horizontal as I can. I find by observation, that the shadow of the top of the Pin or Wire toucheth
the Circle in the forenoon at the Point B, where I make a little mark; and likewise
I observe in the afternoon that it toucheth the said Circle in the Point A: Then I
measure the half thereof from B or A to X, and drawing a Line through the Center to
X, as KCX, you shall have the Meridian Line exactly described KCX. Lastly, I take
the Distance ZX, which I apply to my Scale of Chords, and find the Arch thereof 18
deg. 10 min. and so much is the Declination of th [...] Plane EDNS, which you may see by the Meridian Line XK to be towards the East; therefore
it is a South Plane declining West 18 d. 10 m. This way is the most easie way, and requires time for the making the two Observations;
therefore I will lay down some other ways, that may resolve at one moment, or at
one observation.
How to find the Declination by the Needle, whether the Air be clear or not.
APply the North side of the Instrument wherein the Needle is placed unto the Wall,
and hold it Horizontally as neer as you can, that the Needle may have liberty to
play to and fro; and when it stands, observe upon the Limb of the Chard over which
it moves, upon what Degree the Needle stands; for that is the Declination of the
Plane, reckoned from the South Point of the Needle: And if you would know the Coast,
observe, That if the Needle stand upon the East side of the Meridian Line, then is
the Declination West; but if it stand on the West side of the Meridian Line, the Declination
is East. By the Sea-Compass described Book V. as it hangs in the Box, you may also
find the Declination. Set the slit of the Brass Diameter North and South, as before
directed; then set the square side of the Compass-box next the Plane, reckon outwards
180 deg. and set the Index to it; so reckon the number of Degrees betwixt 180 the Index, and
the Meridian, and that number of Degrees is the Declination of the Plane required;
and by the Chard you may see what Coast it is, that is, whether he declines from the
North or South Eastward or Westward. And note, That all Lines parallel to any Horizontal
Line be Horizontal, and all Lines parallel to Vertical Lines be also Vertical.
CHAP. XI. How to make a Declining Horizontal Dial, or South erect declining from the South
Eastwards 32 deg. 30 min. in the Latitude of 51 deg. 30 min.
HEre three things are required; for besides the Distance of the several hours from
12, and the Elevation of the Gnomon, which are requisite to the making of all direct
and regular Dials, we must here also know the Declination of the Gnomon, which some
call the Distance of the Substile from the Meridian, or the distance of the Meridian
of the Plane from the Meridian of the Place. For in all Dials the Noon-line in the
Meridian of the Place, projected on the Dial, and in all Horizontal or Mural Dials,
not reclining or inclining, the Noon-line is a Perpendicular cutting the Center of
the Dial, how much soever they decline.
But declining Dials which look awry from our Meridian, have a Meridian of their own,
which is called the Meridian of the Plane and the Substile (because the Stile or Gnomon
stands upon it) and is indeed the Meridian of that Place where this Declining Dial
would be a Vertical Dial, and where the Substile would be Noon-line; and to this Substile,
the Hours of the Plane are always so conformed, that the neerer they be to the Substile,
the narrower are the Hour-spaces; and contrarily, because the Meridians do cut every
Oblique Horizon, that is thickest neer the Meridian of the place; and this Declining
Dial being a Stranger with us, followeth the fashion of his own Country, and so hath
his narrowest Hour-spaces neer his own Meridian, rather than ours: And now, as that
is the Meridian of our place, which cutteth our Horizon at Right Angles, passing through
his Poles, Zenith, and Nadir; so the Meridian of any Plane is that which cutteth the
Plane at Right Angles, and passeth through his Poles.
Before we draw the Hour-lines in these sort of Dials, it will be very convenient to
shew a general way for all Latitudes in a Diagram by it self, and how to find the
Substiler Distance from the Meridian or 12 a clock Line, and the height of the Gnomon
or Stile above the Plane. First, Draw the Horizontal-line AB, and upon the Center
at C, take off your Scale with your Compasses a Chord of 60 Degrees, describe the
Semicircle ADB, and with a Chord of 90 you may lay from A to D, and from B to D, so
shall you draw CD from the Meridian-line of 12 a Clock; Then take the Complement of
the Latitude 38 deg. 30 min. and lay from D to E, and so draw EF parallel to the Horizon AC; next take the Declination
of the Dial 32 d. 30 m. and lay from D to G, drawing the Radius OG thereon, you must lay the Distance EF from the Center at C, as CH. Now with the
neerest distance from H to the Meridian CD, as HI, make FL; and drawing a Line from
C through L, it will cut the Limb in the Point M; so measuring DM on the Line of Chords,
you shall have the Substiler Distance 23 deg. 8 min. all which you may see in this Scheme following.
By Calculation,
As the Radius 90 deg.
10
Is to the Sine of the Declination SE 32 deg. 30 m.
973021
So is the Co-tangent of the Latitude 51 d. 30 m.
990060
To the Tang. of the Substiler Dist. from the Meridian 23 d. 8 m.
963081
For the height of the Stile, take the neerest Distance from H to the Horizon K, and
lay the same from L to cut the Arch in N: So measure MN, you shall have on the Line
of Chords the Height of the Stile neerest 31 deg. 40 min.
By Calculation, viz.
As the Radius 90 deg.
10
Is to the Co-sine of the Latitude 51 deg. 30 min.
979414
So is the Co-sine of the Declination 32 deg. 30 min.
To find what Hour or how much Time the Substiler is distant from the Meridian or Inclination
of Meridian.
[geometrical diagram]
TAke the neerest Distance from M to FE, and lay it on the Meridian from F to O: Then
take the Distance from O unto G, and lay it from O unto P on the Meridian; so the
Distance from P to M, measured on a Line of Chords, will be found to be 39 deg. 9 min. or thereabouts; which in time, allowing 15 deg. for an hour, and four Minutes to a Degree, you shall have 2 ho. 36 min. 36 sec. which is the distance of the Substile Line from the 12 a Clock Line, which in this
Dial is between 9 and 10 of the Clock in the morning. And by Calculation,
As the Radius 90 deg.
10
Is to the Sine of the Latitude 51 deg. 30 min.
989354
So is the Co-tangent of the Declination 32 deg. 30 min.
1019581
To the Co-tangent of the Inclination 39 deg. 9 min.
1008935
Thus is shadowed a Geometrical way, and by Calculation, for any Latitude: But for
one particular Latitude, Mr. Philip Staynred, which first composed the Scale and Gnomon Line, and Inclination of Meridians, and
the greater and lesser Pole on the Dialling Scale, for 37 years since, as I have seen
by him calculated, and the Projection Geometrical in his Study: he hath for the more
ease set two Lines upon the Dialling Scale, as he usually makes, to find the Substile
for the Latitude of 51 deg. 30 m. against the Lines stands the Letters Sub or Stile joyned with it; so if you take from off the Substile-line the Declination of the
Dial, and lay it from D unto M, which in the last Example was 32 deg. 30 min. you shall find it to reach in the Diagram from D unto M, as in the Line of Chords
23 deg for the Substile, as before. Also, the other Line noted with the word Stile, you shall likewise take from thence the Declination 32 deg. 30 min. which you shall find to reach in the Diagram from M unto N, or in the Line of Chords
31 deg. 40 min.
CHAP. XII. How to draw the Hour-Lines in a Declining Horizontal-Dial, or South erect, declining
32 deg. 30 min. from the South Eastward, the Latitude being 51 deg. 30 min.
FIrst draw the Horizontal Line AB, and on the Center at C describe the Semicircle
AEB, with the Chord of 60 deg. and from A and B lay down 90 deg. unto E; so shall you draw CE the Meridian Line or Hour of 12; then in [Page 19] the former Diagram take the Substile distance DM 23 deg. and lay the same in the Dial following from E unto F, and from the Center C through
F you shall draw CFK the Substiler Line. Next take the Chord of 90 deg. and lay it from F both ways upon the Arch, so shall you draw the Gnomon Line GH,
whereon from C, with the Stiles Altitude before found, 31 deg. 40 min. taken from the Gnomon Line, you shall make CG and CH. Then take the whole Line of
6 hours, and with the same distance from G describe an Arch at K, and with the like
from [...] cross the same Arch, and draw the Lines GK and HK, which last Line cuts the Meridian
at N. Now if you measure KN on the Hour-lines, you shall find it neer 2 ho. 36 m. ½ as you found in the last Diagram. Then take one Hour more, which is 3 ho. 36 min. and lay the same from K unto M; and so increasing one Hour more, you shall have the
Hour points l and i; also diminishing one Hour less than 2 ho. 36 min. which is 1 ho. 36 min. the same will reach from K to O, and so 38 min. from K to P. Now as you have divided KH, the very same distance as is from K towards
H, must be from G towards K; so drawing the Lines from the Center C through those
Points, you shall have the Hour-lines, as you see in the Dial following.
[geometrical diagram]
By Calculation,
As the Radius 90 deg.
10
Is to the Sine of the Stiles or Gnomons Height. 31 d. 40 m.
972013
So is the Tangent of the Dist. of an Hour from the Subst. 9 d. 9 m.
920701
To the Tangent of the Hour-Arch from the Substile 4 d. 50 m. betwixt the 10 a clock Line, and the Subst. Line on the Arch
and by the same you may make one for any Latitude, and for any Declining Dial; and
you may by it prove your former Work: for if you prick from the Substiler Line F the
Chord of 4 deg. 50 m. and draw a Line from the Center, it will be the Hour-line of 10; and prick the Chord
of 3 deg. 5 min. from the Substile, and draw a Line through that Point to the Center, and it will
be the Hour-line of 9 a clock; and so of the rest, as you find them in the last Column.
Note, That the Height of the Stile FS being equal unto MN in the former Diagram,
which is the Chord of 31 d. 40 m. now because the Plane declines East, therefore the Gnomon shall decline West: for
the Dial being such a Projection of the Sphere, wherein all the usual Lines cross
in the Nodus of the Gnomon, and thence disperse themselves again towards the Plane;
therefore that which is East in the Sphere, will be expressed West on the Plane, and
contrarily, as was said Chap. 2. Theorem 2. Also I consider, that howsoever the Plane be turned East or West, the Gnomon place
is fixed, because it is a part of the Axis of the World, or a Line Parallel to it.
Now therefore I turn a South Dial, and make him decline East, and hold the Gnomon
unmovable, the West side of the Dial will approach neerer to the Gnomon, as reason
and sense will require. Likewise the Hours which are found on the same side of the
Meridian or Noon-line with the Substile, must be set the same way with it from the
Noon-line in the Dial.
And if you would draw the North Dial of this Plane, do but prolong those Hour lines,
and the Substile upwards beyond the Center, and you have the North Dial beyond C,
or above the Horizontal Line AB, as the South Dial below it. And note, Because the
Sun sets after 8 a Clock in Summer, therefore the three hours next before and after
midnight, may be left out in this Dial, and all others which must serve in our Latitude.
This is the most ready way to delineate the opposite face of any Dial. Note, That
if a Wall decline from the South Eastwards 32 deg. 30 min. therefore the Plane which lieth 90 deg. from his Pole, is in the 32½ Azimuth from the East Northward.
Note this well: Extend the Compasses as before from K to N, the Intersection of the
Meridian with the Line KH at N, before found to be 2 ho. 36 min. which converted into Degrees, by allowing 15 Degrees to an Hour, and 4 Minutes to
a Degree, it makes 39 deg. 9 min. which 39 deg. 9 min. shew me the Difference of Longitude between our Country and the Country of this Dial.
You may apply this Distance to the Line of Inclination of Meridians, and it will give
you the Distance before 39 deg. 9 min.
Note, I allow this Countries Longitude to be 27 deg. 44 min. at Bristol, to the Eastward of the Grand Meridian Flowers and Calfs one of the Isles of Azores, which added to 39 deg. 9 min. shews the Longitude of the Country of the Dial to be 66 d. 53 min. Eastward, and Latitude 31 deg. 40 min. which I find by my Globe is in the Desarts of Arabia at Asichia neer Soar.
CHAP. XIII. How to observe the Reclination or Inclination of any Plane.
WHat Reclination and Inclination are, hath been shewed Chap. 8. and you will have it following in a Diagram by it self.
All Reclining and Inclining Planes have their Bases or Horizontal Diameters lying
in the Horizontal Diameter of some Azimuth; but the top of the Plane leaneth back
from the Zenith of your place in the Vertical of the Plane (which is the Azimuth cutting
the Plane at Right Angles) so much as the Reclination hapneth to be: and the Pole
of the Plane, on that side the Plane inclines to, is sunk as much below the Horizon,
as the top of the Plane is sunk below the Zenith; and the opposite Pole is mounted
as much.
Let ESWN be Horizon, Z the Zenith, EW the Horizontal Diameter of the Plane and of
the East Azimuth, EOW a Plane not declining but reclining Southwards from the Zenith
by the Arch ZO 45 deg. and his opposite Face inclining to the Horizon according to the Arch OS 45 deg. the Pole of the reclining Face is at P in the Meridian CP, which here is also Vertical
of the Plane, and is elevated 45 deg. in the Arch NP, equal to the Arch of Reclination ZO, the Pole of the inclining Face
is depressed as much on the other side under the Horizon.
To find the Quantity of the Reclination, you shall draw a Vertical Line on the Plane
by Chap. 3. and thereto apply a long Ruler, which may overshoot the Plane either above or
below: to that Ruler apply any Semidiameter of a Quadrant, and the Degrees, between
that Semidiameter and the Plumb-line, shall be the Degrees of Reclination. Or stick
up in the Vertical Line two Pins of equal height, and perpendicular, and placing
your self either above or below the Plane, as you find most easie, direct the Sights
of your Quadrant to the Heads of the two Pins, being in a right Line with your eye;
and the Plummet shall shew the Reclination on the Side of the Quadrant, and the Inclination,
which is always the Complement thereof, on the other.
CHAP. XIV. To draw the Hour-Lines in all Declining, Reclining, Inclining Planes.
IF a Plane shall decline from the Prime Vertical, and incline to the Horizon, and yet not lie even with the Poles of the World, it is then called a Declining,
Inclining Plane. Of these there be several sorts, you may see 19 Planes in the following
Diagram, and directed how to know them in the 25th Chapt r: but to shew the Recliners in order as they come, viz. the North Recliner 45 deg. and South Incliner falls between the Aequator and the Tropick of ♋, as the Circle
EQW; the South Recliner 45 deg. and North Incliner, falls between the Horizon and the Pole, and is represented by
the Circle EAW. The East and West Recliner and Incliner 45 deg. may be seen by the Circle NVS, the Inclination may be Northward 45 deg. and declining 45, as the Circle FKC, or the Plane may decline S 45 Westward, and
recline 45 deg. from the Zenith, as the Circle CLF. If your Inclination or Reclination fall more
or less, you may see the way. Each of these Planes have two Faces, the upper toward
the Zenith, the lower towards the Nadir, wherein having the Latitude of the Place,
and the Declination, with the Inclination of the Plane, you are farther to consider
what must be found before you can draw the Dial, which will follow in order, and is
represented in this Fundamental Diagram; only I will mention the Arches and Angles
in the hardest, which is a South declining West 45 deg. and reclining from the Zenith 45 deg. In such you must consider.
1. The Arch of the Plane between the Horizon and Meridian CO or FO.
2. The Arch of the Meridian between the Horizon and the Plane ON or OS.
3. The Angle of Inclination between the Meridian and the Plane CON or FOS.
4. The Substile-distance from the Meridian OR or OR.
5. The Height of the Pole above the Plane or Stiles Height PR.
6. The Inclination of both Meridians or Angles at P.
7. The Difference of the Hour from the Substile. But first we will describe the Diagram.
[geometrical diagram]
To Describe the DIAGRAM.
THe Description of this Diagram is set down at large by that worthy Mathematician
Mr. Edmond Gunter, in the Use of his Sector, Chap. 3. But for this purpose it may suffice, if it have the Vertical Circle, the Hour
Circles, the Aequator, and the Tropicks first drawn in it; other Circles may be supplied
afterward, as we shall have use of them; and those may be readily drawn, as I have
borrowed of him, in this manner.
Let the outward Circle, representing the Horizon, be drawn and divided into four equal
parts, with SN the Meridian, EW the Vertical, and each fourth part into 90 deg. That done, lay a Ruler to the Point S, and each Degree in the Quadrant EN, and note
the Intersections where the Ruler crosseth the Vertical; so shall the Semidiameter
EZ be divided into other 90 deg. and from thence the other Semidiameters may be divided in the same sort; those may
be numbred with 10, 20, 30, from E towards C; and for variety, with 10, 20, 30, from
C towards W: But for the Meridian, the South part would be best numbered according
to the Declination from the Aequator, and the North part according to the Distance
from the Pole.
Then with respect unto the Latitude, which here we suppose to be 51 deg. 30 min.[Page 23] open the Compasses unto 38 deg. 30 min. from C toward W, and prick them down in the Meridian from C unto P; so this Point
P shall represent the Pole of the World, and through it must be drawn all the Hour-circles.
Having three Points EPW, find their Centers, which will fall in the Meridian, a little
without the Point S, and draw them in a Circle EPW, which will be the Circle of the
Hour of 6.
Through this Center of the Hour of 6 draw an occult Line at length parallel to E W,
so this Line shall contin [...]e the Centers of all the other Hour-circles: where the Circles of the Hour of 6 crosseth
this occult Line, there will be the Centers of 9 and 3 their Hour-circles.
The distance between these Centers of 9 and 3, will be equal to the Semidiameters
of the Hour-circles of 10 and 2: where these two Circles of 10 and 2 shall cross this
occult Line, there will be the Center of 7 and 5. And again, take with your Compasses
off the Diameter EW 75 deg. under W, and turn the Compasses three times over on the occult Line, from the Center
of the Hour of 6, and you have the Center of the Hours of 1 and 11. Again, take 78
deg. from E towards C, and lay it both ways from the Center of the first Hour-circle of
6, on the occult Line, and you have the Center of the Hour-circle of 4. So practice
for any other Latitude.
The Hour-circles being thus drawn, take 51 deg. 30 min. from C toward W, and prick them down in the South part of the Meridian from C unto
A, and bring the third Point EAW into a Circle; this Circle so drawn shall represent
the Aequator.
The Tropick of Cancer is 23 deg. 30 min. above the Aequator, and 66 deg. 30 min. distant from the Pole; and so in this Latitude it will cross the South part of the
Meridian at 28 deg. from the Zenith, and the North part of the Meridian 15 deg. below the Horizon. Take therefore 28 deg. from C towards W, and prick them down in the Meridian from C unto D, so have you
the South Intersection: Then lay the Ruler to the Point W and 15 deg. in the Quadrant NW, numbred from N toward W, and note where it crosseth the Meridian;
so shall you have the North Intersection. The half way between these two Intersections
in the Meridian Line is the Center of the Tropick of Cancer: Which being truly drawn, will cross the Horizon before 4 in the Morning, and after
8 in the Evening, about 40 deg. Northward from E and W, according to the rising and setting of the Sun at his entrance
into Cancer.
The Tropick of Capricorn is 23 deg. 30 min. below the Aequator, and 113 deg. 30 min. distant from the North Pole; so that in this Latitude it crosseth the South part
of the Meridian at 75 deg. from the Zenith, and the North part of the Meridian at 62 deg. below the Horizon. Take therefore 75 deg. towards W, and prick them down in the Meridian from Z unto ♑; so have you the South
Intersection: Then lay the Ruler to the Point W, and 62 deg. in the Quadrant NW, numbred from N towards W, and note where it crosseth the Meridian,
so shall you have the North Intersection: the half way shall be the Center, whereon
you may describe the Tropick of Capricorn ♑. This Tropick will cross the Horizon after 8 in the morning, and before 4 in the
Evening, about 40 deg. Southward from E and W, according to the rising and setting of the Sun at his entrance
into Capricorn. Now we will proceed to draw the Hour-lines in a North Recliner and a South Incliner,
and shew the Height of the Stile above the Plane on the Meridian, and a South Recliner,
and a North Incliner; and so in order to the rest.
CHAP. XV. How to make a North and South Reclining Dial.
THe Base or Horizontal-line of such a Dial lieth in the East Azimuth, and his Pole
in the Meridian, as you may see Chap. 14.
The Plane of the 14 Chapter was a North Plane reclining Southward 45 deg. the Zenith is distant from the North Pole 38 deg. 30 min. the Complement [Page 24] of the Latitude 51 deg. 30 min. toward the South, and I see the Reclination is 45 deg. more Southward, because I see my Plane reclines so much that way. I add the Complement
of the Latitude 38 deg. 30 min. and the Reclination 45 deg. together, and I see then by the same the North Pole is elevated 83 deg. 30 min. which is the Height of the Stile above the Plane on the Meridian; which 83 deg. 30 min. taken off the Gnomon Line of the Scale, you may proceed and draw the Dial, and lay
that on the Line WE, and work as you did in the other Dials.
The opposite face to this is the South Incliner; and if you would draw it, do but
prolong those Hour-Lines, as was said in Chap. 13. and you have the South Recliner below the Horizontal Line WE.
Note, Had this Reclination been 51 deg. 30 min. and the Complement of the Latitude added to it would have made 90 deg. then it would have fallen into the Plane of the Aequinoctial, and so the Dial would
have been a Polar Dial, and all the Hours would have had equal spaces, and the Gnomon
would have stood perpendicular, which are the properties of a Polar Dial, as hath
been shewed Chap. 5.
[geometrical diagram]
d.m.51304500630As for the South reclination, which is 45 deg. which substract from the Latitude 51 deg. 30 min. and you have the height of the Gnomon or Stile above the Plane, which by reason the
Hour-lines will be so neer together, continue them, and cut them off as they may fit
your Plane, by leaving out one of the Hours, or more, as you will; so will you have
the Stile a handsom height above the Plane, and the Hour-circles a good distance asunder:
But for the North Incliner, his Lines and Stile must be drawn from the Center of the
Plane,d.m.5130312030 although they do come neer. If the Reclination had been 31 deg. then you should have substracted the Reclination from the Latitude 51 deg. 30 min. and the Remainder would have been 20 deg. 30 min. the Height of the Stile or Gnomon above the Plane.
CHAP. XVI. How to make an East or West Reclining or Inclining Dial.
AS it hath been shewn Chap. 15. That the Base or Horizontal-line of a South Recliner lieth always in the East
Azimuth; so the Base of an East Recliner lieth always in the Meridian of the Place:
And as all Declining Planes lie in some Azimuth, and cross one another in the Zenith
and Nadir, by Chap. 13. So these Reclining Planes lie in some Circle of Position, and cross one another
in the North and South Points of the Horizon; which being considered; these East Recliners,
West Incliners, and West Recliners, and East Incliners, shall be made as easily as
the former.
For these East Recliners be in very deed South Decliners to those that live 90 deg. from us Northward or Southward, and have one of those Poles elevated as much as the
Complement of our Latitude; for the Perpendicular or Plumb-line of those People is
parallel to the Horizontal Diameter of our Meridian.
EXAMPLE.
I Have an East Plane reclining 45 deg. which I would make a Dial.
In the former Diagram I number 45 deg. from E to F, and then lay a Ruler from N to F, and it will cut the Semidiameter ZW
in 45 deg. in V. And then draw the Arch SVN, which Circle shall represent the Plane proposed.
Then the Arch of the Plane between the Horizon and the Substiler Distance is represented
in the Diagram by NQ, and may be found by resolving the Triangle QN P, wherein the
Angle at Q is known to be Radius, and the Angle at N to be Reclination, and the Angle
at P the Latitude. Then work thus.
As the Radius or Sine of 90 deg. Q
1000000
Is to the Sine of Reclination 45 deg. N
984948
So is the Tangent of the Latitude 51 deg. 30 min. PN
1009939
To the Tangent of the Substile QN 41 deg. 38 min.
994887
Or upon Gunter's Ruler, Extend the Compasses from the Sine of 90 deg. to the Sine of 45 deg. the same will reach from the Tangent of the Latitude 51 deg. 30 min. to neer 41 deg. 38 min. as before, in the Line of Sines; and such is the Substiler distance.
Secondly, The Height of the Pole above the Plane may be represented by the Arch PQ,
and may be found, by which we have given in the Triangle QNP: For,
As the Sine of 90 Q
1000000
To the Sine of 51 deg. 30 min. PN
989354
So is the Sine of Reclination 45 deg. N
984948
To the Sine of the Stiles Height 33 deg. 36 min. or Pole above the Plane PQ
974302
Extend the Compasses from the Sine of 90 deg. to the Sine of 51 deg. 30 min. the same Extent will reach from the Sine of Reclination 45 deg. to 33 deg. 36 min. as before, which is the Height of the Stile.
Thirdly, The Inclination of Meridians (or indeed you may call it Longitude) is here
represented by the Angle PQN; for having drawn the Arch of the Meridian of the Plane
SQN, or let fall a Perpendicular PQ, and that from the Pole unto the Plane, this Perpendicular
shall be the Meridian of the Plane; so that from Q to N is the Distance of Inclination
of both Meridians, which will be found as before: For, [Page 26]
As the Sine of 51 deg. 30 min. PN
989354
To the Sine of 90 deg. Q
1000000
So is the Sine of the Substiler Distance 41 deg. 38 min.
To the Sine of Inclination of both Meridians
971594
which will be found to be 58 deg. 40 min. NPQ.
Extend the Compasses from the Sine 51 deg. 30 min. to the Sine of 90; the same Extent will reach from 41 deg. 38 min. the Substiler Distance, to 58 deg. 40 min. and such is the Angle PQN of the Inclination, between the Meridian of the Place and
the proper Meridian of the Plane: which resolved into time, doth make about 3 ho. 54 min. and so the Substiler must be placed neer the Hour of 8 in the morning.
[geometrical diagram]
For to draw the Hour-lines on the Plane, first draw the Horizontal-line SN: Then take
off your Line of Chords with your Compasses the Chord of 60 deg. from your Scale, and sweep the Semicircle: Then take off your Line of Chords with
your Compasses the Substiler Distance, and lay it from N on the Arch to A; Then draw
through the Center and A in the Arch, the Substiler Line, crossing it in the Center
at Right Angles with the Line KF: Then take off with your Compasses the Height of
the Pole above the Plane, or the Stiles Height 33 deg. 36 min. from the Gnomon Line of the Scale, and lay it from the Center of the Dial both ways
from K to F. Then take the whole Line of 6 Hours, and sweep the two small Arches from
K towards G, and the like from F toward G: Then draw the Lines KG and FG: Then extend
the Compasses from G to N, and apply it to the Hour-line, and you shall find the Inclination
of Meridians to be as before 3 ho. 54 min. Then take 4 ho. 54 min. and lay it to O, and the same Distance from K unto O toward G; and the like do with
2 ho. 54 min. and make such marks on the Line GF and KG as you see in the Figure to [Page]
[geometrical diagram]
[Page][Page 27] draw the Hour-lines by; and then take off the Line of Chords 33 deg. 36 min. the Stiles Height, and lay from A to B, so drawing the Hour-lines, and you have done.
And then you may see as in a Glass the West Recliner, the opposite Face, as you were
shew'd before Chap. 13. that is, strike the Substiler Line, and all the Hour Lines through the Center,
and the same Figures to every Hour beyond the Center, which you had on the first side,
and set the Gnomon upon the Substile downwards, to behold the South Pole, and you
have done both: So have you on the back side, looking through the Paper, the West
Recliner and East Incliner, if you draw in the like manner, or prick on the back side,
for 11 in the East 1 in the West Recliner, and so contrarily of the rest.
CHAP. XVII. How to find the Arches and Angles that are requisite for the making of the Reclining
Declining Dial.
BEfore you can intelligently make a Reclining Declining Dial, which is the most irregular
of al, having two Anomalies, viz. Declination and Reclination, you must be acquainted with those three Triangles in
the Sphere, wherein certain Arches and Angles lie, which are needful to be known.
I advise you first to draw, though it be but by aim, an Horizontal Projection of the
Sphere, such as here I have drawn for a South declining West 45 deg. and reclining from the Zenith 45 deg. in the Latitude of 51 deg. 30 min. which shall be our Example. The same also is shewed in the Fundamental Diagram; only
I shew you this, to let you know, there is several ways to the Wood besides one.
In this Figure the Arch FLC is the Plane, ZL the Reclination thereof, FE the Base
or Horizontal Line of the Plane, and AE n the Vertical of the Plane, cutting it right
at L, and cutting the Pole thereof at H: for n is the Pole of a Plane erected upon
FE; but the Pole of the Reclined Plane FLE is H n AE; or S [...]n is the Declination of the Plane, PH m the Metidian of the Plane, cutting the North
Pole at P, the Plane at Right Angles at R, and the Pole thereof at H.
In the first Triangle FNO you have given FN 45 deg. the Complement of the Planes Reclination N, the Right Angle of our Meridian with
our Horizon F, the Complement of Reclination 45 deg. whereby you may find FO the Oblique Ascension, or the Arch of the Plane between
the Horizon and our Meridian, that is, how many Degrees the Noon-line shall lie above
the Horizontal-line. Also you may find NO the Perpendicular Altitude of the Noon-line,
or the Inclination of the Noon-line of the Dial to the Horizon, which taken out of
51 deg. 30 min. remains the Arch of the Meridian between the Pole and the Plane. But note, That when
this Altitude of the Noon-line NO is equal to NP the Elevation of the Pole, then is
the second Triangle PRO quite lost in the Point P, and the Plane becometh then a Declining
Aequinoctial Plane: Also you may find the Angle at O, called the Angle of Inclination
between the Meridian and the Plane. In the second Triangle ORP you have given O as
before, R the Right Angle of the Plane with his Meridian, OP the Position Latitude,
that is, the Latitude of the Place wherein the Reclining Plane ORLEQ shall be a Circle
of Position; this is given if you substract NO, the Altitude of the Noon-line, as
before, out of the Latitude, and hence may be found OR the Declination of the Gnomon
or Substiler Distance, or Distance of the Meridian of the Plane from the Meridian
of the Place; RP the Elevation of the Pole above the Plane, in the Planes own Meridian
or Stiles Height; P the Angle between the Meridian of the Plane, and the Meridian
of the Place. This Angle is called the Difference of Longitude, because it shews how
far the Places are distant from us in Longitude; wherein this Dial shall be a Direct
Dial, without Declination, having his Gnomon in the Noon-line of the Place, and shews
also how many Degrees of the Plane comes between the said Meridians. Let this be well
observed by Learners.
Hence may be found, if you will, the third Triangle PZH: You have given PZ the Complement
of our Latitude, ZH the Complement of the Planes Reclination, Z the Supplement of
the Planes Declination.
Also hence may be found HP, whose Complement is PR the Elevation of the Pole above
the Plane, the Difference of Longitude H, whose measure is RL, the Arch of the Plane
between the Meridian of the Plane or Substile, and the Vertical Line of the Plane;
the Complement thereof is RF, the Substiles Distance from the Horizontal Line of the
Plane.
Every Arch and Angle is given and may be found by the Problems of Spherical Triangles,
as before; but we will make short our business.
First find the Arch of the Plane between the Horizon and Meridian FO.
As the Sine of 90 N
1000000
To the Sine of Reclination for the Zenith F 45
984948
So is the Co-tangent of Declination 45 NF
1000000
To the Co-tangent of FO 36. 16
984948
taken out of 90, there remains for FO 54. 44. which is the Arch of the Plane between
the Horizon and Meridian.
Extend the Compasses from 90 to 45; the same Extent will reach from the Co-tangent
45, to 35. 16, as before, by the Ruler.
Secondly, The next to find is the Arch of the Meridian between the Pole and the Plane;
but we will by opposite Work find it thus.
Or, Extend the Compasses from the Sine of 90 deg. to the Sine of 54 and 44; the same Extent will reach from the Sine of 45 deg. to 35. 16 NO, which Substracted from the Latitude 51 deg. 30 min. remains the Arch of the Meridian between the Pole and the Plane OP 16 deg. 14 min.
Thirdly, Now you must find the Angle NOF, which will be ROP, which is called the Angle
of Inclination between the Meridian and the Plane, thus:
As the Sine of FO 54. 44
008805
To the Radius or Sine of 90
1000000
So is the Sine of the Side FN 45
984948
To the Sine of the Angle ROP 60. 0
993753
Extend the Compasses from the Sine of 90, to the Sine of 54. 44; the same Extent
will reach from the Sine of 45, unto 60, as before-found, the Angle of Inclination
between the Meridians and the Plane.
Fourthly, The next to be found is the Substile Distance, or the Meridian of the Plane
from the Meridian of the Place.
As the Sine of the Angle at R 90
1000000
To the Co-sine of the Angle at O 30. 0
969897
So is the Side of the Tangent PO 16. 14
946412
To the Tangent of the Substile OR 8 deg. 17 min.
916309
And thus extend the Compasses from the Sine of 90 deg. to the Co-sine of 30 deg. the same Distance will reach from the Tangent of 16. 14, to the Tangent of the Substile
from the Meridian 8 deg. 12 min.
Fifthly, Next find the Elevation of the Pole above the Plane or Stiles Height: For,
As the Sine of 90 deg. at R
To the Sine of his Opposite PO 16. 14
944645
So is the Sine of the Angle at O 60
993753
To the Sine of the Stiles Height above the Plane 14. 1
438398
Or, Extend the Compasses from the Sine of 90 deg. to the Sine of 16 deg. 14 min the same Distance will reach from the Sine of 60 the Angle at O, to the Height of
the Pole above the Plane 14 deg. 1 min. or Stiles Height, as before.
Sixthly, Now for the Difference of Longitude or Angle at P, or Inclination of the
Meridian of the Plane to the Meridian of the Place, it is thus.
As the Sine of the Arch PO 16. 14
944645
Is to the Sine of 90 deg. R
1000000
So is the Sine of the Substile Distance 8 deg. 17
915856
To the Angle at P or Difference of Longitude 31 deg. 1 min.
97 [...]211
Extend the Compasses from the Sine of 16 deg. 14 min. to the Sine of 90 deg. the same Extent will reach from the Substile Distance 8 deg. 17 min. to the Inclination of both Meridians 2 ho. 4 min. as before.
Seventhly, The Distance of the Hours from the Substiler are here also represented
by those Arches of the Plane which are intersected between the proper Meridian and
the Hour-circles. The Angle at R, between the Pole and the proper Meridian, is a Right
Angle; the Side RP is the Height of the Pole above the Plane; and then the Angles
at the Pole between the proper Meridian and the Hour-circles may be gathered into
a Table: For,
As the Sine of 90 R,
To the Sine of the Height of the Pole above the Plane 14. 1.
So is the Sine of the Angle at the Pole 16. 14.
To the Sine of the Hour-line from the Substiler 3 deg. 50 min.
Extend the Compasses from the Sine of 90, to the Sine of 14. 1; the same Extent will
reach from 16. 14, to 3 deg. 50 min.
The Arch of the Plane between the Horizon and the Meridian FO 54 deg. 44.
The Arch of the Meridian between the Horizon and the Plane ON 35. 16.
The Angle of Inclination between the Meridian and the Plane FON 60.
The Substile Distance from the Meridian OR 8 deg. 17 min.
The Height of the Stile above the Plane PR 14 deg. 1 min.
The Inclination of both Meridians or Angle at P 31 deg. 1 min.
The Difference of the Hour from the Substile 3 deg. 50 min.
CHAP. XVIII. How to draw the Reclining Declining Dial.
NOte, You have got the three principal Arches for the drawing the Hours.
1. The Arch of the Plane between the Horizon and the Meridian 54 deg. 44 min.
2. The Substile Distance from the Meridian of the Place OR 8 d. 17 m.
3. The Height of the Pole above the Plane or Stiles Height PR 14 d. 1 m.
How to draw the South declining West 45 deg. and reclining from the Zenith 45 d. is thus. First draw the Horizontal Line Fe, with your Compasses take off the Scale
and Line of Chords the Radius 60 deg. and sweep the Circle FORPe; then take
[geometrical diagram]
off the same Line of Chords 54 deg. 44 min. the Arch of the Plane between the Horizon and the Meridian, and lay it from F on
the Arch to O: Then take off your Line [Page 31] of Chords 8 deg. 17 min. the Substile Distance, and lay it from O to R; then from the Center draw the Substile
Line CG; then take of the Quadrant 90 deg. of the Line of Chords, and lay it both ways from R to M and N, and draw that obscure
Line; then take off the Poles Height above the Plane or Stiles Height 14 deg. 1 min. from the Gnomon Line of the Scale, and lay it both ways from C to K; and then with
your Compasses take off your Scale the whole Line of 6 Hours, and from K toward G,
and from I toward G, strike the two small Arches; then draw the Lines KG and IG; then
extend the Compasses from G to the Meridian Line at L, and apply that Distance to
your Scale and Hour-line, and you will find it to be 2 ho. 4 m. which is the Inclination of Meridians or Angle at P, as before: Then lay that Distance
from L toward G, and take (as before directed in Chap. 17.) one Hour less, and lay it off in the same manner from G toward K, and from I
toward G; and the like do with 3 ho. 4 min. until you have put all the 6 Hours on the Line from G to K, and from I to G. Then
draw the Hour Lines according to your Plane, whether it be a Triangle, or Circle,
or Square, or what shape soever; then lay off the Height of your Stile 14 deg. 1 min. from R to P, taken off the Line of Chords, and draw it from the Center, and cut it
fit to your Plane in what shape you will, as you may see in the Figure, and your Dial
is done. The opposite Face or Incliner to the Horizon is but to continue the Stile
and Substile and Hour-lines through the Center, as you may see, and you have it. In
like manner you may draw the South East Decliner 45 deg. and Reclining from the Zenith 45 deg. if you once draw on Paper this before, and the Hour-lines over the same, and the
like the Substile, and the Stile to stand upon the Substile upright, as of the rest:
And let the highest part of the Stile be towards the North Pole, pointing upwards;
and where the Hour of 10 is in the South West Recliner, on the back side put 2 a
clock, and for 11 put 1, and for 1 put 11, and for 2 put 10, and so contrary all the
rest. And if you observe your Work, you have the South West and the South East Recliners,
and the North West and North East Incliners; or you may draw them by what was given
in the first, in the same manner.
CHAP. XIX. How to find the Horary Distance of a Reclining Declining Dial.
YOu have seen Chap. 17. how easily East and West Reclining Dials are to be made; and by the Figure in
Chap. 18. how they fall out to be Circles of Position, as you may see by PORCQ.
I will shew you how all reclining Dials may be reduced to East or West Recliners,
for some Latitude or other; and so the Hour-distance found by the Method of Chap. 17.
The Circles of Position, as have been shewed, do all cross one another in the North
and South Points of the Meridian: Now therefore by the Point O, where the Plane cuts
our Meridian, draw a new Horizon, as OBQC, and then shall you see your Plane in that
Horizon to be a very Circle of Position.
But now we are gotten into a new Latitude OP, called before Chap. 18. the Position Latitude; and we have here a new Reclination: for whereas this
Plane reclineth in our Latitude ZFL 45 deg. his Position Reclination is O, viz. ZOL or POR 60 deg. In the making of this Dial therefore you shall forget your own Latitude, and the
Planes Reclination in your Horizon; and with this new Latitude and Reclination make
the Dial after the manner of the East Recliner, Chap. 17. not regarding the Declination at all: for the Base of this Plane is now fallen
into the Horizontal Line of the Meridian; and his Declination being a Quadrant, he
is become a Regular Plane, and neither his Declination nor Reclination shall much
trouble you.
How to place your Noon-line from the Horizontal or Vertical Line of the Plane, you
have found already.
Note, Your new Latitude is PO 16 d. 14 m. then you know your Plane is the 60 d. Azimuth from the Axis, because POR is 60 deg. as before 90 deg. farther from the said Azimuth you have the Pole of the Plane, and therefore is the
Meridian of my Plane, and shall make the Substile of my Dial OPR 3 deg. 1 min. his Distance from the Meridian of the Place in that Aequinoctial, and is therefore
the Difference of Longitude, as before.
Then have you the Side PR the Height of the Stile 14 deg. 1 min. or Elevation of the Gnomon, as before: Likewise have you OR the Declination or Substile
Distance from the Meridian 8 deg. 17 min. as before, and you may proceed to draw the Dial in like manner as you have been directed
in Chap. 16.
[geometrical diagram]
CHAP. XX. To draw the Proper Hours of any Declining Dial.
EVery declining Plane, whether it recline or not, hath two great Meridians much spoken
of.
1 The Meridian of the Plane, which is the proper Meridian of that Country to whose
Horizon the Plane lieth Parallel.
2 The Meridian of the Place, which is the Meridian of our Country in which you set
up this Declining Plane, to shew the Hours; and so either of these Meridian Dials
may be conformed.
How to draw the Hours of our Country on such a Plane, is the harder work, because
[Page 33] the Plane is irregular to our Horizon: yet I suppose I have made the way very easie
in the former Chapters. But to draw the Hours of the Country to which the Plane belongs,
is most easie; for if you take the Substiler for the Noon-line, and the Elevation
of the Pole above the Plane for the Latitude, you may make this Dial in all points
like the Vertical Dial, after the Precept of Chap. 8.
CHAP. XXI. To know in what Country any Declining Dial shall serve for a Vertical.
IF the Dial decline East, add the Difference of Longitude found in Chap. 13. & 18. to the Longitude of the Place, and the sum or the excess above 360 is the number
of the Longitude sought. If the Dial decline West, substract the said Difference of
Longitude out of the Longitude of your Place and the Difference is the Longitude inquired:
but when the Longitude of your place happens to be less than the Difference of Longitude,
you must add to it 360 deg. before you substract the Difference of Longitude. Note, The Elevation of the Pole
above the Plane, or Stiles Height, is the Latitude of the Place inquired.
Example. The Declining Plane of Chap. 13. will be a Vertical Plane in the Longitude of 66 deg. 46 min. in the Desarts of Arabia neer Zoar: and the declining reclining Plane of Chap. 18. & 19. is parallel to the Horizon of those that sail in Longitude 357 deg 41 min. and North Latitude 14 deg. that is, as the Terrestrial Globes and Maps shew me, between Bonavista, one of the Cape Verd Islands, and Barbadoes.
CHAP. XXII. How to find the Arches and Angles which are requisite in a North Decliner Recliner,
and a South Decliner Incliner.
I Could not pass by this Example of the North Recliner Decliner, and South Incliner
Decliner, although it is shewed in the Fundamental Diagram; but it may be too obscure,
and harder to be apprehended by the Industrious Practitioner there; therefore I would
advise him to draw a Scheme of the Dial, as was shewed Chap. 15. & 18. or draw the Fundamental Diagram in Paper, and with a small Needle prick the Hour-lines,
Horizon, and Meridians; Aequinoctial, and the Tropicks; and then you have a Figure
ready to be stamped with a little Charcoal-dust as often as you have occasion: Or
if you apprehend your Work in any manner, the Figure following, or the like, may serve
your turn, to shew you the Angles you are to find, and Arches for the making of your
Dial. I shall be short in this, and refer you to Chap. 15. & 18.
The Circle ESWN is our Horizon, as before; NS our Meridian, FLC the Plane, ZL the
Reclination thereof, FC the Base or Horizontal Line of the Plane, AEN the Vertical
of the Plane, cutting it right at L, and cutting the Pole thereof at H: for N is the
Pole of a Plane erect upon FC; but the Pole of the Reclining Plane FLC is H; SE or
nN the Declination of the Plane.
Now you see your three Triangles all adjoyning in this Scheme, viz. FSO and ORP rectangled at S and R, and PZH obtuse angled at Z.
It is true, That the two first may do the Work, and so we will be brief. Observe,
you are to find as followeth.
1. The Arch between the Horizon and the Plane FO.
2. The Arch of the Meridian between the Horizon and the Plane SO.
[Page 34]3. The Arch of the Meridian between the Pole and the Plane PO.
4. The Angle of Inclination between the Meridian and the Plane FOS.
5. The Angle of Inclination between both Meridians OPR.
6. The Substile Distance from the Meridian OR.
7. The Height of the Stile above the Plane PR.
[geometrical diagram]
1. To find the Arch of the Meridian between the Horizon and the Plane, it is thus.
As the Sine of 90 at S
1000000
To the Sine of Reclination at F 45
984948
So is the Co-tangent of Declination FS 45
1000000
To the Complement Tangent of FO 35 deg. 16 min.
984948
Or thus: Extend the Compasses from the Sine of 90, to the Sine of 45; the same Extent
will reach from the Tangent of 45 deg. to the Sine of 35 deg. 16 min. as before; which substracted from 90 deg. there remains 54 deg. 44 min. the Arch of the Plane between the Horizon and Meridian FO.
2. To find the Arch of the Meridian between the Pole and the Plane, first find the
Arch of the Meridian between the Horizon and the Plane SO thus.
As the Sine of 90 at S
1000000
To the Sine of the Arch of the Plane between the Horizon and Plane FO 54. 44
991194
So is the Tangent of Reclination at F 45
1000000
To the Tangent of the Arch of the Meridian between the Horizon and Plane SO 39 deg. 14 min.
991194
Or thus: Extend the Compasses from the Sine of 90, to the Sine of 54 deg. 44 m. the same Extent will reach from the Tangent of 45 deg. to the Tangent of SO 39 d. 14 min. by the Tables found before: which 39 deg. 14 min. taken out of 90 deg.[Page 35] there remains 50 deg. 46 min. the Arch of the Meridian between the Zenith and the Plane OZ; which being added to
the Complement of the Latitude ZP, there will be 89 deg. 16 min. for the Arch of the Meridian between the Pole and the Plane PO.
3. For the Angle between the Meridian and the Plane FOS or ZOL, it is
As the Sine of the Arch FO 54 deg. 44 min.
991194
To the Radius or Sine of 90 deg. S
1000000
So is the Sine of the Declination FS 45 deg.
98494 [...]
To the Sine of the Angle at O 60. 0
993754
Or, by Gunter's Rule, Extend the Compasses from the Sine of 54 deg. 44 min. to the Sine of 90 deg. the same Extent will reach from the Sine of 45 deg. O, to the Sine of 60 deg. 0, the Angle between the Meridian and the Plane.
4. Before we find the Inclinations of both Meridians, or the Difference of Longitude
or Angle at P, we will find the Substiler Distance OR.
As the Sine of the Angle at R 90 deg.
1000000
To the Co-sine of the Angle at O 60 deg.
969897
So is the Tangent of the Side PO 89 deg. 16 min.
1189279
To the Tangent of the Substile OR 88 deg. 32 min.
1159176
Extend the Compasses from the Sine of 90 deg. to the Sine of 30 deg. the same Extent will reach from the Tangent of 89 deg. 16 min. to the Tangent of the Substile 88 deg. 32 min.
5. Now for the Inclination of both Meridians, or Difference of Longitude, or Angle
at P, it is thus. For,
As the Sine of the Arch PO 89 deg. 16 m.
999996
To the Radius or Sine of 90 at R
1000000
So is the Substiler Distance OR 88 deg. 32 min.
999985
To the Sine of the Angle of Inclinations between both Meridians 88 deg. 45 min. Difference of Longitude
999989
Which converted into Hours, by allowing 15 deg. to one Hour, 4 Minutes to a Degree will be 5 ho. 55 min. as you may find by the Line of Inclination and Hours on your Scale.
6. For the Height of the Pole above the Plane, or Stiles Height,
As the Radius or Sine of 90 deg.
1000000
To the Sine of the Arch PO 89 deg. 16 min.
999996
So is the Sine of the Angle at O 60 deg.
993753
To the Sine of the Stiles Height PR 60 deg.
993749
the Height of the Pole above the Plane or Stiles Height. And thus if you observe
what was laid before in the South Decliners Recliners, and now for the North Decliners
Recliners Incliners, you have the Propositions for any sorts of Reclining Declining
Inclining Dials, and how to find the Arches and Angles, as before, fitting for the
making of them. Now we will proceed and draw the Dial.
CHAP. XXIII. How to draw the Declining Inclining Dial.
AS was said Chap. 18. the three principal Arches for drawing the Hour-lines in a South declining West
and inclining to the Horizon, and a North Recliner Decliner, as before, or South
East Incliner, or North West Recliner, are these three.
1. The Arch of the Plane between the Horizon and Meridian 54 deg. 44 FO.
2. The Substile Distance from the Meridian of the Place 88 deg. 32 OR.
3. The Height of the Pole above the Plane, or Stiles Height 60 deg. PR.
And so follow the Directions of Chap. 19. and draw the Dial as you see. In what shape soever the Plane is, proportion the
Hours and Substile to your Plane, and let the Gnomon or Stile stand upright on the
Substile Line; and so have you the lower Face the South West Incliner to the Horizon
45 deg. Recliner the like North Face, and declining 45 deg. as before, and your Dial is done.
Note, This South West Incliner Recliner is parallel to the Horizon of those that live
on the South Land in the South Sea at Terra vista Decleros, in Longitude 297, and South Latitude 60, neer the Straights of Magellanicum.
[geometrical diagram]
And the North declining East and Recliner will be parallel to the Cossacks, neer Tartaria, in Longitude 114 deg. and Latitude North 60 deg. as the Globe shews me, and I have shewed you how to know the like in Chap. 21.
And observe, The 5 ho. 55 min. or 88 deg. 45 Difference of Longitude, as before, shews that the Sun rises and comes to the
Meridian, or 12 a Clock with us at Bristol, 5 ho. 55 min. before it doth with those Inhabitants in the South Sea, as before: And on the contrary,
the Sun is risen or on the Meridian with the Cossacks to the Eastward of us, the like time as the Sun rises with us before it doth with
those of Terra Vista Decleros in the South Sea: So that you see of what use the Line of Inclination of Meridians
on your Scale is, as likewise of all Declining Dials.
CHAP. XXIV. How to know the several sorts of Dials in the Fundamental Diagram.
THese several sorts of Planes take their denomination from those Great Circles to
which they are Parallels, and may be known by their Horizontal and Perpendicular Lines,
of such as know the Latitude of the Place, and the Circles of the Sphere.
1. An Aequinoctial Plane, parallel to the Aequinoctial, which passeth through the
Points of East and West, being right to the Meridian, but inclining to the Horizon,
with an Angle equal to the Complement of the Latitude; this here is represented by
EOW.
2. A Polar Plane, parallel to the Hour of 6, which passeth through the Pole and Points
of East and West, being right to the Aequinoctial and Meridian, but inclining to
the Horizon, with an Angle equal to the Latitude; this is here represented by EPW.
3. A Meridian Plane, parallel to the Meridian the Circle of the Hour of 12, which
passeth through the Zenith, the Pole, and the Points of South and North, being right
to the Horizon, and the Prime Vertical; this is here represented by SZN.
4. An Horizontal Plane, parallel to the Horizon, here represented by the outward Circle
ESWN.
[geometrical diagram]
5. A South and North erect direct Dial, parallel to the Prime Vertical Circle, which
passeth through the Zenith, and the Points of East and West in the Horizon, and [Page 38] is right to the Horizon and Meridian; that is, makes Right Angles with them both:
this is represented by EZW.
6. A South Declining Plane Eastward is represented by BD.
7. A South Incliner and North Recliner is represented by EQW.
8. The South Recliner and North Incliner is represented by EAW.
9. A Meridian Plane, which is the East and West Incliners and Recliners, and from
the Zenith parallel to any Great Circle which passeth through the Points of South
and North, being right to the Prime Vertical, but inclining to the Horizon; this is
represented by SVN.
10. A declining, reclining, inclining Plane, which is parallel to any Great Circle
which is right to none of the former Circles, but declining from the Prime Vertical,
reclining from the Zenith, inclining to the Horizon and Meridian, and all the Hour
Circles; this may here be represented either by FLC or FKC, or any such Great Circles
which pass neither through the South and North, nor East and West Points, nor through
the Zenith nor the Pole.
Each of those Planes, except the Horizontal, and South inclining 23 deg. hath two F [...]ces whereon Hour-lines may be drawn; and so there are 19 Planes in all. The Meridian
Plane you see hath one Face to the East, and the other to the West: Remember, that
it is an East and West Dial. The other Vertical Planes have one to the South, another
to the North; and the rest, one to the Zenith, and another to the Nadir. What is said
of the one, may be understood of the other.
CHAP. XXV. How other Circles of the Sphere besides the Meridians may be projected upon Dials.
THe Projection of some other Circles of the Sphere besides the Meridians (though it
be not necessary for finding the Hours, yet) may be both an Ornament to Dials, and
useful also for finding the Meridian, and placing the Dial in its due situation, if
it be made upon a movable Body, as shall hereafter be shewed.
The Circles fittest to be projected in all Dials for those purposes, are, the Aequator
with his Tropicks, and other his Parallels, which may be accounted Parallels of Declination,
as they pass through equal Degrees, as every 5 or 10 of Declination: Or Parallels
of the Signs, as they pass through such Degrees of Declination as the Sun declineth
when he entreth into any Sign, or any notable Degree thereof; or Parallels of the
Length of the Day, as they pass through such Degrees of Declination wherein the Sun
increaseth or decreaseth the Length of the Day by Hours or half Hours.
Also the Horizon, with his Azimuths and Almicantars, are as an Ornament to Horizontal
and Vertical Dials, and are likewise useful for projecting the Aequator and his Parallels
in all Dials. My intent is to be brief in this Treatise of the Furniture here following,
because I will have a President of some other Country Dials: I shall therefore think
it sufficient if I shew you one way to furnish any Dial with the Circles of the Sphere,
leaving you to devise others which I could have shewn.
CHAP. XXVI. How to describe on any Dial the proper Azimuths and Almicantars of the Plane.
FRom any Point of the Gnomon taken at pleasure let fall a Perpendicular upon the Substile;
that Perpendicular shall be part of the Axis of the Plane, and shall be reputed Radius
to the Horizon of your Plane. The top of this Radius in the Gnomon is called Nodus, because there you must set a Knot, Bead, or Button, or else cut there a Notch in
the Gnomon, to give shade; or cut off the Gnomon in the place of the Nodus, that the end may give the shadow for those Lineaments. Let not your Nodus stand too high above the Plane, for too great a part of the Planes day; nor let it
stand too low, for then the Lineaments will run too close together: a mean must be
chosen.
At the foot of this Radius take your Center and describe a Circle of the Plane,Tropicks and other Circles of Declination is shewed in Chap. 3. in the Aequinoctial Dial, where they be calculated and drawn by help of a Scale
of Equal Parts. and divide it into equal Degrees, and from the Center draw Lines through those Degrees
infinitely, that is, so far as your Dial Plane will bear; these Lines shall be the
Azimuths of the Horizon of the Plane, and shall be numbred from his Meridian or Substile.
Divide any of these Azimuth Lines into Degrees, by Tangents agreeable to the said
Radius; and having made a prick at every Degree, through every of these pricks you
shall draw parallel Circles, which shall be Almicantars or Parallels of Altitude,
to be numbred inwards; so that at the Center be 90 for the Zenith, and from the Center
outwards you shall number 80, 70, 60, until you come within 10 or 5 deg. of the Horizon; for the Plane is too narrow to receive its own Horizon, or the Parallel
neer, if the Nodus have any competent Altitude.
CHAP. XXVII. How to deal with those Planes where the Pole is but of small Elevation, and how to
enlarge the Stile thereof.
SUch Planes whose Stiles or Gnomons lie low, cannot have their Hour-lines distinctly
severed, unless the Center of the Dial be placed out of the Plane, as you may see
in Chapter 18. Now to inlarge the Stile in such Dials, there are two Lines or Scales in the
large Mathematical Scale, which I call Polar or Tangent Lines of two several Radius's,
the larger of them marked thus ✚, and the lesser of them marked -, as you may see
in the Second Book, Chap. 3. The Use whereof I shall here shew you.
In Chapter 18. of this Book there is descibed a Plane that declines from the South, Westward
45 deg. and reclines from the Zenith 45 deg. Now suppose it were required to inlarge that Stile and Dial.
First you shall find that there is given the Arch of the Plane between the Horizon
and the Meridian, 54 deg. 44 min. Secondly, The Substiler Distance from the Meridian to the Substile is 8 deg. 17 min. Thirdly, The Height of the Pole above the Plane is 14 deg. 1 min. Fourthly, The Difference of Longitude or Inclination of Meridians is 31 deg. 1 min. as you may see in Chap. 18. These being found, you must thus proceed in the delineation of your Dial.
First with your Compasses take from off your Scale a Chord of 60 deg. and on the Center C sweep the Arch HLRP, and draw CH the Horizontal Line blindly.
Then from H set off the several Distances before found, upon this Arch of a Circle,
viz. First, HL 54 deg. 44 min. for the Arch of the Plane between the Horizon and the Meridian. Then set off LR 8
deg. 17 min. for the Distance of the Substile from [Page 40] the Meridian, and there draw the Line of the Substile CRB. Thirdly, From this Point
R set off RP, which is 14 deg. 1 min. for the Height of the Stile above the Substile, and draw the prick'd Line CPK for
the Stile or Axis.
Now this Stile being but somewhat low, for the enlarging hereof first chuse some convenient
place in your Substiler Line, as in this Example at B, and there draw the Line FBA
squire wise to the Substiler Line. Then take with your Compasses from off your larger
Polar Line marked ✚ the Distance of three Hours, and prick it down in this Line from
B to I; then from this Point I take with your Compasses [...]e neerest distance to the Line of the Stile, and with that distance draw the Line
I▪ parallel to the Stile, and this will be the Line of the Stile inlarged.
[geometrical diagram]
Now to set on the lesser Polar Line, that so you may draw the Hour-lines, first take
from off your lesser Polar Line marked -, the distance of 3 Hours, and with that distance
draw a little white Line parallel to the Line of the Stile, and note where it cuts
the Line of the Stile inlarged, which is in the Point R. Mark this Point R well, and
through this Point R draw the Line OG parallel to the former Polar Line FBA. And now
if you take the distance ER in this Line, and measure it on your Scale, if you have
done your work right, you shall find it just equal to 3 Hours upon the lesser Polar
Line. And now by these two Polar Lines you must draw the Hour Lines after this manner.
First you must consider the Inclination of the Meridians, which in this Example is
31 deg. 1 min. which reduced into time, is 2 ho. 4 min. or 2 of the Clock and 4 min. in the afternoon, from whence you may frame a Table for your direction in placing
the Hours, after this manner.
Now according to this Table, take the distance of these Hour-lines first out of your
larger Polar Line ✚, and prick them down in the Line FBA, from the point B towards
F and A. Thus the Line of II must be set 4 min. from the Substile B towards F; the Line of I must be set 1 ho. 4 min. from the Substile B toward F; the Line of XII must be set 2 ho. 4 min. from B towards F. And so on the other side of the Substile, the Line of III must
be set 56 min. from the Point B toward A; the Line of IIII must be set 1 ho. 56 min. from B toward A. And so you must do till you have set down all the Hour-lines upon
the Line FBA, taking them out of the larger Polar Line upon your Scale.
Having thus set down the Hour-lines in the Line FBA, you must set them also in the
Line OEG, taking their Distances off from your lesser Polar Line -, as before you
did from the larger Polar Line, making use of the Table to direct you, as before.
And thus having made marks for the Hour-lines in both these Tangent-lines, you must
draw the Hour-lines through these Marks; and so shaping your Dial into a Triangle,
Square, or Circle, as your Plane will best allow, number the Hour-lines with their
proper Figures, and so finish your Dial, as the Scheme will direct you better than
many words.
CHAP. XXVIII. Another Example, How to Inlarge the Stile in a South Dial, reclining 45 deg. from the Zenith Northward.
[geometrical diagram]
THE Stile or Gnomon in this Example is very low, lying very neer to the Pole of the
World, as you may see before in the Fundamental Diagram. This Dial only reclines from
the Zenith; and therefore to know the Stiles Height, [Page 42] you need only substract the Reclination from the Latitude of the Pole; which being
51 deg. 30 min. the Reclination 45 deg. substracted out of it, there remains 6 deg. 30 min. for the Height of the Stile.
This Dial hath no Declination, and therefore the Substile must be the Meridian-line;
draw that Line therefore first about the middle of the Plane, and then with a Chord
of 60 deg. describe a short Arch RP, and from R prick off 6 deg. 30 min. according to the Height of the Stile before-found, and thereby draw the prickt Line
CP from the Center C, representing the lesser Stile of the Plane.
Now make choice of some fit place upon the Substiler Line, as B, and there cross the
Meridian at Right Angles with the Line FBA: Then take from off your larger Polar Scale
✚ the distance of 3 Hours, and prick it from B to q. Then take the neerest distance
from this Point q, to the Line of the Stile or Axis, and therewith draw a Line parallel
to the Stile, and this shall be the Line of the Stile inlarged. Then take the distance
of 3 Hours off from your lesser Scale marked -, and with that distance draw a blind
Line parallel to the Meridian or Substile, and mark where it crosseth the Line of
the Stile Inlarged, which is in the Point ♈: Then take the neerest distance from this
Point ♈, to the former Polar Line FBA, and so draw the Line OEG parallel thereunto,
through the Point ♈: This shall be your lesser Polar Line.
Now to set off the Hour-lines upon these two Lines, first take with your Compasses
the distance of one Hour off from your ✚ Polar Scale, and prick it both ways from
the Point B toward A and F: Likewise do the same for 2, 3, and 4 Hours, and prick
them down in the Line FBA.
Then take the distances of each Hour also off from the lesser Polar Scale -, and prick
them down from E on both sides the Meridian Line, in the Line OEG. Then draw through
these Points the Hour-lines by their several marks, as you may see in the Figure,
and put the numbers of the Hours thereunto; so your Dial will be finished, as in the
Figure. And if you well understand this, you may do the like in any other Dial which
shall need inlarging.
CHAP. XXIX. How to make a Vertical Dial upon the Cieling of a Floor within Doors, where the Direct
Beams of the Sun never come.
THe greatest part, and as much as you shall use of the Vertical or Horizontal Dial,
described Chap. 8. may by reflection be turned upside down, and placed upon a Cieling; but the Center
will be in the Air without doors.
The first thing to do is to fasten a piece of Looking-glass, as brood as a Groat or
Sixpence, set level; or a Gally-pot of Fair-water, which will set it self level being
placed upon the Sole of the Window, shall supply the use of the Nodus in the Gnomon; and the Beams of the Sun being reflected by the Glass or Water, shall
shew the Hours upon the Cieling.
Before you can draw a Figure for this Dial well, I would advise you first to find
the Meridian of the Room, which may be done thus.
Hang a Plumb line in the Window, directly over the Nodus or place of the Glass; for the shadow which the Plumb-line gives upon the Floor at
Noon, is the Meridian-line sought; and by a Ruler, or a Line stretched upon it, you
may prolong it as far as you shall need.
Then take the perpendicular Height thereof from the Glass to the Cieling of the Room,
which suppose it be 40 Inches, as DB, the Glass being fixed at D: Now from B draw
the Meridian-line upon the Cieling, which shall be represenred both ways continued,
as ABCK, and from D erect a Perpendicular to DC: Or let a stander by stop one end
of a Thred on the Glass at D; extend the same to the Meridian-line, moving the end
of the s [...]ring shorter and longer upon the Meridian, till [Page 43] another holding the Side of a Quadrant, shall find the Thred and Plummet to fall
directly upon the Complement of the Latitude, which in this Example is 38 deg. 30 min. and that is the Intersection of the Aequinoctial. Then raise the Perpendicular,
as DA; take a Chord of 60 deg. and from A sweep the Arch at P, and from Play down the Latitude 51 deg. 30 min. to q, and draw the Line AD.
[geometrical diagram]
Then let fall a Perpendicular from C, as CEFGHI, which is the Aequinoctial Line; and
so likewise draw a Parallel Line to the Meridian of 40 Inches at D. Now note, That
the Hours must be drawn all one as the Horizontal Dials are.
Then draw a Line Parallel to the Aequinoctial, as KO, at what distance you think convenient,
on the Cieling of the Room, which let be here 50 Inches, as CK, as you may measure
by the Scale. Now for the placing of the Hours on the Cieling of the Room, you must
measure how much by the Scale of Inches each Distance on the Aequinoctial Line is
from C to E, and from C to F, and from C to G, H, and I, and likewise on the Parallel
from K, collecting them into a Table; so will it be ready to transport on the Cieling.
So by this Table you shall find the Distance CE, which is from 12 to 1 in the afternoon,
or 11 in the morning, to be 17 Inches; which you may prick upon the Cieling. Likewise
KL on the Parallel, between 12, 11, and 1, will be found to be 27 Inches 7/10 parts
of an Inch in 10 parts; which Hour-line mark out upon the Cieling. Then draw a straight
Line through those two Points L and E; this Line continued shall be the first Hour
from the Meridian, which is 11 in the morning, or 1 in the afternoon: So do for all
the rest of the Hours.
Now by this you may know how far the Center is without the Window; measure it, and
you will find it 31 82/100 Inches from A to B, and from B to C 52 Inches, and from
C to K 50 Inches, as before. I hope now I have given the Practicioner content, in
making this so easie to be understood, although I may be condemned by others.
I will give you one Example more, to find the Meridian Line on the Cieling, which
is this. Fit a plain smooth Board, about a Foot square, to lie level from the Sole
of the Window inwards; then neer the outward edge thereof make a Center in the Board,
in the very place of Nodus, or a little under it: Then by Chap. 3. get the Meridian Line from the Glass on the Board; after you have drawn the Line
on the Board upon the Center, describe as much of a Circle as you may with the Semidiameter
of your Quadrant, which Circle shall be Horizon; then from the Meridian you may with
your Degrees on the Quadrant graduate your Horizon into Degrees of Azimuths both
ways as far as you can.
Next you may devise to make your Quadrant stand firm and upright upon one of his straight
Sides, which I will call his Foot for this time; and that you may thus do, take a
short space of a Ruler or Transom, and saw in one side of it a Notch perpendicularly,
in which Notch you may stick fast or wedge the heel of the toe of your Quadrant,
in such sort as his Foot may come close to the Board, and the other Triangular Side
or Leg may stand perpendicular upon it. Let the Foot be round, and with your Compasses
strike a Circle round it: when you have fitted the Diameter of the Foot on the Meridian
Line on the Board, draw a Circle round the Glass, that so you may set the edge of
the Circle according as you may have need, for to lay off the Suns Altitude at every
Hour. Now to find the Meridian on the Cieling, you may make a Table for the Suns Altitude
every Hour of the Day, in this manner as here is for the Latitude 51 deg. 30 min. and place the Foots Diameter directly on the Meridian of the Board, and elevate the
Quadrant to the Tropick of Capricorn, which in this Latitude is 15 deg.
A Table for the Altitude of the Sun in the beginning of each Sine, for all the Hours
of the Day, for the Latitude of 51 deg. 30 min.
Hours.
Cancer.
Gemini. Leo.
Taurus. Virgo.
Aries. Libra.
Pisces. Scorpio.
Aquari. Sagitta.
Capric.
12
62 0
58 45
50 0
38 30
27 0
18 18
15 0
11 1
59 43
56 34
48 12
36 58
25 40
17 6
13 52
10 2
53 45
50 55
43 12
32 37
21 51
13 38
10 30
9 3
45 42
43 6
36 0
26 7
15 58
8 12
5 15
8 4
36 41
34 13
27 31
18 8
8 33
1 15
7 5
27 17
24 56
18 18
9 17
0 6
6 6
18 11
15 40
9 0
5 7
9 32
6 50
11 37
4 8
1 32
21 40
Let a stander by stop on the Glass a Thred, and extend the other part straight on
to the Cieling, the Thred touching only the Plane of the Quadrant, and making no Angle
with it, but held parallel; and where the Thred thus extended touches the Cieling,
make a Point; then the Quadrant unmov'd, elevated to 62 deg. of Altitude, and extend the Line, and make another Point as before; and between those
two Points draw a straight Line, and that shall be your Meridian, and shall be long
[Page 45] enough for your use. Then elevate the Quadrant to 38 deg. 30 min. and hold the Thred to the Meridian on the Cieling, and where he touches mark; and
cross the Meridian at Right Angles with an Infinite Line, which shall be the Aequator:
So you may do as you did before. but if the Plane of the Cieling of the Wall is interrupted,
and made irregular by Beams, Wall-plates, Cornishes, Wainscot, or Chimney-piece,
and such like Bodies, I will shew you the Remedy to carry on your Hour-lines over
all.
Extend the Thred from any Hour-line to the Tropick of Cancer in the Cieling, as you were taught before, and fix it there; and extend another Thred
in like manner to the Tropick of Capricorn, where-ever it shall happen beyond the middle Beam, or quite beyond the Cieling upon
the Wall, and fix the Threds also. Then place your eye so behind these Threds, that
one of them may cover another; and at the same instant where the upper Line to your
sight or Imagination cuts the Cieling, Beam, Wall, or any irregular Body, about the
end of the lower Line, there shall the Hour-line pass from Tropick to Tropick: Direct
any By-stander to make Marks, as many as you shall need, and by those Marks draw the
Hour-lines according to your desire. This is in Mr. Palmer, pag. 202.
If the Arch of the Horizon, between the Tropicks, be within view of your Window,
you shall draw the same on the Wall to bound the Parallels. The Horizon Altitude is
nothing, and therefore it will be a level Line: and the Suns Azimuth when he riseth,
commonly called Ortine Latitude, is in Cancer 40 deg. East Northward, and in Capricorn as much Southward; and these will be reflected to the contrary Coasts on the Dial.
CHAP. XXX. How to make an Ʋniversal Dial on a Globe; and to cover it, if it be required.
A Globe, saith Euclid, is made by the turning about of a Semicircle, keeping the Diameter fixed. This Dial, if Universal, will want the aid of a Magnetical Needle to set it, and
it must move on an Axis in an Horizon, as the usual Globes do; whose Aequator let be divided into 24 Hours,
the proportion of the Day Natural.
You may see the Figure on the top of the Dial in the Title, but that you cannot see
the two Poles, and the Semicircle, and the Horizontal Circle.
You may imagine this Globe set to the Elevation of the Pole, as that is, with two
Gnomons of the length of the Suns greatest Declination, proportioned to the Poles
Circle, with the 24 Hours, according to the 24 Meridians, and serves for a North and
South Polar Dial.
But in the Meridian let be placed the 12 a Clock Line; then turn the Semicircle till
it cast no shadow: then doth it cross the Hours, which Hours are drawn from the Pole
to each of the 24 Divisions, as before.
If you desire to cover the Globe, and make other Inventions thereon, first learn here
to cover it exactly. With a Pair of Compasses bowed towards the Points (like a Pair
of Calapers the Gunners use) measure the Diameter of the Globe you intend to cover;
which being known, find the Circumference thus.
Multiply the Diameter by 22, and divide the Product by 7, and you have your desire.
Let the Circumference found be the Line EF, which divide into 12 Equal parts; draw
the Parallel AB and CD, at the distance of three of those Parts from E to A and from
F to C; then by the outward Bulks of those Arches draw the Line AB and CD.
And to divide the Circumference into 12 parts, as our Example is, work thus.
[geometrical diagram]
Set your Compasses in E, and make the Arch FC: The Compasses so opened set again
in F, and make Arch E A; then draw the Line from A to F, and from E to C. Then your
Compasses opened at any distance, prick down one part less on both those slanting
Lines, than you intend to divide thereon; which is here 11, because we would divide
the Line EF into 12: Then draw Lines from each Division to his opposite, that cuts
the Line EF in the parts or Division. —But to proceed, It is Mr. Morgan's Conceit, page 116. Continue the Circumference at length to G and H, numbring from E towards G 12
of those Equal Parts, and from F towards H as many, which shall be the Center for
each Arch; so those Quarters so cut out, shall exactly cover the Globe, whose Circumference
is equal to EF.
Thus have you a glance of the Mathematicks, striking at one thing through the side
of another: For here one Figure is made for several Operations, to save the Press
the charge of Figures.
CHAP. XXXI. How to make a Direct North Dial for the Cape of Good Hope, in South Latitude 35 d. and Longitude 57 d. to the Eastward of Flores and Corvo.
THis Dial is made all one as the South Dial you may see Chap. 9. Only observe this, That you are 35 deg. to the Southward of the Aequinoctial, and that the Sun, when he is on the Tropick
of Capricorn, wants 11 ½ Degrees of the Zenith of that Place Northward: As the Sun goes always
to the Southward of us in England, so it goes to the Northward of them; therefore must the Stile or Gnomon point downward
in the North Face, and upward in the South Face. So likewise as in our South Dial
the afternoon Hours are put on the East side the Dial Plane, and [Page 47] the morning Hours on the West side; so in their North Dials they will stand contrarily,
by reason the Sun casts a Shadow (as the Plane must stand there) in the morning to
the West side, and in the afternoon to the East: So you see the Plane is only turned
to face the Sun. If you do but conceive in your mind how the Sun
[geometrical diagram]
casts his shadow, you may as easily make all sorts of Dials on the South side of
the Aequinoctial, as on the North side: but that the People there have neither Order,
Policy, Religion, nor Understanding in Mathematical Arts or Sciences. The Africans at the Cape of Good Hope are of a swarthy dark colour, and made black by daubing themselves with Grease and
Charcoal; they are so wedded to superstition, that some adore the Devil in the form
of a bloody Dragon, others a Ram, a Goat, a Leopard, a Bat, an Owl, a Snake, or Dog,
to whom they ceremoniously kneel and bow. So much for Aethiopia, and for Dials for them; only you see the manner, and that the former Rules serve
for any Latitude.
CHAP. XXXII. How to find the Time of the Night by the Moon shining upon a Sun Dial.
HAving the Age of the Moon by the Epact; as for Example, the 9th. day of August 1665. the Epact 23, to which add 9 the day of the Month, and 6 the Months from March, makes 38; from it substract 29 a whole Moon, with 12 ho. 44 min. which multiply the remaining 9 by 4, makes 36; that divide by 5, and you have 7 ho. and 12 min. for the odd Unite, and 24 min. for the half day, or 12 hours added together, makes 7 ho. 36 min. for the Moon being South.
Now having the Moons coming to the South by the former way, add this Southing of the
Moon and the Shadow of the Moon upon the Dial together, and that is the time of the
Night. If the Sum exceed 12 Hours, take only the overplus.
Moon 9 da. old. So. at 7 h. 30 m. ☉ Di. 301030 at Night.Or thus you may do: If the Moon be 9 days old at Noon, she will be 9 days and an half
at night; therefore you may add about a quarter or half an hour thereunto, as it is
more early or late in the night, and add the Southing of the Moon, which makes 7 ho. 30 min. added to the Shadow of the Moon 3 ho. upon the Sun-dial, it makes 10 ho. 30 min. for the time of the night: So you see there is 6 min. difference betwixt these two ways, which cannot well be estimated; but either way
will give neer enough satisfaction for the time of the Night.
CHAP. XXXIII. How to find the Hour of the Day or Night by a Gold Ring and a Silver Drinking Bowl,
or Glass, or Brass, or Iron, or Tin Vessel.
HAving a Gold Ring and a Silver Drinking Bowl, take a small Thred or Silk and measure
the compass of the top of the Silver Bowl, Glass, or other Vessel, which will be
a convenient length for your use: Then put this Thred through the Ring, and tie the
ends thereof together, taking up as little as you can with the knots. Put this Thred
over your Thumb, where you feel the Pulse beat, upon the lower Joynt it may be; then
stretch our your hand, and hold it so that the inside of your Thumb may be upward;
and hold your Hand so over the Bowl, that the Ring may hang as neer the midst of the
Bowl as you can guess: and you shall see that the beating of your Pulse (holding your
Hand a while as still as you can) will give a motion to the Ring, causing it to swing
cross the Bowl by Degrees more and more, till at last it will beat against the Sides
thereof.
Now mark when it begins to strike, and tell the strokes as you would do a Clock; for
it will strike what Hour of the Day or Night it is, and then leave off striking, and
swinging also by degrees: Which hath been approved of by the experience and judgment
of many.
A Good Observation.
WE may take notice, That there is no Dial can shew the exact time, without the allowance
of the Suns Semidiameter, which in a strict acceptation is true. But hereto Mr. Wells hath answered in page 85. of his Art of Shadows, where saith he, Because the Shadow of the Center is hindered by the Stile, the Shadow of the Hour-line
proceeds from the Limb, which always precedeth the Center one minute of time, answerable
to 15 minutes the Semidiameter of the Sun: which to allow in the Height of the Stile were
erroneous; but there may be allowance in the Hour-line, detracting from the true Aequinoctial
Distance of every Hour or 15 degrees, 15 minutes. But I will go no further with this Subject, to put the Learners in doubt of the true
Hour; for this is as neer a way which I have shewed you, as any projected upon Dial
Planes. You may see a Geometrical Figure of it in my Problems of the Sphere.
CHAP. XXXIV. How to Paint the Dials which you make.
ALthough I never saw any man Make a Dial, nor paint one, but what I made, painted,
and guilded my self; as you may see the Piece in the Title, on which I made 26 Dials,
and put in the Brass Gnomons or Stiles into the Freestone with Lead, and guilded them
with all the Figures, and on the Globe drew the Aequinoctial Circle, the Ecliptick,
the Tropicks, and Polar Circles, and the 24 Meridians, with such Constellations in
the North and South Hemisphere, and Stars, in such Colours as was fit to set out
the Dial, with Pole Dials, and Globe Dials, Chap. 32.
To Paint and Finish the Dials, ready to be set up in their Places.
FOr to fasten the Gnomon to the Plane, be it of Wood or Freestone, you must have a
small thin Chisel, or Googe, or Gimblet, as is fit for the Stile, be it round as a
Rod of Iron, or a piece of Brass, let in with a Foot an inch and half, or more or
less, as you will; and in the Wood make such little Mortises as just the breadth and
length of the Foot of the Stile; and if it comes thorow to clinch it on the other
side, then it is fast.
If it is in Freestone, your Dial drawn first in Paper, lay it upon the Plane [...]s it should be; then cut out the Substile-line as neer its breadth as you can, an [...] [...]ly leave so much as will just hold it together. The Paper laid as before on the Plane,
with a Black-lead Pensil, or such like, draw the Substile Line where it stood in the
Paper, and with a small Chisel make such Mortises in that Line as are answerable to
the Foot of the Stile; and crook his Foot, and put it into its place, with a small
Ladle and some Lead melted, put the Stile perpendicular with the Plane, and pour in
the Lead into the Mortise until it is full; and when it is cold, then with a blunt
Chisel harden the Lead in one Inch side of the Stile or Gnomon: And if the Mortise
should be too wide, or broken, and not even enough with the Plane, then wet some Flower
of Alabaster, as you may have it fit for that purpose at any Masons, and as soon as
'tis wet make a Plaster, and so smooth it, and spread it even and plain with the Plane;
it is presently dry. Now have you the Stile or Gnomon as fast as if it grew there.
To Paint them, you must first Prime them: The Prime is made thus. Take an equal quantity
of Bole Armoniack and Red Lead, well ground together with Linseed Oyl, and well rubb'd
in with a Brush or Pensil into the Plane; that being dry, for the outside Colour,
it is White Lead or Ceruse well ground together with Linseed Oyl. How to know the
best. Buy the White Lead, and grind it to a Powder, and put it into Water until it
become as thick as Pap, and let it dry; then it is for your use.
For the Hour Lines a Vermillion, and a part Red Lead, well ground together with Linseed
Oyl, with a small quantity of Oyl of Spike, or Turpentine that will dure, and make
the Lines shine.
For a Gold Border, Rub the Border well with the white Ceruse Paint; be sure it be
very thick in the Border: Then with Blew Smalts strew very thick the Border while
it is wet; and when it is dry, wing that which is loose off, and save it in a Paper;
and for the rest that clings, it is fast enough.
Take Red Lead and White Lead, and as much Red Lead again as White, or Yellow Oker,
well ground with Oyle of Spike or Turpentine; this is the Sise: Then draw with that
the Figure you would have in Gold, and when it is so dry that it will not come off
on your Fingers by a slight touch, lay on the Gold; and when it is thorowly dry, wing
it off.
How to make a good Black, to shadow or make Figures. Grind well with Linseed Oyl Lam-black,
with some Verdigrease, and that is a firm Black. The like you may do with all other
Colours, as you fancy for such Work.
FIrst, Steep one penny-worth of Brazeel Wood all night in Piss or Urin; then boil
it well and strain it; then bruise two penny-worth of Cochineel, and boil it, and
put in it the bigness of a Hens Egg of Roch Allom, that brings it to a colour, and
then it is for your turn.
To Paint Freestone, wash the Stone with Oil, and it will last; and then all the Colours
before may be used, as directed.
How to cleanse a Picture.
TAke blew Smalts, temper it in Water, and rub the Picture with it, and after wipe
it with a Linnen Cloth, which Cloth should be dipp'd in Beer, or otherwise with a
dry Cloth, and it is clean.
To cleanse a Gold Border.
WAsh it with Beer, and dry it, and then cleanse it with Linseed Oyl.
Masticous is a fine Yellow, ground with some Oyl of Spike or Turpentine.
Bice is a good Blew Colouring, to be ground with Linseed Oyl and Red Lead.
And Spanish Brown will make a lasting Colour for Course Work.
To grind Gold to Write and Paint.
TAke as many Leaves of Gold as you please, Honey three or four drops; mix and grind
these, and keep it in some Bone Vessel. If you will write with it, add some Gum-water,
and it will be Excellent.
Some Ʋses of the following Tables of Logarithmes, Sines, and Tangents.
AMongst the many admirable ways that have been from time to time invented for propagating
the Arts Mathematical, and especially that of Trigonometry, Logarithmes, invented by the Right Honourable the Lord Napier, Baron of Marcheston, may challenge the priority, and the Tables of Artificial Sines and Tangents, composed by Mr. Edmund Gunter Professor of Astronomy in Gresham-College London; for that they expedite the Arithmetical Work in most Questions; Multiplication being
performed by Addition, and Division by Substraction, the Square Root extracted by
Bipartition, and the Cubique Root by Tripartition: So that by help of these Numbers,
and the aforesaid Sines and Tangents, more may be performed in the space of an Hour, than by Natural Numbers or by Vulgar
Arithmetick can be in six. Now of what frequent use the Doctrine of Triangles, both
Plain and Spherical, is in Astronomy (for the Resolution of which the Tables following
chiefly serve) let the precedent Work testifie. And as Mr. Newton in his Mathematical Institutions, or in the same form as Mr. Vincent Wing in his Harmonicon Coeleste, are these Tables following: And therefore I think it not amiss here in this place
to insert some few Propositions, to shew the Use of the Canon and Tables of Sines and Tangents following.
PROBL. I. How to find the Logarithmes of any Number under 1000.
EVery Page in the Table of Logarithmes is divided into 11 Columns; the first of which Columns, having the Letter N at the
Head thereof, are all Numbers successively continued from 1 to 1000: So that to find
the Logarithmes of any Number, [Page 51] is no more but to find the Number in the first Column, and in the second Column
you shall have the Logerithme answering thereunto.
Example. Let the Number given be 415, and if it is required to find the Logarithme thereof, in the Table of Logorithmes, in the first Column thereof, under the Letter N, I find the Number 415, and right
against it in the next Column I find 618048, which is the Logarithme of 415. In the same manner you may find the Logarithme under 1000; as the Logarithme of 506 is 704151, and the Logarithme of 900 is 954243, &c.
But here is to be noted, That before every Logarithme must be placed his proper Characteristick; viz. If the Number consist but of one Figure, as all Numbers under 10, then the Characteristick
is 0; if the Number consist of two Figures, as all Numbers between 10 and 100, then
the Characteristick is 1; if the Number consist of three Figures, as all Numbers betwixt
100 and 1000, then the Characteristick is 2; and if the Number consist of four Figures,
as all between 1000 and 10000, the Characteristick must be 3. In brief, the Characteristick
of any Logarithme must consist of an Unit less than the given Number consisteth of Di [...]its or Places: And by observing this Rule, the Logarithme of 415 will be 2.618048, and the Logarithme of 506 is 2.704151, and the Logarithme of 900 is 2.954243, &c.
PROBL. II. A Logarithme being given, to find the Absolute Number thereunto belonging, by the
former Observation; the Characteristick will declare of what Number of Places the
Absolute Number consisteth.
Example. Let the Logarithme given be 2.164353; now because the Characteristick is 2, I know by it the Absolute
Number consisteth of three places, and therefore may be found in the second Column
of the Logarithme Tables, having 0 at the top thereof, against which I find 146, which is the Absolute
Number answering to the Logarithme of 2.164353.
PROBL. III. How to find the Logarithme of a Number that consisteth of four Places.
You must find the three first Figures of the given Number in the first Column, as
before, and seek the last Figure thereof amongst the great Figures in the head of
the Page; and in the common Area or meeting of these two Lines is the Logarithme you desire, if before it you add or prefix its proper Characteristick.
Example. Let it be required to find the Logarithme of 5745; I find 574, the three first Figures, in the first Column, and 5, the last
Figure, in the head of the Table; then going down from 5 in the head of the Table,
until I come against 574 in the first Column, there I find 759290, before which I
place 3 for the Characteristick, which is 3.759290, and that is the Logarithme sought for.
PROBL. IV. Any Number of Degrees and Minutes being given, to find the Artificial Sine and Tangent
thereof.
Admit it were required to find the Sine of 21 deg. 24 min. I turn to the Sines in the Table, and in the head thereof I find Degrees 21; then
in the first Column (under M) I find 24, and right against it is 9.562146 for the
Sine, and 9.593170 for the Tangent of 21 deg. 24 min. But suppose it were required to find the Sine or Tangent of 56 deg. 35 min. look for all else under 45 deg. are found in the head, and the odd min. in the left hand; and all above 45 deg. are found in the foot of the Table, and the min. in the last Column toward the right hand; as in this Example the Sine of 56 deg. 35 min. is 9.921524, and the Tangent is 16.180590.
PROBL. V. If any Sine or Tangent be given, to find what Degrees and Minutes answer thereunto.
Suppose 9.584663 were a Sine given, I look for the Number in the Table of Sines, and
I find it stand against 22 d. 36 m. and therefore is the Sine thereof. As admit 9.624330 were a Tangent given, look for
the Number in the Column of Tangents, and I find stand against it 22 d. 50 m. The same must be done for Sines and Tangents in the foot of the Tables.
Ovid. lib. Met.
Cuncta fluunt omnisque vagans formatur Imago.
Ipsa quoque assiduo labuntur tempora motu.
All things pass on: Those Creatures which are made
CANON TRIANGULORUM LOGARITHMICUS. OR, A TABLE of ARTIFICIAL SINES and TANGENTS to
every Degree and Minute of the QƲADRANT. The Common Radius being 10,000000. By Capt. SAMUEL STURMY.
The Description of the TRIANGLE.
Let ZPS represent the Zenith, Pole, and Sun, ZP being 38 deg. 30 min. Complement of the Latitude, PS the Complement of the Suns Declination 70 deg. and the Complement of the Suns Altitude ZS 40 deg. 00 min. the Angle at Z shall shew the Azimuth, and the Angle at P the Hour of the Day from the Meridian: Then if from Z to PS we let down a Perpendicular, as ZR, we shall reduce the Oblique Triangle into two Rectangled Triangles ZRP and ZRS: If from S to ZP we let down a Perpendicular SM, we shall reduce the same ZPS into two other Triangles, as SMZ and SMP Rectangled at M. Whatsoever is said of any of these Triangles, the same holdeth for all other Triangles
in the like Cases.
CHILIADES DECEM LOGARITHMORUM, SHEWING The LOGARITHMES of all NUMBERS Increasing by
Natural Succession from an Unite to 10000: Whereby the Logarithmes of all Numbers under 1000000 may be speedily deduced. First Calculated by that Excellent Mathematician
Mr. HENRY BRIGGS, Professor of Geometry in the University of Oxford. And their Use now Amplified, By Capt. SAMUEL STURMY.
A TABLE of PROPORTIONAL PARTS, Whereby the Intermediate Logarithmes of all Numbers, and the Numbers of all Logarithmes, from 10000 to 100000, may more readily be found out by the foregoing Table of Logarithmes.
The Ʋse of the Table of Proportion, for the more ready finding out of any Logarithme, from 10000 to 100000.
WHen you have any Logarithme or Number above 10000, you may find it out as before, by the Differences which are
in the last Column of the Tables: But for your more easie and ready performing it,
this Table is of great use; wherein you have all those Differences ready divided,
and cast into 10 parts: So that between each of the 10000 Logarithmes in the Table, you may easily know the ten Intermediate Logarithmes, by the Proportional Part of the Difference for any of them.
Thus in the Table the Logarithme of 2000 is
3.301029
The next Logarithme, being the Logarithme of 2001, is
3.301247
Alter the Characteristicks of these Logarithmes,
So have you the Logarithme of 20000,
4.301029
And the Logarithme of 20010,
4.301247
The Difference between these two Numbers is 218; which for the ten Intermediate Logarithmes must be divided into 10 Equal Parts, which is ready done in the Table of Proportion,
after this manner.
D
1
2
3
4
5
6
7
8
9
318
31
63
95
127
159
190
222
254
286
So that the Logarithme of 20000 being
4.301029
The Logarithme of 20001, by adding 31, is
4.301060
The Logarithme of 20002, by adding 63, is
4.301092
And so for the rest, to 20010.
Or, on the other side, Let your Logarithme given be 4.301251, and you desire to know what Number answers to it; the next Number
less in the Tables is 301029, which is the Logarithme of 20000: but this is 222 more, and the Common Difference in the Table is about
218; turn therefore to this Difference in the Table of Proportion, and there you shall
see that 222 makes your Number 7 more: So that 4.301251 is the Logarithme of 20007.
And thus you save the Labour of multiplying and dividing the Differences in the Table
of Logarithmes, they being here ready done to your hand.
A SUMMARY OF SUCH PENALTIES and FORFEITURES As are Limited and Appointed by several
ACTS OF PARLIAMENT Relating to the Customs & Navigation. AS ALSO, For the EXPORTING
and IMPORTING of PROHIBITED GOODS. TOGETHER WITH The several STATUTES whereupon they
are Grounded: Being duly Compared with the Statutes at Large, and the Abridgment to
this present Year, 1664. USEFUL FOR Merchants, Factors, for all Officers belonging
to the Customs, Masters of Ships, Pursers, and Boatswains, Mariners, Wharfingers,
Lightermen, and Watermen. In what case both Ship and Goods are Forfeited upon Importation.
TO ALL Merchants and Factors, and Commanders or Masters of Ships; AND To all other
Officers and Mariners: And to all other Honest-minded Men whom this may Concern. SAMUEL STURMY Wisheth Prosperity, Courage, and Wisdom in all Your Lawful Undertakings.
GENTLEMEN,
FOR Your sakes I have inserted this following Abridgment or Summary of the Laws, and
Penalties, and Forfeitures, as are limited and appointed by several Acts of Parliament,
relating to the Customs and Navigation; and in what Cases both Ships and Goods are
forfeited, upon Exporting or Importation of Prohibited Goods; together with the several
Statutes whereupon they are grounded, being duly compared with the Statutes at large,
and the Abridgment to 1664.
I have been provoked to annex this Summary of Custom-House Laws, the more out of a
Principle of good will I have to you, and by knowing some of your defects, or want
of knowledge in these [Page] things, by my own in times past, when I was a Commander my self. Through ignorance
of these Laws your Goods have been seised and lost, and Ships stopp'd and hindred
in their Voyages, to the great loss and damage of the Merchants, and Owners, and Mariners:
Whereas if all concerned had but the knowledge of what they should know, they might
prevent this loss and damage, and walk safely, without any detriment to themselves
or Goods, by the Officers; whereas otherwise, without this knowledge, your Ignorance
is the Officers Advantage, and he will make you pay for it. And as I would advise
every man to follow our Saviour's Counsel Matthew 22.21. Render therefore unto Caesar the things that are Caesar's, and unto
God the things which are God's: as likewise Rom. 13.6, 7. Render therefore to all their dues, Tribute to whom Tribute is due,
Custom to whom Custom, Fear to whom Fear, Honour to whom Honour: So likewise I do advise all Officers to go discreetly on in their Business with all
men, and not hinder, lett, or abuse, in word or action, any one with whom they have
Business, without just cause; nor those that fall into their Hands; nor to take the
just rigour of the Law of England, lest the Ʋniversal God should take the Law of Heaven upon us for our Errors and
Failings: but take reasonable Satisfaction. This is my advice to both Merchants and
Officers, and all others concerned. I am their Well-wisher, and so remain to be,
A SUMMARY OF SUCH PENALTIES and FORFEITURES As are Limited and Appointed by Several
ACTS of PARLIAMENT Relating to the CUSTOMS and NAVIGATION.
First,ALL manner of Goods Imported into his Majesties Plantations, or Exported out of his
Majesties Plantations, in Forreign Shipping, both Ship and Goods are forfeited. Vide Statute of Navigation, 12 Caroli 2.18.
Secondly, All Goods that are of the growth of Asia, Africa, and America, Imported in Forreign Shipping, are forfeit, per id. Stat.
Thirdly, All Goods of the growth, production, and manufacture of Asia, Africa, and America, shall be Imported from the place of their growth, production, or manufacture; otherwise
both Ship and Goods are forfeited, per id. Stat.
Except the Goods of the Spanish Plantations may be brought from Spain, and the Goods of the Portugal Plantations may be brought from Portugal, and East India Commodities may be brought from any Port on the Southward or Eastward of Cape Bona Speranza, and the Commodities of the Levant Seas may be brought from any Port within the Straights; Provided that all these Goods may be Imported in English Shipping, otherwise both Ship and Goods are forfeited, per id. Stat.
Fourthly, All Goods of Forreign growth, production, or manufacture, shall be Imported from
the place of their growth, production, or manufacture,In this Case a Dutch Ship called the Grass Mower, laden with Salt and Brandy, was seised by me, for bringing Goods from Rochel to a Bristol-Merchant; and the Ship and Goods was bought again for 220 l. by a Letter of Licence. or from such place where they are usually first Shipp'd for Transportation only,
and only in English Ships, or in Ships truly belonging to such place where such Goods are lawful to be
Shipped; otherwise both Ship and Goods are forfeited, per id. Stat.
Fifthly, All Goods carried from Port to Port (in England, Ireland, Wales, or Berwick) in Forreign Shipping, whereof the Owners or Part-owners are not all English, as also the Master and three fourths of the Mariners, both Ship and Goods are forfeited,
per id. Stat.
Sixthly, All Goods of the growth, production, or manufacture of any of his Majesties Plantations,
shall be first landed in England, Ireland, Wales, or Berwick, before they can be transported; otherwise both Ship and Goods are forfeited, per id. Stat.
Seventhly, All manner of Wines, except Rhenish; all Spicery and Grocery, Tobacco, Pot-ashes, Pitch, Tar, Rosin, Salt, Deal Boards,
Fir Timber, or Olive Oyl, that shall be imported from the Netherlands or Germany, are forfeited, as also the Ship in which they are Imported. Vide Stat. 14 Car. 2.1 [...]. intituled, An Act to prevent Fraud [...], &c. in his Majesties Customs.
Eighthly, All Fresh Herring, Fresh Cod or Haddock, Cole-fish or Gul-fish, that shall be Imported
into England or Wales in Forreign Shipping, both Ship and Goods are forfeited. Vide Stat. 15 Car. 2. 5. intituled, An Act for Encouraging of Trade.
Goods forfeited for being Imported into England or Wales, without any Penalty upon the Ship; these Goods being all English Manufactures.
ALL manner of Tine and PewterTin and Pewter. Manufactures made in Forreign Parts are forfeited. Vide Stat. 25 Hen. 8. 14.
Officers may search and seise Wares brought into the Realm contrary to the said Act,
and none shall withstand the search of Brass,Brass. Tin, and Pewter, on the forfeit of five pounds, per id. Stat.
Several Commodities forfeited.Woolen Clothes, Woolen Caps, Ribbons, Fringes of Silk and of Thred, Laces of Silk
and of Thred, Silk Twine, Embroidered Laces of Silk or Gold, Saddles, Stirrops, or
any Harness belonging to Saddles, Spurs, Bosses for Bridles, Andirons, Gridirons,
any manner of Locks, Hammers, Pincers, Fire-tongs, Dripping-pans, Dice, Tennis-balls,
Points, Purses, Girdles, Gloves, Harness for Girdles, Iron, Latten, Steel, Tin, or
Alchymy, or any Wrought or any Tawed Leather, any Tawed Furs, Biskin Shoos, Galloshes,
or Cork, Knives, Daggers, Wood-knives, Bodkins, Sheers for Taylors, Scissors, Razors,
Chess-men, Playing Cards, Combs, Pattens, Pack-needles, any Painted Wares, Forsers,
Caskets, Rings of Copper or of Latten, guilt Chafingdishes, Hanging Candlesticks,
Caffing Balls, Sacring Bells, Rings for Curtains, Ladles, Scummers, counterfeit Basons,
Ewers, Hats, and Brushes, Cards for Wooll, black Iron, Thred called Iron Wyre, or
whited Wyre, are forfeited if any such be Imported into England or Wales. Vide Stat. 4 Edw. 4.
Prohibited Goods forfeited.All Iron Wyre, Card-wyre, or Wool-cards, that shall be Imported into England or Wales, are forfeited. per Stat. 39 Eliz. 14. 14 Car. 2. 19.
Prohibited more forfeited.All manner of Girdles, Harness for Girdles, Points, Leather, Laces, Purses, Pouches,
Pins, Gloves, Knives, Hangers, Taylors Sheers, Scissors, Andirons, Cobbards, Tongs,
Fire-locks, Gridirons, Stock-locks, Keys, Hinges and Garnets, Spurs, painted Glasses,
painted Papers, painted Forcers, painted Images, painted Clothes, beaten Gold or Silver
wrought in Papers for Painters, Saddles, Saddle-trees, Horse-Harness, Boots, Bits,
Stirrops, Chains, Buckles, Latten Nails with Iron Shanks, Curvets, Hanging-candlesticks,
Holy-water, Stops, Chafing-dishes, Hanging Lavers, Curtain-rings, Cards for Wool,
Roan Cards, Sheers, Buckles for Shoos, Broaches for Spits, Belts, Hawk-bells, Tin and
Leaden Spoons, Wyre of Latten and Iron, Candlesticks, Grates, Horns for Lanthorns,
or any of these, being Imported into England, are forfeited, or the value thereof, betwixt the King and the Prosecutor. These may
be sued for in any Corporation where they are. Vide Stat. 1 R. 3. 12.
All Girdles, Harness for Girdles, Rapiers, Daggers, Knives, Hilts, Pummels, Lockets,
Blades, Handles, Scabbards, Sheaths for Knives, Saddles, Horse-Harness, [Page 3] Stirrops, Bits, Gloves, or Points, Leather Laces, or Pins,Goods prohibited. that shall be Imported into England or Wales, shall be forfeit. 5 Eliz. 7.
All manner of Silk wrought by it self, or with any other Stuff, in any place out of
the Realm, Ribbons, Laces, Girdles, Corses called Corses of Tissue, or Points, shall
be forfeited, per Stat. 19 Hen. 7. 21.
All Forreign Bone-lace, Cut-work, Fringe, Embroidery, Bandstrings, Buttons,Goods prohibited 100 l. or Needle work, made of Silk or Thred, or either of them, being Imported into England, Wales, or Berwick, shall be forfeited, besides the Forfeiture of one hundred pound. 14 Car. 2. 13.
All manner of Woollen Cloth that shall be Imported into England, Ireland, or Wales, from beyond the Sea, shall be forfeited. Vide Stat. 2 Edw. 3. 3. & 4 Ed. 4. 1.
In what Cases Goods are forfeited for Ʋndue Shipping or Landing.
ALL Goods that shall be Shipped or Landed before the Custom paid or agreed for in
the Custom-house, are forfeited.Goods shipp'd or landed before Custom paid are forfeit.Tide Stat. 12 Car. 2.4. intituled, The Act for the Tonnage and Poundage.
All Goods that shall be Shipped or Landed,Unlawful time or place lose. or put into any other Vessel to be Shipped or Landed, at any unlawful time or place,
are forfeit, or the value of them. 1 Eliz. 2. & 14 Car. 2.11.
All Goods that shall be put into any Lighter, Boat, or any other Vessel, to be Shipped
or Landed, without Warrant from the Custom-house, and the presence of one or more
Custom-house Officer, are forfeited, as also the Lighter or other Vessel in which
they are found to be Shipped or Landed. Vide Stat. 14 Car. 2. 11.
If any Master of Ship, Purser, Boatswain, or other Mariner,Persons consenting to the discharge of Goods inward bound without Warrant, the
penalty.Officers concealing Custom, the penalty.The penalty to ship out less than is expressed. knowing or consenting to the discharge of Goods inward bound, without Warrant from
the Custom-house, or the presence of one or more Custom-house Officer, shall forfeit
the value of the said Goods so unshipped. Vide Stat. 14 Car. 2. 11.
Every Customer, Collector, and Comptroller, that doth conceal his Majesties Customs;
being duly Entred, shall forfeit treble the value thereof, per Stat. 3 H. 6. 3.
If any Goods having paid Custom at the Importation, and ought to have allowance at
the Exportation; if the Merchant Ship out less in quantity than is expressed in his
Certificate, shall be forfeited, or the value of them. 14 Car. 2. 11.
If the said Goods be Landed again in England, Wales, or Berwick,Landing Goods, unless made known in Custom-house, forfeited.Goods carried from Port to Port without Warrant forfeited. Quantity and quality must
be expressed. except they be made known in the Custom-house, shall be forfeit, per id. Stat.
If any Goods be put on board a Ship to be carried from Port to Port, without Warrant
from the Custom-house, all such Goods shall be forfeit, per id. Stat.
If the true content of Quantity and Quality be not mentioned in the Certificate, under
the Customers Hand in the Port where they are Shipped first to pass for another Port,
all such Goods not certified or discharged before the said Certificate delivered,
and the Goods viewed, shall be forfeit, per Stat. 3 H. 7. 7.
Goods exported, and not discovered unto the Officer, forfeit double the value.England and Scotland, Goods that pass.All manner of Goods, Wares, or Merchandize, that shall be Exported, and escape undiscovered
unto the Officers of the Customs, the Owner or Proprietor shall forfeit double the
Value, according to the Book of Rates; Except for Coals, for which they shall forfeit double the Custom. Vide Stat. 14 Car. 2.
All Goods, Wares, and Merchandize, that shall pass by Land betwixt England and Scotland, shall pass by and through Berwick and Carlisle, and pay Custom at one of those Ports, otherwise be forfeited, per Stat. 14 Car. 2. 11.
In what Cases Ship and Goods are forfeited upon Exportation of Goods.
Licence to be granted for Passengers to pass beyond the Sea.IF any Woman, or other person under the Age of twenty one years, except Ship-boys,
Saylors, or Merchants, Apprentices, or Factors, shall pass over the Sea, without Licence
from the King, or 6 of the Privy Council, the Ship in which such Person shall so pass
shall be forfeit. Vide Stat. 1 Jac. 4.
If any Person shall Transport, or Ship to be Transported, Leather, Tallow,Leather, Tallow, &c. or Raw Hides, to any Place beyond the Sea, all such Goods shall be forfeit, as also
the Ship wherein they are Exported. Vide Stat. 18 Eliz. 9.
If any Hoy or PlatHoys or Plats, &c. cross the Seas beyond Norway Eastward, or Caen in Normandy Southward, they shall be forfeit. Vide Stat. 1 Eliz. 13. 5 Eliz. 5. 13 Eliz. 15.
If any Corn, or other Victual,Corn or Victual. be Transported, exceeding the Prices mentioned in the Act for Encouraging of Trade; or if any Wood shall be Trasported, they shall forfeit the Vessel in which it shall
be Exported, and also double the value of the Goods. Vide Stat. 1, 2 Phil. & Mar. 5. the Masters and Mariners all their Goods, and a years Imprisonment.
English Manufactures to be exported in English Shipping.If any Goods of the growth, production, or manufacture of Europe be transported into his Majesties Plantations, except from England, and in English built Shipping, both Ship and Goods are forfeited. Vide Stat. 15 Car. 2. 5.
If the Master shall suffer any Goods to be Landed before a due Entry made within
twenty four HoursDue entry to be made in 24 ho. after arrival in the said Plantations, both Ship and Goods are forfeited, per id. Stat.
If any Ship shall set out to Fishing,Ships for fishing, the time to set out. or other Vessel shall set out for the West Country or Iseland Fishing, before the tenth day of March in any Year, such Vessel shall be forfeit. Vide Stat. 15 Car. 2. 14. intituled, An Act for the Fishing Trade.
If any Sheep or Wooll,Sheep, Wool, &c. Wooll-fells, Wooll-flocks, Mortlings, Shorlings, Yarn made of Wooll, Fullers Earth,
Fulling Clay, shall be Exported, all such Goods are forfeit, as also the Ship wherein
they are Exported. Vide Stat. 12 Car. 2.32.
If any Silver or GoldSilver & Gold, &c. be Exported without Licence, it shall be forfeited. Vide Stat. 5 R. 2. 2. & 9 Edw. 3. 1. & 2 Hen. 4. 5. & 2 Hen. 6. 6.
None but Merchant Strangers shall Transport Wooll, Wooll-fells, Leather, and Lead
beyond the Seas, upon the Forfeiture of the said Goods. Vide Stat. 27 Ed. 3. 3. 14 Rich. 2. 5.
If any Skin, tann'd or untann'd,Skins tann'd or untann'd. of any Ox, Steer, Bull, Cow, or Calf (except Calve-skins of four pound weight apiece,
or under) and Sheeps Skins dressed without the Wooll of such Skins or Hides, which
are for the Ships necessary Provision, shall pass out of England beyond the Seas, or into Ireland or Scotland, or the Islands belonging to England, shall be forfeited. Vide Stat. 14 Car. 2. 7.
If any of the Hides or Skins aforesaid,Hides transported, except from Ireland, the penalty. that shall be taken off of any Beast in any of the Islands belonging to England, except Ireland, shall be Transported into any Place except England, the Offender shall forfeit double the value for every Offence, per id. Stat.
All manner of Ammunition may be prohibited at his Majesties pleasure.Ammunition may be prohibited. 12 Car. 2. 4.
If any Sheep shall be Exported, the Offender shall forfeit 20 s. for every Sheep.Sheep exported, the penalty.Vide 12 Car. 2. 32.
If any Wooll, Wooll-fells, Wooll-flocks, Mortlings, Shorlings,Fullers Earth & Fulling clay. Yarn made of Wooll, Fullers Earth, Fulling Clay, shall be shipp'd to be Exported,
the Offender shall forfeit three shillings for every pound weight.
If any Master of a Ship, or other Mariner,The penalty of giving consent to the shipping of the said Goods. be knowing and consenting to the Exportation of the Goods aforesaid, he shall forfeit
all his Goods and Chattels, per idem Stat.
These Offences are also made Felony per Stat. 13, 14 Car. 2.18.Felony to export Sheep. except such Weather-sheep, Wooll, or Wooll-flocks, as are for necessary proportion
for the Ships use.
If any Wooll, Wooll flocks, or Yarn made of Wooll,Wooll, &c. prest with any Engine, &c. penalty. shall be pressed with any Engine into any Sack, Pack, or other Wrapper, or shall
put, press, or steeve Wooll or Woollen Yarn into any Pipe, But, or Hogshead, Chest,
or other Cask or Vessel, or carry or lay any such Wooll, Wooll-flocks, or Yarn made
of Wooll, neer to the Sea or any Navigable River, all such Wooll, Wooll-flocks, and
Yarn made of Wooll, shall be forfeited. Vide Stat. 13 & 14 Car. 2.18.
If any Wooll, Wooll-fells, Mortlings, Shorlings, Yarn made of Wooll,Fullers earth, &c. Tobaccopipe clay. Wooll-flocks, Fullers-Earth, Fulling-clay, or Tobaccopipe-clay, being in any Pack,
Sack, Bag, or Cask, shall be carried upon any Horse, Cart, or other Carriage, except
in the day time, viz. from the first of March to the twenty ninth of September betwixt the Hours of four in the morning and eight at night, and from the twenty
ninth of September until the first of March between the Hours of seven in the morning and five at night, otherwise to be forfeited,
per id. Stat.
If any Tobaccopipe-clayTobacco pipe clay. be Exported beyond the Sea, the Officer shall forfeit three shillings for every pound
weight, per id. Stat.
If any manner of Sheep-skins, Wooll-fells, Mortlings, Shorlings,Sheep skins, or the Leather, &c. penalty. or the Skins of any Stag, Buck▪ Hind, Doe, Goat, Fawn, or Kid, or the Pelts or Skins
of any of them, or the Leather made of any of them, be put on board any Vessel to
be exported, they shall be forfeited, as also two shillings six pence for every Fell,
Shorling, Mortling, Pelt, or Skin, so Shipped to be Exported. Vide Stat. 5 Eliz. 22.
All great Cattel, except of Scotland,Great Cattel imported, the penalty. that shall be Imported into England or Wales betwixt the first of July and the twentieth of September in any year; and all great Cattel of Scotland that shall be brought in betwixt the twenty fourth of August and the twentieth of December in any year, shall forfeit for every Head forty [Page 6] shillings; and for every Sheep brought in betwixt the one and twentieth of August and twentieth of December, ten shillings, per Stat. 15 Car. 2.5.
If any Goods be entered in any other mans NameGooods entred in another mans name. than the true Owner and Proprietor, they shall be forfeit: And if the Officer conceal
any Offence in the said Act, he shall forfeit one hundred pounds. Vide Stat. 1 Eliz. 11.
If any man, being free of the Prisage or Butlerage of Wine,Prisage and Butlerage of Wines. shall Enter another mans Wines in his Name, whereby the King loseth his Butlerage,
all Wines so Entred are to forfeit double the value of the Customs thereof. Vide 1 Hen. 8.5.
If any man offend contrary to the Stat. 1 Hen. 8.5. he shall forfeit all his Goods. Vide Stat. 2 & 3 Edw. 6.22.
If any Officer of the Customs shall suffer or give any Warrant for any Sugar, Tobacco,
Ginger,Sugar, Tobacco, Ginger. Cotton-wooll, Indico, Speckle-wood, Jamaica-wood, Fustick, or any other Dying-wood, of the growth of any of his Majesties Plantations,
to be conveyed into any Parts beyond the Seas, before they are Landed in England or Wales, for every Offence he shall forfeit the value of the said Goods. Vide Stat. 15 Car. 2.5.
All the Goods of an Alien MerchantGoods of Alien Merchants. or Factor in any of his Majesties Plantations are forfeited. Vide Stat. 12 Car. 2.18.
If any manner of Copper, Brass, Latten, Bell-mettle,Copper, Brass, Bell-mettle, &c. Pan-mettle, Gun-mettle, or Shroof-mettle, shall be put on board any Vessel to be
transported, the Offender shall forfeit double the value, to be divided betwixt the
King and the Prosecutor. Vide Stat. 33 Hen. 8.7.
10 l. penalty.And also ten pounds more for every thousand pound weight, per Stat. 2 & 3 Edw. 6.37.
The Customer not performing duty, the penalty.The Customer shall take Bond in double the value of the said Goods, when they shall
be transported from Port to Port, and also 10 l. over and above for every thousand pounds weight, and give Bond; which Bond, if it
want a Date, the Customer shall forfeit the value of the said Goods, and also his
Place, per id. Stat.
To grant a false Certificate, the penalty.If any Customer grant a false Certificate for the said Goods, he shall forfeit his
Place, and the value of the Goods so concealed. 33 Hen. 8.7.
The penalty of the not discovering.If any Master of a Ship, Owner, Purser, or Boatswain, knowing such Mettles to be Shipp'd,
and do not disclose it within three days, he shall forfeit double the value of it.
Vide Stat. 2 & 3 Edw. 6.37.
Not seising the said Goods, the penalty.If any Officer of the Custom-house, knowing such Mettles to be Shipp'd to be Transported,
do not seise it, he shall lose his Office, and the value of the Goods so Shipp'd.
Vide Stat. 2 & 3 Edw. 6.37.
The said Goods not to be shipp'd but where there is a Customer.If any Person Ship any of the said Mettles at any Place, except where there is a Customer,
he shall forfeit the value of the Goods, and also ten pounds for every 1000 pounds
weight, per id. Stat.
Penalty 1000 l. to a Governour not doing his duty.If the Governour of any Plantation belonging to his Majesty, do not his duty justly,
according to the Act for Encouragement of Trade, he shall forfeit his Place and 1000 l. per Stat. 15 Car. 2.15.
Every Person that shall be found guilty of Transporting of Leather,Transporting Leather. shall for every Offence forfeit 500 l. Vide Stat. 14 Car. 2.7.
Every Customer, or other Officer, that shall neglect his duty,Officer neglecting his duty, penalty 100 l. or connive at the Transportation of Leather, shall for every Offence forfeit 100
pounds. Vide Stat. 1 Jac. 22.
Every Customer, or other Officer, that shall make a false Certificate of the Landing
of Leather, shall forfeit 100 l. per id. Stat.To make false Certificates of landing Leather, penalty 100 l. Goods shipp'd at any unlawful hour, the penalty 100 l.
If any Goods or Merchandise shall be Shipp'd or Landed at any unlawful time or place,
for every Offence the Master, Owner, or Purser shall forfeit 100 l. 1 Eliz. 11. & 14 Car. 2.11.
Lawful times are only from the first of March until the first of September,
What lawful times are by Order.
betwixt Sun-rising and Sun-setting; and from the first of September until the first of March betwixt seven a clock in the Morning, and four a clock in the afternoon. The Port
of Hull is here excepted.
If the Captain, Master of a Ship, or Purser outward bound,Goods taken in before entry made 100 l. shall take in any Goods before Entry, he shall forfeit 100 l. Vide Stat. 14 Car. 2.11.
If he go away before cleared on Oath in the Custom-house,To go before cleared on Oath, penalty 100 l. giving a true Accompt of his Lading, &c. he shall forfeit 100 l. per id. Stat.
If any Captain, Master of a Ship, or Purser, do not bring his Ship to the Port,Master of Ship to bring his Ship with as much convenience, &c. and make Entry with as much speed as Wind and Water will permit, for every Offence
he shall forfeit 100 l. Vide Stat. 14 Car. 2.
If he permit any Goods to be taken out of the Ship, to be Landed,To suffer Goods taken out before a general entry, penalty 100 l. before he hath made his general Entry upon Oath in the Custom-house, for every Offence
he shall forfeit 100 l. Vide Stat. 1 Eliz. 11.
If any Captain, Master of a Ship, Purser, or Boat-swain,Goods imbezeled, &c. 100 l. or other Person taking Charge of the Ship, shall permit any sort of the Package therein
to be opened, imbezeled, or altered, for every Offence he shall forfeit 100 l. Vide Stat. 14 Car. 2.11.
Men of War to be liable to the Rules that Merchant Ships are subject to. Liberty to
go on board and take out Prohibited and Uncustomed Goods. The Commissioners and their
Deputies to enter on board, and bring on shore Goods outward and inward bound. The
Officers may stay on board until the Goods be discharged. 14 Car. 2.
If any Goods be found concealed aboard the Ship,Goods concealed 100 l. when the Officers of the Customs have cleared the Ship, the Master, or other Person,
shall for every Offence forfeit 100 l. per id. Stat.
If any Wharfinger, Crane-keeper, Searcher, Lighterman, or other Officer,Wharfinger, &c. not disclosing, &c. penalty 100 l. knowing any Offence contrary to the Statute, do not disclose it to the Customer,
he shall forfeit▪ 100 l. Vide Stat. 1 Eliz. 11.
If any Wharfinger or Crane-keeper shall take up or Land, or suffer to be Landed,To suffer Goods to be taken up or landed at unlawful times. or Ship off, or suffer to be Water-bound, any Wares or Merchandises, at any unlawful
time, or without the presence of, or notice given to an Officer at the Custom-house,
he shall forfeit for every Offence 100 l. per Stat. 14 Car. 2.11. The Port of Hull excepted.
100 l. penalty for taking of Bribes.If any Officers of the Customs shall directly or indirectly receive any Bribe, Becompense,
or Reward, or shall connive at any false Entry of Goods, he shall forfeit 100 l. per id. Stat.
50 l. penalty to give a Bribe.If any Merchant, or other Person, shall give such Bribes, for every Offence he shall
forfeit 50 l. per id. Stat.
If any Packet Boat, or other Vessel appointed to carry Letters, shall Import or Export
any Goods or Merchandise, for every Offence the Master shall forfeit 100 l.100 l.per id. Stat.
Forreign Bone-lace, Cut-work, penalty 50 l.If any Person offer to sale any Forreign Bone-lace, Cut-work, Imbroidery, Fringe,
or Needle-work made of Silk or Thred, for every Offence he shall forfeit 50 l. per id. Stat.
If any of the said Goods be Imported into England or Wales, the Offender shall forfeit 100 l.100 l.per id. Stat.
False Certificate, penalty 50 l.If any Officer of any Port shall make a false Certificate, he shall forfeit 50 l. per id. Stat.
To falsifie any Warrant 100 l.If any Person shall falsifie any Custom-house Warrant, he shall forfeit for every
Offence 100 l. per id. Stat.
The Officer to make a true report, on penalty of 100 l.Every Officer appointed to perfect an Entry, shall make report thereof under his Hand,
unto the Chief Officers of the Customs, the next day, upon the Penalty of 100 l. unless there be cause of longer time to be allowed by the Chief Officers of the Customs,
per id. Stat.
French Vessels not to put Goods on shore, not paying 5 s. per Tun, the penalty 10 l.If any French Vessel put any Goods or Passengers on Shore, or into any Boat, to be conveyed on
Shore, and not pay the five shillings per Tonnage due upon French Vessels, upon their return they shall forfeit ten pounds, and pay all the former
Duty, per id. Stat.
If any Pilot or Waterman shall go out from any Port, to bring in any Goods or Passengers
from aboard of any French Vessels, for every Offence he shall forfeit 40 l. per id. Stat.Pilot or Waterman going out of any Port brings Goods, &c. penalty 40 l.
Officers not giving attendance, penalty 100 l.If any Customer, Comptroller, or Searcher, or their Deputies, do not give their attendance
at the Custom house at such time and places as are appointed by Law, and also do not
their utmost diligence in their respective Places, for every Offence the Offender
shall forfeit 100 l. Vide Stat. 1 Eliz. 11.
Not resident on his Place, penalty 100 l.If any Customer, Comptroller, or Searcher be not resident upon his Place and Office,
for every Offence he shall forfeit 100 l. Vide Stat. 1 Hen. 4.13. 4 Hen. 4.20, & 21.13 Hen. 4.5.
No Custom Officer to be Owner of a Ship, or use Merchandise, &c. penalty 40 l.If any Custom-house Officer fraight any Ship, or use any Merchandise, or keep any
Wharfe, or hold any Hostery, or Tavern, or be Factor, or Attorney, or Host to any
Merchant, for every Offence he shall forfeit 40 l. for every six Months, to be divided betwixt the King and the Prosecutor. Vide Stat. 20 Hen. 6.5.
If any Customer, Comptroller, or Searcher, be a Common Officer,No Custom Officer to be a common Officer, penalty 40 l. or Deputy to a Common Officer, in any City, Burrough, or Town, upon the penalty of
40 l. for every six Months he shall so officiate both Offices together. Vide Stat. 3 Hen. 7.1.
Quere, Whether this Statute be in force, or Repealed by the Statute of 1 Hen. 8. 5.
English Shipping is either English built, or bought bona fide by English Money,English-built shipping, what is meant. whereof every Owner or Part-owner are English, Irish, Welch, or of his Majesties Plantations.
A COMPENDIUM OF FORTIFICATION, BOTH Geometrically and Instrumentally, BY A SCALE,
The Making whereof is shewed by the Tables, and their Use, both of the Tables and
the Scale, for speedy Protracting of any Fort consisting of 8 Bulwarks, whose Bastion-Angles
shall not exceed 90 Degrees; and so the like for Bastion-Angles of 12 Bulwarks. WRITTEN
BY PHILIP STAYNRED Professor and Teacher of the MATHEMATICKS in the City of BRISTOL.
JER. XVI. 19.
The Lord is my strength, and my fortress, and my refuge in the day of affliction.
FDG the Angle forming the Flank, commonly 40 Degrees.
GAO the Inward Flanking Angle.
APB the Outward Flanking Angle, and APO half the same.
Note, That the Bastion or Flanked Angle HAG must never be less than 60 Degrees, neither
above 90 Degrees; but as neer as you can to an Angle of 90 Degrees: So that it may
be defended from the Flank and Curtain on either side.
The longest Line of Defence KA not to be above 12 score Yards, that is, 720 Foot,
being within Musket-shot; and the breadth of the Rampire to resist the Battery 100
Foot.
To Describe a Fort of Five Bulwarks, or any other; so that the Bastion, or Flanked
Angle of 8 Bastions or Bulwarks exceed not 90 Degrees by the Line of Chords.
FIrst, Draw an Obscure Line, as AB; and upon A, as a Center, with the Chord of 60
deg. describe an Arch, as CDE; and from C lay down half the Polygon Angle (which in the
Table following the Figure you shall find to be) 54 deg. as CD; also the same again from D to E, and draw the Line AE. Now divide the half
Polygon Arches CD and DE each into three equal parts, as in FHIG, and from two of
those parts from D, as F and G, draw Lines unto the Bastion Point at A. Then take
any convenient Distance, and lay the same on those Lines from A unto K and L, which
shall make the Front or Face of the Bulwark. Next, from the shoulder at K let fall
a Perpendicular to AB, as KM; and on the Center at K describe an Arch of 60 deg. from M towards N, and from M lay down on the same Arch 50 deg. or more exact 49 deg. 24 min. and so draw KN, which will cut the Semidiameter of the Polygon in the Point O; so
shall AO be the Capital Line of the Bastion. Then from O draw a Line parallel to AB,
as OP; so shall you have OR for the Gorge Line, and RK the Flank. Now for the Curtain,
take half the Front AK, as AT, and lay it down three times from R towards P, which
will fall in S; so is RS the Curtain. Then on the Point at S erect a Perpendicular,
as SV, equal to RK, which shall be the Flank of another Bastion; and so the Front
KA being laid from V, shall cut the first Line AB in B; so drawing VB, you have the
Front of the same Bastion.
Lastly, Divide AB in the middle, as in W, and from W let fall a Perpendicular to AB,
which will cut the Semidiameter of the Polygon in the Point D; so is D the Center
of the Polygon. And with the same Semidiameter DA you must describe a whole Circle,
of the which AB is the ⅕ part thereof, which Distance will reach from B unto X, and
from X unto Y, and so to Z, and your first Point at A, where you begun your Work.
For the other Bastions, they may be easily transported from the first Bastion. And
note, That if your Fort exceed 8 Bulwarks, you must add 15 deg. to half the Polygon Angle, so have you the Bastion Angle; and then work as before.
But in the Forts that exceed not 8 Bulwarks, where the Bastion Angle will not be above
90 deg. you must take the 2/3 part of the Angle of the Polygon.
The longest Line of Defence is from A unto Q, and should not exceed 720 Foot (because
of being within Musket-shot) the Curtain RS about 420 Foot, and the Front AK 280 Foot:
And for the Flank RK, and Gorge RO, their proportion commonly is as 6 to 7: but the
Angle KOR forming the Flank is about 40 gr. by which the Proportion is neer as 5 to 6.
The ½ Bastion Angle is here found by taking the ⅔ of the ½ Polygon Angle: So the Bastion
Angle will be an Angle of 90 degr. in the Octagon. And no more must the Bastion Angle be in any Polygon.
A Table for 12 Bastions.
Polygons.
½ Angle of the Polygon.
½ Angle of the Bastion.
Degrees.
Degrees.
4 Tetragon
45
30
5 Pentagon
54
34 ½
6 Hexagon
60
37 ½
7 Heptagon
64 2/7
39 9/14
8 Octagon
67 ½
41 ¼
9 Enneagon
70
42 ½
10 Decagon
72
43 ½
11 Undecagon
73 7/11
44 7/12
12 Dodecagon
75
45
The ½ Bastion Angle is here found by adding 15 d. to ½ the Polygon Angle, and take the ½ thereof: So the Bastion Angle will be an Angle
of 90 deg. in a Dodecagon.
Of the Works that are in or about Forts of most Importance.
[geometrical diagram]
AB the Breadth or Walk on the Rampire
40 Feet.
BC the Breadth of the Parapet of the Rampire, with the Fausse-bray, and Parapet thereof,
each 20 Foot; in all
60
DE the Breadth of the Moat, Ditch, or Trench
120
EF the Coridon, or Covert-way of the Counterscarp
20
FG the Argin or Parapet thereof, being
50 or 60
H the Ravelines; I the Half-moons, with their Parapets
20
There may be sometimes an occasion in Forts to raise Mounts, Cavaliers, Platforms,
or Batteries, to command all the other Works, and to view the Country about; which
may be raised upon the Bastions, if you have room withal to make use of the Flanks:
Otherwise let them be raised on the Curtains, a little within the Rampire, so that
you may have room left for the Walk.
To Draw the Platform of a Fort, beginning with the Capital (or Head) Line; And also
to draw the Horn-works.
[geometrical diagram]
LEt the Fort be an Hexagon, that is, of six Bastions or Bulwarks. First draw the Line AB, and upon A describe
an Arch of a Circle, as BDC, whereon lay down half the Polygon-Angle, which in the
former Table you shall find to be 60 deg. as from B unto D, and thence to C; and draw AC and AD. Now the ⅓ part of the half
Polygon Angle is BC and CF; then draw the Obscure Line AF and AG. Next you shall make
choice of the Capital Line, of any sufficient length, which let be AE; and from E
draw a Line parallel to AB, as EH, continued; and upon the Point E, as a Center, describe
KI, making it an Angle of 40 deg. as KEI; so shall EI cut out the Front in L, as AL: So from L let fall the Perpendicular
LM, which shall be the Flank; and MN the Curtain shall be as formerly the whole length
of the Front, and a half more. For the rest of the Work, you must proceed as formerly.
YOu must continue the Flunkers ML and NO unto P and Q; then take the longest Line
of Defence AN, and lay it thereon from the shoulders at L and O, unto P and Q; drawing
the Line PQ, dividing it into three equal parts; and from those parts 1 and 2, draw
parallels unto PL and QO: also from those Points P and Q draw parallels to the Fronts
AL and BO, those will cut the former Parallels in R, S, T, and V, which Intersections
will limit the Fronts, Flanks, and Curtains, as you may easily perceive; unto which
you must make the Rampire, Parapet, &c. as in the former Works.
Now follow two Tables; the one for 12 Bastions, and the other for Forts of 8 Bastions: Whereby you may trace out any Fort by help of a Line of Equal Parts, which
shall divide the Side of the Outer Polygon into 10000 parts.
The First Table for 12 Sides.
Number of Sides
4
5
6
7
8
9
10
11
12
The Side of the Outer Polygon
10000
10000
10000
10000
10000
10000
10000
10000
10000
The Capital Line
2428
2592
2756
2926
3086
3136
3148
3180
3204
The Gorge
1088
1264
1378
1470
1538
1640
1722
1792
1842
The Front
2914
2952
2986
3014
3024
3054
3070
3082
3094
The Flank
970
1128
1246
1360
1516
1526
1536
1542
1546
The Second Table, for 8 Sides, whose Bastion Angle then shall make 90 Degrees.
Number of Sides
4
5
6
7
8
The Side of the Outer Polygon
10000
10000
10000
10000
10000
The Capital Line
2396
2498
2602
2695
2778
The Gorge
1120
1327
1480
1599
1698
The Front
2914
2939
2959
2975
2987
The Flank
940
1113
1242
1342
1423
The Ʋse of these Tables.
LEt it be required to draw the Proportional Dimension of a Regular Fort of 6 Sides:
As for Example, in the fourth Figure, whose Side AB must be divided into 100 equal
parts, and each part supposed to be subdivided into 10 parts; so have you 1000 parts,
which shall suffice. Now proceeding according to former Directions, until you come
to make choice of your Capital Line, you shall here find in the second Table, which
is best for the purpose, under the Figure 6, and right against the word Capital in the first Column, 2602, but 260 will serve: Take the [Page 7] same from the Scale of Equal parts, and lay it from the Bastion Point at A, and it
falls in the Point E, which will be the Center of the Bastion. From thence you may
lay down the Gorge Line out of the Table, which is 148 unto M: So will the Front AL
be 296, and the Flank ML. The Curtain, being once and a half the length of the Front,
will be MN 444. Thus you may do for any of the rest. These Tables are useful for
Irregular Forts; But first I will shew you the Height, Breadth, and Scarpings of the
Rampire, Parapet, Ditch, &c. of these Sconces, as they are represented in the Profile, or Section, as followeth.
[geometrical diagram]
The Breadth of the Rampire may be 24, 30, or 40 Foot; but here AB is but
32
The Inward Scarp AC
6
The Height of the Rampire CD
6
The Breadth of the Walk of the Rampire DE
10
The Breadth of the Bank or Foot-pace of the Parapet EF
3
and the Height of the same Foot-pace
1 ½
The Inward Scarp of the Parapet FG
1
The Inward Height of the Parapet GH
6
The Breadth of the Parapet at the Foot FI
10
The outward Scarp of the Rampire BK
3
The Inward Scarp of the Parapet IL
2
The Outward Height of the Parapet LM
4
The Thickness of the Parapet at the top MN
7
The Brim of the Ditch BO
3
The Breadth of the Ditch at the top OP
32
The Scarp of the Ditch OQ
6
The Depth of the Ditch QR
6
The Breadth of the Ditch at the Bottom RS
20
The Profile or Section of a Fort with a Fausse-Bray and Counterscarp; also Subtrenched.
[geometrical diagram]
CD is the Fausse-bray, and DM his Parapet: EFGH is the Subtrench, and IK the Coridor,
or Covert-way. Lastly, KL is the Argin or Parapet of the [Page 8] Counterscarp. Note, That the Height of the Rampire AB ought to be raised 15 or 18
Foot above the Terra Plana, although here it is but 12 Foot, which is somewhat too low to command the Trench
or Ditch: But if the Trench be made broader, then it will command the bottom thereof.
Of Irregular Fortification.
IN the seventh Figure following let ABCDE be an Irregular Fort, containing 5 Bastions,
or Bulwarks. First we will make a Bastion on the Angle at A, which do thus: Divide
the Polygon Angle in half with the Line AF, and draw the Bastion-Angle as formerly,
being ⅔ of the Polygon-Angle, as AH, and AI continued, being the Sides whereon the
Fronts must be laid down.
Now upon some spare Paper you shall make the half Polygon-Angle GAF, as you may see
underneath this seventh Figure, as LKM: Then make choice of the Capital Line, as
before; let it be of any convenient length (larger than you think your Bastion will
be in the seventh Figure) as underneath KN, and from N the Center of the Gorge draw
a Parallel to KL, continued to O, as NP; and so proceeding as before, you shall find
the Point of the second Bastion at O: So have you the Proportion of your Bastions,
whereby you may gain those in the seventh Figure.
Now to reduce it from the Bastion Point at A, you must take AB the shortest side,
and lay it from O unto Q; and from Q draw a Line parallel to the Capital Line KN continued,
as QR. Lastly, drawing a Line from N to O, it shall cut QR in the Point S; so is QS
the length of the Capital Line sought for, which must be laid down on the seventh
Figure from A unto T; so is T the Center of the Gorge. Then for the Front take KV,
and lay it on the Capital Line from K to W: so a Ruler being laid from W to O, it
shall cut the Line QR in the Point at X; so is QX the length of the Front, to be laid
down in the seventh Figure from A unto H and I. Thus shall you finish your Bastion,
when you have let fallen your Flanks perpendicular on the ends of your Curtains, as
you see. The like Method you are to observe for the other Bastions.
And when you have finished your Fort, you must observe whether your Curtain Lines
(that is, from the Centers of the Gorge) be parallel to the outward Sides AB, &c. which if they are not, you must correct them; and by your Judgment, by help of the
Lines of Defence, you may as you see occasion widen the Necks of the Gorges, and also
the Bastion Angles; but not above 90 Degrees: And so let the Flankers be as neer proportional
as the Rules (or Ocular Demonstration directeth) which commonly the Gorge Line to
the Flank bears proportion as 7 to 6.
Much more could I write of Irregular Fortification: but my purpose at this time is
but to make a small Treatise, or an Epitome thereof.
The seventh Figure, of an Irregular Fort, containing 5 Bastions; being the Platform of the Royal Fort sometimes on St. Michael's Hill, on the North West Side of the City of Bristol.
THis may be performed Geometrically by observing the former Instructions, whereby
you may gain the length of every Line: but it will be sooner done, and more easie,
by these Tables following.
First Table, for 12 Sides.
Numb. of Sides
4
5
6
7
8
9
10
11
12
Semi. Out. Pol.
1000
1000
1000
1000
1000
1000
1000
1000
1000
Semi. Inn. Pol.
661
700
731
756
777
795
810
823
834
The Front
412
347
299
261
232
209
190
174
160
The Gorge
158
151
141
132
123
115
108
101
96
The Flank
133
127
119
111
103
97
90
85
80
Second Table for 8 Sides.
Number of Sides
4
5
6
7
8
Semi. Outer Polyg.
1000
1000
1000
1000
1000
Semi. Inner Polyg.
661
706
740
766
787
The Front
412
346
296
259
229
The Gorge
158
156
148
138
130
The Flank
133
131
124
116
109
Make your Scale of a sufficient length, that may hold both Lines, the one for 12 Sides,
and the other for 8. Make choice within the breadth of the Scale, between the Borders,
any sufficient breadth, as CD; from whence draw Parallels to the Sides, and divide
CD into [...]0 Equal parts, and begin your Account from C with 45: so shall the end at D be 75
degr.
Now make choice of the length of the outer Polygon, which here I make three Inches;
and divide a Line by the Side thereof, equal thereto, in 100 equal parts, and suppose
each part into 10; so have you 1000 parts, agreeable to the Tables. The next thing
is to draw Parallels to CE, according to the Polygon half Angles, as you may see
in the Tables under the Pentagon Fort, being the second Figure: So [...]om the Scale CD you have for the half Angle of a Pentagon 54 Degrees, whereby you
may draw the Pentagon parallel FG; and so in the lower Scale of 8 Bastions. In the
like manner you may do for all the rest.
Now to draw the cross Lines for the Semidiameters of the Inner Polygons, as also
the Lines for the Fronts, Gorges, and Flanks, you shall work thus. First, you must
note, That the Semidiameter of the Outer Polygon is the Radius or whole Line of 1000
[Page 11] equal parts, and that is drawn at Right Angles, or a cross at F: But for the Semidiameter
of the Inner Polygon, look in the Table of 12 Sides in the second Column under 4,
you have 661. Take the same Number off your Scale of Equal Parts, and lay it from
E to H: Then in the third Column under 5 you have 700 parts; lay the same down from
G to I, and make there a prick or point: Do the like for the Hexagon and Heptagon, as at I and K; proceeding along with all the rest, unto the Dodecagon. And lastly, draw a Line through all those Points: So have you the Arch Line HM for
the Semidiameters of the Inner Polygon. In the same manner work for the Front, Gorge,
and Flank Lines. The Scale of 8 Sides is the same Method.
I have also inserted on the left side of the Scale a Line of Chords, whose Radius
(or Arch of 60 Degrees) is three Inches; and on the left side, a Line for the Sides
of the Polygons. The Hexagon, or six Sides, is equal to the Radius: And for the Tetragon, or four Sides, it is equal to the Chord of 90 Degrees. So having described a whole
Circle with the Chord of 60 Degrees, you shall find, that if you take from the Center
N unto 5, it shall divide the Circle into 5 equal parts, for drawing the Figure of
a Pentagon, which in the second Figure of a Pentagon Fort will reach from A to B, and so to X, Y, Z, and A. Now DA in the same second
Figure you shall find to be the Semidiameter of the Outer Polygon, which in the Scale
is GF; and taking GI off the Scale for 12 Bastions, or GO on the Scale of 8 Bastions,
it will give the Semidiameter of the Inner Polygon, as DE in the same Figure. So likewise
GP on the Scale will be equal to the Front AK in the Pentagon Fort. The like you may understand for laying down the Gorge and the Flank. And for
the Curtain, as before, you must make RS 1 ½ the length of the Front AK.
This Scale will also be of good Use in Irregular Fortification. As for Instance. In
the Irregular Fort, the seventh Figure, you shall find the half Bastion Angle GAF
to be 58 Degrees, which falls on the Scale between the Pentagon and Hexagon, from whence you may draw a Bastion on some spare place, and from thence proportionate
the same unto the outer side of the seventh Figure AB. The rest I leave to your own
practice.
How to Fortifie a long Curtain with Bulwarks, or a strait Town Wall.
[geometrical diagram]
LEt the Curtain be AB. First take 200 Foot from the Scal [...] of Fortification, accounting 10 for 100; and lay the same from A unto C, and from
C unto D: So shall AD be the breadth of the Neck of the Gorge: and upon the Point
C erect a Perpendicular, as CF. Then take 420 Foot off your Scale, and lay the same
from D to E, which shall be the length of the Curtain. Next you must take 720 Foot,
and lay the same from E, to cut off the Capital Line at F; so shall EF be the longest
Line of Defence, and CF the Capital Line; which Line CF must be laid down from C unto
G: and draw GF for the shortest Line of Defence. Lastly, upon D erect a Perpendicular,
which will cut the same Line in H; so have you DH for the Flank, and HF the Front.
Thus have you finished half the Bastion, from whence you may transport all the rest
of the Bastions, were they ever so many.
Note, That these Bastions must not exceed 720 Foot, that so it may not be without
Musket-shot. But if you will defend the Front with Cannon, you may make the Line of
Defence almost twice so much. The like for the Curtain, which may be 800 Foot; and
in the middle of the Curtain you may make a Spur, or Point of a Bastion, as at K,
which will be necessary for Musket-shot, beside the Cannon; which in the Line drawn
about the City of Bristol I have seen many of them.