CHAP I.

HE that seeketh Shadow in its predicaments, seeketh a reality in an imitation, he is rightly answered, umbram per se in nullo praedicamento esse, the reason is thus rendred as hath been, it is not a reality, but a confused imi­tation of a Body, arising from the objecting of light,

So then there can be no other definition then this, Shadow is but the imitation of sub­stance, not incident to parts caused by the interposition of a substance, for, Umbra non potest agere sine lumine. And

[Page 8]And it is twofold, caused by a twofold motion of light, that is, either from a direct beam of light, which is prima­ry, or from a secondary, which is reflective: hence it is, that Sun Dials are made where the direct beams can never fall, as on the seeling of a Chamber or the like.

But in vain man seeketh after a shadow, what then, shall we proceed no farther? surely not so, for qui semper est in suo officio, is semper orat, for there are no good and lawful actions but doe condescend to the glory of God, and espe­cially good and lawfull Arts.

And that shadow may appear to be but dependant on light, it is thus proved, Quod est & existit in se, id non ex­istit in alio: that which is, and subsisteth in it selfe, that subsisteth not in another: but shadow subsisteth not in it selfe, for take away the cause, that is light, and you take a­way the effect, that is shadow.

Hence we also observe the Sun to be the fountain of light, whose daily and occurrent motions doth cause an ad­mirable lustre to the glory of God; seeing that by him we measure out our Times, Seasons, and Years.

Is it not his annuall revolution, or his proper motion that limits our Year?

Is it not his Tropicall distinctions that limits our Sea­sons?

Is it not his Diurnall motion that limits out our Dayes and Houres?

And man truly, that arch type of perfection, hath limi­ted these motions even in the small type of a Dyall plane, as shall be made manifest in things of the second notion, that is, Demonstration, by which all things shall be made plain.

CHAP II.

WE have now with the Philosopher, found out that common place, or place of being, that is, the World, will you know his reason? 'tis rendred, Quia omnia reliqua mundi corpora in se includit.

I'le tell you of no plurality, not of planetary inha­bitants, such as the Lunaries▪ lest you grabble in darkness, in expecting a shadow from the light without interposition, for can the light really without a substance be its own Gno­mon? surely no, neither can we imagine our earth to be a changing Cynthia, or a Moon to give light to the Luna­ry inhabitants: For if our Earth be a light (as some would have it) how comes it to passe that it is a Gnomon also to cast a shadow on the body of the Moon far lesse then it selfe, and so by consequence a greater light cannot seem to be darkned on a lesser or duller light, and if not darkned, no shadow can appear?

But from this common place the World with all its parts, shall we descend to a second grade of distinction, and come now to another, which is a proprius locus, and divide it into proper places, considering it as it is divided into Coelum, Solum, Salum, Heaven, Earth, Sea, we need not so far a distinction, but to prove that the earth is in its own proper place, I thus reason: Proprius locus est qui proxime nullo alio interveniente continet locatum: but it is certain that nothing can come so between the earth as to [Page 10] dispossesse it of its place, therefore it possesseth its proper place, furthermore, ad quod aliquid movetur, id est ejus locus, to what any thing moves that is its place: but the earth moves not to any other place, as being stable in its own proper place.

And this proper place is the terminus ad quem, to which (as the place of their rest) all heavie things tend, in quo mo­tus terminantur, in which their motion is ended.

Hence we reason, that a Mil-stone, or any ponderous thing cannot passe the center of the earth, because the cen­ter of the earth is the farthest from the Heavens, for we all know that no man can dispose any heavie thing in the aire to rest, neither can any one force a stone to passe through the earth if there were a way perspicu­ous, for ha­ving passed the center of the earth, 'tis as it were to throw astone upward which will a­gain return to its center. For example, sup­pose the great Circle ABC D to be the cō. cave superficies of the Firma­ment, which all men know to compasse the earth on all

[Page 11] sides, and let the earth be the smaller Circle within that, the center whereof is E. Now if a line and plummet at A be let fall towards the earth it will tend in a right line towards E: also, if another plummet be let fall from B or any other place as C or D, that will fall also in a right line towards E, so that they would all meet in the center E, which point they cannot passe, for then they return as it were upward, and so must fall in the Skie or nowhere, contrary to all ponderous motions, whence is proved the earth to be the center of the World, and what manner of center shall be shewn in the next ensuing Chapter.

CHAP III.

A Center is either to be understood Geometrically or Optically, either as it is a point, or seeming a point.

If it be a point, it is conceived to be either a center of magnitude, or a center of ponderosity, or a cen­ter of rotundity: if it be a seeming point, that is increased or diminished according to the ocular aspect, as being som­time neerer, and somtime farther from the thing in the vi­suall line, the thing is made more or lesse apparent.

A center of magnitude is an equal distribution from that point, an equality of distribution of the parts, giving to each end alike, and to each a like vicinity to that point or center.

A center of ponderosity is such a point in which an un­equall thing hangs in equi libra, in an equall distribution [Page 12] of the weight, though one end be longer or bigger than the other of the quantity of the ponderosity.

A center of rotundity is such a center as is the center of a Globe or Circle, being equally distant from all places.

Now the earth is to be understood to be such a center as the center of a Globe or Sphear, being equally distant from the concave superficies of the Firmament, neither is it to be understood to be a center as a point indivisible, but either comparatively or optically: comparatively in respect of the superior Orbs; Optically by reason of the far di­stance of the one from the earth; as that the fixed Stars being far distant seeme, by the weaknesse of the sense, to be conceived as a center indivisible, when by the force and vi­gour of reason and demonstration, they are found to ex­ceed this Globe of earth much in magnitude; so that what our sense cannot apprehend, must be comprehended by reason: As in the Circles of the Coelestiall Orbs, because they cannot be perceived by sense, yet must necessarily be imagined to be so. Whence it is observable, that all Sun Dials, though they stand on the surface of the earth, doe as truly shew the houre as if they stood in the center.

CHAP IV.

OCular observations are affirmative demonstrations, so that what is made plain by sense is apparent to reason: hence it so happeneth, that we imagine the Earth to move as it were on an axis, because, both by ocular [Page 13] and Instrumentall observation, in respect that by the eye it is observed that one place of the Skie is semper apparens, neither making Cosmicall, Haeliacall or Achronicall rising or setting, but still remaining as a point, as it were, im­moveable, about which the whole heavens are turned. These yet are necessary to be imagined for the better de­monstration of the ground of art; for all men know the heavens to be supported only by the providence of God. Thus much for the reason shewing why the World may be imagined to be turned on an Axis, the demonstration proving that the earth is the center, is thus, not in main­taining unlikely arguments, but verity of observation; for all Gnomons casting shadow on the face of the earth, cast the like length or equality of shadow, they making one & the same angle with the earth, the Sun being at one and the same angle of height to al the Gnomons. As in example, let the earth be re­presented by the small circle within the great circle, marked ABCD, and let a Gnomon stand at E of the lesser Circle, whose horizon is the line AC, and let an other gnomon of the same length be set at I, whose horizon

[Page 14] is represented by the line BD, now if the Sun be at equall angles of height above these two Horizons, namely, at 60 degrees from C to G, and 60 from B to F, the Gnomons shall give a like equality of shadows, as in example is ma­nifest. Now from the former appears that the earth is of no other form then round, else could it not give equality of shadows, neither could it be the center to all the other in­ferior Orbs: For if you grant not the earth to be the mid­dle, this must necessarily follow, that there is not equality of shadow. For example, let the great Circle re­present the heavens, and the lesse the earth out of the cen­ter of the grea­ter, now the Sunne being a­bove the Hori­zon AC 60 d. and a gnomon at E casts his shadow from E to F, and if the same gnomon of the same length doth stand till the Sun come to the opposite side of the Horizon AC, and the Sun being 60 degrees above that Horizon, casts the shadow from E to H, which are un­equall in length; the reason of which inequality proves that then it did not stand in the center, and the equality of the other proves that it is in the center. Hence is also most forceably proved that the earth is compleatly round in the respect of the heavens, as is shewed by the equality

[Page 15] of shadows, for if it were not round, one and the same gnomon could not give one and the same shadow, the earth being not compleatly round, as in the ensuing discourse and demonstration is more plainly handled and made manifest.

And that the earth is round may appeare, first, by the Eclipses, when the shadow of the earth appeareth on the body of the Moon, darkning it in whole or in part, and such is the body such is the shadow. Again, it appears to be round by the orderly appearing of the Stars, for as men travell farther North or South they discover new Stars which they saw not before, and lose the sight of them they did see. As also by the rising or setting of the Sun or Stars, which appear not at the same time to all Countries, but by difference of Meridians, and by the different observations of Eclipses, appearing sooner to the Easterly Nations then those that are farther West: Neither doe the tops of the highest hils, nor the sinking of the lowest valleys, though they may seeme to make the earth un-even, yet compared with the whole greatnesse, doe not at all hinder the round­nesse of it, and is no bigger then a point or pins head in comparison of the highest heavens.

CHAP V.

MAn is the perfection of the Creation, the glory of the Creator, the compendium of the World, the Lord of the Creatures.

He is truly a Cosmus of beauty, whose eye is the Sunne of his body, by which he beholds the never resting motions of the heavens, contem­platively to behold the place of motion; the place of his eternall rest. Lord, what is man that thou shouldest be so mindefull of him, or the son of man that thou so regardest him? thou hast made a World of wonder in his face. Thou hast made him to be a rationall creature, endowed▪ him with reason, so that his intellect becomes his Primum mobi­le, to set his action at work, nevertheles, man neither moves nor reigns in himselfe, and therefore not for himselfe, but is born not to himselfe, but for his Countrey; therefore he ought to employ himselfe in such Arts as may be, and prove to be profitable for his Countrey.

For he hath given me certain knowledge of the things that are, namely to know how the World was made, and the operations of the Elements, the beginning, ending, and midst of times, the alte­ration of the turning of the Sun, and the change of Seasons, the circuite of years and position of Stars, Wisd. 7. 17.

CHAP I.

COSMUS, the World, is divided by Microcosmus the little World, into substantiall and imaginary parts: Now the substantiall are those materiall parts or substance of which the World is compacted and made a Body, by the inter-folding of one Sphear within another, as is the Sphear of Saturn, Jupi­ter, Mars, Sol, &c. And these of themselves have a gentle and proper motion, but by violence of the first mover, [Page 20] have a racked motion contrary to their own proper moti­on: whence it appears, that the motion of the heavens are two, one proper to the Sphears as they are different in themselves, the other common to all.

Will you know how many days doe constitute a year, he telleth you who saith,

The just period of the Suns proper revolution.

The Imaginary part traced out by mans imagination, are Circles, such is the Horizon, the Equator, the Meridi­an, these Circles have of themselves no proper motion, but by alteration of place have an accidentall division, di­viding the World into a right Sphear, cutting the paral­lels of the Sun equally or oblique, making unequall dayes and nights: whence two observations arise:

First, Where the parallels of the Sun are cut equally, there is also the dayes and nights equall.

Secondly, Where they are cut oblique, there also the dayes and nights are unequall.

The variety of the heavens are diversly divided into Sphears, or severall Orbs, and as the Poets have found [Page 21] out a Galazia, the milkie way of Juno her brests, or the way by which the gods goe to their Palaces, so they will assigne to each Sphear his severall god.

Heralts.

Caliope in the highest Sphears doth dwell,

Astrologie.

Amongst the Stars Urania doth excell,

Philosophie.

Polimnia, the Sphear of Saturn guides,

Gladnesse,

Sterpsicore with Jupiter abides.

Historie,

And Clio raigneth in mans fixed Sphear.

Tragedic.

Melpomine guides him that gvids the year

Solace.

Yea, and Erata doth fair Venus sway.

Loud Instruments.

Mercury his Orbe Euturpe doth obey.

Ditty.

And horned Cynthia is become the Court Of Thalia to sing and laugh at sport.

Where they take their places as they come in order.

The Sphear is said to be right where the Poles have no elevation, but lie in the Horizon, so that to them the E­quinoctiall is in the Zenith, that is, the point just over their heads.

The Sphear is oblique in regard of its accidentall divi­sion, accidentally divided in regard of its orbicular form; orbicular in regard of its accidentall, equall variation orbi­cular, it appears before in the Praecognita Philosophicall, his equall variation is seen by the equall proportion of the earth answering to a Coelestiall degree, for Circles are in proportion one to another, and parallel one to another are cut equally, so is the earth to the heavens; having considered them as before, we will now consider another sort of Sphear, which is called parallel.

[Page 22]This parallel Sphear is so that the parallels of the Sun are parallel to the Horizon, having the Poles in their Ze­nith, being the extream intemperate, colde, and frozen Zone: Ovid in his banishment complaines thus thereof.

CHAP II.

THe greatest Circle of a Sphear is that which divides it in two equall parts, and that be­cause it crosseth diametrically, and the dia­meter is the longest line as can be struck in a Circle, and therefore the greatest, which great Circles are represented in the follow­ing figure, representing the Circles of a Sphear in an ob­lique Latitude, according to the Latitude or elevation of the Pole here at London, which is 51 deg. 32 min. being North Latitude, because the North Pole is elevated.

The Horizon is a great Circle dividing the part of hea­uen seen, from where we imagine an Antipodes, the inha­bitants being to us an Antipheristasin, our direct opposites, so that while the Sun continues visible to us, it is above our Horizon, and so continues day with us, while it is night with our opposites; and when the Sun goes down with us it appears to them, making day with them while it remain­eth night with us, and according to the demonstration, is expressed by the greot Circle marked NSEW, signifying [Page 23] the East, West, North, and South parts of the Horizon. So now if you imagine a Circle to be drawn from the Suns leaving our sight, through those Azimuth points of heaven, then that Circle there imagined is the Horizon, and is accidentally divided as a man changes his place, and divides the World in a right or oblique Sphear.

The Meridian is a great Circle scituated at right angles to the Horizon, equally passing between the East and West points, and consequently running due North and South, and passeth through the Poles of the World, being sted­fastly fixed, it is represented by the great Circle marked NDSC, and is accidentally divided, if we travell East [Page 24] or West, but in travailing North or South altereth not, & when the Sun touches this Circle, it is then mid-day or Noon: Now if you imagine a Circle to passe from the North to the South parts of the Horizon, through your Zenith, that Circle so imagined is your Meridian, from which Meridian we account the distance of houres.

The Aequinoctiall likewise divides the World in two equall parts, crossing at right angles between the two Poles, and is therefore distant from each Pole 90 degrees, and is elevated from the Horizon on the contrary side of the Poles elevation, so much as the Pole wants of 90 deg. elevation, demonstrated in the Scene by the Circle passing from A to B, and is accidentally elevated with the Poles as we change our Horizon, and when the Sun touches this Circle, the dayes and nights are then equall, and to those that live under this Citcle the dayes and nights hang in equilibra continually, and the Sun doth move every houre 15 degrees of this Circle, making the houre lines equall, passing 15 degrees in one houre, 30 degrees in two houres, 45 degrees in three houres, 60 degrees for four, and so in­creasing 15 degrees as you increase in houres. This I note to the intent you may know my meaning at such time as I shall have occasion ro mention the Aequinoctiall distances.

The Axis of the World is that which the Stile in every Diall represents, being a line imaginary, supposed to passe through the center of the World, from the South to the North part of the Meridian, whose outmost ends are the Poles of the World, this becomes the Diameter, about which the World is imagined to be turned in a right Sphear having no elevation, in an oblique to be elevated above the Horizon and the angle at the center, numbred on the arch of the Meridian between the apparent Pole and the Hori­zon, [Page 25] is the elevation thereof, represented by the streight line passing from E to F, the arch EN being accounted the elevation thereof, which according to our demonstration is the Latitude of London.

The Stars that doe attend the Artick or North Pole, are the greater and lesser Beare, the last star in the lesser Bears tale is called the Pole Star, by reason of its neerness to it: this is the guide of Mariners, as appeareth by Ovid in his exile, thus

The stars that attend the Southern Pole is the Cross, as is seen in the Globes.

Video Coelos opera manuum tuarum, lunam & stellas que tu fundasti.

CHAP III.

DYals are the dayes limiters, and the bounders of time, whereof there are three sorts: Hori­zontall, Erect, Inclining: Horizontall are al­wayes parallel to the Horizon: Erect, some are erect direct, others erect declining: Incli­ning also are direct or declining: for more explanation the figure following shall give you better satisfaction, where the Horizon marked with diverse points of the Compasse shall explain the demonstration: Now if you imagine Cir­cles to passe through the Zenith A, crossing the Horizon in his opposite points, as from SW through the verticall point A, passing to the opposite point of South-west to North-East, those, or the like circles, are called Azimuthes, parallel to which Azimuthes all erect Sciothericals doe stand.

[Page 27]Those Planes that lie parallel to the Horizontall Circle are called Horizontall planes, and his Style makes an angle with the Pole equall to the elevation thereof; then the ele­vation of the Pole is the elevation of the Style.

Erect Verticals are such which make right angles with the Horizon, and lie parallel to the Verticall point, and these, as I told you before, were either direct or declining.

Direct are those that stand in a direct Azimuth, behold­ing one of the four Cardinall Quarters of the World, as either direct East, West, North, or South, marked with these letters NEWS, or declining from them to some other indirect Azimuth or side-lying points.

[Page 28]Erect North and South are such as behold those Quar­ters, and cuts the Meridian at right angles, so that the planes crosse the Meridian due East and West, and the Poles are their Styles, equally elevated according to the aequino­ctiall altitude, being the complement of the Poles ele­vation. For in all North Faces, Planes, or Dials, the Style beholds the North Pole, and in all South faces, the Style beholds the South Pole: therefore, where the North Pole is elevated, there the North Pole must be pointed out by the Style, and where the South Pole is elevated vice versa.

The second sort of Verticals are declining, which ate such that make an acute angle with the Quarter from which they decline; for an acute angle is lesse then a right angle, and a right angle is 90 degrees: these declining Planes ly­ing in some accidentall Azimuthe.

For supposing a Diall to turn from the South or North towards the East or West, the Meridian line of the South declines Eastward, happening in these Azimuthes or be­tween them.

• South declining East
• South declining West

• S by E	11	15	Or to these points of the West decli­ners, or be­tween them.	S by W	11	15
  S S E	22	30		S S W	22	30
  S E by S	33	45		S W by S	33	45
  South-East	45	00		South West	45	00
  S. E by E	56	15		S W by W	56	15
  E S E	67	30		W S W	67	30
  E by S	78	45		W by S	78	45
  East	90	00		West.	90	00

Again, North decliners, declining toward the East and West, doe happen in these Azimuthes or between them.

• [Page 29]North declining East
• North declining West

• N by E	11	15	Or to these points of the West decli­ners, or be­tween them.	N by W	11	15
  N N E	22	30		N N W	22	30
  N E by N	33	45		N W by N	33	45
  North-East	45	00		North West	45	00
  N E by E	56	15		N W by W	56	15
  E N E	67	30		W N W	67	30
  E by N	78	45		W by N	78	45
  East.	90	00		West.	90	00

By which it appeareth that every point of the Compasse is distant from the Meridian 11 degrees 15 minutes.

The third sort of planes are inclining, or rather reclining, whose upper face beholds the Zenith, and in that respect is called Reclining, but if a Diall be made on the nether side, and thereby respect the Horizon, it is then called an incliner, so that the one is the opposite to the other.

These planes are likewise accidentally divided, for they are either direct recliners, reclining from the direct points of East, West, North; and South, and in this sort happens the direct Polar and Aequinoctiall planes, as infinite more according to the inclination or reclination of the plane, or they are as erect planes doe become declining recliners, which looke oblique to the Cardinall parts of the World, and obtusely to the parts they respect.

Suppose a plane to fall backward from the Zenith, and by consequence it falls towards the Horizon; then that represents a Reclining plane, such you shall you suppose the Aequinoctiall Circle in the figure to represent, reclining from the North Southwards 51 degrees from the Zenith, or suppose the Axis to represent a plane lying parallel to it, which falls from the Zenith Northward reclining 38 de­grees, one being Aequinoctiall, the other a Polar plane.

[Page 30]But for the inclining decliners you shall know them thus, forasmuch as the Horizon is the limiter of our sight, and being cut at right angles representeth the East, West, North, and South points, it may happen so that a plane may lie between two of these quarters in an accidentall Azimuth, and so not beholding one of the Cardinall Quar­ters is said to decline: Again, the said plain may happen not to stand Verticall, which is either Inclining or Recli­ning, and so are said to be Inclining Decliners: First, be­cause they make no right angle with the Cardinal Quarters: Secondly, because they are not Verticall or upright.

There are other Polar planes, which lie parallel to the Poles under the Meridian, which may justly be called Me­ridian plains, and these are erect direct East and West Dials, where the poles of the plane remain, which planes if they recline, are called Position planes, cutting the Horizon in the North and South points, for Circles of position are nothing but Circles crossing the Horizon in those points.

CHAP IV.

A Meridian Line is nothing else but a line whose outmost ends point due North and South, and consequently lying under the Meridian Circle, and the Sun comming to the Meridian doth then cast the shadow of all things Northward in our Latitude; [Page 31] so that a line drawn through the shadow of any thing per­pendicularly eraised, the Sun being in the Meridian, that line so drawn is a Meridian line, the use whereof is to place planes in a due scituation to their points respective, as in the definition of this Circle I shewed there was acci­dentall Meridians as many as can be imagined between place and place, which difference of Meridians is the Lon­gitude, or rather difference of Longitude, which is the space of two Meridians, which shews why noon is sooner to some then others.

The Meridian may be found divers wayes, as most com­monly by the Mariners compasse, but by reason the needle hath a point attractive subject to errour, and so overthrow­eth the labour, I cease to speake any further.

It may be found in the night, for when the starre called Aliot, seems to be over the Pole-starre, they are then true North, the manner of finding it, Mr. Foster▪ hath plainly laid down in his book of Dyalling, performed by a Qua­drant, which is the fourth part of a circle, being parted into 90 degrees.

It may also be fouhd as Master Blundevile in his Booke for the Sea teacheth, being indeed a thing very necessary for the Sea, which way is thus: Strike a Circle on a plain Superficies, and raise a wire, or such like, in the center to cast a shadow, then observe in the forenoon when the shadow is so that it just touches the circumference or edge of the Circle, and there make a mark; doe so again in the afternoon, and at the edge where the shadow goes out make another mark, between which two marks draw a line; which part in halfe, then from that middle point to the center draw a line which is a true Meridian.

Or thus, Draw a great many Circles concentricall one [Page 32] within another, then observe by the Circles about noone when the Sun casts the shortest shadow, and that then shall represent a true Meridian, the reason why you must observe the length of the shadow by circles & not by lines is, because if the Sun have not attained to the true Meridian it wil cast its shadow from a line, and so my eye may deceive me, when as by Circles the Sun casting shadow round about, still meetes with one circumference or other, and so we may observe diligently. Secondly, it is proved that the shadow in the Meridian is the shortest, because the Sun is neerest the Verticall point. Thirdly, it is proved that it is a true Meridian for this cause, the Sun, as all other Lumi­nous bodies, casts his shadow diametrically, and so being in the South part casts his shadow northward, and is there­fore a true Meridian.

But now to finde the declination of a wall, if it be an erect wall draw a perpendicular line, but if it be a declining reclining plane, draw first an horizontall line, and then draw a perpendicular to that, and in the perpendicular line strike a Style or small Wyre to make right angles with the plane, then note when the shadow of the Style falleth in one line with the perpendicular, and at that instant take the altitude of the Sun, and so get the Azimuthe reckoned from the South, for that is the true declination of the wall from the South. The distance of the Azimuthes from the South, or other points, are mentioned in degrees and minutes in the third Chapter, in the definition of the seve­rall sorts of planes: or by holding the streight side of any thing against the wall, as is the long Square ABCD, whose edge AB suppose to be held to a wall, and suppose again that you hold a thrid and plummet in your hand at E, the Sun shining, and it cast shadow the line EF, and at the [Page 33] same instant take the alti­tude of the Sun, thereby getting the Azimuthe as is taught following, then from the point F, as the center of the Horizon., and from the line FE, reckon the distance of the South, which suppose I finde the Azimuthe to be 60 degrees from the East or West, by the propositions that are delivered in the end of this Booke, and because there is a Quadrant of a Circle between the South, and the East or West points, I substract the distance of the Azimuthe from 90 degrees, and it shall leave 30, which is the declination of the wall, equall to the angle EFG: but to finde the inclination or reclination, I shall shew when I come to the use of the Universall Qua­drant, or having first found the Meridian line, you may prick down the Azimuthe.

CHAP V.

LIght being the cause primary of shadows, sha­dows being but the imitation of the secondary cause, that is substance, doth delineate unto us the passing away of time, by receiving light on the substance casting shadow.

[Page 34]Neverthelesse, substance receives not light, if either they want the immediate beame or reflecting light, which is the reason that some Dials are vacant of diverse houres, or else are vacant for a certain Season of the year, wherefore we will shew some reasons why the Sun beams cannot be re­ceived

on diverse planes, which is caused by the acciden­tall scituation thereof, which we will consider by this fi­gurative demonstration, by the Analemma, described in Mr. Gunters Book.

The Sun, though he never moves from the line Eclip­tique wherein he hath his annuall or yearly motion, yet [Page 35] have a declination from the Aequinoctiall North or South, making his diurnall or daily motion, altering the dayes and nights according to all the diversities thereof: for the Sun being in the Aequinoctiall hath no declination, but in his diurnall motion still declyning from the Aequinocti­all makes his progresse towards the North or South, descri­beth many parallel Circles, being parallel to the Aequino­ctiall, whose farthest distance from either side is 23 deg. 30 minutes, so that so many degrees that the Sun is distant from the Aequinoctiall, so much is its declination.

Now if you imagine the Circle before described to re­present the Meridian Circle which crossed diametrically, which diameter shall represent the Aequinoctiall, then lay­ing down the greatest declination, on either side of it, drawing two lines at that distance, on either side of the Aequinoctiall, parallel to it, represent the Tropicks, the upper representing the Tropick of Cancer, marked with GE, the other the Tropick of Capricorn, marked with HI: and if from each severall degree you draw parallels too, they doe represent the parallels of the Sun, which shall shew the diurnall motion of the Sun: now if you crosse these parallels with a line from E to H, that then represents the Ecliptique; now if you crosse the Aequino-Ctiall at right angles with another line, that line represents the Axis of the World: then if you lay down from the Poles the elevation thereof, to wit, the North and South Poles, according to the elevation of the North Pole down­ward, where the number of degrees end make a mark; then account the same elevation from the South Pole up­ward, and there also make a mark, from which two marks draw a right line, which shall represent your Horizon, and cuts the parallels of the Sun according to the time of his abiding above the Horizon.

[Page 36]As in example, they that live under the Aequinoctiall have their dayes and nights equall, for under the Aequi­noctiall the Poles lie in the Horizon, and have no elevation, so that you see the Axis AB cuts the Aequinoctiall at right

angles, and then must needes cut the parallels of the Sun equall, so that the continuing of the Sun above the Poles or Horizon, is equall to his continuance under the Poles or Horizon; so that there is represented a right Sphear where the dayes and nights are equall. But if the Pole hath ele­vation, [Page 37] as here at London, 51 degrees 30 minutes, then the Horizon is represented by CD, where you see that then the horizon cuts the parallel of the Sun oblique, represent­ing an oblique Sphear, so that now the lines grow longer while the Sun declines in them towards the North pole A, then the day is represented by the parallel EF, and the night by FG, when the Sun is in his greatest North decl­nation, so that you see that then in the night the Sun is no lower under the Horizon then from C to G, and then 'tis twi-light all night. Again, the Sun having his greatest de­clination towards B the South Pole, then he continues but the arch ID above the Horizon, then the day is represent­ed by KI: and the night by HK: Thus you see the reason of the dayes and nights inequality in an oblique Sphear, and equality in a right, you may likewise perceive by those parallels, why the Sun cannot shine on all Diall Planes, as we will now shew.

First, An East and West Diall lies parallel to the Me­ridian, therefore the Sun in the Meridian cannot shine on them; neverthelesse, though an East and West Diall cannot have the houre of 12 on it, yet an East or West position may, because it crosseth the Horizon in the North and South.

Secondly, A direct North Diall can have but morning and evening houres on it, and then of no use but when the Sun hath North declination, for then his Amplitude or distance from the East and West is Northward, and so at morning or night shines on the face thereof.

Thirdly, A North reclining may shew all the houres all the year, if it recline from the North Southward, the quantity of the complement of the least Meridian altitude, but if but the complement of the elevation of the Aequi­noctiall, [Page 38] and so become a Polar Plane, it can then but shew while the Sun is in the North Signes, for the Dyall lying parallel to the Aequinoctiall while the Sun is in South de­clination cannot shine on the plane because it lies under.

All upright planes declining from the South may have the houre line of 12, so also may all North decliners, but not in the Temperate Zone, which is contained between the degrees. South incliners also may have the line of 12, whose upper face is not below the least Meridian altitude, as also if greater then the greatest Meridian altitude, then doth the upper face want it.

Fifthly, all North recliners reclining more then the great­est meridian altitudes complement, may have all the houres but will shew but one part of the yeare.

Sixthly, All South declinets or recliners may have the line of 12 on them. And now having proceeded thus far in some theoricall demonstration or grounds of Dials for the Geometricall projection, we will in the next Chapter lay down the theoricall demonstration for the Arithmeticall Calculation, and so proceed to our practicall way of opera­tion as ensueth.

CHAP VI.

THe Arithmeticall part for calculation is thus to be understood, there are certain right lines from e­very degree of a Quadrant, named by certain words of art, which for illustration we will con­sider in their nomination and definition, whose names are [Page 39] these, Sines, Tangents, and Secants, which have certain arches of degrees and minutes of a Quadrant answering thereto, and howsoever the Question is propounded, are resolved by these numbers as in the Golden Rule, by still adding the second and third number together in stead of Multiplication, and substracting the first instead of Divisi­on, doth leave the arch of the Question as it was propoun­ded, which well considered, nothing shall seeme difficult.

A right Sine is halfe the subtense of the double arch, which subtenses are represented by the lines passing from D to D▪ and from E to E, and from F to F, the halfe of which lines subtending the arches are the right Sines.

A Tangent is a right line without the peripherie to the extremity of the Secant to the Radius being perpendicu­lar eraised, such is represented by the line BC.

A Secant is a right line drawn from the center through the circumference to the Tangent, such is represented by the line AB, the Semidiameter of the same Circle is called the Radius.

[Page 40]You may furthermore for very convenient uses have those lines placed on a Ruler, for if from one degree of one Quadrant of a Semicircle you draw lines to the same degree of the other Quadrant, cutting the line GA, that line so cut shall be a line of Sines, and if from the centre you draw lines to the Tangent line through every degree of the Quadrant, that line so cut is a Tangent line, whose use is most exquisite and infinite for the solution of many excel­lent propositions.

CHAP VII.

THe Style being the representation of the Axis of the World, doth become the Gnomon or substance casting shadow on all Planes lying parallel to some Circle or other, as to circles of Azimuthes in all Verticall Dials.

So that the figure following is a representation of divers semidiameters, doth plainly shew the theoricall ground of the practick part hereof.

Where the line in the demonstration, noted the semidi­ameter of the Horizon, signifies the Horizon, for so sup­posing it to represent an Horizontall Diall, the style or Axis must be elevated above it, according to the Poles ele­vation above the Horizon, and then the semidiameter or Axis of the World represents the style or Axis casting sha­dow being the line AC.

CHAP VIII.

SEeing the Zenith makes right angles with the Horizon, and a right angle consisteth of 90 degrees, the middle point betwixt both is 45 degrees, the Sun being at that height, the shadow of all things perpen­dicularly raised, are equal to their bodies, so also is the Radius of a Circle equall to the Tangent of 45 degrees: and if the Sunne be lower then 45 degrees it must necessary follow the shadow must exceed the substance, because the Sun is nigh the Horizon, and this is called the adverse or contrary shadow.

Contrarily, if the Sun exceed this middle point, the substance then exceeds the shadow, because the Sun is neerer [Page 43] the verticall point. Mr. Diggs in his Pantometria laying down the manifold uses of his Quadrant Geometricall, doth there shew, that having received the Sun beams through the Pinacides or Sights, that when the Suns alti­tude cuts the parts of right shadow, then the shadow ex­ceeds the substance erected casting shadow as 12 exceeds the parts cut: But in contrary shadow contrary effects.

CHAP IX.

TO give you Orontius his words, it is con­venient to take the beginning from the greatest obliquation of the Sun, because on that almost the whole harmony of all Astronomicall matters seeme to depend, as shall be manifest from the discourse of the succeeding Canons.

Wherefore prepare of commodious and elect substance, a Quadrant of a Circle parted into 90 equall parts, on whose right angled Radius let be placed two pinacides or sights to receive the beams of the Sun.

Then erect it toward the South in the time of the Solsti­cials, either in Cancer the highest annuall Almicanther, or in Capricorn the lowest annuall▪ meridian altitude, also ob­serve the equilibra, or equality of day and night in the time of the Aequinoctials, from the Meridian altitude thereof substract the least Meridian altitude, which is, when the Sun enters in the first minute of Capricorn, the remainer is the Declination, or substract the Aequinoctiall altitude from [Page 44] the greatest Meridian altitude, the remainer is the Decli­nation of the greatest obliquity of the Sun in the Zodiaque.

The height of the Sun is observed by the Quadrant when the beames are received through the sights by a plum­met proceeding from the center, noting the degree of alti­tude by the thrid falling thereon.

You may also take notice that for the continuall variati­on of the Suns greatest declination it ought to be observed by faithful Instruments: for as Orontius notes that Claudius, Ptolomie found it to be 23 degrees 51 minutes and 20 se­conds, but in the time of Albatigine the same number of degrees yet but 35 minutes, Alcmeon found it of little lesse, to wit 33 minutes, Purbachi and some of his Disciples doe affirme the same to be 23 degrees only 28 minutes, yet Jo­hanes Regiomontan. in the tables of Directions, hath alotted the minutes to be 30, but since Dominick Maria an Italian, and Johannes Varner of Norimburg testifie to have found it to be 29 minutes, to which observation our works doe ex­actly agree. Albeit all did observe the same well neere by like Instruments, neverthelesse, not justly by exact constru­ction, or by insufficient dexterity of observation some small difference might happen, but not so much as from Ptolo­mie to our time.

Having this greatest Declination, to finde the present Declination is thus, by calculation: As the Radius, is to the Sine of the greatest Declination; so is the Sine of the Suns distance from the next Aequinoctiall point, that is Aries or Libra, to the declination required: wherefore in the Naturall Sines, as in the Rule of Proportion, multiply the second by the third, divide by the first, the Quotient is the Sine of the Declination. Or by the naturall Sines, adde the second and third, and substract the first, the remainer is the Sine of the present Declination.

[Page 45]

But I have here ad­ded a Table of Decli­nation of the part of the Ecliptique from the Aequinoctiall, the use whereof you may dis­cern is very plain, for if you finde the Signe on the top, and the de­grees downward, the common angle shall be the Declination of the Sun that day. As if the Sun being in the 10 degree of Taurus or Scorpio, the declina­tion shall bee 14 de­grees 50 minutes, and if you finde the Signe in the bottome, you shall seeke the degrees on the right hand up­ward, so the 20 de­greee of Leo or Aqua­rius hath the same de­clination with the for­mer.

CHAP I.

FRom the Theoricall Demonstra­tion before, take the semidia­meter of the Horizon with your Compasses, then draw the line AB, representing the Meridian or line of 12, and setting one foot in A, describe the Qua­drant CAB, and CA must be at right angles to AB, to which Quadrant draw the tangent line FA, which is the line of contingence, then take from the Theorical demonstration the semidiameter of the Aequator, [Page 48] and placing that on the line AB desctibe a quadrant touch­ing the line of contingence also within the other, represent­ed by the Quadrant H e I which divide into 6 parts, and a Ruler laid to the center e, make marks where the Ruler toucheth the line of contingence, which must be continued beyond F, that so the houre lines may meet with the line BF, where it crosseth that line make marks: then remo­ving the Ruler to the center A of the horizontall Semi­circle, draw lines through each mark of the line of contin­gence which shall be the houres, number the morning houres from the Meridian towards your left hand, and eve­ning or afternoon houres towards the right. The Style must be an angle equall to the elevation of the Pole, the 12 houre must lie under the Meridian Circle.

CHAP II.

EVery plane hath a Verticall point, and for the making of a Verticall Diall for the Latitude of London, out of the theoricall demonstrati­on Chap. 7. Praecog. Astron▪ take the semidia­meter of the Verticall, and with that, as with the semidiameter of the Horizon, describe a Quadrant, & draw the tangent line FG, and with the semidiameter of the Aequator finish all as in the Horizontall: the Style must proceed from the center A, and be elevated from the Meridian line AF, so much as is the complement of the E­levation of the Pole, and must point toward the invisible Pole, viz. the South Pole, and hath but 12 houres on it.

CHAP III.

THe North Diall is but the back side of the South Diall; and differeth little from it, but in naming of the houres, for account­ing the sixth houre from the Meridian in the direct South verticall, to be the same in the direct North Verticall, and account­ing the first houres on the East side of the South, on the West side of the North plane, and so vice versa, the first houres on the West side of the South, on the East side of the North plane, as by the figure appeareth.

And because the North Pole is elevated, the Style must point up toward it the visible Pole.

It must have but the first and last houres of the South plane, because the Sun never shines but at evening or morning on a North wall in an oblique Sphear, and but in Somer, because then the Sun hath North Declination, but in a right Sphear, it may shew all the houres as a South Di­all, but for a season of the yeare.

But if you will make the Verticall plane or Horizontall in a long angled Parallelogram, you shall take the Secant of the elevation of the Pole, which is the same with AC in the fundamentall Diagram, and make that your Meridi­an line, and shall take the Sine of the elevation of the Pole above the Meridian, which in a direct South or North is equall to the elevation of the Aequinoctiall, and in the Fun­damentall [Page 53] Diagram is the line DE, and prick it down from A and C at right angles with the line AC, and so inclose the long square BADBCD, it shall be the boūds of a direct North or South Diall; lastly, if from the fundamental di­agram you prick down the several tangents of 15, 30 45, from Band D on the lines BB and DD, & the same distances from C toward B and D, & lastly if from the cen­ter A, you draw lines to every one of those marks, they shall be the houre-lines of an erect direct South Diall.

To make an Horizontall Diall by the same projection you shall take the Secant of 38 deg. 30 min. the elevation of the Equator, which in the fundamentall Scheme is the line AF, for the Meridian, and the Sine of the elevation of the Pole, which in the fundamentall diagram is the same with DA, and [Page 54] prick that down from the Meridian at right angles both wayes, as in the former planes, and so proceed as before from the six of clock houre and the Meridian, with the se­verall Tangents of 15, 30, 45, you shall have constituted a Horizontall plane.

I have caused the pricked line that goes crosse, and the other pricked lines which are above the houre line of six, to be drawn only to save the making of a figure for the North direct Diall, which is presented to you if you turn the Book upside down, by this figure, contained between the figures of 4, 5, 6, the morning houres, and 6, 7, 8, the evening. And because the North pole is elevated a­bove this plane 38 deg. 30 min. the Axis must be from the center according to that elevation, pointing upward as the South doth downward, so as A is the Zenith of the South, C must be in the North.

The Arithmeticall calculation is the same with the for­mer, also a North plane may shew all the houres of the South by consideration of reflection: For by Opticall de­monstration it is proved, that the angles of incidence is all one to that of reflection: if any be ignorant thereof, I pur­posely remit to teach it, to whet the ingenious Reader in labouring to finde it.

CHAP IV.

FOr the effecting of this Diall, first draw the line AD, on one end thereof draw the circle in the figure representing the Equator; then draw two touch lines to the Equator, parallel to the line AD, these are they on which the houres are marked: divide the Equator in the lower semi­circle in 12 equall parts, then apply a ruler to the center, through each part, and where it touches the lines of con­tingence make marks; from each touch point draw lines to the opposite touch point, which are the parallels of the houres, and at the end of those lines mark the Easterly houres from 6 to 11, and of the West from 1 to 6. These planes, as I told you, want the Meridian houre, because it is parallel to the Meridian. Now for the placing of the East Diall, number the elevation of the Axis, to wit, the arch DC, from the line of the Equator, to wit, the line AD: and in the West Diall number the elevation to B; fasten a plummet and thrid in the center A, and hold it so that the plummet may fall on the line AC for the East Diall, and AB for the West Diall, and then the line AD is parallel to the Equator, and the Dial in its right position. And thus the West as well as East, for according to the saying, Contrariorum eadem est doctrina, contraries have one manner of doctrine.

Here you may perceive the use of Tangent line, for it is evident that every houres distance is [...]t the Tangent of the Aequinoctiall distance.

CHAP V.

I Call this a direct parallel Polar plane for this cause, because all planes may be called by their scituation of their Poles, and so an Aequinoctiall parallel plane, may be called a Polar plane, because the Poles thereof lie in the poles of the World.

For the projecting of the parallel polar plane, first draw the horizontall line AB, then with Scale and Compasses protract an angle of 15 degrees, at each end crossing as in the following figure: which crossing is the center of the E­quator, and the least distance between the horizontall line AB and the center of the Equator, is the semidiameter of the Equator circle, then with the turning of the compasses one foot fixed in the center, describe an obscure circle, at the length of the diameter of that circle, draw a parallel line [Page 58]

to AB, which is CD, now for the houre distances, the Equator must be divided into 24 parts, the houres of a na­turall day, for the performance of which, you shall thus doe it easilie. First, take the semidiameter of the Equator with your Compasses, and place one foot, resting at that opening, in either the angle ABC or D, which in example is at B, then with the other foot describe an arch from the horizontall line to the line BC, which is the small pricked arch EF, which is the 24 part of the Equator, which parts prick about the circle, numbring from the diameter 12 each way; then lay a ruler through each division and the center, and where it crosseth the horizontall lines AB and BC make marks, then from each opposite marks draw lines pa­rallel to the diameter, which are the houre lines, as in the second figure.

The Gnomon must be a quadrangled Parallelogram, whose height is equall to the semidiameter of the Equator, as in the East and West Dials, so likewise these houres are Tangents to the Equator.

CHAP VI.

AN Aequinoctiall plane lyeth parallel to the Aequi­noctiall Circle, making an angle at the Horizon e­qual to the elevation of the said Circle: the poles of which plane lie in the poles of the world. The making of this plane requires little instruction, for by drawing a [Page 60] Circle, and divide it into 24 parts the plane is prepared, all fixing a style in the center at right angles to the plane.

As the Radins, is to the sine of declination, so is the co­tangent of the Poles height, to the tangent of the distance of the sub-stile from the Meridian.

If you draw lines from 7 to 5 on each side, those lines so cut shall be the places of the houre lines of a parallel polar plane, now if you draw to each opposite from the pricked lines, those lines shall be the houre lines of the for­mer plane.

CHAP VII.

IF you will work by the fundamentall Diagram, you shall first draw a line, such is the line AB, representing the Meridian, then shall you take out of the fundamentall diagram the Secant of the Latitude, viz. AC, and prick it down from A to B, and at B you shall draw a horizontall line at right angles, such is the line CD, then you shall continue the line AB toward i, and from that line, and where the line AB crosseth in CD, describe an arch equall to the angle of Declination toward F if it decline Eastward, and toward G if the plane decline Westward. Then shall you prick down on the line BF, if it bean Easterly declining plane, or from B to G if contrary; the Secant complement of the Latitude, viz. AG in the fundamentall Diagram, and the Sine of 51 degrees, viz. DA, which is all one with the semidia­meter of the Equator, and therewithall prick it down at [Page 61]

[Page 62] right angles to the line of declination, viz. BF, from B to H and G, and from F towards K and L, then draw the long square KIKL, and from B toward H and G, prick down the severall tangents of 15, 30, 45, and prick the same di­stance from K and L towards H and G: lastly, draw lines through each of those points from F to the horizontall line CD, and where they end on that line to each point draw the houre lines from the point A, which plane in our ex­ample is a Verticall declining eastward▪ 45 degrees, and it is finished. But because the contingent line will run out so far before it be intersected, I shall give you one following Ge­ometricall example to prick down a declining Diall in a right angled parallelogram.

Now for the Arithmeticall calculation, the first opera­tion shall be thus: As the Radius, to the co-tangent of the ele­vation, so is the sine of the declination, to the tangent of the substiles distance from the meridian of the place. then,

CHAP VIII.

FOr an Horizontall Diall enter the Table with the elevation of the Pole on the left hand, and the arches noted against the houres and the e­levation found, are the distance of the houres from the Meridian.

For a Verticall or direct South or North, enter the Ta­ble with the complement of the elevation on the right side, and the common meeting of the houres at top, and the com­plement of elevation, is the distance of the houres from the Meridian in the said plane. For every horizontall plane is a direct Verticall in that place whose Latitude or distance of their Zenith from the Aequator, is equall to the comple­ment of the elevation of the Horizontall planes Axis or style.

As to make an Horizontall Diall for the Latitude of 51 degrees, I enter the Table and finde these Arches for 1 and 11, for 2 and 10, &c. Now the same distances are the distances of the houre lines of a direct South plane, where the Pole is elevated the complement of 51 degrees, that is 39 degrees, for 51 and 39 together doe make 90.

So to make a Verticall diall, I enter the Table with 39, the complement of the elevation of the pole, and finde the arches answering to 1 and 11, to 2 and 10, &c. Thus much in generall of the use of the Table, now followeth the use in speciall.

CHAP IX.

LEt the great Circle ABCD represent the Meridian, A the North, and C the South, then the line EF represents a South reclining plane, while it fals back from the South North­ward, and represents an inclining plane while it respects the Horizon. This is sufficiently discussed be­fore.

[Page 73]So much as the plane reclines northward beyond the complement of the elevation of the Pole, so much is the North pole elevated above the plane, as here the plane is represented by EF, the elevation of the style or Axis the arch EG, therefore in this case substract the complement of the reclination of the plane from the elevation of the elevated pole, and the remainer is the arch of the poles elevation above the plane, with which elevation enter the Table in the left margent, and there are the houre arches from the meridian. If the reclination of the plane be lesse then the complement, as is IK, substract the arch of recli­nation from the complement of the elevation, there is left the elevation of the South pole above the plane, and with the complement of the elevation of the pole above the plane enter the table on the right margent, and there shall you finde the distance of the houres: and herein Mr. Faile failed, for instead of substracting one from the other, he addeth one to another, causing a great errour. The distance of every houre of the North incliner on the back side of the South incliner as much are equall, saving that the houres on the North side must be named by the complement houres to 12, and as the North pole is above one plane, so is the South pole above the other, you may also conceive the like in making of all South incliners and recliners, by framing the position of the plane on the South side as the figure is on the North: and in North recliners lesse then the eleva­tion of the pole, adde the reclination of the flat, which is the elevation of the North pole above the plane: herein Mr. Fail failed also, as depending on the former, following the doctrine of contraries, which formost well examined would have saved the opening of a gap to this second errour: With the said elevation found enter the Table for the Ho­rizontal [Page 72] [...] [Page 73] [...] [Page 74] arches, and thereby make a Horizontall▪ plane as is shewed, so is the Diall also prepared.

If it recline that it lie between the Horizon and the Equa­tor, then to the elevation of the Pole adde the complement of the reclination, which is the height of the style above the plane, and finish it as a Horizontall plane for that latitude, and not as a Verticall, as Mr. Faile would have it, because every reclining plane is a Horizontall plane where the pole is elevated according to the style.

CHAP X.

SO then, saith he, Si axis, &c. If the axis be oblique to the plane, as the foregoing are, as in any plane oblique to the Equa­tor many of the houre-lines doe concur at the axis with equal angles, but they are easily found thus.

From any point of the axis, as it were from the center of the World, for example in the meridional declining to the right hand 30 de. From the point E of the axis, let be drawn a line at right angles, as it were any radius of the Equator EF, which right line where it doth fall in the me­ridian of the plane, whether the same be the meridian of the place or not, thereby the meridian of the plane let another [Page 85]

right line be drawn at right angles FQ, which shall be the common section of the Equator with the plane, the Ra­dius of the Equator FE let be set again in the meridian of the plane, as is FG, and from G, as from the center of the world, let the circle of the Equator be described as great as you please, and again, let another Radius of the Equator [Page 86] be drawn from the center G to the intersection of the line of the Equator with the meridian of the place, or with the 12 houre line, which Radius, where it shall cut the Equator Circle, which here is in the point K, from thence the begin­ning being made, let the opposite semicircle of the Equator be divided into 12 parts, in the common sections of the Equator and plane, and by every one of the divisions from the center, let right lines be drawn, which may be put out, such are GP, GQ, &c. which when they shall touch the common section of the Equator, then the houre lines an­swering to those divisions necessarily shall passe, and so of every one of the houre lines to be drawn two points shall be obtained, one in the center L of the Diall, where they all meet, the other in the common section of the Equator and the plane, or in the line FQ, by which they all passe, by which two points, if the right lines LQ and LP be drawn, those so drawn are the houre lines. The like may be done of any other.

But because Pitiscus is mute in defining which part he takes for the right hand and which the left, we must search his meaning.

Pitiscus was a Divine is evident by his own words in his dedication, Celsitudini tuae tota vita mea prolixe me excusa­rem quod ego homo Theologus▪ &c. If we take him as hee was a Divine, we imagine his face to be towards the East, then the South is his right hand, and the North is his left hand.

That he was an Astronomer too, appeareth by his Books both of proper and common motion, then we must ima­gine his face representing the South, the East on his left hand, which cannot be, as shall appear.

[Page 87]Neither must we take him according to the Poets, whose face must be imagined toward the West.

In short, take him according to Geographie, represent­ing the Pole, and this shews the right hand was the East, and left the West, as is evident by the Diall before going, for it is a plane declining from the South to the right hand 30 degrees, that is, the East, because it hath the morning houres not the evening, because the Sun shines but part of the afternoon on the plane. Thus in briefe I have run throngh all planes, and proceed to shew you farther con­clusions: But I desire the Reader to take notice that in these examples of Pitiscus. I have followed his own steps, and made use of the Naturall Sines and Tangents.

CHAP XI.

HAving prepared a Horizontall Diall as is taught before: on the 12 houre, as far distant as you please from the foot of the style, draw a line perpendicular to the line of 12, on that describe a Semicircle, plasing the foot of the Compasses in the crossing of the lines, this Semicircle divide into 180 parts, each Quadrant into 90, to number the declination thereon, let the arch of the Semicircle be toward the North part of the Diall. [Page 88] Then prepare a plane slate, such as will blot out what hath been formerly made thereon, and make it to move perpen­dicularly on the horizontal plane on the center of the semi­circle, which wil represent any declining plane by moving it on the semicircle. Now knowing the declination of the plane turn this slate towards the easterly part, if it decline towards the East, if contrary to the West, if toward the West, and set it on the semicircle to the degree of declina­tion, then taking a candle and moving the Diall till the sha­dow fall on all the houres of the horizontall plane, mark al­so where the shadow falls on the declining plane, that also is the same houre on the plane so scituated, drawn from the joyning of the style with the plane. It is so plain it needs no figure.

So may you doe in all manner of declining reclining, or reclining and inclining Dials, by framing your instrument to represent the position of the plane.

Note also that the same angle the axis of the Horizontal Dial makes with the plane, the same elevation must the axis of that plane have, and where it shadows on the representing plane when the shadow of the horizontal axis is on 12, that is the meridian of the place.

By the same also may you describe all the conclusions Astronomicall, the Almicanthers, circles of height: the parallels of the Sun, shewing the declination: the Azi­muthes, shewing the point of the Compasse the Sun is in: and all the propositions of the Sphere.

Seeing this is so plain and evident, nay a delightful con­clusion, I will not give you farther directions in a matter of so great perspicuity, as to lay down the severall wayes for projecting the Sphere on every severall plane, but proceed to shew the making of a general Dial for the whole World, [Page 89] which we will use as our Declinatorie to finde the scituati­on of any wall or plane, as shall be required to make a Diall thereon, as followeth in the next Chapter.

CHAP XII.

THis Universall Diall is described by Clavi­us in his eighth Book de Gnomonicis: But because the Artists of these times have found out a more commodi­ous contrivance of it in the fa­brique, I shall describe it ac­cording to this Figure.

You shall therefore of com­modious matter prepare in sil­ver or brasse a concave Cross, such as is represented to your eye, whose three limbes EFG shall be equall, then shall you on the limbe 12 E describe as before a parallel Polar plane, as accounting that the height of the style, again, on the side

[Page 90] of the Crosse AF you shall describe an Orientall or East Diall, and on the other side a Western Diall, accounting the arms of the Crosse as the height of the style, so have you prepared the plane: you shall again provide a Box with a magneticall needle to be within the body of the Crosse, which shall be contrived to move on the shoulders of the Crosse, so as by the helpe of the degrees of a Qua­drant numbred on the limbe of the Box; you may depress or elevate the end of the Crosse E according to the com­plement of the latitude of the place.

Now to know the houre of the day, you shall turn the plane by the helpe of the needle, so as the end A shall be toward the North, and E toward the South, and elevate the end E to the complement of the elevation, then bring­ing the Box to stand in the Meridian, the shoulder of the Crosse shall shew you the houre.

Upon this also is grounded the Universall Quadrant hereafter described, which Instrument is made in Brasse by Mr. Walter Hayes as it is here described. Prepare a Qua­drant of Brasse, divide it in the limbe into 90 degrees, and at the end of 45 degrees from the center draw the line A B, which shall represent the Equator, divide the limbe into 90 degrees, as other Quadrants are usually divided, then number both wayes from the line AB the greatest declina­tion of the Sun from the North and South, at the termina­tion whereof draw the arch CD which shall be the Tro­picks, then out of the Table of declina­tion, pag. 45, from B both wayes let there be numbered the declinatiō of the Signes according to this Table.

First, for Taurus and Virgo, and his opposite Signe 11 deg. 30 m. namely, Scorpio and Pisces, [Page 91]

[Page 92] whereby divide the arch CD, which if you should draw a line from the center A, one should be the beginning of Taurus and Virgo, the other the beginning of Scorpio and Pisces. Again, if from B both wayes you number 20 de. 30 min. the end of which shall be the beginning of Gemini and Leo, or Sagittarius and Aquarius, number it as is de­scribed, together with a letter for the beginning of each moneth. So you have a quarter of the Zodiaque, and a quarter of a year projected on it.

Now the plane it selfe is no other then an East or West Diall, numbred on one side with the morning houres, and on the other with the evening houres, the middle line AB representing the Equator. And to set it for the houre, you shall project the Tropicks and other intermediate parallels of the Signes upon them as is hereafter shewed, but that the plane may not run out of the Quadrant you shal work thus, opening the Compasses to 15 degrees of the Quadrant, prick that down both wayes, at which distance draw pa­rallels to the line AB, and with the same distance, as if it were the semidiameter of the Equator, describe the semi­diameter of the Equator on the top of the line AB, which divide into 12 parts, and laying a ruler through the center and each of those divisions in the semicircle to those paral­lel lines on each side of AB, marke where they cut, and from side to side draw the parallel houre lines as is taught in the making of an East and West Diall, make those pa­rallel lines also divided as a tangent line on each side AB, so if this Quadrant were held on an East or West wall, and a plummet let fall from the center of the Equator where the style stands (which may be a pin fitted to take out and in, fitted to the height of the distance between the line A B and the other parallels, which is all one with the Ra­dius [Page 93] of the small Circle) it shall I say, be in its right sci­tuation on the East or West wall if you let the plummet and threed fall on the elevation of the Pole in that place.

But because we desire to make it generall, we must de­scribe the Tropicks and other parallels of declination upon it, as is usuall to be done on your Polar and East and West Diall, which how to doe is thus.

Having drawn the houre lines and Equator as is taught from E the height of the style, take all the distances between it and the houre lines where they doe crosse the line AB, and prick them down on the line representing the Equator in this figure from the center B. Then describe an occult arch of a Circle, whereon describe a Chorde of 23 degrees 30 minutes, with such other declinations as you intend on your plane. Then on the line representing the Equator, no­ted here with the figures of the houres they were taken from, 6, 7, 8, 9, 10, 11, at the marks formerly made, that was taken from E the height of the style, and every of the houres, from these distances I say raise perpendiculars to cut the other lines of declination, so those perpendiculars shall represent those houre lines from whence they were taken, and the distances between the Equator and the se­verall lines of declination shall be the same distances from the Equator, and the other parallels of declination upon your plane, through which marks being pricked down upon the severall hourelines from the Equinoctiall.

[Page 94]

If you draw those Hyper­bolicall lines, you shall have described the parallels of de­clination required.

But if you will performe the same work a second and easie way, worke by this Ta­ble following, which is uni­versall, and is composed out of the Table of Right & Ver­sed shadow.

Put this Table before thee, & for the point of each houre line whereby the severall pa­rallels of the Signes shall pass worke thus.

The style being divided into known parts▪ if▪ into 12, take the parts of shadow out of the Table in the same known parts by which the style is divided, & prick them down on each houre line as you finde it marked in the Table answering the houre both before and after noon. As suppose that a Polar plane I finde when the Sun is in Aries or Libra at 12 a clock the shadow hath no latitude, but at 1 and 11 it hath 3 parts 13 min. of the parts of the style, which I prick from the foot of the style on the houres of 1 and 11 both above and beneath the Equator: and for 2 and 10 I finde 6 parts 56 min. which I prick down also from the center to the houre lines of 10 and 2, and so of the other houre lines and parallels, through which if I draw [Page 95] those lines they shall represent the parallels of the Decli­nation.

If you would finde the houre of the day by this Qua­drant, you shall fix the style being of a small pin of the height of the semidiameter of the Equator in the center, then holding up the Quadrant till the thred and plummet fall on the degrees of the latitude of the place on the Tan­gent lines either on the morning or evening houres, and withall turning the Quadrant till you have caused the sha­dow to fall on the declination of the Sun for that day a­mong the parallels, the thred still falling on the elevation, the shadow of the Gnomon shall then give the houre of the day: the declination may be sought in the limbe of the Quadrant answering to the day of the moneth, and this Quadrant may be made use of diverse wayes for very good uses, especially if you cause to be drawn two lines more between the Equinoctiall line and tangent line, divided on either side, which shall be very apt for the making of any sort of Diall: but if you desire you may have the same [Page 96] on a Ruler, together with the use thereof, by the Instru­ment maker before mentioned: the manner of making whereof is as followeth▪ and the use whereof is in part laid down in the Chapter intreating of Declining planes. Let there be a right angled parallelogram ABCD, of what

magnitude you please, on the longer side BC let be cut off a line equall to the shorter AB, which shall be BE, and at E as a center describe the arke of a Circle FG, whereon prick down the Radius FG, then draw the line EA cut­ting the arch in H, the arch FH shall be the eight part of the Circle or the halfe of the Quadrant or right angle, and FG shall be foure parts of the whole Circle divided into 24, whereof FH shall be three of those parts of which H G is one part, so that if you do divide from H to F in three parts the arke FH shall be divided into three equall houres, let there be drawn from E occult lines cutting the side AB in L and M, the same distances shall be placed in the oppo­site side CD, so the right lines CO and CN shall be e­quall to BM and BL, then draw the right lines MO and LN, which are parallel to the sides BC and AD, so BC shall be the houre of 6 and 12, but the line MO of 11 & 1, [Page 97] and of 5 and 7, then the line LN is the hours of a and 10, as also 4 and 8, and AD the hours of 3 and 9; so as BC is the hour of 12 from mid-night, that is mid-day; M O 1 from the meridian, LN 2, AD 3, and again LN 4, MO 5, BC 6; and again NO 7, LN 8, AD 9; and again LN 10, MO 11, and BC 12; for the halfe hours and quarters, you may divide the Arches HI, IK, KF, into 2 or 4 equall parts, now if you divide any side, as the tangent in the quadrant is divided to the radius AB; this ruler shal be of excellent use, which also you may very well put upon a diagonall Scale without prejudice to the other work▪ the use of this Ruler you have in Clavius his Gnomonices. But to conclude this Chapter by reason I have shewed very many plain ways, I shall now endeavour because I have had occasion to speak in this Chapter of the parallels of declination, and have shewed how they are drawn in meridian and polar planes, I shal here insert a ta­ble of Shadows, by which you may be able to do the same on any other plane, but I would have you note that wherein my example was last upon a polar plane, by the table of the latitude of shadow, the work is however the same on a meridian plane, onely accounting the hours in this plane from the 6 a clock houre, as in the polar from the meridian, this quadrant as other quadrants is used, to take the inclination, or reclination of planes, which I will mention with a few cautions hereafter following.

Having promised in the description of the use of this Instrument, to shew how to finde the inclination and re­clination of a plane, I shal proceed to give you some cau­tions; First then, the quadrant is divided in the limbe, as other quadrants are into 90 degrees, by which is measured the angles of inclination or reclination, for if it be a decli­ning plane onely, the declination is accounted from the [Page 98] North or South toward the East or West, if it decline from the North, the North Pole is elivated above it, and the meridian-line ascendeth, if it decline from the South, the south pole is elivated above that plane, if it decline from the South Eastward, then is the style and sub-style refered toward the west side of the plane, if to the con­trary the contrary, and may have the line of 12 except north decliners in the temperate Zone, you may make use of the side of the quadrant to finde the declination, as is taught before page 33, observing the angle as is cut by the shadow of the thred held by the limbe, & through the cen­ter, and that side that lieth perpendicular to the Horizon­tal line which shal be the angle, as is before taught: And if the south point is between the poles of the plane and the Azimuth, then doth the plane decline Eastward, if it be the afternoon you take the Azimuth in, if it be the fore­noon you take the Azimuth in, and the south point be between it and the Poles of the planes horizontal line, it doth decline Westward, if contrary it is in the same quar­ter where the sun is: For an inclining plane, which is the angle that it maketh with the Horizon▪ draw a Horizon­tall line and crosse it again with a square, or verticall line, then apply the side of the quadrant to the vertical line at the beginning of the numeration of the deg. on the qua­drant, and the angle contained between the thred & plum­met, and the applyed side is the inclination; in all north incliners the north part of the meridian ascendeth, in south incliners the south part, and in east and west incliners, the meridian lyeth parallel with the Horizon.

And for the reclination it being all one with the inclina­tion, considered as an upper and under face of the same plane, if you cannot apply the side of the quadrant, you may set a square or ruler at right angles with the verticall [Page 99] line drawn on the upper face and apply the side of the quadrant to the edge of the ruler, and measure the quanti­ty of the angle by the thred and plummet: but this is of di­rect, howsoever these are subject to another passion of de­clining and inclining together, which must be sought se­verally, and such are those whose Horizontal line declin­eth toward the north or south and inclination from north or south, towarde the east or west, which must be sought severally.

CHAP XIII.

THe Table you see consisteth of 11 columns, the first being the minutes of the Suns altitude, and the greater figures on the top are the de­grees of altitude, all the other columns consist of the parts of shadow, and minutes of shadow, noted above with S for shadow, and p m for parts and minutes of shadow, answerable to a gnomon divided into 12 equall parts, and it is, As the sine of a known altitude of the sun, is to the sine complement of the same altitude; so the length of the Gnomon in 10 or 12 parts, to the parts of right shadow: or for the versed shadow, as the sine complement of the given altitude of the sun, to the right sine of the same altitude; so the style in parts, to the length of the versed shadow So if we enter the Table with the given altitude of the Sun in the great figures, and if we seeke the mi­nutes in the sides, either noted with horizontall or verti­call shadow, according as your plane is, it shall give you the length of the shadow in parts and minutes in the com­mon angle of meeting together. As if we look for 50 de. 40 m. the meeting of both in the Table shall be 9 parts 50 min. for the length of the right shadow on a horizontall plane: But for the versed shadow, take the complement of the altitude of the Sun, and the minutes in the right side of the Table, titled verticall shadow, and the common area of both shall give your desire. By this Table it appeareth [Page 106] first, that the circles of altitude either on the horizontall or verticall planes are easily drawn, consicering they are no­thing else but circles of altitude, which by knowing the al­titude you will know the length of the shadow, which in the horizontall Diall are perfect circles, and have the same respect unto the Horizon, as the parallels of declination have to the Equator, but in all upright planes they wil be conicall Sections, and by having the length of the style, the altitude of the Sun may be computed by the foregoing Table with much facility, but for the more expediating of the work in pricking down the parallels of declination with the Tropicks, I have here added a Table of the altitude of the Sun for every houre of the day when the Sun enters into any of the 12 Signes.

This Table is in Mr. Gunters Book, page 240 which if you desire to have the point of the Equinoctiall for a Ho­rizontall [Page 107] plane on the houre of 12, enter the Table of sha­dows with 38 de. 30 m. and you shall finde the length of the shadow to be 15 parts 5 m. of the length of the style di­vided into 12, which prick down on the line of 12 for the Equinoctiall point, from the foot of the style. So if I desire the points of the Tropick of Cancer, I finde by this Table that at 12 of the clock the Sun is 62 de. high, with which I enter the Table of shadows, finding the length of the shadow, which I prick down on the 12 a clock line for the point of the Tropick of Cancer at the houre of 12. If for the houre of 1, I desire the point through which the parallel must pass, looke for the houre of 1 and 11, in this last table under Cancer, and I finde the Sun to have the height of 59 de. 43 m. with which I enter the table of sha­dows, and prick down the length thereof from the bottome of the style reaching till the other foot of the Compasses fall on the houre for which it was intended. Doe so in all the other houres, till you have pricked down the points of the parallels of declination, through which points they must be drawn Hyperbolically. Proceed thus in the making of a Horizontall Diall, but if it be a direct verticall Diall, you shall then take the length of the verticall shadow out of the said Table, or work it as an Horizontal plane, only account­ing the complement of the elevation in stead of the whole elevation.

For a declining plane you may consider it as a verticall direct in some other place, and having found out the E­quator of the plane and the substyle, you may proceed in the same manner from the foot of the style, accounting where the style stands to be no other wayes then the me­ridian line or line of 12 in a Horizon whose pole is elevated according to the complement height of the style above [Page 108] the substyle, and so prick down the length of the shadows, from the foot of the style, on every one of the Houre lines, as if it were a horizontal or Verticall plane.

But in this you must be wary, remembring that you have the height of the sun calculated for every houre of that Latitude in the entrance of the 12 signes, in that Place where your Plane is a Horizontall plane, or otherwayes, by con­sidering of it as a horizontall or Verricallplane in another latitude

For the Azimuths, or verticall circles, shewing one what point of the compasse the sun is in every houre of the day it is performed with a great deale of facility, if first, when the sun is in the Equator, we doe know by the last Table of the height of the sun for every houre of the day and by his meridian altitude with the help of the table of shadows, find out the Equinoctiall line, whether it be a Horizontall or upright direct plane, for having drawn that line at right angles with the meridian, and having the place of the Style, and length thereof in parts, and the parts of shadow to all altitudes of the sun, being pricked down from the foot of the Style, on the Equinoctiall line, through each of those points draw parallel lines to the meridian, or 12 a clock line on each side, which shall be the Azimuths, which you must have a care how you denominate according to the quarter of heaven in which the sun is in, for if the Sun be in the easterly points, the Azimuths must be on the Western side of the plane, so also the morning houres must be on the opposite side.

There are many other Astronomical conclusions that are used to be put upon planes, as the diurnall arches, shewing the length of the day and night, as also the Jewish or old [Page 109] unequal houres together with the circles of position, which with the meridian and horizon distinguisheth the upper he­mispheare into 6 parts commonly called the houses of Hea­ven: which if this I have writ beget any desire of the reader, I shall endeavour to inlarge my self much more, in shewing a demonstrative way, in these particulars I have last insisted upon. I might heare also shew you the exceeding use of the table of Right and versed shadow in the taking of heights of buildings as it may very wel appear in the severall uses of the quadrant in Diggs his Pantometria, & in Mr. Gunters qua­drant, having the parts of right and versed shadow gradu­ated on them, to which Books I refer you.

CHAP XIV.

IT is an Axiom pronounced long since, by those who have writ of Opticall conceipts of Light and Shadow, that Omnis reflectio Luminis est secundum li­neas sensibiles, latitudinem habentes. And it hath with as great reason bin pro­nounced by Geometricians, that the Angles of Inci­dence and Reflection is all one; as appeareth to us by Euclides Catoptriques; and on this foundation is this conceipt of which we are now speaking.

[Page 110]Wherefore because the direct beams cannot fall on the face of this plane, we must by help of a piece of glasse apt to receive and reflect the light, placed somwhere horizon­tally in a window, proceed to the work, which indeed is no other then a Horizontall Diall reversed, to which required a Meridian line, which you must endeavour to draw and finde according as you are before taught, or by the helpe of the Meridian altitude of the Sun, your glasse being fixed marke the spot that reflects upon the seeling just at 12 a clock, make that one point, and for the other point through which you must draw your meridian line, you may finde by holding up a threed and plummet till the plummet fall perpendicular on the glasse, and at the other end of the line held on the seeling make another mark, through both which draw the Meridian line. Now for so much as the center of the Diall is a point without, and the distance be­tween the glasse and the seeling is to be considered as the height of the style, the glasse it selfe representing the cen­ter of the world, or the very apex of the style, wee must finde out those two Tangents at right angles with the Me­ridian, the one neere the window, the other farther in, through severall points whereof we must draw the houre­lines. Let AB be the Meridian line found on the seeling, now suppose the Sun being in the highest degree of Cancer should shine into the Glasse that is fixed in C, it shall again reflect unto D, where I make a mark, then letting a plum­met fall from the top of the seeling till it fall just on C the glasse, from the point E, from which draw the line A B through D and E, which shall be the Meridian required, if you do this just at noon: Now if you would finde out the places where the hour-lines shall crosse the Meridian, the Center lying without the window EC, you may work thus

[Page 111]Divide the line EC into 12 parts, then may you en­ter the Table of right & versed shadow, seeking the degrees and minutes of Elevation of the Equator, viz. 38 deg. 30′, to which answereth 15 parts 5 minutes of the parts of the Style, which prick down from E to F, and at that point draw the line FG, at right angles to the Meridian A

B, which shall represent the Equator. Now to know the center of the hour-lines, if you look again in the Table of shadows 51 deg. 30 min. the elevation of the Pole, you shall finde 9 parts 33 min. of shadow, which added to FE, 15 parts 5 m. shall give the center at B, making the whole line from F to B to be 24 parts 38 primes, and at 6 shall the center of the hour-lines meet. Lastly, supposing the former work to be done upon the floare, for the more ea­sie working, and having drawn a line representing the Me­ridian AB if from B you prick down 9 parts 33 min. of the radius EC, and at E draw it perpendicular to AB, & 15 parts 5 min. from E to F, where if you draw the line FG perpendicular to AB, also, if from the Center B you shall [Page 112] proceed to make a horizontal Dial as is above taught: it shall divide the lines EC and FG in such parts by draw­ing the hour-lines from the Center through those lines, which if they are transferred from thence to the seeling on each side the Meridian, upon the same line EC and FG, you have finished, for FB shal be the same as in the funda­mental Diagram I call the Semidiameter of the Horizon; and FC shal be the semidiameter of the Equator: by the same meanes also may be made hour-lines, on a Wall re­ceiving it's light only through some hole in a glasse win­dow: as also very recreative Dials on the sides of Build­ings, having water or glasse so placed, as that it may reflect the Beames according to several Azimuths: but the best way for the placing the glasse is, by preparing the hour­lines first, and by a known altitude of the Sun at some known houre of the day, to mark out on that hour-line where the reflection ought to fal, and expect that houre precisely, and depresse or elevate the glasse til it fal on that spot or mark assigned.

CHAP X.

THis may be used on a Staff or other round, made like a Cylinder being drawn as is here descri­bed, where the right side represent the Tro­picks, and the left side the Equinoctial: or it may be used flat as it is in the Book; the Instrument as you see, is divided into months, and the bottom into [Page 113] signs, and the line on the right side is a tangent to the radius of the breadth of the Pa­rallelogram, ser­ving to take the height of the Sun, the several Paral­lels downward running through the pricked line, in the midle, are the lines of Alti­tude, and the Pa­rallels to the E­quator are the Parallels of De­clination, num­bred on the bot­tom on a Sine of 23 de. and a half.

CHAP XVI.

A Globe (saith Euclide) is made by the turning about of a semicircle, & the Dia­meter fixed; this Dyal if universal, will want the aide of a magnetical Needle to set it, and it must move on an Axis in an horizon as the usual Globes doe; in whose Equator, let be divided into 24 houres, the proporion of the day natural, then on the Axis let a semicircle move. The figure is plain in the top of the Title, supported by the boy: this Globe set to the ele­vation of the Pole, & in the Meridian, let be placed the 12 [Page]

12 a clock line: then turne the Semicircle till it sha­dow not: then doth it crosse the houre: which houres are drawn from the poles to each of the 24 divisions.

If you desire to cover the Globes, and make other inventions thereon, first learn here to cover it ex­actly, with a pair of com­passes bowed toward the points, measure the Dia­meter of the Globe you intend to cover, which had, finde the Circumference thus; Multiply the Dia­meter by 22, and divide that product by 7, and you have your desire.

That Circumference, let be the line A B, which di­vide into 12 equal parts, and at the distance of three of those parts, draw the Parallel C D, and E F,

A Parallel is thus drawn, take the distance you would have it asunder, as here it is; three of those 12 divisions: set one foot in [Page 117] A, and make the Arch at E, & another at B, and make the Arch with the other foot at F, the compasses at the wide­ness taken, then by the outward bulks of those Arches, draw the line E F, so also draw the line C D.

And to divide the Circumference into parts as our ex­ample is into 12, work thus, set your compasses in A, make the Ark B F, the compasses so opened, set again in B, and make the Ark A C, then draw the line from A to F, then measure the distance from F to B, on the Ark, and place it on the other Arch from A to C, thence draw the line C B, then your compasses open at any distance, prick down one part less on both those slanting lines; then you intend to divide thereon, which is here 11: because we would divide the line A B into 12, then draw lines from each division to the opposite, that cuts the line A B in the parts of division.

But to proceed, continue the Circumference at length, to G and H, numbring from A toward G9 of those equal parts, and from B toward H as many, which shal be the Centers for each Arch.

So those quarters so cut out, shall exactly cover the Globe, whose Circumference is equal to the line A B. Thus have you a glance of the Mathematicks, striking at one thing through the side of an other: for I here made one figure serve for three several operations, because I would not charge the Press with multiplicity of figures.

CHAP XVII.

THis is done by erecting a gnomon horizontal, and at 3 times of the day to give a mark at the end of the [Page 118] Shadows: now it is certain, that represents the Parallel of the Sunne for that day; then take three thin sticks or the like, and lay them from the top of the gnomon, to the places where the shadows fell, and on these three so standing, lay a board to ly on all three flat, and a gno­mon in the midle of that board points to the Pole: be­cause every Parallel the Snn moves in, is parallel to the Equinoctial, and that is at right Angles, with the pole of the World.

Now the Meridian passeth through the most elevated place of that board or circle so laid, neither can the Sun's Declination make any sensible difference in the so small proportion of 3 or 4 houres time.

CHAP XVIII.

WIth your Compasses describe the Circle A B C D place it horizontal, with a gnomon in the Cen­ter, crosse it with two Diameters; then turn the board till the shadow be on one of the Dia­meters, at the end of the shadow, mark, as here at E, lay down also, the length of the gonmon from the Center on the other Diameter to F, from E to F drawe a right line: then take your Compasses, and on the

[Page 119] chord of 90, take out the Radius the Ark of 60, set the compasses so in E, describe an Arch, then take the di­stance between the line E F, and the Diameter D B; which measure on the chord of 90, and so many degrees as the compasses extend over; such a quantity is the height of the Sun, in like manner any Angles being given, you must measure it by the parts of a circle.