ANNO SALUTIS NOSTRAE J616.
ARTIFEX
Arithmetica, Geometria
THE SVRVEYOR in Foure bookes by AARON RATHBORNE
Thesaurū & talentam ne abscondas in agro.
Inertia stremia
LONDON Printed by W: Stansby for, W: Burre. 1616.
TO THE HIGH AND MIGHTY PRINCE CHARLES, Prince of WALES, Duke of CORNWALL, YORKE, ALBANY and ROTHSAY: Marques of ORMONT, Earle of ROSSE; and Baron of ARMANOCH: High Seneschall of SCOTLAND: Lord of the ILES; and Knight of the most noble Order of the GARTER.
IF in former ages (most noble Prince) the studies Mathematicall were held meet for Princes; I doubt not but in these, they may implore (by your fauourable admittance) the approbation and defence of your gracious patronage; which emboldens me the rather, to presume the dedication of these my labours to your Highnesse protection. Wisedome is defined by CICERO to be diuinarum at (que) humanarum rerum scientia, of the former part of which definition [Page](being the most absolute) I will leaue to speake vnto those who can better write; yet will thus much auerre, that no man shall obtayne the absolute perfection thereof, being absolutely ignorant of the rules, rudiments and principles of Mathematicall discipline, as the due consideration of that sacred and mysticall Vnitie and Trinitie, may well approue. And how auaileable and important they are, for the attayning to that humanarum rerum scientiam, in Peace or Warre, is worthily witnessed by PLATO, VEGETIVS, LIVIVS, and other Authors; who testifie of LYCVRGVS, and that famous Syracusean ARCHIMEDES; by the one, what excellent Lawes and Ordinances were established and ministred, in the time of peace; and by the other, what more then wonderfull deuises and stratagems were wrought against the inuincible forces of MARCELLVS, in the time of Warre, which they worthily impute to this their scientiae mathematicarum. But should my weakenesse here vndertake to illustrate the excellency of that worth, which all worthy men admire; and that to your Highnesse, whose iudgement is best able to discerne; were but to depraue the one and the other, and rest in mine owne reproch. Wherefore, assuring my selfe, of your Highnesse loue and affection to these Artes, and your respect to the Professors thereof; with your power and abilitie of defence, against the malignant courses of malicious detractors, I presume in all humilitie to intreate your Patronage of these my labours, which in all dutie I prostrate at your Highnesse feet; with continuall inuocation to the Prince of Princes, euer to preserue your Princely dignitie.
THE HIGHE & MIGHTIE PR. CHARLES PR. OF WALES. D. OF CORN: YOR: ALB: & ROTHS: MARQ: OF ORM: &c.
The Preface.
I Am not ignorant (friendly Reader) that hitherto, in writing, neuer any man pleased all; nor will I expect to be the first. To perswade the courteous, were causelesse, for they are naturally kind; and to disswade the captious, were bootlesse, for they will not be diuerted: Let the first make true vse of these my Labours, and they shall find much pleasure and profit therein; let the last (if they like not) leaue it, and it shall not offend them. To make apologie or exornation, in defence or commendation of the subiect whereof I treat, were needlesse; it being alreadie, in the worlds opinion, of sufficient habilitie and reputation, both to defend and commend it selfe: Onely of my manner and order in handling the same, I will say somewhat, as briefely as I may, for thy better instruction and vnderstanding thereof.
In generall, I haue disposed and digested the same into foure Bookes; whereof, the two former tend specially to the principles and rules of Geometrie, with performance thereby of many vsefull and necessarie conclusions; and the two later, to matter of suruey, with many instrumentall conclusions, tending as well to that, as diuers other purposes.
More particularly, in my first Booke, I begin with the Matters, Grounds, and Elements of GEOMETRIE, as the definitions and termes of art belonging thereunto; most fitting first by the Practicioner to be learned and well vnderstood: then next haue I placed therein diuers Geometricall THEOREMES, as the foundations, grounds, and reasons whereon the practike part dependeth.
In the second (hauing formerly laid the foundation) I shew the means and practike operation of manie necessarie conclusions and Geometricall PROBLEMES, as the distinction, application, and diuision of Lines and Angles; and the description, reduction, addition, inscription, transmutation, diuision, and separation of all formes and kinds of superficiall Figures, with their seuerall dimensions. And considering, that as well in the THEOREMES of the first Booke, as the PROBLEMES in the second, I wholly omit (for breuities sake, and auoiding confusion to [Page]the learner) their seuerall demonstrations, vsing onely explication and construction; I therefore expresse in the Margent against those THEOREMES and PROBLEMES, where, how, and in what place of EVCLID, RAMVS, and other Authors, to find their seuerall demonstrations at large: and likewise, at the end of each construction, I haue inserted the like numbers and notes of reference, from the THEOREMES in the first Booke, to the PROBLEMES in the second; and the contrarie: whereby most plainely and readily is found, and had, as well the reason and ground of any PROBLEME proposed, as the effect and operation of any THEOREME deliuered.
In the third Booke I begin with the description of the seuerall Instruments fit and vsuall for Suruey; and of their seuerall vses: wherein somewhat haue I spoken (though too sparingly) concerning their abuse; being now growne shamefully generall, by the multitude of simple and ignorant persons (vsing, or rather abusing, that good plaine Instrument, called the Plaine Table) who hauing but once obserued a Surueyor, by looking ouer his shoulder, how and in what manner be directs his sights, and drawes his lines thereon; they presently apprehend the businesse, prouide them of some cast Plaine Table, and within small time after, you shall heare them tell you wonders, and what rare feats they can performe; yea, and will vndertake (or I will for them) that for tenne groats a day, and their charges defrayed, they shall be able to vndoe any man they deale with; or at leastwise, to doe him such wrong and preiudice, as perhaps he might, with more ease, and lesse losse, haue giuen ten pounds a day to one that would haue spoken lesse, and performed more. But what should I say more of them then thus, Monoculi inter caecos oculissimi sunt, and so will I leaue the blind, with tumbling the blind into the myre.
I further describe in this third Booke, the composition and vse of an Instrument of mine owne, which I call the Peractor, and of a Chayne, which I call the Decimall Chayne, with the diuisions and parts thereof: which rightly vnderstood and practised (I dare boldly say without ostentation) is farre more vsefull and absolute for speed and exactnesse, then any yet euer vsed: And I will maintaine by sufficient demonstration, that no man (vsing not the same, or the like) shall attaine to the same or the like perfection, for precise exactnesse in any dimension, as I will thereby performe.
And I further shew therein, the best, speediest, and exactest meanes, for the suruey and instrumentall mensuration of a Mannor, or any other superficiall content whatsoeuer, by diuers and seuerall meanes; with [Page]manie extraordinarie obseruations and courses, therein to be had and taken, not vsually knowne or practised; as by the argument of that Booke more particularly appeareth.
Wherein by the way J would aduise the Reader, who desireth to make vse thereof, and to profit himselfe thereby, in reading and practising; to take the Chapters before him as they lye in order; for that I haue striued to place them in such an orderly and methodicall forme, as the one necessarily follow the other in vse and practise; well knowing that disorder and irregularitie in this kinde, breedeth not a little trouble and confusion to the weake practicioner.
The fourth and last Booke, consisteth of the legall part of Suruey; wherein I first shew what a Mannor is and the seuerall parts thereof, with the appendants thereunto; how the same is created and maintayned, and how and by what meanes destroyed and discontinued: also the seuerall sorts and kindes of estates whereby any lands or tenements may be holden, and the seuerall tenures, rents and seruices depending on those estates: I further shew therein the order and manner of keeping Courts of Suruey, with the entrie of the tenants euidence and estates; and the orderly and artificiall manner of ingrossing the same, with many other necessary rules and obseruations tending to those purposes, as more at large also appeareth by the argument of the same Booke.
And here, as before, would I aduise the practicioner, to obserue the like course in reading and practising the rules and instructions of this Booke, as I haue formerly directed for the third, for that J haue strictly obserued the like decorum in placing the Chapters each after other, as of necessity they are to be vsed and practised.
Now might I here much inlarge and protend this Preface, in explicating the wonderfull vse of the two former Bookes, in the performance of infinite conclusions Geometricall, so farre passing this subiect of Suruey, as it in it selfe exceedeth the meanest matter of dispose, which (to auoide prolixitie) J will here forbeare, leauing the consideration thereof to thine owne iudgement, when thou shalt finde therein by thy diligent practise the sweet and pleasing taste of such sense-beguiling fruit.
And further might I amplifie the same, not only in declaring the great and infinite pleasure, with no lesse profit, which the true knowledge, vse, and vnderstanding of the two later Books may bring (aswell to Surueyors, as all owners and occupyers of land in generall) but also of the antiquitie and necessitie of Suruey; howsoeuer sleighted by many, who will not bestow a penny in points, or two pence in tape, or the like, but they will number the one, and measure the other, before they pay for [Page]either, and yet will disburse manie thousands in a purchase, without the certaine knowledge of either quantitie, qualitie, or value thereof: (and these are those which are called penny-wise, &c.) whereby it often happens (as I haue often knowne) that a valuable purchase being made, within few weekes after, the money hath beene raised of the woods, and the lands perhaps immediately sold for much more then the money disbursed; and the same againe vented at the third hand, hath yeelded a double value: and all this vnseene, and vnsurueyed, with what disaduantage to the first vendor, I will leaue to the consideration of my young Master, who hath thus offended in selling all, and resteth now in repentance, with full resolution not to offend in the like. And the like haue I knowne of a purchase made, when a moitie of the charge could scarcely be raised.
But to spend time to this purpose, were to little end, and therefore will I end this purpose; only intreating thee gentle Reader, that as I haue thus imployed mine idle houres, to find thee houres of imployment; if thou reape either pleasure or profit by these my paines, to affoord me thy good opinion (for Virtus laudata crescit, & honos alit artes) which is all I craue.
THE SURVEYOR The first Booke.
THis first Booke consisteth of two parts; the former whereof intreateth onely of the first Matters, Grounds, and Elements of GEOMETRIE, as the distinction of Lines, Angles, Triangles, and other Figures, with their DEFINITIONS, shewing what they are: The second part containeth diuerse Geometricall THEOREMES, tending chiefely and most fitly to the subiect and matter prosecuted in the subsequent Bookes, whereby the ingenious Practitioner may readily conceiue and apprehend the ground and reason of the Precepts, Rules, and Problemes therein deliuered.
THE FIRST PART.
DEFINITION I. A Point is that which is the least of all Materialls, hauing neither part nor quantitie.
BEtweene Vnitie in ARITHMETIKE, Euc. 1. Def. 1. and this Point in GEOMETRIE, there is a neere resemblance: but that, more simple and pure; this, materiall; and (although the least that can be imagined) requireth position and place, as this Point A.
DEFINITION II. A Line is length, without breadth-or thicknesse.
THis is the first quantitie in GEOMETRIE, Euc. 1. Def. 2. and may be diuided into parts, in respect of his length, but admitteth no other diuision or dimension; and hath for his termes and limits that Geometricall point formerly spoken of. And of these lines are there two sorts; namely right, as the line A. and crooked, or sphericall, as the line B.
DEFINITION III. A Superficies is that which hath onely length and breadth.
THis is the second quantitie in GEOMETRIE, Euc. 1. Def. 5. hauing two seuerall dimensions, namely, length and breadth, without depth or thicknesse (for that belongs to a solid or bodie, being the third quantitie in GEOMETRIE, and impertinent to this place, whereunto is attributed all three dimensions, as length, bredth, and thicknesse.) And as a line is limited with points, so is a Superficies with lines, and a Solid or Bodie with Superficies. As the figure A. being a Superficies, hath for his length B. C. or D. E. and for his breadth B. D. or C. E. which foure lines are the bounds, limits, and termes of the same Superficies.
DEFINITION IIII. An Angle is the congression or meeting of two lines in any sort, so as both make not one line.
GEnerally of Angles (in respect of their lines) there are three sorts; Euc. 1. Def. 8. namely, right-lined, sphericall, and mixed. As the Angle A. is called a right-lined Angle, being composed of two right lines; the Angles B. and C. are sphericall, or crooked Angles; and the Angle D. a mixed Angle, being caused of both.
DEFINITION V. Jf a right line fall on a right line, making the Angles on either side equall, each of those Angles are called right Angles: And the line erected is called a Perpendicular line vnto the other.
AS in generall there are three sorts of Angles, in respect of their lines, Euc. 1. Def. 10. as aforesaid; so in particular, of right-lined Angles there are three sorts, in respect of their quantities; namely, an Orthogonall, or right Angle; an Ob [...]use, or blunt Angle; and [...]n Acute, or sharpe Angle: whereof, ac [...]rding to this definition, the line A. B. [...]ling on the line C. D. and making the Angles on either side equall, namely, the Angle A. B. C. on the one side equall to the Angle A. B. D. on the other side; those two Angles are called right Angles: And the line A.B. erected on C.D. without inclination to either side, is called a Perpendicular, or Plumbe line. And here note further, that vsually an Angle is described by three letters; of which, the second or middle letter representeth alwayes the Angle intended. PROB. 4.5.6.9.
DEFINITION VI. An Angle which is greater then a right Angle, is an obtuse Angle.
EVerie Angle in generall (not being a right Angle) whether greater or lesser, is called an Oblique Angle: Euc. 1. Def. 11. but particularly, if greater then a right Angle, it is called an Obtuse Angle; if lesser, an Acute Angle: As the Angle B. D. C. (being greater then the right Angle A. D. C.) is an Obtuse Angle; for it containeth it, and also the Angle A. D. B.
DEFINITION VII. An Acute Angle is that which is lesse then a right Angle.
THis Definition is manifest by the former Diagram, wherein the Angle A. D. B. is an Acute Angle, Euc. 1. Def. 12. being lesse then the right Angle A. D. E. for the same right Angle containeth it, and also the Angle B. D. E. being likewise an Acute Angle.
DEFINITION VIII. A Figure is that which is contained vnder one o [...] many limits.
AS these three figures, A. B. and C. whereof the first is contained vnder one limit; Euc. 1. Def. 14. the second, vnder three; and the third, vnder foure; and the like of others. Wherein is to be noted, that of two right lines, no Figure can be contained.
DEFINITION IX. A Circle is a plaine Figure, and contained vnder one line, which is called the Circumference thereof.
A Circle, Euc. 1. Def. 15. R [...]m. 1.15. of all other Figures, hath the prioritie, being of all most perfect and absolute; and therefore most fitting first to be defined: as the Figure A. in the last Diagram.
DEFINITION X. The Centre of a Circle is that point which is in the middest thereof: from which point, all right lines, drawne to the Circumference, are equall.
AS the point B. in this Figure is the Centre thereof, Euc. 1. Def. 16. from whence the lines B. A. B. C. and B. D. being drawne to the Circumference (and as manie others as are s [...] drawne) are all equall, and are semidiameters to the same Circle. PROB. 34.1.
DEFINITION XI. The Diameter of a Circle is a right line, passing by the Centre through the whole Circle, and diuideth the same into two equall parts: Either [Page 5]halfe of which Diameter, is called the Semidiameter of the same Circle.
AS the line A. B. C. in this circle is the diameter thereof, Euc. 1. Def. 17. for that it passeth by the centre B. through the whole circle, as from A. to C. and also diuideth the circle into two equall parts, the one halfe towards D. the other towards E. Either halfe of which Diameter, as A. B. or B. C. is called the Semidiameter of the same Circle. PROB. 34.
DEFIN. XII. A Semicircle is a Figure contained vnder the Diameter of a Circle, and the semicircumference of the same Circle.
AS supposing the circle A. B. C. the Diagram of the last DEFINITION, Euc. 1. Def. 18. to be diuided into two equall parts by the diameter thereof A. C. whereby two Figures are proiected, namely, A. D. C. and A. E. C. Now by this DEFINITION, each of these Figures are Semicircles, for that the one of them is contained vnder the diameter of the same circle A. C. and the semicircumference A. D. C. and the other vnder the same diameter, and the semicircumference A. E. C.
DEFIN. XIII. A Segment, Section, or Portion of a Circle, is a Figure contained vnder a right line, and part of the circumference, either greater or lesse then the Semicircle.
AS the diameter of a circle passing by the centre thereof, Euc. 1. Def. 19.3. Def. 5. diuideth the same into two equall parts; so any right line, drawne from any one part
of the circumference to any other part thereof (and not passing by the centre) diuideth the circle into two vnequall parts, which are called Segments, Sections, or Portions of a circle: As in this circle, the figure A. C. B. because it is contained vnder the right line A. B. and the lesser part of the circumference A. C. B. is called a lesser Segment, Section, or Portion of a circle; and the figure A. D. B. because it is contained vnder the right line A. B. and the greater part of the circumference A. D. B. is called a greater Segment, Section, or Portion of a circle. Here note also, that these parts and such like of the circumference so diuided, are commonly called Arches, or arch lines. And all lines (lesse then the diameter) drawne [Page 6]and applied as the line A. B. are called Cordes, or cord lines, of those Arches which they so subtend; or Subtenses, because they subtend both segments. THEOR. 73. PROB. 34.
DEFIN. XIIII. Equall Circles are such as haue equall Diameters, or whose lines, drawne from their Centres, are equall.
AS these two circles A. and B. are equall, Euc. 3. Def. 1. their diameters being equall, namely, C. D. and E. F. or their Semidiameters, (which, according vnto this DEFINITION, are lines drawne from the centres vnto their circumference) as A. C. or A. D. and B. E. or B. F.
DEFIN. XV. A right line is said to touch a circle, which touching it, and being extended or produced, doth not cut the circumference thereof.
AS the right line A. B. being drawne by the point C. doth there onely touch the circle, Euc. 3. Def. 2. and being produced vnto B. cutteth not the circumference thereof. This line is commonly called a Tangent, or Contingent line: whereof there is great and infinite vse, in manie Conclusions, Geometricall and Astronomicall, especially in the mensuration and resolution of Triangles, as well right-lined, as sphericall, by the Canons and Tables of Synes, Tangents, and Secants, in that behalfe calculated.
DEFIN. XVI. An angle of a Section, or Segment, is that which is contained vnder a cord line, and the arch line of the same Section.
AS the angles A. B. C. and B. A. C. in the lesser segment of this circle, Euc. [...]. Def. 6. are angles of a Section, because they are contained vnder the cord line A. B. and the arch line A. C. B. Also the angles D. B. A. [Page 7]and D. A. B. in the greater Segment are angles of the same Segment, by the like reason. And all angles of this kind are called mixed angles, because they are contayned vnder a right line and a crooked. Of which two Segments, the lesser hath alwaies the lesser angle; and the greater, the greater angle.
DEFIN. XVII. An angle in a Section, or Segment, is when two right lines are drawne from any point in the arch line, to the ends or extreames of the cord line; the angle in that point of the arch line is called an angle in a Section or Segment.
AS the angle A. B. C. in the lesser Segment is an angle in a Section, Euc. 3. Def. 7. or Segment, by reason that the two right
lines B. A. and B. C. are drawne from the point B. in the arch line to the ends or extreames of the cord line A. C. And also the angle A. D. C. in the greater Segment is an angle in a Section, or Segment, because the two right lines D. A. and D. C. are drawne from the point D. in the arch line to the ends or extreames of the cord line A. C. And here note, the greater Section hath in it the lesser angle, and the lesser Section the greater angle, contrarie to the mixt angles in the prece [...]ent DEFINITION mentioned. And here also is to be noted, by the declaration of this and the former DEFINITION, the difference betweene an angle of a Segment, and an angle in a Segment; the first being called a mixt angle, and this a right lined angle.
DEFIN. XVIII. If two right lines be drawne from any one point in the circumference of a Circle, and receiue any part of the same circumference, the angle contained vnder those two lines is said to belong and to be correspondent to that part of the circumference so receiued.
AS the angle B. A. C. contained vnder the right lines A. B. and A.C. drawne from the point A. and receiuing the circumference B. D. C. by this DEFINITION is said to belong, Euc. 3. Def. 8. subtend, and pertaine vnto the circumference B. D. C. And if right lines be drawne from the [Page 8]centre to the former points B. and C. then is that angle said to be in the centre of a circle, as the angle B. E. C. Which angle likewise subtendeth the same circumference B. D. C. and is alwayes double in quantitie to the former angle, drawne from the circumference. And the circumference B. D. C. is also the measure of the quantitie or greatnesse of the angle B. E. C. in the centre.
DEFIN. XIX. A Sector of a Circle is a figure contained vnder two right lines, drawne from the centre of a Circle, and vnder part of the circumference receiued of them.
AS in the last DEFINITION, Euc. 3. Def. 9. Ram. 16.3. the figure B. E. C. is the Sector of a circle, because it is contained vnder the two right lines E. B. and E. C. drawne from the centre E. and vnder part of the circumference, namely, B. D. C. receiued of them.
DEFIN. XX. Right lined figures are such, as are contained vnder right lines, of what number soeuer, aboue two.
AS those which follow, Euc. 1. Def. 20. being contained vnder three, foure, fiue, or more sides, who take their denominations, as well of the number of their angles, as of their sides; so a figure contained vnder three lines, in respect of his sides, is called a three sided figure; and in respect of his three angles is called a Triangle: and so of the rest. Where is to be noted, that euery right lined figure hath as many angles as it hath sides.
DEFIN. XXI. An Equilater Triangle is that, which hath three equall sides.
AS a Triangle is the first of all right lined figures (for vnder lesse then three right lines can no figure be contained: Eus. 1. Def. 24. Ram. 8.8.) So
of all Triangles, the Equilaterall Triangle is most simple and absolute, hauing equall lines and equall angles; containing euery of them a Sextans of a Circle, which is 60 degrees. Triangles haue their denominations, differences, and appellations, as well of their angles as their sides: As this Triangle A. in respect of [Page 9]his three equall sides, is called an Equilater Triangle; and in respect of his sharpe angles, is tearmed an Oxigontum, or acute angled Triangle. But this appellation (in mine opinion) to this kind of Triangle, is needlesse; for that the name of an Equilaterall Triangle doth imply the same, seeing it can containe no other but acute angles: for if in any Triangle there be either a right or an obtuse angle, the sides cannot be all equall, and therefore by this DEFINITION no Equilater.
DEFIN. XXII. An Isosceles is a Triangle, which hath onely two equall sides.
THis is the second sort of Triangles, Eue. 1. Def. 25. and hath two sides onely of one length, the third being either longer or shorter: As these three Triangles A. B. and C. haue euerie of them two equall sides, and are therefore called Isosceles. But in respect of their angles, the Triangle A. is called an Orthigonium, or right angled Isosceles, for that his angle at the point D. is a right angle. Also the Triangle B is called an Ambligonium, or an obtuse angled Isosceles, for that his angle at the point G. is a blunt or obtuse angle. And likewise the Triangle C. is called an Oxigonium, or acute angled Isosceles, because all his angles are acute or sharpe. Also this Triangle is called an Equicrurall Triangle, in respect of his two equall sides.
DEFIN. XXIII. A Scalenum is a Triangle which hath all his sides vnequall.
THis is the third kind of Triangles, Euc. 1. Def. 26. and hath all his sides of seuerall lengths: As these three Triangles A. B. C. haue euerie of them all their sides vnequall, and therefore called Scalena. But in respect of [Page 10]their angles, the triangle A. is called an Orthigonium, or right angled Scalenon, for that his angle at the point D is a right angle. Also the triangle B. is called an Ambligonium, or an obtuse angled Scalenon, for that his angle, at the point I. is a blunt or obtuse angle: And lastly, the triangle C. is called an Oxigonium, or an acute angled Scalenon, because all his angles are acute or sharpe. It is to be noted generally in all Triangles, that in comparison of any two sides of a Triangle, the third side is called the Base; as of the Triangle A. in respect of the two lines E. D. and E. F. the line D. F. is the base: In regard of the two lines F. D. and F. E. the line E. D. is the base; and in respect of the two lines D. E. and D. F. the line E. F. is the base.
DEFIN. XXIIII. A Square, or Quadrat, is a foure sided figure, whose sides are all equall, and angles all right angles.
AS the figure A. B. C. D. is a Square, Euc. 1. Def. 30. Ram. 12.2. 2. Con. 1. or Quadrat, because all the lines thereof are equall, and all the angles right angles. These foure sided figures likewise, as well as Triangles, take their appellation partly of their sides, and partly of their angles; as by their seuerall DEFINITIONS hereafter appeareth.
DEFIN. XXV. A long Square is that whose angles are all right angles, and whose opposite sides onely are equall.
THis figure differeth little from the Square, Euc. 1. Def. 31. Ram. 1.13. or Quadrat, last defined, hauing all equall angles like vnto it; but the sides are vnequall. As in this figure A. B. C.D. all the angles are right angles, and the opposite sides onely are equall, as the length A. B. is equall to the length C. D. and the breadth A. C. to the breadth B. D. but compare them otherwise, and they are vnequall.
DEFIN. XXVI. A Rhombus (or Diamond) is a figure with foure equall sides, but no right angle.
AS this figure A. B. C. D. is a Rhombus, hauing all his sides equall, and likewise the opposite angles; Euc. 1. Def. 32. Ram. 8.14. but the angles at A. and D. are acute angles, and those at B. and C. obtuse. Betweene a Square, or Quadrat, [Page 11]and this figure, is much resemblance, either kind hauing all
sides equall; and likewise their angles in generall quantitie; but different in particular qualitie; that hauing foure right angles, this two obtuse; and two acute angles; yet are they in generall quantitie equall to foure right angles: for by how much the two acute angles are defectiue or wanting of two right angles, by so much are the obtuse angles abounding or exceeding. This figure is described by the connexion of two Equilater Triangles, by any two of their sides, as appeareth by the prickt Diagonall line B. C. which being omitted and left out, this figure remaineth perfect, and hath his acute angles equall to those of an Equilater, namely, 60. degrees, and the obtuse angles double thereunto. PRO. 57.
DEFIN. XXVII. A Rhomboydes (or Diamond like) is a figure, whose opposite sides and opposite angles are onely equall, and bath no right angles.
AS this figure A. B. C. D. is a Rhomboydes, Euc. 1. Def. 33. Ram. 9.14. and hath his sides A. B. and C. D. opposite and equall, and likewise A. C. and B. D. but hath no right angle: For the angles at the points A. and D. are acute, opposite, & equall; and likewise the angles, at the points B. and C. are obtuse, opposite, and equall.
Note here, that the foure figures last before defined, namely, a Square, a figure of one side longer, a Rhombus, and a Rhomboydes, are commonly called Parallelograms; of which foure, the two former are called right angled Parallelograms. PROB. 90.
DEFIN. XXVIII. All other foure lined figures, besides those formerly defined, are called TRAPEZIA, or Tables.
AS all figures, of foure sides, Euc. 1. Def. 34. Ram. 10.14. which are made at aduenture, without respect or regard of equalitie, or inequalitie, or obseruation of order, either in their lines or angles; which are therefore called irregular figures: as these figures A. and B. are.
DEFIN. XXIX. Manie sided figures are those which haue more sides then foure.
OF these, Euc. 1. Def. 23. Ram. 11.14. may infinite sorts be described, by addition of lines: but if they containe aboue foure sides, they are generally called Polygona, but particularly according to the number of their sides: As the figure A. is called a Pentagon, because it is contained of fiue sides; the figure B. a Sexagon, being contained vnder six lines; and C. is called a Septagon, because it is contained vnder seuen sides. And the like of others. PROB. 62.95.
DEFIN. XXX. Either of those Parallelograms, which are about the diameter of a Parallelogram, together with the two supplements, is called a Gnomon.
RIghtly to conceiue this DEFINITION, Euc. 2. Def. 2. it is requisite first to vnderstand, what those Parallelograms are which are said to be about the diameter of a Parallelogram; and likewise, what supplements [Page 13]are. For the first, those are said to be parallelograms about the diameter, which
haue for their particular diameters part of that which the whole parallelogram hath: And supplements are such, as are without the whole diameter, the diameter passing betweene them, and cutting them not. As in the parallelogram A. B. E. D. the particular parallelograms H. K. E. F. and A. C. H. G. are said to be about the diameter, because they haue for their particular diameters part of the whole diameter A. E. as A. H. and H. E. And the supplements are the two parallelograms C. B. K. H. and G. D. F. H. because they are without the whole diameter A. E. which passeth betweene them, and cutteth them not. Now take away either of those particular parallelograms, which soeuer it be, and the other remaining, together with the two supplements, is that which by this DEFINITION is called a Gnomon. PROB. 101. THEOR. 5.
DEFIN. XXXI. That r [...]ht lined figure is said to be inscribed in another right lined figure, which hath euerie angle touching euerie side of the figure wherein it is inscribed.
AS in these two figures, the Triangle A. B. C. is
sayd to bee inscribed within the Triangle D. E. F. because euerie of his angles A. B. and C. doth touch euerie side of the Triangle D. E. F. Likewise, the square G. H. I. K. is said to be inscribed within the greater square L. M. N. O. because euery of his angles G. H. I. K. toucheth euerie side of the same greater square. The like consideration is to be had of circumscribing one right lined figure about another.
DEFIN. XXXII. A right lined figure is inscribed within a circle, when euerie angle of the inscribed figure toucheth some part of the circles circumference.
ALl the angles of a regular right lined figure, inscribed in a circle, or the sides of the like figure circumscribed about a circle, Euc. 4. Def. 3. may easily touch the circumference thereof, by reason of the perfection and vniformitie of a circle. As the Triangle A. B. C. is inscribed in the circle A. B. C. [Page 14]and also the square D.E.F.G. is circumscribed about
the circle A. for that euerie angle of the Triangle inscribed, and euerie side of the square circumscribed, toucheth some one point of the circumference of the circle. And the like consideration is to be had of circles inscribed or circumscribed within or about any right lined figure. PROB. 112.113.
DEFIN. XXXIII. The altitude of a figure, is the parallell distance betweene the top of a figure and the Base.
AS the height or altitude of this Triangle A.B.C. is the space or distance betweene the line D.E. drawne parallell to the Base B.C. by the highest point of the same Triangle, Euc. 6. Def. 4. as by A. and the same Base B.C. which parallell distance is equall, and the same thing with the perpendicular A. F. And the like of the rest.
DEFIN. XXXIIII. Parallell lines are such, as being drawne on any plaine Superficies, and produced either way infinitely, doe neuer meet or concurre.
AS these right lines A. and B. Euc. 1.35. Ram. 2.11.5.11. which being produced and drawne forth infinitely, by reason of their equall
and parallell distance the one from the other, will neuer meete or concurre; and therefore are called parallell lines. PROB. 2.3.
DEFIN. XXXV. A right line is said to be diuided by extreame and meane proportion, when the lesser part, or segment thereof, is to the greater, as the greater is to the whole line.
As the line A. B. being so diuided in the point C. that the lesser part, or segment, C. B. hath the same proportion
to the greater part, or segment, A. C. as the same greater part hath to the whole line A.B. then is the same line A. B. diuided by an extreame and meane proportion. The meanes how to diuide a line in such sort, is hereafter taught by the 20. PROB. of the next booke. This line is of wonderfull and infinite vse in manie Geometricall operations, as appeareth manifestly almost through the whole thirteenth booke of Euclid. PROB. 20.
DEFIN. XXXVI. The power of a line is the square of the same line, or any plaine figure equall to the square thereof.
AS the power of the line A. B. is the square of the same line, namely, the figure
A. B. C. D. or any other plaine figure equall thereunto. And so great power and abilitie is a line said to haue, as the quantitie of the square it makes: As this line A. B. containing 4. the power thereof is 16. In this kind is the Diagonall or diameter of a square (as the line A. D.) said to be double in power to the side of the same square, for that a square made of the Diagonall, is double in quantitie to the square made of the side. And likewise the line which subtendeth the right angle in an Orthigonall Triangle, is said to be equall in power to both the containing sides: as the line A.D. which subtendeth the right angle A. C.D. in the Triangle A.C.D. is equall to both the squares made of the two containing sides, namely of A.C. and C.D. PROB. 23.
DEFIN. XXXVII. To diuide a giuen line in power, is to finde two other lines, whose squares together shall be equall to the square of the giuen line, [Page 16]but the square of the one to the square of the other, to be in any proportion required.
AS if A. B. were a line giuen, and it were required
to diuide the same line in power, according to the proportion of 2. to 3. It is hereby intended to finde two other lines, as C. A. and C. B. whose squares together are equall to the square of the giuen line A.B. but the square of the one, namely C. A. is to the square of the other C. B. in such proportion, as 2. to 3. that is, the square of C. B. containeth the square of C. A. once and a halfe. The meanes how to performe the same, is hereafter taught in the 23. PROB. of the 2. Booke. PROB. 23.
DEFIN. XXXVIII. To inlarge a line in power, is to find another line, whose square shall haue any proportion required (of the greater inequalitie) to the square of the giuen line.
AS suppose the line 2 B in the former Diagram were giuen to be inlarged in power, as 3. to 5. It is hereby intended to find out another line, as the line C. B. whose square shall beare such proportion to the square of the giuen line 2 B. These lines are of infinite vse in many Geometricall Conclusions. as 5 to 3. which is a proportion of the greater inequalitie, and is called Superbipartiens tertias: that is, as 5 containes 3. so the square of C. B. the line sought for, containeth the square of 2 B. 1 ⅔· The working whereof is taught in the 24. PROB. of the 2. booke.
DEFIN. XXXIX. A meane proportionall line is that, whose square is equall to the right angled Parallelogram, or long Square, contained vnder his two extreames.
A Meane proportionall line
is so tearmed, in respect of the relation it hath to two other lines, which are called his extreames, for of a meane without extreames, or extreams without a meane, there is no comparison. As in this Diagram, the perpendicular A.C. of the right angled Triangle A. B. D. is a meane proportionall line betweene the two segments of [Page 17]the Base B.C. and C.D. his extreames, because the square of the same line A. C. namely, A. E. F. C. is equall to the long square, contained vnder the lines B. C. and C. D. (for the line C. G. is equall to the line B. C.) for as B. C. is to A. C. so is A. C. to C. D. Also the line A. B. is a meane proportionall betweene B. C. the segment of the Base lying next vnto it, and B.D. the whole Base; for as B. C. the lesser segment of the Base, is to B. A. so is B. A. to B. D. the whole Base. And lastly, the line A.D. is a meane proportionall, betweene C. D. the segment of the Base, lying next vnto it, and B.D. the whole Base; for as C. D. the greater segment of the Base, is to A.D. so is A. D. to B. D. the whole Base.
DEFIN. XL. Like right lined figures are those, which haue equall angles, and proportionall sides about those equall angles.
AS in these two right angled Parallelograms, Euc. 6. Def. 1. the angle in the point
A. of the greater, is equall to the angle E. of the lesser; likewise the angle B. to the angle F. and C. to G. and D. to H. And moreouer, the side A. B. hath that proportion to the side A. C.
as E. F. hath to E.G. and A.C. to C.D. as E. G. to G. H. and so of the rest: Wherefore these two Parallelograms are called like right lined figures: and so of Triangles and all other figures, of what kind soeuer. PROB. 45.
DEFIN. XLI. Reciprocall figures are such, as haue the sides of either to other mutually proportionall.
As the Parallelograms A. and B. haue
their sides mutually proportionall; Euc. 6. Def. 2. that is, as the side C. D. of A. is to the side G. H. of B. so is the side G.I. of B. to the side C. E. of A. and therefore are they called Reciprocalls: for as 20. is to 15.
an antecedent of A. to a consequent of B. so is 12. to 9. an antecedent of B. to a consequent of A.
DEFIN. XLII. The quantitie or measure of an angle, is the arch of a circle, described from the point of the same angle, and intercepted betweene the two sides of that angle.
As in the Triangle A. B. C. the measure or quantitie of the angle B. A. C. is the arch B. D. or E. F. for the
circumference of euerie circle (whether greater, or lesser) is diuided into 360 equall parts, which are called degrees, and euerie degree into 60 scruples or minutes, and euerie minute into so manie seconds, &c. Which parts or degrees are greater or lesser, as the circles, whose parts they are, are greater or lesser; and those arches which containe the same number of parts or degrees in equall circles, are equall; and in vnequall circles, they are called like arches; as the arches B. D. and H. G. are equall; but the arches B. D. and E. F. are like arches: for as B.D. is 50 degrees in the greater circle, so is E. F. 50 degrees in the lesser circle. And the like of others. PROB. 8.
DEFIN. XLIII. The Quadrant of a circle is the fourth part thereof, or an arch containing 90 degrees.
AS the arch K. B. D. in the former Semicircle is a Quadrant of that whole circle, or a fourth part thereof, and containeth 90 degrees.
DEFIN. XLIIII. The Complement of an arch, lesse then a Quadrant, is so much as that arch wanteth of 90 degrees.
As the Complement of the arch B. D. 50 degrees in the former Semicircle, is the arch K. B. 40 degrees.
DEFIN. XLV. The Excesse of an arch, greater then a Quadrant, is so much as the said arch is more then 90 degrees.
AS the Excesse of the arch H. K. B. D. 130 degrees, is the arch H. K. 40 degrees more then a Quadrant, that is more then K. D.
DEFIN. XLVI. The Complement of an arch, lesse then a Semicircle, is so much as that arch wanteth of a Semicircle, or of 180 degrees.
AS the arch H.K. B.D. is an arch lesse then a Semicircle, and containeth 130 degrees, and the Complement thereof to a Semicircle, is the arch H.G. 50 degrees, which is so much as the arch H. K. B. D. wanteth of a Semicircle, or of 180 degrees.
DEFIN. XLVII. The Complements of Angles are as the Complements of Arches.
AS the arch K. B. is the complement of the arch B. D. to a Quandrant, and the arch B. D. of the arch K. B. So the angle K.A.B. 40 degrees is the complement to a right angle of the Angle B.A.D. 50 degrees; and likewise the same angle B. A. D. of the same angle K.A.B. And in this sence is the third angle of any Triangle said to be the complement of the other two, to two right angles, or a Semicircle: For the three angles of any Triangle are equall to two right angles; as is hereafter declared.
❧ The second Part. Jnstructions concerning this Part.
THis second Part consisteth of diuerse Geometricall THEOREMES, or approued Truths; which are the Foundations, Grounds, and Reasons, whereon the Practike part dependeth. For as in the generall course and tract of all designes, before the vndertaking or execution of whatsoeuer action, the fittest meanes for an orderly performance, is iudicially to consider; first, the Propertie, Passion, Nature, and kind of the intended enterprise; then, the best and most immediate meanes how to effect the same, and the Causes, Grounds, and Reasons, why, by those meanes such effects may be wrought; and afterwards, to put in execution: So before we enter into the Practike part, I will first here premise diuerse THEOREMES concerning this subiect, whereby the ingenious practitioner may most euidently conceiue and vnderstand the ground and reason of all the Rules and Problemes in the following Bookes contained. Wherein I vse onely Explication and Construction, omitting (for breuitie sake, and auoiding confusion to the Learner) their seuerall Demonstrations; yet with such ample notes of direction in the Margent, as the Reader may readily find in EVCLID, RAMVS, and other Authors, their Demonstrations at large. And for their further ease and helpe, I haue at the end of euery Construction inserted the like notes of Reference from these THEOREMES to the following PROBLEMES, and the like from those to these; that hauing here the reason or cause, hee shall there most readily find the effect; or seeing there the effect, hee may as speedily vnderstand the cause or reason thereof, Scire enim, proprie est, remper causam cognoscere.
THEOREME I. Jf any two right lines cut the one the other, the opposite or verticall angles are euer equall; and both the angles, on one and the same side of either line, are either of them right angles, or (being both taken together) are equall to two right angles.
SVppose that B. E. and C.
F. are two right lines, Euc. l. 1. p. 13.15. Ram. 5.8. Con 2. Ceul. 2.9. which cut the one the other in the point A. Then I say, first, that the opposite or verticall angles are equall, namely, the angle B. A. C. to the angle F. A. E. and the angle B. A. F. to the angle C. A. E. for they are euerie of them right angles: Si ouotcunque recte in eodem puncto m [...]o sese in [...]ersecent omnes in communi sectione quatuor rectis aequabuntur. and let the right line D. G. be likewise drawne, cutting the line B. E. in A. Then I further say, that both the angles, taken together on one and the same side of either line, is equall to two right angles, as the angles B. A. D. and D. A. E. on the vpper side of the line B. E. and also the angles B. A. G. and G. A. E. on the nether side of the same line, are respectiuely equall to two right angles; for they consist of the right angles formerly mentioned. And the like of the angles on either side of the line D. G. PROB. 114, 117, 118.
THEOREME II. A right line, falling on two parallell right lines, maketh the outward angles on contrarie sides of the falling line equall; and likewise the inward and opposite angles on the contrarie sides of the same line; and also the outward angle, equall to the inward and opposite angle on one and the same side of the falling line; and the inward angles on one and the same side equall to two right angles.
LEt the right line E. F. fall on the
two parallell right lines A. B. and C. D. Then sayth this THEOREME, Euc. 1. p. 29. Ram. 7.9. Pit. 1.38. first, Lineae eidem parallelae inter se sunt parallelae. that it maketh the outward angles on contrarie sides of the falling line, namely, the angles A. H. E. and F. G. D. to be equall; and likewise the inward and opposite angles on the contrarie sides of the same line, as the angles A. H. F. and E. G. D. And also, that the outward angle, as A. H. E. is equall to the inward and opposite angle, [Page 22]on one and the same side of the falling line, namely, to E. G. C. And lastly, that the inward angles on one and the same side, as the angles E. G. C. and A. H. F. are equall to two right angles. PROB. 50.
THEOREME III. If a right line be diuided into two equall parts, halfe the square of that whole line is double to the whole square of halfe the same line.
SVppose A. B. to be a right line, and
let the same be diuided into two equall parts in the point C. Then I say, that halfe the square of that whole line, namely, the Parallelogram A. B. D. E. (for the whole square is A. B. F. G.) is double to the whole square of halfe the same line, namely, to the square A. C. D. H. as is manifest by the Diagram.
THEOREME IIII. A right line being diuided by chance, the square of the whole line is equall to both the squares made of the parts, and also to two rectangle figures, comprehended vnder the same parts.
LEt the right line A. B. be diuided
by chance in the point C. Then I say, Euc. 2.4. that the square of the whole line, namely, A. B. D. E. is equall to both the squares made of the parts, namely, to the squares A. C. G. H. and H. K. F. E. (for H. K. is equall to C. B.) and also to the two rectangle figures, comprehended vnder the same parts, namely, to the rectangle figures C. B. H. K. and G. H. D. F.
THEOREME V. The Supplements of those Parallelograms which are about the diameter in euery Parallelogram, are alwaies equall the one to the other.
SVppose the figure A. B. D. E. in the former THEOREME, Complementa sunt aequalia. Euc. 1.43. Ceul. 2.81. be a Parallelogram, whereof the diameter is A. E. and let the Parallelograms about the same diameter (according to the declaration of the 30. DEFINITION) [Page 23]be A. C. G. H. and H. K. F. E. Then I say, that the supplements of those Parallelograms, namely, the supplements C. B. H. K. and G. H. D. F. are equall the one to the other. PROB. 18.87.88.105.
THEOREME VI. In right angled Triangles, the square of the side subtending the right angle, is equall to both the squares of his containing sides.
In triangulo rectangulo figura ad besin descripta aequatur fi [...]uris ad crura similibus similiterque sitis.LEt the Triangle A. B. C. be a
right angled Triangle, whose angle, at the point B. is a right angle; and let the line A. C. be the side subtending the same right angle, and B. A. and B. C. his containing sides. Then I say, Euc. 1.47. Pit. 1.50. that the square of the side subtending the right angle, which is the square A. C. D. E. is equall to both the squares of his containing sides, namely, to the squares A. G. F. B. and B. H. K. C. PROB. 23.24.25.30.36.38.52.65.99.100.101.102.104.106.
This former THEOREME, and the two next following, Nota. are of infinite and wonderfull vse in most Geometricall Conclusions; especially in TRICONOMETRIE, or the supputation of Triangles, by the Canons thereof; as those excellent Tables of Logarithimes, or those of Synes, Tangents, and Secants, in that behalfe calculated; and therefore especially to be regarded; and the most excellent properties and passions thereof to be well vnderstood and practised.
THEOREME VII. In obtuse angled Triangles, the square of the side subtending the obtuse angle, is greater then both the squares of the containing sides, by two rectangled figures, comprehended vnder one of the containing sides (being continued) and the line of continuation, from the obtuse angle to a perpendicular let fall thereon.
LEt the Triangle A. B. C. be an obtuse angled Triangle, In triangulis obtusangulis basis plus potest cruribus duplici rectangulo ex allero crure & eius continuatione ad verticis perpendicularem E. 2. p. 12. whose angle at the point C. is obtuse; and let the line A. B. be the side subtending the same obtuse angle, and A. C. and C. B. his containing sides; whereof, let A. C. be the side continued, and C. L. the line of continuation from the obtuse angle at the point C. to the perpendicular let fall thereon B. L. Now [Page 24]I say, that the square of the
side, subtending the obtuse angle, namely, A. D. E. B. is greater then both the squares of the containing sides, namely, B. K. C. H. and A. C. F. G. by two rectangle figures, (which is all one, with one twice taken) comprehended vnder one of the containing sides (being continued) and the line of Continuation, namely, C. L. G. M. being twice taken. PROB. 44.
THEOREME VIII. In acute angled Triangles, the square of the side subtending the acute angle, is lesse then both the squares of the containing sides by two rectangle figures, comprehended vnder one of the containing sides (whereon a perpendicular falleth) and that segment of the same side which is betweene the perpendicular and the acute angle.
Theorema generale est ad inuestigationem perpendicularis intra triangulum cadentis data trium laterum quantitate.LEt A. B. C. be an acute
angled Triangle, hauing the angle at the point A. acute; let B. C. be the side subtending the same angle, and A. B. and A. C. the containing sides: also let B. L. be the perpendicular, Euc. 2.15. Ceul. 2.84. A. C. the side whereon it falleth, and A. L. the segment thereof betweene the perpendicular and the acute angle A. Now I say, that the square of the side subtending the acute angle, namely, B. D. C. E. is lesse then both the squares of the containing sides, which are F. B. G. A. and A. C. H. K. by two rectangle figures (being all one, with one twice taken) comprehended vnder one of the containing sides A. C. (whereunto A. H. is equall) and the segment A. L. namely, A. L. H. I. twice taken. PROB. 41.
THEOREME IX. In rectangle Triangles, if from the right angle a perpendicular be let fall vnto the Base, it shall diuide the Triangle into two Triangles, like vnto the whole, and also the one like vnto the other.
LEt A. B. C. be a recte angle
Triangle, Euc. 6.8. Ram. 5.12.13. whose angle at the point B. is a right angle; from whence, let the perpendicular B. D. be let fall to the Base A. C. Then I say, the perpendicular so falling, shall diuide the Triangle into two Triangles, that is, A. B. D. and B. C. D. like vnto the whole Triangle A. B. C. and also the one Triangle like vnto the other; which is (according to the 40. DEFINITION) with equall angles, and proportionall sides about those equall angles. PROB. 19.23.24.25.30.38.
THEOREME X. An Isosceles, or a Triangle of two equall sides, hath his angles at the Base equall; and the equall sides being produced, the angles vnder the Base are also equall.
LEt A. B. C. be an Isosceles, Euc. 1.5. Ceul. 2.3.4. or a
Triangle, whose two sides A. C. and A. B. are equall, and let A. C. be produced to D. and A. B. to E. I say then, that-his angles at the Base, namely, A. B. C. and A. C. B. are equall; and that the angles vnder the Base, as E. B. C. and D. C. B. are also equall, the one vnto the other. PROB. 40.
THEOREME XI. All equiangle Triangles haue their sides, containing equall angles proportionall, [Page 26]and their sides subtending equall angles, are of like proportion.
Hoc Theorema praecipuum est totius Trigonometriae fundamentum. SVppose
A. and B. to be two equiangle triangles, that is, Euc. 6.4. Ram. 5.12.7, 9. Pit. 1.46. Ceul. 2.62. hauing the angle D. equal to the angle G. and C. to F. and E. to H. Then I say, they haue their sides, which containe those equall angles proportionall; as D. C. and D. E. in the Triangle A. are proportionall to G. F. and G. H. in the Triangle B. because they containe equall angles, namely, D. and G. for as D. C. is to D. E. so is G. F. to G. H. and the like of the rest: also their sides subtending equall angles, are of like proportion, as D. C. and G. F. subtending equall angles E. and H. and C. E. and F. H. subtending equall angles D. and G. are of like proportion: for as D. C. is to C. E. so is G. F. to F. H. And the like of the other sides and angles. PROB. 30.38.45.65.
THEOREME XII. In any two Triangles compared, if two sides of the one be equall to two sides of the other, and the Base of the one to the Base of the other; they shall also haue the angles contained vnder their answerable equall sides, the one equall to the other in either Triangle.
LEt A. B. C. and D. E.
F. be two Triangles compared, Euc. 1.8. hauing two sides of the one equall to two sides of the other, as A. B. and A. C. of the one, equall to D. E. and D. F. of the other, and also the Base B. C. of the one, equall to the Base E. F. of the other. Now I say, they shall haue their angles contained vnder answerable equall sides (as the angle A. contained vnder A. B. and A. C. equall to the angle D. contained vnder the answerable equall sides D. E. and D. F.) to be equall the one to the other. And the like of the rest.
THEOREME XIII. If any side of a Triangle be continued, the outward angle made by that continuation, is equall to the two inward and opposite angles: And the three inward angles of any Triangle are equall to two right angles.
LEt A. B. C. be a Triangle, Euc. 1.32. Ram. 6.9. Pit. 1.48.49. Ceul. 2.20.
whereof let any of the sides be produced, as B. C. to D. Then I say, that the outward angle, made by that production or continuation, as the angle A. C. D. is equall to the two inward and opposite angles, namely, the angles C. A. B. and C. B. A. And also, that the three inward angles of any Triangle, as C. B. A. B. A. C. and A. C. B. are equall to two right angles. PROB. 111.114.117.118.
THEOREME XIIII. In euerie Triangle, two of his angles, which two soeuer be taken, are lesse then two right angles.
AS in the Diagram of the former THEOREME, take any two angles, Euc. 1.17. as those at the points A. and C. or C. and B. or B. and A. and they are lesse then two right angles; for by the same former THEOREME all three of them are equall to two right angles.
THEOREME XV. In euerie Triangle, two sides thereof (which two soeuer be taken) are greater (being ioyned together as one line) then the third side remaining.
LEt A. B. C. (the Diagram of the 13. THEOREME) be a Triangle, whereof take any two of the sides, as A. B. and A. C. I say, Euc. 1.20. those two sides being taken and ioyned together as one line, are greater then the third side remaining, namely, B. C. And the like of any other two, taken together. Whereby it is manifest, that vnder all three lines (without respect of quantitie) a Triangle cannot be contained. PROB. 42.
THEOREME XVI. Jn all Triangles, the greater side subtendeth the greater angle, and the lesser side subtendeth the lesser angle.
Let A. B. C. be a Triangle, Trianguli maius latus subtendit maiorem angulu [...]s. hauing
the side A. C. greater then the side A. B. and lesse then the side B. C. Then I say, Euc. 1.18.19. Ram. 6.11. Euc. 1.47.48. Pit. 1.5. Ceul. 2.19. that the angle A. B. C. being subtended by the greater side A. C. is greater then the angle A. C. B. being subtended by the lesser side A. B. And also, that the angle A. B. C. being subtended by the lesser side A. C. is lesser then the angle B. A. C. subtended by the greater side B. C.
THEOREME XVII. If two sides of one Triangle be equall to two sides of another Triangle, and the angle contained vnder the equall sides of the one, be greater then the angle contained vnder the equall sides of the other; then the Base also of the one (namely, of that which hath the greater angle) shall be greater then the Base of the other.
LEt there be two Triangles, A. B. C. and D. E. F. which haue
two sides of the one Triangle, as A. B. and A. C. equall to two sides of the other Triangle D. F. and D. E. and let the angle F. D. E. contained vnder the equall sides of the one be greater then the angle B. A. C. contained vnder the equall sides of the other. Euc. 1.24. Then I say, that the Base F. E. of the one (namely, of that which hath the greater angle) is greater then B. C. the Base of the other.
THEOREME XVIII. If a Triangle be equicrurall, or hauing two equall sides; a perpendicular let fall from the angle contained vnder those equall sides to the Base, and continued, shall diuide as well the same Base and angle, as also the measure of that angle, into two equall parts: Et contra.
LEt A. B. C. be a Triangle, Pit. 1.23. whose sides A. B. and A. C. are equall, and let fall a perpendicular from the angle, included by those equall sides, as A. E. to the Base B. C. and let the same be continued to D. Now I say, that a perpendicular fo let fall, shall diuide as well the same Base B. C. and angle B. A. C. as also the measure thereof, namely, the arch line B. D. C. into two equall parts. PROB. 10.11.40.
THEOREME XIX. If a Triangle hath two equall sides, the power of one of those equall sides exceedeth the power of the perpendicular let fall on the Base from the angle it subtendeth, by the power of halfe the Base.
LEt A. B. C. be a Triangle, hauing
two equall sides, B. A. and B. C. and let B. F. be a perpendicular let fall to the Base A. C. from the angle it subtendeth A. B. C. Then I say, that the power of one of those equal sides, namely, the square A. B. D. E. exceedeth the power of the perpendicular, namely the square B. H. F. G. by the power of halfe the Base, namely, the square F. C. K. L. PROB. 36.40.41.64.
THEOREME XX. If the power of one side of any Triangle be equall to both the powers of the other two sides, the angle contained vnder those two other sides, is a right angle.
THis THEOREME is the conuerse of the 6 THEOREME, and therefore the explication and construction thereof serueth here. Euc. 1.48.
THEOREME XXI. If a right line diuide any angle of a Triangle into two equall parts, and if also the same line diuide the Base, the segments of the Base shall haue such proportion the one to the other, as the other sides of the Triangle haue: Et contra.
LEt A. B. C. be a Triangle, and let the right line A. D. diuide the angle B. A. C. of the same Triangle into two equall parts; Euc. 6.3. Ceul. 2.61. and also let the same line diuide the Base B. C. Then I say, the segments of the Base, namely, B. D. and D. C. shall haue such proportion the one to the other, as the other sides of the Triangle haue, namely, A. B. and A. C. for such proportion as B. D. hath to D. C. the same hath A. B. to A. C.
THEOREME XXII. If a right line be drawne parallell to any side of a Triangle, the same line shall cut the sides of that Triangle proportionally.
LEt A. B. C. be a Triangle, Euc. 6.2. Ram. l. 6. p. 9.5. p. 13. Con. 1.2. & 3. Pit 1.47.45. Ceul. 2.27. vnto one of the sides whereof A. B. is drawne a parallell line D. E. Wherefore the same line doth cut the sides of that Triangle A. C. and B. C. proportionally: for first, as A. E. is to E. C. so is B. D. to D. C. also, as A. E. is to B.D. so is E. C. to D. C. and as A. C. is to A. E. so is B. C. to B. D. PROB. 12.13.14.15.16.22.98.
THEOREME XXIII. The superficiall content of euery right angled Triangle, is equall to halfe that right angled Parallelogram, which hath his length and breadth equall to the containing sides of the right angle; or whose length is equall to the subtending side, and breadth to the perpendicular, drawne from the right angle to the same side.
LEt A. B. C. be a rectangle Triangle, Euc. 1. D. f. 27. Ram. 8.2. whose angle at
the point C. is a right angle, whereof the containing sides are A. C. and C. B. the subtending side A. B. and the perpendicular drawre from the right angle to the same side, is C. E. Now I say, the superficiall content or Area of this right angled Triangle is equall vnto halfe that right angled Parallelogram (namely, A. D. C. B.) which hath his length C. B. and breadth A. C. equall to the containing sides of the right angle; or whose length A. B. is equall to the subtending side, & breadth A. F. to the perpendicular line, drawne from the right angle to the same side, as the Parallelogram A. B. F. G. PROB. 39.52.92.102.106.
THEOREME XXIIII. The Area or superficiall content of euery Equilater Triangle, is equall to halfe that long square, whose length and breadth is equall to one of the sides and the perpendicular.
LEt A.B.C. be an Equilater Triangle, and A. E. the perpendicular thereof. Now I say, that the superficiall content thereof is equall to halfe that long square D. F. B. C. whose length B.C. and breadth B. D. is equall to one of the sides, and the perpendicular. PROB. 37.
THEOREME XXV. All Triangles, of what kind soueur, are equall in their superficiall content vnto halfe that right angled Parallelogram, whose length and breadth is equall to the perpendicular, and the side whereon it falleth.
Let A. B. C. be a Triangle, whose perpendicular is A. F. and the
side whereon it falleth B. C. I say, that this Triangle A. B. C. is equall in his superficiall content vnto halfe the right angled Parallelogram D. E. B. C. whose length B. C. and breadth B. D. are equall to the perpendicular, and the side whereon it falleth. PROB. 39.41.44.72.77.99.
THEOREME XXVI. Triangles which consist on one and the same Base, or on equall Bases, and in the same parallell lines are equall the one to the other.
LEt A. B. C. D. B. C. and E. B. C. be three
Triangles, Triangula in aequali besi & intra easem parallelas sunt aequalia. consisting on one and the selfe-same Base B. C. (or on equall Bases, Euc. 1.38. which is all one thing) and in the same parahell lines A. E. and C. B. Now I say, that all those three Triangles, A. B. C. D. B. C. and E. B. C. and as manie moe as may be drawne on the same Base, or a Base equall thereunto, and in the same parallell lines, are all equall the one to the other. PROB. 26.27.28.29.46.47.73.74.75.79.80.81.93.103.107.110.
THEOREME XXVII. If Triangles and Parallelograms haue one and the same Base, or [Page 33]equall Bases, and be in the same parallell lines, the Parallelograms shall be double to the Triangles.
LEt B. C. D. and F. C. D. be two Triangles, Euc. 1.41. Ceul. 2.25. and
let A. B. D. C. and B. E. D. C. in this same Diagram be two Parallelograms, which Triangles and Parallelograms haue one and the same Base C. D. and are in the same parallell lines A. F. and C. D. Now I say, that either of those two Parallelograms are double to either of those two Triangles. PROB. 76.77.92.110.
THEOREME XXVIII. If a Triangle hath his Base double to the Base of a Parallelogram, and that they are both in the same parallell lines, then are they both equall the one to the other.
LET A. B. C. D. be a Parallelogram, whose Base is C. D. and let A. E. D. and B. E. D. be two seuerall Triangles, whose Bases E. D. are double to the Base of the Parallelogram (for E. C. and C. D. are equall) and who are within the same parallell lines with the Parallelogram A. B. C. D. Then I say, that either of those Triangles are equall to the same Parallelogram. PROB. 29.78.91.110.
THEOREME XXIX. The power of the side of an Equilater Triangle, is to the power of the perpendicular thereof let fall from any angle to the subtendent side, in proportion Sesquitertia, or as 4. to 3.
LEt A. B. C. be an Equilater Triangle, Euc. 13.12. whose perpendicular is A. E. let fall from the angle B. A. C. to the subtendent side B. C. Now I say, that the power of the side of the same Triangle, namely, B. C. G. H. which is the power or square of the side B. C. is to the power of the perpendicular thereof, namely, A. D. E. F. (which is the power or [Page 34]square of the perpendicular A. E.) in proportion Sesquitertia, or as 4. to 3. For of what parts the line B. C. or B. A. containeth in power 8. of such parts B. E. (which is the halfe of B. C.) containeth in power 2. Wherefore the perpendicular A. E. being the residue, containeth in power of such parts 6. (for the squares of the lines A. E. and B. E. are by the 6. THEOREME equall vnto the square of the line A. B. whereunto B. C. is equall.) Now 8. to 6. is Sesquitertia: wherefore the power of the line B. C. is to the power of the line A. E. in Sesquitertia proportion. So is the square A. D. E. F. ¾· of the square B. C. G. H. PROB. 36.
THEOREME XXX. The Diagonall line, or Diameter of any Square, is double in power to the side of the same Square.
LEt A. B. C. D. be a Square, whose Diagonall line, or Diameter, is the line A. D. Now I say, that the same line A. D. is double in power to the side of the same Square, that is, the Square A. D. E. F. is double to the Square A. B. C. D. PROB. 99.102.106.
THEOREME XXXI. A Square, whose side is equall to the Diameter of any other Square, is double in content or superficiall quantitie to that other Square.
THe explication hereof, is manifest by that of the last: For let A. D. E. F. in the last Diagram be a Square, whose side A. D. is equall to the Diameter of another Square, as the same line A. D. is the Diameter of another Square, namely, A. B. C. D. Wherefore I say, that the Square A. D. E. F. is double in content, or superficiall quantitie, to that other Square. PROB. 102.106.
THEOREME XXXII. All parallelograms haue their opposite sides, and angles equall one to another; and their Diameters diuide them into equall parts.
AS these two parallelograms
A. and B. haue their opposite sides and angles, Eu. 1.34. Ram. 10.6. Ceul. 2.25. equall one to another, as in the figure A. the sides C. D. and E. F. are opposite & equall, and likewise D. F. and C. E. Also the angles thereof at the points C. and F. are opposite and equall, and likewise those at D. and E. And moreouer, their diameters diuide them into equall parts, as the diameters C. F. and D. E. doe either of them diuide the parallelogram A. into two equall parts: And the like explication and construction is to be made of the figure B. PROB. 115.116.
THEOREME XXXIII. Parallelograms which consist on one and the same base, or on equall bases, and in the same parallel lines are equall the one to the other.
LEt A. B. C. D. and E. A. C. D. bee two parallelograms, Eu. 1.35, 36. which consist on one & the same base, namely C. D. (or on equal bases which is one and the same thing) and in the same parallel lines, namely E. B. and C. B. Now, I say, that those two parallelograms are both equall the one to the other. PROB. 89.90.
THEOREME XXXIIII. Euery Rhombus and Rhomboydes is equall to the long square, whose length is one of the sides, and breath equall to the parallel distance.
LEt G. H. I. K. be a Rhombus, and A. B. C. D. a Rhomboydes. I say, the Rhombus G. H. I. K. is equall to the long square G. H. L. M. whose length is one of the sides G. H. and breadth the parallel distance [Page 36]
H. M. And also that the Rhomboydes A. B. C. D. is equall to the long square E. F. C. D. whose length is one of the sides C. D. or A. B. and breadth, the parallel distance E. C. or F. D. PROB. 58, 59, 60, 61, 89, 90.
THEOREME XXXV. Parallelograms and Triangles, within the same parallels, are in such proportion the one to the other as their bases are.
LEt A. B. C. D. and B. E. D. F. be two parallelograms, Triangula vel parallelogramma aequealta sunt vt basis. Eu. 6.1. Ram. 10.13. Ceul. 2.26.
within the same parallel lines A. E. and C. F. and let also B. C. D. and B. D. F. bee two triagles within the same parallel lines. Then, I say, as the base C. D. is to the base D. F. so is the parallelogram A. B. C. D. to the parallelogram B. E. D. F. And so also is the triangle B. C. D. to the triangle B. D. F. PROB. 29, 110, 123, 124, 125, 126, 127, 128, 129.
THEOREME XXXVI. A right line being first equally, and then vn-equally diuided; The square which is made of the part lying betweene those sections, together with the right angled parallelogram, contayned vnder the vn-equall parts of the whole line; are equall to the square of halfe the whole line.
LEt A. B. bee a right line diuided, Hereby is demonstrated that equation, of the greatest and least karectes or numbers, and their equal [...]tie to the midd [...]e. Of g [...]eat vse is th [...] [...] in the rules of Algebar. Eu. 2. [...]. first equally
in the point C and then vn-equally as in the point D. The square, I say, which is made of the part lying betweene those sections C. and D. namely the square E. F. G. H. together [Page 37]with the right angled parallelogram A. D. F. I. contayned vnder the vnequall parts of the whole line as A. D. and D. B. are equall to the square C. B. K. H. being the square of halfe the whole line, as of A. C. or C. B.
THEOREME XXXVII. Two right lines being drawne in a circle, and the one intersecting the other, either equally or vnequally howsoeuer; The rectangle figure contayned vnder the parts of the one line, shall be equall to that, contayned vnder the parts of the other.
LEt A. C. and B. D. be two right lines drawne in the circle A. B. C. D. and let the one intersect the other vnequally at all aduentures in the point E. I say, The wonderfull properties of a circle here by appeareth. And many strange conclusions Geomertricall from hence may be gathered. Eu. 3.35. Ceul. 2.50. that the rectangle figure contayned vnder the parts of the one line, namely vnder A. E. and E. C. being the parts of the line A. C. shall bee equall to that contained vnder the parts of the otherline, namely vnder the parts B. E. and E. D. of the line B. D. And the like if those lines had intersected the one the other equally.
THEOREME XXXVIII. In all right angled parallelograms, the length thereof being infolded in the breadth, produceth the Area or superficiall content of the same.
LEt A. B. C. D be a parallelogram right angled, I say, the length there of A. B. 40. being infolded in the breadth A. D. 20. produceth the Area or superficiall content of the same 800. PROB. 51.55.
THEOREME XXXIX. Euery regular Poligon is equall to the long square, whose length and breadth is equall to halfe the perimeter, and a perpendicular drawne from the center to the middle of any side of the same.
LEt the Sexagon A. B. C. D. E. F.
be a regular Polygon, whose three sides (being halfe the Perimeter) contayne 18. and the perpendicular 6. G. 5⅕· This Polygon is equall to the long square H. 6. G. I. whose length H. 6. or I. G. is equall to halfe the Perimeter, and breadth H. I. or 6. G. to the perpendicular 6. G. PROB. 64.95.
THEOREME XXXX. If two or more right lines, are cut by diuers parallel right lines; the intersegments of those lines so cut shall bee proportionall the one to the other.
LEt A B. and A C. be two right lines, Pit. 1.39. being cut by
diuers parallel right lines as Q E. S H. T L. and the rest; I say, the Intersegments of those lines so cut, as A F. and A G. A I. and A K. F M. and G N. and the rest are proportional the one to the other; that is to say, if A F. be ¼· of A B. then is A G. ¼· of A C. and if A M. bee ¾· of A B. then shall A N. bee ¾· of A C. The reason is, because the right line S H. cutteth off ¼· of the whole parallelogram Q E P R and the right line V O. ¾· thereof; and consequently the like parts, from all lines drawne ouer-thwart those parallels. And the like consideration is to bee had of all the other intersegments so by those lines cut out.
THEOREME XLI. Three right Lines being proportionall, a Square made of the Meane, is equall to the right angled figure, contained vnder the Extreames.
LEt A. B. and C. be three right lines proportionall in continuall proportion, Euc. 6.17. & l. 7.20. Ram. 12.4. Pit. 1.43. so that as A. is to B. so let B. be to C. Then I say, that the Square, namely, D. E. F. G. made of the Meane B. shall be equall to the right angled figure, namely, H. I. K. L. contained vnder the two Extreames A. and C. as appeareth by the Diagram. PROB. 79.83.84.85.86.88.95.99.109.129.130.131.
THEOREME XLII. Foure right lines being proportionall, the right angled Parallelogram, contained vnder the two Meanes, is equall to the right angled Parallelogram, contained vnder the two Extreames.
LEt A. B. C. and D. be foure right lines proportionall, Euc. 6.16. Pit. 1.42. so that as A. is to B. so let C. be to D. Then I say, that the right angled Parallelogram, namely, E. F. G. H. contained vnder the two Meanes B. and C. shall be equall to the right angled Parallelogram, namely, I. K. L. M. contained vnder the two Extreames A. and D. as appeareth by the Diagram. PROB. 49.56.87.88.93.
THEOREME XLIII. Of any three proportionall right Lines, the Square which is made of the Meane, and that which is made of either of the Extreames, haue such proportion the one to the other, as the two Extreames haue.
LEt A.B.C.D. and E. F. bee three
right lines, Eu. 12.2. D.P. 1. Ceul. 2.75. in continuall proportion: so that as A. B. is to C. D. so let C. D. be to E. F. Then I say, that the Square (namely, C. D.I.H.) which is made of the meane C.D. and that which is made of either of the extreames A. B. or E.F. haue such proportion the one to the other (respectiuely) as those extreames haue: For the same proportion as the greater extreame A. B. hath to the lesser extreame E. F. the same hath the square A. B. G. H. to the square C. D. H. I. and that, to the square E.F. I. K. which in this Diagram is Dupla sesquiquarta, as thereby appeareth. And the like consideration is to be had of the proportion of Circles, whose Diameters are so proportionable. PROB. 79.81.82.94.129.130.131.
THEOREME XLIIII. If a rationall right line be diuided by an extreame and meane proportion, either of the segments is an irrationall residuall line.
LEt the line A. B. be a rationall right line, Euc. 13.6. and let the same be diuided by
an extreame and meane proportion in the point C. Then I say, that either of the segments, namely, A.C. and C. B. is an irrationall residuall line. PROB. 20.
THEOREME XLV. If a right line be diuided by extreame and meane proportion, the whole line hath the same proportion to the greater segment, as the same greater segment hath to the lesser.
LEt A. B. be a right line, diuided by extreame and meane proportion,
as in the point C. I say then, that the whole line A. B. hath the same proportion to the greater segment A. C. as the same greater segment hath to the lesser segment C. B. for as A. B. is to A. C. so is A. C. to C. B. PROB. 20.
THEOREME XLVI. If a right line be diuided by extreame and meane proportion, the Rectangle figure, comprehended vnder the whole line, and the lesser segment, shall be equall to the Square made of the greater segment.
LEt A. B. be a right line,
AB. 10.
AC. 15—√ (powerof2) 125.
CB. √ (powerof2) 125-5.
and let the same be diuided by extreame & meane proportion in the point C. Then, I say, the Rectangle figure, namely, A. C. G. H. comprehended vnder the whole line A. G. (being equall to A. B.) and the lesser segment A. C. shall be equall to the Square, namely, C.B.E.F. made of the greater segment C.B. DEF. 35. PROB. 20.
THEOREME XLVII. Two right lines being drawne in an equilater Equiangle Pentagon, in such sort as they subtend any two of the next immediate angles, those two lines by their intersections shall diuide the one the other by an extreame and meane proportion: and the greater segments of either of them shall be equall to the side of the Pentagon.
LEt A. B. and C. D. be two right lines, Eu. 13.8. drawne in the Equilater Equiangles
Pentagon, A. D. B. E. C. and let the line A.B. subtend the angle A. D. B. and the line C. D. the angle C. A. D. being two of the next immediate angles. I say then, that those two lines, by their intersection in the point F. shall diuide the one the other by an extreame and meane proportion. And the greater segments of either of them, as the segments F. B. and F. C. shall be either of them equall to the side of the Pentagon A.D.B.E.C. PROB. 20.21.48.62.63.119.120.
THEOREME XLVIII. Like Triangles are one to the other in double proportion that the sides of like proportion are.
LEt A.B.C. and D.E.F. be
two like Triangles, Eue. 6.19. and let the angle A. of the one be equall to the angle D. of the other; the angle B. of the one, to the angle E. of the other; and the angle C. of the one to the angle F. of the other; and as the side A.B. is to the side B. C. so let the side D. E. be to the side E. F. so are B. C. and E. F. sides of like proportion. Now I say, that the proportion of the Triangle A. B. C. vnto the Triangle D. E. F. is double the proportion of the side B. C. to the side E. F. PROB. 45.
THEOREME XLIX. All like right lined figures whatsoeuer, are the one to the other in double proportion, that the sides of like proportion are.
LEt A. and B. bee two
right lined figures like, Euc. 6.20. hauing the angle at the point C. equall to the angle at the point H. and the angle at the point D. equal to the angle at the point I. the angle E. to the angle K. and so of all the rest. And also as the side D. E. is to E. F. so let I. K. be to K. L. &c. so are the sides E. F. and K. L. sides of like proportion. Then, I say, that the proportion of the figure A. vnto the figure B. is double, the proportion of the side E. F. to the side K. L. PROB. 45.
THEOREME L. All angles in equall circles, whether they are in the centers or circumferences, haue the same proportion one to the other as the circumferences haue wherein they consist: And so are the sectors, which are described on the centers.
LEt A.B.C. and F. G. H. bee two equal
circles wherof let D. and E. be their centers; Euc. 6.33. and let the angles which are in their centers be B. D. C. and H. E. G. and the angles which are in their circumferences B. A. C. and H. F. G. and let the sectors described on their centers be D. B. C. and E. H. G. Then, I say, that the angles B. D. C. and H. E. G. in the centers, and the angles B. A. C. and H. F. G. in the circumferences, haue the same proportion one to the other, as the circumferences haue wherein they consist, that is, as the circumference B.C. hath to the circumference H. G. And the same proportion also hath the sector D. B. C. to the sector E. H. G.
THEOREME LI. If on the end of the Diameter of a circle, a perpendicular bee raysed, it shall fall without the circle, betweene which, and the circumference, another right line cannot be drawne to the Diameter, and the angle within the circle is greater, and that without the circle is lesser, then any acute angles made of right lines.
LEt A. B. C. be a circle, Euc. 3.16. whose diameter is A. C. and on the same diameter
let the perpendicular D.C. bee raysed. Then, I say, first, that the same perpendicular D. C. shall fall without the circle; and that betweene the same perpendicular and the circumference B. C. another right line cannot be drawne to the diameter A.C. And also that the angle within the circle, namely A. C. B. is greater, and that without the circle, namely B.C.D. is lesser then any acute angles made of right lines.
THEOREME LII. If a right line bee a tangent or touch line to a circle, and another right line bee drawne by the center to the point of touch, it shall bee a perpendicular to the tangent: And if a perpendicular bee let fall from the center to the tangent, it shall fall in the point of touch.
LEt the right line A. B. bee a tangent, Euc. 3.18. or touch line to the circle D. E. C. let the
point of touch be C. and the center of the circle E. and let another right line, as D.C. bee drawne by the center E. to the point of touch C. Then, I say, that the same line so drawne by the center to the point of touch, shall bee a perpendicular to the tangent A. B. And that the perpendicular E. C. being let fall from the center E. to the tangent A. B. shall fall in the point of touch C. PROB. 31.33, 114, 115, 116, 117, 118.
THEOREME LIII. If a right line be drawne in a circle and not by the center thereof, another right line bysecting the same by right angles shall passe by the center of the same circle. And if from the center a perpendicular be let fall on a right line drawne in the same circle not by the center; the perpendicular shall diuide the same line into two equall parts.
LEt A. B. D. C. be a circle whose center is E. and let B. C. be a right line drawne in the same circle, and not by the center
thereof, and let another right line as A. D. bysect the same by right angles in the point F. Then, I say, that the same line A. D. shall passe by the center of the circle. Also from the center E. let fall the perpendicular E. D. on the right line B. C. drawne in the same circle not by the center; Then, I say, further that the perpendicular E D. shall diuide the same line B.C. into two equall parts.
THEOREME LIIII. If one angle be placed in the circumference of a circle, and another in the center thereof, and are both subtended by one part of the circumference. That angle in the center shall bee double to that in the circumference.
LEt A. B. C. be a circle, and let one angle bee placed in the circumference thereof,
Angulus in centro duplus est anguli in peripheria, in candem peripheriam insistentis. Euc. 3.20. as the angle B. A. C. and another in the center thereof, as the angle B. E. C. and let them both bee subtended by one part of the circumference as B. D. C. Then, I say, that the angle B.E.C. in the center, shall be double to the angle B. A. C. in the circumference.
THEOREME LV. All angles consisting in one and the same segment of a circle are equall the one to the other; If in a semicircle, they are right angles; If in a lesser segment, they are greater then a right angle; If in a greater [Page 46]segment, they are lesser. And also the angle of a greater segment, is greater then a right angle, and the angle of a lesser segment is lesse then a right angle.
Anguli in cadem sectione sunt aequales.LEt A. B. C. D. E. F. bee a circle, and let A. E. be the diameter thereof, which diuideth
the same into two semicircles or equall segments; Then, I say, that the angles A. C. E. and A. D. E. consisting in one and the same segment, Eu. 3.21.31 are equall the one to the other, and being in a semicircle they are both right angles: Ceul. 2.46. Let also the line or cord A. C. diuide the same circle into two vnequall segments, as A. B. C. the lesser segment, and A. F. E. D. C. the greater, Note here the difference betweene an angle in a segment, and an angle of a segment, See Def. 16.17. I say, the angle A. B. C. in the lesser segment is greater then a right angle, and the angle A. E. C. in the greater segment is lesse then a right angle. And also F. A. C. an angle of the greater segment is greater then a right angle, and the angle B. A. C. being an angle of the lesser segment, is lesse then a right angle. PROB. 19.23.24.25.30.38.52.65.
THEOREME LVI. If a right line be a tangent to a cirle, and another right line be drawne from the touch (crossing the circle) to what point soeuer in the circumference; the angles caused by intersection or meeting of those two lines, are equall to the angles consisting in the alternate segments of the circle.
LEt A. B. C. D. be a circle, and the right line E. F. a tangent to the same circle; Euc. 3.32.
and let another right line as B. D. bee drawne from the touch, namely, the point D. crossing the circle to what point soeuer in the circumference as the point B. Then, I say, the angles caused by intersection or meeting of those two lines E. F. and B. D. are equall to the angles consisting in the alternate segments of the circle; that is, the angle B. D. F. shall be equall to the angle B.A.D. and the angle B. D. E. to the angle B. C. D. in the alternate segments. PROB. 32.33.111.
THEOREME LVII. If from a point without a circle, two right lines be so drawne, that the one be a tangent to the circle, and the other diuide the same circle into two equall or vnequall parts: The rectangle figure contayned vnder the whole line which diuideth the circle, and that part thereof lying betweene the vtter circumference and the point, is equall to the square made of the tangent line.
LEt B. C. F. be a circle, and without the same, take a
point at all aduentures, as the point A. from whence let two right lines be so drawne, that the one bee a tangent to the circle, as A. C. and the other diuide the same circle, as A. B. F. Then, I say, that the rectangle figure contained vnder the whole line A. F. and that part of the same line, lying betweene the vtter circumference & the point, as B. A. namely, the rectangle figure A. H. G. F. is equall to the square made of the tangent line A. C. namely, to the square A. E. D. C.
THEOREME LVIII. If from a point without a circle, two right lines be drawne to the concaue circumference of the circle, they shall be reciprocally proportionall with their parts taken without the circle. And another right line drawne from the point as a tangent to the circle, shall bee a meane proportionall betweene either whole line, and the vtter segment thereof.
LEt B. E. F. D. be a circle, and without the same circle take a point at all aduentures, as at A. and from that point to the concaue circumference of the circle, draw the two right lines A.E. and A. F. And let another right line be drawne from the same point as a tangent to the circle, as the line A.D. Then, I say, first, that these two lines A.E. and A.F. are reciprocally proportionall with their parts taken without the circle, that is, as A. E. is to A. F. so is A. C. to A. B. And moreouer, that betweene the lines A. F. and A. C. or betweene the lines A.E. and A.B. the tangent A. D. is a meane proportionall.
THEOREME LIX. Euery circumference of a circle, is more then triple his Diameter, by such a proportion as is more then 10/71· and lesse then 1/7 of the same, the neerest rationall proportion whereof is 22. to 7.
IN the former Diagram, Corol: Euc. 12.1. let B. C. D. F. E. be the circumference of a circle, and C. F. the Diameter thereof; I say, the same circumference is more then triple the Diameter C. F. by such a proportion as is more then 10/71· and lesse then 1/7· of the same; and that the neerest rationall proportion thereof, is 22. to 7. Wherefore to know the quantitie of the circumference, multiplie the Diameter by 22. and diuide the Factus by 7. the Quotient resolueth the question. PROB. 34.67.
THEOREME LX. Euery circle is neere equall to that right angled Triangle, of whose sides (containing the right angle) the one is equall to the semidiameter, and the other to the circumference of the same circle.
Corol. Euc. 12.1.
The precise squaring of a circle was neuer yet found out; and therfore in this and the 4. next Theoremes following, this word (Neere) is vsed. But all Conclusions hereby wrought, are without any apparant error.LEt A. C. D. be a circle, and A. E. B. a right angled Triangle, whose angle at the centre B. is a right angle, and whose sides containing the right angle, namely, A. B. and E. B. the one is equall to the Semidiameter, as A. B. and the other to the circumference, as E.B. Then I say, that the Triangle A. E. B. is neere equall vnto the same circle. PROB. 68.68.
THEOREME LXI. The Square made of the Diameter of a Circle, is in that proportion [Page 49]to the circle (very neere) as 14. to 11. And therefore euery circle is neere 11/14· of the square about him described.
LEt A. B. C. D. be a circle, Corol. Euc. 12.1. and E.F.G.H. a square made of the diameter A. C. or B. D. Then, I say, that the square E. F. G. H. is very neere in the same proportion to the circle A. B. C. D. as 14. to 11. And theree fore, the circle very neere 11/14· of the same squar-about him described. PROB. 68.
THEOREME LXII. Euery circle is neere equall to the long square, whose length and breadth are equall to halfe the circumference, and halfe the Diameter; or to the whole Diameter, and 11/14· thereof.
LEt A. C. D. be a circle; Corol. Euc. 12.1. And let H. A. B. G. be a
long square, whose length H. A. is equall to halfe the circumference, and breadth A. B. to halfe the diameter; And let also K. A. D. E. bee another long square, whose length is the whole diameter A. D. and breadth 11/14· thereof, namely, K. A. Then, I say, that either of those two long squares are neere equall to the circle A. C. D. PROB. 68.
THEOREME LXIII. Euery semicircle is neere equall to the long square, whose length and breadth is equall to halfe the arch line, and the semidiameter.
LEt E. B. F. be a semicircle, Corol. Euc. 12.1. whose semidiameter is the
line B. C. and the halfe of whose arch line is equall to the line A. B. or D. C. Then, I say, that the long square, namely, A. B. C. D. (whose length A. B. is equall to halfe the arch, and [Page 50]whose breadth is B. C. the semidiameter) is neere equall to the semicircle E. B. F. PROB. 69.
THEOREME LXIIII. Euery sector of a circle, is neere equall to that long square, whose length and breadth is equall to the semidiameter, and halfe the arch-line of the same sector; or the halfe semidiameter, and the whole arch line.
LEt B. A. C. bee the sector of a circle, Cor. Eu. 12.1. and let D. B. A. E. bee a long square,
whose length B. A. or D. E. is the semidiameter or equall thereunto, and whose breadth D. B. or E. A. is equall to halfe the arch-line B. C. And let also H. F. A. G. be another long square, whose length H. F. or G. A. is equall to the whole arch-line B. C. and whose breadth F. A. or H. G. is halfe the semidiameter, or equal thereunto. Then, I say, that either of those two long squares is neere equall to the sector B. A. C. PROB. 70.
THEOREME LXV. All circumferences of circles, haue the same proportion the one to the other, as their diameters haue.
LEt A. and B. be two circles, This Theor. is of excellent vse in the forming of Mill-wheels, clocks, cranes, and other engines for water-workes, &c. wherof let C. D. and
E. F. bee their seuerall diameters, I say, that the same proportion that the diameter C. D. of the circle A. hath to the diameter E. F. of the circle B. the same proportion hath the circumference of A. to the circumference of B. PROB. 65.66. Cor. Eu. 12.1.
THEOREME LXVI. All circles haue the same proportion the one to the other, Omnes figurae fimiles circulis inscriptae sunt, vt quadrata à diametris circulorum quibus inscribuntur. Euc. 12.2. as the squares of their Diameters haue.
LEt A. and K. bee two circles, and let the squares circumscribed about them, bee the seuerall squares of their Diameters. Then, I say, that the circle A. hath the same proportion to the circle [Page 51]K. as the squares of their Diameters haue, namely, as the square B. C. D. E. hath to the square F. G. H. I. PROB. 65, 66, 104.108.
THEOREME LXVII. If in a circle be described a quadrilaterall figure, the opposite angles thereof shall bee equall to two right angles: and being intersected with two diagonalls, the right angled figure made of those diagonalls, is equall to the two right angled figures, comprehended vnder the opposite sides of the quadrilaterall figure.
LEt A.B.C.D. be a circle, Exempla illustrissima babebis, Pit. lib. 2. p. 32.33.35.36.37.38 Anguli oppositis sect onibus aequantur duobus rect [...]s. and let therein bee described the quadrilaterall figure A. B C. D. let also the same figure
be intersected with the two diagonals A. C. and B. D. Then, I say, first, that the opposite angles at the points A. and C. are equall to two right angles, Eu. 3.22. Pit. 1.54. and likewise the opposite angles at the points B. and D. And also that the right angled figure made of the diagonalls A. C. and B. D. is equall to the two right angled figures (taken together) comprehended vnder the opposite sides A. B. and D. C. and vnder A. D. and B. C. This Prop. is of very great vse in trigonometric.
THEOREME LXVIII. The power of the side of an equilater triangle inscribed in a circle, hath to the power of the semidiameter of the same circle triple proportion.
LEt A. B. E. C. be a circle, whereof F. B. is the semidiameter, and let A. B. C.
be an equilater triangle inscribed in the same circle. Then, I say, that the power of the side of the equilater triangle A.B.C. namely, the square H. A C. G. hath to the power of the semidiameter F.B. namely, the square F.B.D.E. triple proportion, that [Page 52]is, as 3. to 1. For the square H. A. C. G. contayneth the square F. B. D. E. three times.
THEOREME LXIX. A triangle inscribed in a circle, hath euery of his angles equall to halfe the arch, opposite to the same angle.
LEt D. be a circle, Pit. 1.53. and let A. B. C. be a triangle, inscribed at all aduentures in the same circle. Then, I say, that the
triangle A. B. C. hath euery of his angles equall to halfe the arch, opposite to the same, as the angle at the point A. is equall to halfe the arch B. F. C. opposite thereunto, the angle at the point B. is equall to halfe the arch A. G. C. and the angle at the point C. is equall to halfe the arch A. E. B. For, the whole of euery circle is 360. degrees, whereof the halfe is 180. and the three inward angles of euery right lined triangle, is equall to two right angles, which is 180. degrees. THEOR. 13. PROB. 48.119.120.
THEOREME LXX. If in a rectangle-triangle a perpendicular bee drawne from the right angle to the base, the same perpendicular is a meane proportionall betweene the sections of the base: And the side annext to either section, shall bee a meane betweene the same section and the whole base.
LEt A. B. C. be a rectangletriangle, Euc. Coroll. 6.8. Ceul. 2.63. right angled at B. from whence let the perpendicular B. D. bee drawne to the base A. C. Then, I say, that
the same perpendicular B. D. is a meane proportionall betweene the sections of the base, namely, betweene A. D. and D. C. And also that the side A. B. annext to the section A. D. is a meane proportion betweene A. D. and the whole base A. C. and that the side B. C. annext vnto the section D. C. is a meane proportion betweene the [Page 53]same section D. C. and the whole base A. C. For, as A. D. is to A. B. so is A. B. to A. C. &c. PROB. 17, 19, 23, 24, 25, 30, 38, 43, 65, 66, 109.
THEOREME LXXI. If in equall parallelograms, one angle of the one, be equall to one angle of the other, the sides which contayne those equall angles, shall bee reciprocall.
Parallelogramma aequiangula aequalia, sunt lateribus reciproca: & contra.LEt A. B. C. D. and E. F. G. C.
bee two parallelograms, equall the one to the other, Euc. 6.14. and let the angle B.C.D. of the one bee equall to the angle E. C. G. of the other. Then, I say, that the sides which contayne those equall angles, are reciprocally proportionall, that is, as D. C. to C. G. so is E. C. to B. C. PROB. 88.93.
THEOREME LXXII. In rectangle-triangles, the figure which is made of the subtending side of the right angle, is equall vnto both the figures made of those sides, which contayne the right angle, so as those three figures are like, and in like sort described.
LEt A. B. C. be a triangle, Euc. 6.31. whose angle at the point C. is a right
angle. Then, I say, Ceul. 2.79. that the equilater triangle E. B. A. which is made of B. A. the subtending side of the right angle C. is equall vnto both the equilater triangles, made of the contayning sides B. C. and C. A. namely, to the triangles B. F. C. and A. C. G. taken together: And the like of squares, and all other like figures, in like sort described.
THEOREME LXXIII. In all plaine triangles, the sides are in proportion the one to the other, as the subtenses of the angles opposite thereunto; or as the sines of the angles opposite to those sides.
LEt D. be a triangle, and let there be circumscribed about the same triangle the circle A. B. C. by meanes whereof the
side A. B. is made the subtense of the angle A. C. B. that is, of the arch A. E. B. which is opposite to the same angle A. C. B. Also the side B. C. is made the subtense of the angle B. A. C. that is, of the arch B. F. C. which is opposite to the same angle B. A. C. and lastly, the side A. C. is made the subtense of the angle A. B. C. that is, of the arch A. G. C. which is opposite to the same angle A. B. C. Then, I say, that the side A. B. is in proportion to the side B. C. as the subtense of the angle A. C. B. to the subtense of the angle B.A.C. for the sides and subtenses, are one and the same. And likewise of the sines of those angles; which sines are the one halfe of their subtenses, and what proportion the whole hath to the whole, the same hath the halfe to the halfe. CHAP. 14.3.
THEOREME LXXIIII. Euery right lined figure, or plat, consisteth of more sides by two; then the number of triangles, whereof the same figure is composed.
LEt A.B.C.D.E.F.G. be a right lined figure, Ram. 10.1.
and let the triangles, whereof it is composed, bee 1, 2, 3, 4, and 5. Then, I say, that the same right lined figure consisteth of more sides by two, then the number of triangles, whereof it is composed. For the sides thereof are seuen, and the triangles fiue; as is apparant by the Diagram. CHAP. 37.3.
THE VSE AND OPERATION OF THE FORMER THEOREMES.
The second Booke.
THIS Booke consisteth of diuers Conclusions, or Geometricall PROBLEMES, here duly placed, by obseruation of naturall course; the cause being formerly, in the first Booke, amply expressed, and here the effect as fully made manifest, hauing either to other due relation. This Booke is diuided into foure Parts, wherein most plainely, briefely, and methodically, is expressed the practicke operation of the precedent THEOREMES; as the Distinction, Application, and Diuision of Lines and Angles, and the Description, Mensuration, Reduction, Addition, Inscription, [Page 56]Transmutation, Diuision, and Seperation of all sorts and formes of superficiall Figures, according to their seuerall kindes.
THE FIRST PART. Of the Properties, Passions, Dispositions, Applications, and Diuisions of Lines and Angles.
PROBLEME I. Two right lines giuen, being vnequall; to take from the greater a line equall to the lesser.
LEt a. and B. be two vnequall right lines
giuen, Euc. 1.3. whereof let A. be the greater; from which it is required to take a line equall to the lesser.
First, ioyne the two giuen lines together in such sort, as thereby they make any kind of angle, as C. D. E. and making the centre D. and the space D. E. (the length of the line B.) describe the arch line F. E. which shall cut off from the greater line, the line F. D. equall to the lesser line B. which was required to be done. DEF. 10.
PROBLEME II. To a right line giuen, to draw a parallell line at any distance required.
SVppose the right line giuen, Euc. 1.31. to be A. B. vnto which line it is required to haue a parallel drawn:
Open your Compasse to the distance required, and setting one foot in the end A. strike an arch line on that side the giuen line, whereon the parallell is to be drawne, [Page 57]and the like in the end B. as the arch lines C. and D. and by the conuexitie of those arch lines, draw the line C. D. which shall be parallell to the giuen line, as was required. DEF. 34.
PROBLEME III. To performe the former Proposition at a distance required, and by a point limitted.
LEt A. B. be a right line giuen, Euc. 1.31. whereunto it is
required to haue a parallell line drawne at the distance, and by the point C. Place one foot in C. from
whence take the shortest extention to the line A. B. as C. E. at which distance, place one foot in the end B. and with the other strike the arch line D. by the conuexitie of which arch line, and the limitted point C. draw the line F. G. which shall be a parallell to the giuen line A. B. the thing required. DEF. 34.
PROBLEME IIII. To erect a perpendicular on any part of a right line giuen.
LEt A. B. be a right line giuen,
and let C. be a point therein, Euc. 1.11. Ceul. 2.7. whereon it is required to erect a perpendicular, Open the Compas to any conuenient distance, and setting one foot in the point C with the other marke on either side thereof, the equall distances C. E. and C. F. Then opening the Compas to any conuenient wider distance, with one foot placed in the points E. and F. strike two arch lines, crossing each other, as in D. from whence draw the line D. C. which shall be the perpendicular required. DEF. 5.
PROBLEME V. To raise or let fall a perpendicular to a line giuen, from a point either aboue or beneath the same line.
LEt A. B. Euc. 1.11. be a line giuen, and let D. be a point aboue the same line: It is required from
the point D. to let fall a perpendicular line to the giuen line A. B. At any indifferent distance placing one foot in the point D. describe an arch line, intersecting the giuen line twice, as the arch line H. intersecteth the giuen line, in the points E. and F. Then either with the same, or some other conuenient distance, by placing the one foot in those points E. and F. strike two arch lines, crossing each other, as in G. By which point, and the giuen point D. draw the line D. C. which shall be a perpendicular to the giuen line. And the like construction is to be vsed, if the point were beneath the giuen line. DEF. 5.
PROBLEME VI. Ʋpon the end of a line giuen to raise a perpendicular.
LEt A. B. be a right
line giuen, and let B. be the end thereof, whereon it is required to rayse a perpendicular line. Open your Compas to a conuenient distance, and with one foot in B. draw the arch line F. G. D. Then placing one foot in F. at the same distance marke the arch in G. and on G. draw the arch E. D. Then on D. crosse the last arch in E. still keeping the same distance; from which intersection draw the line E. B. which shall be the perpendicular required. DEF. 5.
[Page 59] To performe the same another way.
LEt A.B. in the former Diagram be a right line giuen, and B. the end thereof, whereon a perpendicular is to be raysed. From the end B. pricke out any fiue equall distances, and opening the Compas to 4. of them, with one foot in B. strike an arch line towards C. Then opening the Compas to all 5. Diuisions, with one foot in the third Diuision, crosse the same arch line in C. from whence draw the line C. B. which shall be the perpendicular required.
PROBLEME VII. To diuide a right line giuen, into two equall parts.
LEt A. B. be a right line giuen, Euc. 1.10. Ceul. 2.6. which is to be diuided
into two equall parts: Open the Compas to more then halfe of the giuen line, and placing one foot thereof in either of the ends A. or B. with the other strike an arch line towards D. and another towards C. then place one foot in the other end of the giuen line, and with the same distance crosse the two former arches in D. and C. by which intersections, draw the line D. C. which shall diuide the giuen line A. B. as was required.
PROBLEME VIII. Vpon a right line giuen, on a point therein limited, to make an angle equall to an angle giuen.
LEt A. B. be a right line giuen, and C.
a point therein limited, and let H. F. G. be an angle giuen: It is required on the right line A. B. and on the point therein C. to describe an [Page 60]angle equall to the angle giuen H. F. G. At any conuenient distance setting one foot in F. the giuen angle, Euc. 1.23. strike the arch line H. G. and at the same distance placing one foot in the limitted point C. make the arch line D. E. Then take the distance from G. to H. and place that distance on the last drawne arch line from E. which endeth in D. by which point draw the line D. C. which shall include the angle D. C. E. vpon the giuen line A. B. on the point therein limitted C. being equall to the giuen angle H. F. G. the thing required. DEF. 42.
PROBLEME IX. To make a right angle vpon a line giuen, and on a point in the same line limitted.
LEt A. B. be a line giuen, and let B. be a point therein limitted. It is required on the line A. B. and to the point in it limitted B. to describe a right angle. By the sixt PROB. on the point B. raise the perpendicular C. B. which with the giuen line shall make the right angle A. B. C. on the line A. B. and to the point therein limitted B. which was required. DEF. 5.
PROBLEME X. To diuide an angle giuen into two equall parts.
LEt A. B. C. be an angle giuen,
to be diuided into two equall parts. Euc. 1.9. Ceul. 1.5. Hauing opened the Compas to any conuenient distance, place one foot in B. and with the other crosse the two lines B. A. and B. C. in the points D. and E. vpon which two points, strike two arch lines, at any equall distance, crossing one the other, as at F. From whence to the angle B. draw the right line B. F. which shall diuide the giuen angle into two equall parts, as was required. THEOR. 18.
PROBLEME XI. To diuide a right angle giuen into three equall parts.
LEt A. B. C. be a right angle giuen,
to be diuided into three equall parts. At any conuenient distance, with one foot in B. crosse the line B. C. as at E. and at the same distance on the points B. and E. strike two arch lines, crossing one the other in the point D. by which point, and the angle B. draw the line B. G. Then, by the tenth last before going, diuide the angle G. B. C. into two equall parts, with the line F. B. So shall those two lines F. B. and G. B. diuide the right angle giuen into three equall parts; which was the thing required. DEFINITION 21. THEOREME 18.
PROBLEME XII. To diuide a right line giuen into diuers equall parts, as many as shall be required.
LEt B. C. be a right line giuen,
and let it be required to diuide the same line into sixe equall parts. First from B. draw a line at all aduentures, making an angle of any quantitie with the giuen line, as the line B. A. making the angle A. B. C. Then by the 8. of this booke make the angle D. C. B. equall to the angle A. B. C. and from B. towards A. and likewise, conuenient distance, make 5. equall spaces (that is, one alwayes lesse then the number of parts required) and from point to point respectiuely draw lines, intersecting the giuen line: So shall you diuide the same into sixe equall parts required. THEOR. 22.
PROBLEME XIII. To diuide a right line giuen proportionally, according to any proportion required.
LEt A. bee a
right line giuen, Euc. 6.10, 11, 12. Ceul. 2.67. and let it be required to diuide the same into two such parts, that the greater may bee in proportion to the lesser, as the line B. is to the line C. First, make an angle of any quantitie, as E. D. F. whereof make the side F. D. equall to the giuen line A. then place on the other side the line C. from D. to H. and the line B. from H. to E. from E. to F. draw the line E. F. and lastly, by the 3. of this Booke, draw a parallel line to E. F. by the point H. as H. G. cutting F. D. in G. So shall you diuide F. D. (being equall to the giuen line A.) in the point G. in such sort that the greater segment F. G. hath the same proportion to the lesser G. D. as the line B. hath to the line C. which was required. THEOR. 22.
PROBLEME XIIII. From a right line, giuen to cut off any parts required.
LEt A. B. bee a right line giuen; Euc. 6.9. Ceul. 2.66.
and let it bee required to cut off from the same line 4/9 parts there of. Make any angle on the end B. by drawing the line E. B. on which line from the point B. make any 9. equall distances (or so many alwayes as the Nomen importeth, which is here 9.) From the ninth distance, draw the line E. A. and to that line by the fourth distance from B. draw the parallel line D. C. by the 3. of this Booke, cutting the giuen line A. B. in C. So haue you cut off the segment C. B. which is 4/9· parts of the giuen line A. B. the thing required. THEOR. 22.
PROBLEME XV. To finde a third line in continuall proportion vnto two lines giuen.
LEt A. and B. be two lines giuen; and let it be required to finde a third line, to be in such proportion to A. as A. is to B. Make an angle of any quantitie, as H. E. C. then place the line A. from E. to D. and the [Page 63]line B. from E. to F. and
draw the line D. F. place also the line A. from E. to G. and lastly, Euc. 6.11. Ceul. 2.68. by the 3. of this Booke, by the point G. draw the line G. C. parallel to F. D. So shall E. C. bee a third proportionall line to the two giuen lines, the thing required. THEOR. 22.
PROBLEME XVI. To finde a fourth proportionall line to three lines giuen.
LEt A. B. and C.
bee three lines giuen; and let it be required to find a fourth line, hauing such proportion to A. as B. hath to C. Make an angle of any quantitie, as D. G. K. And seeing it is the greater extreme, which is sought, Euc. 6.12: Ceul. 2.69. place first the lesser extreme C. from G. to H. and the lesser meane B. from G. to F. then draw the line F. H. and place the greater meane A. from G. to I. by which point I. draw the line E. I. parallel to F. H. which cutteth D. G. in E. So haue you E. G. the fourth proportionall line required. THEOR. 22.
PROBLEME XVII. To finde a meane proportionall line betweene any two lines giuen.
LEt A. and B. be two lines giuen, Euc. 2.14. & 6.13. Ram. 16.19. Ceul. 2.64.
betweene which it is required, to finde a meane proportionall line. First, ioyne the two giuen lines together, so as they make both one right line, as C. F. D. meeting in the point F. then describe thereon the semicircle C. E. D. and on the point F. by the 5. of this Booke erect a perpendicular to cut the circumference in E as F. E. which shall be the meane proportionall required. DEF. 39. THEOR. 70.
[Page 64] Note here, that if from the two points C. and D. to any one point in the limbe be drawne two right lines ( which by the 55. THEOR. make a right angle) as the lines C. E. and D. E. meeting in the point E. Then are those two lines meane proportionalls, that is, the line C. E. betweene C. F. and C. D. and the line D. E. betweene D. F. and D. C. THEOR. 70.
PROBLEME XVIII. To finde two meane proportionall lines, betweene any two right lines giuen.
LEt A. and B. be two right lines giuen, Euc. 1.43. between
which it is required to finde two meane proportionall lines. On the end of the line A. by the 6. of this book place the line B. perpendicularly, making a right angle, as the angle E. C. D. then drawe the Diagonall line E. D. whereon describe the semicircle E. G. D. making F. the center, then inlarge the lines C. E. and C. D. towards L. and M and taking in your compasse the giuen line B. the lesser extreme, place one foot in D. and with the other strike through the limbe of the semicircle in G. and on the point G. lay your ruler, turning it vp and downe on that point till by drawing the line H. K. you may cut the two lines C. L. and C. M. equidistantly from the center F. So shall E. H. and D. K. bee two proportionall lines, betweene the giuen lines A. and B. as was required. THEOR. 5.
PROBLEME XIX. To finde out in a line giuen, the two extremes of a meane proportionall giuen: So as the same meane bee not greater then halfe the giuen line.
LEt the meane proportionall giuen be A. and the line giuen B. C. It is required in the line B. C. to finde two extremes, betweene which the line A. shall bee a meane proportionall. Vpon the giuen line B. C. describe the semicircle B. F. C. then at the distance of the giuen meane, by [Page 65]the second of this booke draw a line parallel to B. C. (which
of necessity must either touch or cut the semicircle) as the line D. E. cutting the semicircle in F. From which point F. by the 5. of this booke, let fall the perpendicular F. G. which shall so diuide the giuen line B. C. in the point G. that the line giuen A. shall be a meane proportionall betweene the two segments B. G. and G. C. the thing required. THEOR. 9.55.70. PROB. 83.
PROBLEME XX. To diuide a line giuen by an extreme and meane proportion.
LEt A. B. bee a
AB. 12.
CB. 12.
BD. 6.
EB. [...] 180.—6.
AE. 18— [...] 180.
line giuen to be so diuided. Euc. 2.11.6.30. Ram. 14.10. Ceul. 2.36. By the 9. PROB. make of the giuen line A. B. a right angle, as A. B. C. setting C. B. equall to A. B. then increase the giuen line by halfe the length therof to D. and opening the compasse to the distance C. D. with one foot in D. strike through the giuen line at E. which shall diuide the same line by extreme and meane proportion, whereof the greater segment is E. B. and the lesse A. E. As was required. DEF. 35.1. THEOR. 44, 45, 46, 47.
To performe the former Prob. arithmetically.
LEt the number giuen to be diuided by extreme and meane proportion be 12. First, infold the square thereof in 5. the factus is 720. diuide that by 4. the quotus is 180. from the square roote whereof deduct halfe the giuen number, the remainder is [...] 180-6. and this is the greater portion or section, which being deducted from the giuen number, there remayneth 17— [...] 180. for the lesser portion; both which taken together makes 12. the giuen number.
PROBLEME XXI. To find the lesser segment of a line, diuided by extreame and meane [Page 66]proportion, when onely the greater is giuen; and consequently, to find the whole line.
LEt A. B. be the greater segment giuen. Euc. 12.2. It is required to find the lesser segment, and consequently,
the whole line. With the giue line A. B. and halfe the same line, by the 9. PROB. make on the point B. the right angle A. B. C. Then continue the giuen line infinitely towards D. and at the distance A.C. with one foot in A. cut the continued line in D. from whence cut off D. E. equall to halfe the giuen line: So is E.A. the lesser segment, and consequently, E.B. the whole line; as was required.
Or otherwise, thus.
LEt A. D. be the greater segment giuen, it is required to finde out the lesser, and consequently, the
whole line. On the giuen line A. D. by the 63. PROB. hereafter expressed, describe the Pentagon A. D. B. E. C. and against any two, the next immediate angles, draw subtending lines, which shall intersect the one the other, as the lines A.B. and D. C. in the point F. So shall F. A. or F. D. be the lesser segment; F. B. or F. C. the greater; and consequently, A. B. or D. C. the whole line, which was required. THEOR. 47.
PROBLEME XXII. Hauing the greater or lesser segment of a line diuided by extreame and meane proportion giuen, to find the other segment, and so the whole line.
LEt M. be the lesser segment of such a line giuen; Euc. 6.2. It is required to find the other segment,
and so the whole line. First, by the 20. PROB. diuide any line by extreame and meane proportion, as the line A.B.C. Then make an angle of any quantitie, as E. D. F. and place the lesser segment of the diuided line A. B. from D. to [Page 67]K. and the greater B.C. from D. to I. and draw the line I. K. Then place the lesser segment giuen M. from K. to H. and so working forwards, as by the 16. PROB. is taught, you shall produce I. G. the greater segment, which added to K. H. makes the whole line required. And the like course is to be held if the greater segment were giuen, and the rest sought. THEOR. 22.
PROBLEME XXIII. To diuide a right line giuen in power, according to any proportion giuen in two right lines.
LEt the right line giuen be C. D. and the proportion giuen, Eu. 1.47.3.31.6.4.6.8.6.20. Cor. 2. that betweene the two right
lines A. and B. It is required to diuide C. D. in power, according to the proportion of A. to B. First, by the 13. PROB. diuide the giuen line C. D. in the point F. proportionally, as A. to B. Then on the same giuen line describe the semicircle C. E. D. and from the point F. rayse a perpendicular, to cut the lymbe in E. from which point draw the lines E.C. and E.D. which two lines together, shall be equall in power to the giuen line, and the power of the line E. D. shall be in such proportion to the power of the line E.C. as A. to B. which was required. DEF. 36.37. THEOR. 6.9.55.70.
To performe the same another way, according to any proportion required, betweene two numbers giuen.
LEt A. B. bee a right line giuen, and let it be required
to diuide the power thereof in proportion, as 2. to 3. First, adde the two giuen termes together, which make 5. wherefore diuide the giuen line A. B. into 5. equall parts, then describe thereon the semicircle A. C. B. and on the second part erect the perpendicular 2. C. to cut the limbe in C. from which point draw the line C. A. which shall be 2/5· parts of the power of the giuen line; and the line C. B. which shall be 3/5· parts of the power of the same giuen line A. B. which was the thing required. DEFIN. 36, 37. THEOR. 6, 9, 55, 70.
PROBLEME XXIIII. To inlarge a line in power, according to any proportion required.
LEt C. D. bee a line giuen, Eu. 1.47.3.31.6.4.6.8.6.20. Cor. 2. to bee inlarged in power, according to
the proportion of A. to B. Seeke first by the 16. PROB. a line, which shall beare the same proportion to the giuen line C. D. as B. to A. which will be found to be C. E. and thereon describe the semicircle C. F. E. and, on the point D. erect a perpendicular, to cut the limbe in F. and lastly, draw the line C. F. which shall bee in power to the giuen line C. D. as B. to A. as was required. DEFIN. 38. THEOR. 6, 9, 55, 70.
To performe the same another way.
LEt A. B. be a line giuen; Euc. 10. Def. 5.1.47. and let it be required,
to encrease the same line A.B. in power, as 2. to 5. By the ninth PROB. with the giuen line make the right angle D.A. B. placing D. A. equall to A. B. Then draw the line D.B. and by the 23. PROB. before going, take halfe the power of the giuen line, which is A. E. and place it perpendicularly on the line D. B. as B. F. And lastly, draw the line D. F. which shall haue such proportion in power to the giuen line A. B. as 5. hath to 2. as was required. DEFINITION 38. THEOREME 6, 9, 55, 70.
PROBLEME XXV. To diuide the circumference, or find out all the cord lines of a circle, not exceeding the tenth.
LEt A.B. C.D. be a circle giuen; whereof it is required to find the cord lines. First, the Diameter A. B. being drawne, diuideth the circle into two equall parts: The next is found by opening the Compas to the [Page 69]semidiameter, and with one foot in A. strike through the lymbe at F. and E. and draw the
line F. E. which will diuide the circumference into three equall parts. Then draw the line C. D. diuiding the Diameter A. B. into two equall parts at right angles, Euc. 1.47.2.6.4.6.4.15.13.9, 10. and draw the line C.A. which shall be the side of an inscribed square. Then setting one foot in I. and at the distance I. C. crosse the Diameter in G. and draw the line C.G. which shall be the fift part, or the side of a Pentagon. The sixt part, or the cord of a Sextans, is the semidiameter. The seuenth part, or the side of an inscribed Septagon, is halfe the third, as F.I. or I. E. The eight being the side of an inscribed Octogon, is the line B.H. which is the cord of halfe the Arch or Quadrant B.H.D. The ninth is a cord of the third part of the arch F.L.A.E. as F.L. And the tenth is the line K.G. which is the greater segment of the semidiameter, diuided by extreame and meane proportion. THEOR. 6, 36, 55. PROB. 118, 119, 121.
PROBLEME XXVI. To draw a line from an angle in a Plot giuen, which shall take in as much as it cuts off.
LEt A.B.C.D.E. be a Plot giuen. It is required by a line drawne-from an angle, Euc. 1.37. in the
same Plot, to take in as much as shall be cut off by the same line. Increase the line C. D. at pleasure towards F. Then from the point A. draw the line A.D. making the Triangle A.E. D. and by the point E. draw the line E.F. parallell to A.D. cutting the increased line in F. from whence draw the line F. A. which shall take in the Triangle H. and cut off the Triangle G. each equall to other, as was required. THEOR. 26.
PROBLEME XXVII. To draw a line in such sort, as to retaine the superficiall quantitie of a giuen Triangle, and yet alter the Base to any possible length required.
LEt A.B.C. be a Triangle giuen, Euc. 1.37. and let it be required to draw a line in such sort, as to retaine the superficiall quantitie of the same Triangle, and yet alter the Base to the length of the line F. First, increase the [Page 70]Base B.C. to D. making B. D. equall to the giuen line F. Then draw the line A. D. and by the 3. PROB. by the point C. draw a parallell line to A. D. as E.C. cutting the side A.B. in E. And lastly, draw the line E.D. including the Triangle E. B. D. which retaineth the superficiall quantitie of the giuen Triangle A. B. C. and yet the Base altered, to the length required. THEOR. 26. PROB. 47.74.
PROBLEME XXVIII. To draw a line in such sort, as to retaine the superficiall quantitie of a giuen Triangle, and yet alter the altitude to any possible height required.
LEt A.B.C. in the former Diagram be a Triangle giuen; Euc. 1.37. and let it be required to draw a line in such sort, as to retaine the superficiall quantitie of the same Triangle, and yet alter the altitude to the height of the line G. At the distance of G. draw the line E. H. parallell to the Base B.C. which shall cut the side A. B. in E. from which point draw the line E.C. Then continue the Base at pleasure towards D. and from the point A. draw a line parallell to the line E.C. as A.D. cutting the line of continuation in D. from which point draw the line E.D. which shall make the Triangle E.B.D. retaining the superficiall quantitie of the giuen Triangle A.B.C. and yet the altitude altered to the height required. THEOR. 26.
PROBLEME XXIX. To find two right lines in such proportion as two figures giuen.
LEt the Trapezium, Euc. 1.17.6.1.1.38.1.41.
A.B.C.D. & the Parallelogram F.G.K. H. be two figures giuē. It is required to finde 2. right lines in such proportion the one to the other, as those two giuen figures are. By the 26. PROB. reduce the Trapezium A.B.C.D. into the Triangle A. C. E. and let fall the perpendicular A.O. Then increase the Base of the Parallelogram at [Page 71]pleasure towards M. and place on that increased line K.L. equall to the Base K.H. and draw the line F. L. including the Triangle F. L. H. equall to the Parallelogram F.G. K.H. Then by the 27. PROB. reduce the Triangle F. L.H. to the Triangle N.M.H. making the Base thereof M.H. equall to the Base C.E. of the first reduced Triangle; and lastly, let fall the perpendicular N.P. So haue you the two perpendiculars N. P. and A. O. in such proportion the one to the other, as the Parallelogram to the Trapezium giuen. THEOR. 35, 26, 28.
PROBLEME XXX. To find two right lines in such proportion the one to the other, as two giuen Squares.
LEt A.B.D.C. and E.F. G. D. be two Squares giuen. It is required, Euc. 6.4.8. Ac Cor. 19. Euc. 12.2. This Prob. is generall for all figures reduced into Squares. Which may also be thus wrought. Seeke a third proportionall to the sides C. D. and D.G. which shall haue such proportion to either of them, being the other extreame, as the squares haue the one to the other. Euc. 12.2.
to find two right lines in such proportion the one to the other, as the two giuē squares. Continue the side D. G. of the greater square giuen to H. making G. H. equall to the side of the lesser square giuen: Then draw the line F. H. subtending the right angle F.G.H. from which right angle let fall a perpendicular, cutting the line F.H. in K. So shall H.K. and K.F. be two lines in such proportion, as the two giuen squares, as was required. THEOR. 6, 9, 11, 55, 70.
PROBLEME XXXI. To draw a touch line to a circle giuen, from a point assigned.
LEt A. be the circle giuen, Euc. 3.17. and B. the point assigned. It is required
from the point B. to draw a touch line to the circle A. Draw the line A.B. from the centre to the point assigned, and diuide the same into two equall parts in the point E. and at the distance E.A. or E. B. crosse the lymbe in C. by which point and the point assigned B. draw the line B.D. which shall be a touch line to the circle A. as was required. THEOR. 52.
PROBLEME XXXII. To apply a line vnto a circle giuen, in such sort, as thereby to cut off a segment, wherein may be placed an angle, equall to an angle giuen.
LEt A. be a circle giuen. Euc. 3.34. Ceul. 2.48.49. It is required, to apply thereunto a line in such sort, as to cut
off a segment from the same circle, wherein may be placed an angle equall to the angle B. By the PROB. last before going, draw the touch line C. E. D. and on the point E. by the eight PROB. make the angle F. E. D. equall to the giuen angle B. by drawing the line F.E. So shall the segment F.G.E. containe an angle, drawne from any point in the arch thereof, equall to the giuen angle B. as the angle E. G. F. the thing required. THEOREME 56.
PROBLEME XXXIII. To describe, vpon a line giuen, such a segment of a circle, as shall containe an angle, equall to an angle giuen.
LEt the line giuen be A. B. and it is required to
describe thereon such a segment of a circle, Euc. 3.33. as shall containe an angle equall to the giuen angle F. To the end B. of the giuen line A.B. draw the line G. B. making by the 8. PROB. the angle A. B. G. equall to the giuen angle F. Then on the point B. erect a perpendicular to the line G.B. by drawing the line D.B. and on the end A. of the giuen line make the angle E.A.B. equall to the angle D. B. A. and draw the line A.E. which shall cut the perpendicular D. B. in C. which shall be the centre: Therefore on the point C. at the distance C.A. or C.B. describe the arch A. D. E. B. which with the giuen line shall make the segment of a circle, containing an angle equall to the giuen angle F. For if two lines be drawne from [Page 73]any one part of the circumference to the ends of the giuen line, they shal containe such an angle, as those at the points D. and E. which was required. THEOR. 52.56.
PROBLEME XXXIIII. A segment of a circle being giuen; to find out the centre, and consequently the diameter and the whole circle.
LEt A. B. C. be the segment of a circle giuen. It is required to find out the center thereof. Take a point at pleasure
with most conueniencie in the arche A. B. C. as B. and on the point B. at any meete distance describe the arch E. F. also one towards G. and another towards H. then remoue the compasse to A. and with the same distance crosse the next two arches in G. and E. and with the same distance on the point C. crosse the two next arches in H. and F. And lastly, Eu. 3.25. by those intersections, draw the lines G. P. and H. O. intersecting in D. which shall be the centre required; and consequently B. M. the Diameter, and A. B. C. M. the whole circle. DEF. 10.11.13.
To find out Arithmetically the diameter of the whole circle, (the segment A. B. C. being giuen) and consequently the other parts; worke thus: Suppose the corde line A. C. to be 12. and the perpendicular B. 2.4. Square halfe the corde line, which makes 36. which deuided by the perpendicular 4. quoteth, 9. whereunto adde the same perpendicular, which makes 13. the length of the whole diameter, whereby the rest of the parts are easily knowne.
To finde the extention of the arche line A. K. B. C. and to deliuer the same in a right line, worke thus; Deuide the corde A. C. into foure equall parts, and place one of those parts on the arche from A. to K. and from K. draw a line to the third part in the corde line, as K. 3. which taken double, shall be equall to the arche line, A. K. B. C. THEOR. 59.
The second Part.
OF the making and description of all sorts of superficiall figures, with their seuerall and particular mensurations.
PROBLEME XXXV. To make an equilater triangle, the side thereof being giuen.
LEt A B. be a right line giuen, and it is required to
make an equilater triangle, whose side shall be equall to the same line. Euc. 1.1. At the distance A. B. setting one foot in A. strike an arche line towards D. and at the same distance with one foot in B. crosse the same arche line in D. and from the intersection draw the lines D. A. and D. B. which with the giuen line A. B. shall make the equilater triangle A. B. D. as was required. DEF. 21. PROB. 36.
PROBLEME XXXVI. To find the perpendicular of an equilater triangle Arithmetically, the side being giuen.
IN the former Diagram let the side be giuen 8. It is required to find the perpendicular. By this Prob. the perpendicular of an Isoseeles is also found, the side and base being giuen. Square the side 8. makes 64. then square the halfe base, 4. makes 16. which deduct from 64. rests 48. whose square roote √ (powerof2). 48. neere rationall 6. 9/10 is the length of the perpendicular D C. required. THEOR. 6.19.29.
PROBLEME XXXVII. The perpendicular and side of an equilater triangle being giuen, to finde the Area or superficiall content.
IN the former Diagram the perpendicular √ (powerof2). 48. and the side 8. is giuen, and the Area is required, Multiplie the whole of eyther by halfe, the other as √ (powerof2). 48. by 4. the product is √ (powerof2). 768. neere rationall, 27. 39/54· the superficiall content required. THEOR. 24.
[Page 75] Or thus, without the perpendicular.
MVltiply 4. the halfe of one of the sides squarely, it makes 16. and the same product by the former halfe, Ram. 8.8. makes 64. and that by halfe the perimeter which is 12. the product is 768. from whence extract the square roote, which is neere rationall 27. 39/54· the superficiall content as before.
PROBLEME XXXVIII. To make a right angled triangle, the two containing sides being giuen.
LEt A. and B. bee two right lines giuen, for the containing sides of a right angled triangle required to bee made. By the 9. PROB. of the two giuen sides A. and B. make a right angle, as the angle C. F. E. then draw the subtending line C. E. Euc. 3.31. So haue you included the right angled triangle C. F. E. with containing sides equall to the giuen lines A. and B. which was required. THEOR. 9.6.11.55.70.
PROBLEME XXXIX. The perpendicular and base of a right angled triangle giuen to find the superficiall content.
IN the former Diagram the perpendicular F. G. 12. and the base C. E. 25. is giuen, and the content required. Multiplle the whole of eyther by halfe the other, as 12. by 12. ½· or 25. by 6. the product is 150. the superficiall content required. Or multiplie the whole by the whole, the Product is 300. whereof take halfe being 150. as before. THEOREME. 23.25.
Or thus, without the perpendicular.
MVltiply the containing sides 15. by 20. the product is 300. whereof take halfe for your demaund. Or multiply the halfe of the one in the whole of the other, the Product is 150 as before. THEOREME. 23.
PROBLEME XL. To make an Jsoseeles triangle on a right line giuen.
LEt the right line giuē be A. B. wheron it is required to describe an Isosceles triangle. Open the compasse at pleasure, and placing one foote in A. with the other strike an arch towards C. and at the same distance placing one foote in B. crosse, Euc. 1. Def. 25. the former arch in C. and draw the lines C. A. and C. B. which shall include the Triangle required. THEOR. 10.18.19.
PROBLEME XLI. The perpendicular and base of an Isosceles Triangle, giuen to find the area, or superficiall content Arithmetically.
IN the former Diagram, let C. E. the perpendicular √ (powerof2). 336. neere rationall 18. ⅓· bee giuen (or found by PROB. 36.) and let the base giuen be 16. It is required Arithmetically to finde the content. Multiplie the whole of eyther by halfe, the other as √ (powerof2). 336. by 8. the production is √ (powerof2). 21504. neere rationall, 146. 47/73· the superficiall content required. Or multiplie the whole by the whole, as √ (powerof2) 336. by 16. the Product. is √ (powerof2) 86016. whereof take the halfe which is √ (powerof2). 21504. and neere rationall 146. 47/73· as before. THEOR. 8.19.25.
Or otherwise without the perpendicular, thus.
IN the same former Diagram. Add all three sides together, which makes 56. whereof take halfe which is 28. then take the difference of each side from that halfe, as 8.8. and 12. And infold those 3. each in other makes, 768. which multiplie by 28. the former halfe, the Product is, 21504. whereof take the square roote, which is neere rationall, 146. 47/73· for the superficiall content as before, which rule is generall for all right lined Triangles whatsoeuer.
PROBLEME XLII. To make a Triangle of three vnequall sides, the lines being giuen, so as the two shortest together bee longer then the third line.
LEt A. B. and C. bee three lines giuen, whereof a Triangle is required to bee made. Euc. 1.22. Place the line A. from E. to F. then taking in the compasse, the line B. with one foot in F. make an arch towardes D. and [Page 77]at the distance of the third line, and with one foote in E. crosse the arch line in D. from which intersection to E. and F. draw the lines D. E. and D. F. So shall you include the Triangle D. E. F. whose sides are equall to the giuen lines A. B. and C. as was required. THEOR. 15.
PROBLEME XLIII. To finde the perpendicular of any Triangle Arithmetically, the sides being giuen.
IN the former Diagram let the sides giuen be E. D. 6. D. F. 8. and F. E. 10. It is required to finde the perpendicular. Square the three sides seuerally, which make 36. 64. and 100. then adde the square of the base, E. F. 100. to the square of one of the sides, as to 36. the square of the side F. D. which makes 136. from whence subtract the square of the other side D. F. 64. rests 72. whereof take the halfe 36. which deuide by the base 10. producing 3. ⅗· for the lesser segment of the base E. G. Euc. 1.47. The square of which segment 12 24/25· being deducted from the square of E. D. 36. first added, the remainder is 23. 1/25 whose radix is 4. ⅘· the length of the perpendicular D. G. required. THEOR. 70.
PROBLEME XLIIII. The perpendicular and base of any Triangle being giuen, to finde the area or superficiall content thereof Arithmetically.
IN the same former Diagram let the perpendicular D. G. 4. ⅘· & the base E. F. 10. be giuen, and let it be required to find the Area or superficiall content of the Triangle D. E. F. Multiply the whole of eyther by halfe the other, as the whole perpendicular 4. ⅘· by the halfe base 5. the Product shall bee 24. the superficiall content required. Or multiplie the whole by the whole, as 4. ⅘· by 10. the Product is 48. whereof take halfe, which is 24. as before the superficiall content. THEOR. 25.7.
Or thus without the perpendicular.
ADde the three sides together 6. 8. and 10. making 24. whereof take 12. the halfe, and then the difference of each side from that halfe, as 6. 4. and 2. and infold those differences each into other, which [Page 78]bringeth 48. that multiplied by the former halfe 12. produceth 576. The radix whereof is 24. the superficiall content as before.
This Proposition holdeth generall in all Triangles, and is the fittest and most meete for the mensuration of all plots and irregular figures; Regula generalis. whatsoeuer, being first reduced into Triangles, by drawing lines from angle to angle after the vsuall mannner.
PROBLEME XLV. To make a Triangle vpon a line giuen, like vnto another Triangle giuen.
LEt the Triangle A. B C. and the line E. F. be giuen. It is required on the line E. F. to make a Triangle like vnto the Triangle giuen. Euc. 1.37. Ceul. 2.78. Vpon the point E. by the 8. PROB. describe an Angle equall to the Angle A. B. C. and on the point F. describe another Angle equall to the Angle A. C. B. in the giuen Triangle, whereby shall bee included the Triangle D. E. F. vpon the line giuen E. F. like vnto the Triangle giuen A. B. C. that is with equall angles and lines proportionall, as was required. DEF. 40. THEOR. 11.49.
PROBLEME XLVI. To make a Triangle equall to another Triangle giuen, vpon the same Base, hauing an Angle equall to an Angle giuen.
LEt A. B. C. be a Triangle giuen, whose base is B. C. and let the
Angle giuen be D. It is required vpon the base B. C. to describe a Triangle equall to the Triangle giuen A. B. C. hauing an Angle equall to the giuen Angle D. Euc. 1.28.1.37. By the point A. by the 3. PROB. draw a parallel line to the base B. C. as E. A. and on the point C. describe an Angle equall to the Angle D. whereof the base to be one of the containing sides, and draw the line E. C. till it intersect, the parallel line in E. And lastly, from the point E. draw the line E. B. which shall include the Triangle E. B. C. equall to the giuen Triangle A. B. C. vpon the [Page 79]same base, and hauing the Angle thereof E. C. B. equall to the giuen Angle D. as was required. THEOR. 26.
PROBLEME XLVII. To make a Triangle: equall to another Triangle giuen, with a base or perpendicular limited.
VVOrke this Probleme in all respects according to the doctrine taught in PROB. 27. and 28. THEOR. 26. Euc. 1.37,
PROBLEME XLVIII. To make an Isosceles Triangle vpon a line giuen, whose Angles at the base, shall be eyther of them double to the third Angle.
LEt A. B. be a line giuen, whereupon it is required to describe a Triangle, whose Angles at the base shal be eyther of them double, to the third Angle. Suppose the giuen line A. B. to be the greater segment, Euc. 4.10. of a line deuided by extreame and meane proportion, and by the 21. PROB. finde out the whole line, at the distance of which line so found, with one foot in A. strike an arch line towards C. and at the same distance with one foot in B. crosse that arch line in C. from which intersection, draw the lines C. A. & C. B. which shal include the Triangle required. THEOR. 47.69.
PROBLEME XLIX. To make a Triangle equall to a Parallelogram giuen, vpon a line limited, and with an Angle equall to an Angle giuen.
LEt the Paralelogram
giuen bee A. B. C. D. the line giuen E. C. & let the Angle giuen be M. It is required vpon the line giuen to make a Triangle equall to the parallelogram giuē, Euc. 1.23.6.12.6.16. hauing an Angle equall to the giuen Angle M. Take the giuen line E. C. for [Page 80]the base, and by the 16. PROB. reason thus reciprocally. If halfe the giuen line E.C. yeeld H. D. the bredth of the parallelogram: what giues A.B. the length thereof: The answere shall bee F. G. the perpendicular of the Triangle to be made; at which distance by the 2. PROB. draw the line K. F. parallel to the base E. C. and on the end E. by the 8. PROB. describe an Angle equall to the giuen Angle M. and draw the line E. F. which shall cut the parallel line in F. from which point of intersection draw the line F. C. which shall include the Triangle E. F. C. equall to the parallelogram giuen, vpon the line E. C. giuen and hauing the Angle F. E. C. thereof equall to the giuen Angle M. as was required. THEOR. 42.
PROBLEME L. To make a Square vpon a line giuen for the side thereof.
LEt C. D. be a line giuen, whereon it is required to describe a Square. By the 6. PROB. on the end of the giuen line C. erect the perpendicular B. C. equall to the giuen line C.D. at which distance, Eu. 1.46. with one foote in B. strike an arch line towards A. then with one foote in D. crosse that arch line in A. And lastly, draw the lines A. B. and A. D. which shall include the square A. B. C. D. vpon the line giuen for the side thereof C.D. as was required. THEOR. 2.
PROBLEME LI. The side of a Square being giuen, to finde the Area or superficiall content Arithmetically.
LEt A. B. 16. be the side of a square giuen, whereof the superficial content is required. Multiplie the line giuen 16. in it selfe, the product is 256. the superficiall capacitie of the Square A. B. C. D. required. THEOR. 38.
PROBLEME LII. To make two Squares which shall be equall the one to the other; and also to two vnequall Squares giuen.
LEt the right lines A. and B. be the two sides of two vnequall squares giuen; and let it be required to make two other squares, which shall bee equall vnto them, and also the one of them equall to the other. By [Page 81]the 9. PROB. of the two giuen lines A. and B. make a right Angle, Euc. 1.6.1.32. as the Angle C. E.D. and draw
the subtending line C. D. on which line describe the semicircle C. E. F. D. and on the middle of C. D. erect the perpendicular F. G. to cut the lymbe in F. From which point draw the two lines F. C. and F. D. which shall be the two sides of two squares, equall to the two giuen squares, and also the one of them equall to the other; as was required. THEOR. 6.23, 55.
PROBLEME LIII. To describe a Square in such sort as it shall passe by any three points giuen.
LEt A. B. and C. be three points giuen, by which it is required to
make a square to passe. First, by the two neerest points, Euc. 1.12.1.31. as B. and C. draw a line at length as the line B. G. then by the third point A. by the 3. PROB. draw the line D. E. parallel to the line B. G. and on the point B. raise the perpendicular B. D. to cut the line D. E. in D. then at the distance D B. marke the points E. and F. betweene which draw the line E. F. which shall include the square D. E. F. B. passing by the three giuen points, as was required. PROB. 3.6.
PROBLEME LIIII. To make a long Square or right angled Parallelogram, the length and bredth being giuen.
LEt A. and B. be two right lines giuen for the length and bredth, it is required to make a right Angled parallelogram, whose length shall be A. and bredth B. By the 9. PROB. of the two giuen lines A. [Page 82]and B. make a right Angle, as the Angle C. F.E. and at the distance of the line A. with one foot in C. make an arch line towards D. and at the distance of the line B. with one foot in E. crosse, the former arch line in D. from which intersection draw the lines D.E. and D. C. which shall include the right angled parallelogram C. D. E. F. hauing the length and bredth equall to the giuen lines; as was required. THROR. 55.
PROBLEME LV. The length and breadth of a right angled Parallelogram or long Square being giuen, to finde the Area or superficiall content thereof Arithmetically.
IN the former Diagram let the length giuen be A. 20. and the bredth B. 10. Multiplie 20. by 10. the Product will bee 200. for the Area or superficiall capacitie of the Parallelogram, C. D. E. F. which is the thing required. THEOR. 38.
PROBLEME LVI. To make a Parallelogram, whose length is limited, equall to a Triangle giuen, with two opposite Angles each equall to an Angle giuen.
LEt the Triangle giuē
be F. E. C let the length limited be D. C. & let the Angle giuen be M. It is required vppon the giuen line to make a Parallelogram equall to the Triangle giuen, Euc. 1.23.6.12.6.16. Ceul. 2.82. hauing two opposite Angles each equall to the Angle giuen. By the 16. PROB. reason thus reciprocally; If the giuen line D. C. yeeld the perpendicular of the Triangle F. G. what giues halfe the base E. C. the answere shall be H. D. the bredth of the Parallelogram to be made; at which distance [Page 83]draw the line A. B. Parallel to the giuen line D. C. then on the point C. by the 8. PROB. describe an Angle as the Angle B. C. D. equall to the giuen Angle M. and draw the line B. C. which shall cut the Parallell line A. B. in B. from whence marke out the line A. B. equall to the giuen line D. C. and lastly draw the line A. D. which shall include the Parallelogram, A. B. C.D. equall to the Triangle giuen vpon the line D. C. giuen, and hauing two opposite Angles, namely, A. and C. each equall to the Angle giuen M. as was required. THEOREME 42.
PROBLEME LVII. To make a Rhombus, the side being giuen.
LEt the line giuen be A. B. wheron it is required to describe a Rhombus. Euc. 1.1. At the
distance A. B. with one foote in B. describe the arch line D. C. & at the same distance setting one foot in A. Crosse, the Arch line in D. on which point placing the compasse at the former distance crosse the arch line in C. And lastly, draw the lines D A. D C. and C B. which shall include the Rhombus, A. B. C. D. on the giuen line A. B. as was required. DEFINITIOM 26.
PROBLEME LVIII. The side of a Rhombus being giuen to finde out the Area or superficiall content thereof Arithmetically.
IN the former Diagram, let the side A B. or D C. 16. be giuen, and let it be required to find the Area or superficiall content thereof. By the 36. PROB. finde out the perpendicular D. E. √ (powerof2). 192. and neere rationall, 13.23/26· and multiply the same by the giuen side, 16. the Product shall be √ (powerof2). 49152. and neere rationall, 221.311/442 for the superficiall content required. THEOR. 34.
PROBLEME LIX. To make a Rhomboydes the length and bredth being giuen in two right lines.
LEt the length giuen bee the line D. C. and the bredth, the line G. of which length and bredth it is required to describe a Rhomboydes. At the distance of the giuen breadth G. and from one of the ends of the [Page 84]giuen length, as from C. choose a point as B. and
at the same distance with one foot on the other end of the giuen length, Euc. 1.31. as on D. strike an arch towardes A. then at the distance of the giuen length with one foot in the point B. crosse the former arch in A. And lastly, draw the lines A. D. A. B. and B. C. which shall include a Rhomboydes of the length and bredth giuen. PROB. 3.
PROBLEME LX. To make a Rhomboydes with lines limited, hauing two opposite Angles, equall to an Angle giuen.
LEt the limited lines for the length & bredth be D.C. & G. as in the former Diagram, and let the Angle giuen be F.C. B. and let it be required to make a Rhomboydes of such length and bredth, and with two opposite Angles equall to that giuen. Euc. 1.23.1.31. On the end C. of the giuen length D.C. by the 8. PROB. protract an Angle equall to the giuen Angle, as D. C. B. making the line B. C. equall to the giuen bredth G. And so worke forward in all respects as in the former Probleme. PROB. 3.8.
PROBLEME LXI. A Rhomboydes giuen to finde the superficiall content Arithmetically.
LEt the Rhomboydes giuen bee that in the former Diagram, It is required to find the superficiall content thereof. Take the giuen length D. C. or A. B. 20. and seeke out (as hath beene formerly taught) the parallel distance or perpendicular B. E. 12. which multiply the one by the other, the Product is 140. the Area of the giuen Rhomboydes required. THEOR. 34.
PROBLEME LXII. To describe a Pentagon, hauing sides and Angles equall.
LEt any obscure line be drawne as the right line A. B. and at any conuenient distance place thereon foure equall parts or diuisions, as from C. to 4. and at the distance of two of them, on the second part, as a centre, [Page 85]describe an obscure circle,
on which centre rayse the perpendicular D. 2. to cut the limbe in D. then at the distance D. 1. with one foot in 1. crosse the line in K. and with the distance D. K. marke the limbe of the Circle in the points D. E.F. G. and H. And lastly, draw lines from point to point, which shall include the Pentagon required. DEF. 29. THEOR. 47. PROB. 20.21.25.
PROBLEME LXIII. To describe a Pentagon vpon a line giuen.
LEt A. B. be a right line
giuē, Euc. 13.8. wheron it is required to describe a Pentagon. Suppose the line giuen, to bee the greater segment of a right line diuided by extreame and mean proportion. And by the 21. and 22. PROB. finde the whole line; which let be A. C. and at the distance of A. C. with one foot in B. strike an arch towards H. and another towards K. and with the foote in A. strike one towards L. then take the distance of the giuen line, and on A. and B. crosse the Arches at H. and L. and on H. or L. crosse the Arch in K. And lastly, draw lines from each intersection to other, which shall inclose the Pentagon, as was required. THEOR. 47. PROB. 20.21, 22, 25.
PROBLEME LXIIII. The side of a Pentagon being giuen, to finde the superficiall content Arithmetically.
IN the former Diagram, let the giuen side be 10. It is required to finde the Area of that Pentagon. By the 36. PROB. (supposing an Isosceles described on any side of the Pentagon, whose top is the centre, as A. D B.) [Page 86]seeke the perpendicular D. G. 6.22/25· which multiplied in halfe the perimetrie 25. produceth 172. the Area required. This rule is generall in all kind of regular polligons, A generall Rule. of how many sides so euer; aswell for their superficiall content, as finding their perpendicular. THEOR. 19.39. PROB. 36.
PROBLEME LXV. To make two like Figures, bearing the one to the other, any proportion assigned in two right lines.
LEt A. and B. be two right lines giuen, and let it be required to make two
like Triangles, Squares, Circles or other like figures, hauing such proportion the one to the other, Eue. 6.4.6.8. as A. to B. make of the two giuen lines one right line, as D. E. and describe thereon the Semicircle C.D.E. and on the point F. where the 2. giuen lines meete, erect the perpendicular C. F. to cut the limbe in C. from whence draw the lines C. D. and C. E. which two lines shall bee the sides of two equilater Triangles, or of two Squares, or other like figures or the diameters of two circles, bearing such proportion the one to the other, as the two giuen lines; which was required. THEOR. 6.11, 55, 66, 70, 65.
PROBLEME LXVI. Two Circles being giuen, to make one Circle equall to them both.
LEt A. and B. be the Diameters of two Circles giuen. It is required
to make one Circle equall to them both. With the lines A. and B. by the 9. PROE. make a right Angle, as D. C. E. then draw the subtending line D. E. And lastly, on the line D. E. describe the Circle D. C. E. which shall bee equall to the two giuen circles as was required. THEOR. 65.70, 66.
PROBLEME LXVII. The Diameter of a Circle being giuen, to find the circumference thereof Arithmetically.
IN the former Diagram, let the Diameter giuen be 14. It is required to finde the circumference thereof. Multiply the Diameter giuen 14. by 22. the Product is 308. which diuided by 7. bringeth 44. the circumference required. Or multiply 14. the Diameter by 3. 1/7· the Product is 44 as before. THEOREME. 59.
If the circumference bee giuen, and the Diameter required, It appeareth by this rule, that the circumference 44. being multiplied by 7. and the product diuided by 22. bringeth 14. the Diameter.
PROBLEME LXVIII. The Diameter and Circumference of a Circle being giuen, to find the Area, or superficiall content thereof Arithmetically, diuers wayes.
IN the former Diagram let the Diameter of the Circle D. C. E. be 14. and the Circumference thereof 44. It is required to find the superficiall content. Multiply the Semicircumference 22. by the Semidiameter 7. the Product will be 154. the superficiall content required. THEOR. 62.
Or multiplie the whole Circumference 44. by the Semidiamenter 7. the Product will be 308. whereof take halfe, which is 154. as before. THEOREME. 60.
Or multiply the square of the Diameter 196. by 11. the Product will be 2156. which diuided by 14 bringeth 154. as before. THEOR. 61.
PROBLEME LXIX. The Diameter and Arch-line of a Semicircle giuen, to find the Area thereof.
LEt A. B. C. be a Semicircle giuen,
A B. √ (powerof2). 98.
A. E. √ (powerof2). 24.½·
E. D. √ (powerof2). 24.½·
F. E. 7.—√ (powerof2). 24.½·
whose Diameter is A. C. and the Arch line A.B.C. It is required to finde the Area of the Semicircle. Multiplie halfe the arch line 11. by the Semidiameter 7. The Product will be 77. for the Area required. THEOREME. 63.
PROBLEME LXX. The Semidiameter and Arch line of a Sector of a Circle giuen to find the Area.
IN the former Diagram, let B. C. D. be the Sector of a Circle, whose Semidiameter is D. C. or D. B. and the arch line B. C. and it is required to finde the Area. Multiply the Semidiameter 7. by halfe the arch line B. C. 5.½· the Product is 38.½· for the Area required. THEOR. 64.
PROBLEME LXXI. Any Segment or part of a Circle being giuen, to finde the superficiall content thereof.
IN the former Diagram, let A. F. B. E. be the Segment of a Circle, the content whereof is required. By the 34 PROB. finde out the Centre, and then draw the lines D. A. and D. B. and cast vp the whole Figure A. F. B. D. as in the last Probleme which will be 38.½· then finde the superficiall content of the Triangle, A.B.D. by the 41. PROB. which is 24.½ and deduct the same out of the whole content 38.½ resteth 14. for the superficiall content of the giuen Segment as was required.
By this rule (obserued with discretion) may all manner of Segments or parts of a Circle, whether greater or lesser then a Semicircle, be easily measured without further instruction.
But here is to bee noted, that the precedent rules concerning the mensuration of Circles, Note. and their seuerall parts, are not exactly true: for that the proportion betweene the Diameter and circumference is irrationall; and the squaring of a Circle or the meanes thereof (other then mechanically) not yet discouered or found out; yet of such sufficient precisenesse as no notable or apparant error can be made or found in the conclusions thereby wrought.
PROBLEME LXXII. An irregular plotte or Figure being giuen, to finde the Area or superficiall content thereof.
LEt A. B. C. D. E. F. G. H. I. be an irregular plotte or Figure, whose superficiall capacitie is required. Reduce the same into as many Trapezias as it will containe, as first the Trapezium A. B. C. D. Secondly, A. D E F. Thirdly, A. F. G. I. and there remaineth the Triangle, I. G. H. In which three Trapezias draw the Diagonals B D. D F. and [Page 89]F. I. which shall serue as
the common bases to each Triangle on eyther side, on which bases let fall the perpendiculars from the seuerall Angles at A.C.E. and G. then in euery Trapezium take the length of the base by it selfe, and the length of the two perpendiculars theron falling, ioined together in one number by it selfe, then multiply the halfe of the one in the whole of the other, the Product is the Area of that Trapezium, which reserue by it selfe, and working in the like sort with the rest: And lastly, the Triangle I. G. H. collect all their Products together, which shall shew the superficiall content required. PROB. 44. THEOR. 25.
This I hold the best manner of Mensuration of plots, aswell for expedition, as exactnes in auoyding errours, often hapning by multiplicitie of numbers, and many multiplications.
THus in this second part, haue I taught the Description and Mensuration of all manner right lined superficiall Figures, according to the strict and precise rules and precepts of absolute Arte: yet seeing that dispatch and expedition in businesse of import, is much more requisite then needlesse nicenesse; I would not haue my Surueyor ignorant or vnfurnished of such other ready and perfect helpes (though more mechanike) as may yeelde him ease, and saue much labour in furthering his intents. To which end, for the speedie and exact mensuration of all superficiall figures, J would haue prouided a Protractor in Brasse, whose Scale should containe in length about 8. or 10. ynches with equall Diuisions on the edge of eyther side, of 12 in an ynch on the one side, and 11. on the other, being numbred by tennes after the vsuall manner of those kind of works (which for mine own part is the Scale, I neuer vse in all my first drawne plots, whether the quantitie be small or great, well knowing the inconueniencie of smaller Scales) hauing placed thereon a Sextans of a Circle most excellent for many vses, as the speedy laying downe of any Angle required, or the ready finding of any Angle giuen, &c. The order and making whereof is well knowne to Master Elias Allen, who for [Page 90]my selfe and friends hath made of them. By this Scale with helpe of the middle line thereof, Orthigonally drawne to the edge; you shall readily rayse a perpendicular, and as instantly receiue the length thereof, and most speedily obtaine the base of any Triangle, or the side of any Figure giuen by applying the edge of the Scale thereunto: which is much more facile and speedy then the former Precepts; and to bee preferred for exactnes, speede and perfection, before the ordinarie course with Scale and Compasse. But doubting to exceede the Scale and Compasse of my intended purpose, I will here conclude the second part of this my second Booke.
The third Part.
OF the Reduction and Translation of all manner of superficiall figures, from one forme vnto another, retayning still their first quantitie.
PROBLEME LXXIII. To reduce one triangle into another, on the same base, but hauing an angle equall to an angle giuen.
LEt A. B. C. be a triangle giuen, Euc. 1.37. and let
the angle giuen be E. and the base of the giuen triangle B.C. on which base it is required to reduce the giuen triangle to another, hauing an angle equall to the angle E. From the point A. by the 3. PROB. draw a parallel line to the base B. C. as the line A. D. then on the point C. by the 8. PROB. make the angle D. C. B. equall to the giuen angle E. and draw the line D. C. cutting the parallell line in D. and lastly, draw the line D. B. which shall include the triangle D. B. C. equall to the giuen triangle A. B. C. vpon the same base B. C. and hauing an angle equall to the giuen angle E. as was required. THEOR. 26.
PROBLEME LXXIIII. To reduce one triangle into another, vpon a base equall to a base giuen.
LEt the triangle giuen be A. B. C. and F. the giuen base, Eu. 1.37.
whereon it is required to reduce the giuen triangle. Take the giuen base F. and place it from B. to E. and from the point E. to the top of the giuen triangle A. draw the line A.E. then increase the side A. B. of the giuen triangle towards D. and from the point C. by the 3. PROB. draw a parallel line [Page 92]to A. E. as D. C. cutting the increased side in D. and lastly, draw the line D. E. which shall include the triangle D. B. E. equall to the giuen triangle A. B. C. vpon the base B. E. equall to the giuen base, as was required. THEOR. 26. PROB. 27.
If the giuen base be greater then the base of the giuen triangle, worke in all respects, as is taught in PROB. 27.
PROBLEME LXXV. To reduce one triangle to another, of any possible height required.
LEt the triangle A. B. C. in the former Diagram be giuen, Ceul. 3.5. and let the height required be the line G. of which height it is required to reduce the giuen triangle into another of the same quantitie At the distance of the giuen height G. by the 2. PROB. draw the line H. K. parallel to the base B. C. then increase the side A. B. till it cut the parallel line H. K. in D. from which point draw the line D. C. then by the point A. draw the line A. E. parallel to D. C. cutting the base in E. and lastly, from the point E. draw the line E. D. which shall include the triangle D. B. E. equall to the giuen triangle A. B. C. and of the height G. as was required. THEOR. 26. PROB. 28.
If the giuen height be lesse then the height of the giuen triangle, worke in all respects, as is taught in PROB. 28.
PROBLEME LXXVI. To reduce a triangle giuen into a square.
LEt A. B. D. be a triangle giuen. Euc. 1.31. It is required to reduce the same into a Geometricall square. By the 17. PROB. finde out a meane proportionall line between the base B. D. and halfe the perpendicular A. C. which shall be the line E. D. on which line by the 50. PROB. describe the square E. F. G. D. which shall be equall to the giuen triangle. THEOR. 25.27.
PROBLEME LXXVII. To reduce a triangle giuen, into a right angled parallelogram.
LEt A. B. C. be a triangle giuen. Euc. 1.42. Ceul. 2.28. & 2.80. It is required to reduce the same into a right angled parallelogram. From the angle A. let fall to the base B. C. the perpendicular A. F. then take halfe thereof for the breadth, and [Page 93]the whole base B. C. for the length, with which breadth and length by the 54. PROB. describe the right angled parallelogram D. E. B. C. which shall be equall to the giuen triangle, as was required. THEOR. 25.
PROBLEME LXXVIII. To reduce a tringle giuen, into a parallelogram, hauing an angle equall to an angle giuen.
LEt A. B. C. be a triangle giuen, and let
the angle giuen be D. It is required to reduce the same triangle into a parallelogram, hauing an angle equall to the angle D. By the 3. PROB. from the point A. draw the line E. A. parallel to the base B. C. then diuide the base B. C. into two equall parts in the point G. on which point by the 8. PROB. describe the angle F. G. B. equall to the giuen angle D. and draw the line F. G. cutting the parallel line in F. from whence draw the line F. E. equall to G. B. and lastly, draw the line E. B. which shall include the parallelogram E. F. G. B. equall to the giuen triangle, hauing an angle F. G. B. equall to the giuen angle D. as was required. THEOR. 28.
PROBLEME LXXIX. To reduce a triangle giuen into a Rhombus.
LEt the triangle giuen be A. B. C. and it is required to reduce the
same into a Rhombus. Euc. 1.37. Cor. 6.19. By the 3. PROB. draw the line A. H. parallel to the base B. C. then diuide the base B. C. into two equall parts in the point E. on which point at the distance E. B. or E. C. strike an arch towards D. and on the point C. at the same distance crosse the former arch in D. by which intersection [Page 94]and the point C. draw the line H. C. to cut the parallel line in H. then by the 17. PROB. find out a meane proportionall line betweene C. D. and C. H. which is C. K. vpon which line C. K. by the 57. PROB. describe the Rhombus G. K. C. F. which shall be equall to the triangle giuen, as was required. THEOR. 26.41.43.
PROBLEME LXXX. To reduce a square giuen into a triangle, hauing an angle equall to an angle giuen, and that on a line giuen.
LEt the square giuen be A. B. C. D. the
angle giuen E. and let the giuen line be F. Euc. 1.37. On which line it is required to reduce the giuen square in to a triangle, hauing an angle equall to the angle E. First, continue the side of the giuen square D. C. to G. making G. D. equall to D. G. and draw the line G. B which shall include the triangle B. G. C. equall to the giuen square; then take the giuen line F. and lay it downe from C. to O. and by the 74. PROB. make the triangle M. O. C. then by the third PROB. draw the line M. N. parallel to the base G. C. and on the point C. protract an angle equall to the giuen angle E. as N. C. O. and draw the line C. N. to cut the parallel in N. and lastly, from the point N. draw the line N. O. which shall include the triangle N. O. C. equall to the giuen square, hauing an angle as N. C. O. equall to the giuen angle, and that on the line O. C. equall to the giuen line F. as was required. THEOR. 26.
PROBLEME LXXXI. To reduce a square giuen into a triangle, with angles equall, and lines proportionall to a triangle giuen.
LEt the giuen square
be A. B. C. D. and let the giuen triangle be E. F. G. and let it be required to reduce the same square into a triangle, Euc. 1.37. Cor. 6.19. with angles equall, and lines [Page 95]proportionall to the giuen triangle. According to the first Part of the last PROB. make the triangle B. H. C. equall to the giuen square, then continue the side A. B. of the giuen square towards K. and on the point H. protract the angle M. H. D. equall to the angle F. in the giuen triangle, drawing the line M. H. at length to cut the continued side A. B. in K. then on the point C. protract the angle M. C. H. equall to the angle G. in the giuen triangle, and draw the line C. M. to cut the line M. H. in M. which shall include the triangle M. C. H. with equall angles to the giuen triangle, but of greater content then the giuen square; wherefore by the 17. PROB. finde out a meane proportionall line betweene H. K. and H. M. which is H. N. and from the point N. by the 3. PROB. draw the line N. O. parallel to M. C. which shall include the triangle N. O. H. equall to the giuen square, and hauing equall angles and lines proportionall to the giuen triangle, as was required. THEOR. 26.43,
PROBLEME LXXXII. To reduce a square into an equilater triangle.
LEt the square giuen be A. B. C. D. and it is required to reduce
the same into an equilater triangle. Euc. 1.37. Cor. 6.19. Double the side D. C. by increasing the same to G. and draw the line G. B. to include the triangle B. G. C. equall to the giuen square, then at the distance of the side of the square, with one foot in C. strike an arch towards I. and at the same distance with one foot in D. crosse the same arch in I. and by the intersection and the point C. draw the line K. C. at length to cut A. B. in H. and take C. H. and place the same from C. to E. then by the 17. PROB. finde out the meane proportion betweene C. E. and C G. which is C. F. at which distance describe the equilater triangle K. F. C. which shall be equall to the giuen square, as was required. THEOR. 26. [...]5.43
PROBLEME LXXXIII. To reduce a square giuen, into a right angled parallelogram or long square, the length and bredth being limited in a right line: So as the side of the square exceede not halfe the line giuen.
LEt the square giuen be A. B. C. D. and let the right line giuen be F. C. Euc. 1.47.6.13. It is required to reduce the same square into a long square, whose length and bredth together shall be equall to F. C. Vpon the giuen line F. C. place the giuen square, as in the Diagram, then describe the semicircle [Page 96]F. E. C. to cut the side A. B. of the giuen square in E. from which point let fall the perpendicular E. G. to cut the giuen line in G. So shall F. G. be the length, and G. C. the bredth of the long square to be made, of which length and bredth by the 54. PROB. describe the parallelogram F. G. H. K. which is equall to the giuen square, and of the length and bredth required. THEOR. 41. PROB. 19.
PROBLEME LXXXIIII. To reduce a square giuen into a long square, whose bredth is limited in a right line giuen.
LEt the square giuen be A. B. C. D. and let F. be the right line
giuen. Euc. 1.47, 6, 13. It is required to reduce the same square into a long square, whose bredth shall be equall to the giuen line F. Continue the side C. D. of the giuen square towards K. and place the giuen line F. from D. to G. and from the point G. draw the line G. A. which diuide equally in N from which point draw the perpendicular N. H. to cut the line of continuance in H. on which point at the distance H. G. describe the semicircle K. A. G. to cut the line of continuance in K. so shall K. D. be the length sought for, with which length, and the giuen bredth F. or D. G. by the 54. PROB. describe the parallelogram L. M. D. K. which shall be equall to the giuen square, as was required. THEOR. 41. PROB. 19.
PROBLEME LXXXV. To reduce a square giuen into a long square, whose length is limited in a right line giuen.
IN the former Diagram, Euc. 1.47, 6, 13. let that square be the square giuen, and the length giuen the right line P. First, continue D. C. as before towards K. and make K. D. equall to the giuen line P. and draw the line K. A. vpon the middle whereof S. raise the perpendicular S H. to cut the line of continuance in H. on which point describe the semicircle, as before, to cut D. C. in G. so shall D. G. be the bredth sought, of which bredth and the giuen length, make the long square, as before. THEOR. 41. PROB. 19.
PROBLEME LXXXVI. To reduce a long square giuen into a geometricall square.
LEt the long square giuen be A. B. C. D. and it is required to
reduce the same into a geometricall square. Euc. 2.14. Ceul. 2.37. Continue the side D. C. of the long square giuen towards H. and let the bredth B. C. of the long square be placed on the line of continuation from C. to H. then on D. H. describe the semicircle D. F. E. H. and increase the bredth of the long square C. B. till it intersect the limbe in E. so shall E. C. (being the meane proportionall betweene D. C. and C. H.) be the side of the square sought, wherefore, on the line E. C. by the 50. PROB. describe the geometricall square F. E. C. G. which shall be equall to the long square giuen, as was required. THEOR. 41. PROB. 19.
PROBLEME LXXXVII. To reduce one long square giuen into another, whose length or bredth is limited in a right line giuen.
LEt the long square giuen
be A. B. C. D. and let the bredth giuen be E. and let it bee required to reduce the long square giuen, Euc. 1.43. into another long square of the giuen breadth E. Continue the bredth of the giuen square D. A. towards F. making A. F. equall to the giuen bredth E. also continue the length D. C. of the giuen square towards G. and by the point F. and B. draw out the line F. G. to cut the line of continuance last drawne in G. so shall G. C. be the length sought for, with which length and the giuen bredth E. by the 54. PROB. make the long square K. H. C. G. which shall be equall to the giuen long square, and of the limited bredth, as was required. THEOR. 5.
Or otherwise, thus.
IN the former Diagram, let it be required to reduce the long square there giuen, into another long square, Euc. 6.1 [...]. whose bredth shall be the giuen line E. By the 16. PROB. reason thus. If E. the giuen bredth giue A. D. the bredth of the giuen square, what giues A. B. the length thereof, the answere [Page 98]shall be G. C. or K. H. the length sought for, with which length and the giuen bredth, make the long square K. H. C. G. as before. THEOR. 42.
PROBLEME LXXXVIII. To reduce one long square giuen into another, whose length and bredth shall haue proportion the one to the other, as two giuen lines.
LEt the long square giuen be A. B. C. D.
A B. 72.
B C. 60.
E F. 90.
E G. 48.
and let the proportion giuen be such, Euc. 6.12.13.14. as that betweene the two right lines M. and N. It is required to reduce the same long square into another long square, whose length and bredth shall haue proportion the one to the other, as those two giuen lines. By the 17. PROB. seeke out the meane proportion betweene the two giuen lines M. and N. which is Q. seeke also the meane proportion betweene A. D. and A. B. the bredth and length of the long square giuen, which is the right line P. then by the 16. PRO. reason thus. If Q. giues P. what giues N. the answere shall be E.F. for the length sought; and againe, if Q. giues P. what giues M. the answere shall be E. G. for the bredth sought; of which length and bredth E.F. and E.G. by the 54. PROB. make the long square E. F. C. G. which shall be equall to the long square giuen, and the bredth to the length, in such proportions as the line M. to the line N. as was required. THEOR. 5, 41, 42, 71.
PROBLEME LXXXIX. To reduce a Rhombus into a geometricall square.
LEt the Rhombus giuen be A. B. C. D. and let it
be required to reduce the same into a geometricall square. Euc. 1.36. First, by the 17. PROB. finde out a meane proportionall line betweene D. C. the side of the giuen Rhombus, and the parallel distance, or perpendicular line B. E. which shall be the line F. C. vpon which line, by the 50. PROB. describe the geometricall square F. G. H. C. which shall be equall to the giuen Rhombus A. B. C. D. as was required. THEOR. 33, 34.
PROBLEME XC. To reduce a Rhomboydes giuen, into a Geometricall Square.
LEt the Rhomboydes giuen be A. B. C. D. and it is required to reduce the same into a Geometricall Square. Euc. 1.36. Let fall the perpendicular B. E from the Angle B. to the Base D. C. betweene which Base and Perpendicular, by the 17. PROB. finde out a meane proportionall line, which shall be F.C. vpon which line by the 50. PROB. describe the Square F. G. H. C. which shall be equall to the giuen Rhomboydes as was required. DEF. 27. THEOR. 33.34.
PROBLEME. XCI. To reduce a Rhomboydes giuen into a Triangle, hauing an Angle equall to an Angle giuen.
LEt the Rhomboydes giuen be A. B. C. D. and let the Angle giuen be E. F. G.
It is required to reduce the same Rhomboydes into a Triangle, Eu. 1.41. hauing an Angle equall to E. F. G. First, increase the line A. B. towards E. and also the base D. C. towards F. and make C. F. equall to D. C. then on D. protract the Angle E. D. F. by the 8. PROB. equall to the giuen Angle E. F. G. and draw the line D. E. to cut the increased line in E. And lastly, from E. draw the line E. F. which shall include the Triangle E. D. F. equall to the Rhomboydes giuen, and hauing an Angle equall to the giuen Angle as was required. THEOR. 28.
PROBLEME XCII. To reduce a Trapezium giuen into a right angled Parallelogram, or into a right angled Triangle.
LEt the Trapezium giuen be A. B. C. D. which is to be reduced into a right angled Parallelogram, or into a right angled Triangle. First, Euc. 1.41. Ceul. 2.29. draw the Diagonall line B. D. then by the 2. PROB. by the points A. and C. [Page 100]draw the lines E. F. and H. G. Parallel to B. D. and by the point B. and
D. draw E. H. and F. G. to cut the two last lines Orthigonally, so shal you include the Parallelogram E. F. G. H. which is double to the Trapezium giuen, wherefore diuide the same into two equall parts, by drawing the line K. L. So haue you the right angled parallelogram E. F. L. K. or K. L. G. H. equall to the Trapezium giuen. And the Diagonall line E. G. being drawne, shall include the right angled Triangle, E. H. G. or E. F. G. likewise equall to the same Trapezium as was required. THEOREME 23.27.
PROBLEME XCIII. To reduce a Trapezium giuen into a Triangle, vpon a line giuen, and hauing an Angle equall to an Angle giuen.
LEt the Trapezium giuen be A. B. C. D. the giuen line E. and let the Angle giuen be F. It is required to reduce the same Trapezium into a Triangle, Euc. 1.23.1.13.1.37.1.44.6.12. on the giuen line E. and hauing an Angle equall to the giuen Angle F. First, increase the base D. C. at
length towards H. then draw the line A. C. and from the point B. by the 3. PROB. draw the line B. H. parallel to the line A. C. to cut the increased line in H. and then draw the line A. H. which shall include the Triangle A. D. H. equall to the Trapezium giuen, but not hauing the Angle nor Base required; wherefore take the giuen Base E. and place it from D. to Q. and by the 16. PROB. reason thus; if D. Q. the giuen Base giues D. H. what giues A. G. the perpendicular; the answere will be the line P. for the perpendicular of the Triangle sought, at which distance draw the line M. O. parallel to the line D. H. then vpon the point D. protract an Angle equall to the giuen Angle F. as N. D. Q. and draw the line D. N. to cut the parallel line in N. And lastly, draw the line N. Q. which shall include the Triangle N. D. Q. equall to the Trapezium giuen, vpon a line giuen, and hauing an Angle equall to an Angle giuen, as was required. THEOREME 26.42.71. PROBLEME 3.8.16.74.
PROBLEME. XCIIII. To reduce a Trapezium into a Triangle, which shall be like vnto another Triangle giuen.
Euc. 1.37.1, 23. Cor. 6.19. Ceul. 3.1.LEt the Trapezium giuen be A. B. C. D. and let the Triangle
giuen be E. F. G. It is required to reduce the same Trapezium into a Triangle, which shall be like vnto the giuen triangle E. F. G. First, by the 93. PROB. last before going; reduce the Trapezium giuen into the triangle A. D. H. which is equall thereunto, then by the 81. BROB. reduce the same triangle A. D. H. into the triangle N. D. K. which shall be equall to the Trapezium giuen, and like vnto the giuen triangle E. F. G. as was required. THEOR. 26.43. PROB. 38.16.93.
PROBLEME XCV. To reduce an equiangled Poligon giuen, into a Geometricall Square.
Euc. 2.14. Ceul. 2.30.LEt A. B. C. D. E. be a Pentagonall Poligon giuen, to be reduced into a Geometricall
square. By the 17. PROB. find out the meane proportionall line betweene halfe the perimetrie of the giuen Poligon, and the perpendicular thereof M. N. being let fall from the Centre to the middle of any side, This rule is generall for the reducing of all rectangle poligons. which meane proportionall is the line G. H. whereon by the 50. PROB. describe the square F.G.H.K. which shall be equall to the giuen Poligon, as was required. DEFINITION. 29. THEOREME 39.41.
PROBLEME XCVI. To reduce a plotte giuen into a Triangle, with lines drawne from an Angle assigned.
LEt A. B. C. D. E. be a plotte giuen, and let the Angle assigned be A. It is required from the Angle A. to reduce the same plot into a Triangle. First, increase the side C. D. of the giuen plot, of conuenient [Page 102]le [...]gth both wayes towards F. Caeterorum polygonorum rectangulorum reductio & demonstratio buic est similima: semper enim bac reductione sigillat im vnum latus detrabitur. Euc. 1.37. Ceul. 3.1.2. and G. then draw the line A. D. and by the 3.
PROB. by the point E. draw the line E. G. parallel to the line A. D. to cut the continued line in G. from which point, draw the line G. A. then draw the line A. C. and by the point B. make B. F. parallel to A. C. to cut the line of continuance in F. from which point, draw the line F. A. which shall include the Triangle A. F. G. equall to the giuen plot, with the lines A. F. and A. G. drawn from the Angle A. assigned as was required. THEOR. 26.
PROBLEME XCVII. To reduce a Figure giuen, into a Lunula or Figure of a Lunular forme.
LEt A. B. C. D. be a square giuen, and let it be required to reduce the
same into a Lunula. Draw the Diagonall A. C. and on C. the end therof by the 6. PROB. erect the perpendicular E. C. equall to A. C. then continue the side A. B. to E. and on the point B. at the distance B. A. or B. E. describe the Semicircle A. F. E. And lastly, on the point C. at the distance C.A. or C.E. describe the Arch line A.G.E. which shal include the Lunula A.F.E.G. equall to the giuen square, as was required.
PROBLEME XCVIII. To reduce an irregular Figure giuen, into a greater or lesser forme, according to any giuen proportion.
LEt A.B. C.D.E.F. be an irregular
Figure giuen, and let the proportion giuen bee that betweene M. and N. It is required to reduce the same Figure into a lesser, to be in such proportion to that giuen as M. to N. First reduce the giuen Figure into Triangles, by drawing right lines from any one angle, as from F. to all the opposite Angles, as B.C. and D. then by the [Page 103]23. PROB. diuide one of the sides as F.E. in power, as M. to N. so that the power of F. L. may be to the power of F. E. as M. to N. Then by the point L. draw the line L. K. parallel to E. D. to cut F. D. in K. and in like sort proceede with the rest, as K. I. I. H. and H. G. drawing them parallel to their answerable sides; so shall you include the Figure F. G.H. I.K.L. being like vnto the Figure giuen, and in proportion to it, as the line M. to N. as was required. But suppose the lesser plot were giuen, and let it bee required to reduce the same into a greater, according to the proportion of N. to M. then increase all the lines from F. towards A. B. C. D. and E. and by the 24. PROBLEME inlarge the line F. L. in power as N. to M. which set from F. to E. and by the point E. draw the line E. D. parallel to L. K. to cut F. D. in D. and in like sort proceede with the rest. So shall you include the irregular Figure A.B.C.D.E.F. like to that giuen, and of the proportion, as was required. THEOREME. 22.
PROBLEME XCIX. To reduce an irregular Figure giuen, into a Geometricall Square.
LEt A.B.C.D.E.F.G. bee an irregular Figure giuen, to bee reduced into a Geometricall Square. First, draw the lines B. F. and C. E. diuiding the giuen Figure thereby into two Trapezias, and one Triangle, namely, A.B.F.G. B.F.E.C. and C.D.E. then crosse those Trapezias with the Diagonall lines B. G. and C. F. and let fall perpendiculars theron from the Angles A. F. E. and B. and likewise from D. to the Base of the Triangle, then by the 17. PROBLEME finde out the meane proportionall, betweene halfe the Diagonall B. G. and the two perpendiculars thereon falling, which shall be the line P. Also the meane proportionall betweene halfe the Diagonall C. F. and the two perpendiculars thereon falling, Euc. 2.14.6.13.1. 47. which shall be the line Q. and likewise the meane proportionall betweene halfe the Base C. E. and the perpendicular thereon falling from D. which shall be the line R. then by the 9. PROBLEME describe a right Angle at pleasure, as H. I. K. and take the line P. and place the same, from I. to L. and also the [Page 104]line Q. from I. to M. and draw the line L. M. which line place from I. to O. and also the line R. from I. to N. And lastly, draw the line N. O. which shall be the side of a Square, equall to the giuen Figure as was required. THEOREME 6.25.41.30.
Here might I fitly insert the Reduction and manner of translation of large and spacious plots, from one Scale to another, diuers wayes, with many other works of this nature, fitte to bee knowne, which for some speciall reasons, I will referre vnto the later end of my next Booke. And in the meane space will here conclude the third part of this Second Booke.
The fourth Part.
HOw diuers superficiall figures, of seuerall formes, are brought into one figure, and one forme: Also to subtract one figure from another, and thereby to know how much the one exceedeth the other in quantitie; and likewise, hereby is taught, the inscription and circumscription of one figure within and without another; and the diuision and separation of figures, into any parts required.
PROBLEME C. Two Geometricall squares being giuen, to adde them together into one square.
LEt the two giuen squares
be A. B. C. D. and D. E. F. G. and let it be required to adde them together into one square. Euc. 1.47. First take a side of either of the giuen squares, as A. D. and D. G. and by the 9. PROB. make thereof the right angle A. D. G. (as they are already placed in this Diagram) then draw the diagonal line A. G. and on that line by the 50. PROB. describe the square A. K. H. G. which shall be equall to the two giuen squares, as was required. THEOR. 6.
PROBLEME CI. Two Geometricall squares being giuen, to adde them together in such sort, as the one shall be a Gnomon vnto the other.
LEt the two giuen squares be A. B. C. D. and E. F. G. H. and let it be required to adde them together, Euc. 1.47. Ceul. 2.38. in such sort that the square E. F. G. H. shall be a Gnomon vnto the other. Increase the side C. D. of the greater square to M. making D. M. equall to the side of the lesser [Page 106]square, then draw the subtending line A. M. which take and lay downe from C. to N. and thereon by the 50. PROB. describe the square O. P. C. N. which shall performe what was required. DEF. 30. THEOR. 6.
PROBLEME CII. To adde diuers squares together into one geometricall square.
LEt A. B. C. D. and E. be the sides of fiue squares giuen, Euc. 1.47.6.31. and
let it be required to adde them together into one geometrical square. Diuers figures of what forme or kinde soeuer, being by the [...]ormer rules reduced into squares, may hereby instantly be added to gether into one. First, by the 9. PROB. make a right angle at pleasure▪ as F. G. H. then (beginning with the least sides first) take the line E. and place it from G. to N. and the line D. from G. to O. and draw the line N. O. whose square shall be equall to both the squares of E. and D. then take N. O. and place it from G. to P. and the side C. from G. to M. and draw the line M. P. which line place from G. to Q. and the side B. from G. to L. and draw the line L. Q. which line place from G. to R. and the line A. from G. to K. and lastly, draw the line K. R. and thereon by the 50. PROB. describe a square which shall be equall to all the fiue squares, whose sides were giuen, as was required. THEOR. 6.23.30.31.
PROBLEME CIII. To adde two giuen triangles together into one, which new composed triangle, shall haue his perpendicular, equall to that of one of the giuen triangles.
LEt the two giuen triangles bee A. B. C.
and E. F. G. and let it be required to adde those 2. triangles into one, Euc. 1.37.1.38. which shall haue his perpendicular equall to that of the triangle A. B. C. By the 75. PROB. reduce the triangle E. F. G. to the triangle H. F. K. of equall height to the other giuen triangle, then increase the base B. C. of the triangle A. B. C. from C. to D. making C. D. equall to F. K. the base of the reduced triangle H. F. K. and lastly, draw the line A. D. which shall include the triangle A. B. D. equall to both the giuen triangles, and hauing the same perpendicular A. N. as the giuen triangle A. B. C. as was required. THEOR. 26. PROB. 28.
PROBLEME CIIII. Diuers Circles being giuen, to adde them together into one Circle.
LEt A. B. and C. D. bee two Circles giuen, Euc. 1.47.12.2. and
let it be required to adde them both into one Circle, or to make one Circle, which shall be equall to them both. Take the Diameter A. B. and by the 6. PROB. rayse it perpendicularly on the end of the other Diameter C. D. as E. C. then draw the subtending line E. D. on which line as a Diameter describe the Circle E. D. which shall bee equall to the two giuen Circles as was required. And in like sort, by helpe of the 102. PROB. may be added as many Circles as shall be required; For Circles are added by their Diameters, as Squares by their sides. THEOREME 6.66.
PROBLEME CV. Two long Squares being giuen, to adde them together into one long Square, whose breadth shall be equall to that of one of the long Squares giuen.
LEt the two giuen long Squares be
A. B. C. D. and E. F. G. H. and let it bee required to adde those two Squares together into one Square, Euc. 1.43. whose breadth shall be equall to the giuen long Square A. B. C. D. First, increase the side G. F. of the greater long Square giuen towards K. making F. K. equall to the breadth of the lesser giuen Square, and so working on by the 87. PROB. reduce the giuen long square E. F. G. H. into the long square M. N. H. L. then on the Line M. L. by the 54. PROB. describe the long square O. M. L. P. equall to the giuen long square A. B. C. D. So shall the long Square O. N H. P. bee equall to the [Page 108]two giuen long Squares, and the breadth thereof O. P. or N. H. equall to the breadth of the lesser long Square giuen, as was required. THEOREME. 5.
PROBLEME CVI. Two Geometricall Squares being giuen, to subtract the one out of the other, and to produce the remainder in a third Square.
LET the two giuen Squares be A. B. C. D. and E. F. G. C. and let
it be required to subtract the lesser out of the greater, Euc. 1.47.6.31. and to produce the remainder in a third square. Continue out at length the side C. G. of the lesser giuen square towards M. and at the distance of the side of the greater giuen Square with one foot in the Angle F. of the lesser square strike an Arch line through the line of continuance in M. And lastly, by the 50. PROB. on the line G. M. describe the Square H. K. M. G. which shall be the remainder of the greater giuen Square, the lesser being subtracted from the same, as was required. THEOREME. 6.23.30.31.
PROBLEME CVII. Two Triangles being giuen to subtract the one out of the other, and to leaue the remainder in a Triangle of equall height to one of the giuen Triangles.
LEt the two Triangles giuen be A. B. D. and E. F. G. Euc. 1.37.1.38. And let it be
required to subtract the Triangle E. F. G. out of the Triangle A. B. D. and to leaue the remainder in a Triangle of equall height to the Triangle A. B. D. By the 75. PROB. reduce the Triangle E. F. G. to the triangle H. F. K. of equall height to the other giuen triangle, then take the Base F. K. of the same reduced triangle, and place it from D. to C. And lastly, draw the line A. C. which shall include and subtract the Triangle A. C. D. (equall to the lesser giuen triangle E. F. G) from [Page 109]the greater giuen triangle A. B. D. and leaue the remainder in the triangle A. B. C. of the same height of the giuen triangle A. B. D. as was required, THEOR. 26. PROB. 28.
PROBLEME CVIII. Two circles being giuen, to subtract the one out of the other, and to produce the remainder in a third circle.
LEt the two circles giuen be A. B. and
B. C. and let it be required to subtract the circle B. C. out of the circle A. B. and to produce the remainder in a third circle. Euc. 1.47.12.2. Take the diameter B. C. and by the 6. PROB. raise it perpendicularly on the point B. as D. B. then at the distance of the diameter A. B. with one foot in D. strike an arch through the diameter B. C. in E. then at halfe the distance of B. E. describe the circle F. G. on the point B. So haue you subtracted the circle B. C. out of the circle A. B. and produced the remainder in a third circle F. G. as was required. THEOR. 6, 66.
PROBLEME CIX. A geometricall square and a triangle being giuen, to subtract the triangle from the square, and to produce the remainder in a square.
LET the Geometricall square giuen be A. B. C. D. and let the triangle giuen be E. C. F. and let it be required to subtract the triangle from the square, Eu. 1.47.2.14.6.18. and to produce the remainder in a square. By the 17. PROB. find out the meane proportionall, betweene the perpendicular of the giuen triangle E. N. and halfe the base C. F. which shall be the line O. P. which place from C. to H. on which point H. at the distance of the side of the giuen square, strike an arch through the base C. F. as at M. and lastly, vpon the line C. M. describe by the 50. PROB. the square G. K. M. C. So haue you subtracted the giuen [Page 110]triangle E.C.F. from the giuen square A.B.C.D. and produced the remainder in the square G.K.M.C. as was required. THEOREME 6, 41, 70.
PROBLEME CX. A triangle and a long square being giuen, to subtract the long square from the triangle, and to produce the remainder in a triangle of equall height, to the triangle giuen.
LEt the giuen triangle bee A.B.C. and
the long square giuen D. E. F. C. and let it berequired to deduct the long square from the triangle, Euc. 1.38.1.41.6.1. and to produce the remainder in a triangle, of equall height to the triangle giuen. First, increase the side F. E. of the long square towards K. and placing E.F. from E. to H. draw the line H.C. including the triangle H. C. F. equall to the long square giuen; which triangle by the 75. PROB. reduce into the triangle K.G.F. of equall height to the giuen triangle A.B.C. then take the base G.F. of the last found triangle, and place it from C. to M. on the base of the giuen triangle, and lastly, draw the line A.M. to include and subtract the triangle A.M.C. (equal to the giuen long square) from the giuen triangle A.B.C. & producing the remainder in the triangle A.B.M. of equall height to the triangle giuen, as was required. THEOR. 26, 27, 28, 35.
PROBLEME CXI. Within a circle giuen to inscribe a triangle, with angles equall, and lines proportionall, to a triangle giuen.
LEt the circle giuen be A. B. C. and the triangle
giuen D.E.F. and let it be required within the giuen circle to inscribe a triangle, Euc. 3.32.4.2. Ceul. 2.53. with equal angles and lines proportionall to the giuen triangle By the 31. PRO. draw the giuen circle, the right line G. H. touching the same in the point C. vpon which point vnto the line C.H. describe the angle A. C. H. equall to the angle D. E. F. and likewise on the same point, to the line G. C. describe the angle B. C. G. equall to [Page 111]the angle E. D. F. and lastly, draw the line B. A. which shall include the triangle A. B. C. inscribed within the giuen circle, with equall angles and lines, proportionall to the triangle giuen, as was required. THEOR. 56, 13.
PROBLEME CXII. To describe a Circle about a Triangle giuen.
LEt the Triangle giuen be A.B.C. about which it is required to describe a Circle, Euc. 4.5.
vpon the middle point of any two sides of the Triangle by the 5. PROB. erect perpendiculars, which being produced will intersect each other, as E. D. and F. D. in the point D. which point of intersection shall be the Centre; whereupon at the distance from thence to any one of the Angles describe the Circle A.B.C. which shall circumscribe the Triangle as was required. DEF. 32.
Or otherwise the Centre may bee found as is taught in the 34. PROB.
PROBLEME CXIII. To inscribe a Circle within a Triangle giuen.
LEt the Triangle giuen be A. B.C. within which Triangle it is required
to inscribe a Circle. By the 10. PROB. diuide any two Angles of the giuen Triangle into two equall parts as A. B.C. and A. C.B. by drawing the lines B. D. and C. D. intersecting in D. which point of intersection shall be the Centre, Euc. 4.4: Ceul. 2.55. whereon at the neerest distance from thence to any side of the Triangle describe the circle D. which shall be inscribed within the Triangle, as was required. DEF. 32. PROB. 10.
PROBLEME CXIIII. To describe a Triangle about a Circle giuen, which shall be like vnto a Triangle giuen.
LEt the Circle giuen be H. K. M. and let the Triangle giuen be E. F. G. It is required to describe a triangle about the giuen Circle, Euc. 4.3. Ceul. 2.54. like vnto the giuen triangle. First, continue the Base F. G. of the giuen [Page 112]Triangle both wayes towardes
N. and O. making the two outward Angles E. F. N. and E. G. O. then from the centre D. of the giuen circle draw to any part of the lymbe a semidiameter as D.M. to which line vpon the centre D. describe the angle H. D. M. equall to the angle E.F.N. and also the Angle K.D.M. equall to the angle E.G.O. then by the points H. K. and M. draw right lines, (making right angles with the three semidiameters) which will intersect each other in the points A.B. and C. including the triangle A. B. C. like vnto the giuen triangle, and circumscribed about the giuen circle, as was required. THEOR. 1.13.52.
PROBLEME CXV. To describe a Square about a Circle giuen.
LEt the circle giuen be A. B. C. D. and let it be required to describe a
square about the same circle. Euc. 4.7. Ceul. 2.57. Draw the two Diameters A. C. and B. D. cutting each other at right angles in the centre 1. then by the 3. PROB. by the points A. and C. draw parallel lines to the Diameter B. D. And likewise by the points B. and D. draw parallel lines to the Diameter A. C. which shall intersect each other in the points E.F.G.H. and include the square as was required. THEOR. 32.52.
PROBLEME CXVI. Within a Square giuen to inscribe a Circle:
LEt the square giuen be E.F.G.H. in the former Diagram, within which it is required to inscribe a circle. Draw the Diagonall lines E. G. and F. H. interfecting each other in 1. the center, on which point at the distance of the shortest extention to any side, describe the circle A B C D. within the square as was required. THEOR. 32.52.
PROBLEME CXVII. About a Square giuen to circumscribe a Circle.
LEt the Square giuen be A. B. C. D. in the former Diagram, about which it is required to circumscribe a Circle. Draw the Diagonall lines A.C. and B. D. intersecting each other at right Angles in the point I. the centre, Euc. 4.9. on which point, at the distance from thence to any of the points A.B.C. or D. describe the Circle A.B.C.D. which shall circumscribe the Square as was required. THEOREME 1.13.52.
PROBLEME CXVIII. To inscribe a Square within a Circle giuen.
LEt the Circle giuen be A.B.C.D. in the former Diagram, and let it be required to inscribe a square within the same Circle. Draw the Diameters A. C. and B. D. crossing each other at right Angles in the centre I. then betweene the points A.B.C. and D. draw the right lines A B. B C. CD. and D A. including and inscribing the square A.B.C.D. within the giuen Circle as was required. THEOREME. Euc. 4.6. 1.13.52. PROBLEME 25.
PROBLEME CXIX. To inscribe a Pentagon, within a Circle giuen.
LEt A. B. C. D. E. be a Circle giuen, Euc. 4.11. Ceul. 2.58. within which it is required to inscribe
a Pentagon. By the 25. PROBLEME Finde out the fift corde line of a Circle, at which distāce passing through the limbe of the circle note fiue marks as at the points A. B. C. D. and E. And lastly, frō point to point draw fiue right lines, which shall include and inscribe the Pentagon A. B. C. D. E. as was required. THEOR. 47.69. PROB. 48.
PROBLEME CXX. About a Circle giuen to circumscribe a Pentagon.
LEt the Circle giuen be A.B.C.D. in the former Diagram, about which, it is required to circumscribe a Pentagon. Let first a Pentagon be inscribed as before, and from the Centre N. draw right lines to euerie [Page 114]Angle of the inscribed Pentagon, Euc. 4.12. as to A.B.C.D. & E. on which fiue points, draw lines Orthigonally to those lines issuing from the centre, which will intersect each other in the points F. G. H. I. K. and circumscribe the Pentagon about the giuen circle as was required. THEOR. 47.69.
PROBLEME CXXI. To inscribe a Sexagon within a circle giuen.
LEt the circle giuen be A. B. C. D. E. F. Eu. 4.15. Ceul. 2.63. within which it is required to inscribe
a Sexagon. By the 25. PROB. finde out the sixt corde of a Circle, which is alwayes the semidiameter of the same circle; wherefore at the distance of the semidiamenter; Diuide the limbe of the circle into sixe equall partes, as in the points A.B C.D.E. and F. and then from point to point, draw right lines, which shal include and inscribe the Sexagon A.B.C. D.E.F. within the giuen Circle, as was required. PROB. 25.
PROBLEME CXXII. To circumscribe a Sexagon about a Circle giuen.
LEt the Circle giuen be A.B.C.D.E.F. in the former Diagram, about which it is required to circumscribe a Sexagon. First, diuide the limbe of the Circle into sixe parts, as was taught in the last PROB. in the the points A. B. C. D. E. and F. and draw Diameters from each opposite point to other, making in all three Diameters, then at euery end of those Diameters draw lines Orthigonally vnto them, which will intersect each other in the points G.H.I.K.L. and M. and include the Sexagon circumscribed about the giuen circle as was required. PROB. 25.
PROBLEME. CXXIII. To diuide a right lined Triangle giuen, into any number of equall parts required, from a point limited in any side of the same Triangle.
LEt A. B. C. be a right lined Triangle giuen; let the limited point be D.
in the Base B. C. and let it be required from the same limited point D. to diuide the giuen triangle into fiue equall parts. First, diuide the Base B. C. into fiue equall parts, as in the points E. F. G. and H. (or into [Page 115]as many as shall bee re quired) then from the limited point to the opposite angle, draw a right line as A D. vnto which line, by the points E.F.G. and H. draw Parallel lines, as E I. F K. G L. and H M. And lastly from the points I. K. L. and M. to the limited point D. draw the lines I D. K D. L D. and M D. which shall diuide the giuen triangle into fiue equall parts from the limited point D. as was required. THEOREME. 26.35.
PROBLEME CXXIIII. To diuide a giuen Triangle by a line issuing from an angle assigned, in
any proportion required.
LEt the Triangle giuen be A.B.C. the angle assigned A. and the proportion that betweene the two right lines F. and G. Euc. 6.1. Ram. 10.13 Ceul. 3.8. It is required to diuide the same triangle into two parts by a line issuing out of A. the one part hauing proportion to the other, as F. to G. By the 13. PROB. diuide the Base B C. as F. to G. the point of which diuision will fall in D. From which point draw the line D. A. So is the giuen Triangle diuided into two parts, hauing proportion the one to the other, as F. to G. For as the line F is to the line G. so is the triangle A. C. D. to the Triangle A. B.D. as was required. THEOR. 35. PBOB. 13.
A. E. 64.
B. C. 80.
PROBLEME. CXXV. A Triangle being giuen, and the Base thereof known, to diuide the same into two parts by a line from an angle assigned, according to any proportion giuen in two numbers.
LEt the triangle giuen be A. B. C. in the former Diagram, whose Base is B.C. 80. and the angle assigned A. and let it be required to diuide the same into two parts, the one in proportion to the other Sesquidtera, [Page 116]that is, as 3. to 2. First, adde together alwayes the Termini rationis, which is 3/2· makes 5. then multiply the Base 80. by 3. the greater terme, the factus will be 240. which diuided by the former 5. quoteth 48. for the greater segment of the Base D C. Euc. 6.1. Ra. 10.13. and that deducted from 80. the whole Base resteth 32. the lesser segment B D. from which point D. draw the line D A. which shall diuide the giuen Triangle into two parts, as 3. to 2. that is, the greater A C D. shall be 3/5· parts, and the lesser A B D. 2/5· of the whole giuen Triangle, as was required. THEOR. 35.
PROBLEME CXXVI. To diuide a Triangle of knowne quantitie giuen, into any two parts, from an Angle assigned, according to any number of Acres, Roodes and Perches required.
LEt the Triangle be A B C. in the former Diagram, whose quantitie is 16. Acres; let the Angle assigned be A. and let it be required to diuide the same into two parts twixt H. and I. viz. to H. 9. Acres. 2. Roodes, 16. Perches: and to I. 6. Acres, 2. Roodes, 24. Perches. First, measure the Base B C. 80. Perches; then reduce the whole quantitie of the Triangle into Perches, Euc. 6.1. Ra. 10.13. by multiplying 160. (the number of Perches in an Acre) by 16. (the numbet of Acres in the Triangle giuen) the Product will be 2560. Reduce also one of the parts into Perches, as the greater part 9. Acres, 2. Roodes, 16. Perches; the reduction will be 1536. Perches. And then by the rule of proportion reason thus. If 2560. (the whole quantitie) require 80. (the whole Base; whatpart of the Base will 1536. Perches (being the part of H.) require. Multiply 1536. by 80. and diuide the Product by 2560. the quotus will be 48. the part of the Base sought, which placed from C. to D. and the line D A. drawne, will diuide the giuen Triangle into the parts required. THEOR. 35.
Or to lay out one of the parts giuen (whereby the other is knowne) worke thus.
LEt it be required to lay out the lesser part, which containeth 1024. perches, diuide that number 1024. by halfe, the perpendicular A E. 32. the quotus will be 32. for the part of the Base to be cut off for that part; which placed from B. to D. and the line A. D. drawn, performeth the worke. Or double the quantitie giuen 1024. and diuide that double by the whole Base 64. the worke will be the same.
PROBLEME CXXVII. To diuide a Triangle giuen into two parts by a line drawn, from a point limited in one of the sides, in any proportion required.
LEt A B C. be the Triangle giuen, the point limited E. in the Base B C. and let the proportion be that betweene I. and K. It is required from the point E. to draw a line, which shall diuide the same Triangle into [Page 117]two parts, hauing
A G. 40. Euc. 6.1. Lam. 10.13 Coul. 3.9.
B C. 42.
B D. 12.
D C. 30.
B E. 30.
F H. 16.
proportion the one to the other, as I. to K. First, from the giuen point E. to the opposite angle A. drawe a right line, as A. E. then by the 13. PROB. diuide the Base B. C. as I. to K. the point of which diuision will fall in D. from whence draw the line D F. parallel to A. E. cutting the side A B. in F. And lastly, from thence draw a right line to the limited point E. as F E. which shall diuide the giuen Triangle into two parts, and include and separate the Triangle F E B. from the Trapezium A F. E C. hauing such proportion one to the other, as the two giuen lines I. and K. as was required. THEOR. 35.
PROBLEME CXXVIII. To diuide a Triangle giuen into two parts, according to any propertion giuen in two numbers from a point limited in any side thereof, Arithmetically.
LEt the Triangle giuen bee A B C. in the former Diagram. Let the point limited be E. in the Base B C. and let it be required to diuide the same into two parts, in proportion one to the other, dupla sesquialtera, which is, as 5. to 2. First, let the Base B. C. be diuided according to the proportion giuen thus; Adde together the two giuen termes of the proportion 5. and 2. makes 7. then multiply the Base of the Triangle B. C. 42. by 2. the lesser terme, the Product is 84. which diuided by 7. the summe of the termes quoteth 12. for the lesser Segment of the Base B. D. which deducted from 42. the whole Base resteth 30. the greater segment D. C. then (considering the lesser part is to bee layde towards B.) measure the distance from the giuen point E. to B. which admit 30. and by the rule of Proportion reason thus: If the distance E. B. 30. giues B. D. 12. Eu. 6.1. Ram. 10.13. the lesser segment what giues A G. 40. the perpendicular of the giuen Triangle, and multiplying 40. by 12. and diuiding the Product by 30. the answere will be 16. at which distance draw a parallel line to B E. cutting the side A B. in F. from which point draw the line F E. which shall diuide the giuen Triangle in such sort that the Triangle F B E. shall be 2/7· parts, and the Trapezium A F. E C. 5/ [...]· parts thereof, that is the Trapezium, containing the Triangle F B E. twice and a halfe, according to the proportion required. THEOR. 35.
PROBLEME CXXIX. To diuide a Triangle of any knowne quantitie giuen, into two parts, from a point limited in any side thereof, according to any number of Acres, Roodes and Perches.
LEt the Triangle giuen be A B C. in the last Diagram, whose quantitie let be 5. Acres, 1. Roode, 0. Perch. Let the limited point bee E. in the Base B C. and let it be required from the same point to diuide the Triangle into two parts betweene M. and N. viz. to M. 3. Acres, 3. Roodes, 0. Perches thereof, and the residue to N. 1. Acre, 2. Roods, 0. Perch. First, reduce the quantitie of N. being the lesser, 1. Acre, 2. Roods, 0. Perch, into Perches, Euc. 6.1. Ra. 10.13. which makes 24. Perches, then (considering on which side of the limited point this part is to bee layde, as towards B.) measure that part of the Base from E. to B. 30. Perches, wherof take halfe, which is 15. and therby diuide 240. the part of N. the quotus will be 16. the length of the perpendicular F. H. at which parallel distance from the Base B C. cut the side A B. in F. from whence draw the line F E. which shall include the Triangle, F. B. E. containing 1. Acre, 2. Roods, 0. Perch, the part of N. and so shall the Trapezium A F E C. containe the residene, namely, 3. Acres, 3. Roodes, 0. Perch, the part of M. as was required. THEOR. 35.
PROBLEME CXXX. To diuide a giuen triangle by a parallel line, to one of the sides, according to any proportion giuen.
LEt A. B. C.
A D. 48.
B C. 54.
B E. 24.
B F. 36.
E C. 30.
H G. 32.
be a triangle giuen, and the proportiō that between I. and K. and let it be required to diuide the same triangle into 2 parts, Euc. 6.10.6.13. Cor. 6.19. Ceul. 3.10. by a line drawne parallel to the side A. C. the one to be in proportion to the other, as the line I. to the line K. By the 13. PROB. diuide the line B. C. in E. as I. to K. then by the 17. PROB. finde the meane proportion betweene B. E. and B. C. which let be B. F. from which point by the 3. PROB. draw the line F. H. parallel to the side A. C. cutting the side A. B. in H. which line shall diuide the triangle giuen into two [Page 119]parts, the trapezium A. H. F. C. hauing such proportion to the triangle H. B. F. as the line I. to the line K. which was required. THEOREME 43, 41. PROBLEME 13, 17.
PROBLEME CXXXI. To diuide a giuen triangle by a parallel line, to one of the sides, according to any proportion giuen in two numbers Arithmetically.
LEt the triangle giuen be A. B. C. in the former Diagram, and let it be required to diuide the same by a parallel line, Euc. 6.10.6.13. Cor. 6.19. to one of the sides into two parts, to be in proportion the one to the other, Sesquiquarta, that is, as 5. to 4. First, let the base B C. 54. be diuided, according to the proportion giuen, as is taught in the 127. PROB. so shall the lesser segment be B E. 24. and the greater segment E C. 30. then finde out a meane proportionall betweene B E. 24. and the whole base B C. 54. by multiplying 54. by 24. whose product will be 1296. the squa e roote whereof is 36. the meane proportionall sought, which in the former Diagram is B F. then by the rule of proportion, reason thus: If B F. 36. giues B E. 24. what A D. 48. the answere is H G. 32. at which distance draw a parallel line to the base, to cut the side A B. in H. from whence draw the line H F. parallel to A C. which shall diuide the giuen triangle into two parts, in such sort, as the trapezium A. H. F. C. shall be 5/9· parts, and the triangle H. B. F. 4/9· thereof, that is the trapezium, contayning the triangle H. B. F. once and a quarter, as 5. doth 4. which was required. THEOR. 43, 41. PROB. 13, 17.
PROBLEME CXXXII. To diuide a triangle of any knowne quantitie, giuen into two parts, by a parallel line, to one of the sides, according to any number of Acres, Roodes, and Perches required.
LEt the triangle giuen be A. B C. in the last Diagram, Euc. 6.10.6.13. Cor. 6.19. whose quantitie is 8. Acres, 0. Roods, 16. Perches, and let it be required to diuide the same (by a parallel line to one of the sides, as the side A. C.) into two vnequall parts, betweene M. and N. viz) to M. 4. Acres, 2. Roods, 0. Perches, thereof, and the residue to N. being 3. Acres, 2. Roods, 16. Perches. First, reduce both quantities into perches, which will be 720. and 576. then reduce both those numbers by abbreuiation into their least proportionall termes, which is 5/4· and according to that proportion diuide the base B C. 54. of the giuen triangle, as is taught in the two last Problemes, which will be in E. then seeke the meane proportionall betweene B E. and B C. as is taught in the last Probleme, which is B F. 36. of which 36. take the halfe, and thereby diuide 576. the lesser quantitie of perches, the quotus will be H G. 32. at which parallel distance from the base, cut the line A B. in H. from whence draw the line H F. parallel, to the side A C. which shall diuide the triangle giuen into two parts; the trapezium A. H. F. C. contayning the part of M. and the triangle H. B. F. the part of N. as was required. THEOR. 41, 43.
[Page 120] HEre might I now much inlarge and amplifie this latter part, for the diuiding of superficiall figures of all formes and kindes: But seeing that all irregular figures and plots are most conueniently to be reduced into triangles, before the contents thereof can be had, or any diuision thereof made; and generally in matter of Suruey (whereunto my purposes chiefly tend) all figures are found irregular; I will content my selfe (and intreat my Surueyor to be likewise satisfied) with these few former instructions, which being well vnderstood, with due obseruation of the precedent rules, will serue his turne to whatsoeuer purpose can be required. But who so desireth further satisfaction, and more varietie in this kinde, I referre him to M. Iohn Speidels Booke, intituled his Geometricall Extraction, lately by him diuulged; (wherein he hath taken more paynes to excellent purpose, then this age, I feare, can afford him recompence) and to himselfe M. Henry Brigs, M. Thomas Bretnor, M. Io. Iohnson, and others, who are here amongst vs Professors, and excellent Teachers of these Arts. And thus, I conclude, this second Booke.
THE EXACT OPERATION OF INSTRVMENTALL DIMENSIONS BY DIVERS MEANES.
The third Booke.
THis Booke tendeth chiefly to matter of Suruey, wherein is first described and declared the seuerall Jnstruments, fit for that purpose (with their vse in practise) as the Theodelite, the Playne table, and Circumferentor, whereunto J haue added an absolute Instrument, which J call the Peractor, together with the making and vse of the Decimal Chayne, vsed only by my selfe; then is taught the vse of a necessarie Fieldbooke; the taking, protracting, and laying downe of angles diuers wayes; the reducing of customarie measures [Page 122]into Statute measure, and the contrarie, shewing their difference; the vse of the table of Synes, and the diuided sights on the Circumferentor, withsupply of those defects in the ordinary vse of the Playne table, by ignorant persons; the meanes to take Altitudes, Longitudes, Latitudes, and inaccessible Distances, aswell by synicall computation, as diuers other meanes; the Dimension and plotting Instrumentally of all manner superficiall figures, and irregular plots, by diuers and sundrie wayes, with their seuerall protractions accordingly: how with the Playne table exactly to take a plot of the largest Forrest on one sheet of paper, without altering thereof: the means of Survaying and plotting of a Lordship or Mannor, with the orderly handling of the same: the wayes and meanes Instrumentally, to reduce Hypothenusall, to Horizontall lines, and the contrarie: with the best and exactest courses to be held in the dimension, and plotting of Mountaynous, and vn-euen grounds: the manner of inclosing, diuiding, and laying out of Commons, wasts, or common Fields, into any parts required: the ordering of a plot after the protraction thereof: the reducing of plots from a greater to a lesser forme, and the contrarie: and lastly, the speedie reduction of perches into Acres, and those againe into perches: Most exactly and artificially wrought, by the best and most immediate meanes for those purposes.
CHAP. I. Of the seuerall Instruments in vse, meete for Suruaie; which of them are most fit for vse, and somewhat concerning their abuse.
BEfore we enter the fieldes to suruaie, I hold it necessarie wee prouide vs of fitting furniture for the purpose, lest by our neglect therein, those by whom wee haue imployment receyue no lesse losse and preiudice, then our selues shame and reproch: Wherefore let vs first consider what Instruments are most vsuall, and then of those, what most fit for our present purpose. The Instruments now most in vse are the Theodelite, Plaine-table and Circumferentor, whereunto I will adde one more, which I call the Peractor, vsed onely by my selfe, and certaine friends by my directions; Of all which I will hereafter make briefe and particular descriptions, aswell concerning their seuerall parts and composition, as of their vse in practise. Nor will I exclude or wholly neglect the Familiar Staffe of M. Iohn Blagraue, and the Geodeticall staffe and Topographicall glasse of M. Arthur Hopton (though now together dead) or any other Instrument which are, or hereafter may be by the artor industry of any man, artificially inuented or composed for his owne, or others vse. For, as I tye not my selfe to the vse of any one Instrument, at all times, nor on all occasions, but for a large and spacious businesse vse the Circumferentor or Peractor (as for many reasons most necessary and conuenient) and for a lesser (where many small Inclosures and Town-ships are) the Plaine-Table; although eyther of the other will well serue for performance of eyther kind (and therefore if possibly you may, in one and the same businesse, vse euer one and the same Instrument, for auoyding many inconueniences:) So will I not limit my Surueyor to the vse of any one Instrument, but referre him to all or any at his pleasure, well knowing they are all composed and framed on one and the same Theoricall ground: and although in performance and dispatch, one may be more speedy then another (wherein I finde much difference, with the same exactnesse) yet all or any of them artificially handled, are to excellent vse, aswell in suruey of lands, as the performance of many other excellent conclusions Geometricall. Wherein I cannot by the way but condemne the folly of diuers, who (deeming themselues more wise and skilfull, then any other shall haue iust cause to conceiue) hauing some small superficiall vse or insight into some one of these Instruments, are onely wedded to that, and ignorant of the others vse, will condemne them as vnmeet and insufficient, the defect consisting in their owne vnderstanding. But I must needs acknowledge (which could I as easily reforme, I should deserue well at the hands of many, who pay deerely for it) that by the ignorance of diuers, vsing (or abusing) these instruments, infinite grosse and palpable errors and abuses are dayly committed, to the great losse and preiudice [Page 124]dice of many, who receiue secret and insensible blowes (found in their estates, though vnfelt in themselues) who for my part shall passe vnpittied (saue only for their weakenesse) the fault being chiefly their owne, when out of a penurious sparing, to yeeld better satisfaction to those which better can; they can be well contented blindly to swallow frogges for flyes, and to pay treble damage (nay perhaps ten fold) another way, so they neyther feele nor find it in their open view. And thus much concerning the seuerall Instruments in generall. Next of their Definitions, Parts, and Compositions particularly.
CHAP. II. Of the Theodelite and his seuerall parts, with the Description and Composition thereof.
THis Instrument consisteth of foure seuerall parts; As first, the Planisphere or Circle, whose limbe is diuided into 360. equall parts or diuisions, called degrees, without which it is fitting equidistantly to draw and describe sixe concentricke lines or circles with crosse Diagonals, by whose intersections are had the parts of a degree: vpon which Planisphere, and within the diuisions before specified, there is described the Geometricall square, being the second part thereof both which together, or eyther of them seuerally serue to excellent purpose, for the dimension of lengthes, bredthes and distances. The third part is a Semicircle or Quadrant perpendicularly raysed, and artificially placed on the former Planisphere (or more properly on the Index thereof) to be moued about circularly at pleasure on all occasions; which semicircle or quadrant hath the limbe thereof equally diuided into parts or degrees, with like concentricke Circles and Diagonals to those of the Planisphere: And the fourth and last part is a Scale, described within the same Semicircle or Quadrant, whose sides are diuided into diuers equall parts, the more the better, and to best purpose; which two later parts serue chiefly for the mensuration of altitudes and profundities; All which together composed, make an excellent Instrument meete for many purposes, especially for the description of Regions and Countries, or other spacious workes. This Instrument would aske a farre more exact and particular description, which for three principall reasons I will here omit; First, for that I hold it (although a generall and fit Instrument for all mensurations, yet in particular) for this our businesse of Suruey, not altogether so fitting and commodious as the rest before named, by reason of the multiplicity of Diuisions therein contained, which will bee so much the more troublesome in vse and protraction: Secondly, for that M. Thomas Digges in his Pantometria, hath made a large and particular description thereof: And thirdly and lastly, for that M. Allen, who makes of these and all other Geometricall Instruments in mettall, is so well acquainted with the parts and composition thereof, that whosoeuer purposeth to vse the same, needeth only to bespeake it of him, without further instruction or direction to him. The vse and imployment whereof [Page 125]shall be hereafter touched. And thus much for this Instrument and the description thereof.
CHAP. III. The Playne table and his seuerall parts, with the description and composition thereof.
THis Instrument for the playnenesse and perspicuitie thereof, and of his easie vse in practise, receiueth aptly the name, and appellation of the Playne Table. A most excellent and absolute Instrument, for this our purpose in Suruey; but with all, by reason of this his playnenesse (offering it selfe at first view, in some measure, to the weake vnderstanding of meanest capacities, inciting them thereby to the practise thereof) is more subiect to abuse, then all, or any of the rest: For notwithstanding that these, by the common and ordinary vse and practice of this Instrument, may easily attayne to a reasonable truth in Dimension and plotting of regular and euen Playnes; yet, if they come to irregular and vn-euen formes, as Hills and Dales, they are so farre vnfitting for the true mensuration thereof, that many vsing this Instrument, neglect the meanes, and those proper parts of the same Instrument (being the sights hereafter specified, whereof they know no vse) which is their onely helpe and aide in this kinde. But, lest this relation should be held digression, lets returne to the purpose. This Table it selfe is diuided into fiue parts, or small bords, whereof three are in the bredth, and the other two imployed as ledges in the length, to keepe the rest together, whereunto a ioynted frame is artificially applyed, for the fastning and keeping playne thereon, an ordinarie sheet of paper for vse in the fields; of which length and bredth, or rather lesse, as 14½· inches in length, and 11. in bredth, the whole Table together should consist. But for my purpose, I would haue made of these three boords in bredth, with helpe of ledges, to be thereon glued, to the back-side thereof, and strong ioynts or gemmowes, fastning them together, an artificiall boxe; which at any instant being opened, and the ledges fastned on, is fit for vse in the Fields, and afterwards those ledges taken away, may be as instantly turned back wards, and inclosed as before, fit for the keeping of loose papers and small tooles, till further occasion. Which if any desire to vse, M. Iohn Thomson in Hosier-Lane, will artificially fit him. To this Instrument doth also belong a Ruler with two fights thereon, which by diuers men are vsed of diuers kindes, but by the best Playne Table men (as occasion serues) those of M. IOHN GOODWYNS inuention, that excellent and honest Artist, whose liuing Name (though himselfe be dead) I cannot remember without good respect. This Ruler is to contayne in length about 16. inches, or as long or longer then the table, for drawing parallel lines on the paper fastned thereon, by the equall and opposite diuisions on the frame; it is likewise to contayne in bredth about two inches or vpwards, and in thicknes halfe an inch; the sights thereof are double in length the one to the other, the longer contayning about 12. inches, and the shorter halfe as much; on the head or top [Page 126]of which shorter sight, must be placed a wyer or brasse pin; and to this sight there must be fastned a thred and plummet, to place the Instrument Horizontally; through the longer sight must be made a slit, extending almost the whole length of the sight. These two sights thus prepared, are to be placed perpendicularly on the Ruler, by square mortesses, made to that purpose through the same, in such sort as the brasse pin on the shorter sight, and the slit through the longer sight, be precisely ouer the fiduciall edge of the Ruler; either sight being equi-distantly placed from each end of the Ruler, and the space betweene the sights to be exactly the iust length of the longer sight, or to speake more properly of the diuided part of the same sight; (which is alwayes to be vnderstood when I speake of the length thereof.) Vpon this longer sight there is to be placed a vane of brasse, to be remoued vp or downe at pleasure; through which a small sight hole is to be made, answerable to the slit in the same sight, and the edge of the vane.
By these sights thus placed on the Ruler, there is proiected a Geometrical square, whose side is the diuided part of the long sight, or the distance betweene the two sights. In the middle of which long sight, ouerthwart the same, there is drawne a line called the line of leuell, diuiding the side of the proiected square into two equall parts, also the same side is on this sight diuided into 100. equall parts, which are numbred vpwards and downewards, from the line of leuell, by fiues and tens to 50. on either side, which diuisions are called the Scale.
There is also on the same sight expressed another sort of diuision, representing the Hypothenusall lines of the same square, as they increase by vnits, and are likewise numbred vpwards and downewards from the line of leuel, from one to 12. as 1, 2, 3, &c. signifying 101, 102, 103, &c. which declare how much an Hypothenusall or slope line, drawne ouer the same square (that is, from the pinnes head to any such diuision) exceedeth the direct horizontall line, being the side of the same square.
There is moreouer a third sort of diuisions on this sight, representing the degrees of a quadrant (or as many as can well be expressed on the same sight being 25.) which are (as those before) numbred, from the line of leuell vpwards and downewards, by fiues and tens to 25. which diuisions are called the quadrant.
Likewise there may be placed on the surface, or vpper part of this Ruler, the table of Synes, mentioned in the description of the Circumferentor, next hereafter following; very necessary for vse, as shall hereafter partly appeare.
Yet would I further aduise concerning this Instrument; that in playne and leuell grounds, where is no vse of reducing Hypothenusall lines; instead of those long sights formerly expressed, to haue vsed such double sights as are hereafter mentioned in the description of the Peractor, which are farre more readie, and lesse cumbersome for vse, especially in rough and boysterous weather. And likewise, when occasion shall be offered, by irregular and vn-euen grounds, to reduce those Hypothenusall lines, to vse in stead of those long sights, the quadrant hereafter likewise specified in the same description, which when need requires may be easily fixed to the Ruler; and as soone laid by, hauing no further vse, for of all other meanes in taking of heights, and especially for reducing of those Hypothenusall lines, I hold [Page 127]that quadrant artificially handled to be most readie and exact, all conclusions being speedily wrought therewith without Arithmetick, or other collaterall computation whatsoeuer.
To this Instrument belongeth diuers other ordinary parts, whereof I shall not neede particularly to speake, seeing most men (if not too many) already know them, being of ordinarie vse; as the Socket, the Boxe and Needle with the staffe or foote thereof, &c.
CHAP. IIII. The Circumferentor with his parts, description, and composition.
THis Instrument for expedition and portabilitie, exceedeth far the rest, and nothing inferior to any for exactnesse, if care and arte be vsed; but not so vulgarly vsed (though much more generall for vse) as the playne Table is; the full and perfect vse thereof, not lying so open and apparant to all mens vnderstanding, as the other is.
It is made and framed of well seasoned boxe, contayning in length about eight inches, in bredth halfe as much, and in thicknesse about 4/3· of an inch; the left side whereof is besyled, and diuided into diuers equall parts, most fitly of 12. in an inch, to be vsed as the Scale of a Protractor, the Instrument it selfe seruing well to protract the plat on paper, by helpe of the needle, and the degrees of angles and lengths of lines, taken in the field, and entred into your field Booke, as shall be hereafter shewne.
About the middle in the surface thereof, a round hole is to be turned of the depth of halfe an inch, whereof the Diameter to be about 3½· inches, to place a carde and needle therein, to be couered ouer with cleere glasse. The best carde for this purpose is that, diuided in the limbe into 120. equall parts or degrees, with a Dyall according to the Azimuths of the Sunne, wherein the houres are numbred, and the moneths named, seruing very aptly to shew the time of the day.
Besides, on the surface hereof is placed the table of Synes, calculated from the quadrant of a circle so diuided, as the carde before specified; the arch of this quadrant being 30. degrees, and the semidiameter or totall syne thereof, is diuided into 1000. equall parts, and numbers placed accordingly, answering to euery degree and halfe degree, seruing to expresse the length of euery right syne for the Dimension and Supputation of Triangles, as shall be partly hereafter shewne, and in the meane space the Table is here expressed.
| ½ | 26 | 334 | 609 | 824 | 959 | ||||
| 1 | 52 | 7 | 358 | 13 | 629 | 19 | 839 | 25 | 969 |
| 78 | 383 | 649 | 853 | 972 | |||||
| 2 | 104 | 8 | 407 | 14 | 669 | 20 | 866 | 26 | 978 |
| 130 | 430 | 688 | 879 | 983 | |||||
| 3 | 156 | 9 | 454 | 15 | 707 | 21 | 891 | 27 | 987 |
| 182 | 477 | 725 | 902 | 991 | |||||
| 4 | 208 | 10 | 500 | 16 | 743 | 22 | 913 | 28 | 994 |
| 233 | 522 | 760 | 923 | 997 | |||||
| 5 | 259 | 11 | 544 | 17 | 777 | 23 | 933 | 29 | 998 |
| 284 | 566 | 793 | 943 | 999 | |||||
| 6 | 309 | 12 | 588 | 18 | 809 | 24 | 951 | 30 | 1000 |
There is also hereunto belonging two sights, double in length, the one vnto the other, the longer contayning about seuen inches or vpwards, being placed, and diuided in all respects, as those formerly mentioned in the description of the playne Table; the shorter sight of these hauing one propertie and vse, which that (as needlesse) hath not, being this, in the edge thereof, towards the vpper part is placed a small wyer, representing the center of a supposed circle, the semidiameter whereof is that part of the same sight, betweene the wyer, and a perpendicular point on the edge of the Instrument vnderneath the same; which part is imagined to be diuided into 60. equall parts, and according to those diuisions is the right edge of the Instrument, diuided and numbred from the perpendicular point by fiues and tens, as 5.10.15. &c. And also from the same point on the surface and vpper edge of the Instrument is perfected, the degrees of a Quadrant supplying the residue of those which could not be expressed on the long sight from 25, to 90. by tens, &c. For the vse of these Diuisions is there also belonging hereunto an Index or small Rule, at the one end whereof is a Centre hole to place on the Wyer in the edge of the shorter sight, and at the other end a sight is placed, answerable to the fiduciall edge of the same Index, which edge is likewise diuided according to those diuisions on the edge of the Instrument. This short sight is to haue a plummet of lead hanging in a fine thred, seruing to place the Instrument horizontally. Where note, that these sights, and their seuerall diuisions serue onely for Altitudes, Profundities, and the reducing of Hypothenusall lines to horizontall, which is to excellent purpose, and full of vse. But when there is no occasion or vse of these (as seldome there is in respect of other vse) I would alwayes haue ready, such double sights as are hereafter expressed in the description of the Peractor, which doubtlesse are of excellent vse, as I there explaine; with this caution, that you alwayes carefully keepe one and the same part of your instrument forwards, and reccon euer your degrees in one and the same end of your Needle.
The foot of this Instrument is that with three staues ioynted in the head, [Page 129]and to be taken a sunder in the middle with brasse sockets, according to the vsuall order, most fitte for all Instruments, except in such cases, as in the description of the Peractor is excepted. And these are the seuerall parts of this Instrument, and thus is it composed.
CHAP. V. The Peractor, with his seuerall parts Description and Composition.
AS I will not with the cunning Wine-taster forbeare commendations, fearing too many partners; So will I refraine wooing any to vse what I much affect, further then reason, and their owne iudgement shall rule them therein. But thus much will I boldly speake and maintaine of this Instrument, that for generall vse, perspicuity, speed and perfection, it may well compare with any hitherto in vse. It consisteth of a Planisphere in brasse, much like vnto that of the Theodelite, but where the limbe of that is diuided into 360. parts or degrees; this is only into 120. (so that each of these containeth three of those) and these subdeuided into halfes and quarters towards the limbe thereof, without which diuisions, there are drawne and described three concentricke Circles, being crossed with Diagonals, by whose intersections are exactly expressed the third part of euery degree; whereby, and by tripling the former degrees cut (if occasion require) is had exactly the degrees of the Theodelite considering that ten of these contain thirty of those, wherewith in matter of Suruey we shall little need to trouble our selues. And here haue we large and spacious degrees with their exact parts to ¼· (which in others wee want) by meanes whereof, and with helpe of such a chaine, as I alwayes vse (which shall be next hereafter expressed) I will boldly approue and maintaine, to worke with much more facilitie and exactnes, and come neerer the precise truth then any other can possibly doe, not vsing the same or the like, which will appeare most manifest in practise to all mens vnderstanding.
There is also an Index hereunto belonging, fixed on the Centre as that of the Theodelite, with two sights thereon placed, with sliding loopes, eyther of them alike, and of like length, and eyther of them double sighted, the one hauing a slitte beneath, and a thred aboue, the other a thred beneath, and a slit aboue, seruing to looke backwards and forwards at pleasure, without turning about or stirring the Instrument, when the Needle is at quiet; whereby I saue neere halfe the labor, and halfe the time that any man shall spend with other sights, for that hereby I need plant mine Instrument but at each other Angle, which is no small helpe for expedition, and such a meanes for exactnesse rightly handled, as few will imagine without due proofe; and that without trouble of sending one before, and leauing another behind, as is vsually accustomed. Neyther vse I these sights with this Instrument onely, but with all others as the Theodelite, Plaine-Table, and Circumferentor.
These sights I alwaies vse in plain and euen grounds; but if occasion serue for the reducing Hypothenusall, to Horizontall lines; I then euer vse this [Page 130]
Quadrant here expressed most fitte for the speedy taking of all maner of altitudes and profundities, and the suddaine reduction of those lines without Arithmeticall calculation, or other collaterall account, as is manifest in vse. This Quadrant (the other sights taken away) is on any occasion speedily placed on the Index with scrue pinnes, and as suddainely taken off, if no further vse, and the sights replaced.
[Page 131]Then is there also hereunto belonging other vsuall parts, as a boxe of brasse on the Centre to containe a Carde and Needle therein, such as is formerly expressed for the Circumferentor, to be likewise couered ouer with cleere glasse, and close stopped with red waxe about the edge thereof, to defend the Needle from Winde, Wether, and Ayre, the onely enemies thereunto.
Also a brasse Socket to bee scrued on with foure scrue pinnes on the backe of the Instrument, which Socket ought to be precisely turned, and the head of the staffe therewith, (I meane the brasse part thereof) which will neuer otherwise turne euenly and nimbly about as it ought, the one within the other, without iercking and starting, which much troubleth the Needle in finding his naturall point and place of rest. And if any doubt the truth of his Needle, let him take backe sights for his better satisfaction therein.
And lastly, for this as for the rest the like staffe is to be prouided as before is spoken of, which for all purposes is the absolute best, except onely for water leuels, and the works thereunto belonging, wherein it is necessary at all times, and at each seuerall stacion to keepe the instrument at one and the same horizontall distance, which otherwise may breede much cumber, and no lesse incertainety in those conclusions; wherefore for those and the like occasions, a foot with one staffe, hauing three iron pikes therein, after the old order is best, and to best purpose.
The making of this Instrument and the rest in brasse are well knowne to M. ELIAS ALLEN in the Strand; and of those in wood to M. IOHN THOMPSON in Hosyer Lane.
And thus haue I briefly described these seuerall Instruments with their particular parts, laying them before my Surueyor to take his choyse as his fancie leades him; But in mine opinion all are better then any; so shall hee best know what is best for his purpose.
And now let vs consider, what other necessaries are yet to bee prouided, before wee beginne our businesse, for feare wee are to seeke when occasion serues for vse. As the Chaine, Protractor, Field-booke, and the Scale and Compasse whereof we will further speake.
CHAP. VI. The making and diuision of a Chaine, called the Decimall Chaine.
THis is the chaine before spoken of in the Peractors description; which for conueniencie in carryage, and auoyding casualties often happening to breake it (though made of a full round wyer) I would aduise should containe in length but onely two statute Poles or Perches, or three if you please at the most. In the diuiding whereof it is to bee considered, that the statute Perch or Pole, (which here we call an vnite, or (Comencemente) containeth in length 16 ½ feete. which is, 198. ynches: This quantitie is first to be diuided into 10. equall parts called Primes, so shall euery of these Primes containe in length 19. ynches, and ⅘· of an ynch: And then these Primes to bee euery of them [Page 132]subdiuided into other 10. equall parts, which wee will call Seconds: and so euery of these Seconds shall containe in length one ynch, and 49/50· parts of an ynch. And thus is the whole Perch vnite or comencement diuided into 100. equall parts or linkes called Seconds.
Which Chaine so diuided is thus to bee distinguished and marked: First, at the end of euery fifth Prime, or fiftieth second or lincke, which is the end of euery halfe Pole, let a large curtaine ring be fastned, so shall you haue in the whole Chaine (if but two Poles) three of those rings; the middlemost being the diuision of the two Poles, which in a Chaine of this length is easily and readily discerned from those rings of the halfe Poles, though all of one greatnes. Then at the end of euery Prime, that is, at the end of euery tenth second, or linke, let a small curtaine ring be placed, and not those rings of brasse wyer, as is vsuall in other chaines, which, with euery bush and twig are continually broken off, and lost.
By those distinctions this Chaine is now diuided into these three termes, Vnites, Primes and Seconds, whose Characters are these 0.1.2. So that if you would expresse 26. Vnites, 4. Primes, and 5. Seconds, they are thus to be written 26 0· 4 1· 5 2· or together thus, 26 0 4 1 5 2· or more briefly thus, 26 4 [...] 5 [...]. making prickes or points onely ouer the Fractions, whereby the rest may be conceyued to be Vnites, or Intigers, and the first point Primes, and the next seconds.
But besides these diuisions for mine owne vse, I alwayes at the end of euery 2 ½· Primes, which is the 1/4 · of a Pole, sowe on a small red cloth, or the like, (thrust through the ring of the chaine, and at euery 7 ½· Primes being the ¾· of a Pole; the like of yellow, or some other apparant colour, where with being once acquainted, and thereunto inured, you shall most speedily at the first view reckon the quantitie of euery ring, remembring that if it bee the next ring, short of the red, it is two Primes, if the next ouer, three, if the next short of the yellow, it is 7. Primes, if the next ouer, 8. if the next short of a great halfe ring, it is 4. the next ouer, 6. And lastly, if the next short of the middle great ring, it is 9. and if the next ouer, 1. and so of the rest; Wherin is to bee noted that your Chaine thus marked, is alwayes to be vsed with one and the same end forwards.
This Chaine thus diuided and marked, you haue euery whole Pole equal to 10. Primes, or 100. seconds, euery ¾· of a Pole equall to 7 ½· Primes, or 75. seconds, which is ¾· of 100. euery halfe Pole equall to 5. Primes, or 50. seconds, which is ½· of 100. and lastly euery ¼· of a Pole equall to 2 ½· Primes, or 25. seconds, which is ¼· of 100.
And here is to bee noted, that in the ordinarie vse of this Chaine for measuring and platting; I obserue onely Vnites and Primes (but on necessity) which is much more exact then the ordinary vse; but hauing occasion to make diuision or separation of lands, or for the dimension of common fields in their seuerall parts by furlongs or wents and rigges, I vse my Seconds; wherein, what exactnes and most excellent vse I find, I will referre to those who can descerne the difference betweene a portion lesse then two ynches, the length of my second, and that of 6. ynches, and 3/ [...]6· of an ynch, the least part of the best and exactest Chaine now commonly vsed; but with those of the last of long twelues I will not meddle.
[Page 133]But here me thinkes I heare the Aduersarie question, to what purpose serues this nicenesse of ynches in Instrumentall obseruation, when comming to your protraction with a small Scale, you are not able to distinguish feet? I answere (and to purpose) thus: If by your ordinary Chaine you take obseruation in your Field-bookes of ¼· ½· and ¾· and few or none otherwise, (or if they doe, to small purpose, as they afterwards handle the matter) then I say, I taking my obseruation of 1. 2. 3. or 4. Primes, or of 6. 7. 8. or 9. Primes, can in my Protraction with a small Scale and Protractor (yet mine I must confesse is none of the least) easily distinguish and expresse how much lesse or more then ¼· ¾· or ¾· those quantities are, as may easily appeare, with due obseruation of my former notes. Yet may it bee further sayd, What is all this to purpose, if there be not as exact a meanes to obtaine or get the true superficiall content in casting vp the plat, it being thus exactly layde downe? I may answere againe, Better one mischiefe then many; neyther will I suffer this; for be well assured, I will not be so carefull in that, and altogether carelesse in this; the meanes whereof in due time I may hereafter shew, being vnfit for this place; hauing already inlarged my Chaine in length more by a Pole, then first I meant; and therefore purpose now to be no longer chained therein.
The making of this Chaine is well knowne to M. CHRISTOPHER IACKSON at the Signe of the Cocke in Crooked-Lane, who by my directions hath made of them for me, and hath the scantling thereof.
CHAP. VII. Of the Protractor and the Scale thereof.
PRouide for this Protractor a fine thinne piece of brasse well polished, in forme of a long square, as the Figure E F K I. which (for conueniencie in vse) ought to containe in length from G. to H. about 5 ¼· inches, and in bredth from L. to B. somewhat better then [Page 134]3 ½· ynches, whereupon draw two lines as G H. and L B. cutting each other precisely at right Angles in the point D. diuiding the Square into foure equall parts, then on the point D. as a Centre, at the distance of D L. or D B. describe the Semicircle A B C. (for it is not materiall, or of necessitie, that the Diameter thereof should agree with the Diameter of the Carde in the Instrument, as M. Hopton would haue it in the 62. Chap. of his Top. glasse) then diuide the limbe of the Semicircle A B C. into 60. equall parts or degrees, numbring them by fiues and tens in the outward space to 60. and in the inward space from 60. to 120. as in the figure, the first numbers seruing for the East side, and the later (being the opposite degrees) for the West side of the whole Circle, so is a labour saued of diuiding the other side which serues to no purpose; then let the edges of the Scale as E G. E F. and F H. be somwhat besiled, and made very smooth, and precisely parallel to the first drawne lines respectiuely, and aboue the rest, let care be had that the line L B. bee made exactly perpendicular to the edge E F. of the Scale, or otherwise great errors may insue in the vse; then diuide the Parallel degrees at eyther end of the Scale, betweene E G. and F H. And let the Scale of 12. bee placed on the edge E F. and the sale of 11. on the edge of the backe side, which are most necessarie and meetest for vse of any other; and lastly, cut out the square about the Centre D. and likewise that between the Semicircle & the pricked lines, hauing care that the line G H. be left perfect, and euen with the Diameter A C. being the meridian line, and the guide of the rest. And so is this worke finished.
Yet would I haue besides in some spare place of this Protractor, or on the backe side thereof, the Sextans described, which is mentioned in the next Chap.
Here is it to be noted, that this PROTRACTOR serueth without alteration or any difference, aswell for the PERACTOR as the CIRCVMFERENTOR. But if you would haue it for the THEODELITE, then must the limbe of the Semicircle bee diuided into 180. equall parts, and numbers placed accordingly, which is all the difference.
And here also would I not haue you forget to prouide that long Protractor formerly mentioned in the conclusion of the second part of my second Booke, which will stand you in good stead.
Also to these there belongs a protracting pinne made of a needle (according to the Centre hole of the Protractor) to be placed in the end of a small turned sticke; or of Iuorie, as best likes you. And so are you thus farre fitted: wherefore to the rest.
CHAP. VIII. Of the ordinarie Scale with the Sextans thereon described, very necessarie for vse.
FOr this purpose let a Ruler of Brasse or Boxe (but brasse the better) be prouided, as the Figure A B C D. which let containe in length about 7. or 8. ynches, and in bredth about two ynches, or somewhat lesse, whereupon on the one side let bee placed two Scales, the one of 11. the other of 12. in an ynch; and on the same side, let there be also described a Sextans or the sixt part of a Circle, whose chorde E F, (which is alwayes equall to the Semidiameter of the same Circle) let containe in length about two ynches or lesse, and let the limbe be diuided into 60. equall parts or degrees, and numbred by fiues and tens, as in the Figure. On the other side thereof there may be placed (after the order of these) diuers other Scales, as of 16. 20. 24, &c. as you thinke fitting. So haue you a necessary Instrument for many purposes. And this Sextans also would I haue described on some spare place of your Protractor.
To this must you haue prouided a neate payre of Compasses of brasse, with fine steele points, which must alwayes bee ready seruing for infinite occasions.
Besides these ordinary Compasses, it is very fitting to bee also prouided of a payre of Callem Compasses, with scrues to alter the one legge at pleasure, wherein to fasten a penne, blacke leade, a steele point, or the like, very fit for many purposes.
CHAP. IX. Of a Ruler, for the reducing of Plats.
ALthough wee are not as yet fitte for the vse of this Rule, yet seeing our businesse now in hand, is to prouide vs of necessaries: It is no ill rule to take our businesse before vs: Wherefore repayre to Master IOHN THOMPSON in Hosier lane, who without further Instructions [Page 126]will furnish you, onely this before you goe: Let the Ruler bee made of drie boxe, if you may, of a yard in length, and let the equall diuisions thereon bee of 12. in an ynch, to bee numbred with double numbers, as he vseth for me. So will it serue you to good purpose, aswell for casting vp of large plats. &c.
CHAP. X. The order of making of a necessary and fitting Field-booke, seruing aswell for the Peractor and Circumferentor, as for the Theodelite, with the ordering and vse thereof in the Fields.
THis Booke may consist of halfe a quire of paper, to bee bound (most aptly for vse) in a long Octauo: Let it be ruled towards the left margent of euery side, with foure lines, so shall you describe three Collums, the first seruing for the degrees, the [...] cond (according to my Chaine) for Vnits: and the third and last for Primes; or according to the accustomed vse, for degrees, poles, and parts of a Pole.
The order of vsing it is thus: Suppose you are to suruey the Mannor of Beauchampe, and are to beginne with these fiue seuerall parcels numbred in this plat or figure, with 1.2.3.4.5. being seuerall grounds of seuerall Tenants, and of seuerall natures, whereof you are to make seuerall obseruation, as appeareth. First, for the title of your Booke begin thus: Maner. de Beauchampe in Com. Ebor. Pro Rege incip. 24. Iunij, 1616.
Then begin with the first field, at A. writing the Tenants name, the name of the field, of what nature, and of what Tenure thus: Io: Iacksons Hall Close, Meadow, Free: and expresse in the Margent of your Booke No(1) signifying the first Close, then hauing placed your Instrument at A. directing your sight to B. you finde the degree cut 65. which place in the first Collum towards the left hand, then measuring the distance betweene A. and B. you shall finde 20. Vnites, and 2. Primes, or 20. Poles, and a little lesse then a quarter, which place seuerally in the two next Collums towards the right hand; then place your Instrument at B. and directing your sight to C. obserue and expresse the degrees and length there found, as 6 [...] [...]·16 0·6 1· and considering that at C. you are to leaue the bounder you formerly went against, strike there a single line ouerthwart your booke, and aboue that line against the degrees and quantities taken, write what was the bounder you went against, whether it be of this or any other Mannor, as here in this example thus, the Mannor of Barnsey; and thus proceed with the rest of this Field, till you come to inclose at A. where you first began; and then strike a double line▪ and place this marke ⊕ or the like in your booke, at the end of the inclosing line, signifying to inclose. And so is that Field finished. Then consider which Close is next sitting to be taken in hand, which let be (2.) and withall at what Angle thereof it is meetest to begin, which let bee at C. and here (for your helpe when you come to protraction) you must expresse in the Title of this second Close, at what Angle you begin the same (vnlesse [Page 127]
| Maner. de Beauchamp. in Com. Ebor. pro Rege incip. 24. Iunij. 1616. | Iohn Iacksons Hall, Close, Mead-Free. | d | 0 | 1 | ||||||
| (1) | 65 | 20 | 2 | |||||||
| 68 | 16 | 6 | ||||||||
| The Mannor of Barnsey. (1) | ||||||||||
| 36½ | 22 | 5 | North Breading. (2) | |||||||
| 27¼ | 12 | 3 | ||||||||
| 106 | 14 | 1 | ||||||||
| 21½ | 18 | 2 | ||||||||
| Vpper barne Close. (3) | ||||||||||
| 112 | 21 | — | South-Field. | |||||||
| 32¾ | 20 | 2½ | ⊕ Where began. | |||||||
| (2) | William Bensons North-breading, Close, Pasture, Cop. from No. (1.) | |||||||||
| 55½ | 14 | 5 | The Mannor of Barnsey. | |||||||
| 65 | 18 | 9 | ||||||||
| 41½ | 16 | 2 | ||||||||
| to | 5¼ | 35 | 4 | ⊕ Vpper Barne Close, at (2.) | ||||||
| (3) | Thomas Brownes vpper Barne Close, arr. Lease for yeeres. | |||||||||
| 31½ | 10 | 3 | ||||||||
| 44¾ | 14 | 5 | North Field. | |||||||
| 20 | 16 | 2 | ||||||||
| 4½ | 34 | 6 | Oake Close. | |||||||
| 46¾ | 27 | 3 | ⊕ At (3.) East garth. | |||||||
| (4) | His East garth, Meade, Free. | |||||||||
| 28 | 22 | 2 | South Field. | |||||||
| 46½ | 16 | 6 | ||||||||
| to | 78½ | 23 | 2 | ⊕ Oake Close. | ||||||
| (5) | His Oake Close-Pasture, Lease for three liues. | |||||||||
| 26¾ | 18 | 5 | South field. | |||||||
| 74 | 15 | — | ||||||||
| 63 | 20 | 4 | East Field. | |||||||
| 75 | 19 | 5 | ||||||||
| 84¾ | 19 | — | North Field. ⊕ On vpper barn close. | |||||||
The Mannor of Barnesey
The North field.
The east field
The south fielde
you had begunne it where you ended the last, and then it is not materiall) wherefore seeing now you are to begin at C. looke in your booke on the worke of your last Close, at what place the line B C. ended, as at the end of 68. degrees, 16 0· 6 1· and there expresse No. (1.) on the right side of your down right lines, & then write your title for this Close thus: W. Bensons north breading close Pasture Copie from No. (1.) so shall you readily know when you come to Protraction, where to begin with this Close, and in the Margent place N. (2) for the number of this your second Close; and then worke on as before is taught for the last Close, till you come to K. where noting, that you want but one line as K D. to inclose this second Close, and also that this Angle K. is the sittest place to begin your third close at, placing here your Instrument, and directing your sight to D. where you are to inclose, write in the margent of your Booke against the degree here taken, this word, (to) as thus, to 5¼· degrees, 35 0· 4 1· which serues to put you in mind, when you come to Protraction, that you are to seeke no other place, to begin your third Field at, but where now you are at the Angle [Page 128]K. where you tooke your last sight. And then hauing inclosed this second Close, proceede to the third, from K. in the title whereof you need not make mention at what place you begin, for the former reason, but working in that, and all the rest, as before is taught, you shall with great ease, and without confusion, if there were infinite seuerall parcels, distinguish the worke of the one from the other, and be able by the onely helpe of your Field booke, thus formally ordered (if need were) to protract, and draw a perfect plat thereof many yeeres after: remembring alwayes that those numbers in the Margent of your Booke, are to be placed seuerally in your platte, in those Closes they seuerally represent; so shall you not need to write in the plat the names of the Tenants, or of the Closes, nor the quality or tenure thereof, but onely the number, which will alwaies direct you to your booke, where you may find it at large; for much writing in your plat (as is alwayes vsed in working with the Plaine Table) breedeth confusion, and causeth much cumber and trouble in casting vp the contents.
But seeing that practise is much more instructiue (in works of this nature) then many words; I will referre the rest to your owne trauell; which by comparing those former notes of the Field booke with the plat, and often protracting the same according to those degrees and lengthes, the whole course is very easily vnderstood. The manner and order of which protraction is hereafter taught in the 39. Chapter of this Booke.
And here note further that in practise you shall find many helpes, which are too tedious here to expresse, as the taking in of diuers seuerals together, when they lie in such sort diuided with regular lines and hedges, that by onely taking true notice of their seuerall ends, as you passe by them, you shall most easily and speedily seuer them on your plat: All which with many others (to auoide prolixitie) I must refer to your owne finding out by diligence and practise.
And thus are we now reasonably well furnished of necessary implements for our purpose, and therefore fitting to prepare vs to practise: but yet before wee goe into the Fields, we will consider of some necessary conclusions and obseruations, fit to bee knowne and remembred.
CHAP. XI. To lay downe an Angle of any quantitie required; or to find the quantitie of any Angle giuen, by the Sextans and the Scale.
SVppose it be required to lay downe an Angle of 40. degrees.
First draw any line at pleasure, as A B. then opening the Compasse to the distance E F. the chorde of your Sextans (described in the 8. Chapter) and placing one foot in A. with the other, describe the Arch C B. cutting the line A B. in B. then extend the Compasse on the Sextans to 40. degrees, and with [Page 129]one foot in B. crosse the Arch line in C. and draw the line A C. So shall you describe the Angle C A B. containing 40. degrees, as was required.
Againe, suppose C A B. be an angle giuen, & let it be required to know the quantitie thereof. Extend the Compasse to the chorde of the Sextans as before, and at that distance with one foot in A. describe the Arch line C B. to cut both sides of the giuen Angle, as in C. and B. then opening the Compasse to C B. and applying them at the same distance, to the degrees in the Sextans. It will appeare that the quantitie of the Angle is 40. degrees, the thing required.
If the Angle giuen or required happen to be more then the whole Sextans, or aboue 60. degrees, yet take still the chorde of the same Sextans, and describe the Arch line as before; and first place the whole Sextans, (which is the chorde thereof) on the Arch line from B. so farre as it will extend beyond C. and thereunto on the same Arch line, adde so many degrees more as the Angle giuen or required, containeth degrees aboue 60. So shall you performe what was required.
CHAP. XII. To lay downe an Angle of any quantitie required, or to finde the quantitie of any Angle giuen, by the Protractor.
SVppose it be required to lay
downe an Angle of 30. degrees with the Protractor. First, draw any right line at length, as A B. then on any point thereof, as on C. place the Centre of the Protractor, on which point moue it about, by helpe of the protracting pinne held in the Centre, till the Meridian line of the Protractor lye precisely on the line A B. (the semicircle of the Protractor being vpwards) and by the edge of the Semicircle at the diuision of 30. degrees, marke with the protracting pinne the point D. and draw the line D C. which shall include the Angle D C B. containing 30. degrees as was required.
Againe, suppose D C B. in the former Diagram bee an Angle giuen, and let it be required to know the quantitie thereof. Place the Centre of the Protractor as before in the Angle C. and the Meridian line thereof on the line C B. and hauing the semicircle vpwards, note what degree on the edge of the Protractor is cut by the line D C. which you shall find to be 30. Shewing that the giuen Angle D C B. containeth 30. degrees, the thing required.
But here is to be noted, that these degrees thus taken by the Protractor belonging to the Peractor or Circumferentor, are not the true degrees of a [Page 130]Circle; for one degree of a Circle, is but the 360. part thereof, and one of these degrees thus taken are the 120. part, so that one of these containeth three of those; Wherefore if you are to take the quantity of an Angle (according to the degrees of a Circle) by those Protractors, take alwayes a third part vpon the Semicircle, of the number giuen or required; as in the former example, where 30. is giuen, take 10. and so shall you finde the Angle E C. B. in the last Diagram, to be an Angle of 30. degrees, and to be a third part of the Angle D C B. As may be proued, if you apply thereunto a Protractor belonging to the Theodelite; yet notwithstanding these Protractors and degrees in all our occasions in the vse of the Circumferentor and Peractor are alwayes to bee vsed, which will tend to one and the same purpose.
CHAP. XIII. The reducing of statute measure into Acres of any customary measure required, and the contrary, shewing the difference betweene them.
BY the Statute of 33. Ed. 1. It was ordained that an Acre of ground should containe 160. square Perches, to be measured by the Pole of 16½· feete, which is the measure now receyued, and generally allowed of, and is commonly called Statute-measure: yet notwithstanding in many places of this Kingdome, there are diuers other sorts retayned and claimed as customary, wherof some are greater, and some lesse then that by Statute. Wherefore I hold it very fitting, and a maine point belonging to a Surueyor, readily to reduce these quantities from the one to the other, whereby the difference may appeare; whereof in practise hee shall find often vse; which to effect worke thus. Suppose there are 5. Acres, 2. Roodes, 20. Perches, of 18. feet to the Pole giuen (called Wood-land measure; and let it be required to know the quantitie thereof by statute measure, being of 16 ½· First, finde out the least proportionall termes, betweene 18. and 16 ½· which by their abbreuiation, by 1 ½· you shall finde to be 12. and 11. then reduce your giuen quantity into the lowest denomination, which is Perches, so shall your 5. Acres, 2. Roodes, 20. Perches, be 900. Perches. And considering that the same proportion which the square of 12. bears to the square of 11. the like proportion beares the Acre of 18. foot Pole to that of 16 ½· therefore square those two termes 12. and 11. which produceth 144. and 121. then multiply the giuen quantitie 900. Perches by 144 the greater square (because the greater measure 18. is to be reduced into the lesser 16 ½· the Factus is 129600. which diuided by the lesser square 121. quoteth 1071 9/121· Perches; which reduced into Acres, is 6. Acres, 2. Roodes, 31. Perches, and 9/121· parts of a Perch; for the quantitie required in statute measure, whose difference by deducting that from this, appeareth to be 1. Acre, 0. Roode, 11. Perches 9/121·
But suppose the giuen quantitie had beene statute measure, and the same required to be reduced into Wood-land measure; then should you haue multiplied the 900. Perches giuen by 121. the lesser square (because the lesser measure 16½· were to be reduced into the greater 18.) whereof the Product [Page 131]is 108900. which diuided by 144. the greater square quoteth 756 ¼· Perches, which reduced into acres, is 4. Acres, 2. Roodes, 36 ¼· Perches, for the quantitie by Wood-land measure; whose difference by deducting this from that appeareth to be 0. Acre, 3. Roode, 23 ¾· Perches.
And the like course is to be held in all respects, with all other quantities of what proportion soeuer; as those of 12.20.24½· and 28. foot to the Pole, of all which seuerall sorts I haue found in diuers places, whose difference; [...] euery Acre, from that of 16 ½· appeareth by this Breuiat following.
An Acre measured by the Pole of these feet,
- 12
- 18
- 20
- 24
- 24½
- 28
Containeth of Statute measure
- 0 — 2 — 4 76/121·
- 1 — 0 — 30 50/121·
- 1 — 1 — 35 85/1089·
- 2 — 0 — 18 62/121·
- 2 — 0 — 32 133/174·
- 2 — 3 — 20 4/ [...]·
And here is it not amisse to note the benefite and vse of your two Scales of 11. and 12. in an inch formerly described in the 8. Chapter, which will serue you now to purpose. For if in your Surueyes (as often hapning) you meete with Wood-land grounds, whose quantities are required to be of the Acre of 18. foot Pole, and yet platted with the rest: In such case you may measure those Wood-lands with the Pole of 16½· and likewise plat the same with the Scale of 12. as the rest, but to cast vp the contents of those Wood-land grounds by the Scale of 11. which will produce the desired quantitie; By reason that if 11. Perches be measured in a right line with the 18. foote pole, the same length containeth 12. Perches measured with the 16½ foote pole. But if you are constrained in the measuring of your Wood-lands, to vse the Pole of 18. foot; then must you protract and lay downe the same in your plat by the Scale of 11. which otherwise will not ioyne with your other works; and the same likewise to be cast vp by the same Scale of 11. as before. So shall you obtaine the true quantitie thereof in Acres, after the measure of the 18. foot pole required.
CHAP. XIIII. Of the Table of Sines expressed on the Circumferentor.
THis Table (as is specifyed in the description of the Circumferentor (Chap. 4.) serueth for the calculation, resolution and dimension of Triangles; not in respect of the Area or superficiall content thereof; but for the finding out of the vnknown sides and angles of the same; by meanes whereof, all manner of quantities in mensuration of altitudes, profundities, longitudes and latitudes are exactly knowne and discouered, considering that none of these can be had or obtained instrumentally, without description of Triangles.
[Page 142]Wherefore let it first bee considered, that by the 73. THEOREME of the first booke, the sides in all plaine Triangles are in proportion the one to the other, as the Sines of the Angles opposite to those sides.
And then that in euery Triangle, there are sixe termes, that is, 3. sides and 3. angles, whereof any three being knowne (so one be aside) the other three are had, by meanes of this Table, and the rule of proportion thus. Suppose B C. in the Triangle
A B C. to be a Tower or perpendicular altitude giuen, and let it be required to finde the height thereof, and the length of the Hypothenusall line A B. First, measure the distance from A to C. 40. then by any Instrument planted at A. take the quantitie of the Angle B A C. which let be 10. degrees, then consequently by the 13. THEOREME of the First, the Angle A B C. shall be 20. degrees, for that the Angle A C B, is a right Angle: So is there here alreadie had foure parts or termes of the sixe before mentioned, namely, the three Angles, and the length of the line A C. Wherefore now repayre to the table, and find there the Sines of those Angles which are these, of the Angle A B C. 20. degrees, the Sine is 866. of the Angle B A C. 10. degrees, the Sine is 500. and of the Angle A C B. 30. degrees the Sine is 1000. the totall Sine: Then by the rule of proportion reason thus. If A C. 866. the Sine of the Angle A B C. giues B C 500. the Sine of the Angle B A C. what giues A C. 40, and multiplying 500. by 40. the Product is 20000. which diuided by 866. quoteth 23 49/433· for the height of A C. required.
Again, if A C. 866. the Sine of the Angle A B C. giues A B. 1000. the Sine of the Angle A C B. what giues A C. 40. and multiplying 1000. by 40. the Product is 40000. which diuided by 866. quoteth 46 82/433· for the length of A B. the Hypothenusall line, as was required.
And thus much for a small taste onely of this little Table, which may serue to induce and incite a willing minde, not onely to the vse and exercise thereof; but to the further consideration and practice of the infinite vse of those most excellent Tables and workes de BARTHOLOMAEO PITISCO GRVNBERG. now partly translated into English by M. RAPH HANDSON; and of those Tables, and more then admirable inuention of LOGARITHMES, by that diuine and noble Writer, the Lord MARCHISTON, whose name and honour will neuer out.
CHAP. XV. Of the congruitie in vse betweene the Peractor and Circumferentor; and the meanes to find the quantitie of an Angle by eyther of them.
THese two Instruments in vse differ little or nothing, considering the degrees of eyther are equally numbered, although those of the Circumferentor are placed on the Carde, and these of the Peractor, on the limbe of the Planisphere, whereby they are so much the larger, and thereby the fitter for vse: onely herein they differ, the degrees obserued and taken by the Peractor, are euer cut by the edge of the Index, moued about till through the sights thereon, the obiect be found, the Needle being alwayes kept on one degree, and that most fitly on the Meridian line in the Carde, the North end (being that with the Crosse) lying euer ouer the Flower deluce, and the south end pointing to the beginning of the degrees: and the degrees obserued and taken by the Circumferentor, are alwayes cut in the Carde by the South end of the Needle, playing about at pleasure. whilest the Instrument and the sights thereof are directed to the purposed marke. By meanes of which diuersitie, there is a diuers meanes to be vsed, in taking the true quantitie of an Angle, by these two Instruments, as followeth.
By the 49. DEF. of the First. The quantitie or measure of an Angle, is the Arch of a Circle, described from the point of the same Angle, and intercepted betweene the two sides of that Angle, which is found by the Circumferentor thus.
Suppose A B C. be an Angle giuen in the
Fields, and let A B. and B C. be two hedges or other lines, containing the same Angle; and let it be required, to take the quantitie thereof by the Circumferentor. First, place your Instrument in the Angle at B. and turning the same about on the staffe, direct your sight towards A. to bee alwayes parallell to the line B A. where obserue what degree the south end of the Needle cutteth, which let be 40. degrees, then direct your sight towards C. and there making like obseruation, let the degree cut bee 18. But yet haue you not the quantitie of this Angle, but onely the degrees cut at each obseruation. Wherefore now you are to deduct 18. the lesser from 40. the greater terme, the remainder will bee 22. the true quantitie of the giuen Angle. But had this remainder exceeded 60. (which is two right Angles, or a Semicircle) then must you haue deducted that remainder out of 120. the whole circle, and the remainder of that last deduction had beene the quantitie required; which circumstance in the Peractor needeth not, wherewithall worke thus.
Place your Instrument at B. in the former Diagram. (the Index standing on the Diameter where the degrees commence) then turne about the Instrument [Page 144]on the staffe (the Index remayning) towards A. till your sight bee parallel to the line B A. and there your Instrument fixed, remoue the Index, directing your sight towards C. to be likewise parallell to B C. where obserue what degree the edge of the Index cutteth, which will be 22. the quantitie required. And here note an exquisite dispatch.
CHAP. XVI. To take any horizontall distance at two stacions, by Sinicall computation.
SVppose A and B. be two places
giuen, & let it be required to know the distance from eyther of them to C. First, place your Instrument at A. where directing your sight first to C. and then to B. take the quantitie of that Angle, as was taught in the last Chap. which suppose to be 26. degrees, then measuring from thence to B. let the length there found be 20. where placing your Instrument, take likewise the quantitie of that Angle as before, which let be 19. degrees. Now to finde the quantitie of the third Angle at C. (Forasmuch as by the 13. THEOREME of the first, the three inward Angles of euery Triangle is equall to two right Angles) adde together the quantitie of those two Angles at A. and B. already found, 26. and 19. which makes 45. and that deducted from 60. (which is two right Angles or a Semicircle) the remainder is 15. the quantitie of the Angle at C. So haue you gotten the quantitie of euery Angle, and the length of one of the sides, namely, the Stationary distance A B. Now to euery of those Angles in the Table of Sines, finde out their seuerall Sines, whereof make collection in your Field booke, or otherwise thus.
Then by the Rule of Proportion, reason thus. If A B. 707. the Sine of 15. degrees yeelde 20. the stationary distance, what A C. 839. the Sine of B. 19. degrees, and multiplying 839. by 20. and diuiding the Product by 707. the quotus will bee 23. 519/707· for the distance A C. And againe, if A B. 707. the Sine of C. 15. degrees, yeeld 20. the stationary distance, what B C. 978. the Sine of A. 26 degrees, and multiplying 978. by 20. and diuiding the Product by
| d | s | |
| A. | 26 | 978. |
| B. | 19 | 839. |
| C. | 15 | 707. |
| A B. | 20. Perches. |
[Page 145]707. the quotus will bee 27 471/707· the distance B C. required.
The Theoricall ground and reason of this worke dependeth on the 13. and 73. THEOREMES of the first booke.
Where note in all works of this nature, that if any of the three Angles bee an obtuse Angle, containing aboue 30. degrees, then (seeing the Table of Sines exceedeth not 30.) deduct the excesse of the obtuse Angle aboue 30. out of 30 ( as if it were 44. the excesse whereof aboue 30. is 14. which deduct out of 30. there remaineth 16. and of that remainder seeke the Sine in the Table, which serues the turne. The reason hereof is, because the right Sine of the Arch in the greater or lesser Quadrant are all one and the same thing.
Likewise note alwayes in your working by the golden Rule, that the Sine of the Angle opposite to the Stacionary line (as in this example 707.) must bee your first proportionall number; and most fitly ( though it may be otherwise, transpositis terminis medijs) the distance betweene the two stacions the second; and the Sine of the Angle, opposite to the side, whose length you seeke the third.
And note also, that not onely this, but all other the like Propositions are to be performed, aswell by the Peractor and Theodelite, as the Circumferentor, the Table of Sines being had in any voyde paper, or much rather those small Tables of Logarithmes, or of Pitiscus, which are imprinted by themselues in small volumes, being most excellent pocket-companions for infinite Conclusions, aswel Geometricall as Astronomicall.
And if any desire the performance of this Proposition, or the like by protraction; let him diligently obserue the doctrine of the next.
CHAP. XVII. To take the distance aswell betweene diuers seuerall places remote from your place of being, as betweene your being, and those seuerall places, by the helpe of two stacions.
SVppose A.B. and C. bee three places giuen remote from your place of being, which let be at D. and let it bee required at D. to finde the seuerall distances, aswell betweene A B. and B C. as betweene D A. D B. and D C. First place your Instrument at D. and directing your sight to A. obserue what degree is there cut, eyther by the Needle of the Circumferentor, or by the Index of the Peractor, which let be 32 ½· degrees, to be noted for your first obseruation, thē turning your sight to B. make the like, where you finde 21. degrees, and the like towards C. obseruing 15. degrees. Then your second Stacion, (not being limited) make choyse thereof with such discretion (if the place will affoord it, as at E.) that your Stationary distance bee no lesse at the least then ⅛· of the other distances from you, how much greater (with reason) makes no great matter; and as neere as you may, let it make a right Angle with the first obseruation of your first Stacion, then for the last worke of that stacion direct your sight to E. obseruing the degrees cut 120. then take vp your Instrument, and leauing a marke at D. measure from D. to E. the stationary distance, which suppose 48. then at E. plant your Instrument precisely as at D. vsing the helpe both of your Needle and backe sights herein, looking backe to your marke at D. whereof speciall care is to bee had, or [Page 147]maine errors may ensue: which done, direct your sight first (as at the first stacion) to A. obseruing the degree there cut 45. the like to B. 31 ½· degrees, And lastly, to C. 15. degrees. So haue you finished; if you omit not the collection of your seuerall obseruations, which in your Field Booke or otherwise are thus to be expressed.
Then prouide a cleane sheete of paper, and according to these collections layde before you, protract the seuerall angles or degrees here obserued, as is taught in the next.
Obs.
- 1.
St.
d 1 32 2 21. 3 9¼· 4 120. - dist. p. 48.
- 2.
St.
1 45. 2 31 ½· 3 15.
CHAP. XVIII. To protract any number of Angles or degrees taken by the Peractor, Theodelite or Circumferentor, at seuerall obseruations.
LEt the Angles or degrees taken be those expressed in the former Chapter, and let it bee required to protract the same, whereby the quantitie of each seuerall distance there sought for, may appeare. First, on your paper prouided, draw a right line at pleasure, as F G. in the former Diagram; then laying your Field-Booke before you, with the former obseruations, make choyse of any point in the line F G. to represent your first Stacion, as at D. then applying the Scale of your Protractor to that line, lay downe your stacionary distance 48. Perches from D. to E. representing the place of your second stacion; and placing your Protractor with the Centre on the point D. (the Semicircle vpwards) turne it about thereon, till that degree on the Protractor which was taken from the first to the second stacion (which in this example is 120.) lye precisely on the line F.G. and then looke in your Field-Booke for the degrees obserued at your first stacion, which were 32 ½· 21. and 9 ¼· (for the fourth 120. that is supposed alwayes to fall on the first drawn line F G.) and against those seuerall degrees on the limbe of your Protractor, by the edge thereof with your protracting pinne, make seueral pricks or points, as at H. I. and K. then by the point D. and those three seuerall pricks with the scale of your Protractor, and protracting pinne, draw out at length the lines D H. D I. and D K. so haue you finished your first obseruations; then place your Protractor on the point E. in all respects as before, at D. and there mark the degrees of your second obseruations, as 45. 31½ and 15. as before, at the points L. M. and N. whereby, and by the point E. draw out at length the [Page 148]lines E L. E M. and E N. till they intersect the three former lines, drawne from D. as in the points A.B. and C. by which intersections from point to point, draw the lines A B. and B C. So haue you finished your Protraction. And by applying the scale of your Protractor, (whereby the stacionary distance was laide downe) to any line or distance, the seuerall quantities will appeare to be as they are expressed in the Diagram, on the seuerall lines therof, as was required.
But here is to be noted, that if those former obseruations were made and taken by the Theodelite, then this Protraction is to be made and layde downe by the Protractor belonging to the Theodelite, being diuided into 360. degrees, as is before mentioned, which is to be performed in all respects, according to the rules and instructions before deliuered.
CHAP. XIX. To take any accessible altitude by the Circumferentor or Plaine Table with the diuided sights.
SVppose A B. to bee a perpendicular
height, & let it bee required from C. (whose distance to B. is accessible) to know the altitude thereof. First, place your Instrument at C. precisely horizontall, by helpe of the plummet hanging on the shorter sight; whereunto great care must bee had aswell in this, as all other works of this nature, or maine errors will arise; then moue the vane on the longer sight vpwards and downwards, till through the hole therein, and by the pinnes head on the toppe of the shorter sight, you espie the Summitie of the giuen height at A. where note the equall diuisions cut on the longer sight by the edge of the vane, which suppose to be 40. then measure the distance from C. to B. which let be 110. And the same proportion which 40. the part cut hath to 100. the side of the square proiected on the Instrument, the like hath the altitude A. B. to the measured distance, 110. Wherefore multiply 110. the distance by 40. the parts cut, and diuide the Product by 100. the quotus will be 44. the altitude required.
But it hapneth oftentimes that the altitude required is of that height, that you cannot produce the vane low inough, to see the summitie of the height, as before. In which case you are to vse the Index to be placed on the wyerpinne in the edge of the shorter sight, and turning it vp and downe close by the right edge of the Instrument, till through the sight thereof, and by the wyer pinne you espie the summytie of the giuen height, and then note the parts cut on the same edge of the Instrument, by the fiduciall edge of the Index. For the same proportion which the parts cut, beare to 60. (the imagined parts on the edge of the shorter sight) the like hath the measured distance [Page 149]to the altitude required. Wherefore, multiply the same measured distance by 60. and diuide the Product by the parts cut, the quotus sheweth your demand. And if you desire to know the Visuall or Hypothenusall line, multiply the measured distance by the parts cut on the edge of the Index, and diuide the Product by the parts cut on the edge of the Instrument; the quotus sheweth what you desire. For what proportion the parts cut on the edge of the Instrument, beare to those cut on the edge of the Index, the like doth the measured distance to the Visuall line.
And here is to be noted, that of this later worke the plaine Table hath no vse; and therefore of all other Instruments most vnfit for these purposes of Altitudes and Profuncities, without helpe of the quadrant specified in the description thereof, CHAP. 3.
Or with the Circumferentor, by Protraction, thus.
PLace your Instrument at C. as before, and there obserue the quantitie of the Angle of altitude, which being gotten, protract and lay downe the same as hath beene taught, and on the base line from C. to B. lay downe the measured distance 110. at the end whereof, as on the point B. either by the 6. PROBLEME of the second Booke, or with helpe of your Protractor erect a perpendicular line as A B. to cut the other side of the protracted Angle, as in A. and with applying your Scale thereunto, the altitude appeareth, as was required.
CHAP. XX. To take any accessible altitude diuers wayes, by the Peractor and the Quadrant thereof.
SVppose M N. to be a perpendicular height giuen; and let it be required from O. to find the Altitude thereof. Place your Instrument precisely Horizontall at O. as before is taught, then mooue your Quadrant vp and downe, till through the small round hole, in the end of the sight towards you at B. on the Quadrant (as it is described in the 5. Chapter) and by the pin in the great round hole of the other end at A. you espie the Summytie M. of the giuen height, where letting your quadrant rest, measure the distance O N. which suppose to be 48. then looke on the side C D. of the quadrant, for the 48. line, reckoning from the Centre P. and passing downe by that line to the edge of the handle or Index (which suppose to stand now on the line P F. drawne from the Centre of the quadrant) [Page 150]note what line (passing from the other side A.B. of the quadrant) the former line 48. meeteth and intersecteth on the edge of the Index, supposed as before the line P. F. and you shall find it somewhat more then 34. wherefore I conclude, that the altitude A B. is so much: and noting what part of the Index is there cut, you shall find it somewhat more then 58 4/5· the length of the visuall line, or hipothenusall O. M. Where is to be noted, that in all workes wrought with this quadrant, the side thereof A. B. representeth the perpendicular height, the side C. D. the horizontall distance, and the Index or handle, the hipothenusall or visuall line.
Or Sinically thus.
HAuing placed your Instrument as before at O. By the degrees on the limbe of the quadrant, obserue the angle of altitude MON. 35. degrees 1′9. and measuring the distance ON. 48. as before (by the helpe of Pitiscus, or any other Canon) Reason thus: If ON. the radius 100000. yeeld 48. the measured distance; what M N. 70848. the tangent of the Angle MON. 35. degrees, 1′9. and multiplying the tangent 70848. by 48. the measured distance, you shall produce 3400704. which parted by the radius 100000. quoteth 34 704/100000 or in lesser termes, 34 22/3125· the altitude M N required.
Againe, if O N. the radius 100000. giues 48. the measured distance, what O M. 122554. the secant of the Angle M O N. 35. degrees, 1′9. and multiplying the same secant 122554. by 48. and parting the Product by the radius 100000. you haue 58 [...]2592/100000· or in lesser termes, 58 2581/3125· being somewhat more then 4/5· the length of the visuall or hipothenusall line O.M. as before.
Or by Protraction, thus.
PLacing your Instrument at O. as before, obserue the Angle of altitude M O N. and measure the distance from O. to N. And then proceed to protracting thereof, as is taught in the later part of the last Chapter before going; and the worke is finished.
CHAP. XXI. To find out any inaccessible height by the Peractor, Theodelite, or Circumferentor.
IT may oftentimes happen that inaccessible heights may be required, when by reason of waters, trenches, danger of shot, or many other impediments, a man cannot approch to the Base of the altitude required; yet of necessity to bee had and knowne, which to performe, in a most absolute and exact manner, worke thus.
Suppose B C. to be a perpendicular height giuen, vnto the Base, whereof C. (by some impediment) you may not approch neerer then D. yet the altitude is required; wherefore place your Instrument at D. precisely horizontall, and obserue the Angle of altitude, as is before taught, which let be 53. degrees, 8′. then looking backwards, make choyse in a right line from C. by D. of a second station, which let be A. and measure the distance from D. to A. which suppose 40. then at A. place your Instrument as before, and, likewise obserue there the Angle of altitude, which suppose 36. degrees, 2′. so is your Instrumentall worke already finished. Then repayring to your Canon of Triangles, finde the complements of the Tangents of those two Angles taken, which of 53. degrees, 8′. the angle first obserued, is 74991. and of 36. degrees, 2′. the Angle of your last obseruation, is 137470. betweene which two complements take the difference, by deducting the lesser from the greater, which will bee 62479. and then (for as much as the same proportion which the difference of the complements 62479. beareth to the radius 100000. the like hath the measured distance betweene your two stations D A. 40. and the required altitude) multiply the radius 100000. by 40. [Page 152]the measured distance, the Product is 4000000. which parted by 62479. the difference of the complements quoteth 64 1244/1000300· for the required altitude.
And if it bee required, to haue the length of a scaling ladder to extend from D. to B. or the length of the visuall line A B. or the inaccessible distance betweene D. and C. by respectiue obseruation of what was taught in the last Chapter, they are easily resolued.
To performe the same by Protraction.
HAuing taken both the Angles of altitude, and measured the stationary distance A D. 40. as before; protract the same thus; Draw a right line out at length, as E F. in the former Diagram, then on any point thereof, as at D. protract the first Angle taken, 53. degrees, 8′. (as before taught) then from D. towards E. lay downe the stationary distance 40. to end in A. on which point A. protract your other Angle 36. degrees, 2'. and continue forth the vpper sides of those two Angles, till they interfect each other as in B. from which point B. by the 6. of the second, or by helpe of your Protractor, let fall a perpendicular, to cut the line E F. as B C. in C. which line B C. shall represent the giuen altitude, the height whereof, and the seueall lengthes and distances in the whole worke contained, is speedily had by applying thereunto the same scale, whereby your measured distance was laid downe.
And here is to be noted, that in all workes of this kind, it is very requisite to take your stationary distance as large as conueniently you may, for that otherwise by reason of the acuitie of the Angle, as here of A B D. you shall hardly discerne the true point of intersection by the lines B A. and B D. whereby, or from whence you may precisely let fall the perpendicular B C. as before in his due place: by neglect whereof mayne errors may insue.
Wherefore a most excellent, absolute and exact course is that in the former part of this Chapter, for the performance of all manner of conclusions of this kind, and to be preferred before all others.
And here now might I much inlarge this worke, by inserting seuerall Propositions for the taking and finding out of distances in heights, with the mensuration of profundities diuers wayes; all which and infinite other conclusions are fully included within the limits of these few former instructions, and with diligent obseruation and practice thereof may bee well vnderstood and performed; for whoso can take one height artificially, may performe another, and by deducting the one from the other, may decerne the difference: and he that can skilfully take an altitude, by the same rule may performe a profunditie, being the one a direct conuersion of the other, without alteration or any difference, either in the Theoricall ground, or practike operation thereof: wherefore to make great shewes, or accumulation of needelesse varieties to one and [Page 153]the same purpose, were but expence of time vnto my selfe, and cause of confusion to the learner, seeing fewest Precepts (so effectuall) are fittest, aswell for apprehension as retention.
But before I passe further, let this bee remembred, that in all the former obseruations in taking of heights: the height of your Instrument is alwayes to be added to the altitude taken.
CHAP. XXII. To take the plot of a Field at one stacion; taken in any part thereof, from whence you may view all the Angles, and measuring from the stacion to euery Angle.
SVppose ABCDEFG. to be a Field, whereof it is required to take the Plot. First, cause papers or other markes to be placed directly in euery Angle; and then make choise of some such conuenient place within the same, as from whence you may best view the seuerall Angles thereof, and there as at X. Plant your Instrument; if it be the Peractor, Theodelite, or Plaine-Table, fasten the Instrument to the staffe with the scrue-pinne, that it stirre not till your worke be finished, the needle standing on the Meridian line of the Carde, if the Circumferentor, that care is already taken; but admit the Peractor; then direct your sight, by turning the Index to any one Angle at your pleasure, as first to A. and obserue the degree there cut by the edge of your Index, which let be 10. and with your Chaine measure from your station to that Angle, which suppose 30. then direct your sight to B. and performe the like, and so to C.D. E.F. and G. till you haue finished; still entering as you passe your seuerell [Page 154]obseruations, aswell of degrees as lines into your Field-booke, as was formerly taught in Chap. 10. which when you haue finished shall appeare to bee thus.
Which being layde before you, shall most speedily and exactly be protracted and layde downe, as is taught in the next.
| d | o | 1 |
| 10 | 30 | — |
| 26 | 33 | 2 |
| 39 ½ | 41 | — |
| 51 | 36 | — |
| 66 | 47 | 5 |
| 81 | 18 | 8 |
| 105 ½ | 41 | 5 |
CHAP. XXIII. To protract and lay downe the obseruations made in the last Chapter, or any other taken in the like sort.
TAke a cleane sheete of paper, and draw a right line thereon at pleasure as P Q. in the Diagram of the last Chapter, representing the meridian Line; and laying your Field-booke before you with the former obseruations; First, place the Centre of your Protractor (the Semicircle vpwards) on any point of the line as on X. with the Meridian line of the Protractor directly ouer that on the paper; where keeping your Protractor fixed, note all the degrees vnder 60. taken in your Field-booke, as 10.26.39 ½· and 51. against which seuerall degrees on the limbe of your Protractor by the edge thereof, make seuerall prickes or points with your protracting pinne, as at H.I.K. and L. Then considering that those degrees are vnder 60. and therefore to lye on the East side of your plot or meridian line P Q. lay the edge of the scale of your Protractor on the Centre X. and euery of those points H.I.K. and L. and draw from the Centre X. vnder the meridian line P Q. the seuerall lines to A.B.C. and D. laying them downe of the seuerall lengthes, belonging to them, according to your Field-booke, as X A. 30 0 X B. 33 0· 2 1· X C. 41 0· and X D. 36 0· making prickes or points at the end of euery seuerall length, as at A.B.C. and D. and then from point to point, draw the lines A B. B C. and C D. so haue you finished the worke on the East side of your Meridian. Then place againe your Protractor in all respects as before, and note all the other degrees being aboue 60. as 66.81. and 105 ½ which by the edge of the Semicircle marke out as before, as at M. N. and O. and seeing they belong to the West side of your worke, draw your lines from the Centre X. vpwards towards those three seuerall points, laying them downe with their seuerall lengthes obserued in your booke, as X E. 47 0· 5 1· X F. 18 0· 8 1· and X G. 41 1· 5 1· making points as before at the end of euery length; and lastly from point to point draw the lines A G. G F. F E. and E D. So shall you inclose the Figure A B C D E F G. with equall Angles and proportional lines to the measured field as was required.
CHAP. XXIIII. To take the plot of any Field at one stacion in any one Angle thereof, from whence may bee seene all the other Angles of the same Field; and measuring from the station to euery Angle.
SVppose A B C D E F G. to be a field, the plat whereof is required to be taken. First, cause whites or markes to be placed directly in euery Angle, then make choyse of the most conuenient Angle, from whence you may best view all the rest, as at A. where place your Instrument as before is taught, and directing your sight to one of the next Angles on eyther hand as to B. obserue the degrees cut by the edge of the Index, which let bee 24 ½· degrees and measure that line A B. 33 0· 4 1· then turning your Index to the next, as to C. make the like obseruation of 13 ¼· degrees. and measure the length A C. 55 0·2 1· and in like manner proceed to the rest, as to D.E.F. and G. still expressing in your Field-booke your seuerall Angles and Lines as before is taught, which hauing finished will thus appeare.
And then protract, and lay downe the same in all respects, according to the instructions of the last Chapter. The order whereof appeareth by the Diagram,
| d | o | 1 |
| 24 ½ | 33 | 4 |
| 13 ¼ | 55 | 2 |
| 6 | 47 | — |
| 116 | 69 | 2 |
| 104 | 65 | 6 |
| 95 ½ | 41 | 6 |
CHAP. XXV. To take the Plot of a Field at one station in any Angle, from whence the rest may be seene, and by measuring the sides of the Perimeter.
SVppose A B C D E F G. be a Field to be plotted. First; set vp markes as before, and chuse an Angle fittest for your purpose, from whence you may see all the rest, and there plant your Instrument as at A. then direct your sight to one of the next Angles, on eyther hand, as to B. and note the degree there cut, 24 ½· and measure the length of that line A B. 33 0· 4 1· then direct your sight to C. and note the degree there cut 13 ¼· and measure the line from the Angle B. to the Angle C. 33 0· and in like manner worke forwards to D.E. and F. and then note (hauing finished at F.) that you haue yet remayning two lines to measure, namely, F G. and G A. and but one degree to bee taken, as from A. to G. (The reason whereof dependeth on THEOR. 74.1.) wherefore measure the line F G. 33 0· 5 1· and expresse the same in your Booke without any degree; and lastly, directing your sight to G. obserue the degree cutte 95 ½· and measuring the line A G. place the length thereof in your Booke 41 0· 6 1· against the last degree taken. So haue you finished your Field-worke, and your obseruations stand thus.
Which are to be protracted and layde downe as is taught in the next.
| d | o | 1 |
| 24 ½ | 33 | 4 |
| 13 ¼ | 33 | — |
| 6 | 21 | — |
| 116 | 36 | 5 |
| 104 | 41 | 8 |
| — | 33 | 5 |
| 95 ½ | 41 | 6 |
CHAP. XXVI. To protract and lay downe the obseruations had, according to the worke in the last Chapter, or any other taken by the like meanes.
THe seuerall degrees and lengthes so had and taken as before, and expressed in your Field booke, being layde before you, worke thus. First, according to your degrees taken, and as is taught in Chap. 23. draw out at length your seuerall lines A B. A C. A D. A E. A F. and A G. as in the former Diagram, then opening your Compasse on your Scale, take therein the first length 33 0· 4 1· at which distance with one foote in A. strike with the other an Arch through the line A B. cutting the same in B. then take your second length 33 0· at which distance with one foot in B. crosse the line A C. in C. and draw the line B C. then take the third length 21 0· and with one foote in C. crosse the line A D. in D. and draw the line C D. and in this manner worke forwards, laying downe euery length, and drawing each line till you haue inclosed the Figure A B C D E F G. which shall be a like Figure to the measured Field.
CHAP. XXVII. To take the plot of a Field at two stations, where all the Angles cannot be seene at one, and measuring as in the 22. Chapter.
SVppose this Figure be a Field to bee plotted, which lyeth in such sort, as from no one place all the Angles thereof can be seene. In such case make choyse of a place for your first station, where may be viewed as many Angles thereof as possibly you may, which let be M. where you may see the seuerall [Page 152]Angles at A.B.C.D.E. and F. then plant your Instrument in M. and there obserue all those Angles, and measure the seuerall lines, beginning from M. to A. and ending from M. to F. as is taught in the 22. Chapter, so haue you finished the worke of your first station. Then (before you remoue your Instrument) make choice of some other conuenient place for your second station, from whence you may see all the other Angles not formerly seene, as those at I.H. and G. which let be N. vnto which place direct your sight, and obserue the degree cut 65. then measure the stationary distance M. N. 40 0· and leauing a marke at M. remoue now your Instrument to N. where place it precisely as it stood at M. with helpe of your needle and backe sight, then obserue your seuerall degrees, and measure the seuerall lengthes from N. your second station to I.H. and G. as before, and your Field-worke is finished. So as you remember alwayes to expresse your obseruations in your Field-booke, which shall thus appeare.
To be protracted and layde downe as is taught in the next.
| d | o | 1 |
| 83 ¾· | 29 | — |
| 109 ½· | 19 | 2 |
| 1. Sta. 10 ¼· | 26 | 8 |
| 28 ½· | 30 | 8 |
| 41 ¼· | 31 | 5 |
| 55 ½· | 16 | 4 |
| sta. dist. 65 | 40 | — |
| 74 ¼· | 26 | — |
| 2. Sta. 46 ½· | 29 | — |
| 33 ¼· | 27 | 5 |
CHAP. XXVIII. To protract and lay down any obseruations taken, according to the work of the last chapter.
ON a fayre sheete of paper draw a right line at length in any conuenient place thereof, as K.L. in the Diagram of the last Chapter, which line is not to be taken as before for a meridian line, but supposed to be drawn according to the degree taken from the first to the second station; and therefore call it the stationarie line, and thereon lay downe your stationary distance 40 0· as from M. to N. which two points shall represent your two stations, then on M. the point of your first station, place the centre of your Protractor (the Semicircle vpwards) turning it about till the degree 65. on the limbe of your Protractor (being the degree taken from the first to the second stacion) cut precisely the stacionary line drawne, and there keeping it firme and immoueable, marke out the seuerall degrees of your first stacion; according to your Field-booke; and so worke on in all respects as is taught in Chap. 23. which effected, remoue your Protractor to N. the point of your second station, [Page 153]where placed precisely, as at the first, worke forwards with the degrees and lines of your second station, as before; and so haue you finished.
CHAP. XXIX. To take the Plot of a Field at diuers stacions in diuers Angles, where all cannot be seene from one, and to measure as in the 24. Chapter.
SVppose this Figure be a Field to be plotted, whose angles cannot bee seene from any one Angle thereof, wherefore imagine you are now standing in the Angle A. from whence you view and consider what Angles may there bee conueniently seene and taken, which you find to be those at B. C. D. and E. then directly in those Angles cause markes to bee placed, and planting your Instrument as before is taught in A. direct your sight first to B. then to C. after to D. and lastly to E. noting the seuerall degrees cut towards each seuerall Angle, and measuring as in the 24. Chapter, from your station to euery of those Angles seuerally, and your worke of that station is finished. Now for that you ended your last worke at E. remoue your Instrument to that Angle, and there plant it precisely, as at the first stacion, vsing both your needle and backe-sight for your helpe therein; And here consider what Angles from hence may be perfectly seene and taken, which on view had, you finde to be all the residue not formerly taken, as F.G.H.I.K.L. and M. wherefore hauing your markes placed, take your seuerall obseruations and measure your seueral lines to euery of these Angles, as to those of the first station, taking them in order as they lye, and you haue finished. But suppose at this station, you could haue seene onely those Angles at F. G. H. and I. [Page 154]then here must you haue finished those, and remoued your Instrument to I. for a third station, and there to haue performed the rest, or as many as there you might, and if any remaining, to take a fourth, and a fift station, &c. till you haue finished, wherein many words are needlesse, the matter being apparant. Your obseruations of this worke are these.
A generall Note. And let it be noted, that where for breuitie sake in mine instructions, aswell here as else where, I omit to expresse particularly, the seuerall degrees and lengthes obserued and taken betweene each station and the seuerall Angles, (which would be no lesse tedious, then troublesome in breeding confusion) I obserue a due order in the placing of these obseruations of degrees and angles, according to the order of the letters about the plot, as the first degree and length 116 ½· 28 0· 4 1· belong to the first line A B. the second to the second A C. and the like of all the rest, which are to bee protracted and layde downe, as appeareth in the next.
| d | o | 1 |
| 116 ½ | 28 | 4 |
| 101 | 41 | 8 |
| 85 ¼ | 55 | 8 |
| 74 ½ | 41 | 8 |
| 86 | 22 | — |
| 75 ½ | 32 | 8 |
| 69 1/ [...] | 22 | 6 |
| 59 | 31 | 5 |
| 45 ¼ | 42 | 5 |
| 36 ½ | 44 | 4 |
| 23 | 30 | 4 |
CHAP. XXX. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
THe speediest and exactest course for protracting works of this nature, consisting of diuers stations is thus. First, draw a right line at length on your paper, to represent the meridian line, as N O. in the last Diagram, whereon placing your Protractor, worke in all respects as is taught in Chap. 23. for the obseruations of your first station; so shall you finish as much of this worke as is included by the lines A B. B C. C D. and D E. the worke of your first station; then by the point E. where you left, draw another meridian line as P Q. which (by PROB. 3.2.) make Parallel to the first line, N O. and then on the point E. place your Protractor in all respects, as at the first, & worke with the rest as before, whereby you shall finish the worke of your second station, and perfect the Figure A B C D E F G H I K L M. with equall Angles, and proportionall lines to the measured field. And if there were more stations to be vsed in the Field-worke, then at the point where the worke of each station endeth, you are to draw another meridian line parallel to the rest. Or before you begin your protraction, you may draw diuers parallell lines on your paper, representing so many meridians, and by helpe of your parallel diuisions, placed at eyther end of the Scale of your Protractor, you shall on any point falling either vpon or besides those Meridians place your Protractor parallel as you please. And this kind of protraction may be vsed in stead of that taught in the 28. Chapter, as the better, though eyther will serue, and both tending to one end.
[Page 155] Hitherto haue I taught after a perfect and exact manner the mensuration of seuerals (by diuers meanes) where one field or close onely is to bee taken by it selfe: But if many seuerals (as a whole Lordship or Mannor) were to bee measured and plotted together: I hold not these former courses the fittest: but rather those which shall bee hereafter taught. But first I will deliuer some few directions and examples, for the dimension of seuerals after another order, by intersection of lines at seuerall stacions as followeth.
CHAP. XXXI. To take the plot of any field at two stations, so as all the Angles may be seene from both stacions, by measuring onely the stationarie distance.
SVppose A B C D E F G. be a Field, the plot whereof is required to be taken. First, make choice of two such conuenient places for your Stations, as from whence you may see all the Angles about the Field; with these further considerations, that the distance betweene your stations bee of conuenient length, the longer the better, that they lye towards the middle of the field; and that neyther of them lye interposed in a right line betweene the other, and any Angle of the Field; but to be chosen with such discretion, as all lines drawne from eyther station to the seuerall Angles, may intersect ech other with as large angles as you may, which let be the two points H. and I. and causing marks to be placed in euery Angle, plant your Instrument at H. as is before taught, and directing your sight to A. obserue the degree there cut; and the like to B. C. D. E. F. and G. and also to I. the second station, [Page 156]then take vp your Instrument, leauing a marke at H. from whence measure the stationarie distance to I. where placing your Instrument precisely as at H. obserue likewise all the degrees cut by your Index, directed to each seuerall Angle as before: Of all which seuerall obseruations keepe notice in your Field-booke as hath beene often mentioned; wherewith on a cleane sheete of paper by the directions of the 18. and 30. Chapters, the plot thereof is speedily protracted, and your businesse fully finished.
And here note the accuitie of diuers of those Angles in the Diagram, caused by the intersection of the pricked lines, notwithstanding all care had therein; and what inconueniences may hereby grow, without good regard; and yet are these Angles vsually drawne by many, who make a poore shift therewith.
CHAP. XXXII. To take the plot of any Field remote from you at two stations, when eyther by opposition you may not, or some other impediment you cannot come into the same.
SVppose A B C D E
F. bee a Fielde, whereinto by no meanes you can bee suffered to enter, yet of necessity must the plot thereof be had. In such case make choise of any two places, eyther neere hand, or further off, all is one; so from thence you may well decerne the seuerall Angles of the same Field; and let your stationary distance bee the full length of the Field, at the least if possibly you may; which two places let be G. and H. First plant your Instrument at G. and by directing your sight in order one after other to A.B.C.D.E. and F. the seuerall Angles of the Field, obserue the seuerall degrees there cut, as is before taught; then turne your sight to H. your second station, and note the degree there cut; which done, take vp your Instrument, leauing a marke at G. and measure from thence to H. your stationary distance, and there plant your Instrument in all respects as before, and make the like obseruations to all the seuerall Angles of the Field, as formerly at G. So haue you finished your Field-worke, which is to be protracted and layde downe according to your Instructions of the 18. Chapter.
And here note, by reason of the length of the stationary distance, how [Page 157]excellently the lines issuing from thence, intersect each other, which of necessitie makes the conclusion absolute.
CHAP. XXXIII. To take the Plot of any Field by making obseruation at euery Angle, and measuring onely one line, but no part of the perimeter.
SVppose A B C D E. be a
Field, & it is required to plot the same as before. First, place your Instrument at A. and obserue some notable mark towards the middle of the Field; if there be none such, cause one to bee so placed as it may bee seene from euery Angle about the Field, which let be F. then direct your sight to F. and note the degree there cut 109. and (your Instrument remaining) measure the distance from A to F. 21 0· 7 1 ½· (which shall bee all the mensuration you shall need to vse in this worke) then direct your sight to B. obseruing there the degree cut 118 ½· then remoue your Instrument to the Angle at B. and there first direct your sight to F. taking that degree 10 ½· and then to C. the next Angle 88 ½· which done, remone your Instrument to the Angle at C. and there as before, first direct your sight to F. taking that degree 106. and then to D. the next angle 74 ½· and so proceed from angle to angle, first taking your sight to F. and then to the next angle, till you haue finished, Or take here the right degrees and protract the opposite. wherein by the way obserue this course; when you take your degrees to F. reckon those on the North end of your needle, and when you expresse them in your Booke, make in the margent against them a pricke, or some other marke, whereby it may bee apparant, when you come to Protraction which are those degrees, from the rest, which are to bee reckoned on the South end of the Needle, after the vsuall manner, and belong to the perimeter: so haue you very exactly finished your field-worke, and your obseruations are these.
| d | o | 1 |
| 109 | 21 | 7 ½ |
| 118 ½ | ||
| 10 ½ | ||
| 88 ½ | ||
| 106 | ||
| 74 ½ | ||
| 93 | ||
| 49 | ||
| 75 ½ | ||
| 35 ¼ |
Which is to be protracted, as is taught in the next.
CHAP. XXXIIII. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
THis kinde of protraction is somewhat different from all the rest formerly taught, wherefore obserue it thus. First, draw diuers parallel lines ouer all your paper, of conuenient distance one from another (not exceeding the bredth of the Scale of your Protractor) which shall represent the Meridians; then with your obseruations layde before you, take a point in any conuenient place of your paper, whether vpon or besides any parallel line, it is not materiall, as at A. in the last Diagram, then thereon place the Centre of your Protractor (the Semicircle vpwards) turning it about on your protracting pinne, till you find the match or opposite parallel diuisions on eyther end of the scale, to lye eyther precisely vpon any one line, or equally distant ouer or beneath the same; then looke in your booke what are the first two degrees, which is 109. and 118½· against which two degrees, by the edge of your Protractors Semicircle, make seuerall prickes on your paper, whereby, and by the point A. draw the two right lines A F. and A B. out at length, and from A. to F. lay downe your measured length 21 0· 7 1 ½· then on the point F. place the Centre of your Protractor precisely as before, by helpe of your diuisions on the end of the Scale, and find in your Field-booke what degrees you haue (more then the first already done) marked with points or other markes in the margent (as was taught in the Field-worke) which are 10 ½· 106.93. & 75 ½· against all which degrees on your Protractor make seuerall prickes as before, whereby, and by the point F. draw out at length the seuerall lines F B. F C. F D. and F E. remembring (as you are taught in Chap. 23.) alwayes to draw those lines proceeding from all the degrees vnder 60. downwards or towards you from the point F. and those aboue 60. vpwards from the point F. Then note, that by drawing out the line F B. you haue intersected the line A B. (formerly drawne) in the point B. on which point now place your Protractor as before, and finde in your Booke the second degree vnmarked in the margent (for the first A B. is alreadie done, and likewise all those which are marked) which is 88 ½· against which, on your Protractor make a pricke, and thereby, and by the point B. draw the line B C. till it intersect the line F C. in C. then place your Protractor on the point C. as before, and finde in your booke the next degree vnmarked, which is 74 ½· and against that degree on your Protractor make a pricke, whereby, and by the point C. draw out at length the line C D. to cut the line F D. in D. and in like manner proceed with the rest; so shall you include the Figure A B C D E. like vnto the measured Field.
Where note, that now the points in the margent serue you to purpose, at an instant distinguishing those degrees taken at each angle towards the point F. from the others vnmarked, representing those of the Perimeter.
This kind of worke well handled, is very exact and artificiall.
CHAP. XXXV. To take the Plot of any Field at diuers stations, measuring onely the stacionary distances.
SVppose this irregular Figure to represent a Field, whose plot is to be taken. First, cause markes to bee placed in euery Angle thereof, and then (remembring those cautions and considerations in this behalfe deliuered in Chap. 31.) make choyce of your first station, which let be at N. where planting your Instrument, direct your sight to as many Angles seuerally as lyes there within your view, as to A.B.C. and D. and take obseruation of the seueral degrees there cut, and then (before you remoue your Instrument) make choyce of your second station, from whence you may not onely see all those former angles, but as many more as possibly you may, which let be at O. then directing your sight to O. obserue the degree there cut, and taking vp your Instrument, leaue there a marke, and measure the distance from thence to O. where againe plant your Instrument as before at N. and then directing your sight to those former Angles A. B. C. and D. note seuerally what degrees are cut, and you haue finished for those Angles. Now consider what other Angles you can here espie as those at E. M. L. and K. and thither likewise direct your sight seuerally, and make your seuerall obseruations as before, which done (your Instrument remayning) make choyce of your third station, (from whence you may not onely see those Angles at E. M. L. and K. but as many other vnfinished Angles as you may) which let be at P. then directing your sight to P. obserue your degree thither, leaue a marke at O. take vp your Instrument, and measure from O. to P. where againe plant your [Page 160]Instrument, and make your obseruations of those former Angles E. M. L. and K. so haue you thus farre wrought. And now againe considering what other Angles you can here espie, you shall finde within your view F. G. H. and I. being all the residue vnfinished; wherefore here also make your seuerall obseruations of these last Angles, and choyse of your last station, as at Q. whither directing your sight, and making obseruation, take vp your Instrument, and with a marke at P. measure the distance to Q. where lastly plant your Instrument, and making your obseruations of those last seuerall Angles at F. G. H. and I. as before; your Field-worke is finished. And by the directions of the 18. and 30. Chapters, you may protract the same accordingly.
CHAP. XXXVI. To take the Plot of a Forrest, or any spacious Common or Waste, of whatsoeuer quantitie, by the Plaine Table, on one sheete of paper without altering thereof.
SVppose A. B. C. D. E. F. G. H. I. to bee some large irregular ground, and it is required to plot the same. First, a sheete of paper being placed on your Plaine Table after the vsuall manner, Take a point at all aduenture about the middle thereof as at Q. in the Diagram, and thereon describe foure concentricke [Page 161]Circles of conuenient distance, the vttermost extending neere the bredth of the Table, whereby you shall produce three seuerall circular spaces, which are thus to bee employed; In the vttermost, expresse the seuerall lengthes of all such lines, which in passing forward in your worke shall bee found declining or extending outwards, as the lines D E. G H. and I A. which lines alwayes issue from an Angle greater then a Semicircle, the inclination whereof causeth the declination of the line: the second which is the middlemost space, let serue for the number of each line; and in the third and innermost space, expresse the seuerall lengthes of all such lines, as in passing forwardes shall be found inclining or extending inwards, as all the other lines in the figure not formerly mentioned.
And these thus vnderstood, proceede forwards in this manner. Plant your Instrument after the vsuall order in the Angle, where you purpose to begin, which let be A. and let the Centre Q. on your Table, alwaies in your whole worke represent your place of standing, then lay your Ruler on the Centre Q. and turning it about, thereon direct your sight to I. and by the edge of the Ruler, draw the line Q. K. (which shall be your inclosing line) extending it to the vttermost Circle, and in the middle space, on that line (which is to beare no number) place the Ciphero. then keeping still the Centre point Q. turne about your Ruler, and direct your sight to B. and by the edge of your Ruler draw the line Q N. and in the middle space on that line expresse the Figure 1. the number of the first line you goe on, so haue you on the Table the Angle K Q N. equall to the Angle I A B. in the field, then take vp your Instrument, and measure the line A B.29 0· 2 1 ½· which length expresse on your first line Q N. and for as much as the line is inclining, place the same in the innermost space, as in the Diagram; then place your Instrument precisely at B. and turne it about on the staffe (your Ruler lying on your first line Q N. till you finde the same line to lye parallel with the line A B. of the Field, hauing Q. towards you, and N. towards the Angle A. (for the Centre Q. in euery Angle must represent the point of the same Angle) then your Table there fixed, direct your sight to C. and draw your line which shall bee the line Q R. and expresse thereon in the middle space the figure 2. for the second line, so haue you also the Angle N Q R. on the Table, equall to the second Angle A B C. in the Field: then take vp your Instrument, and measure the line B C. 12 0· 5 1· which place on the second line Q R. in the innermost space, and planting your Instrument at C. direct your last line Q R. towards B. as you did, the first towards A. and then take the quantitie of this Angle, as before, and you shall haue the Angle R Q L. on your Table, equall to B C D. in the Field. And thus proceede from angle to angle, making alwayes the last side of the last Angle to bee the first of the next, with all their points concurring in the Centre Q. and measuring and expressing the seuerall lengthes of each line, as before; you shall obtain the quantitie of euery Angle, and the length of euery line throughout the whole worke: whereby you may speedily protract and lay downe the same, as is taught in the next.
But in the meane space it is to be noted, that if in your former worke it happen at any time, that one line fall directly on any other, formerly drawn on your Table (as in the former Diagram, the seuenth line Q R. falleth on [Page 162]the second line formerly drawne) then in such case expresse the number of the same last line in the middle space, where the first is numbred, with a stroke betweene them; but place the numbers, expressing the length of the same last line, without the vttermost Circle, if it be declining (as in the former example 22 0· 2 1·) and within the innermost, if it be inclining.
This kind of mensuration of spacious works with the Plaine Table (which likewise may be wrought with any other Instrument according to Chapter 15.) is wonderfull necessary both for speed and exactnesse, if artificially handled: but if it be required to haue notice taken of the seuerall lands and grounds abutting and confining hereon, you must then haue prouided a Field-booke for that purpose.
CHAP. XXXVII. To protract and lay downe any obseruations taken, according to the worke of the last Chapter.
HAuing before you the paper of your Field-worke as you wrought on the Table: If you imagine that one sheete of paper will not serue the turne, you may with mouth-glue, lay as many together as you please; and then (considering which way your worke will extend) draw a right line accordingly on your paper, whereon, with your Scale and Compasse, lay down the length of your first line Q N. 29 0· 2 1 ½· as the line A B. in the Diagram of the former Chapter, and then on the end B. of that line, by PROB. 8.2. protract an Angle, equall to the Angle N Q R. as the Angle A B. C. and on the side B C. of that Angle, place by your Scale the length of your second line R Q. 12 0· 5 1· from B. to C. and on the point C. where your last length ended, protract another Angle, equall to the Angle R Q L. your third Angle taken, as the Angle B C D. and on the side C D. extended, place the length of your third line L Q. 14 0· 9 1· from C. to D. And so proceede from Angle to Angle, protracting your Angles equal (and in order by the number of lines) to those answerable in your concentricke Circles, and laying downe duly by your Scale on each line, the length thereunto belonging, you shall produce the Figure A B C D E F G H I. like vnto the measured quantitie.
And hauing thus finished; if you doubt or make question, whether you haue wrought exactly or not, and desire to be resolued therein, make approbation thus. Collect the quantitie of all your Angles in your whole worke, and adding them together, note the totall thereof; which in this former worke is 1260. then multiply 180. (the number of degrees in a Semicircle) by a number lesse by 2. then the number of Angles in your worke, If your degrees be those of 120. as obserued by the Peractor, or Circūferentor, then in stead of 180. you must take 60. which here is 7. (for the number of Angles is 9. as appeareth by the line O Q. which sheweth the number of the last line, and consequently of the Angles) and if the product of that multiplication agree with the former totall; then by THEOR. 74.1. you haue done rightly, otherwise not; as the Product of 180. by 7. is 1260. agreeing with your first number; and therefore may you confidently affirme to haue wrought exactly, which rule is generall for all other plottes, and superficiall figures whatsoeuer.
[Page 163] Thus hitherto haue wee dealt in the plotting and dimension of seueralls; and that by such seuerall, exact and artificiall wayes and meanes, as may most sufficiently serue aswell for the absolute performance thereof, as (if well vnderstood and practised) of many other excellent conclusions. But were you to suruey and plot great quantities, and many seuerals together as a whole Lordship or Mannor, or to deale in mensuration of impassible wood-grounds, wherein you are debarred from crossing ouer or working within the same; I cannot aduise you therein to the vse of these former Precepts, (though otherwise to excellent purpose) but rather to vse and obserue the meanes and courses prescribed and taught in the next following Chapter, which of all other is the most generall, absolute and exact, for the mensuration of all manner superficiall Figures, of what forme, quantitie, or number soeuer; and therefore to bee obserued with good respect.
CHAP. XXXVIII. To take the plot of a Lordship or Mannor, consisting of diuers seuerals, of what nature or kind soeuer, whether Wood groundes or other.
THe precepts and instructions taught, and deliuered in the 10. Chapter of this booke, concerning the description and vse of a necessarie Field-booke, might well serue with diligent obseruation for the performance of this worke: But seeing that there (according to the proposed matter) my chiefe indeuours tended rather to the explanation thereof, then to the forme and order of mensuration, wee will here make vse of the Figure there expressed; and by inserting certaine necessarie obseruations there omitted (as needlesse to that purpose) referre you for a full satisfaction, to the consideration and due obseruation of those, and these conioyned. Wherefore suppose that the figure or Diagram there expressed, were a Mannor, or part of a Mannor to bee measured and plotted. First, write your Title, as there is mentioned then planting your Instrument in A. direct your sight to B. and hauing obserued the degree there cut, take vp your Instrument, and measure to B. entring your degree and length into your booke, then plant not your Instrument at B. but onely measure from thence to C. and there place your Instrument, & direct ypur sight backwards to B. obseruing your degree; but with this speciall regard, that in taking your backe sights, you alwayes reckon the degree cut by one and the same end of the Index, as you reckon on, when you direct your sight forwards; or otherwise you take the opposite degree to that you should, which will much trouble you in protraction: then here consider, that you are to leaue the bounder, which you went against from A. where you began, to this place, & therfore draw a single line, and as you are [Page 164]there taught, write the bounder past, as in the example of your booke; then direct your sight to D. and obseruing your degree, measure thither, which hauing entred, make there another single line, for that here you leaue that bounder also, which let be expressed; then measure from D. to E. and there plant your Instrument, and as before direct your sight backewards to D. and obserue the degree with the former caution, which done, turne your sight to F. and hauing your degree, measure from E. to F. and likewise from F. to G. expressing in your Booke those lengthes seuerally, then plant your Instrument at G. and taking your degree from thence backwards to F. as before, here strike a single line, and write your third bounder, then take your degree to H. and measure thither, where also (that being your last Angle) you must plant your Instrument, and worke as before to A. where you began, and then strike a double line, signifying you haue finished that Field. And in like manner proceed with all the rest from one Close to another, till you haue finished the whole worke, as you are most plainely directed in the 10. Chapter, remembring alwayes to measure euery line; and to place your Instrument at each other Angle, taking your backe-sights to that very point or marke, whereunto you directed your sight last before. So shall you most exactly, and with great expedition performe your desire. And your worke is to bee protracted as is taught in the next.
CHAP. XXXIX. To protract and laye downe a Plot of many seuerals, of what quantitie or number soeuer.
ACcording to the quantitie of your plot, or the largenesse you suppose it will be of, glue papers together; but if very large, lay first together but 4. or 8. sheetes onely, and rule them all ouer with parallel lines, representing Meridians of such conuenient distance, as they exceede not the bredth of your Protractors Scale. Then laying your Field-booke before you, suppose you are to protract the obseruations mentioned in the tenth Chapter, and considering towards what point of the Compasse your worke will most incline or extend, begin your protraction accordingly; as in that example; it inclineth towards the East, and North-east from the place of beginning; wherefore beginne your protraction towards the Southwest part of your plot; and there make a point, whereon place the Centre of your Protractor (with the Semicircle eyther vpwards or downewards, as you best fancie) and holding your protracting pinne in that point, moue about your Protractor thereon, till you finde one and the same parallel diuision on eyther end of your Protractors Scale, to lye eyther directly vpon any one parallel line, or equidistantly aboue or beneath the same; and there, with your left hand keepe firm your Protractor, whilest you finde in your Field-booke the first degree 65. against which, on the limbe of your Protractor place the point of your protracting pinne, and there keepe it, bringing the edge of your Protractors Scale thereunto [Page 165]with the first diuision of the Scale on your chosen point, and then draw a line by the edge of your Scale of your first length in the booke 20 0· 2 1· as the line A B. But with this respect, that (as before is taught) the lines belonging to euery degree vnder 60. bee drawne from the Centre point downewards, or inclining towards you, and the lines belonging to euery degree aboue 60. (as the last line A B.) bee drawne from the same point vpwards, or reclining from you: Then place your Protractor on the point B. (being the end of your last line) in all respects as before at A. and finding your next degree and length in your booke to bee 68. 16 0· 6 1· against 68 degrees on your Protractor, place the point of your protracting pinne, and applying the Scale of your Protractor thereunto, with the beginning of the diuisions thereof on the point B. draw your line of the length 16 0· 6 1· as B C. and here (considering you are to leaue the bounder you went against) make a small stroke or marke at C. with your penne, and finding in your Booke, at the end of your last length the figure (1) place that at C. with blacke lead (the vse whereof shall partly appeare) then place your Protractor on the point C. (the end of your last line) as before, and find your next degree and length 36 ½· 22 0· 5 1· and against that degree on your Protractor, place the point of your protracting pinne, and bringing the edge of your Protractor thereunto as before, from the point C. draw the line C D. of the length 22 0· 5 1· which seruing to a degree vnder 60. is to bee drawne from the point C. downwards as before. And so proceede with your seuerall degrees and lines, in order as you finde them in your booke, till you come to the point A. where you first began, and hauing wrought truly, you shall there iustly inclose your first seuerall. Then looke in your Booke where you are to beginne your next inclosure; and you shall be thereby directed to beginne from N o. (1.) Wherefore seeke in your last protraction where you placed that number, which you shall finde at C. and there you are to beginne your second parcell; wherewith, and with the rest proceed in all respects as with the first. Small practice with good obseruation (whereunto I will leaue you) is much more auaileable then many words. And therefore will I cease to spend further time herein.
CHAP. XL. The order and meanes of measuring and taking the seuerall and particular quantities in common fields, with a briefe instruction concerning the vse of my Chaine.
THe whole plot and quantitie of common fields are to be taken and plotted as they lye among other the adiacent grounds, according to the directions of the 38. Chapter, without regard of the seuerall and particular quantities therein contained; which afterwards are to be had and obtained after this maner.
Let a Booke bee purposely prouided, which call your Common-Field-Booke to bee ruled as in the example, containing eight Collums. The first [Page 166]towardes the
| Tenants names. | Bredth. | Length. | Quantity | ||||||||
| 0 | 1 | 2 | 0 | 1 | 2 | a. r. p. | |||||
| (1) Io. Woods from the Church lane east wards free. | 1 | 3 | 2 | 16 | 2 | — | 0-0-21 | ||||
| (2) Wil. Browne by Copie. | 6 | 8 | 2 | 16 | 2 | — | 0-2-30- | ||||
| (3) Fra. Jacksons for 3. liues. | 8 | 4 | 1 | 14 | 2 | 3 | 0-3-0 | ||||
| (4) Tho. Coakes for yeeres. | 7 | 5 | 2 | 15 | 4 | 2 | 0-2-36 | ||||
| (5) [...]il. Iones at will. | 5 | 6 | 3 | 15 | 4 | 2 | 0 2-7 | ||||
left hand, seruing for the Tenāts names and the tenure whereby they hold the same lands; the next three for the bredth of euery parcell intituled with these Signes, or Characters, 0.1.2. signifiing Vnites, Primes and Seconds, as is taught in the 6. Chapter of this Booke; the next three for the length with the like Title; and the eighth or last towards the right hand, for the content of each seuerall parcell, the length and bredth being multiplied together. In this worke there is no vse of any other Instrument, then your chaine onely: And beginning with any one furlong or went, expresse first in your booke the name of the field, & then of the same your first furlong, and so the rest of the Title, as in the example, then in the first Collum write the Tenants name, whose land you first measure, & withall from what place you begin, & on what point of the compasse you passe from thence, wherby you shalbe able afterwards (obseruing the same course in the beginning of euery furlong) to abut and bound euery parcell if need require, & likewise in the same first collum, expresse by what tenure it is held, thē cōsider how the whole furlong lyeth if al of one length, then need you take the length but once for all, although there were twentie Tenants lands in the same; but if irregularly, as in some places shorter, and others longer; then at euery second or third bredth, (or oftener if occasion require) take the length thereof, and expresse the same vnder your title of length, as for the seuerall bredthes, you may onely crosse ouer the whole furlong about the middle therof, taking euery mans bredth, and entring the same as you passe, vnlesse you finde extraordinary difference betweene the bredthes at eyther end; if so, then measure the bredth of both ends, adding those two bredthes together, whereof take halfe for your bredth, & enter it in your booke, or you may enter both bredthes, and take halfe thereof when you cast vp the contents. And thus proceede from one furlong to another, till you haue finished the whole field. And when you haue done (or at any time after at your pleasure) by multiplying those lengthes and bredthes together (which is most speedily and exactly performed, as hereafter followeth according to the order of decimall multiplication) you haue your seueral contents to bee expressed in the last Collum. And lastly, number all the seuerall [Page 167]parcels in the whole booke by Figures in the margent, from 1. forwards as in the example, which will serue you to good purpose, in the collecting of euery mans parcels together, as shall be hereafter declared.
And here shal you find (and in all other workes of this nature) most excellent vse of my decimall chain described in the sixt chapter of this booke.
But lest you should bee absolutely ignorant of the manner and order of casting vp of the seuerall contents, according to the lengthes and bredths so taken and obserued as before, and consequently the chaine with the seuerall parts and fractions thereof may stand you in little stead, I will here briefly touch the order thereof, in two examples thus.
Suppose your length taken to bee 16 0· 2 1· and your bredth 1 0· 3 1· 2 2· being those first specified in the former example. First, let your numbers bee placed in all respects, as if they were whole numbers to bee multiplied the one by the other thus; but ouer euery Fraction of your multiplicand, place a pricke or point, being in this example onely one, as that ouer the figure
2. and at the end of your multiplier, place as many prickes or points as there are Fractions in that number, being in this example two, then multiplie the one number by the other after the vsuall manner, as if they were whole numbers, and your worke will stand thus; The Product of your multiplication being 21384. and considering there are belonging to
your two first numbers three prickes, representing so many fractions, you are therefore to cut off from the Product of your multiplication, with a stroke, three Figures towards your right hand, which shall bee the Numerator of a Fraction, whose denominater is so many Ciphars, with an Vnite placed before them, and the other two figures towards your left hand are the Intigers of your multiplication: So shall the Product thereof (being the content sought for) be 21. Perches, and 384/1000· parts of a Perch, which is somewhat more then ⅓· and your worke standeth thus. But to expresse the quantitie of these fractions, which you find in the Product of your
multiplication, were needlesse nicenesse, in busines of this nature; yet hauing wrought your multiplication, if you find your fraction to bee neere an vnite, increase your intire number by an Vnite, and let passe the Fraction as not to bee regarded.
Againe for your better practice, let your measured length be 15 0· 4 1· 2 2· and your bredth 7 0·5 1·2 2· being the fourth sort expressed in your former example in the beginning of this Chapter. Of which length & bredth you desire the
content by multiplication: first, as before, place your numbers, as if they were intigers to bee multiplied, the one by the other thus, as in this exāple: The product whereof is 1159584. and considering there are foure points or prickes belonging to the two first numbers, that is, to eyther of them two, signifying [Page 168]so many fractions; you are therefore to cut off foure Figures from the Product towards the right hand, which is the numerator of your fraction as before, and those three Figures towards your left hand are so many Perches: so shall the Product of your multiplication (being the content sought for) bee 115. Perches, and 9524/10000· parts of a Perch, which being very neere an Vnite, I therefore adde an Vnite to the former number of Perches, which makes 116. Perches; that is, by reduction 2. roodes. and 36. Perches.
And after this manner (with due obseruation) may you most easily and aptly apply this chaine, and the seuerall parts and fractions thereof to all the ordinarie rules of Arithmetike, as Addition, Subtraction, Multiplication, and Diuision, working any dimension thereby, as if they were Intigers, or whole numbers.
And thus much for a taste onely of the necessarie and infinite vse of this Chaine thus diuided.
All this time hitherto in our former mensurations haue wee walked in plaine and euen leuelles, wherein the Plaine Table artificially handled, (whereof many vsing it, are to seeke) is an excellent Instrument. But suppose wee are now trauelling into Wales, or any other place where are mountainous and vneuen grounds; then must wee of necessitie, eyther leaue that Instrument behind vs, or vse those meanes, or the like expressed in the description thereof, Chap. 3. Vnlesse we haue insight in that excellent art, which many plaine plaine Tablemen haue (wanting those meanes) at an instant to conuert the highest mountains to plaine and leuel grounds, pressing them downe, and inforcing them on a Plaine sheete of paper to lye leuell with the rest; which they easily performe by onely thrusting out their bordering grounds from their due and naturall place, where, ab initio they haue remained.
Yet let vs consider of some better meanes for the performance thereof, which shall be hereafter taught.
CHAP. XLI. To reduce Hipothenusall to Horizontall lines by the Peractor.
SVppose
A B C D. be a hill, or mountaine to be protracted and layde downe in your plotte amongst your other grounds: It is apparant by the Figure, that the hipothenusall lines A B. and B C. cannot be layd down exactly in a right line betweene the other grounds which bounder on this hill at the points A. and C. Wherefore we are to find the true leuell and horizontall distance betweene A. and C. which is a right line, extending ouerthwart [Page 169]the ground whereon the hill standeth. Which to performe, worke thus. Plant your Instrument at A. the foot of the hill precisely horizontall, by help of your plummet (wherin great care must be had) then cause a marke to be placed on the top of the hil at B. to be of equal height from the ground, with the Centre of your Quadrant, whereunto direct your sight, mouing the Quadrant vp and downe, till you perfectly decerne the same, where letting it stand, measure the Hipothenusall line A B. which suppose to bee 6/30· then seeke 30. on the Index of your quadrant, and note what line issuing from the left side C D. of the Quadrant is cut by the same diuision or number of 30. on the edge of the Index, which you shall finde to be 18. and that number is to be obserued, and kept for the horizontall line A D. to be protracted and layde downe in stead of the hipothenusall A B. And if the same hill from B. continue not plaine and horizontall, but descendeth againe on the other side, as this from B to C. then must that hipothenusall line B C. bee likewise taken, by planting your Instrument at B. and causing a marke to be placed at C. as before, and then direct your sight to the marke, and measure the hipothenusall line B C. which suppose to be 40 0· then note as before what line cutteth 40 on the Index, as 32. and take that for your horizontall line C D. which added to the former 18. maketh 50 0· for the whole line A C. which is to be protracted and layde downe for the two lines A B. and B C.
And if at any time it hapneth (as often it may) that the measured distance of the hipothenusall line exceedeth the greatest number on the Index: In such case take halfe, or a third part of the measured distance, and finding that number on the Index, note what line from the left side of the quadrant intersecteth therewith on the edge of the Index, and the double or triple of the number of that line, is your horizontall line sought for.
CHAP. XLII. To reduce hipothenusall to horizontall lines by the Circumferentor, or by the Plaine Table, with vse of those meanes expressed in the description thereof.
PLant your Instrument as before, at the foote of the hill, and let a marke be placed in the toppe thereof, in manner as is directed in the last, then directing your sight to that marke, moue the vane vp and downe on the longer sight, till through the small hole thereof, and by the pinnes head in the shorter sight, you espie the marke, then note among the hipothenusall diuisions, what is then cut by the edge of the Vane, which suppose to be 7. signifying (as is expressed in the description, Chap. 3) 107. Then measuring the hipothenusall line, which suppose to be 40. by the rule of proportion, reason thus. If 107. the hipothenusall in the Instrument, yeeld 100. the side of the square thereby proiected, what 40. the hipothenusall measured and by increasing 100. by 40. and parting the Product by 107. your answere will be 37. and very neere ⅖· the length of your horizontall line sought for. And [Page 170]the same course in all respects which you haue here helde in this Angle of ascention, the like is to bee obserued in all works whatsoeuer for Angles of descention.
Or otherwise it may be performed thus.
BY any Instrument take the quantitie of the Angle, eyther ascending or discending as before taught, and then by the 11. or 12. Chapter protract the same Angle, and on the one side, from the point thereof, lay downe the measured length of your hipothenusall line, and from the point where those measures end, by PROB. 5.2. let fall a perpendicular on the other side of the Angle; and the Segment of that other side, betweene the intersection of the perpendicular, and the point of the Angle shall bee your horizontall line required.
And thus hauing shewed the meanes for the reducing of these lines; let vs now consider of the application and vse thereof; which for our present purpose is to finde and expresse the true content of irregular and vneuen grounds, and withall notwithstanding their irregularitie, in protraction to plot and lay them downe in such sort, as neyther in themselues they exceede the bounds of their owne Perimeter (but may truly inclose though expressed in Plano) nor displace or thrust out of order the grounds adiacent. And seeing it is impossible, and against the rules of art and nature, precisely to expresse and limit a solid, or bodie within the bounds or termes of one visuall superficies, which is comprised and composed of many: It is therefore not to be expected, wee should truely expresse the irregular capacitie of mountainous and vneuen grounds in a plaine sheete of paper: for if the plot bee exact and answerable to the rest; the superficiall content must needes be wanting, or if the true content of lines and angles bee expressed, the plot of necessitie must bee erronious: yet notwithstanding we are now to resolue of some direct and immediate course, aswell for the obtaining of the true superficiall content, as for the orderly expressing and laying downe of such disordered Figures, which shall be amply and plainely taught and deliuered in the next following Chapter.
CHAP. XLIII. The best and exactest means for the dimension & protraction of mountainous and vneuen grounds, and the obtaining of their true Contents by the Plaine Table.
SVppose A B C D E F G. bee a mountainous and vnleuelled ground to bee measured and plotted: First, plant your Instrument at A. and directing your sight to B. and measuring the line A B. draw the same by the edge of your Ruler, and after the vsuall manner finding the line Horizontall, place theron your measured length found, from A. to B. 10 0 8 1. then place your Instrument at B. where finding an ascent to C. direct your sight to a marke there placed as is taught in Chapter 40. and hauing drawne your line at length towards [Page 171]
C. as if it were leuell ground, measure the Hypothenusal distance frō B. to C. 16 0· 2 1· ½· but withal finde out the horizontall distance as before is taught 15 0· 6 1· of which length lay down your line B C. but expresse theron in figures your measured Hypothenusall length 16 0· 2 1½· so haue you the horizontall truely as it is laid down & your hypothenusal length, for the obtaining of the true content expressed thereon; then plant your Instrument at C. and finding from thence a descent to the next angle D. direct your sight thither to a marke there placed as before, and drawing your line at length, measure the Hypothenusall distance, which found to be 30 0· 6 1· expresse it on the line drawne; but searching out the horizontal distance 21 0· 4 1· make your line C D. of that length, and in like sort should you proceede if there were diuers other ascending and descending lines: but seeing all the rest in this figure are horizontall and leuell lines, proceede with your worke in all respects as is taught in Chap. 38. So shall you exactly inclose with your other grounds this irregular figure.
But yet are we further to cōsider, that notwithstanding we haue obserued the difference betweene the Hypothenusall & horizontall lines happening in the Perimeter of this figure, wherby we are able to place the same in his due scituation, yet are there within the compasse of this Perimeter many hils & dales wherof we haue hitherto taken no notice, sauing only in cōpassing thē about: And if we should, with these lines alreadie had, cast vp the superficial content after the vsual maner, we should come far short of the true quantitie thereof; which to redresse, worke thus. Before you take your work from the table, reduce the same into the largest Trapezium you may, by drawing the lines A M. M H. H K. & K A. as in the former Diagram, then crosse the Trapezium with the diagonall line A H. and theron let fall the Perpendiculars M N. and K L. then by direction of your Instrument or otherwise let those lines be exactly measured with the chaine ouer Hils & Dales in a right extention▪ which by reason of the vneuennesse of the ground you shal find to cōtaine much more in length then your lines alreadie laide downe; as the line A▪ H. in the plot [Page 172]containing but 51 0· 4 1· by measure is found to bee 60 0· 2 1· also M N in the plot but 22 0· 3 1· by measure 24 0· 4 1· and L K in the plot but 24 0·8 1· by measure 28 0·2 1· which seuerall lengths so found by measure expresse in figures seuerally on the lines in the plot whereunto they properly belong; and by those lengths so expressed cast vp (after the vsuall manner) the superficiall content of the Trapezium, whose true quantitie you shall finde to be 9. Acres, 3. Roodes and 23. Perches. But let the same be cast vp according to those lines protracted it will containe thereby but 7. Acres, 2. Roodes, and 10. Perches. Whereby is manifest the vsuall errors committed by omission of these meanes. Then cast vp the seuerall contents of those small triangles about the Perimeter, adding those left out, and abating those taken in by the Trapezium, and your worke is finished.
Yet were it necessary to distinguish these frō the rest in your plot, shaddowing them off with Hils, &c. and likewise to expresse therein the true content, with some note of instruction concerning the same, for that otherwise a stranger not herewith acquainted, applying your Scale to the plot, may causelessely taxe you of errors committed.
And here note that notwithstanding I haue directed this worke to be wrought with the plaine Table; (by reason that these former courses and obseruations are more vsually omitted in the vse of that Instrument then any other) yet I doubt not but hee who can performe the same by that, (with due obseruation of what hath beene formerly deliuered) will be able to effect the same by any other Instrument. For your obseruations being had and taken in the field, and then laide downe accordingly; you shal thereby afterwards measure your Diagonall and perpendicular lines in the field; as before is directed.
CHAP. XLIIII. To diuide a Common of pasture, or a common field into any parts required.
SVppose A B C D E F G H bee a stinted pasture, or a common field in the vse and occupation of three men, as A. B and C. and let it be required to diuide the same betweene them proportionally, either according to their seuerall stints (beeing a stinted pasture) or according to their seuerall quantities, being an arrable field in common: First measure and plot the whole quantitie, and as is taught in the second part of the second Booke, finde the superficiall content thereof which suppose to bee 4268. Perches, or by reduction 26. acres 2. roodes 28. perches, then seeke the content of the high wayes A E and H E passing through the same, which let be 341. perches to be deducted out of the whole quantitie: and there resteth 3927. Perches, which to diuide according to their seuerall portions, reason thus with the rule of proportion. If all the whole number of beast-gates of A. B and C. together yeeld the whole quantitie 3927. what the number of those belonging to A. and the answere will be his part: & working thus seuerally for their three seuerall parts suppose them thus, to A. 924. Perches, to B. 1798. and to C. 1205. Then as [Page 173]
is taught by the ten last Problemes of the second Booke, diuide your plot accordingly, by the lines I M. and K L. laying euery mans part in such place as they shall mutually agree: which beeing performed on your plot, you are then to effect the like on the ground, wherein you are onely to lay out the lines I M. and K L. in their due places, which is thus to be performed. Finde out by your Scale the distance on the plot betweene the next angle, and the beginning of your first line as G M 3 0·6 1· then find also the angle G. in the field, and with your chaine measure out that length from G. on the line G F. to end in M. where place your instrument, and finding by your Protractor on what degree the line M I. passeth on the plot, on the same degree of your Instrument place the Index: and looking through your sights, cause dooles and markes to be made and placed in a rig [...]t line from M. to I. and for the more exactnesse, and your better satisfactio, note in your plot also what angle is next vnto the ende I. of the line M I. as A. from whence take your distance by the Scale on your plot to I 18 0· 8 1· and finding the Angle A. in the field, from thence measure by your chaine that length 18 0· 8 1· on the line A B. which if you finde to ende in I. is an infailable assurance you haue truely wrought; if not, reforme it by reducing the ende I. of your line M I. to that place. And by the like meanes, you are to lay out the line K L. and more, if more required. And after this sort shall you lay out and diuide any common or waste whatsoeuer; into what parts soeuer the same is required, to be deuided, and layde out.
CHAP. XLV. To know the houre of the day by the Peractor or Circumferentor with the Sunne.
PLace the Index of your Peractor on the Meridian line of the Planisphere, and direct the North part of the Instrument towards the Sunne, turning it about on the Staffe til the shaddow of the thrid, or shining of the Sunne fall directly through the sight, on the fiduciall edge of the Index or Meridian line: then obserue in the Carde the next concentrick circle aboue the present moneth, and note what houre-line (which are those sphericall lines issuing from the centre) intersecteth that concentrick circle, vnder the North ende of the needle; which point of intersection, lying directly vnder that end of the needle, sheweth directly the houre of the day, expressed at the vpper ende of that houre-line, or being short or ouer, it sheweth how much the time of the day is short or ouer that houre.
And the like is performed by the Circumferentor with turning the North ende of the Instrument towards the Sunne, till through the sight, the Sunne shineth on the Meridian line; and making your obseruations as before.
CHAP. XLVI. The ordering of a plot after the protraction thereof.
THe order and course of protraction and laying downe of plots according to the diuersitie of obseruations taken, is alreadie in this Booke sufficiently declared; and likewise the meanes of obtaining the true superficiall content of all sorts of Figures by the Second part of the Second booke; (the conclusion of which part, I would aduise you to make vse of for your further ease therein.) Now it resteth (the plot being thus protracted and cast vp) next to consider whether it were fitting, to draw it fayre of the same Scale and scantling wherein it is, or first to haue it reduced into a lesser forme. If the businesse be not so large, but that with conueniencie it may bee drawne in the first Scale wherewith it was plotted and cast vp, then prouide you a piece of ordinarie new linnen cloth of a reasonable finenesse, and thereon paste you cleane paper, according to the order of Maps (but those for the most part are too course) of the largenesse of your plot; which beeing well washed with Allome water, and dryed, fasten your rough plot, slenderly thereon with mouth glue at each corner thereof; and with a bodkin or pin of brasse (which I hold the best for these purposes) trace out all your lines with a reasonable hand; for a light hand with discretion wil make sufficient impression on your new plot, whereby you may well discerne to draw your lines either with Pen or pensill as you please; and then expresse your houses, buildings, woods, riuers, waters, wayes, and all other remarkeable things in their due proportion perspectiuely; not placing your houses and trees euery way, whereby [Page 175]here the tops and there the bottomes shall seeme standing vpwards, as is vsually accustomed; and then garnish your plot about with some neate border, and within with Copartments for your Scale and Title; and in some conuenient place thereof describe a Carde, shewing the scituation according to the points of the Compasse; and then let it be neately coloured about the seuerall lines, lightly washed off, and not dawbed all ouer, as some painting Surueyors vse. But aboue the rest, forbeare much writing in your plots either of names or quantities (beeing absurd and grosse) but onely numbers of reference to your ingrossed booke concerning the same, if you make any; if not, then it may serue, as Chalke on a trencher.
But if it bee required to haue your first plot reduced into a lesser or greater forme; vse the helpe of the next Chapter.
CHAP. XLVII. To reduce any plot from a greater to a lesser quantitie, and the contrarie.
IN the third part of my Second booke I haue at large declared the manner and meanes as well of reducing and translating of all superficiall Figures from one forme into another, retaining the same quantitie; as one quantitie into another, retaining the same forme. But those being of particular figures, are not so meet or necessarie for the Reduction or translation of a plot consisting of many (being often in vse.) Wherefore I will here deliuer you a very speedie and exact meanes for the performance thereof. According to the proportion whereunto you would reduce your first plot, lay papers together with mouth-glue; as if you would reduce it into a fourth part, make your new plot to be in quantitie a fourth part thereof. Then place your new made plot on your rough plot in such sort, and with such discretion, as the middle of the one may be about the middle of the other, whereby all the worke of the one may be reduced the better into the other, and there fasten the one to the other slenderly at the corners with mouth-glue, in such sort, as when occasion serueth, any one of those glued places may be easily taken asunder, and the paper folded in; then take your reducing ruler mentioned in Chap. 9. and fasten the same with a needle or blue pin striken into the table about the middle of the plots through the centre hole of the same rule, in such maner as both the plots together may bee turned about at pleasure vpon the table; wherein you must take great care of renting or tearing out your centre point in the plots; which to preuent, would require to bee strengthened with a small piece of a Carde or past-boord, to bee glued thereon, vnderneath the first plot. And being thus prepared, you shall finde good part of the worke in your first plot, to lye without the vtter edges of your cleane paper, which let be first reduced; wherein (hauing resolued into what proportion you will make your Reduction, as into ¼· of the first) worke thus; bring the edge of your Ruler to any Angle in your first plot, and note what number of diuision on the edge of your Ruler is there cut, which admit 40. then take halfe [Page 172]therof which is 20. and against that diuision by the edge of your Ruler make a pricke on your cleane paper, then remoue your Ruler to the next Angle in your first plot, and note the diuision there cut, whereof take likewise the halfe, and against that halfe by the edge of the Ruler make another prick; and betweene these two prickes draw a line, which shall represent the line betweene those two Angles of your first plot; and so proceede from Angle to Angle, and from Close to Close, till you haue reduced all the worke on your olde plot, lying without the edges of the new, or so much as lies without the same towards any one side or corner thereof; and then vngluing one of your corners or sides, fold in backwards towards your olde plot, that part of your new plot as is wrought; so shall you come to worke that which formerly lay vnder the same; and thus by folding in, and working one side after another to the centre; whilst the other sides are fastned together with the olde plot, you shall speedily reduce the whole plot into your desired proportion: For in taking the halfe of euery line in your first plot your new by THEOR. 49.1. shall be ¼ of the olde.
And this kinde of Reduction hold I the speediest and exactest of all other: Which by small practice, you shall much better finde and vnderstand, then with many words of Relation.
And after this manner may you make your Reduction, into any other proportion, or reduce your plot from a lesser to a greater forme in any proportion required; by increasing your second number, proportionally, as in this worke you decreased the same.
CHAP. XLVIII. To reduce any number of Perches giuen into Acres, and the contrarie.
SVppose 5496. Perches were giuen to bee reduced into Acres. First after the vsuall manner (considering that a Statute Acre containeth 160. square Perches) diuide the giuen number by 160. the quotus will be 34. and the remainder 56. which remainder diuided by 40. (the number of Perches in a Roode) quoteth 1. and the remainder 16. So is the whole Reduction 34. Acres, 1. Roode, 16. Perches.
Or more briefely thus.
THis rule is much briefer in operation, though not in demonstration then the former; which is thus. Frō your giuen number cut off with a stroke, the first Figure towards the right hand thus, 54916. then diuide the other three Figures by 4. and your worke will
stand thus. the 137. being so many Roodes, and 16. Perches remaining, then diuide likewise that 137. by 4 and your worke is finished and will stand thus. Where note that your first remainder with the Figure cut off, are alwaies the odde Perches; and the last remainder (if any be) are Roodes.
[Page 17]And if you would reduce these Acres, Roodes, and Perches, into their least Denomination, as into Perches. First multiply your number of Acres 34. by 160. the product is 5440. then multiply the number of Roodes 1. by 40. produceth 40. which together with the number of Perches 16. added to 5440. makes 5496. the first number. And the like of all others.
Here might I now much inlarge this worke by shewing many other necessarie conclusions fit for a Surueyor to know, as the mensuration of Timber, Boorde, Glasse, Pauements, and the like; also the seuerall waies and meanes of plotting of countries and large continents; of carrying of mynes and trenches vnder ground; of Water-workes, and the conueying of the water from any Fountaine to appointed places of whatsoeuer possible distance; of the taking and making of the formes and models of Plot-formes, Forts, Castles, Houses, and the like. But seeing that not onely these, but infinite other Conclusions Geometricall, may be easily performed and wrought by the former rules and instructions well vnderstood and practised; and for that they are without the scope and limits of Suruey, whereunto I chiefly bend the subiect of this Booke; I will leaue them to your owne indeuours, and diligent practice. And hauing thus performed at large the Mathematicall part of Suruey in generall; we will next consider of the Legall; and in the meane space here conclude this Third Booke.
THE LEGALL PART OF SVRVEY.
The fourth Booke.
I Would not bee mistaken, or haue it vnderstood; that J here vndertake (as a Lawyer) to instruct or teach the rules or Jnstitutions of the Law (being out of mine element) but as a Surueyor, briefely and truely to expresse and deliuer herein what J hold fit and meete for a Surueyor to know and vnderstand. As first what a Mannor is, and the seuerall parts and members thereof, with the appendants therevnto: Next, the perquisites, casualties and profits of Court, and their seuerall natures: Then the diuersitie of estates, whereby any Lands or Tenements may bee holden, occupied or enioyed; and the seuerall tenures depending on those estates; with the Rents and seruices incident and belonging to those tenures: Also what reprises, paiments, and deductions may bee issuing out of a Mannor, and the [Page 176]considerations thereof to bee had: Likewise what courses are to bee obserued and taken, before the beginning of a Suruey: The order and manner of keeping those Courts: The entrie of the Tenants euidence and estates; and the orderly manner of ingrossing the same: with other briefe and necessarie Rules, and Obseruations tending to those purposes.
CHAP. I. Of a Mannor with his seuerall parts, and of the name and nature therof: how made and maintained, and how discontinued and destroyed.
AS in my workes concerning the Mathematicall part of Suruey, comprised in the three former Bookes, I premise the definitions, principles and grounds thereof: so in this Legal part I hold it answerable to order, and a good decorum, (before we abruptly enter on the Suruey of a Mannor) first to consider what a Mannor is, and the seueral parts thereof (lest being questioned of our present imployment, wee discouer our owne weakenesse in vndertaking we know not what) and then to informe our selues of the seuerall natures, qualities and conditions, of the estates, tenures, and seruices of land; and of the seuerall profits, rents, and commodities therevnto incident and appertaining; with such other meete and necessarie obseruations, as are most fitting for a Suruey or to know and vnderstand, before he assume and take vpon him the name, or at least the office or function to a Surueyor belonging. Of all which in order; and first of a Mannor what it is, and of the parts thereunto belonging.
Mannor whence deriued.Concerning the Deriuation or Etymologie of the word, I will not stand, whether it be of Maneo manere to remaine in a settled place; or of Mano manare to proceede or spread abroad out of the bountie of those Princes liberalitie who in the beginning bestowed them; or of Manuarius gotten by labor of the hand; which I hold the best; because there is more skil in getting then keeping; and with Manerium I will not meddle, seeing (as I take it) Mannors were created before the word was made: But from whencesoeuer deriued, A Mannor is now that which hath therunto belōging, What a Mannor is. Parts of a Mannor. messuages, Lands, Tenements, Rents, Seruices & Hereditaments; wherof part are Demeanes, being those which anciently and time out of minde, the Lord himselfe euer vsed, occupied and manured with the Mannor house; the residue are Free-holds, [Page 177]Farmes, and customarie or coppihold tenements; and these haue commonly diuers seruices besides their rents properly belonging thereunto, whereof I will hereafter speake.
Parts of a Mannor.There is moreouer belonging to a Mannor a Court Baron, and to diuers a Court Leere; which is of more worth and efficacie, and is alwaies granted from the King, or held by prescription. To these Courts, and consequently to the Mannor is there vsually belonging; Fines, issues, amerciaments, heriots, waiues, straies, excheates, reliefes and other perquisites and profits of Court; whereof likewise I will further speake.
Appendants to a Mannor.Besides, those there are often appendant and belonging to a Mannor (which are not of necessitie to be taken as the proper parts thereof) Wards, Marriages, aduowsons, patronages, free-gifts or presentations of parsonages, Vicarages, Chappels, Prebends, &c. also Commons of Pasture, Moores, Marishes, free Warrens, Customes, Liberties, Franchises, and Priuiledges; likewise yeerely Rents, suites of Court, tenths and seruices issuing and reprised out of other Mannors. And of these, a Mannor is neither made by them; nor destroyed or marred for want of them; wherefore they are termed rather appendants then parts of a Mannor.
No present meanes to make a Mannor.Neither doe those parts formerly named, properly of themselues make a Mannor: For should any man at this day alott and appoint out any competent quantitie of Land, and diuide the same into demeanes and tenement Lands; in feoffing Tenants in Fee of some part, and granting others by copie of Court-Roll, and perfecting the rest which before is said to belong vnto a Mannor; yet all this will not make a Mannor; for that it is the office of time by long continuance to make and create the same. Continuance of time may perfect a Mannor.
How a mannor may be destroyea and dismembredBut a Mannor at this day may bee dismembred, and vtterly destroyed both in name and nature, by escheating the Free-holds, and Copie-holds; for if of Free-holds or Copie-holds there are not two at the least, then are there no Sutors, and if no Sutors, no Court, and consequently no Mannor, and then may it bee termed a Seigniorie, which can keepe no Court Baron at all.
How one Mannor may be diuided, and made diuers Mannors.Also it is to be vnderstood, that one Mannor may bee diuided into diuers Mannors; whereof wee haue diuers examples at this day; as where a Mannor descendeth to co-heires, and they make diuision and partition thereof; allotting to euery of them demeanes and seruices; whereby euery of them hath a seuerall Mannor, and may keepe seuerall Courts Baron thereon, as if anciently entire.
How diuers Mannors may be reduced into one Mannor.And in like manner two distinct and seuerall Mannors may bee conioyned and made one entire Mannor, if formerly the one held of the other; and that Mannor so holding of the other doe escheate; but otherwise not. And thus much concerning the name, nature, and parts of a Mannor.
CHAP. II. Of Perquisites Casualties and profits of Court, and their seuerall natures.
IN the former Chapter I declare, that (among other things) there is belonging to a Mannor a Court Baron at the least, and to some a Leete or Law-day, commonly called the view of Franke pledge: Now herein will I shew what perquisites, casualties, and profits are incident and belonging to those Courts; wherein I would haue it vnderstood, that it is not of necessitie, that all these hereafter mentioned, must be in euery Mannor, but that they may be in any. And first of Fines.
Fines of Land.
Fines of post mortem.FInes of lands are of diuers kindes; As first, if a man holding to him and his heyres, or otherwise certaine Lands and Tenements, by Copie of Court Roll, according to the custome of the Mannor dye, his heyre vpon his admittance by the Lord, shall pay a Fine for such his admittance: And these Fines are of two sorts, either certaine, or arbitrable; if certaine, as one or two yeeres rent, or the like; there is then no other question to bee made, but the Lord by his Steward is to admit him, and hee to pay such certaine Fine accordingly; if vncertaine or arbitrable, then is the Tenant to vndergoe what Fine the Lord shall in reason impose or require; and these are called Fines of post mortem.
Fines of alienation.Also a Tenant by Copie of Court Roll, hath not power to Alien or sell his estate or interest vnto any other man, without he surrender the same into the Lords hands to the vse of him vnto whom he shall so sell the same; for which Alienation the Lord is also to receiue a Fine, which in some Mannors are likewise certaine, and in others arbitrable, but being arbitrable, they are vsually rated at a lower and more reasonable value then those after death, and these are commonly called Fines of Alienation.
Likewise if a customarie Tenant let or set his lands vnto another for any terme of yeeres, not warranted by the custome, hee is first to obtaine licence of his Lord in this behalfe; and is to pay a Fine in respect thereof.
And moreouer if the Lord of a Mannor grant a Lease of any lands vnto a Tenant for any number of yeeres or for life or liues; and besides his annuall Rent, make composition for a Fine to be in hand paide; this is also a Fine of Lands.
Also in some places, the Custome is, that if a customarie Tenant alien and make surrender of his whole estate hee shall pay and yeeld vnto the Lord the best beast hee hath, Asarewell paid. or a certaine piece of money, in name of a Farewell. And in some places as well Free-holders, as customary Tenants on euery alienation shall pay a certaine summe of money for a Fine in name of Offare, Offare & onfare and onsare, and all these and the like are Fines of Land.
Americiaments.
A Merciaments are also perquisites of Court, whereof there are diuers sorts; which in generall are such Fines, penalties, and amerciaments as by the homage or afferers of the Court Leete, or otherwise are imposed on such Tenants as are found offenders within the Mannor; As if the Free sutors, Copie-holders, or other Tenants, make default or bee absent from the Lords Court, they are therefore amerced. Wherein is to be noted, that many Free-sutors make composition, Common Finet quid. and are at their Fine certaine in respect of their seruice of suite of Court; and these are called common Fines.
Heriots.
AN Heriot is properly the best beast which any heriotable Tenantis possessed of at the time of his death, whether it be Horse, Oxe, Cow or the like; for which in many places a summe of money is paide by ancient composition, and in some places for default of liue Cattell (or the best beast not being to the Lords minde) it is in his choise to take the best of any other goods, implement, or commoditie the Tenant hath at the time of his decease.
Of these Heriots there are two sorts, Seruice and Custome: Heriot Seruice. Heriot seruice is commonly mentioned and expressed in the Tenants grant: and therefore the Land answerable for satisfaction thereof; Heriot Custome. and Heriot custome is that which time out of minde hath beene euer paide vpon and after the death of any Tenant dying seised of any such heriotable Lands: And these Heriots of either kinde, are by the homage of euery Court to bee presented as they fall due; and seised by the Lords Bay liffe accordingly.
And it is to bee vnderstood, One Tenant may be chargeable with diuers Heriots. that if a Tenant dieth seised of diuers tenements or lands, which haue beene anciently charged with diuers Heriots; the Lord at the time of the death of such Tenant shall receiue so many seuerall Heriots, as those lands at any time then-to fore were anciently charged or chargeable to yeeld.
Heriotable lands diuided, are seuerally chargeable.And moreouer, if any heriotable Tenement shal be seuered & diuided into diuers parts, amongst seuerall Tenants; the Lord shal haue of euery such Tenant particularly a seuerall Heriot, for and in respect of those seuerall heriotable parcels; which the Lord may seise and take, Lib. Ass. 27.24. wheresoeuer hee shall finde the best for his best aduantage.
Reliefes.
REliefes are like wise accompted amongst perquisites of Court: but seeing it is a speciall seruice tyde to the tenure of Lands; I will here omit to speake thereof; referring you for your satisfaction therein to the Title of Warde, Marriage and Reliefe, in the 4. Chap. following.
Escheates.
Escbcates what they are.THese are likewise perquisites of Court; and are such as if a Free-holder or Copie-holder of inheritance, commit any manner of Felonie, and be thereof attainted; his Lands are escheated and forfeited to the Lord of the Mannor of whom they are holden; but the Lord shall not immediately enter thereinto; for the King is first to haue annum diem & vastum; after which time expired, it then remaineth to the Lord and his heyres for euer.
Also if any such Tenant dye without heyre generall or speciall, all his Lands and Tenements shall fall vnto the Lord by escheate, to remaine vnto him and to his heyres for euer.
Forfeitures.
Forseituret of diuers k [...]ndes.FOrfeitures are of diuers kindes; As if a Copie-holder or customarie Tenant deny, or will fully refuse the paiment, doing or performance of his Rents, Seruices and Customes; or if hee fell or cut Timber on his Copie-hold Lands contrarie to the custome; or doe or commit waste in the houses or otherwise; or if he grant or sell his Copie-hold estate by deede; or alien or let the same without licence of the Lord, beyond limitation of the Custome; In all or any of these, the customarie Tenant shall forfeite his Copie-hold estate into the Lords hands: Which offences are to bee found and presented by the homage at the next Court; and thereupon seisure made accordingly.
Also Tenant for terme of yeeres, life, or liues, may forfeit his estate for making a larger estate of Free-hold then he hath, or for not performance of such prouisoes and conditions, as are expressed in his Lease or Deede, if any be.
Waiues.
IF any man feloniously steale or take any goods or chattels of what nature or kinde so euer; and by earnest prosecution he is inforced in flying to leaue the same behinde him; these goods are called Waiues or waiued goods; and in what place soeuer they are so left and waiued, they shall be taken and seised for the vse of the Lord of that Mannor, if by his grant, charter or prescription, they belong vnto him (or otherwise they are the Kings) and being so seised by the Bailiffe or other Officer, they are to bee presented and found at the next Court by the Homage there. Fresh suite. Waiued goods restored. But if the right owner make fresh suite after the thiefe, and attaint him at his suite for stealing thereof, hee shall haue his goods againe, although they be waiued. And the like in all respects is if any goods be taken by an officer, from any whom hee suspecteth to haue stolne the same, though there be no pursuite made or prosecuted.
Estraies.
EStraies are when a Horse, an Oxe Sheepe, or any other Cattell of what kinde so euer come into a Lordship or Mannor, no man knowing from whence, nor the owner thereof; such are to bee seised to the Kings vse, or to the vse of the Lord of the Mannor who hath the same by grant or prescription; and if the owner come and make claime within a yeere and a day, then hee shall haue the same againe, paying for the charge thereof; or else after such time expired, the propertie thereof shall be to the King; or the Lord of the Mannor hauing the same by graunt or prescription; So that Proclamation bee thereof made in the next markets and the Parish Church, according to the Lawes in that behalfe.
Pleas and Proces of Court.
THese are where the Lord of a Mannor in the Court Leete, or Court Baron, holdeth plea of his Tenants for actions of debt, of trespasse or other causes, not exceeding the value of xl s. debt and damage.
And vnder this title of perquisites is comprised all other casualties whatsoeuer, which may happen to grow or arise within any Mannor; as profits arising by mines of Copper, Tynne, Leade, Cole, and quarries of Stone; also by sale of Woods, Turbarie, and Pannage; likewise profits of Fayres and Markets, Fishing, Casualties may become certaine. Fowling and the like. All or any of which may become certaine; by beeing letten and disposed of for yeerely rents.
CHAP. III. Of the diuersitie of estates, and their seuerall natures.
HAuing alreadie shewed in the two former Chapters what a Mannoris, with the seuerall parts thereof, and the appendants thereunto: I hold it fitting here now to consider of Estates; as how and by what meanes a man may bee estated either in a Mannor or any other Lands or Tenements: wherein it is to bee vnderstood that all estates in generall consist of two principall kindes as Free-holds and Chattels: which more particularly are sub diuided into diuers other parts or branches; as first Feesimple and Feetaile, which are termed Free-holds of inheritance: also estates After possibilitie of issue extinct; By Curtesie; In dower; and for terme of life: Which foure last mentioned, are called Free-holds, but not of Inheritance: Likewise estates by Copie of Court Roll, being claimed & held by custome, and are diuided into the like parts, as Free-holds at the common Law: and lastly estates For terme of yeeres, and at will, which two last are Chattels. Of all which briefely in order as followeth.
1. Fee-simple.
FEe simple of all other estates is the most large, ample, and absolute that wee haue in this Kingdome, Desined. or that can by our Lawes bee inuested or made; and is that which is granted to any man and his heyres for euer, without any further or other limitation of vse or vses; and therefore if such Tenant hath issue of his bodie, the land descendeth to him, if not, to the next of kinne within the degrees of limitation hereafter specified.
But if a man purchase in Fee-simple to him and his assignes for euer, omitting this word heyres; here hath he but an estate for terme of life; for heires is the word which carrieth the inheritance.
Yet it is otherwise if lands be so deuised by Will; for the Law intendeth that learned counsell cannot alwaies be present in such cases; and therefore is such deuise construed for the best, according to the Testators meaning and intention, and not to the strict letter of the Will.
Also if Lands be granted to any man with a woman in Franke-marriage, this word implyeth an estate of inheritance without mention or addition of the word Heyres; Or to a man and to his bloud the like.
And here is to bee considered, who are those which are said to bee within the degrees of limitation before spoken of; that is, who are vnderstood to be a mans heires by the common Law. Suppose A. B. dieth seised of a state of inheritance without issue of his bodie: Neither his brother, or sifter of the halfe bloud, The halfe bloud. Abastard no beire. Lineall and collater all descent. nor their issue shall be his heyre; nor his bastard; nor his father, mother, grandfather, or grandmother; for inheritance may lineally or collaterally descend, but by no meanes lineally ascend by our Lawes; but the brother or sister of the father of A.B. (which is called a collateral descent) shal be his heyre; and then they dying seised without issue, the father of A.B. shall haue the land as heyre to his vncle or aunt, but not as heyre vnto him.
Likewise it is to be vnderstood, that by the lawes of this Realme the eldest sonne is wholly to inherite; and he dying without issue, the second son, and so of the rest; Coparceners. and if no sonnes but daughters, they shall ioyntly inherite as coparceners; but if no issue neither sonne nor daughter, then shall the eldest brother be heyre, for want of such all the sisters; and in default of them, the vncle by the fathers side, if the Land came by the father, or bee of the purchase of him so deceased: But if there be no heyre of the fathers side, the purchased lands goe to those of the mothers side; But if none such, then all those lands shall Escheate to the Lord of whom they are holden. Escheate.
2. Fee Tayle.
Fee taile of two kindes. Generall tayle.THis estate of Fee-taile is diuided into two kindes or sorts Generall and speciall. The first, being Fee-tayle generall, is when Lands or Tenements are granted vnto any man, and the heyres of his body begotten without limitation, or expresse mention made by what woman; wherefore if such Tenant marrieth diuers wiues, and hath issue by them seuerally; they shall all be capable of the Inheritance of those Lands. But if it be mentioned and expressed in the graunt by what woman, his Heyres shal proceed [Page 187]or be begotten, as if the gift be made to A. B, and to the heyres of his body lawfully begotten on Chis wife, this is an especial In-taile, for any of his issue begotten by another woman, Speciall taile. shall not inherite by force or meanes of this graunt or tayle. And the like in all respects if Lands bee granted to a woman in the like kinde.
Also if Lands be granted vnto AB. and C. his wife, and to the heyres of their two bodies lawfully begotten; here are the man and his wife Ioyntpurchasers, and this is also a speciall tayle both in him and her.
Likewise if any man grant Lands or Tenements to another man with his Daughter in Franke-marriage, this is also a speciall tayle; Franke mariage. and both the man and woman shall be here Tenants in the speciall tayle, for the word Frankemarriage implyeth as much.
Also if Lands be granted to a man and the heyres Males of his bodie; Descent by beires males. this is an estate tayle, and here the Female shall not inherite.
3. After possibilitie of issue extinct.
IF Lands or Tenements be granted to a man and to his wife, Free-bolds. and to the Heyres of their two bodies lawfully begotten, and either of them dye without such issue betweene them; then is he or she suruiuing Tenant in tayle of those Lands, but are without all hope and past possibilitie of hauing such Heyre to inherite those Lands as was limitted in the graunt; & therefore is he or she so suruiuing and ouer-liuing the other, called Tenant in tayle after possibilitie of issue extinct: and from and after the death of him or her so suruiuing; the estate taile so made and granted vnto them, shal be vtterly voide, extinct and dead, as if the same were neuer granted; and the estate of inheritance of and to those lands, shal reuert and turne vnto the first Donor thereof and his heires.
4. By courtesie.
IF a man marrie a wife, being an Inheritrix, and hath issue by her, and she die; by our lawes he shall hold, occupie, and enioy such lands as his wife died seised of either in Fee-simple, or Fee taile, during his naturall life; and he is called Tenant by courtesie of England, because no other Nation admitteth the like estate. Wherin the Law requireth that such issue be vital, & brought forth into the world aliue, although it immediately die, and also it is requisite that the husband bee in actuall and reall possession of those lands, and seised of them in the right of his wife, at the time of her death, or otherwise he shall not be admitted Tenant by courtesie thereunto.
But if any such Tenant by courtesie commit or suffer any stripe or waste, he is punishable in that behalfe, by action of waste.
Also it is to be vnderstood, that no man can bee Tenant by courtesie of a reuersion: for if a woman solye seised in Fee, granteth a Lease to A. B. for terme of his life, and afterwards marry and hath issue, and then dye; the Tenant or Leasee for life suruiuing; her husband in this case shall not be Tenant by courtesie.
5. In Dower.
BY the common Law of this Kingdome, if a man marry a wife, and at any time during the time of couerture he be directly and lawfully seised, either by purchase or descent of any lands or tenements, either in fee-simple or fee-tayle, and being so seised die, his wife shall be indowed of a full third part of all those lands and tenements, during her life; Dower at the common Law. Dower by custome. and being thus indowed, she is called Tenant in Dower, and this is by the common Law.
Besides this, there is Dower by custome, for in some places the woman shall haue a moytie, and in some places more, and in others lesse during her life, of all the lands her husband was seised of at any time during the couerture, according to the custome of the place.
But if the wife be not aboue the age of nine yeeres, at the time of her husbands death; the common Law will not permit her indowment.
A Woman shall haue no dower.And for diuers causes a woman may be defeated of her dower; as if shee or her husband, commit treason, murder, or felony, and be thereof attaynted (yea, though they haue their pardon:) also if she forsake her husband and liue incontinently; and be not againe reconciled without constraint of law; or if shee detayne and with-hold the deedes and euidences from the heire, of those lands, whereof shee claymeth dower, and the like.
And some things there are, whereof a woman is not capable of indowment; as of Commons, Annuities, Estouers sans number, Homages, Seruices, and the like.
There are besides these other kindes of dowers, as one called dowment, Ex assensu patris: another termed dowment ad ostium Ecclesiae: and a third, de la plus bel part: As appeareth at large by our common Lawes; whereunto I referre you.
6. For terme of Life.
A Tenant for terme of life, is he, who holdeth lands or tenements, either for terme of his owne life, or for terme of another mans life; but for distinctions sake, he who holdeth for his owne life, is termed barely Tenant for life, and hee that holdeth for anothers life, is called tenant for terme d'autervie, that is of anothers life.
And if either of these kinde of tenants commit or suffer waste, Waste. the leasor or he in reuersion, shall bring and maintayne against him an action of waste, and thereby recouer treble damages.
7. By Copie of Court roll.
THese tenants are such, as in diuers Mannors hold lands and tenements to them and to their heires, some in the nature of fee-simple, others in fee-tayle, or for terme of life or liues, at the will of the Lord, according to the custome of the Mannor; and in some Mannors they hold by copie for terme of yeeres: And all these haue no other euidence to shew concerning the tenure of their lands, saue only the copies of the rolls of their Lords Court; and therefore are they called tenants by Copie of Court roll.
Alienation of estate.And if any of these tenants alien or sell his lands or estate by deede, he shal absolutely forfeit the same into the Lords hands: wherefore if he will alien [Page 189]his copy-hold estate, he must come into the Lords Court and there surrender the same into the Lords hands, to the vse of him vnto whom he alienateth the same.
But in diuers Mannors the surrender may be made out of Court, Surrender out of Court. vnto any copy-hold tenant, in presence of two of the homage (to the vse as aforesaid) who are to present the same vnto the Steward at the next Court, and admittance made accordingly.
And these tenants can neither sue or be sued, in any of the Kings Courts, by writ or otherwise, for these lands so holden. But they must implead and sue for the same, by way of plaint, in the Lords Court.
And some are of opinion that these tenants are but in the nature of tenants at will of the Lord; who at his pleasure may displace them, Tenants at will. and they without remedie, but by the Lords fauour. Yet others are of a contrarie minde, who maintayne, that if any such customarie tenant (but those for yeeres, their terme being expired; Remedie for copy-holder pat off. paying and doing their seruices) shall without iust cause be eiected and displaced by the Lord; he may bring and maintayne his action of trespasse against him, at the common Law.
And if any of these cut timber, growing on his lands, Waste. without licence of the Lord (but only for repaire of his tenement) it is a waste, and an absolute forfeiture.
And in most Mannors if any such tenant shall farme or let out his land, for any longer time then a yeere, without the Lords licence, Forfeiture. it is likewise a forfeiture vnto the Lord. But of these and many other the like, we are to be guided according to the custome of the Mannor, where such tenants are.
And generally these tenants, Base tenure. for that they haue no free-hold at the common Law, but by custome, are termed tenants of base tenure.
And thus much concerning free-holds and estates of inheritance; and next of Chattels.
8. Terme of yeeres.
A Tenant for terme of yeeres, is he, vnto whom an estate is granted of lands, for any number of yeeres agreed vpon, betweene the Lord and Tenant; which terme is alwayes expressed in the lease so granted.
On which lease there is vsually reserued some annuall rent, Rent reserued. payable either halfe yeerely or quarterly, according to their contract. For the recouerie and obtayning of which rent, if it happen to be ariere and vn-paid, Distresse or action of debt. the leasor is at his choyce; whether he will enter and distrayne, or bring his action at the common Law for the same.
And in these leases for terme of yeeres, whether by writing or otherwise, No liuery of seisin. there neede no liuery of seison, but the tenant may immediately enter by vertue of his lease without further ceremonie. But in leases for terme of life or liues, it is otherwise.
Also if this tenant commit or suffer waste; Waste. the leasor may bring his action of waste against him; wherein he shall recouer locum vastatum, and his treble damages.
And if this tenant shall grant vnto any other man a greater or larger estate [Page 190]in those lands he holdeth, then he hath therein himselfe; whereby hee conueyeth the fee-simple to himselfe, hee shall forfeit his lease and the state, and terme of yeeres therein granted.
9. Tenant at will.
A Tenant at will, is he, vnto whom lands or tenements are granted, to hold at the will of the Lord or leasor, by whom they are granted. And this tenant may bee displaced or put out at any time, Displaced at pleasure. without further notice, at the Lords pleasure; yea, although he hath tilled and sowne his grounds. Libertie to take his corne and goods. Yet in this case the law alloweth him free libertie of ingresse, egresse, and regresse, aswell to take, cut, and carry away his corne when it is ripe; as to take and carry away his goods, and house-hold-stuffe, within conuenient time; without punishment of committing trespasse, or otherwise; for that be knew not his Lords intention or time of entrance. But with tenant for terme of yeeres it is otherwise.
Like remedie for rent.And the Lord or leasor here hath the like remedie against this his tenant at will, for his rent, if it be behinde and vn-paid, as he hath against the tenant for terme of yeeres, last before mentioned.
And it is to be noted, that this tenant at will is not by the law charged or chargeable with reparations, No action of Waste. as is the tenant for yeeres; and therefore no action of waste lyeth against him; vnlesse he wilfully committeth waste, by pulling downe the buildings, or felling of timber, &c. In which case it is held, Trespasse. that the leasor may bring his action of trespasse, and recouer his losse sustayned.
And thus much briefly concerning estates in land; whereof the two later kindes are termed Chattells reall; and all moueable goods are called Chattels personall; as appeareth by this breuiate following.
An ANALYSIS or briefe resolution of all estates in generall.
They consist of
-
Free-holds,
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At the common Law.
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Fee-simple,
- By purchase.
- By descent.
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Fee-tayle,
- Generall.
- Speciall.
-
Free-hold,
- After possibilitie of issue extinct.
- Curtesie of Eng.
- Dower.
- Terme of life.
- Terme d'auter vie.
-
Fee-simple,
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By Custome.
- These are diuided as free-holds at the common Law.
-
At the common Law.
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Chattells,
-
Reall.
- Terme of yeeres.
- Wardship of lands.
- Tenure at will.
-
Personall.
- Corne, Cattell, Money, Plate,
- House-hold-stuffe, &c.
-
Reall.
CHAP. IIII. Of the diuersitie of Tenures, and their seuerall natures, with the seruices belonging.
HAuing informed our selues, as before, of the diuersitie of estates (for all lands whatsoeuer, consist of some of those, formerly mentioned) let vs here, next, consider of the seuerall tenures and seruices thereunto particularly belonging; which are maine and principall obseruations to be had and vsed in suruey of a Mannor, and most meet and necessarie for a Surueyor to know and vnderstand. Whereof in order as followeth.
1. Knights seruice.
THis tenure of Knights seruice includeth Homage and Fealtie, and commonly Escuage; and whoso holdeth any lands or tenements by this seruice, is bound by the Lawes of this our Realme, to doe vnto his Lord Homage and Fealtie; Homage. being a seruice of the greatest humilitie and respect, that can be performed by a tenant vnto his Lord: and for the most part, he is to pay Escuage, called Scutagium, that is, seruice of shield; which is to be assessed by authoritie of Parliament, as shall be hereafter declared.
How the tenant shall doe homageWhen the tenant shall doe homage to his Lord, the Lord shall sit, and the tenant kneeling downe before him, on both his knees, and holding both his hands betweene his Lords hands, shall say thus: I become your man, from henceforth, of life and member, and of earthly honor, and to you will be faithfull and true, and faith to you shall beare for the lands I hold of you (sauing my faith which I owe and beare vnto our Soueraigne Lord the King) and then the Lord so sitting, shall kisse him.
How fealtie is to be done.Fealtie is as much to say, as Fidelitas, or fidelitie. In doing whereof, the tenant shall lay his right hand on a booke, and say thus: I will be vnto you my Lord, faithfull and true, and faith to you shall beare, for the lands and tenements which I claime to hold of you, and duly shall doe and performe vnto you the customes and seruices which I ought to doe, at the termes assigned, as helpe me God; and then he shall kisse the booke. And it is to be noted, that Homage must be done vnto the Lord himselfe personally; but Fealtie may be made to the Steward of the Court, or to the Baylife thereof. Also, tenant for terme of life shall doe Fealtie, but not Homage.
What dutie belongeth to Escuage.As concerning Escuage: hee that holdeth his lands by a whole fee of Knights seruice; when the King goes in person to the warres, he is bound to be with him by the space of fortie dayes, sufficiently appointed for the warres: and he that holdeth by the moitie of the fee of Knights seruice, is bound by his tenure to be with the King in such sort as before, by the space of twentie daies; and so proportionally, according to the quantitie and rate of his tenure.
[Page 192]And it is to be vnderstood, that after the Kings returne from the warres, a Parliament is vsually called; Parliament. by authoritie whereof, it is assessed, what and how much, in money, euerie man that holdeth by a whole fee of Knights seruice (who was neyther personally, nor any man for him, with the King) shall pay vnto the Lord of whome he holdeth his land, by Escuage: and according to such assessement, euerie tenant shall pay to his immediate Lord, What Escuage to be paid. after the rate and proportion of his tenure: and this money thus assessed, is called Scutage, or Escuage; for which, the Lord vnto whom the same is due, may distraine for non-payment thereof.
Yet some tenants by custome, Custome. are not otherwise bound, but to pay a moitie or third part of what shall be assessed, as aforesaid.
And in some places the custome is, that whatsoeuer be assessed by Parliament, yet their Escuage is certaine, and they pay neither more nor lesse, but such a summe of money, Escuage certaine as fiue shillings, or the like; and this is called Escuage certaine: and this tenure is Socage tenure, and not Knights seruice; the Escuage whereof is alwayes vncertaine, and so called. And this Escuage vncertaine (belonging alwayes to Knights seruice) draweth vnto it Ward, Marriage, and Reliefe, as hereafter appeareth.
2. Ward, Marriage, and Reliefe.
AS formerly appeareth, Knights seruice (the tenure last before mentioned) draweth vnto it Ward, Marriage, and Reliefe; and therefore I hold it fitting here next to treat thereof. Wherefore, first it is to be vnderstood, that if a man hold any lands or tenements by this tenure, and dieth, his heire male being within the age of one and twentie yeares; the Lord of whome those lands are holden, shall haue the Ward, that is, the custodie and keeping of those lands so holden of him, Ward. to his owne vse and behoofe, without account, vntill the heire come to the full age of one and twentie yeares: for it is intended by the Law, that vntill he attaine to that age, he is not fit or able to performe such seruice, as by this tenure is required.
And if at the time of the death of such tenant the heire be vnmarried, Marriage. the Lord shall not onely haue the wardship of his bodie and lands, but the bestowing of him in marriage.
And if a tenant by Knights seruice die, and leaue an heire female, being of the age of foureteene yeares, Heire female. or vpwards; then the Lord shall not haue the ward of such heire, The full age of a woman. neyther of her bodie nor lands; because a woman of that age may haue a husband able to performe the seruices required by this tenure.
But if such an heire be vnder that age of foureteene yeares, and vnmarried at the time of her ancestors death, then shall the Lord haue the wardship of her lands so holden of him, till she attaine to the age of sixteene yeares; by force of an act of Parliament, in the Statute of Westminster 1. Cap. 12.
And note here the great difference betweene the ages of Males and Females: [Page 193]for the Female hath these seuerall ages appointed vnto her by the Law. First, Diuersitie of the ages of women. at seuen yeeres of age the Lord her father may distrayne his tenants for aide to marry her. Secondly, at nine yeeres of age shee is dowable. Thirdly, at twelue yeeres of age shee is held able to assent to matrimonie: Fourthly, at foureteene she is able to haue her land, and shall be out of ward, (if shee be of this age at the death of her ancestor.) Fiftly, at sixteene shee shall be out of Ward, though shee were vnder foureteene yeeres of age at the death of her ancestor: and sixtly, at one and twentie yeeres shee is able to make alienations of her lands or tenements. Mens ages. But the Law limiteth to the Male, only two ages, that is, at fourteene yeeres to haue his lands holden in Socage: and at one and twentie to make alienations.
As concerning Reliefe, Reliefe. if a man hold his lands by Knights seruice and dyeth, his heire male being of the age of one and twentie yeeres, or his heire female of the age of fourteene yeeres, the Lord of whom such lands are holden, shall haue reliefe of the heire.
Also it is to be noted, that all Earles, Barons, or other the Kingstenants (holding of him in chiefe by Knights seruice) if they dye, their heire being of full age, as aforesaid, they ought to pay the old reliefe for their inheritance, What reliefes are to be paid, and how. that is, the heire of an Earle for a whole Earledom an hundred pounds; the heire of a Baron for a whole Baronie a hundred markes; the heire of a Knight fiue pounds, and he that hath lesse shall giue lesse, according to the old custome of Fees. And the like is to be vnderstood and obserued of all others, that hold such lands immediately of any other Lord.
And also a man may hold lands of a Lord by two Knights fees, and then the heire being of full age at the death of his ancestor, shall pay to his Lord for reliefe ten pounds.
3. Castle garde.
IT is also to be vnderstood, that a man may hold lands by Knights seruice, Defined. and not by escuage, nor pay escuage for the same: But hee may hold by Castle garde, which is to keepe a tower or some other place of his Lords Castle, on a reasonable warning, when the Lord heareth of the approch of his Enemies.
This is likewise Knights seruice, and draweth thereunto ward, marriage and reliefe, in all respects as the common Knights seruice doth, before mentioned.
4. Grand Sergeantie.
ALso there is another kinde of tenure in Knights seruice, Defined. which is called Grand Sergiantie, and this is where a man hold any lands and tenements of the King, by some such certayne seruices, as he ought to performe in proper person, as to beare the Kings Banner, or his Speare, or to conduct his Host, or to be Sewer, Caruer, &c. and such seruice is called Grand Sergeantie, that is, a high or great seruice, because it is the most honorable and worthy seruice that is: for whoso holdeth by escuage, is not tyed [Page 194]by his tenure to performe any other more speciall seruice, then another holding by escuage, but he that holdeth by grand Sergeantie, is tyed to performe some speciall seruice to the King.
And if a man hold land of the King by grand Sergeantie, and dye; his heire being of full age, then shall his heire pay vnto the King, not only fiue pounds, as he that holdeth by escuage, but also the cleere yeerely value of such lands, as he shall hold by grand Sergeantie.
And also in the borders of Scotland, diuers hold their lands of the King by cornage, Cornage. which is to blow a horne, to giue notice to the Countrey of the Enemies approch, which seruice is also a kinde of grand Sergeantie.
No tenure in grand sergeantie, but of the King.And it is to be vnderstood that none can hold by this tenure of grand Sergeantie, of any other Lord saue only of the King.
5. Petie Sergeantie.
PEtie Sergeantie is, Defined. where a man holdeth lands or tenements immediately of the King, and for his seruice in respect thereof, is bound to pay vnto the King yeerely, a Bow, a Speare, a Dagger, or some such other small thing belonging to the warre. And this seruice is in effect no other then Socage, because the tenant is not tyed to performe any personall seruice, but to pay somewhat yeerely, as a rent is paid. Wherefore this seruice of petie Sergeantie, No Knights seruice. is no Knights seruice. Yet can it not be held of any other Lord saue the King only, aswell as grand Sergeantie.
6. Homage Ancestrell.
IF a man and his Ancestors, whose heire he is, haue holden lands or tenements of another, and his ancestors time out of minde, by homage, and haue done vnto him homage, this tenant thus holding, is called tenant by homage ancestrell, by reason of the long continuance which hath beene by title of prescription, aswell concerning the tenancie in the bloud of the tenant, as concerning the Signiorie in the Lord.
This tenure extinct.And it is to be noted, that if this tenant by homage ancestrell, shall at any time alien those lands vnto another, although he immediately, or at any time after, purchase them againe, he shall no longer hold by this tenure, because he hath discontinued, but shall from thenceforth hold it by the accustomed homage.
7. Socage tenure.
TEnure in Socage is, Defined. where a man holdeth lands or tenements of a Lord by certayne seruice, for all manner of seruices; so as the seruice be not Knights seruice: As where a man holdeth of his Lord by fealtie and certayne rent, for all manner of seruices; or else, where a man holdeth by homage, fealtie and certayne rent, for all manner of seruices; for homage by it selfe maketh not Knights seruice. Also a man may hold his lands by fealtie only, which is likewise tenure in socage. For euery tenure that is not tenure in Chiualrie, is tenure in Socage.
[Page 195]These tenants were tyed in ancient time euery of them with their ploughs by certayne dayes in the yeere, to plow and sow their Lords demeanes, for which cause this tenure was called Socagium, or seruicia socae, Why so called. which is the same with Caruca, one Soke or one plow land. But now that seruice is by mutuall consent, betweene the Lord and Tenant, in most places, Conuerted to rent. conuerted to an annuall rent, yet the name of Socage still remayneth.
Escuage certayne.Also if a man holdeth by escuage certayne, as is before spoken, he holdeth in effect but by socage.
And further, it is to be vnderstood, that when a tenant in socage dyeth, the heire is to pay vnto the Lord, of whom those lands are holden, a reliefe, Reliefe. that is to say, the value of one yeeres rent, besides the yeerely rent, for the payment of which reliefe, the Lord may at his pleasure immediately distraine.
8. Franke Almoigne.
THis tenant in Franke Almoigne, or free almes, Defined. is where an Ecclesiasticall person holdeth lands of his Lord, in pure and perpetuall Almes, which tenure beganne in ancient time, thus: If a man being seised of certayne lands and tenements in his demesne, as of Fee, should thereof infeoffe an Abbot and his Couent, or a Prior and his Couent, or any other Ecclesiasticall person, as a Deane of a Colledge, or Master of an Hospitall, or the like, to haue and to hold the same lands to them and their successors for euer, in pure and perpetuall almes, or in franke almes, in these cases the tenements should be holden in franke Almoigne. By force of which tenure, Seruice to be done. those tenants which hold lands thereby, were bound to make Orisons and Prayers, and to doe other diuine seruices, for the soules of their Grantors and Feoffors, &c. and therefore discharged by the Law, to doe or performe any other profane or corporall seruice, as fealtie, or the like.
Otherwise since the statute.But it is now otherwise, since the Statute called Quia emptores terrarum, An. 18. ED. 1. So as now no man can hold in franke Almoigne, but by force of such grants as were made before that Statute.
9. Burgage tenure.
A Tenure in Burgage is, where an ancient Borough is, Defined. whereof the King is Lord, and they which haue tenements within the same Borough, hold of the King, by paying a certayne yeerely rent, which tenure in effect is but socage tenure. And the like is, where any other Lord spirituall or temporall is Lord of such Borough.
And it is to be noted, that for the most part such ancient Boroughs and Townes, haue diuers and sundrie customes and vsages, Diuers customes. which other Townes haue not. For some Boroughs haue a custome that the yongest sonne shall inherit before the eldest, which custome is commonly called Borough English. And in some places the woman by the custome of the Borough there, shall haue all such lands and tenements in Dower, as her husband at any time during the couerture stood seised of.
Diuers customes contrary to the course of common Law.There are diuers other customes in England, which are contrarie to the course of the common law, which being probable and standing with reason, [Page 196]are good and effectuall, notwithstanding they are against the common law.
No custome without prescription.But no customes are allowable, but those, as haue beene vsed by prescription, or time out of minde.
10. Ancient Demesne.
THere is likewise another tenure, Defined. called ancient demesne, and the tenants who hold by this seruice, are Free-holders by Charter, and not by copie of Court roll, or by the verge after the custome of the Mannor, at the will of the Lord.
And these are such tenants as hold of those Mannors, which were Saint EDWARDS the King, or which were in the hands of King WILLIAM the Conqueror, which Mannors are called the ancient Demesnes of the King, or the ancient demesnes of the Crowne of England.
And to such tenants as hold of those Mannors, the Law granteth many large priuileges and liberties, Quit of toll. as to be quit of toll and passage, and such like impositions, vsually demanded and paid of and by other men, for their goods and cattell, bought and sold in Fayres and Markets by them; also to be quit and free of taxe and tallage granted by Parliament, Free of taxe. except it please the King to taxe ancient demeanes, when he thinketh fit, for great and vrgent occasions. And diuers other priuileges are belonging to this tenure, wherein I refer you to our Lawes.
And if such tenant be at any time distrayned, to doe and performe vnto their Lord any such other seruice or dutie, which they or their ancestors haue not beene accustomed to doe, A wit of Monstrauerunt. they shall sue out a Writ, called a Monstrauerunt, directed to the Lord, commanding him that he distrayne them not to doe other seruices or customes, then they haue beene accustomed to doe.
And it is further to be vnderstood, that in the Exchequer there is a booke remayning, Doomes-day booke. called Doomes-day-booke, which booke was made in the time of S. EDWARD the King, and all those lands which were in the seisin and in the hands of the said S. EDW. at the time of the making of the said booke, are ancient demesnes.
And thus much concerning the diuersitie of tenures and seruices. Now next let vs consider of the rents thereon vsually reserued, and the seuerall kindes thereof.
CHAP. V. Of Rents, and their seuerall natures.
COnsidering that on euery tenure there is vsually some rent or other reserued: I hold it not vnfitting to say somewhat here concerning the same. And first, it is to be vnderstood, that as there are diuersitie of tenures, so likewise of rents; as one sort which is called a Rent seruice, Diuersitie of Rents. another Rent charge, and a third Rent seck, or Redditus siccus, a drie rent.
Rent seruice.As concerning rent seruice, it fitly hath the name, for that it is vsually [Page 197]tyed and knit to the tenure; and is, as it were, a seruice, whereby a man holdeth his lands or tenements, or at least, when the rents are vnseparably coupled and knit with the seruice. As for example; where the tenant holdeth his land of the Lord by fealtie and certaine rent, or by homage, fealtie, and by certaine rent, or by any other kind of seruice and certaine rent, this rent is called rent seruice.
Distresse of common right.And here is to be noted, that if at any time this rent seruice be behind and vnpaid, the Lord of whome the lands or tenements are holden, whether in fee simple, fee tayle, for terme of life, for yeares, or at will, may of common right enter and distraine for the rent, though there be no mention at all, nor clause of Distresse put in the Deed or Lease.
The nature of this rent seruice I say is to be coupled and knit to the tenure; and therefore, where no tenure is, there can be no rent seruice: wherefore, if at this day I be seised of lands or tenements in fee simple, and make a Deed of Feoffement thereof vnto another in fee simple, and reserue by the same Deed a rent, this can be called no rent seruice; for that there can be now no tenure betweene the Feoffor and Feoffee. But it is otherwise of Feoffements made before the Statute of Quia emptores terrarum, Anno 18. Ed. 1. formerly mentioned. For before the making of that Statute, if any man had made a Feoffement in fee simple, and had reserued thereon vnto himselfe a certaine rent, although it had beene without Deed, here had beene created a new tenure betweene the Feoffor and Feoffee, and the Feoffee must haue holden of the Feoffor, who by meanes thereof, might of common right haue distrained for such rent: but since the time of that Statute, there can be no such holding or tenure created or begun; and consequently, No rent seruice can be now reserued on gift in fee. no rent seruice can at this day be reserued vpon any gift in fee simple; except in the Kings case; who being chiefe Lord of all, may, and euer might, giue lands to be holden of him.
Thus it is apparant, that at this day no subiect can reserue any rent seruice vnto himselfe, vnlesse the reuersion of those lands, so by him graunted, be still in himselfe: as where hee graunteth them in fee tayle, or maketh but a lease for terme of life, or for yeares, or else at will; for in all these cases, the reuersion of the fee simple remaineth still in him: wherefore, if any rent be here reserued, it is to be called a rent seruice; and of common right is distrainable, although there be no clause of Distresse comprised in the Deed or Lease.
And if a man shall absolutely and wholly graunt away in fee simple anie lands or tenements by him so holden, leauing no reuersion thereof in himselfe; and yet shall reserue vnto himselfe in his Graunt an annuall rent; with a clause of Distresse in his Deed indented, That it shall be lawfull for him to distraine for the same, if need require; this rent (in regard, that the land is therewith charged) is called a rent charge: Rent charge. But hee cannot distraine for this rent of common right, as before for the other, but onely by force and vertue of his Deed indented.
And if there be no such clause of Distresse contained in the Deed, then is this rent so reserued called a rent seck. Rent seck.
Also, if a man standing seised of lands and tenements in fee simple, will graunt either by Indenture, or poll Deed, an yearely rent vnto another, issuing [Page 198]out of the same lands, whether it be in fee simple, fee tayle, for terme of life, for yeares, or at will, with clause of Distresse; then this rent is called a Rent-charge, and he vnto whome this rent is graunted, may for default of payment enter and distraine.
And it is further to be vnderstood, that if a man make a lease vnto another for terme of life, and reserue thereon vnto himselfe an yearely rent, and afterwards graunteth that rent vnto A.B. reseruing the reuersion of the lands vnto himselfe; Rent seck. this rent is but a Rent seck: for that A. B. who hath the rent, hath nothing in reuersion of the land.
And if a man giueth lands and tenements in tayle, and reserue to him and his heires a certaine rent; or else make a lease for terme of life, reseruing certaine rent; if he graunt the reuersion to another, and the tenant attorne accordingly, the whole rent and seruice shall passe by this word reuersion, because the rent and seruice in such case be incident to the reuersion, and passe by the graunt of the reuersion; Rent charge. and here is the rent a Rent-charge. But if hee had graunted the rent onely, it had beene then a Rent seck.
CHAP. VI. Of Reprises and Deductions.
AS we haue formerly vnderstood, what seuerall rents, profits, and commodities may yearely arise or grow out of any Mannor to the Lord thereof; so is it as fitting to consider, what Reprises, Deductions, Payments, Charges, and Duties, may be yearely issuing or going out of any Mannor from the Lord thereof: For otherwise, in the conclusion of our Suruey, or in making a perfect Constat, or Particular, (such duties not being reprised) the true value of the Mannor may oftentimes seeme greater then in truth it is; which would tend much to our shame and discredit.
These Reprises and Deductions are neuer certaine, or in all Mannors alike; but in this more, and in that lesse: yet in one and the same Mannor they are commonly the same, and vsually such as these here following.
Reprises are any manner of Rents, either in Money, Capons, Hennes, Pepper, Cummin seed, or the like, issuing and payd out of one Mannor to another: also, Suits of Court, or annuall fines for the same; and the like may be issuing and payable to a Sherifes Turne or Hundred; also Pensions or Portions to Ecclesiasticall liuings: likewise a rent may be issuing for way-leaue, or some particular Passage; also for Water-courses, or placing of Pipes for conueyance of Water: likewise yearely Fees to Officers, as Stewards, Receiuers, Baylifes, Collectors, Keepers, &c. and also stypends, salaries, or annuities to Chaplins, or the like: All which are euer to be deducted and reprised out of the totall value of a Mannor.
And hauing thus furnished and informed our selues, first of the Mathematicall part of Suruey, by the three former Bookes; and thus farre of the Legall, as to [Page 199]know what a Mannor is, and the seuerall parts thereof; and likewise of all estates in generall; and what Tenures, Seruices, and Rents are thereunto incident, appertaining, and belonging; let vs proceed in an orderly and formall course; supposing we are now to vndertake the suruey of a Mannor, which is to be performed as followeth.
CHAP. VII. Obseruations and courses to be held and taken, before the beginning of a Suruey.
IT is first to be considered, for whome the businesse we vndertake, is to be performed: if for the King, then are we to obtaine Commission from his Maiestie out of such Court or Courts as is requisite, according to the tenure of the lands to be surueyed, as the Exchequer, Duchie, &c. In declaring the forme of which Commissions, I need not spend time, for that they are vsuall, and of ordinarie course (in such cases) graunted out of those Courts. Yet seeing, that for the most part, those Commissions giue power to the Surueyor, by reference to certaine articles annexed; and according to the efficacie and force thereof, the power and authoritie of the Surueyor is limitted; it behooueth to haue those articles as ample, full, & forcible, as you may deuise; not knowing with what people you are to deale (who often proue obstinate) nor the nature, estate, or condition of Tenancie (for the most part variable.) Which articles let be these here following, or the like in effect.
Articles to be inquired of, and courses to be obserued and held by A. R. in this present Commission named, for the better effecting and execution of his Maiesties seruice, in surueying of his Highnesse Honors, Lordships, or Mannors of A. and B. in the Countie of C. and of all Castles, Houses, Parkes, Mesuages, Lands, Tenements, and Hereditaments thereunto belonging and appertaining.
1. FIrst, the said A.R. is to enter into the said Honors, Lordships, and Mannors, and all and singular other the premisses, and euery of them, and into euery part and parcell of them, and euery of them, and to make a suruey of the quantitie, qualitie, and yearely value thereof, and of euery part, parcell, and member thereof respectiuely.
2. Also the said A.R. is to call before him all such as now are or formerly haue bin Stewards, Baylifes, Reeues, or Collectors of all or any his Maiesties issues, rents, reuenewes, and profits within the premisses, and their deputies, and euery or any of [Page 200]them, and to charge them on their oaths to deliuer in vnto him true and perfect Rentalls of all and euery their seuerall collections: and likewise to call before him all and euery such person and persons, as haue or are suspected to haue any Euidences, Court Rolls, Rentalls, Bookes of Suruey, Couchars, Terrars, Escripts, Writings, or Mynuments whatsoeuer, touching or concerning the said Mannors, Messuages, Lands, Tenements, and Hereditaments, and euery or any of them; and all and euery such person and persons to examine vpon their oathes concerning the same writings, and euery of them: and also to demand, require, and receiue of them, all and euery such Bookes, Rentalls, and other Writings, as he shall so find to be in their hands or custodie: And if any shall make refusall of the deliuerie thereof, to certifie his or their name and names, and the reason of such his or their refusall, to the Lord Treasurer of England, and Chancellor of the Exchequer, that speedie and due courses may be therein held and taken accordingly. But this is to be vnderstood of Bookes and Writings not being in the hands of the present Steward or Stewards of any of the premisses, nor in the custodie of any of his Maiesties Officers of his Highnesse Courts of Record at Westminster; whereof he is only to take and extract notes, for his better instruction and information concerning the premisses.
3. Likewise he is to inquire, what are the seuerall limits, butts, and bounds of all and singuler the premisses, and to expresse the same accordingly; and what Lord or Lords are conioyning or boundering thereon; & whether they or any of them haue or do intrude or incroach vpon or within the limits or boūds aforesaid, or the liberties or priuiledges comprised within the same.
4. Also, whether the premisses, or any part thereof, doth lye or extend into any other Mannor; and whether any other Mannors, Messuages, Lands, or Tenements do lye within the limits or bounds of the premisses; whose, and what they are; and to make perfect distinctions thereof particularly.
5. What Castles, and other Mannor or Mansion houses his Maiestie hath within the same; in what estate of reparations the same now are and be; and if decayed or wasted, by whome the same hath bin committed, & to what value; what demesne lands now are, or heretofore haue bin, belonging or appertaining to the said Houses, and in whose tenure and occupation the same [Page 201]now are; by what right or title they clayme or challenge to hold; what seuerall rents they pay in respect thereof; and what is the true quantitie, qualitie, and yearely value of the premisses.
6. What Forrests, Parks, and Chases his Maiestie hath within the premisses; what number and store of Game are in them; what Officers are thereunto belonging; what Fees they receiue in respect thereof; in what estate of reparations, the Houses, Lodges, Walls, Pales, and Fences are; what is the quantitie, qualitie, and yearely value thereof by the acre; what Iuistments, or what Cattell, as Oxen, Kyne, Horses, or the like, are vsually depastured within the same; who hath the disposall thereof; and what is the value of a Beast-gate there.
7. Also, what Moores, Marshes, Heaths, Wasts, or Sheepewalkes, his Maiestie hath of, in, vpon, or belonging to the premisses; what are the seuerall quantities thereof; how many Sheepe may be kept on those walks; and what is a Sheepe-gate worth.
8. He is also to inquire, what Free-holders there are within and belonging to the premisses; what Mannors, Messuages, Lands, or Tenements they hold thereof, and what are their seuerall quantities; and likewise, by what seuerall tenures, rents, and seruices they hold the same.
9. Also, what other estates there are; as tenants for terme of life, or liues, yeares, or at will; what customarie or copy-hold tenants, or what other tenants there are within the premisses; what lands they do seuerally hold, and the true quantitie, qualitie, and yearely value thereof seuerally, and what yearely rents they pay for the same.
10. Also, what are the seuerall customes concerning the customarie tenants; whether their fines vpon death or alienation be certaine, or incertaine, and arbitrable; and if certaine, what Fines they vsually pay on euery death or alienation of Lord or Tenant; and how, and in what manner, doe those customarie lands descend after the death of an ancestor.
11. What Reliefes, Heriots, Fines, or other duties are payd, or answerable, vpon or after the death or alienation of any Freeholder, Copy-holder, or other tenant within the premisses; how and by whom are they vsually collected and disposed of; and what may be the value thereof in Communibus annis.
[Page 202]12. Whether any customarie tenants (whose lands are hariotable) haue seuered, aliened, diuided, or dismembred the same, who hath the vse and occupation thereof, and what are the seuerall quantities, qualities, and yeerely values of the same.
13. Also, what are all and euery the customes in generall of, within, or belonging to the premisses, and how, by what meanes or for what cause, may a copy-holder or customarie tenant, forfeit or lose his customarie estate.
14. What Commons there are, of, within, or belonging to the premisses, whether stinted or vnstinted; if stinted, then how, by what meanes, and according to what rate and proportion, how many beast-gates they contayne, the value of each beast-gate, and the quantitie, qualitie, and value of the whole.
15. What arable fields and meddowes there are, which lie in common, what are their seuerall names, and of the seuerall furlongs and wents therein contayned; also how and in what manner they are kept and vsed; whether is it lawfull for any tenant at his pleasure to inclose any part thereof, without leaue of the Lord; how are they imploied when the corne and grasse is taken away, how stinted, and what is the eatage therof worth by the beast-gate, or sheepe-gate, after the corne and grasse is so taken off, as aforesaid.
16. What woods or wood-grounds his Maiestie hath within the premisses, what grounds haue beene heretofore wood, and now conuerted to other vses, how long since & by whom, what wastes and spoiles haue beene had or made of his Maiesties woods, how long since, by whom, and of what value; whether may any profit by pannage be made orraysed, by, or within the same woods, and what the profit or value thereof may yeerely be.
17. Also what tenants there are within the premisses, who demise or let any part or parcell of their lands or tenements vnto vnder-tenants, either for their whole terme or any part thereof, and what fines and rents haue or doe they receiue for the same.
18. Likewise, what lands, tenements, rents, seruices, or other profits, are concealed or detayned from his Maiestie, how long since, by whom, and what the yeerely value thereof is.
19. What lands, tenements, leases, or other estates of, or in [Page 203]the premisses, haue beene or are escheated, or forfeited to his Maiesty, by whom, when, for what cause, and in whose occupation the same now are, and what is the value thereof.
20. What fines, issues, amerciaments, perquisites of Court, heriots, waifs, strayes, felons goods, and other casualties doe yeerly accrue and grow vnto his Maiestie out of the premisses, by whom the same is, or hath beene collected, gathered, and receiued, and what is and hath beene the value thereof yeerely in communibus annis.
21. Also what inclosures and Incrochments haue beene heretofore made of, in, or vpon any of his Maiesties commons, wastes, or other grounds, how long since▪ by whom, what rents are paid for the same, and what the yeerely value thereof is.
22. What Corne-mills, Fulling-mills, or other mills, his Maiestie hath within the premisses, who hold the same, what rents they pay, what is the yeerly value thereof, what customes are thereunto belonging, and in what estate of reparations are all and euery those Mills.
23. What Markets and Fayres are there within the premisses, on what dayes kept, what tolls are belonging to the same, by whom the same is collected and receiued, and what yeerely profit ariseth thereby vnto his Maiestie.
24. Also what Warrens, Fishings, Fowlings, Hawking, Hunting, or other Royalties his Maiestie hath within the premisses, by whom the same is occupied or enioyed, what rents are yeerely paid for the same, and what is the yeerely value thereof.
25. What quarries of stone, mines of Tinne, Lead, Cole, or other mines his Maiestie hath within the premisses, who hath the vse and occupation thereof, what rents they pay for the same, and what the yeerely value is.
26. What Mosses of peate or turffe, what Broome, Heath, Furze or Flagge, are within the premisses, belonging to his Maiestie, what are the rents and yeerely values thereof.
27. What Aduowsons, Patronages, Free-gifts or presentations of Parsonages, Vicarages, Chappells or Prebends, or what Impropriations, are appendant or belonging to the premisses, who is or are the present Incumbent or Incumbents, who hath the vse of such Impropriation, what rent is paid for the same, and what is the yeerely value thereof.
[Page 204]28. Whether any Tenant or other person or persons whatsoeuer haue plowed vp, cast downe, remoued or taken away any meere-stone, baulke, hedge-row, or land-share, betweene the demeanes of the premisses, and any other messuages, lands, or tenements, or between any the freeholds, and the tenement or customary lands, or betweene any of the premisses, and the lands of other Lords, by whom such offence was committed, and where, and in what place and places those altered bounders ought to stand and remaine.
29. Also what Officers his Maiestie hath within the premisses, what fees doe they yeerely receiue in respect thereof, what rents, deductions, reprises, or other payments or summes of money, are yeerly paid, reprised, or issuing out of his Maiesties reuennues of the premisses, and to whom, for what cause, and to what end and purpose are the same so paid.
30. And lastly, the said A. R. is to make all and euery other such further and other inquirie and inquisition of, for, and concerning all and euery such matters and things whatsoeuer, as in his discretion shall be held fit and requisite, for the better effecting and execution of his Maiesties seruice, in surueying of the premisses.
These Articles or the like, being drawne and faire written in Parchment by the Surueyor (the commission being to be taken out of the Exchequer) a briefe warrant is to be directed to one of the Remembrancers, and written vnder the Articles to this effect.
M. I. O. These are to will and require you immediately to cause a Commission to bee made, and directed to A. R. for the suruey of his Maiesties Honors, Lordships, and Mannors of A. and B. in the Countie of C. and of all Castles, Houses, Parkes, Messuages, Lands, Tenements, and Hereditaments, thereunto belonging or appertayning; wherunto is to be annexed the abouementioned Articles. Whereof fayle you not: and these shall be your warrant in this behalfe. From the Court, &c.
[Page 205] Which Warrant is to be signed by the Lord Treasurer, or Chancellor of the Exchequer, and deliuered to the Remembrancer accordingly.
But if the businesse vnder-taken, be not for the King, but for a priuate man, then in regard that a Surueyor hath no power by any authoritie of Surueyorship, to be granted vnto him by any such priuate man, to minister an oath, or performe such other duties as are requisite, it is fitting either that the Steward of the Mannor, which is to be surueyed, ioyne with him, in calling a Court Baron, and Court of Suruey, to be there held (wherein the Steward is to giue the charge and to deliuer Articles, and minister oathes, aswell concerning the Court Baron as Court of Suruey.) Or otherwise, the Surueyor is to haue a commission, grant, or deputation from the Lord of the Mannor vnder his hand and seale of the office of Steward and Surueyor of his Mannors, Lands, and Tenements, for a certayne terme, or during pleasure: And then may the Surueyor, of himselfe execute all those offices and duties fit and requisite for a Steward and Surueyor, to doe and performe. Which commission, grant, or deputation, let be thus, or to the like effect.
OMnibus ad quos hoc praesens scriptum peruenerit A. B. de C. Comit. E. Armig. salutem. Sciatis me praefat. A. B. tam pro sincero amore & beneuolentia qua iamdudum affectus sum, erga A. R. de cuius prouida circumspectione, pia sedulitate, ac singulari in hac parie prudentia merito plurimum confido, quam pro diuersis alijs causis & considerationibus, ex mera & spontanea voluntate mea dedisse & per praesentes concessisse eidem A. R. Officium Seneschall. siue Seneschalciam omnium & singulorum Domin. Maner. & haereditament. meorum quorumcun (que) in Comit. F. & custod. siue officium tenendi omnes & omnimodas Cur. Baron. Letar. Ʋis. franc. pleg. Dominior. & Maner. praedict. & eorum cuiuslibet, ac gubernationem & superuisionem eorundem. Ac ipsum A. R. generalem ac capital. Seneschall. ac Superuisorem meum omnium Curiarum, Dominiorum, Maneriorum, & haereditament. meorum praedict. facio, constituo, & ordino per praesentes. Habend. tenend. gaudend. exercend. & occupand. Officia praedict. cum pertinentijs, à dat. praesentium durante bene placito meo. Mando insuper vniuersis & singulis Balliuis, Praeposit. Firmarijs, tenentibus & occupatoribus meis praemissor. & eorum cuilibet, quod praefat. A. R. de tempore in tempus, assistentes sint, obedientes, & auxiliantes in omnibus prout decet durant. termin. praed. In cuius ret testimonium huic praesenti scripto meo sigillum meum apposui. Dat. &c.
Or to the same purpose in English.
And being thus authorised, we may now proceede.
CHAP. VIII. What courses are first to be held in the beginning of a Suruey.
COnsidering how precious time is, and withall, how chargeable these imployments are to those whome it concerne; it behooueth a Surueyor (respecting his credit and reputation) so to appoint and dispose of his businesse in an orderly course, as no time be idly lost, or vainely spent therein. Wherefore, first let the Baylife of the Mannor be called, and a Warrant or Precept directed and deliuered vnto him, to summon as well a Court Baron (if need require) as a Court of Suruey; to this, or the like effect.
Branton. A. R. Seneschall. & Superuis. Manerij praed. Balliuo eiusdem, salutem: Tibi praecipio pariter & mando, quod diligenter praemonere facias omnes tenentes infra Maner. praedictum, tam residentes quam non residentes, at (que) omnes tenentes custumarios Manerij praedicti, quod sint coram me in hac parte sufficienter deputato apud Branton pradictam, die Lunae secundo die Septembr. proximè futuro post datum huius, Non solum ad faciend. sectam suam ad Curiam Baron. & Superuis. sed etiam ad producend. & ostendend. omnes Literas, Chartas, Instrumenta, Indentur. copias Cur. Rotul. ac al. euidenc. vnde tenere vendicant seperal. terr. & tenement. suas de Manerio praedicto; & omnia alia quae eis incumbent, & pertinebunt; & haec nullatenus omittas, & habeas ibi hoc praeceptum: Datum sub sigillo meo vicesimo quarto die Augusti, Anno Regni &c.
Or to the like effect in English.
Wherein let a conuenient time be limited, as fiue or sixe daies at the least after notice giuen for the tenants apparance, that they may the better prepare themselues, and be the more inexcusable, if they happen to make default.
Then are you to receiue from the Baylife all such Rentalls as he hath concerning his whole collection, as well such as are ancient, and of former times, as those of his last collection; which you are diligently to compare together, noting the difference: and if the later be lesser, then what decayes of rent there are, and how occasioned; if greater, then what increment of rent, and whereon raysed; which you are carefully to note and expresse, when you come to ingrosse your Rentall.
[Page 207]Next would I haue you to reduce your Rentall to an Alphabeticall forme: wherein, vse all the moderne Tenants names; not omitting the ancient; which will be a great helpe for the speedie dispatch of your entries; and the readie finding of any Tenants name, or rent, as you are to vse them: which would be written thus, or in the like manner.
- Atkinson Thomas, late Brownes,— xx s.
- Armstrong William, late Tomlinsons,— v s.
- Bennet Iohn, late Brights,— xv s.
- Branthwait Edward, late Finches.— xxij s.
And in this sort proceed Alphabetically with all the whole Rentall; which is much auaileable, where manie Tenants are.
And hauing thus prepared your Rentall in a readinesse, against you haue occasion to vse it; you are to spend the residue of the time, vntill the day appointed for your Court, in conferring with the Steward concerning the present estate of the Mannor; and in diligent view and search of the Lords Euidences and Court Rolls; taking them orderly before you, and from yeare to yeare briefely to expresse in a Booke, for that purpose prouided, the seuerall Customes, Estates, Tenures, Rents, and Seruices, and all other remarkable things. So shall you be able fully to informe your selfe of the nature, qualitie, estate, and condition thereof; and to vnderstand, what articles are now most fit and apt to be presented vnto your Homage, or Iurie, to be inquired of, when you haue giuen them their charge; which you haue now also fit opportunitie to write in a readinesse for them against that time. And these articles would I haue to be such as are expressed in the last Chapter, or so manie thereof as you hold fitting and necessarie for the purpose; and (if need require) to adde and insert such others thereunto, as you shall find meet and requisite, for as much as by the last article annexed to your Commission, you haue power and authoritie to make such further and other inquisition, as in your discretion shall seeme fitting.
Also, now haue you conuenient time, either to ride or walke abroad, and to take a respective view of the situation and extent of the Mannor; whereby you shall be able to informe your selfe, where, how, and in what sort you may with most conueniencie begin, continue, dispose of, and performe your Instrumentall mensuration, either by your selfe or seruants, whome you imploy therein: wherein, for manie respects, I would haue nothing done or performed, before the first day of your Court be past; when as you haue read and made knowne your Commission, and settled an orderly course with the tenants, for their attendance, aide, and assistance in that behalfe.
And thus, and in this like manner, may you spend the time to good purpose, till your Court day come.
CHAP. IX. The order of keeping a Court of Suruey.
IF a Court Baron be kept with your Court of Suruey, as is euer most fitting, then are you first to enter the stile of the Court in this manner.
Branton. CVria Baronis & Superuis. A. B. Armiger, ibidem tent. die Lunae, videlicet secundo die Septem. Anno Regni IACOBI, Dei gratia Angliae, Franciae, & Hyberniae Regis fidei defens. &c. xiiij. & Scotiae 50. tent. per A. R. Seneschallum & Superuis.
After the stile of the Court thus entred, you shall cause the Bayliffe, who serueth the Court, to make Proclamation by crying once Oyes, and then shall you will him to say, thus; All manner of persons, who were summoned to appeare here this day, to serue the Lord of the Mannor, for his Court now holden, draw neere and giue your attendance, and euery one answere to his name, as he shall be called, vpon the payne and perill that may fall thereon.
Then by your Rentall, call them all seuerally by their names, marking those which are absent to be amerced. Which done, cause the Bayliffe to make another Oyes, and willing them to draw neere, and keepe silence whilst the Commission be read; let the same be read vnto them, and likewise the Articles thereunto annexed, if it be for the King.
Then out of those tenants which are present, make choise of the most sufficient for your Iurie, wherein your best course is, formerly to informe your selfe, and to take speciall notice, who are most fitting for your purpose, and to haue their names readie written in a paper by themselues, which you may now thereby call accordingly. But being for the King, you haue alwayes a writ of assistance directed to the Sheriffe of the shire, requiring him to returne you a sufficient Iurie: yet may you without him by vertue of your Commission, impannell any Iurie at your owne pleasure. Then direct the Fore-man of the Iurie, to lay his hand on the Booke, and sweare him as followeth, or to the like purpose.
YOu shall diligently inquire and make true presentment of all such matters, as on the Lords behalfe of this Mannor, shall be giuen you in charge, you shall neither for fauour, feare, affection, or other parciall respect whatsoeuer, forbeare to present what you ought to finde, or finde what you ought not to present, you shall herein keepe the Lords counsell, your owne, and your fellowes, and in all things according to a sincere and vpright conscience, you shall present the truth, the whole truth, and nothing [Page 209]but the truth, as by euidence and your owne knowledge you shall be induced, to the best of your power, so helpe you God, and by the contents of the Booke, which he is to kisse.
And after the Fore-man is thus sworne by himselfe, then cause three or foure of the rest of the Iurie, to lay their right hands together on the Booke, and giue them their oath, as followeth.
THe same oath which A. B. your Fore-man before you, for his part, hath made and taken, you and euery of you, for your parts shall truly keepe and performe to the vttermost of your powers, so helpe you God. And cause them seuerally to kisse the Booke.
And in like manner sweare all the rest. And all being sworne, cause the Bayliffe to number them, as you reade their names. Then cause him also to make the third Proclamation, and say thus: All you that be here sworne, draw neere and hea [...]e your charge, and all the rest keepe silence.
Then make your exhortation, and deliuer the charge of a Court Baron, after the vsuall manner. Which being finished, you are to addresse your speech vnto them, concerning the present businesse of Suruey, as occasion shall be offered, whereof to prescribe you any forme or president, were to little purpose, seeing it is to be framed and directed to such ends and purposes, as the present cause requires, which you shall alwayes finde different and variable, and therefore I refer the same to your owne discretion, deeming you now able and fitting sufficiently to performe the same in any kinde.
And then deliuer vnto them the Articles which you haue readie drawne, according to the directions of the last Chapter, which is their charge concerning the businesse of Suruey, relating vnto them, that as they receiue these Articles (whereof they are to inquire) in writing, so are they to answere the same in writing vnder their hands and seales particularly by a day, now to be limitted and appointed, which for many reasons is most fitting to be, about the time of your concluding the businesse. Which day is to be expressed vnder their Articles, and your name subscribed thereunto.
And now are you to take order, and giue speciall directions vnto all the tenants for their attendance, aide, and assistance, in your instrumentall mensuration, appointing them by turnes, how, when, and where you are to vse their helpe and assistance, wherein you are to deale with such discretion, as you neither faile of their helpe, when occasion serueth, nor oppresse them with grieuance by their ouer-much attendance.
And hauing thus farre proceeded, the rest of this day may be spent in entering their seuerall deedes, euidences, and estates, in manner as shall be hereafter declared.
But before you discharge the tenants, you are to consider (according to the number of them) in what time or how many dayes, you shall be able to enter their estates, and if they consist of diuers Towneships, as large and spacious Lordships vsually doe; then your best course is to appoint them seuerall [Page 210]dayes for their attendance, and bringing in of their euidence by seuerall Towneships, for it would be no lesse troublesome to your selfe, then distastefull to the tenants, to require their generall and daily attendance vntill the businesse were wholly finished.
And now may you aiourne the Court vnto the next day, (or such other time as you thinke fitting) by causing the Bayliffe to make proclamation to that purpose, and the like from time to time, till you haue ended your businesse.
The next day you may beginne your mensuration in the fields, either by your selfe or those whom you imploy to that purpose, according to the instructions of the third Booke. But it were fitting for your owne part to be imployed in entering of the tenants estates, vntill you haue finished, or you may spend such time therein, as when the weather is not fitting to stirre abroad, or in the mornings and euenings, as you shall finde meetest for your purpose.
CHAP. X. The order and manner of entering the Tenants euidence, and their seuerall estates.
FIrst, it is to be considered, that most Mannors (as if formerly spoken) consist of diuers Towneships or particular parts, and the tenancie of those Towneships of diuers estates, as Freeholds, Copy-holds, &c. Wherefore, I hold it sitting and an orderly course, that not only euery of those Towneships, but the seuerall estates therein, be entered and taken seuerally and particularly by themselues, that is, all of one and the same Towneship and estate vnder one and the same title, for auoyding of confusion. As, suppose you are to suruey the Mannor of Branton, which consisteth of these seuerall Towneships or parts, Branton, Bodley, and Sutton, and within those Towneships, are diuers tenants, holding their lands by seuerall estates, as Free-hold, Copy-hold, &c. Then would I haue you make your seuerall entries vnder those seuerall titles whereunto they properly belong, as vnder the title of Branton towneship Freehold, enter all those which are of that Towneship, and of that nature: and vnder the title of Branton towneship Copy-hold, enter all the Copy-holds of that towneship, and the like of all the rest.
And these entries I would haue made in loose sheets of paper at large, keeping them alwayes sorted, according to the seuerall Towneships and estates, till you haue finished all your entries, and then to file them together orderly in a Booke, each Towneship following other.
In which seuerall entries obserue this course▪ hauing written your title as before, in the head or top of the sheet, then enter the tenants name, and the very words of grant, as they are in his Deede, Copy, or Leafe, which is to be written from the margent the whole bredth of the sheet, leauing only towards the right hand a space, wherein is to be expressed the rents, and seruices, and in the margent alwayes expresse the tenure.
[Page 211]And considering, that in few or no Deedes, Euidences, Copies, or Leases, the lands are particularly expressed by particular names, closes, and quantities, as the tenant now holdeth the same, and as you shall find them in your instrumentall mensuration; hauing entred the effect of the Deed, Copie, or Lease, according to the purport thereof: it is fitting to question with the tenant, what seuerall parcels he holdeth, for and vnder the seruices and rents contained in euerie Deed, Copie, or Lease particularly; as, what Meadow, what Arable, what Pasture, and their seuerall names and quantities, as he esteemeth them; and if he know not what acres they containe (as most tenants will seeme ignorant thereof) let him expresse of his Meadow how many daies mowing, of his Arable how many daies plowing, and of his Pasture how many Beast-gates, and the like: for although it be not greatly materiall for these their giuen quantities, in respect you measure euery particular; yet this kind of entrie will serue you to good purpose, as shall hereafter appeare. Also, you are to question him concerning his Pastures, as well those in seuerall, as the Pastures in common, what euery Beast-gate is worth by the yeare in euery of them seuerally; whereof you may otherwise also informe your selfe, lest you be deceiued. The forme of which entries let be in this manner.
BRANTON Towne-ship Free-hold.
Socage,
ANTHONIE BORNE holdeth freely to him and his heires for euer, by deed indented, bearing date 14. die Ianuarij, Anno Reg. Reginae Elizabeth. &c. 30. made and graunted by and from WILLIAM BATEMAN, of &c. All that Mesuage or Tenement (expressing the verie words of Graunt) On which Graunt is there a deed of Feoffement of the same date, with liuerie of seisin thereon past accordingly; by the yearely rents and seruices of Fealtie, & iij. s.
- THe mansion-house, out-houses, and scite, consisting of two Orchards, three Gardens, and two Yards or Garths. containing together— 3. r.;
- A Close of Meadow, called Broad Meade, containing— 10. a.
- Another called White-thorne close, containing— 8. a.
- Meadow in the common Meadow, called Long meade, three parcels containing — 12. d.
- A Close of Arable, called Bennets, containing— 7. a.
- Another of Arable, called the High Close, — 15. a.
- Arable in the South field, in six parcels, which containe all together— 4. a. 3. r.
- [Page 212] Arable in the North field, fiue parcels containing— 6. 2. 2. 1.
- One Close of Pasture, called the Oxe Pasture, containing 30. beast. gates, at 13. s. 4. d. le gate, — 30. gates.
- Another called the Calfe Close, containing 12. beast-gates, at 8. s. le gate, — 12. gates.
- In the great common Moore, at 2. s. 8. d. le gate, — 20. gates.
- In the common Cow-pastures, at 6. s. 8. d. le gate, — 10. gates.
- On the Downes depasturing, for 150. sheepe, at 3. d. le gate,— 150. gates.
- Common sans stint on the Moores.
And in this order, and vnder this Title, enter all the Free-holds within the Towne-ship of BRANTON: But withall, obserue this course in all your entries; that these particulars, in the entring of them thus in your rough Booke, exceed not, nor extend past halfe the breadth of euerie sheet or leafe, or little further, because directly after the seuerall contents, before specified, there is particularly to be expressed afterwards, the true quantities of euery parcell found by measure; and after that, the seuerall and particular yearely values thereof, as shall be hereafter shewne.
In like manner, let your tenants for life or liues, in euerie Towne-ship, be entred vnder their due and seuerall titles thus.
BODLEY Towne-ship for liues.
THOMAS HOCKLEY holdeth by Indenture, bearing date 23. die Nouembr. Anno Reg. Reginae Elizabeth. 32. made and graunted by and from A. B. All that Mesuage or Tenement (vsing the words of Graunt) for the terme of the naturall liues of the said THOMAS HOCKLEY now aged 50. yeares; of IANE his wife now aged 42. yeares, and IOHN their sonne now aged 30. yeares, successiuely each after other: and payeth rent per annum— xx. s.
Particular.
ANd here write the particulars in forme as before.
Then vnder the foot of those particulars make a briefe Memorandum of such necessarie obseruations as are to be noted, thus.
There is due vnto the Lord on the death of euery of them dying tenant in possession, the best Beast Nomine Heriot. The tenant is to pay his rent quarterly, or within one and twentie dayes, on paine of xx. s. Nomine poenae; or within fortie dayes, on paine of forfeiture by prouiso. To doe all manner of reparations (except great tymber.) Not to let the whole, or any part, without [Page 213]the Lords licence. The Lord warranteth the premisses against him and his heires.
And the like notes may be expressed vnder all your other entries.
Where note, that speciall care is to be alwayes had in the entrance of these grants for liues, whether they are all ioynt purchasers, as all named in the words of grant, or whether only granted to one during all their liues; wherein there is great difference; which is to bee noted, and the verie words of grant to be euer precisely expressed in your entries as before.
As concerning your Copihold Tenants let them be entred thus.
SVTTON Town-ship Copi-holds.
W. B. holdeth the Copie of Court roll bearing date viij. die Iunij Anno Regni Regis IACOBI Angliae iij. of the surrender of C. D. One messuage or tenement (according to the words of the Copie) late E. F. and before that G. H. To him and his heires at the will of the Lord according to the custome of the Mannor. For which he payd fine XL s. And payeth rent per annum. XXX s.
Particular.
Then write your particular as before: and after that such briefe Memorandums as you shall finde fitting, according to the former directions in that behalfe.
And the like course are you to holde with Tenants for terme of yeeres, and at will; and all the rest: vntill you haue fully finished all your entries. Then place all your leaues in order; and if you bee assured that all are entered (which will easily appeare by your Rentall, if you alwayes crosse the same, as you enter them seuerally) then write a new Rentall according to this your rough Booke, and ioyne the same to the beginning of your Booke, and file all together; But let your Rentall bee alphabeticall; or else if your booke be verie large, make an alphabeticall Index or Table of all the Tenants names, to be placed in the beginning of the Booke, before the Rental, with numbers of reference to the number of leaues; whereby instantly you may turne to any Tenants name therein as you shall haue occasion to vse the same.
And thus haue you finished with the Tenants concerning their seuerall entries; but haue not yet fully perfected your rough Book; whereby you may be able to ingrosse the same. Wherefore proceed therein as is taught in the next.
CHAP. XI. The meanes and order of perfecting the Booke of entries last mentioned, and the due placing therein of the seuerall contents of euerie particular found by measure through the whole Mannor, with the valuation thereof.
WE are now to suppose before wee beginne this worke, that not onely all the Instrumental mensuration throughout the whole Mannor is finished; but also the first plot drawne; and the seuerall contents thereof cast vp, and expressed therein; with the proper and particular numbers, seuerally belonging thereunto, according to the instructions of the Third Booke.
Which being effected, we are first to make an Index or alphabetical Table of al the Tenants names; wherunto is to be added the Lords and the Parsons, the one for Demesnes, the other for gleabe; which Index is thus to bee composed. If one sheet of paper will not serue, you are to take two, three or more, and with mouth-glue fasten them end to end, making thereof a long scrole or schedule of the bredth of the whole sheet. And with a small margent towards the left hand, let it from thence be ruled with blacke or red Inke ouerthwart the whole paper, of the distance of lines in ordinarie writing; betweene which lines downe by the same left margent write all the Tenants names each vnder other alphabetically, leauing all the rest of the ruled Paper to be thus imployed.
First, take your Field-booke, and beginning where you first beganne your worke in the fields, take all the numbers before you expressed in the margent thereof, as appeareth in Chap. 10.3. and noting to what Tenants name they belong in your Field-booke, against the same name place them in your Index betweene the ruled lines; whereby you shall speedily expresse and reduce the seuerall numbers representing the seuerall fields and closes throughout the whole Mannor against euery mans name in the Index to whom they properly belong.
Then take your common Field-booke (mentioned in Chap. 40.3. and with it performe the like; but make a stroke or other marke for distinction betweene these and the former numbers; so haue you also euerie mans particular parcels lying in the common fields expressed against his name. And thus is your Index perfect and fit to be imployed as followeth.
Take now your rough Booke of entries, and turne to the first Tenants particular therein entred; and looke in your Index what numbers are belonging to that Tenant; also take your Field-booke, and comparing those numbers in your Index with the same in the margent of your Field-booke, you shall find therein the seuerall names of the fields and closes belonging to that Tenant; and the like names shall you find in your Booke of entries, in that Tenants particular: wherefore expresse those seuerall numbers against euerie particular parcell in the margent of your entries, whereunto they properly [Page 215]belong: and the like performe in all respects with your common field-booke, for those parcels lying in the common fields. Then lay your rough plot before you, and finding those seuerall numbers in your plot, note the seuerall content and quantitie of euery seuerall parcell of ground, expressed in the plot represented by those numbers; and those quantities expresse and write downe in figures particularly in your booke of entries, to euery parcell whereunto they belong next after the quantities deliuered by the tenant. As for the particular quantities in the common fields, you shall not find them in the plot, but in your common field-booke, according to the direction thereof in Chap. 40.3. before recited; which let thereby be expressed accordingly.
And the like course in all respects is to be holden with all other the entries throughout your whole booke.
But it is to be considered, that one tenant may within this Mannor hold lands of seuerall estates, and by seuerall rents and seruices, as Free-hold, Copy-hold, Tenement Lands, &c. yet are they all comprised and represented within and by those numbers expressed in the Index. In such case you are to compare these numbers with the particular names expressed in your field-booke, and those with the entries; and you shall most easily distinguish the one from the other, and expresse and assigne to euery of them their due and proper number and quantitie, as before.
Also, it is here to be considered, that we haue not yet spoken of any means to expresse each mans particular quantitie in the common stinted pastures, or sheepe-walkes, whereof onely the generall quantitie is taken by measure, and expressed in the plot. Which to performe, worke thus.
Suppose there is in the Mannor a stinted Cow-pasture, wherein euery tenant hath a certaine number of Beast-gates, some more, and some lesse, which are vsually rated and stinted, either according to their rents, or after their quantities of knowne grounds, or their parts in the common arable fields: and imagine this pasture is found to containe by measure 212. Acres, 3. Roods, 20. Perches; first, collect out of your entries all the number of gates in the same pasture; which added together, let containe in the whole 116. Then reduce your measured quantitie into the lowest denomination, as Perches (according to the directions of Chap. 48.3.) whereby you shall find the same to be 34060. Perches: and supposing the tenant, whose quantitie you seeke, hath in the same pasture 10. beast-gates; by the rule of proportion reason thus. If 116. the whole number of beast-gates, giue 34060. Perches, the whole quantitie, what giues 10. gates; and by increasing 34060. by 10. and parting the product by 116. your answer will be 2936. Perches, and a small quantitie more, the proportionall quantitie belonging to 10. gates; which reduced into acres (by Chap. 48.3. before recited) is 18. Acres, 1. Rood, 16. Perches. And in like manner worke with all the rest. And here is to be noted, that hauing thus gotten the proportionall quantitie belonging to a beast-gate, and truely vnderstanding the value of a beast-gate, you shall be able at pleasure, and most certainely, to expresse by the acre the true yearely value thereof.
[Page 216] And thus haue you perfected your entries, for the true and certaine quantities, according to measure; and now resteth the valuation.
The best, speediest, and most certaine meanes for your valuation, in mine opinion, is thus: Let it first be considered, that all grounds generally consist in qualitie of these three kinds, Meadow, Arable, and Pasture; and supposing euery of these kinds likewise to consist of three sorts in value and goodnesse; as the first and best sort; the second and meane; and the third and worst sort: In your instrumentall mensuration, when you write in your field-booke the title of euery field or close, consider with your selfe, which of those three sorts the same field or close consisteth of; if of the best sort, expresse in some place of your title the figure 1. if of the second sort, the figure 2. and if of the third and worst sort, the figure 3. and hauing informed your selfe by the best meanes you can (which I hold not fitting here to relate) of the generall value, what the best sort of Meadow, Arable, and Pasture, is worth by the acre, and the like of the other sorts, throughout the whole Mannor; then, according to those rates, passe ouer your whole booke of entries, and value euery particular parcell by it selfe; which, by finding in each title of your field-booke of what sort they are, is most speedily and exactly performed. And thus are your entries thus farre perfected.
But yet, before we proceed to the ingrossing hereof, or rather before wee finish with the tenants concerning their entries, it is to be enquired and considered, what other profits and commodities, besides these lands and tenements, are demised and granted by the Lord to any tenant within the Mannor for yearely rent, or otherwise; which likewise are to be entred and expressed in your rough booke, with the rents and yearely values thereof: as Mynes of Tynne, Lead, Copper, Coale, &c. Quarries of Stone, Fishing, Fowling, Hawking, and Hunting, Iuistments, Herbage, and Pannage, free Warrens, Customarie workes, or Seruices, profits of Fayres and Markets, and mosses of Peat or Turfe; all or any of which, and the like, may be within a Mannor, and disposed and letten for yearely rents, which by no meanes are to be omitted. All which premisses, and the seuerall quantities, rents, and values thereof, are here to be summed vp, and their seuerall totals expressed.
Then are you to expresse the seuerall reprises issuing out of this Mannor, being such as are mentioned in the sixt Chapter of this Booke. All which being likewise summed vp, the totall thereof is to be deducted from the former value, and the cleare remainder expressed.
Next are you to consider, if any of those profits and commodities, last before named, or the like, are within this Mannor, and not letten by lease, or otherwise, for any certaine yearely rent; and if any such be, then are they to be here mentioned and expressed as casualties, and the yearely value thereof estimated, what they may or are likely to proue worth by the yeare.
Also, the names and quantities of the common Fields, common Meadowes, stinted Pastures, and all other vnstinted Commons, are here to be expressed; and of those vnstinted Commons, how they are accustomed, held, and occupied; whether peculiar to the Lord and tenants of this Mannor; or whether any other Lord or Lords, and their tenants, haue rake, escape, [Page 217]eatage, or other interest therein; and the butts, bounds, and limits thereof seuerally.
Also, what woods and vnderwoods are within the Mannor, and their seuerall values.
Then would there be entred an abstract, in nature of a Custome-roll, shewing briefely all the ancient customes of and belonging to the Mannor. And also a Suit-roll of all the free suitors, &c.
And lastly, a true and perfect description of all the out-bounds and limits of the whole Mannor.
And hauing thus perfected your rough Booke, you may now call your Iurie, and receiue their verdict, as in the next.
CHAP. XII. The manner and order of receiuing the Juries verdict, and the courses therein to be obserued.
WHen you are drawing towards a conclusion of perfecting your Booke, according to the directions of the last Chapter; it were not amisse, that you hasten the Iurie, in perfecting their verdict, lest you be forced to spend time idly in attendance for the same: Which when they haue effected, considering, that for the most part they are vnacquainted with matter of forme, though in effect and substance they may answer your desire and expectation, according to the articles deliuered vnto them; you are to call them before you, and reading, examining, and comparing the articles, with their seuerall and particular answeres thereunto, reduce the same into an orderly forme of an Inquisition, obseruing still the substance of what they haue found and presented: and then hauing read the same vnto them, with their approbation and allowance thereof, cause your Clarke to ingrosse the same accordingly in parchment, and then let them againe consider thereof; and hauing set their hands and seales thereunto, demaund of them, if they are mutually agreed on this their verdict; which when they haue acknowledged, receiue the same from the Fore-man, and dismisse your Court.
And here haue you finished what here need to be performed: and now may you leaue the Mannor of BRANTON; and repairing homewards, may there perfect your plot, as you are formerly directed by the third Booke; and ingrosse your Suruey, as is hereafter declared in the next.
CHAP. XIII. The forme and order of ingrossing a Suruey.
TO prescribe and direct one certaine and setled forme and course herein for all in generall, were impossible, in respect of the varietie of occasions offered, according to the nature of the businesse, and the disposition of those for whome the same is performed: wherefore, the performance hereof must mainely depend vpon the iudgement, skill, and discretion of the Surueyor. For mine owne part, I neuer yet for any two Lordships or Mannors limitted my selfe to one and the same forme; but euer framed my course as the cause required: as in one Mannor, where I find a commixture of other lands and tenements within the same, being holden of other Lords; here of necessitie must I abutt and bound euery seuerall particular thereof: but in another, which I find entire, I hold it needlesse. Againe, I find in one Mannor diuers and seuerall Towne-ships and parts, and those to consist of seuerall estates and tenures; in such case, these are to be seuerally distinguished and diuided, according to their seuerall parts. Another shall you find sole and entire, which is to be ordered accordingly: and many other such like differences shal you find, which will minister occasion to alter any setled forme.
Besides, the will and disposition of him by whom you are imployed, shall often cause you to alter your course: one perhaps approuing of the forme you vse; another will haue it in the nature of your ancient Terrars; a third, in order of a Particular, or Constat; and a fourth, it may be, in a fourth forme; for Quot homines, tot sententiae. And againe, one, for his owne vnderstanding, will haue it in English; and another, of better vnderstanding, will require it in Latine. And certainely it were very requisite, although your rough booke be drawne in English, that alwaies your ingrossed booke be written in Latine; vnlesse the contrarie be specially required. Yet in these mine examples and directions following, I hold it most fitting to deliuer the same in English, for the better vnderstanding of those who haue most need; considering, that a reasonable Surueyor may be lame of that legge.
But notwithstanding such varieties often happen; yet will we for a generalitie propose these rules and directions following; which I hold most meet and fitting to be obserued and held in a formall and well ordered Suruey.
To which purpose, let vs now suppose wee are to ingrosse a Suruey of the Mannor of BRANTON; according to the rough booke thereof, specified in the tenth Chapter of this Booke; wherein first begin with the title, which let be thus, or to the like effect.
[Page 219]An exact and perfect Suruey and view of the Mannor of BRANTON, in the Countie of D. being parcell of the possessions of A. B. who holdeth the same of our Soueraigne Lord the King, as of his Mannor of G. in free and common Socage, and by the yearely rent of xiij. s. iiij. d. Had, made, and taken there, as well by Inquisition, and the oathes of a sufficient Iurie in that behalfe, as by the view and particular mensuration of all and euery the Messuages, Lands, and Tenements, of, within, and belonging to the same. Anno Domini 1616. Annoque Regni Regis IACOBI, Angliae, &c. 14.
By A. R. Superuis.
NExt after this, or the like Title, in the following leafe, are you to write and expresse and Index or Alphabeticall Table of all the Tenants names (as hath beene formerly taught) with numbers of reference against each name, in what leafe or leaues of the Booke each Tenants particular is to be found: But notwithstanding, this Index is to be placed in your Booke first and next after the Title; yet is it most conuenient and fitting, to collect and write the same (and also the Rentall next hereafter following) after the whole Booke be ingrossed; before which time, you shall not know how to place your numbers of reference therein, according to the number of the leaues.
Then after this let next be placed a generall rentall of the whole Mannor, but to be diuided into such towneships and parts, as your Booke is diuided into; wherein first expresse your rents of such demesnes, as are letten in lease, then the rents and seruices of your free-holds of inheritance. Thirdly, of your customarie or copy-hold tenants. Fourthly, of the tenants for life or liues. Fiftly, of those for terme of yeeres: and sixtly and lastly, your tenants at will, wherein let euery of these beare their seuerall titles, and vnder the foot of each kinde, let the totall thereof be collected and expressed, and in the end or foot of the whole rentall, expresse first the totall of euery kinde, and after that the generall and totall summe of all together. And if any rents or other reprises be issuing out of this Mannor, you may here expresse them particularly vnder the title of reprises, which let be deducted out of the former totall, and expresse the cleere remainder.
And thus is our Rentall finished; but to be collected and written (as I formerly noted) after the bodie and substance of the Booke be ingrossed.
[Page 220]And your rentall being thus finished, you may next place (if you please) the out-bounder of the whole Mannor; and if any of the towneships or parts thereof lye dispersed and remote (as in many places you shall finde them perhaps twentie miles distant, from the chiefe and principall part of the Mannor; and sometimes in another Countie) it were very fitting and necessarie, to expresse seuerally the seuerall out-bounders of those towneships and parts. And it is to be noted, that in the expressing of these bounders, a maine and principall care is to be had, that you vse, obserue, and keepe the olde and ancient names of such meeres, markes, and bounds, as haue beene anciently vsed and accustomed; for that innouation in this kinde is very dangerous for many causes; yet if you finde the ancient meeres, markes, and bounds, to very few and slender; or any of them decayed and worne out of knowledge, you may adde as many moe as in discretion you shal finde cause; but by any meanes omit not, or leaue out any of those which are ancient and noted bounds. If you thinke good, these bounders may be placed after, or in the ende of the booke; which being no matter of necessitie whether (so it bee had at all) I leaue to your discretion.
And now are you to begin with the body and substance of the booke; and first of all with the Mannor or mansion house, and the scite thereof; wherein you are to consider, whether the same be in the Lords owne hands and occupation; or whether letten by Lease, or otherwise vnto any Tenant or Tenants, and to enter the same accordingly, as followeth vnder this Title.
BRANTON DEMESNES.
A B. Esquire is Lord of this Mannor, and hath at this present in his owne hands and occupation, the Mannor or mansion house with the scite thereof; and so much of the demesnes thereunto belonging as are hereafter particularly expressed. Which Mannor with all and singuler the appurtenances, he holdeth of our Soueraigne Lord the King, as of his Mannor of G. in free and common Socage, and by the yeerely rent of xiij s iiij d
Particulars.
THe Mannor or mansion house called Branton Hall, being fairely built with free stone, and all offices thereunto belonging, with two Stables, one Oxe-house, and a Doue-house; also the scite consisting of three faire gardens, two orchards, two courts, and three out-yards, lying all together betweene the high streete of Branton south, and the Oxe-pasture hereafter mentioned North; abutting East, on Long meddow, and West on the scite of the Parsonage. And containeth together fiue Acres, two Roodes, and twenty Perches. a. r. p. 5-2-20.
Valet per anum — xx li.
[Page 221]Then next vnto the house and scite, expresse the Parkes (if any be) with the number of Deere therein; what number of Aunteller, and what of rascall Deere; also what number of beasts may bee therein Iuisted without preiudice to the game; and also what pannage; and these may you particularly aburt and bound as before if neede require; which is most easily and speedily done, hauing before you the rough plot of the whole Mannor, and in the conclusion hereof expresse the quantitie and yeerely value as before.
Then after these Parkes, enter particularly all such seuerall fields and closes of the demeanes, as the Lord hath in his owne vse at the time of this Suruey; which you may particularly abutt and bound as before, and expresse the seuerall quantities, and values thereof: But herein for orders sake it were fitting first to enter all the meddow grounds particularly each after other, then the inclosed arable grounds, and next the pastures; and if any of the demesnes are lying in the common fields, then to expresse them particularly with their quantities and values; also you are to expresse what woods the Lord hath; and what right or custome the Tenants haue or claime therein, either for depasturing or otherwise: and lastly, what wastes the Lord hath within the Mannor. And at the foote of this particular, expresse the totall quantitie and value thereof.
But it is to be noted, that all these particulars are to be collected (by help of the numbers in your Index) out of your field booke for the names, and out of the rough plot for the seuerall quantities, for that it is needlesse to enter these lands which are in the Lords hands into your rough booke of entries; and the like course also is to be holden for your Glebe lands.
And if any of the Demesne Lands are in Lease, let them also bee entered vnder the former Title, in this manner.
C. D. holdeth by Indenture of Lease, bearing date the twentith day of Ianuary An. R. Regis Iacobi &c. secundo, made & granted by and from A. B. vnto the said C. D. All those lands, &c. (vsing the very words of graunt) for the terme of one and twentie yeeres, commencing from and after the Feast of the Birth of our Lord God last past, before the date of the same Lease, for and by the paiment of the yeerely rent of — xxx li.
Particular.
THen here enter the seuerall and particular Closes, which you may abutt and bound, as before, expressing the particular quantitie and value of euery seuerall Close, and at the foot of the particular expresse the totall quantitie and value, and if the yeerely value exceede the rent reserued, deduct the rent from the totall value, and expresse the cleere remainder, thus.
Valet ad demittend. — x [...].
[Page 222]And vnder this particular expresse a briefe memorandum of the seuerall couenants, clauses, conditions, and prouisoes in the lease contayned, after this manner.
The Tenant is to pay his rent quarterly, or within one and twentie dayes after euery Feast, on payne of forfeiture, by prouiso to that purpose. He is to doe all manner of reparations (except great timber) not to let or set without licence of the Lord. The Lord maketh speciall warranties against his father, himselfe and his heires, &c.
And the like course is to be held for all other Leases, after the particulars expressed.
And here also vnder this title of Demesnes, are you to enter all such Mills, Mines of Tinne, Leade, Copper, Cole, &c. also quarries of Stone, Slate, and the like; also Fishing, Fowling, Hawking and Hunting; likewise Iuistments, Herbage, Pannage, free Warrens, customarie Workes or Seruices, profits of Fayres and Markets, and also mosses of Peate or Turffe, and the like, as are letten and Markets, and also mosses of Peate or Turffe, and the like, as are letten and demised by the Lord, to any Tenants within the Mannor by lease for yeerely rent or otherwise: all which (being thus letten) are in the nature of Demeanes, and are to be particularly entred and expressed accordingly, with their seuerall Rents and the yeerely values thereof.
But all of those last mentioned (excepting Mills) are to be seuered and distinguished from the Demesne lands, because they are not matters of firme, stable, and certaine perpetuitie: For notwithstanding, that during the termes of the seuerall leases thereof made and granted, the Tenants may be charged and bound to pay seuerall yeerely rents for the same, which for the time being are certaine; yet perhaps at the end and expiration of those termes, they may be of little or no value at all; or on the other side of farre greater worth and value then now they are, as often happeneth by those mines of Tinne, Leade, Copper, Coles, and the like. Wherefore notwithstanding they are entered vnder this generall title of Demesnes; yet for destinctions sake, let them passe more particularly vnder this title of Casualties made certayne.
And after all these demesnes are thus entred and ingrossed, make a briefe conclusion thereof vnder-neath the same, in this or the like manner.
Conclusion of the Demesnes.
The Demesnes of this Mannor, consisteth of
-
Lands, Tenements, & Mills in the vse of
-
The Lord
- Quantitie—1320 a.—o r.—o. p.
- Value—660 li.—o s.—o. d.
-
The Tenants
- No. of Tenants—5.
- Quantitie—163 a.—0 r.—0. p.
- Rent—54 li.—6 s.—8 d.
- Value—81-10-0.
- Ad demitt.—27-3-4.
-
The Lord
-
Casualties made certayne.
- No. of Tenants—6.
- Rent—25 li.—8 s.—6. d.
- Value—94-18-0.
- Ad demitt.—69-9-6.
And after the Demesnes are thus entred and ingrossed, then next vnto it place the Rectorie or Parsonage, and then the Vicarage (if any be) vnder the proper title thereunto belonging, after this manner.
The Rectorie of BRANTON.
A. B. Clarke, being Parson there, holdeth the Rectorie of the gift of the Lord of this Mannor (if it be so, and if otherwise, expresse it accordingly) who hath the gift, nomination and presentation thereof, as in the right of this Mannor, as often as the same shall happen to be void, which is valued in the Kings bookes per annum. Lvi •
Particular.
THe Parsonage or Mansion-house with the out-houses belonging thereunto, as a Barne, Stable, Oxe-house, and a Doue-coate, with the scite thereof, consisting of two Gardens, an Orchard, and three out-Yards, which lye together betweene the high street of Branton South, and a field called the Oxe-pasture North, abutting towards the East on the scite of this Mannor, and West on a Lane there leading out of the high street into the Oxe-pasture aforesaid, and contayneth together, one Acre and three Roodes. 1 a.—3 r.—0. p.
Valet per annum—iij li.
[Page 224]And in this sort let euerie particular parcell of glebe-Land bee expressed with the buttes and bounds thereof, which by helpe of the plot and field-Booke lying before you (being directed thereunto by the numbers in your Index) is instantly and exactly performed: For these glebe-Lands; and the Demesnes which are in the Lords hands, are neuer entred in your rough Booke of entries. Wherein is alwaies to be obserued; that you expresse the true quantitie and yeerely value of euerie particular parcell; and in the foot of the particular, the totall quantitie and value as before. Yet is it not vsuall neither of these nor the [...] holds of Inheritance to expresse any value at all; which I will referre to your owne discretion, and the will and disposition of those by whom you are imployed. And in like maner are you to expresse the Vicarage if any such bee.
And hauing thus finished your Parsonage, Vicarage and glebe-Lands, proceede next vnto the Free-holds within this Towne-shippe; which are to bee entered and ingrossed after this order, and vnder this title following.
BRANTON Free-holds.
Socage.
A.B. holdeth freely to him and his heires for euer by deed indented bearing date xxviij. die Marcij Anno Regni Regis IACOBI Angliae, &c. Sexto made and granted by and from C. D. All that messuage or tenement (expressing the verie words of grant) By the yeerely rents and seruices of — Fealtie & v. s.
Particular.
THe Mansion house, out-houses and the scite thereof consisting of one Garden, two Orchards and three out yards, lying together betweene the high street of Branton North, and the common field called the South-field South; abutting Easton the Church-yard, and West on a lane leading into the South field. And containeth three roodes and thirtie perches. a. 1. p. 0. 3. 30.
And thus proceed [...]h euerie parcell belonging to this free-holder; which being finished, at the foot of this particular expresse the quantitie and value thereof: But as concerning the valuation of Free-holds, vnlesse it be specially required, by reason of some purchase thereof to be made; or a possibilitie of escheat, or the like, you need not trouble your selfe therewith.
And hauing perfected your particular, expresse vnderneath the same, a briefe Memorandum of such necessarie obseruations as you shall find fitting, aswell concerning the Tenants euidence, as what Heriots, Reliefes and other Duties and Seruices the Tenant ought to yeld, doe and performe vnto the Lord on euery death or alienation.
[Page 225]And in like manner vnder the same Title [...]nter all other the free-holders within this towneship; after which, collect and expresse together their seuerall quantities in one totall summe, and likewise their values (if it bee required as before:) But in these and all others, as I haue formerly noted, I would alwaies haue an orderly course holden in placing the particular lands of one and the same nature and qualitie together, as first (after the house and scite) all the meddow grounds, &c.
And thus hauing entered and ingrossed your free-holds of this towneship, let next be entered your copyholds or customarie Tenants, after this manner, and vnder this Title.
BRANTON Copy-holds.
A. B. holdeth by copie of Court roll, bearing date 26 Februarij Anno Regni Regis Iacobi Angliae &c. 4 to. of the surrender of C. D. Al that messuage or tenement &c. (vsing the very words of the Copie) to him and his heyres at the will of the Lord, according to the custome of the Mannor. For which he paid Fine on his admittance v li. And payeth rent for the same per annum, —. xl s.
Particular.
LEt your particulars here be entred in all respects as b [...]fore with the seuerall buts and bounds thereof, expressing the quantitie and value of euery seuerall parcell: and in the foote of the whole particular expresse the totals as before; then out of the totall value (which admit to bee x li.) deduct the Rent, and expresse the remainder thus.
Valet in toto per annum x li.
Viz. ad dimittend viij ii.
Then vnder this particular thus perfected, make (as before) a briefe Memorandum of such necessarie obseruations as are sitting, as what herio [...] (if any be due on the death of the Tenant) what fines on death or aliena [...]; and what other seruices the Tenant oweth, &c. And after this order and vnder this title enter all the rest of the copy-holds within this towneship.
And the like course in all respects is to bee holden in entrie of the Tenants for life, or liues; for terme of yeeres; and those at the Will of the Lord; whereof to make seuerall demonstrations, and to deliuer seuerall examples, were but great labour to smal purpose, seeing they tend all to one and the same end.
[Page 226] Wherefore take this for a briefe and generall rule, that all lands whatsoeuer, and the tenancy thereof, consist of one of these seuen kindes, which in euery Mannor where they are, are to be vsed as your seuerall titles, and ought to be placed in the ingrossing of your Booke, each after other, as here they are expressed, viz.
- 1. Demesnes.
- 2. Gleabe-lands:
- 3. Free-holds.
- 4. Customarie.
- 5. For liues.
- 6. For yeeres.
- 7. At will.
And hauing after this forme and order entred and ingrossed the seuerall Lands and Tenements, lying within this Towne-ship of BRANTON, vnder the seuerall titles last before mentioned, collect your totall of euery kinde, and in the end of this Towne-ship make your conclusion to this or the like purpose following.
Conclusion of the Towne-ship of BRANTON.
This Towne-ship consisteth of
-
Demesnes in the vse of
-
The Lord
- Quantitie—1320 a.—0 r.—0. p.
- Value—660. li.—0 s.— 0. d.
-
The Tenants
-
Lands
- No. of Tenants—5.
- Quantitie-163 a.—0 r.—0. p.
- Rent— 54 li.— 6 s.—8. d.
- Value—81 — 10— 0.
- Ad demitt —27— 3 —4.
-
Casualties made certayne,
- No. of Tenants—6.
- Rent—25 li. —8 s.—6. d.
- Value— 94 — 18 —0.
- Ad demitt.—69 —9— 6.
-
Lands
-
The Lord
- Gleabe lands Quantitie— 56 a—0-0.
-
Freeholds of inheritance,
- No. Tenants.—7.
- Quantitie—230 a.—2 r.—0. p.
- Rent— 13. s.6. d. — 3. II. pepper.
-
Customarie lands,
- No. Tenants.—16.
- Quantitie—340 a.—3 r.—0. p.
- Rent—17-16-5.
- Value—152-13-4.
- Ad demitt.—134-16-11.
-
Tenement lands.
-
For liues
- No. Tenants.— 12.
- Quantitie—432 a.—2 r.—0. p.
- Rent—143 li.—6. s.—3. d.
- Value—220-0-0.
- Ad demitt.—76-13-9.
-
For yeeres
- No. Tenants. — 23.
- Quantitie—624 a.—2 r.—0. p.
- Rent—156 li.—13 s.—9. d.
- Value—310-18-0.
- Ad demitt.—154-4-3.
-
At will
- No. Tenants.—8.
- Quantitie—120 a.—0 r.—0. p.
- Rent—42 li.—5 s.—6. d.
- Value—58-10-0.
- Ad demitt.—16-4-6.
-
For liues
[Page 228]And hauing thus finished this Towne-ship, proceede in the like forme and order in all respects, with all other the towneships and seuerall parts of and belonging to the whole Mannor, obseruing still after euery towneship and part, to make such or the like conclusion, as is last before specified; and in the end of all a full and generall conclusion of the whole Mannor; not forgetting first to enter all reprises, issuing, and going out of the same, which is to be deducted out of the whole value, as is before declared.
And after this conclusion thus perfected, you are to remember and expresse all such necessarie obseruations, as are fitting, according to the directions in that behalfe deliuered, in the latter end of the 11. Chapter of this Booke, and your worke is finished.
Now might I here inlarge and amplifie this worke with many rules and examples, tending to these purposes, but presuming that what I haue formerly deliuered (being well vnderstood and practised) may sufficiently serue a reasonable capacitie; I will forbeare to pester the practicioner in reading, or my selfe in writing of needlesse varieties; and therefore will here conclude my labours, and expose them to thy good liking.
ERRATA.
PAge 35 line 31. for C B. reade C D. p. 38. lin. last, for cut out, r. cut. p. 42. l. 9. for equiangles, r. equiangled. p. 77. l. 19. for F D. r. E D p. 110. l. 39. for draw the, r. draw to the. p. 118. l. 11. for 24. r. 240. p. 127. l. 17. for 4/3 r. ¾▪ p. 133. l. 10. for 3/2· or 5/4· r. ½ or ¾· p. 146. l. 13. for 15. r. 9 ¼· p. 162. l. 38. for O Q r. X. Q. p. 204 l. 16. for euery other such, r. euery such.
In the Diagrams.
Page 109. in Diagr. 109. place K. in the angle opposite to C. In Diagr. of Chap. 29.3. neere vnto K. place O. opposite to N.