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 MELLIFICIUM MENSIONIS: 
 
 OR, THE 
 
 Marrow 0/Meafuring. 
 
 WHEREIN 
 
 A New and Ready Way is fliewn how to Meafure 
 
 Glazing, ‘Painting, Plaiflering , Mafonry, Joyners, 
 Carpenters, and Bricklayers Works. 
 
 AS ALSO, 
 
 The Measuring of Land, and all other Superficies, and 
 Solids, by Vulvar Arithmetic!:, without reducing the Inte- 
 gers into the leaft Denomination ; giving the Content of 
 any Superficie or Solid, confining of Feet, Inches, and 
 Parts of Inches, in a Fourth Part of the Time and Labour 
 required by the ufual Way in Vulgar Arithmetic it. 
 
 TOGETHER 
 
 With fome Choice Principles and Problems of 
 GEOMETRY conducing thereto. 
 
 To -which is Added, 
 
 An APPENDIX, left under the Author’s own Hand. 
 
 The C 0 NT E NT S of -which, fee in the next Page . 
 
 
 llluf rated with Copper C Ill'S. 
 
 
 The like not heretofore Publiflied. 
 
 
 $Cl)c fourth Cuttton, laid) 
 
 
 By VENTERUS MANDEY. 
 
 
 Non gfuot, Jed gfuale-s. 
 
 
 LONDON: 
 
 Printed for Tho. Tebb, in Little-Britain ; Sam. Illibge, 
 under SerW s Gate, Lincoln s- Inn New-Square ; and Tho. 
 King, in Petty-France . M. DCC. XXVII. 
 
 
N this Edition is added an Appendix. 
 Wherein, 
 
 I. The Errors and Disagreements of 
 feveral Meafurers, in meafuring the Bricklayers 
 Work of Chimneys of all Sorts, are reformed : 
 Which Miftakes and Errors are the more pardon- 
 able, if we confider, that the moft Part of the 
 Work relating to Chimneys, is invifible, or hid. 
 
 II. Rules are laid down for the Meafuring of 
 Arches and Vaults, whether they are Semicircles, 
 Segments of Circles, or Ellipfes, together with 
 their Butments. 
 
 III. The Menfuration of fome Surfaces and So- 
 lids, which were not inferted in the firft and fecond 
 Editions. 
 
 IV. Two Tables, one whereby to deter- 
 mine the Price of one, or any Number of Feet 
 of Brickwork, at any Price per Rod : The other, 
 to find the Perpendicular of any Gable End, by 
 having its Bafe. 
 
 V. Some Alterations in the Method of Mealur- 
 ing are propofed, as being more agreeable to Truth 
 than the Ways heretofore pra&ifed. 
 
 VI. The whole Work is corredted and amend- 
 ed, by omitting fbme Things, and adding others, 
 where Neceffity required it. 
 
THE 
 
 Author’s EPISTLE 
 
 TO THE 
 
 READER. 
 
 Courteous Reader , 
 
 HE Subject of the enfuing Treatife, 
 is the Science of Meafuring in a new 
 and brief Method by vulgar Arith- 
 metick, wherein the laborious and 
 troublefome Part of reducing the 
 Feet and Inches into the lead Denomination, t. e. 
 into Parts of Inches, is omitted, and a brief and 
 fpeedy Way directed for the calling up any Di- 
 menfion whatfbever. 
 
 Which Way of Meafuring, confidering that 
 Workmens Rulers are not divided decimally, and 
 that mod Dimenfions are taken by a Ruler divided 
 into Feet, Inches, Half Inches, and Quarters, and 
 for the mod Part, the Contents are required in 
 
 A 3 the 
 
\i To the Reader. 
 
 the fame Kind ; I fay, Confidering this, it Com- 
 eth very little Ihort of the Decimal Way of Mea- 
 furing. 
 
 For if Dimenfions be given in Feet, Inches, 
 and Halfs, or Quarters of Inches (as moft ufually 
 they are ) and the Contents required in the fame 
 Method, then I look upon the reducing thofe 
 Dimenfions into Decimals, and after the Contents 
 are found in Decimals, the reducing them again 
 into Feet, Inches, and Half Inches, to take 
 up more Time than this Way which is taught in 
 the enfuing Treatife. 
 
 Befides, many Men can multiply and divide by 
 vulgar Arithmetick, which do not underftand De- 
 cimals 3 for whofe Sakes chiefly this is written. 
 
 I conceive every Book meets with many critical 
 Cenfures, and I doubt not, but this will partake 
 with the reft ^ and therefore it might (perchance) 
 be expe&ed, that I Ihould excufe myfelf for what- 
 foever any Man fhall be pleafed to object againft 
 in it, which I fhall neglect, only defiring the judi- 
 cious Reader to pals by fome fmall Overfights, 
 which, perhaps, there may be crept into the en- 
 fuing Work, as knowing that all Mens Doings 
 are fubject to Error : But as for grofs Faults, I 
 think there are none ; for I have been as careful 
 as I could, both in writing, and likewife in cor- 
 recting the Printed Sheets in this Edition, 
 confidering my want of Time, occafioned by my 
 other Employment, 
 
 To 
 
To the Reader. 
 
 VII 
 
 To perfuade any one to the Study of this Sci- 
 ence, would be a Folly, fince the Ignorant (which 
 are blind) cannot judge truly ; and to him that 
 already underftands it, the Labour would be ufe- 
 lefs and unprofitable • and to the Averfe and Care- 
 lefs, it would be like the calling of Pearls before 
 Swine ; the exquifite Knowledge whereof, cannot 
 be attained by fuch prejudicated Perfons, altho 3 
 to many that are judicious, the Ufefulnefs hereof 
 is already fufficiently known ^ and to thofe indu- 
 ftrious Perfons which are yet to feek in the Know- 
 ledge of this Subject, and defire to be informed, 
 this Treatife will fully anfwer their Expectations. 
 
 In the following Treatife, I have endeavoured 
 to proceed methodically, and have, to my Know- 
 ledge, omitted nothing, which might tend to the 
 making of a Man an expert Meafurer ; in order to 
 which, there are three Books of Geometry inferred, 
 the firft containing the Rudiments thereof, and 
 the fecond and third containing choice Problems - 
 which two laft, I tranflated from Latin Copies, 
 wherein I have endeavoured to render the Mean- 
 ing of the Author as plain as the Work would 
 permit, and have amended fbme Miftakes (I flip- 
 pofe) of the Printer of the Latin Copies : I have 
 likewife explained the Meaning of fbme difficult 
 Terms, and alfb added a new Diagram belonging 
 to Chap. 6. Lib. 3. Fig* 1. which was wanting in 
 the Copy. 
 
 So that the whole Treatife confifts of fix Books, 
 a general Account whereof follows. 
 
 A 4 The 
 
via 
 
 To the Reader. 
 
 Thefirft Book of this Treatifo contains the Ru- 
 diments of Geometry , confifting of fuch Definitions 
 and Propofitions, as are meet to be known to any 
 Man that intends the Science of Meafuring, and 
 is an Introduction to the other five Books , in 
 which firft Book (perhaps) fome few of the Defi- 
 nitions may feem ftrange, as not agreeing with 
 the Definitions of fome others, yet, in my Opini- 
 on, agreeable both to Reafon and Truth. There 
 is likewife added, the Defcription of Ovals to any 
 Length and Breadth, with a Pair of Compafles j 
 and alfo a Digrefiion concerning Ovaller Arches. 
 
 The fecond Book, being tranflated from a La- 
 tin Author, contains a Garden of Geometrical 
 Rofesj confifting of choice Propofitions in Geome- 
 try , wherein a new Way is ftiewn of cutting right 
 Lines in extream and mean Proportion ; alfo the 
 Divifion of Angles, and finding a ftrait Line, in 
 Length equal to a Circular j as alfo to find the 
 Centre of Gravity of a Semicircle or Quadrant, 
 with feveral other Things not heretofore publilhed 
 in Englijh , that I know of 
 
 The third Book, being like wife tranflated from 
 the fame Author, contains fome Principles and 
 Problems in Geometry , formerly thought defpe- 
 rate, now briefly explained in Englijh , confifting 
 of the SubjeCt, Principles, and Method of the 
 Mathematicks , and of Algebraical Operations ; 
 likewife of a new Method of treating of Solids and 
 their Superficies, by the efficient Caufes, with fe- 
 veral other Things. 
 
: To the Reader. 
 
 IX 
 
 The fourth Book contains the Science of Men- 
 furation , (hewing how to meafure all kinds of 
 Works relating to Building, to wit, Carpenters , 
 Gl after s, Faint ers , Plaljierers , Mafons , and Brick- 
 layers Works, Be. the Contents whereof may be 
 feen more at large in the following Table. 
 
 The fifth Book contains the meafuring of fiiper- 
 ficial Plains , wherein is (hewn how to meafure 
 triangles of all kinds, Ketlangled Figures , whofe 
 Sides are equal or unequal, Circles , Ovals, Pyra- 
 mids , Cones , Be. In brief, it (hews how to find the 
 Content of moft kind of fuperficial Figures in ufe, 
 whether they be regular or irregular, each Propo- 
 fition having a Figure or Diagram belonging to 
 it, for the more eafier underftanding how to re- 
 folve it. 
 
 This fifth Book alfo (hews how to meafure 
 Land lying in any Form, and how to reduce Cu- 
 ftomary Meafure into Statute Meafure, and the 
 contrary. Be. 
 
 The fixth Book contains the Meafuring of / olid 
 Bodies , (hewing how to meafure 5 timber , and 
 Stone , &c. alfo how to find the folid Contents 
 of Pyramids , Cones , Cylinders , Spheres , and other 
 fuch like Solids, each Propofition having a Dia- 
 gram (or Scheme) belonging to it, for the readier 
 attaining the Refolution of the Propofition. In 
 it is alfo contained the Science of Gauging, 
 or meafuring Liquid VeJJels , and how to find 
 their Contents in Wine or Ale Gallons 3 and like- 
 
 wife 
 
X 
 
 To the Reader. 
 
 wife how to find the Contents of Brewers Veflels* 
 Tuns, or Fats, in Gallons, and to reduce them 
 into Barrels : All which, I have drawn into a 
 Pocket Volume. 
 
 This Treatife thus finifhed, I prefen t thee with, 
 defiring thy friendly Cenfure and Acceptance of 
 thefe my Labours ; as alfo to pafs by fuch Faults 
 as may poffibly have efcaped the Prefs, or my- 
 felf, which, I am certain, are but few : And in 
 fb doing, thou wilt oblige him, who is 
 
 A Friend to All , but more especially to thofe 
 that are Mathematically inclined 9 
 
 V. M. 
 
 A TABLE 
 
XI 
 
 A 
 
 TABLE 
 
 ■ To the feveral 
 
 Chapters, Propositions, and 
 Series of the whole TREATISE. 
 
 The Contents of the Firft BOOK, being 
 the Rudiments of GEOMETRY. 
 
 CHAP. I. 
 
 Defin. a hPoint Page r 
 
 Defin. 2. Of a Line , find its Kinds 
 
 ibid. 
 
 Defin. 3. Of a Super fide 2 
 
 Defin. 4. Of the Extreams of fi 
 Superficie ibid. 
 
 Defin. 5. Of Angles in general 3 
 
 Defin. 6 . Of right Angles, and perpendicular Lines > ibid. 
 Defin. 7. Of ohtufe or blunt Angles , alfo of acute or fharp 
 
 ibid, 
 ibid, 
 ibid. 
 Defin. 10. 
 
 t 
 
 Defin. 8. Of a Cdrcle 
 
 Defin. 9. Of the Diameter of a Circle 
 
xii The TABLE. 
 
 Defin. io. Of the Semi diameter -> or Radius of a Circle 
 
 ibid, 
 
 Defin. ii. Of a Semicircle ibid. 
 
 Defin. 12. Of Trilateral, or T hree-fided Figures 4 
 
 Defin. 13. Of an Equilateral Triangle " ibid. 
 
 Defin. 14. Of an JfofcelesTriangle ibid. 
 
 Defin. 1 5. Of a Scalenum Triangle ibid. 
 
 Deiin. 16. Of Triangles, according to the Quality of their 
 Angles ibid. 
 
 Defin. 17. Of the Bafe of a Triangle 5 
 
 Defin. 18. Of four-fid ed Figures in general ibid. 
 
 Defin. 19. Of a Qiiadrate, or Square ibid. 
 
 Defin. 20. Of an Oblong, or long Square ibid. 
 
 Defin. 21. Of a Rhornbus, or Diamond Figure ibid. 
 
 Defin. 22. Of a Rhomb oides , or Diamond-like Figure, ib. 
 Defin. 23. Of a Parallelogram ibid. 
 
 Defin. 24. Of Trapeziums ibid. 
 
 Defin. 25. Of Parallel Lines ibid. 
 
 CHAP. II. 
 
 Prop. 1. TfRom a Point given in a fir ait Line , to raife 
 ^ a Perpendicular 6 
 
 Prop. 2. To raife a Perpendicular on the End of a Line 7 
 The fame to perform another Way ibid. * 
 
 Prop. 3. To raife a Perpendicular on the End of a Line , 
 having little or no Paper beyond the End of the Line 
 whereon you are to raife the Perpendicular 8 
 
 Prop. 4. XJp on any Angle given, to raife a Perpendicular 
 
 ibid. 
 
 Prop. 5. To let fall a Perpendicular upon a Line given , 
 and from a Point above the Line 
 Prop. 6 . By a Point given, to draw a Line, parallel to 
 right Line given, two Ways ibid. 
 
 Prop. 7. To cut a right Line in two equal Parts 10 
 
 Prop. 8. To cut a right-lined Angle in two equal Parts r 1 
 Prop. 9. To make a Triangle of three right Lines ibid. 
 Prop. 10. By a Point in a right Line given, to make a 
 right-lined Angle equal to an Angle given 1 2 
 
 -f Prop. ii„ 
 
 §4 NO 
 
The TABLE. 
 
 xm 
 
 Prop. ii. STi make a Parallelogram equal to a Triangle 
 given , in an Angle , ^ lined Angle given 
 
 ibid. 
 
 Prop. 12. Upon a right Line given, to make a Parallelo- 
 gram \ at aright-lined Angle given* equal to a Triangle 
 given 1 5 
 
 Prop. 13. Parallelograms {landing upon equal Safes, and 
 the fame Parallels , are equal one to the other 14 
 Prop. 14. Triangles {landing upon the fame Safe, and be- 
 tween the fame Parallels, are equal ibid. 
 
 Prop. 1 5. If a Parallelogram have the fame Safe with a 
 Triangle , and be between the fame Parallels, then is the 
 Parallelogram double to the Triangle 1 5 
 
 Prop. 1 6, Upon a right Line given, to make a Parallelo- 
 gram equal to a right-lined Figure given, at a right-lined 
 Angle given 5 and from hence is eajily found the Excefs , 
 whereby any right-lined Figure exceeds a lefs right-lined 
 Figure 1 G 
 
 Prop. 1 7. In right-angled Triangles, the Square which is 
 made from the Side that fubtends the right Angle, is 
 equal to both the Squares which are made of the other 
 two Sides that contain the right Angle 1 7 
 
 Prop. 18. To make one Square , who fe Area or Content 
 {hall be equal to two given Squares , or to three given 
 Squares 18 
 
 Prop. 19. Two unequal right Lines being given, to make 
 
 a Square equal to the Differences of the two Squares of 
 the given Lines 19 
 
 Prop. 20. Any two Sides of a right-angled Triangle being 
 known , to find out the third ibid. 
 
 Prop. 21 .To defcribe a Circumference that [hall touch any 
 three Points given, Jo they be not in a right Line 20 
 
 Prop. 22 .To defcribe an Oval upon a Length given 21 
 
 Prop. 23 .To divide a fir ait Une into as many equal Pans 
 asyoupleafe ibid. 
 
 Prop. 24. To defcribe an Oval equal in Length to the firfi 
 Oval, 7iot rifing fo high iz 
 
 Prop. 25. Another Way to defcribe Ovals 23 
 
 Prop. 26. To defcribe an Oval* according to any Length 
 and Breadth given ibid . 
 
 Prop, 27. 
 
X1V . The TABLE. 
 
 Prop. 27. Lo find the Center and both Diameters of any 
 
 Oval ?4 
 
 Prop. 28. To defcribe an Oval with a Tair of Compaffts, 
 to any Length and 'Breadth given 2 5 
 
 CHAP. HI. 
 
 jj qyigreJTion concerning Ellipfes, or Ov alter Arches , 
 jl fitting how to defcribe them , and make Moulds for 
 
 them, relating to Bricklayers *6 
 
 To defcribe (trait Arches , and make their Moulds 32 
 To find the diminifnng of the fommering Mould by the 
 Rule of Three , and likewife by Geometry 37,38 
 
 The Contents of the Second BOOK, being 
 a Garden of Geometrical Rofes. 
 
 Prop. cutting right Lines in extrcam and mean 
 
 \J proportion 4° 
 
 Prop. 5 . Of Regular ‘Polygons 49 
 
 Prop * . Of the Proportion of crooked Lines , to crooked 
 
 Lines in the Circumferences of Circles 5 f 
 
 Prop. 5 , 6 , 7 , 8 . Of the Magnitude of an Arch of a Circle 
 
 Prop. 9 . Of the Divifion of an Angle given 7 i 
 
 Prop. io. Of Sines , Subtenfes, and other Lines , z« 
 
 *-*.*»*£ 
 tigstber with the Tangent of go Degrees j6 
 Prop. 12. Aright Line which cuts the Safe of art Equi- 
 lateral Triangle from any Vertical Point in the Middle, 
 in Sefquialter of the Tangent of an Arch of 30 Degiee^ 
 
 Prop i2. The Difference between the greater andlcffer 
 Segment of a right Line being divided in extream and 
 mean ‘Proportion , is double the Difference between 
 
The TABLE. 
 
 xv 
 
 fame right Line , and a right Line whofe ‘Power is to it 
 as 5 to 4 79 
 
 Prop. 14. If the Secant of an Arch of 30 Degrees he cut 
 in extream and mean Proportion , the greater Segment 
 1 vcill he equal to the Semidiagonal of a Quadrate made 
 from the Semidiameter 8 1 
 
 Prop. 1 5, 16. ADigreJJion concerning the Difcord between 
 the Computation of Lines , of Superficies , and of Num- 
 bers , Demonflration of Geometricians 8 3 
 
 Prop. 17. ‘The Side of an Icosahedron, is equal to the third 
 Pan of the greatefl Semicircle in its own Sphere 88 
 
 Prop. 18 . Of a Square equal to the Content of a Quadrant 
 
 Prop. ip. Between a right Line given, and the Half of it, 
 to find two mean Proportionals 100 
 
 Prop. 20 .Of the Center of Gravity of the Quadrant of a 
 Circle 103 
 
 Prop. 21 .Of the Center of Gravity of two Lines , one of 
 which is an Arch of a Quadrant , and the other is the 
 Suhtenfc of the fame Arch 106 
 
 The Contents of the Third BOOK, confifting 
 of fbme Principles and Problems in Geometry , 
 formerly thought defperate. 
 
 Chap. 1 . r\F the Suhjeff , Principles , and Method of the 
 ^ Mathematicks no 
 
 Chap. 2, Of Reafon , or Proportion 1 1 7 
 
 Chap. 3. Of Algebraical Operations 122 
 
 Chap. 4. Of fquare Figures , and fquare Numbers 125 
 Chap. 5. Of Angles 132 
 
 Chap. 6. Of the Proportion of a Perimeter to the Radius 
 of the fame Circle 136 
 
 Chap. 7. Of mean Proportionals 146 
 
 Chap. 8. Of the Proportion of a Square , to the Quadrant 
 
 of a Circle inferihed in it 
 Chap. p. Of Solids , and their Superficies 
 
xvi The TABLE* 
 
 Chap. io. Of a new Method, of treating of Solids , and 
 their Superficies, by the efficient Caufes 162 
 
 Chap. 11. Of ‘Demon fir at ion 172 
 
 Chap. 12. Of Fallacies 1 76 
 
 Chap. 13. Of Infinite 182 
 
 The Contents of the Fourth BOOK, con- 
 fifting of Me a f tiring. 
 
 Chap. 1. r\F Menfuration in general 188 
 
 ^ Chap. 2. Carpenters Work, how meafured 
 
 189 
 
 Chap. 3. Of fuperficial or flat Meafure as relating to 
 Carpenters Work , foe wing what a Dimenfion is, and 
 how to caft up Dimenfions in Inches 191 
 
 How to caft tip Dimenfions in Feet , and to reduce them 
 into fuperficial Squares or Tards 193 
 
 How to caft up Dimenfions in Feet and Inches , and the 
 Reafon of multiplying the Inches into the Feet demon - 
 fitrated, both by Definition and Geometry 196 
 
 How to meafure Roofs two Ways 3 like wife the Meafuring 
 of Flooring, and Roofing of a Building that is wider at 
 one End, than at the other 203 
 
 How to meafure a Gable-End , and to find the Length of a 
 Hip-Rafter for any Building $ likewife the Angles 
 which the Rafters, and Hip-Rafters make 20$ 
 
 How to meafure a Hipp'd Roof 208 
 
 Chap. 4. How to meafure Glaziers Work, wherein is 
 foewn how to multiply Farts of Inches into Feet and 
 Inches, by vulgar Arithmetic k, without reducing the 
 Feet and Inches into the leaf Denomination, proved to 
 be true three fever al Ways 212 
 
 How to fet down Dimenfions in a Fccket-Bock 3 alfo how 
 to make a Bill of Meafurement 225 
 
 Chap. 5. How to meafure Joiners Work , and reduce it 
 into Tards 229 
 
 Chap. 6. Of meafuring of Faint ers Work, Flaifierers 
 Work , andMafonry 231 
 
 Chap. 7. 
 
The T ABLE. xvii 
 
 Chap. 7. How to meafure Bricklayers Work , and reduce 
 it to the tiftal T hicknefs 233 
 
 How to reduce Feet into Rods 237 
 
 How to make an Efiimate for a Building from a Defgn 
 given 3 alfo how to take the Dimenfions , and fet them 
 down in a Book , together with the Method of cafling 
 them up 3 likewife how to fuh raff: ( or deduff ) the 
 Windows and Door -ways out of the folid Brickwork 
 
 238 
 
 How to meafure Chimneys the ufial Way 252 
 
 Notes and Examples for the fpeedier cafling up of Dimen- 
 fions 254 
 
 Some Obfervations in the Me a firing of Brickwork 257 
 
 How to know the Price of any Number of Feet ( of Brick- 
 work) at any Rate by the Rod 3 as alfo of Tyling or 
 Carpenters Work , at any Rate by the Square 242 
 
 The Contents of the Fifth BOOK, con« 
 filling of the Meafuring of Superficial Plains . . 
 
 Chap. 1. ]\/f Ea firing qf Plains, what Page 2 66 
 
 J fJ. p r0 p j How t0 meafure or find the 
 Content of a Rhombus , being a Figure like a Quarry 
 of Glafs 26 j 
 
 Prop. 2. How to meafure a Trapezium, or four fide d 
 Figure 269 
 
 Notes concerning Meafuring 270 
 
 Prop. 3. To meafure any Triangle 271 
 
 Prop. 4. The Side of an Equilateral Triangle being given, 
 to find the Perpendicular Arithmetically 272 
 
 Prop. 5. The Perpendicular and one Sire of an Equila- 
 teral being given , to find the Content of that Triangle 
 
 - * *74 
 
 Vrcyp. 6. The Perpendicular andBafe of a right-angled 
 Triangle being given, to find the Content of that Tri - 
 angle 275 
 
xvni The 1 A B L L. 
 
 Prop. 7. The (perpendicular and Safe of an Ifofceles Trr 
 etude being given, to find the Content of that Triangle 
 
 276 
 
 Prop. 8. The Sides of any Triangle being given ■> to find the 
 Perpendicular Arithmetically 2 7 7 
 
 Prop. 9, The Side of a regular Pentagon being given, to 
 find the Content of that Pent agon ffi 
 
 Prop. 10. The Diameter of a Circle being given, to find, 
 the Circumference thereof Arithmetically 2 '9 
 
 Prop. ir. T he Diameter and, Circumference of a Circle 
 being given, to find the Content divert Ways 2 °° 
 
 Prop. 12. The Diameter of a Circle being only given , to 
 find the Content of that Circle _ . 201 
 
 Prop. 1 5. The Circumference of a Circle being given, to 
 fni the Content of that Circle j 
 
 Prop. 14. The Content of a Circle being given , to find the 
 
 Diameter ... , , 
 
 Prop. 1 5. The Content of a Circle being given, to find, the 
 Circumference 1 , 
 
 Prop. 16. The Content of a Circle being given, to find the 
 Side of a Square, the Content cf which Square Jh all be 
 equal to the Content of the Circle given _ *°S 
 
 Prop. 17. The Diameter of any Circle being given, to find 
 
 the Side of a Square, the Content whereof jball be equal 
 to the Content of the Circle, whofe Diameter is given 
 
 feveral Ways „ . 284 
 
 4 new Way to refolve the aforefaid Trofofmon, together 
 with Notes fir multiplying -the Tarts of Inches, into 
 
 Feet and Inches . ' . £ J fho 
 
 Prop. 18. The Content of a Circle being given, to find the 
 
 Tiameter „ . - — . . . £ j 
 
 Prop 19. The Diameter of a Circle being given, tofi.d 
 the Side of a Square, which may be infcribed within 
 
 Cl) A cl@ • ^ 
 
 Prop 2C The ! Diameter and arch Line cf a Semicircle 
 
 being given, to find the Content of that Semicircle ibid 
 
 Prop. 2i. The Semi diameter and arch Line of a Sector 
 of a Circle being give?*, to find the Content thereof z 99 
 
 Prop. 
 
The T A B L E. xix 
 
 Prop. 22. Any Segment or Tart cf a Circle being given* 
 to find the Content thereof 299 
 
 Prop. 23. To find the Diameter of a Circle, by having one' 
 ‘Part of the Diameter given , and the Length of the 
 Chord Line cr offing the Diameter in the given Dart 306 
 Prop. 24. Any Segment of a Circle being given , whofe 
 Chord Line doth not exceed the Chord of the Quadrant 
 of the fame Circle 5 to find the Content without finding 
 the Diameter , and without defending any more cf the 
 Circumference 301 
 
 Prop. 2 5. To meafure or find the Content of an irregular 
 flat or Figure 303 
 
 Prop. 16. Do meafure or find the Content of any Oval 
 
 304 
 
 Prop. 27, Dhe Definition of a Cylinder , and how to find 
 the Content thereof the Diameter being given 3c 6 
 
 Prop. 28. What a Cone is, a\id how to meafure it 308 
 Prop. 29. What a Dyramid is, and how to meafure it 
 
 309 
 
 Prop. 30. How to meafure a Globe or Sphere 310 
 
 Prop. 31. How to meafure a Fragment, or Dart of a 
 Globe 3 1 1 
 
 Meafuring of LAND. 
 
 IfiTAHAL* a Dole or Derch is according to Statute - 
 ** Meafure, and how many Derches are contained in 
 an Acre 3 alfo what kind of Chain to be ufed land how 
 divided ) in taking the Dimenfions 3 1 1 
 
 How to write down the Dimenfions in a Docket-Book^ and 
 how to caft them up 312. 
 
 AnExample how to meafure any Diece of Land lying in 
 the Form of a long Square 313 
 
 An Example how to meafure a Diece of Land lying in the 
 Form of a Dri angle 314 
 
 Another Example how to meafure Land 315 
 
 Cufiomary Meafure, what it is, and how to reduce it into 
 Statute- Meafure 3 and contrari-wife, how to reduce Sta- 
 tute- Meafure into Cufiomary ibid,- 
 
 E * Th£ 
 
XX 
 
 The TABLE. 
 
 The Contents of the Sixth BOOK, confiding 
 
 of the Meafuring of Solid Bodies . 
 
 W 
 
 II AT the Meafuring of Solid Bodies is 324 
 Prop. 1. IV hat a Cube is, and how to meafure it 
 
 325 
 
 Prop. 2. How to meafure a folid Piece of 1 imber or Stone 
 in the Form of a Cube 32 6 
 
 Prop. 3. How to meafare a Piece of Timber, each End 
 thereof being an Oblong , or long Square 328 
 
 Prop. 4. How to meafure Timber or Stone, whofe Ends are 
 equal Quadrates or Squares 329 
 
 An Advert ife went concerning the falfe Way of meafuring 
 the fore f aid kind of Timber 330 
 
 Prop. 5. How to meafure Timber or Stone, whofe Ends 
 are Equilateral Triangles 331 
 
 Prop. 6 . How to meafure Timber or Stone, each End 
 thereof being a regular Tent agon, viz. a Figure having 
 five equal Sides 334 
 
 Prop. 7 How to meafjye fquared Timber or Stone being 
 T ifer, viz. being lefs at one End than at the other 
 
 335 
 
 An Advertifement concerning the falfe Way which many 
 Men life in meafuring fuch Timber or Stone 338 
 
 Prop. 8. What Pyramids are, and how to meafure a 
 S Pyramid , whofe Safe is an Equilateral Triangle 340 
 How to find the Height of any Pyramid 34 r 
 
 Prop. 9. How to meafure a Pyramid, whofe Safe is a 
 Square 344 
 
 Prop. 10. How to meafure a Pyramid, whofe Bafe is a 
 Pentagon 345 
 
 Of 
 
The TABLE. 
 
 xxi 
 
 Of Round Solid Bodies. 
 
 Prop. ii. JJ/ % HA T a Cylinder is, and how to measure 
 it 349 
 
 Prop. 12. What a Cone is, and how to meafure it 35 1 
 Prop. 1 3 . What a Segment of a Cone is, and how to 
 meafure it 353 
 
 Prop. 14. What a Globe or Sphere is, and hew to meafure 
 n 356 
 
 Prop. 15. The folid Content of a Globe or Sphere being 
 given, how to find the Length of the Axis or Diameter 
 
 358 
 
 Prop. 16. A Tort ion or Tart of a Sphere, together with 
 the Segment of the Axis , and the Length of the whole 
 Axis being given, how to meafure that Tortion or Tan 
 
 . . 35 9 
 
 Prop. 1 7. The Segment or Tort ion of a Sphere being given, 
 
 how to find the Length of the Axis 3 61 
 
 Prop. 18. The Axis of a Sphere, being given, how to find 
 the Superficial Content thereof 362 
 
 Prop. 19. A Tortion or Segment op a Sphere being given, 
 to find the Content of the Convex Superficies thereof 
 
 363 
 
 The Truth of the Rejolution to the foregoing Tropofition 
 demonftrated by the Comparifon of Motion 364 
 
 Of the Troportions between a Cube , a Trifna , and a Ty~ 
 ram id, a Cylinder, Sphere, and Cone, whofe Heights and 
 Tafes are alike 3(55 
 
 Gauging or Mea firing of Liquid Vejfels, what it is, and 
 how to meafure any Veffel, and give the Content in 
 Wine or Ale Meafure, See. 3 <57 
 
 B 3 
 
 The 
 
XXII 
 
 The TABLE. 
 
 The Contents of the APPENDIX. 
 
 r H E Me a fur ng of Chimneys reformed 379 
 
 A 'Table of Brickwork, from 20 s. to 50 s. per .Rod 
 
 396 
 
 Another from 4I. 10s. to 6 1. 10 s. per Rod 397 
 
 A Table of Ty ling from 2 s. 6 d. to 40 s. per Square 
 
 398 , 399 
 
 y? of Perpendiculars for Gable -Ends 399 
 
 Oj Me a firing Superficies and Solids 400 
 
 V 
 
 O F 
 
XXUl 
 
 O F 
 
 I 
 
 GEOMETRY in General. 
 
 EOMETRY is a Greek Word, and 
 imports only the Meafuring of Land . 
 and fince feveral Parcels of Lands are 
 divided into various Forms, it behoves 
 him, that would be efleemed a Land-Meafurer, to 
 know how to meafure all Kinds of fuperficial 
 Figures. But though the Word import no more 
 but Land (or fuperficial) Meafuring ; yet under 
 the Name of Geometry is comprifed the 
 Meafuring of all Kinds of Solids. This Science 
 (according to Hiftory) was firit invented by the 
 Egyptians , and the Caufe which put them upon 
 inventing it, was the Overflowing of the River 
 Nilus , the greateft and longeft River in the 
 World, which, when it overflowed, wafhed away 
 
 B 4 their 
 
xxiv Of Geometry in General. 
 
 their Banks and Land-Marks ; and when the 
 Waters were dilperfed, it was a difficult Matter 
 for every Man to know his own Quantity of Land , 
 infomuch that it caufed Quarrelling and Strife 
 amongft them, till at length, through the lnduftry 
 of fome ingenious Perfon orPerfons, this Science of 
 Geometry was found out, which put an End 
 to their Quarrellings, and reftored to every Man 
 the fame Quantity of Land after the Flood, that 
 he had before. 
 
 I n brief. Geometry is a Science, where- 
 by the Quantities of Things not meafured, are 
 determined, by comparing them with other Quan- 
 tities meafured. 
 
GEOMETRY. 
 
 Definition s. 
 
 POINT is a Body whofe Quantity 
 is not confidered ^ if confidered, is 
 that which is not put to account in 
 Demonftration ; and is made with 
 the Point of the Compafs, of a Pen 
 or Pencil, or fuch like, as the Point noted by A. 
 
 II. A Line is a Body whole Length is conlidered 
 without its Breadth, and is made by the Motion of 
 a Point from one Extreme to another. 
 
 Extreme fignifies the Beginning or End of a Line. 
 
 Of Lines are ‘Three Sorts. 
 
 Firft, Crooked or Circular. Secondly, Streight 
 or Right. Thirdly, Mix’d. 
 
 i . Crooked or Circular Lines are thole, which have 
 a Poflibility of didudting or fetting far- . ^ ^ 
 ther afunder their Extremes, as the Line A 
 
 2. Streight 
 
2 
 
 Of GEOMETRY. Book L 
 
 2. Streight or Right Lines are thofe which have 
 no Poffibility of didudting or fetting farther afunder 
 their Extremes ; and therefore is the leaft Length 
 between the two Extremes. Plato defines a Right 
 Line to be that, whole Extremes do 
 
 fhadow all the middle Parts, and is re- B d 
 
 prefented by the Line B C. 
 
 3. Mix’d Lines are compounded of 
 Streight and Circular Lines, as the Line 
 
 III. A Superficie is that which hath Length and 
 Breadth, the Thicknefi not being confidered ; and 
 as a Line is produced by the Motion of a Point, 
 fo a Superficie is produced by the Motion of a Line 
 fuppofed to move tranfverily ; that is to fay, the 
 Line AC being fuppofed to move fide- ways to 
 BD, it will produce a Superficies ABC D. 
 
 IV. The Extremes of a Super- A. 
 
 ficie are Lines. But of a Cir- 
 cular Superficie, a Line is the 
 Extreme. q 
 
 V. An Angle is an Inclination of two Lines, the 
 one to the other, the one touching the other, and 
 not lying ftreight forth at length. And of Angles 
 there are three Sorts, namely. Right-lined, Curve- 
 lined, and Mix’d. 
 
 Note , When an Angle is mentioned by three 
 Letters, the middlemoft Letter fignifies the Angle 
 intended. 
 
 n 
 
 X> 
 
 So 
 

 c D 
 
 T 
 
 B 
 
 Book I. Of GEOMETRY. : 
 
 So A B C is a B E 1‘1- 
 
 right lined Angle A / \ / 
 
 D E F is a curve 
 lined Angle, and 
 GH1 is a mix’d 
 Angle. 
 
 VI. When a right Line BC (land- 
 ing upon a right Line DE, making 
 the Angles on either fide BCD, and 
 BCE equal, they are called right 
 Angles y and the right Line BC, is ^ C V* 
 called a perpendicular Line to that upon which it 
 is eredted, viz. DE. 
 
 VII. An Angle is laid to be obtule or blunt, when 
 
 it is greater than a right Angle • jp 
 
 and acute or (harp, when it is lels 
 than a right Angle: So the Angle 
 ABC is an obtufe Angle, and the 
 Angle ABD is an acute Angle, c 13 
 and the Angle DBC is a right Angle. 
 
 VIII. A Circle is a plain Figure, comprehended 
 by one Line, being generated by the Motion of a 
 Compafs, or other equivalent Means, wherein all 
 right Lines drawn from the Centre to the Circum- 
 ference, are of equal length. The Centre is a Point 
 exactly in the Middle of the Circle. 
 
 IX. The Diameter of a Circle is a right Line, as 
 A B drawn by the Centre C, and being terminated 
 by the Circumference on either fide, divides the 
 Circle into two equal Parts. 
 
 X. The Semi-diameter is half the a, ^ 
 
 Diameter, as A C or C B. ^ 
 
 XI. A Semicircle is one half of the whole Circle- 
 
 XII. Tri- 
 
4 Of GEOMETRY. Book I. 
 
 XII. Trilateral, or three-fided Figures, are thofe 
 which are contained within three right Lines, and are 
 called Triangles 3 becaufe three Lines being joined 
 together at Angles, conftitute three Angles. 
 
 XI II. Of three-fided Figures, that which hath 
 three equal Sides, is called an Equilateral Triangle, 
 as the Triangle ABC. 
 
 XIV. But that which hath two Sides alike equal, 
 is called an Ifofceles Triangle, as CDE. 
 
 XV. That Triangle, whofe three Sides are 
 unequal, is called a Scalcnum, as EFG. 
 
 Equilateral. Ifofceles. Scalenum. 
 
 J 
 
 A 
 
 C c5 EE 
 
 XVI. There are alfo other Triangles, and are 
 named according to the Quality of their Angles. A 
 right angled Triangle, is that which hath one right 
 Angle, as the Triangle A. An Ambligonium, or 
 obt ufe-angled Triangle, is that which hath one 
 Angle obtufe, as the Triangle B. An Oxigonium, 
 or acute angled Triangle, is that which hath three 
 acute Angles, as the Triangle C. 
 
 A 
 
 XVII. 
 
BookL Of GEOMETRY. 5 
 
 XVII. In every Triangle, two of the Lines being 
 taken fortwofides, the third Line remaining is called 
 the Bale, whether it be the lowermoft Line of the 
 Triangle, or not • lb that an^ one of the three Lines, 
 which inclofea Triangle, may be taken for the Bafe. 
 
 XVIII. Of Quadrilateral, or four-fided Figures, 
 there are leveral Sorts. 
 
 XIX. A Quadrate, or Square, is that which hath 
 equal fides, and right Angles, as the Figure A. 
 
 XX. An Oblong hath the oppofite Sides alike, 
 and right Angles, as the Figure B. 
 
 XXI. A Rhombus, or Diamond Figure, hath 
 four equal Sides, and two Angles acute, and two 
 obtule, as C. 
 
 XXII. A Rhomboides, hath oppofite Sides, 
 and oppofite Angles alike, but it is neither equila- 
 teral nor right angled, as the Figure D. 
 
 XXIII. A Paralallelogram is a four-fided Figure, 
 whofe oppofite Sides are parallel, as the Figure E. 
 
 XXIV. All other four-fided Figures are called 
 Trapezia’s or Tables, as F, &c. 
 
 XXV. Parallel Lines are fuch, which being in the 
 lame Superficie, if produced, will not meet, as G. 
 
 Thus much may ferve for Definition ; I pro- 
 ceed now to Practice. 
 
 CHAP. 
 
 I 
 
6 
 
 Of GEOMETRY. Book I. 
 
 CHAP. II. 
 
 Note, Float the Figures belonging to the following 
 Proportions are in the next folded Page. 
 
 PROP. I. Fig. I. 
 
 From a Point given in a freight Line , to raife a 
 Perpendicular , that foall cut the freight Line at 
 right Angles. 
 
 I E T C be the Point propofed in the Line AB, 
 A from which a Perpendicular is to be raifed. 
 From the Point given C, draw at pleafure any Se- 
 micircleEF , then from thePointsEF, you fnuft make 
 a Section thus : Open your Com palfes to the Diftance 
 EF, fetting one Point in E, with the other defcribe 
 the Arch IG ; then remove that Point of your Com- 
 pafs, and fet it in F (keeping the Compafles at the 
 fame Diftance) with the other Point thereof defcribe 
 the Arch IH, which will interfedl or cut through the 
 Arch IG ; from which Interfe&ion at I, a Line being 
 drawm to the Point C, will be perpendicular, and 
 cut the Line AB at right Angles. 
 
 Fhis Prop, may be perfortned , as in Fig. II. 
 
 By opening your Compafs to any convenient Di- 
 ftance, and fetting one of the Points of the Compafles 
 in the Point C, with the other Point make the prick 
 A, then turning the Compafles, keeping one Point 
 ftill at C, with the other Point make the prick 
 B in the Line given j then fetting one Foot in A, and 
 
 opening 
 
Book I. Of GEOMETRY. 7 
 
 opening the Compares to B, defcribe the Arch HI ; 
 alio remove the Compafles, and letting one Point 
 in B, defcribe another Arch HK, from which inter- 
 fed; ion, draw the Line HC, which will cut the 
 Line AB at right Angles. 
 
 PROP. II. Fig. III. 
 
 S > o raife a Perpendicular at the End of a Line . 
 
 L E T the Line give be AB, and on the End 
 B, it is requir'd to raife a Perpendicular. 
 
 Set one Foot of your Compafles above the Line 
 AB at pleafure, as fuppofe at D, and opening the 
 other Point till it flay in B, with this Diftance 
 keeping the Point at D, defcribe a Semicircle 
 FBE ; by the Points F and D, draw the flreight 
 Line FDE j and where the flreight Line cuts the 
 Circle as at E, from thence draw the Perpen- 
 dicular EB. 
 
 So perform this Prop, another way. Fig. IV. 
 
 L E T the Point B, at the End of the Line AB, 
 be the Point from whence a Perpendicular 
 is to be raifed. 
 
 The Compafles being opened at Pleafure, fet 
 one Foot in the Point B, and with the other Foot 
 defcribe the Arch of a Circle CDG, keeping the 
 Compafles at the fame Diftance, fetting one Foot 
 in C, defcribe the Arch BD, then fetting one 
 Foot in D, defcribe the Arch FEB, and from the 
 Point E, draw the Arch ID, and from the Inter- 
 fedion of this Arch, with the Arch FEB, draw 
 the Perpendicular IB, 
 
 PROP. 
 
 \ 
 
8 
 
 Of GEOMETRY. Book I. 
 
 PROP. III. Fig. V. 
 
 raife a Perpendicular at the End of a Line , 
 having little or no Paper beyond that End of the 
 Line , whereon you are to raife the Perpendicular . 
 
 L E T AB, be the Line given, and B the End. 
 
 Opening the Compaffes at Pleafure, Petting 
 one Foot in B, with the other Foot defcribe the 
 Arch CD 3 then remove one Foot of the Compals, 
 and let it in C, and interleft the Arch CD at E; 
 then lay a Ruler from C to E, and ftrike a Line 
 as EF ; then fet one Foot of your Compaffes 
 (being at their firft Diftance) in E, and with the 
 other Foot make a Point or Prick in the Line EF 
 at F, from whence draw the Perpendicular FB. 
 
 PROP. IV. Fig. VI. 
 
 Upon an Angle given to raife a Perpendicular . 
 
 T ET ABC be the Angle given; fetting one 
 
 l Foot of your Compaffes in B, defcribe the 
 
 Arch AC ; then opening the Compaffes a little 
 wider, and fetting one Point in C, delcribe the 
 Arch DE ; then removing the Point to A, inter- 
 feft the Arch D E at F, from whence draw the 
 Perpendicular FB. 
 
 PROP. 
 
9 
 
 Book I. Of GEOMETRY, 
 
 PROP. V, Fig. VII. 
 
 To let fall a Perpendicular upon a Line given 3 and 
 from a Point above the Line . 
 
 L E T the Line given be A B, and the Point 
 given be C ; fetting one Foot of the Com-* 
 pafles in C, defcribe the Arch DE, and from D 
 and E make the Interfedtion at F, from whence 
 draw the Line CF. 
 
 PROP, vt Fig. VIIL 
 
 By a Point given , to draw a Line parallel to £ 
 Right Line given > 
 
 L E T A be the Point, by which we ffiuft draw 
 a Line, which may be parallel to the Line 
 B C. Draw at Pleafure the Oblique (dr Diagonal) 
 Line AD; from the Point A draw the Arch DE 5 
 and from the Point D defcribe the Arch AF, then 
 fetting one Point of the Compaffes in the given 
 Point A, extend the other Point to that Part of 
 the given Line which is interfered by the Arch 
 A F, with this Diftance, letting one Point of the 
 Compalfes at the Interfedlion D, extend the other 
 upon the Arch D E, and where that Point falls 
 upon the Arch as at G, draw a Line from the 
 Point A through the Point at G, and it will 
 parallel to the Line BG 
 
 € 
 
 jriMthsf 
 
Of GEOMETRY. Book I, 
 
 Another Way to draw a Parallel to the Line B C. 
 Fig. IX. 
 
 THE Compares being fet to the Space you 
 intend (hall be between the two Lines (otherwife 
 any Diftance will ferve) fetting one Foot on the 
 End of the Line B, with the other Foot defcribe 
 the Arch DE ; this being done, keeping the Com- 
 paflfes at the fame Diftance, fet one Point on the 
 End of the Line C, and defcribe the Arch F G, 
 then draw the Line HI, juft to touch the upper- 
 moft Part of thefe two Arches, and it will be 
 parallel to the Line BC. 
 
 PROP. VII. Fig. X. 
 
 To cut a right Line AB equally in the Middle. 
 
 T H E Compares being opened to any Diftance 
 ftiorter than the whole Line, and longer than 
 half the Line ; as fuppofe them opened from A 
 to C, fetting one Point of the Compafs in A, with 
 the other defcribe the Arch DE } keeping the 
 Compares at the fame Diftance, fetting one Foot 
 in B, with the other defcribe the Arch F G, and 
 from the Interfecftion (or cutting thro ) of theie 
 two Arches, draw the Line H I, it will divide the 
 Line AB equally in the Middle. 
 
 The Figures of the Propofitions following, yon will 
 find in the next folded Page to this. 
 
 PROP. 
 
Frsrp .X . 
 

 -A 
 
 tfhe Figures of the Propofit ions followings you will 
 find in the next folded Page to this. 
 
 PROP, 
 
Book I. Of GEOMETRY. i i 
 
 Prop. viii. Fig. xi. 
 
 To cut a right lined Angle BAG, into two equal 
 Parts . 
 
 O Pening the Compares at Pleafure Qviz. open- 
 ing them to any Diftance) fetting one Point 
 of them on the Angle A, with the other Point de- 
 fcribe the Arch D E, then from D and E, make 
 the Interledtion at G, through which Interfedlion, 
 and the Angle A, draw the Line A G, which di- 
 vides the Angles B AC into two equal Parts. 
 
 PROP. IX. Fig. XII. 
 
 To make a triangle ABC, of three right Lines 
 (viz. AB,BC, CA) equal to three Lines given 
 D. E. F. of which three Lines , any two being 
 taken and added together , muft he longer than the 
 third Line remaining . , otherwife you cannot make 
 a triangle of them, 
 
 F Irft draw the ftrait Line G H, then upon ,that 
 Line take G A, equal to the given Line D ; 
 alfo upon the fame Line G H, fet the Line E, 
 from A to C ; alio fet the given Line F, from C 
 to D, then opening the Compaffes to GA, with 
 one Point in A, deicribe the Circumference GBC ; 
 this being done, fet one Point of the Compafs at 
 C, and the other being opened to D, defcribe 
 the Circumference DBI, then join the Triangle 
 
 C 2 where 
 
12 Of GEOMETRY. Book I. 
 
 where thefe two Circles interfeft, which is at B, 
 and make the Triangle ABC, which is equal to 
 the three Lines given D. E. F. 
 
 PROP. X. Fig. XIII. 
 
 At a Point A, in a right Line given A B, to make 
 a right lined Angle A, equal to a right lined 
 Angle given D. 
 
 S ET one Point of the Compafles at D, and 
 with the other defcribe the Arch EF, then 
 with the Compafles being at the lame Diftance, 
 fetting one Point at A, with the other Point de- 
 feribe the Arch B C, then take the Chord Line 
 of the Arch E F ( viz . the ftrait Line E F ) be- 
 tween the Points of your Compafles, and fet one 
 Point at B, and where the other Point toucheth 
 the Arch BC as at C, by that Point draw the 
 Line AC, and the Angle A, will be equal to the 
 Angle D. 
 
 PROP. XI. Fig. XIV. 
 
 to make a Parallelogram A BCD, equal to a 'tri- 
 angle given EFB, in an Angle equal to a right 
 lined Angle given G. 
 
 npHrough the Point E, draw the Line E D, 
 JL parallel to the Line FB (by the 6 Prop. ) 
 then upon the Point B raife the Line B D, mak- 
 ing the Angle at B, equal to the given Angle G, 
 
 (as 
 
Book I. Of GEOMETRY. 13 
 
 (as you were taught in the ioth Prop.) then Bifed: 
 (viz. divide in the Middle) the Bale F B, as at 
 A, and draw the Line A C parallel to B D, and 
 the Parallelogram A B C D, will be equal to the 
 Triangle given EFB, and like the Angle given 
 G. 
 
 PROP. XII. Fig. XV. 
 
 Upon a right Line given A, to make a Parallelo- 
 gram F L, at a right lined Angle given C, equal 
 to a 'triangle given B. 
 
 B Y the foregoing Prop, make a Parallelogram 
 FD, equal to the Triangle B, fo that the 
 Angle GFE may be equal to the Angle given C j 
 continue the Line G F, "till F H be produced equal 
 to the given Line A ; then by the Point H, draw 
 the Line IL, parallel to EF, alfo continue the 
 LineDE ’till it touch the Line HI, then draw 
 the Diagonal Line I K, ’till it meet with the Line 
 DG being continued, then through the Point K, 
 draw the Line KL parallel to GH, then extend 
 or continue the Line EF unto M, and the Line 
 I H unto L, then fhall F L be the Parallelogram 
 required • for the Parallelogram FL is equal to 
 the Parallelogram F D, and F D is equal to the 
 Triangle given B, and the Angle MFFI, is equal 
 to GFE, and GFE is equal to the given Angle C. 
 
Of GEOMETRY. Book I. 
 
 PROP. XIII. Fig. XVI. 
 
 Parallelograms BCD A, GHFE, J landing upon 
 equal Bafes BC, GH, and betwixt the fame Pa- 
 rallels A F, B H, are equal one to the other. 
 
 iRaw BE and CF, becaufe BC is equal to 
 
 u GH, and GH equal, to EF, therefore is 
 BCFE a Parallelogram. Whence the Paralle- 
 logram BCD A is equal to BCFE, and that 
 equal to GHFE, which was to be demon- 
 ftrated. 
 
 PROP. XIV. Fig. XVII. 
 
 i triangles BC A, BCD, ftanding upon the fame 
 Bafe B C, and between the fame Parallels B C, 
 E F, are equal one to the other . 
 
 D Raw BE parallel to CA, and CF parallel 
 to BD, then is the Triangle BCA, equal 
 to half the Parallelogram BC AE, and alfo equal 
 to half BDFC, and that equal to the Triangle 
 BCD, which was to be demonftrated. 
 
 PROP, 
 
Book I. Of GEOMETRY. 
 
 PROP. XV. Fig. XVIII. 
 
 Jf a "Parallelogram ABCD, have the fame Bafe 
 B C, with the triangle BCE, and he between 
 the fame Parallels A E, B C, then is the ParaU 
 lelogram ABCD, double to the triangle BCE. 
 
 ET the Line AC be drawn. Then is the 
 
 JLj Triangle BCA equal to BCE: Therefore 
 is the Parallelogram ABCD, equal to two fuch 
 Triangles as BCA, and likewife alfo equal to two 
 fuch Triangles as BCE, which was to be demon- 
 ftrated. 
 
 From hence may the Area (or Content) of any 
 Triangle as BCE be found. For whereas the 
 Area of the Parallelogram ABCD is produced by 
 the Altitude drawn into the Bafe, therefore fhall 
 the Area of a Triangle be produced by Half 
 of the Altitude drawn into its Bafe, or Half its 
 Bafe drawn into its Altitude • as if fb be, the 
 Bafe BC be 8, and the Altitude 7, taking the 
 Half of the Bafe 4, and multiplying it by the 
 Altitude 7, it produceth 28, which is the Area 
 or Content of the Triangle BCE ; or otherwife, 
 if you take the whole Bafe 8, and half the Alti- 
 tude, 3 and an Half, and multiply them, they 
 produce 28 (as before) for the Content. 
 
id Of GEOMETRY. Book I, 
 
 PROP. XVI. Pig. XIX. 
 
 Upon a right Line given F G, to make a Paralleled 
 gram F L, equal to a right lined Figure given 
 A B C D, at a right lined Angle given E. 
 
 R Efolve the right lined Figure given into two 
 Triangles BAD, B C D 5 then make a Paral- 
 lelogram F H equal to BAD, fo that the Angle 
 F may be equal to the Angle E (as you were taught 
 at the 1 2th Prop.) F I being produced to K, make 
 the Parallelogram I L equal to the Triangle BCD. 
 Then is' the Parallelogram FL equal to FH more 
 I L, and therefore equal to the Figure given 
 A BCD, which was to be done. 
 
 SCHOL. Fig. XX. 
 
 F TEnce is eafily found the Excefs HE, whereby 
 JL any right lined Figure A, exceeds a lefs 
 right lined Figure B ; namely. If to fome right 
 Line CD both be applied, and both the Trape- 
 zia's, each of them being divided into two Tri- 
 angles, and working as before is taught, you will 
 find the Parallelogram D F equal to the Trapezia 
 A, and the Parallelogram D H equal to the Tra- 
 pezia B, fo that the Figure A exceeds the Figure 
 B by fo much as the Parallelogram GEFH con- 
 tains. 
 
 PROP. 
 
Book!. Of GEOMETRY. 
 
 PROP. XVII. Fig. XXI. 
 
 In right Angled triangles B A C, the Square B E, 
 which is made of the Side B C that fuhtends the 
 right Angle B A C, is equal to both the Squares 
 B G and C H, which are made of the Sides A B 5 
 A C, containing the right Angle . 
 
 J Oin AE and AD, and draw AM parallel to 
 CE, becaufe the Angle DBC is equal to 
 FBA, add the Angle ABC common to them 
 both, then is the Angle A B D equal to F B C. 
 Moreover A B is equal to F B, and B D equal to 
 BC; therefore is the Triangle ABD equal to 
 FBC. But the Parallelogram BM, is equal to 
 two fuch Triangles as ABD, and the Parallelo- 
 gram or Square BG, is equal to two fuch Tri- 
 angles as FBC (for GAC is one right Line by 
 the Hypothefis) therefore is the Parallelogram 
 B M equal to the Quadrate B G. By the lame 
 Way of Argument, is the Parallelogram CM equal 
 to the Quadrate C H • therefore is the whole Pa- 
 rallelogram (or Square) B D E C equal to the two 
 Quadrates B A G F and A C I H, which was to 
 be done. 
 
 PROP, 
 
Of GEOMETRY. Book I. 
 
 18 
 
 PROP. XVIII. Fig. XXII. 
 
 ‘There are Three Quadrates or Squares given, 
 whereof the Sides are AB, BC, CE, and it is 
 required to make one Square , whofe Area or Con- 
 tent Jhall he equal to the Area of thofe Three 
 Squares. 
 
 M Ake the right Angle F B Z, having the Sides 
 infinite (the Meaning of infinite is to draw 
 the Sides long enough, and of what Length there 
 is no Determination) and on thefe two Sides trans- 
 fer AB and BC ; that is to lay, Take the given 
 Line A B between your Compaffes, and place it 
 from the Angle B to A ; alfo take the Line given 
 BC, and place it from the Angle B to C , join 
 AC {viz. draw the Line AC) then is a Square, 
 whole Side is AC, equal to two Squares made of 
 the two Lines AB and BC, then take the Line 
 A C and place it from B to X j alfo take the 
 third given Line or Side CE, and place it from 
 B to E, then draw the Line E X, then a Square 
 being made whole Side is E X, is equal to three 
 Squares, being made of the three Lines or Sides 
 given, AB, BC, CE, which was to be done. 
 The Truth whereof is manifefted by Arithmetick $ 
 for let the Line A B be 8* Feet in Length, then 
 the Line BC will be 5 Feet, and the Line CE 
 will be 4 Feet, and the Line EX will be ro Feet 
 and 6 Inches : Now the Square of 8 is 64, and the 
 Square of 5 is 25, and the Square of 4 is 16, 
 which three Squares being added together, pro- 
 duce 
 
Book I. Of GEOMETRY. \ 9 
 
 duce 105, for the Area of the three given Squares • 
 therefore the Square of the Line EX being io 
 Feet and 6 Inches, is 105 Feet, and equal to the 
 three Squares given. 
 
 PROP. XIX. Fig. XXIII. 
 
 f two unequal right Lines being given A B, B C • to" 
 make a Square equal to the Difference of the two 
 Squares of the given Lines AB, BC. 
 
 F Rom the Centre B, with the Diftance B A de- 
 fcribe a Semicircle, and from the Point C 
 eredt a Perpendicular C E, meeting with the Cir- 
 cumference in E, and draw BE. Then is the 
 Square of B E (or BA) equal to the Square of 
 BC and CE. Therefore when the Square of 
 B C is taken out of the Square of BA, the re- 
 maining Part of the Square of BA will be equal to 
 the Square of C E, which was to be done. 
 
 PROP. XX. Fig. XXIV. 
 
 Any two Sides of a right Angled 'triangle ABC, 
 being known , to find out the third . 
 
 L ET the Sides AB, AC, encompafling the 
 right Angle, be the one 6 Feet, the other 
 8 Feet : Therefore, whereas the Square of AB is 
 3 6, and the Square of AC is 64, which being ad- 
 ded, make 100, therefore (as you were taught at 
 the j 7th Prop.) the other Side fought for, muft 
 be equal in Power Quiz, being fquared) to the 
 
 two 
 
40 Of GEOMETRY. Bookl. 
 
 two given Sides being fquared, which contain ioo, 
 whole Square Root is io, the Length' of the Side 
 fought B C, which w r as to be done, 
 
 PROP. XXI. Fig. I. 
 
 So defcribe a Circumference that Jhall touch any 
 three Points given, provided they arc not in a 
 right Line ; fuppofe the Points given , to he A. 
 B. C. ( S’ he Figures of this , and the following 
 Propofitions, you will find in the next folded 
 Page.) 
 
 F'l’^Ake the Diftance between A and B with your 
 I Compaftes, and fetting one Point of the Com- 
 pafs on the Point by A, defcribe the Arch D E ; 
 then with the fame Diftance fetting one Point of the 
 Compaft on the Point by B, defcribe another Arch, 
 which will cut the former Arch in the Points by i ; 
 then laying a Pailer to the Points by i, where the 
 Arches interfeft, draw a ftrait Line F G. This be- 
 ing done, take with your Compaftes the Diftance 
 between the other Point C and B, and with this 
 Diftance, fetting one Point in B, defcribe the Arch 
 HI ; then with the Compaftes at the fame Diftance, 
 fetting one Point on the Prick by C, defcribe ano- 
 ther Arch, cutting the former in the Point, and by 
 2, thro 5 which Points draw another ftrait Line, 
 ^till it cut through the firft: ftrait Line, as at G ; 
 
 I fay G is the Centre, from whence the Circum- 
 ference A. B. C. is deferibed, which toucheth the 
 three given Points. 
 
 PROP. 
 
J 
 
Book I. Of GEOMETRY, 
 
 zi 
 
 PROP. XXII. Fig. II. 
 
 Sft defcribe an Oval upon a Length given AB. 
 
 D ivide the given Line into three equal Parts 
 (as the next Propofition following will teach) 
 A C D B, fetting one Point of the Compares at the 
 Point by C, with the Diftance C A, defcribe the 
 Circle A E F, then with the fame Diftance fetting 
 one Point by D, defcribe the Circle B E F, then 
 draw four ftrait Lines thro* the Centres C and D, 
 and the Interfedftion of the two Circles E and F ; 
 then fetting one Point of the Compalfes in E, and 
 extending the other Point to I, defcribe the Arch 
 IH, then with the Compafles at the fame Diftance, 
 fetting one Point on the Interfedlion by F, defcribe 
 the Arch OP, which concludes the Oval. Note, 
 That I A O and HB P, are vulgarly called Hanfes^ 
 and I H and O P, are called Schemes. 
 
 PROP. XXIII. Fig. III. 
 
 fZa divide a ftrait Line given AB, into three 
 equal Parts . 
 
 F Rom the End A, draw at Fleafure the Line 
 A C, making what Angle you will, then from 
 the other End of the Line B, draw the Line BD, 
 parallel to the Line A C ; then opening your Com- 
 pares at Pleafure, fetting one Point in A, turn 
 them three Times over the Line AC, which will 
 make three Divifions, viz, E F G, then with the 
 Compaffes continuing at the fame Diftance, fetting 
 one Foot in B, make three DivifionS on the 
 
 Luie 
 
H Of GEOMETRY. Book L 
 
 Line B D, *i uZj H I K, then draw with a Ruler, 
 and the Point of the Compafs, a ftrait Line from 
 A to K, another from E to I, alfo another from 
 F to H, and another from G to B; and they be- 
 ing drawn, the given Line A B is divided into 
 three equal parts, which was to be done. You 
 may, if you pleafe, divide the fame Line, or any 
 other, into what Number of equal Parts you pleafe, 
 by dividing the two parallel Lines AC and BD, 
 into fo many equal Parts as you would have the 
 given Line divided into. 
 
 PROP. XXIV. Fig. IV. 
 
 So defer lie an Oval equal in Length to the firjl 
 Oval not rifing fo high. 
 
 D ivide the given Line A O into four equal Parts 
 (by the foregoing Prop.) in BCD, then 
 taking one of thofe Parts between the Compaffes, 
 upon the Centres BCD, deforibe three Circles, 
 and thofe 2 Parts of the middlemoft Circle, that is 
 without the 2 other Circles, divide in the Middle at 
 E and F, then from E to the Centre D, draw a 
 ftrait Line, and contine it to the Circumference at 
 4, alfo draw another ftrait Line from E, thro 3 the 
 Centre by B to the Circumference, which will cut 
 it at 3 ; likewife from F thro 5 the fame Centres 
 draw right Lines, which will cut the two Circum- 
 ferences, the one in 2, the other in 1. Then from 
 the Centre F with the Radius (or Diftance) F, 1, 
 deforibe the Arch 1, 2 • alfo from the Centre E 
 with the fame Radius, deforibe the Arch 3, 4, 
 which concludes the Oval. 
 
 PROP, 
 
Book I. Of GEOMETRY. 
 
 PROP. XXV. Fig. V. 
 
 Another Way to defcrihe Ovals. 
 
 Pon the Line given, defcribe two equilateral 
 
 Triangles, join them together with one com- 
 mon Bafe, fo that they make a Rhombus ; then 
 continue (or draw) the Line AC to 3, fo that C 3 
 may be 6, fuch Parts whereof AC is 5, viz. It 
 muft be the Length of A C, and one 5th Part more 
 of it. Alfo draw the Line B D to 2, that it mar 
 be the fame Length with A 3 ; alfo draw B C to 
 1, and AD to 4, being all of one Length : Then 
 from the Centre A, with the Radius A 3, de- 
 fcribe the Arch 3,4, alfo from the Centre B, with 
 the lame Radius, defcribe the Arch 1, 2 ; then 
 from the Centre C, with the Radius C 1, defcribe 
 the Arch 1, 3 ; likewife from the Centre D, with 
 the fame Radius, defcribe the Arch 2, 4, which 
 will inclofe the Oval. From thefe 4 Centres you 
 may defcribe Ovals, greater or leffer as you pleafe. 
 
 PROP. XXVI. Fig. VI. 
 
 defcrihe an Oval according to any Length and 
 Breadth given. 
 
 ET the Length given be AB, and the Breadth 
 
 j CD. 
 
 Apply the 2 given Lines together, fb that they 
 may cut each other into 2 equal Parts, and at right 
 Angles in the Point B ; then take half the Line AB 
 between your Compafles, and fetting one Point of 
 the Compafles in C, extend the other till it touch 
 
 the 
 
‘24 Of GEOMETRY. Book I. 
 
 the Line AB, in K and L, which 2 Points are 
 called the burning Points, or Focus’s. In which 
 Points, drive 2 Nails if you delcribe it upon Boards, 
 but upon Paper, as here, 2 Pins will do 5 the Pins 
 being ftuck firm in the Points K and L, ftick alfo 
 another Pin in the Point C ; then take a Thread 
 and encompafs thefe 3 Pins in Form of a Triangle, 
 pulling the Thread tight, tie the 2 Ends of the 
 Thread together by a Knot at C ^ then taking out 
 the Pin at C, take a Pencil, holding it clofe to the 
 Infide of the Thread, and carrying the Pencil round 
 upon the Paper, about the Pins, with the Thread 
 always ftrait, the Ellipfis or Oval ACBD ftiall be 
 thereby defcribed. 
 
 PROP. XXVII. Fig. VII. 
 
 ? 0 find the Centre and the two Diameters of an Gvah 
 
 L Et S E T D be the Oval whereof the Centre 
 and the Diameters are to be found. 
 
 Within the Oval, draw at Difcretion the Paral- 
 lel Lines EF, GH j cut thefe Lines into 2 equally 
 in I and K, draw the Line I K, cut it into 2 equally 
 in L, which is the middle Centre pf the Oval j up- 
 on this Centre L, deferibe at Pldafure the Circle 
 MN O, cutting the Oval in P and Q, from which 
 Sections, draw the right Line P Q, cut it in the 
 Middle in R, from which, through the Centre L, 
 draw the greater Diameter ST, and from the 
 Centre L, draw the leffer Diameter ELD parallel 
 to the Line P Q, which was to be done. 
 
 HO E 
 
2 5 
 
 Book I. Of GEOMETRY. 
 
 PROP. XXVIII. Fig. VIII. 
 
 To defcribe an Oval with a pair of Compares , to 
 any Length and Breadth given. 
 
 I Shall only defcribe a Semi-oval, and according 
 to the fame Rules, if you will, you may de- 
 fcribe the whole Oval. 
 
 Let the Length given be A B, and one half of the 
 Breadth C D 3 divide A B into feven equal Parts 3 
 then upon one feventh Part from A, as at E, raife 
 a Perpendicular from the Line A B (viz. E G.) 
 Alfo at one feventh Part from B, as at F, raife 
 another Perpendicular F H3 then divide the half 
 Breadth given C D, into fifteen equal Parts, and 
 take eleven of thole Parts and let upon the Perpen- 
 dicular from E to G, and likewife from F to H3 
 then taking the Space between A and G, fetting 
 one Point of the Compares in A, defcribe the Arch 
 G i 3 keeping the Compaffes at the fame Diftance, 
 fet one Point in G, and defcribe another Arch, 
 which will cut the former in the Point by i 3 from 
 which Point, with the Radius A G, defcribe the 
 Sembhanfe A G. This being done, take between 
 your Compaffes the Space B H, and fetting one 
 Point in B, defcribe the Arch I i 3 then remove 
 your Compaffes to H, and interfedi that Arch in 
 the Point by i 3 then fetting your Compaffes on 
 the Point i, with the fame Diftance, defcribe a Part 
 of the Oval B H, which part, as alfo the other part 
 A G, are vulgarly called Semi-haftfes, becaufe it is 
 but a Semi-oval (Semi fignifies half) but if it had 
 been an whole Oval, then tiie Semi-hanfe above 
 tue Line A, and another Semi-hanfe below the 
 
 D Line 
 
2$ Of GEOMETRY. Book I.' 
 
 Line A, being joined, is called an Hanfe, from 
 the Latin Word Hanus> fignifying a great- 
 bellied thing. The other part to be deferibed, 
 from G to H, is called the Scheme, which to de- 
 feribe, continue or draw longer the half Breadth 
 D C, and in that Line find a Centre, whereon, 
 fettiag one Point of the CompafTes, the other Point 
 may touch the three Points G, D, H, as on the 
 Centre I, whereby deferibe the Scheme GDH, 
 which was to be done. 
 
 CHAP. III. 
 
 A DigreJJion concerning EUipfis Arches. 
 
 A N D fince Ellipfis or Semi-oval Arches, be- 
 ing neatly wrought in Brick, fhew very plea- 
 fant ; and are fometimes ufed over Gate- ways, and 
 fometimes over Kitchen Chimneys, inftead of 
 Mantle-trees • I think fit to write fomething con- 
 cerning them, relating to Bricklayers making the 
 Moulds, and dividing the Courfes. 
 
 The Ellipfis you may deferibe to what Length 
 and Heighth you pleafe, either by the laft Propofi- 
 tion, or by the 26th. 
 
 We will fuppofe an Ellipfis Arch to be made 
 over a Chimney, whofe Diameter between the. 
 Jaums is 8 Feet, and the under Side of the Arch,, 
 at the Key, to rife in Height 18 Inches, from the 
 Level of the Place whence you begin to fpring the 
 Arch, The Height or Depth of the Arch, we will 
 fuppofe to be made of the Length of two Bricks, 
 which when they are cut to the Sweep of the Arch, 
 
-tVtrp . -XXL . 
 
. 
 
 
 j r 
 
 will 
 
Book!. Of GEOMETRY. 
 
 will not contain above ^Inches; and perhaps you 
 muft cement Pieces to many of the Courfes in the 
 Hanfe, to make them long enough to contain or 
 hold 14 Inches, efpecially if you intend to make 
 the Courfes of the Hanfe, and the Courfes of the 
 Scheme to feem alike in Greatnefs on the under Side 
 of the Arch. For if you make the Hanfe to come to 
 .a true Sommering for the Scheme, by that time 
 that you have ended the Hanfe, and are ready to 
 fet the firft Courfes of the Scheme, the Mould, and 
 lb likewife eachCourfe in the Hanfe, will be much 
 lefi at the lower Part, or under Side of the Arch, 
 than the Mould or Courfes of the Scheme ; as you 
 may perceive by the Hanfe B K, in the IX Fig. 
 which Way of working thefe kind of Arches ‘is 
 ftronger, than to make the Courfes feem alike in 
 Bignefs in Hanfe and Scheme, altho’ it be not fo 
 plea ling to the Eye. In the IX Fig. I will fliew how- 
 to make one Half of the Arch this Wa\ , and in 
 the other Half Ihew how to make the Courfes in 
 Hanfe and Scheme of a Bignefs. 
 
 Firft, Defcribe the under Side of the Arch (viz. 
 the Ellipfis A D B, whofe Diameter A B is 8 Feet, 
 and the Height C D iS Inches) upon feme fmooth 
 Floor, or ftrait plaiftered Wall' or ftich like ; 
 then continue (viz. draw longer) both the Lines 
 A B, C D, cutting each other at right Angles ; 
 then from A to E ; alfo from B to F ; likewife from 
 D to G, fet 14 Inches, the intended Height of your 
 Arch. Then deferibe another Ellipfis to that Length 
 and Height, after this manner : Lay a ftrait 
 Ruier on the Centre by I, and on the joining of 
 the Hanfe and the Scheme togethet, as at K, and 
 craw the Line K. L5 then let one Point of your 
 
 D 2 Com« 
 
x8 Of GEOMETRY. Book I. 
 
 Compafles in the Centre of the Hanfe at M, and 
 open the other Point of the Compaffes to F, and 
 defcribe the upper Hanfe F L ; likewife fetting one 
 Point of the Compafles in the Centre by I, with 
 the other extended to G, defcribe the Scheme GL; 
 (altho’ I fpeak here of Compafles, yet when you 
 defcribe an Arch to its full Bignefs, you muft 
 make ui'c of Centre Lines, or Rules ; the laft ar? 
 beft, becaufe Lines are fubje<ft to ftretch) then 
 taking between your Compafles the Thicknefs of a 
 Brick, abating fome final! Matter, 'which will be 
 rubb’d off from both Beds of the Brick, with the 
 Compaffes, at this Diftance, divide the upper Hanfe- 
 f, om L to F into equal Parts, and if they happen 
 not to divide it into equal Parts, then open them 
 a fmall matter wider, or fiiut them a final! matter 
 defer, till it doth divide it into equal Parts, and 
 look how many equal Parts you divide the upper 
 Hanfe into, fo many equal Parts you muft divide 
 the lower Hanfe from K to B into likewife (or you 
 may divide the upper Hanfe from the Centre O, 
 making a right Angle from each Sommering Line 
 to the Ellipfis, as is fhewn in deferibing the ftreight 
 Arches following ; and from the Centre O, and the 
 Divifions in the upper Hanfe being thus divided, 
 you may draw the ftrait Lines to the lower Hanfe, 
 and not divide it with the Compafles) thro’ each 
 of which Divifions, with a Rule and Pencil, draw 
 ftrait Lines; then get a Piece of thin Wainfcot, 
 and make it to fit between two of thefc Lines, al~ 
 lowing what thicknefs for Mortar you intend, this 
 will be the Sommering Mould for the Hanfe ; then 
 divide the upper Scheme likewife, with the Com- 
 pafies, at the lame Diftance, into equal Parts, and 
 
 laying 
 
Book I. Of GEOMETRY. i 9 
 
 laying a Ruler on the Centre I, from each Divifi- 
 on in the Scheme G L, draw ftrait lanes to the 
 lower Scheme D K, then make another Som- 
 mering Mould to fit between two of thefo Lines, 
 abating lb much as you intend the Thickneft of 
 your Joints of Mortar to be, which, if you fet 
 very dole Mortars, the Breadth of the Line will 
 be enough to allow : then laying the inner Edge 
 of a Bevil ftrait on the Line K L, bring the 
 Tongue to touch the under Side of the firft Courfe 
 of the Scheme $ then take up the Bevil, and let 
 that Bevil Line upon the Sommering Mould of 
 the Scheme, which Bevil Line ferves for each 
 Courie in the Scheme : but you muft take the Be- 
 vil of each Courfe in the Hanle, and fet them upon 
 your Sommering Mould, and number them with 
 i, 2, 3, 4, &c. becaufe each Courie varies. 
 
 Thus having made your Simmering Moulds, in 
 the next place you muft make the Moulds for the 
 
 n 0 th of your Stretchers, and for the Breadth of 
 the Headers and the Clofiers. A Piece of Wain- 
 fcot 7 Inches long, and 3 Inches and an half broad, 
 wil ferve for the Length of the Stretchers, and 
 the Breadth of the Headers ; the Clofiers^ will be 
 1 Inch ^ broad. 
 
 So the Clofier will be half the Breadth of the 
 Header, and the Header half the Length of tho 
 Stretcher, which will look well. 
 
 It remains now to fpeak fbmething to the other 
 part of the Arch, to wit, A D, whole Courfes both 
 in Hanfe and Scheme run alike upon the Ellipfis 
 Lines, and leem of one Bignels (altho’ perhaps 
 there may be fomedmall matter of Difference, by 
 realon I have not divided the Courles in this Fi- 
 
 D 3 gure 
 
Of GEOMETRY. Book I. 
 
 gure from a right Angle, but every Courfe from 
 the Angle, which it makes with the Ellipfis, which 
 I chofe rather to dp, that fo the Bevil of one 
 Courfe might nopfieem to run more upon the Ellip- 
 fis than the Bevil of another, and the Difference 
 of the Thicknefles being fo inconfiderate, is not 
 difcerned.) 
 
 Having defcribed both the Ellipfis Eincs A D, 
 EG, divide each of them into a like Number of 
 equal Parts, always remembring to make each Di- 
 vifion on the upper Ellipfis Line no greater than 
 the Thicknefs of the Brick will contain, when it 
 is wrought ; then through each Divifion, in both 
 the Ellipfes, draw flrait Lines, continuing them 
 4 or 5 Inches above the upper Ellipfis Line, and 
 as much below the lower Ellipfis Line : 1 hen 
 having provided fbme thin Sheets of fine Pafl- 
 board about 20 Inches fquare, cutting one Edge 
 flrait, take one Sheet, and lay the flrait Edge 
 even upon the Line A E, fo that it may cover 
 both the Ellipfis Lines, and being cut to Advan- 
 tage, it may cover 8 Courfes (° r 9 of the flrait 
 Lines.) Having laid it thus upon the figure of the 
 Arch, flick a Pin or two through it, to keep it ill 
 its Plac£ ; then lay a Ruler upon the Paflboard 
 to the 7th, 8th, or 9th flrait Line of ^the Arch, 
 according as the Paflboard be in Bignefs to cover 
 them, and take a (harp Penknife, laying the Ru- 
 ler upon the. Paflboard true to the flrait Line 
 (whole Ends being continued longer than the 
 Arch is deep, as I dire&ed before, will be feen be- 
 yond the Paflboard) and cut the Paflboard true to 
 the Line ; then take another Sheet and join to it, 
 and put it as you aid the firft, and fo continue till 
 
 you 
 
Book L Of GEOMETRY. 
 
 you have covered the Arch from A E juft to the 
 Line D G, flicking Pins in each Sheet to keep 
 them in the Places where you lay them : Then de- 
 fcribe both the Ellipfis Lines upon the Paftboard, 
 from the fame Centers and Radii that you de- 
 fcribed the Ellipfes under the Paftboard, and either 
 divide the Ellipfis Lines with the Compaffes on the 
 Paftboard, or elfe draw Lines upon the Paftboard 
 from or by the ftrait Lines underneath whole 
 Ends you fee ; but the furer Way is to divide the 
 Ellipfes on the Paftboard, and draw Lines through 
 thole Divisions as you did beneath the Paftboard * 
 then fet 7 Inches, being the Length of each 
 Stretcher, from A towards E, and from D to- 
 wards G, and defcribe, from the former Centres, 
 the Ellipfis 0 0 thro 5 each other Courfe on the Paft- 
 board, as you may fee in the Fig. alfo fet 3 Inches 
 and an Half, being the Breadth of the Header from 
 A towards E, and likewife from D towards G ; 
 alfo fet the fame 3 Inches and an Half from E 
 towards A, and from G towards D, and defcribe 
 thefe two Ellipfis Lines from the fame Centres 
 thro 5 each Courfe which the Ellipfis Line of the 
 Stretchers mifs’d - likewife draw in the fame 
 Courfes, two other Ellipfis Lines one Inch and •! 
 from each of thofe two Lines you drew laft ; 
 which is the Breadth of the Clofiers ; thus one 
 Courfe of the Arch will be divided into two 
 Stretchers, and the next t<? it into three Pleaders, 
 and two Clofiers through the whole Arch. This be- 
 ing done, cut the Paftboard according to the 
 Lines into feveral Courfes, and each other Courfe 
 into two Stretchers, and the Heading Courfe into 
 three Headers, and two Clofiers, exactly according 
 
 D 4 to 
 
 1 
 
p Of GEOM-ETRY. Book I. 
 
 to the Sweep of the black Lead Lines, and mark 
 each Courfe with Figures, marking the firft Courfe 
 of the Hanfe with i, the next with 2, the third 
 with 3, and fo continue till you have marked all 
 theCoarfes to the Key or Middle, for every Courfe 
 differs ; you were beft to mark the lower Clofier in 
 each Courfe with a Cypher on the left Hand of its 
 own Number, that you may know it readily from 
 the upper Clofier, and make no Miftakes when you 
 rorne to fet them ; alfo the middle Headers in each 
 Courfe fhould be marked befides it own Number, 
 the Thicknefs of the upper Header being eafily 
 difcern’d from the lower Header needs no marking 
 befides its own Number : The crofs Joints, and 
 likewife the under fide and upper fide of each 
 Courfe muft be cut circular, as the Paftboards, 
 which are your Moulds, direcft you. 
 
 If you will add a Keyftone and Chaptrels to the 
 Arch, as in the Figure ; let the Breadth of the up- 
 per part of the Keyftone be the Height of the Arch, 
 *viz. 14 Inches, and Sommer, from the Center at I, 
 then make your Chaptrels the fame Thicknefs that 
 your lower part of the Keyftone is, and let the Key- 
 ftone break without the Arch, fo much as you pro- 
 ject or fail over the Jaums with the Chaptrels. 
 
 Other kind of Circular Arches, as half Rounds 
 and- Schemes, being defcribed from one Centre, are 
 fo plain and eafy, that I need fay nothing con- 
 cerning them : But fince ftrait Arches are much 
 ufed, and many Workmen know not the true 
 Way of defcribing them, I fhall write fomething 
 briefly concerning them. 
 
 Strait Arches are ufed generally over Windows 
 and Doors, and according to the Breadth of the 
 
 Piers 
 
Book I. Of GEOMETRY. ^ 
 
 Piers between the Windows, fo ought the Skew- 
 back, or Sommering of the Arch to be ; for if the 
 Piers be of a good Breath, as 3 or 4 Bricks in 
 Length, then the ftrait Arch may be deforib’d (as 
 it’s vulgarly called) from the Oxi, which being but 
 part of a Word is taken from the Word Oxtgoni- 
 um, fignifying an equilateral Triangle with three 
 lharp Angles ; but if the Piers are fmall, as fome- 
 times they are but the Length of two Bricks, and 
 fometimes but one Brick and an Half, then the 
 Breadth of the Window', or more, may be let down 
 upon the middle Line for the Centre, which will 
 give a lefs Skew-back or Sommering than the Cen- 
 tre from an Ox/. I will (hew- how to defcribe them 
 both Ways, and firft from the Oxi. 
 
 Suppole a ftrait Arch x Brick and an Half in 
 Height, to be made over a Window, 4 Feet in 
 Width. [See Fig X.] wherein one Half of the 
 Arch is defcribed from the Oxi, and the other Half 
 from the Width of the Window. Let the Width 
 of the Window be AB; taking the Width between 
 the Compaftes, from A and B as tw'o Centres, de- 
 fcribe the two Arches, interfering each other at 
 P, (though I fpeak here of Compaftes, yet when 
 you defcribe the Arch to its full Bignefs,you mult 
 ufe a Ruler or a Line, fcarce any Compaftes being 
 to be got large enough) then draw another Line 
 above the Line A B, as the Line C D, being pa- 
 rallel to it, at fuch a Height as you intend your 
 Arch to be, as in this Fig. at 12 Inches ; but mod: 
 commonly thefe fort of Arches are but 11 Inches 
 in the Height, or thereabouts, w'hich anfwers to 
 4 Courfos of Bricks, but yon may make them more 
 or lels in Height according as Occanon requires ; 
 
 then 
 
Of GEOMETRY. Book I. 
 
 then laying a Ruler on the Centre P, and on the 
 End of the Line A, draw the Line A C, which 
 15 vulgarly called the Skew-back, for the Arch. 
 
 The next thing to be done, is to divide thofe two 
 Lines A B and C D into fo many Courfes as the 
 Arch will contain, the Thicknefs of a Brick being 
 one of them, which fome do by dividing the upper 
 Line into fo many equal Parts, and from thofe 
 Parts, and from the Centre P, draw the Sommering 
 Lines or Courfes ^ others divide both the upper and 
 lower Line into fo many equal Parts, and make no 
 ufe of a Centre, but draw the Courfes by a Ruler, 
 being laid from the Divifions on the upper Line, 
 to the Divifions on the lower Line, both which 
 Ways are falfe and erroneous ; [but this by Way 
 of Caution.] 
 
 Having drawn the Skew-back A C, take be- 
 tween your Compafles the Thicknefs that a Brick 
 will contain, which I fuppofe to be two Inches 
 when it is rubb’d, and fetting one Point of the 
 Compaffes on the Line C D, fo that when you 
 turn the other Point about, it may juft touch the 
 Line A C in one place, and there make a Prick in 
 the Line C D, but do not draw the Sommering 
 Lines until you have gone over half the Arch, 
 to fee how you come to the Key or Middle ; and 
 if you happen to come juft to the middle Line, or 
 want an Inch of it, then you may draw the Lines, 
 but if not, then you muft open or fhut the Com- 
 pares a little till you do. 
 
 Then keeping one End of the Rule dole to the 
 Centre at P, (the fureft Way is to ftrike a fmall 
 Nail in the Centre P, and keep the Rule clofe to 
 the Nail) lay the other End of the Rule clofe to 
 
 the 
 
Book I. Of GEOMETRY. 35 
 
 the Prick that you made on the Line C D, keeping 
 the Compares at the lame Width Quiz. 2 Inches) 
 fet one Point of the CompalTes on the Line C D as 
 before, fo that the other Point being turned about 
 may juft pals by the Rule, and, as it were, touch 
 it in one Place (you muft remove the Point of the 
 CompalTes upon the Line C D, farther or nearer to 
 the Rule, until it juft touch the Rule in one Place) 
 and fo continue with the Rule and CompalTes until 
 you come to the middle Line, and if it happen that 
 your laft Space want an Inch of the Middle, then 
 the Middle of the Key-courle will be the Middle of 
 the Arch, and the Number of the Courfes in the 
 whole Arch will be odd 3 but if the lall Space hap- 
 pen to fall juft upon the middle Line E F, as it doth 
 in the Fig. then the Joint is the Middle of the Arch, 
 (but if it Ihould happen neither to come even to the 
 Line, nor want an Inch of it, then you muft open 
 or fnut the CompalTes a fmall matter, and begin 
 again till it doth come right) and the Number of 
 the Courfes in the whole Arch is an even Number. 
 
 Note , When the Number of all the Courfes in the 
 Arch is an even Number, then you muft begin the 
 two Sides contrary, viz. A Header to be the lower 
 Brick of the firft Courfe on one Side (or half) of 
 the Arch, and a Stretcher the lower Brick of the 
 firft Courfe on the ocher Side (or Half) of the Arch; 
 And contrariwife, if it happen that the Number of 
 the Courfes be an odd Number, as 25 or 27, or 
 fuch like, then the firft Courfes of each Half of the 
 Areh muft be alike, that is, either both Headers 
 or both Stretchers, at the Bottom. 
 
 Thus having delcribed the Arch, the next thing 
 to be done is to make the Sommer ing Mould, which 
 
 to 
 
3 6 Of GEOMETRY. Book 'I. 
 
 to do, get a Piece of thin Wainfcot (being {trait 
 on one Edge, and having one Side plained finooth 
 tofet the Bevil Strokes upon) about 14 Inches long, 
 and any Breadth above 2 Inches 3 then laying your 
 Ruler, one End at the Centre P, and the other End 
 even in the Skew-back Line, clap the ftrait Edge 
 of the Wainfcot dole to the Rule, fo that the lower 
 End of the Wainfcot may lie a little below the Line 
 A B 3 then take away the Centre Rule, but ftir not 
 the Wainfcot, and laying a Ruler upon the Wain- 
 fcot, juft over the Line CD, ftrike a Line upon the 
 Wainfcot 3 then fet one Point of the Compaffes, be- 
 ing at the Width of aCourfe Quiz. 2 Inches) upon 
 that Line, fb that the other Point being turned 
 about, may juft touch the ftrait Edge of the Wain- 
 fcot (as you did before in dividing the Courfes) 
 then make a Prick on the Line on the Wainfcot, 
 and laying your Centre Rule upon it, and on the 
 Centre P, drawh Line upon the Wainfcot by the 
 Ruler with a Pencil, or the Point of a Compafs, 
 and cut the Wainfcot to that Line, and make it 
 ftrait by (hooting it with a Plane 3 then your Wain- 
 fcot will fit exadlly between any two Lines of the 
 Arch. You may let it want the Thicknefs of one 
 of the lanes, or fome fmall Matter more, which is 
 enough for the Thicknefs ofa Mortar. The Length 
 of your Stretcher in this Arch may be 8 Inches and 
 3, and the Header 3 Inches and 4, but if your Arch 
 be but 1 1 Inches in Height, then make your Stretch- 
 er 7 Inches and an Half long, and the Header 3 In- 
 ches l. One Piece of Wainfcot will ferve both for 
 the Length of the Scretcher, and the Length of the 
 Header, making it like a long Square, or Oblong, 
 whole Sides are 8 Inches and 3 Inches and 
 
Book I. Of GEOMETRY. 
 
 Then take a Bevil, and laying the inner Edge of 
 it (trait with the Line A B, and the Angle ot the 
 Bevil juft over the Angle at A ; take off the Angle 
 that the Skew-back Line A C makes with the Line 
 A B, and let it upon the fmoothed Side of your 
 Sommering Mould for the Bevil Stroke of your 
 firft Courfe ; then drawing your Bevil towards E, 
 ftrait in the Line, until the Angle of the Bevil be 
 juft over the Angle that the fecond Sommering 
 Line makes with the Line A B. When it is lb, 
 draw the Tongue of the Bevil to lie even upon the 
 fccond Sommering Line, (in brief, caule the Bevil 
 to lie exactly on the Line A B, and on the fecond 
 Sommering Line) then take up your Bevil, and lay 
 it on the Mould, and ftrike that Bevil Line on the 
 Mould with the Point of theCompaffes about half 
 a quarter of an Inch diftant from the firft, and that 
 is the Bevil of the Under-fide of the fecond Courfe. 
 Proceed thus, until you come to the middle Line 
 E F, but after you have fet 3 Bevil Lines upon your 
 Sommering Mould, leave about * of an Inch be- 
 tween the 3d and 4th, and fo likewife between the 
 6th and 7th, and the 9th and 10th; which will be 
 a great Help to you in knowing the Number of 
 each Line on the Mould. 
 
 The Moulds for the other half of the Arch, name- 
 ly E B, are made after the fame Manner ; but the 
 Arch is dcfcribed from a Centre beneath P, as Q, 
 which cauleth a lels Skew-back (viz. B D.) 
 
 The diminilhing of the Sommering Mould to any 
 Skew- back may be found by the Rule of Three, by 
 dividing a Foot into 10 equal Parts, and each of 
 thole into 1 o Parts, fo that the whole Foot may 
 contain 100 Parts, Then proceed thus: The upper 
 
 Line 
 
3 8 Of GEOMETRY. Book I. 
 
 Line C F, will be 309, that is 3 Feet and almoft 1 
 Inch, and the lower Line A E will be 252, that is, 
 2 Feet and an Half and ; and the upper Part of 
 the Sommering Mould will be 1 7 almoft, that is, 
 two Inches offuch whereof there are 12 in a Foot; 
 having thele three Numbers (viz. 309, 252, 17) 
 work according to the Rule of Three, and you will 
 find 13 and 6 of 100 Parts, that is almoft 14 (fuch 
 Parts whereof there are 100 in a Foot, Line- 
 meafure) for the Breadth of the lower Part of the 
 Mould. 
 
 Ton may likewifie find it Geometrically thus : 
 
 Having drawn the upper Line and under Line 
 of the Arch, as C F and A E, and drawn any Skew^ 
 back, as fuppofe A C in [Fig. X.] make at Difcre- 
 tion the Angle G C H in [Fig. XI.] then take the 
 upper Line C F, and fet it from C toF; alfo take the 
 lower Line A E, and fet it from C to E, and draw 
 the Line E F ; then take the Thicknefs of your 
 Brick, which fuppofe to be £ Inches, and fet it 
 from F to G, and draw G H parallel to F E ; I 
 lay, F G is the Breadth of the upper Fart of the 
 Sommering Mould, and E FI the Breadth of the 
 lower Part. Then make your Sommering Mould 
 true to thole two Lines, and beginning in the mid- 
 dle Line A C, delcribe the ftrait lanes by the 
 Mould from the Key F E, until you come to the 
 Skew-back A C, and then take off the Bevil Lines, 
 and let them on your Sommering Mould : With, 
 which I conclude this firft Book of Geometry, being- 
 as it were, an Introdu&ion to that which follows. : 
 
p ysa 
 
 F Gr 
 
■F 
 
 A 
 
 GARDEN 
 
 O F 
 
 Geometrical Rofes: 
 
 OR, SOME 
 
 PROPOSITIONS 
 
 Being hitherto hid, are now made known. 
 
 The Second Book. 
 
 Written in Latin 
 
 By THOMAS HOBBES: 
 
 And done into Englijb 
 
 By FEN. MANT>ET. 
 
 LONDON: 
 
 Printed in the Year M.DCC.XXVIl 
 
Geometrical ROSES. 
 
 PROP. I. 
 
 Of cutting a Right Line in extream and mean 
 Proportion. 
 
 E T there be defcribed a Square A B 
 C D, and let each Side be divided in 
 the Middle, in E, F, G, H * and 
 F E, GH, being joined, they will cut 
 each other in the Centre of the Square 
 at I. Likewife from the Centre D, let there be 
 defcribed a Quadrant D A C, cutting F E and 
 G H in K and X. 
 
 Laftly, Let E X be drawn; I fay E X is equal 
 to the greater Segment of the right Line E F, or 
 of the Side A B, being divided in extream and 
 mean Proportion. 
 
 Let F X be drawn, and with that Semidiameter 
 defcribe an Arch of a Circle X z, cutting F E in 
 z. Likewife with the Semidiameter E X, defcribe 
 an Arch of a Circle X y, cutting the fame F E in 
 y, and let the right Lines X z, X y be drawn. 
 
 Now the Angle Ez X is equal to thofe 2 An- 
 gles z F X, F X z, becaule thefe two Angles are 
 
 within. 
 
Book II. Geometrical Roses. 
 
 within, the other is without the Triangle zFX. 
 Again, the Fame Angle E zX for the fame Caule 
 is equal to the 2 Angles X y z, y X z. And X z y 
 and FXz are equal. Wherefore alfo the 2 Angles 
 XFz, and zXy, are equal to one another. 
 
 Therefore the 2 Triangles • F-X z, yXz, have 2 
 Angles equal to 2 Angles -of the Triangle zXy, 
 and by confoquence 3 to 3 : Therefore they are 
 Equiangled. Wherefore, as XF or EX, to Xz ; 
 fo is Xz or Xy, to zy. Therefore Xz (or Xy) 
 is a mean Proportional between EX (or Ey) 
 and zy. 
 
 In the right Lines EX, F X, let there be taken 
 Er, Fs, and either of them equal toEz, or Fy. 
 Let rs, and sz be joined. Then rX, Xs will be 
 equal. Alfo rs will be paralld to the right Line 
 EFj and therefore the Altern Angies rsy, Fys, 
 and likewife the A kern Angles sry, Ezr, are all 
 equal to one another ; and Fsyr, and Ezsr, will 
 be Rhombus’s both alike. Therefore the right' 
 Lines Fs, sz, ry, rE, Ez, Xy, are equal to one 
 another. 
 
 And becaufe it is manifeft, 
 that X y is a mean Proportio- 
 nal betwen Ey and zy, alio 
 Fy (equal to Xy) will be a^ 
 mean Proportional between ^ 
 
 Ey and zy. 
 
 Therefore it will be, as Ey 
 to Fy, fo Fy to zy ; and be- 
 ing compounded, as Ey more 
 y F (that is E F) to F y more 
 yz (that is, Fz, or Ey) fo 
 likewife is E v to E z. There- 
 £ fore 
 
A i GEOMETRICAL Book II. 
 
 fore E F is divided in z and y, fo that the whole 
 Line E F, a Part E y, and the remaining Part E z ; 
 likewise FE, aPartFz, and the remaining Part 
 Fy, are continual Proportionals. Therefore Ey, 
 that is E X, is the greater Segment of the right 
 Line EF, being divided in extream and mean Pro- 
 portion, which was to be demonftrated. Let him 
 reprehend it that can. 
 
 A N l M A D V E K S. 
 
 To him that rightly confiders, it is a Paradox : 
 Neither Euclid , , nor any other, hath taught this. 
 Who ever thought that right Line which joins half 
 one Side of the Square, with one third Part of the 
 Quadrant infcribed within the Square, to be the 
 ^ greater Segment of the Side 
 
 ' cut in extream and mean Pro- 
 portion ? Thole Segments, of 
 all hitherto, have been de- 
 figned after this Manner. 
 Produce the right Line IE' 
 in M, fo that I E, E M, may 
 be equal, and IM equal to 
 the Side. Moreover from the 
 Centre M, with the Radius 
 M H or M G, defcribe an 
 Arch of a Circle, it will cut 
 off a Part from EF (fuppofe 
 Ey) equal to the greater Segment. But that the 
 right Line EX fhould be equal, they never, nor 
 any one did declare. Therefore, Reader, beware, 
 left thou truft too much in an uncertain Thing un- 
 advifedly, forafmuch as we take Things near the 
 
 OTi 
 
 /t 
 
 K 
 
 y 
 
 
 
 i 7 
 
 x / 
 
 / 1 
 
 
 
 A 
 
 Prop J 
 
 E / 1 
 
 
 1 
 
 4 
 
 H 
 
Book rr. roses. 4? 
 
 Truth for true Things. Examine the Demonltra- 
 tion, and confer with true or fiird Numbers or 
 confult Algebraifts. 3 
 
 CoNSECT. I. 
 
 T HE Lines sy, rz, being drawn about, will 
 make the Figure of a Pentagon (viz. afive- 
 lided Figure) Xsyzr, Eq mangled and Equicrural. 
 For the Angles Xsr, X rs, are equal, becaule the 
 right Lines Xs, Xr are equal: and either of thole 
 Angles are equal to the Angle at F, becaufe s r F y 
 are Parallels ; and the Angles rsz, rsy, are either 
 ot them equal to the fame Angle at F, becaule Es 
 Fr, are Rhombus’s alike. Alio the Angle yXz 
 appears to be equal to the Angle at F. Likewile 
 the Angle. Xzr, is equal to the Angle yXz be- 
 caufe rz, Xy are Parallels. Lallly, the Angles 
 Xzs, Xyr, are equal to the fame Angle yXz 
 becaule the Bales X s, X r, y z, are equal. ’ 
 
 C O N S E C T. II. 
 
 T HE Square of the greater Segment Ey, is 
 equal to two Squares ( to wit) to the Square 
 ol the right Sine of 30 Deg. and to half the Square 
 ol lo many Deg. of the Line of Chords. For E I 
 is equal to the right Sine of 30 Deg. And becaufe 
 (.the Chord X K being drawn) the Triangle X K I 
 ismade Equicrural and right Angled, the Square 
 ot X I will be half the Square of the Chord XK • 
 therefore Ey, that is, EX being fquared, con- 
 tains as much as the Square of E I and X I added 
 together. 
 
 E 2 
 
 Co N* 
 
44 
 
 GEOMETRICAL Book II. 
 
 Con sect. III. 
 
 Ky is the Half of FI : cut 
 F I in the Middle at t ; there- 
 fore becaufe Fy more yl, 
 and 1 1 more tF are equal ; 
 „ by how much F y is greater 
 than F t, by fo much 1 1 is 
 greater than y I ; but F y is 
 greater than F t, by lb much 
 j) as t y is. 
 
 Wherefore alfo 1 1, that is 
 Ft, is greater than yl, by 
 the fame Quantity t y. And 
 F t is divided in K, as 1 1 in y. 
 
 For becaufe F y more y I, and I K more K F are 
 equal , by how much F y is greater than I K, lo 
 
 much is F K greater than y I. 
 
 Therefore when as F t is made equal to t y more 
 V I - Ft will be divided in K, as 1 1 in y. 
 
 Wherefore t K is equal to y I ; and K y equal to 
 F K more y I. Therefore K y is equal to the Halt 
 
 oFFI. 
 
 PROP. 
 
45 
 
 Book II. ROSES. 
 
 One Segment being given of a right Line divided in 
 cxtream and mean Proportion , to find the other. 
 
 L E T the right Line given be A B, being the 
 greater Segment of any right Line. Cut A B 
 in the Middle in D, and from the Centre D, with 
 the Interval (or Semidiameter) A D, defcribe the 
 Circle A F B E. Then to the Point A, raife a Per- 
 pendicular A C, equal to the given Line A B ; 
 and through the Centre D, draw a Line to the 
 Concave Periphery C E, to which let A G be made 
 equal. 
 
 I fay, the whole Line A G, hath the fame Pro- 
 portion to A B, that A B hath to B G. For, be- 
 
 E 3 caule 
 
4 6 GEOMETRICAL Book II. 
 
 caufe (by the 36 of the 3d Book of Euclid ) as EC, 
 is to E F ; fo is E F, to E C. Alfo as A G, toge- 
 ther with his Equal E C, is to A B, and his 
 Equal E F ; fo. will A B, be to B G, and his Equal 
 F C. Wherefore (by the Definition of extream and 
 mean Proportion) the right Line A G is divided in 
 extream and mean proportion in the Point B. 
 
 Again, let B G be the leffer Segment of any right 
 Line. Cut B G the given Line in the Middle in C. 
 And from the Centre C, with the Semidiameter CB, 
 defcribe a Circle B M G. In which Circumference, 
 the Quadrant B M being taken, draw by the Point 
 n, the right Line MR, fo that Bn be a third Part 
 of the given Line B G. Then draw B K and G K, 
 and in the Line B K, take a Part B o, which fhall be 
 equal to the right Line B n, draw the Line o, n, to 
 which let there be a parallel Line drawn, BL, cut- 
 ting GK being produced in L. Finally, raife a 
 Perpendicular to G L in the Point L, which will 
 cut G B, being produced in D. From the Centre 
 D, with the Space D B, defcribe the Circle B L A. 
 I fay, G A is to A B, as A B to B G. 
 
 For becaufe the Angle B M K infifts in the Cir- 
 cumference of the Quadrant B M, it will be half a 
 right Angle, and being drawn from the right Angle 
 BKG, it leaves half aright Angle 11KG. And 
 becaufe the Angle BKG is divided by the right 
 Line nK in the Middle, the right Line Kn, will 
 cut the right Line B G, fo that as G n, is to n B ; 
 fo will G K be to K B (by the 3 of the 6 of Euclid'). 
 But G n is by Conftrudtion double to Bn There- 
 fore alfo G K is double to K B. And becaufe B o 
 is put equal to B n, the Angle 011B will be equal 
 to the Angle Bon. And becaufe n o and B L, are 
 
 Parallels, 
 
Book II. ROSES. 47 
 
 Parallels, the Angle LBD will be equal to the 
 Angle o n B. Likewife becaule L D and K B are 
 Parallels and Perpendiculars, the Angles D L B, 
 and Bon, will be equal Likewife Bon, and 
 B n o, are equal. Therefore D L B and B n o are 
 equal. But B n o and D B L are equal, there- 
 fore D L B and D B L are equal. Therefore alfo 
 the right Lines D L, D B, being Chord Lines, are 
 equal. Likewife the Circle defcribed with the Space 
 (or Semidiameter) D B will pals by the Point L. 
 And becaufe D L is perpendicular to G L, G L 
 toucheth the Circle B L A in L. Therefore it will 
 be (by the 36 of the 3 of Euclid ) as G A to G L, 
 fo G L to G B. And A B is equal to the faid G L. 
 (For whereas the right Line G K is double to the 
 right Line BKj fo alfo the right Line G L, will be 
 double to ;he right Line L D). Therefore in the 
 
 E 4 lame 
 
48 GEOMETRICAL Book II. 
 
 fame Manner the right Line B A is double, and for 
 that Caufe equal to the right Line GL. There- 
 fore alfo it will be as G A is to AB ; fo A B to 
 B G, and by confequence, we have added the other 
 Segment of the right Line given, &c. which was to 
 be done. 
 
 A N I M A D VE R S. 
 
 I fee not wherefore he hath invented this new 
 Way of cutting proportional Lines, unlefs, perhaps, 
 from this he thought to find the Greatneft of the 
 Circle, which he fhould feek for, by comparing the 
 Power of the greater Segment, with the Power of 
 the Semidiameter, not before known. 
 
 The Quantities of thefe Segments, nor the 
 Squares of them, cannot be exprelTed accurately by 
 Numbers. But their Proportion comes very nigh to 
 the Proportion of 5 to 3 ; for if the Side A B be 8 
 Parts, the greater Segment will be almoft 5, and 
 the leffer Segment almoft 3 ; for 8, 5, 3 J are con- 
 tinual Proportionals. But they make a Line greater 
 than the right Line A G by J part of the 25th Part 
 of the Line A B : So that the Difference of the 
 whole Line, and his greater Segment, is a fmall 
 matter greater than | Parts of the whole Side, and 
 the greater Segment left than 3 Parts • but how 
 much left is not eafy to be difcovered in the Dia- 
 gram (being it is fmall). 
 
 Alfo becaufe that greater Segment fubtends \ 
 Parts of the Quadrant A C, the Difference of the 
 Segment from the Arch . which it fubtends, could 
 not eafily be diftinguifhed. 
 
 PROP. 
 
Book II. 
 
 4 9 
 
 ROSES. 
 
 Of Regular 
 
 ONS, 
 
 In a Circle given to defer ibe a Regular Heptagon. 
 
 . . x / • [ , - v • i r -• A. 
 
 W Hatloever right Line, fuppofe A B, Jet it be 
 cut in 8 equal Parts, whereof let A C be 7. 
 Then from the Centre A, with the Semidiameters 
 AB, AC, deferibe 2 Circles. Moreover take one 
 eight Part of the Perimeter of the outward Circle 
 (to wit) B D (which is eafily done) and draw the 
 
 Line 
 
50 GEOMETRICAL Book II. 
 
 Line A D, cutting the inner Circle in E, which 
 will cut off the eighth Part of it C E. 
 
 Divide the Arch B D in the Middle in a , and 
 draw the Chord Lines B a, a D ; likewife in the 
 Arch C E apply the right Line C b equal to B a, 
 and again b c equal to the fame B a or a D j for 
 they are equal. 
 
 I fay, A right Line Cc being drawn, is the Side 
 of a Heptagon in the Circle C E. For becaufe A B 
 is to A C, as 8 to 7 ; lo alfo the Perimeter of the 
 Circle BD to the Perimeter of the Circle CE, will 
 be as 8 to 7. 
 
 Alfb becaufe the Sedtors A B D and ACE are 
 alike, and the Triangles A B D, ACE are alike, 
 both the Arch B D, to the Arch C E, and the Chord 
 B D, to the Chord C E, will be as 8 to 7. 
 
 For the fame Caufe the two Chords B a, aD 
 will to C /, i E the two Chords of the Half Arch 
 CE, as 8 to 7. 
 
 Now becaufe (by Conftru&ion) both the Chords 
 B a D, and the 2 Chords C b 9 bc 9 either to other, 
 and between themfelves, are equal, and thofe Chords 
 will be to the 2 Chords of half the Arch C E, as 8 to 
 7. And likewife for the fame Reafon the Arch C c to 
 the Arch CE, as 8 to 7. Alfb take A £, Ac, and 
 draw them thro 5 to the outer Circumference in d 
 and c, thofe 2 Chords B d, de^ will be to the 2 
 Chords B a, a D, or C b c, as 8 to 7. 
 
 Again, cut the Arch B a in the Middle in /, and 
 draw A/, cutting the Arch CE in g, and draw the 
 Chords B /, Cg • likewife cut the Arch B d in the 
 Middle in k , then draw A k cutting the Arch C E 
 in b , the Chord B k will be to the Chord C as 8 
 to 7. And by confequence, 4 Chords C to 4 
 
 Chords 
 
Book II. ROSES. 5 1 
 
 Chords C g, as 8 to 7. Likewife if 2 Arches C b 
 and Cg , be cut in the Middle, and their Bifegments 
 cut ad infinitum , every Chord of thole, to evey Chord 
 of thefe, will be in Proportion, as 8 to 7. But the 
 Chords thus taken infinite in any Arch are (all toge- 
 ther) the Chord of the fame Arch. Wherefore The 
 Arch Cc is to the Arch C E, as 8 to 7. And the 
 Arch C c is the 7th Part of the whole Perimeter by 
 C. Therefore a right Line being drawn C c, is the 
 
 Side of a Heptagon in the Circle by C. Likewife the 
 right Line A e being drawn, will cut the 7th Part 
 of the Perimeter by B. 
 
 For 
 
5 2 
 
 GEOMETRICAL Book II. 
 
 For if every one of the Chords to every one of the 
 Arches which they fubtend, are not equal, thofe 
 feveral Arches, and their Bifegments, may be again 
 bifeded, which is contrary to the Suppofition. 
 Therefore the Side of a Heptagon is found, which 
 was propofed. 
 
 C o n s e c t. I. 
 
 From this Demonftration appears a Method of 
 finding the 7th Part of an Angle given. For if to an 
 Arch given, of what Limits foever, be drawn ftrait 
 Lines from the Centre of the Circle 3 moreover the 
 Semidiameter being divided into 8 Parts, and 7 of 
 thofe Parts being taken, with that Space, and upon 
 the Centre aforefaid defcribe a Circle, the greater 
 Arch will be to the leffer, as 8 to 7. Wherefore 2 
 Arches of the greater Bifeded, to 2 Arches of the 
 leffer Bifeded, will be as 8 to 7. Likewife their 
 Chords will be as 8 to 7. Therefore 2 Chords of the 
 greater Arch being applied to the leffer Arch, will 
 determine the Excefs of the 7th Part of the leaft Pe- 
 rimeter, above the 8th Part of the fame Perimeter. 
 For it is manifeft, that the Arch D c is the 7th Part 
 of the Arch C c. For whether the Arch of the Peri- 
 meter be divided in 7 Parts, or more or left, the De- 
 monftration will always be the fame. 
 
 C o n s e c t. II. 
 
 An Arch being deferibed with the Radius B C, is 
 equal to the 8th Part of a Perimeter of a Circle 
 whofe Radius is A B : And to the 7th Part of the 
 Perimeter of a Circle, whofe Radius is A C : And to 
 l_ * a 6th 
 
Book II. ROSES. 5 j 
 
 a 6th Part of the Perimeter of a Circle, whole Ra- 
 dius is | Parts of the right Line A B, Sc. 
 
 Let us experience this our Method in known Po- 
 lygons, and fee whether or no by this, the Side of 
 an Equilateral Triangle may be found from a Tetra- 
 gon (viz. a Square). 
 
 From the Centre A, 
 defcribe the Circle BCD, 
 whofe Quadrant is B D. 
 
 Then draw the Chord 
 Line B D, which is the 
 Side of a Square infcrib- 
 ed within that Circle. 
 
 From the fame Centre 
 A, with a Semidiameter 
 A E ( which let be to 
 A B, as 3 to 4) defcribe 
 the Circle E F G ; then 
 the Circle by B, will be to the Circle by E, as 
 4 t0 3 - 
 
 Let the Quadrant BD be cut in two in the Mid- 
 dle in C, and let the equal Chord Lines B C, C D, 
 be drawn. Then draw A C, cutting E G in F„ the 
 Arch E F will be the 8th Part of the Perimeter 
 by E, and equal to the Arch F G. 
 
 Therefore both the Arch and the Chords BC, CD 
 will be to the Arch, and to the Chords E F, F G, 
 as 4 to 3. 
 
 Then from the Point E in the Circle drawn by E, 
 apply EH, HI, being the 2 Chords BC, CD, either 
 of thefe being equal to either of thofe. Wherefore 
 the 2 Chords EH, HI, are to the 2 Chords E F, 
 F G, as 4 to 3 i that is, as the Arch BD to the Arch 
 E G. Wherefore as the Arch E I to the Arch E G, 
 
 fo 
 
54 
 
 GEOMETRICAL Book II. 
 
 fb is 4 to 3. Therefore the Arch G I is the 3d Part 
 of the Arch E G, that is, a 12th Part of the whole 
 Perimeter drawn by E; and the Arch El, a 3d 
 Part of the fame Perimeter. The Demonftration 
 is the lame which concerns the Heptagon, which 
 you have before in Pag. 50. * Therefore the Chord 
 Line E I being drawn, is the Side of an Equilateral 
 Triangle in the Circle drawn by E. 
 
 Again let us try the fame Method from a Hexagon to 
 a Pentagon. 
 
 From the Centre A, with the Semidiameter A B, 
 deicribe the Circle BCD; and in that Circumfe- 
 rence from the Point B, apply the Chord Line B D 
 equal to the Semidiameter A B. 
 
 Then the Chord Line BD, will be the Side of a 
 Hexagon drawn in the Circle by B. 
 
 In the Semidia- 
 meter A B, take 
 A E, which muft be 
 to A B, as 5 to 6. 
 And from the fame 
 Centre A, with the 
 Radius AE defcribe 
 the Circle E F G. 
 Then the Perimeter 
 by B to the Peri- 
 meter by E, will be 
 as 6 to 5. Let B D 
 be cut by a right 
 Line AC (cutting 
 the Circle drawn by 
 and Jet the Chords 
 B C, C D be drawn, equal to which apply the 
 
 Chords 
 
 B 
 
Book II. ROSES. > 55 
 
 Chords EH,HI, in the Circle drawn by E. Then 
 are the two Chords EH, HI, to the two Chords 
 E F, F G, as the Arch B D to the Arch E G, that is 
 as 6 to 5 ; and the Arch E I, to the Arch EG, as 6 
 to 5 ; and the Arch G I is a 5th Part of the Arch 
 E G, that is, the 30th Part of the Perimeter ; and 
 the whole Arch El, 6 of thofe 30 Parts (that is, 
 a 5th Part) of the whole Perimeter drawn by E. 
 The Demonftration is the fame as in the Heptagon. 
 Therefore the Chord Line El, is the Side of a 
 Pentagon drawn in the Circle by E. 
 
 Cor. 1. Therefore the Side of a Pentagon may be 
 found, without the Work of cutting the Semidiame- 
 ter in extream and mean Proportion. 
 
 Cor. 2. As from the outward Circle to the inward, 
 the Demonftration hath proceeded hitherto ; fo 
 like wife it may proceed from the inward Circle to 
 the outward. As from a Triangle given, may be 
 found a Quadrat (or Square) and a Pentagon from 
 a Square, and a Hexagon from a Pentagon, and fo 
 of the reft. 
 
 For to the Side of a Pentagon given E I, the 2 
 Chords EH, HI are given. Wherefore if to the 
 Semidiametjer A E, be added a 5th Part of the fame 
 Semidiameter (to wit) E B, and the Arch B D be 
 defcribed ^ the 2 Chords EH, HI, will be equal 
 to thofe 2 Chords BC, CD, and equal each to 
 other, and the Chord B D, the Side of a Hexagon 
 inforibed in the Circle by B. 
 
 PROP. 
 
56 GEOMETRICAL Book IE 
 
 PROP. IV. 
 
 Of the Proportion of crooked Lines , to crooked Lines 
 in the - Circumferences of Circles. 
 
 I. ASa Circle is defcribed from a Semidiameter 
 
 il with one Foot of a Pair of Compaffes being 
 fixed, and the other Foot carried about ^ fo alfo it 
 may be underftood to be defcribed from a right 
 Line given, being bent uniformly, that is, fo as 
 the Angles be always equal ; from which certain 
 Flexion or Bending, if the Angles be conceived to 
 be infinite in Number, is defcribed a Circle. For 
 a Circle in its Nature, differs nothing from a Poly- 
 gon of an infinite Number of Sides. 
 
 And bending is a departing from Straitnefs ac- 
 cording to fbme Angle, which is Crookednefs. 
 
 2. And the Crookednefs of fome to other fome, 
 is greater or leffer ; and therefore Crookednefs is 
 Quantity, and belongs to the Subjedl of Geometrici- 
 ans, and chiefly to thofe who write concerning the 
 Magnitude of a Circle, and the Infcription of Poly- 
 gons in a Circle, altho 3 concerning this Thing, no- 
 thing hath been delivered to us from the Antients. 
 
 3. What Proportion an Angle in a Circumference, 
 to an Angle, in a Circumference in the fame Circle 
 hath, the fame Proportion hath the Crookednefs of 
 the greater Arch, to the Crookednefs of the leffer 
 Arch. For fince Crookednefs is no-other Thing, than 
 a Bowing (by an Angle in a Circumference) from 
 Straitnefs ; it mull: be that the greater Angle makes 
 the greater Crookednefs. From whence it follows, 
 that the Crookednefs of Arches in the fame Circle, 
 be to each other like their Angles. 
 
 '• 4. In 
 
Book II. ROS E S. 
 
 4. In divers Circles the Crookednefs of the greater 
 Perimeter, is lefs than the Crookednefs of the le Her 
 Perimeter, in the Proportion of the Radius to the Ra- 
 dius, or of the Diameter to the Diameter reciprocal. 
 For in great Circles, as in the great Circle of theEarth, 
 no Crookednefs can bedifcerned in a long Space, but 
 in a Ring it is every where difcern’d ; therefore in in- 
 different Circles, the leffer Circles have the greater 
 Crookednefs, for this very Caufe, becaufe the Diameter 
 is leffer • which is manifeft by the Light of Nature. 
 
 5. ( If the Proportion of an Arch in a Circumfe- 
 rence be the fame to the Perimeter that the Radius 
 is to the Radius) Any fame Chord fubtends a 
 greater Portion of his own Perimeter, than of a 
 greater Perimeter, according to the Proportion of 
 the greater Perimeter to the leffer. 
 
 For the Caufe wherefore the fame right Line, fub- 
 tends a greater Part of the lefs Perimeter than of the 
 greater, is the only and effential greater Crookednefs 
 of the fame Line, being bowed in the leffer Peri- 
 meter, than when it is bowed in a greater Perimeter : 
 Therefore if the Crookednefs of a leffer Arch to the 
 Crookednefs of a greater Arch in a Semicircle be as 
 S to 7, the Chord, which fubtends the eighth Part 
 of the greater Arch, will fubtend the feventh Part 
 of the Semiperimeter of the leffer : I fay in a Semi- 
 circle, becaufe the Crookednefs of a Perimeter be- 
 yond a Semicircle proceeds a contrary Way, and be- 
 caufe the Subtenfes of the Arches, which together 
 with themfelves in a Semicircle increafe j but be- 
 yond a Semicircle they all decreafe. 
 
 6. Alfo therefore an Angle in the Circumference of 
 a leffer Perimeter (becaufe the Subtenfes are equal) 
 will be to an Angle in a greater Perimeter, as the 
 greater Perimeter to the leffer. 
 
 F 
 
 PROP. 
 
5 * 
 
 GEOMETRICAL Book II, 
 
 prop. v. 
 
 mean Proportional between the Semidiameter of a 
 Circle , and i Parts vf the fame is equal to -] Parts 
 of the Fourth Part of the Circle. 
 
 D Efcribe a Quadrant of 
 a Circle D AC, and 
 compleat the Quadrat A B 
 CD j in the Side D C take 
 D T, being two fifth Parts 
 of the Side D C, and be- 
 tween D C and D T, let 
 there be taken a mean Pro- 
 portional DR, fo that DC, 
 DR,DT, may be continual 
 Proportionals; alfo deferibe the Quadrantal Arches 
 R S, T V y then the Arch TV is two fifth Parts 
 
 of the Arch C A. . , T • p 
 
 I fay the Arch T V, and the right Line D P_ 
 
 are equal. , . , _ . 
 
 Let it be fuppofed that there is a right Line given 
 
 equal to the Arch A C, and from it deferibe a Qua- 
 drantal Arch (viz. with the right Line given being 
 Radius, deferibe the fourth Part of a Periphery) 
 the Line D C, C A, and the Quadrantal Arch 
 above C A are continual Proportionals. 
 
 Let there be writ apart 
 And under thefe 
 Again under thefe 
 
 DC, C A, the Arch above CA~ 
 DR,RS, the Arch above R S-^f 
 D T, T V, the Arch above TVt* 
 
 Then the Antecedents to the Confcqueuts wil 
 
 be thro’ all the Orders or Ranks forward, as the Se 
 
 midiametc: 
 
Book II. ROSES. 59 
 
 midiamerer to the Arch of a Quadrant, and back- 
 ward, as an Arch of a Quadrant to the Semidiameter. 
 
 Therefore as D C to RS, fo is R S to the Arch 
 above T V ; therefore D C, R S, and the Arch above 
 T V, will be continual Proportionals. And be- 
 caufe DC,CA, and the Arch above C A are like- 
 wife continual Proportionals, and have the firft An- 
 tecedent D C common • the Proportion of the Arch 
 above C A to the Arch above T V, will be (by the 
 23 of the 14 of Euclid ) in Duplicate Proportion of 
 CA to RS; and for the fame Caufe the Arch above 
 RS, is a mean Proportional between the Arch above 
 C A, and the Arch above T V. Now if DC be 
 greater than R S, likewife R S will be greater than 
 the Arch above T V ; and the Arch C A greater than 
 the Arch above R S. Wherefore, when as DC, C A, 
 and the Arch above C A are continual Proportionals, 
 the Arch above T V, and the Arch above R S, and 
 the Arch above C A cannot be continual Proporti- 
 onals^becaufe it is demonftrated to the contrary. 
 Therefore D C is not greater than R S. 
 
 Again let R S be fuppofed to be greater than 
 D C ; then the Arch above R S will be a mean Pro- 
 portional between the Arch above T V, and the 
 greater Arch above C A. Therefore the Inconve- 
 nience will return. 
 
 Wherefore the Semidiameter D C is equal to the 
 Arch RS • and by Cqnfequence the Arch FV, that is 
 two Fifths of the Arch C A, and the right LineD R are 
 equal, that is, a mean Proportional between the Se- 
 midiameter and two fifth Parts of the fame, is equal 
 to two fifth Parts of the fourth Part of the Circum- 
 ference of the Circle: Which waS to be demon- 
 ftrated 
 
 F 2 
 
 PROP. 
 
6o GEOMETRICAL Book II. 
 
 PROP. vi. 
 
 E T there be defcribed a Quadrat A B C D, 
 and let it be cut in the Middle both Ways by 
 
 EF, G H ; alfo let it be cut by the Diagonals AC, 
 B D (concurring in the Centre I) four Ways. 
 
 Moreover, between D C and two fifth Parts there- 
 of, let there be taken a mean Proportional D R, and 
 join A R, cutting E F, in a, let it be produced until 
 it meet with the Side B C being produced in b. 
 
 I fay that the right Line B b is Quintuple to QviZ. 
 five times as long as) the right Line E or the fifth 
 Part of the Arch A C (or it may be thus render’d, 
 the right Line B b is equal to the Quadrantal Arch 
 
 A C.) 
 
 Becaufethe Triangles A D R, A B b are alike, 
 and the Arch of a Quadrant defcribed from D R is 
 equal to the Side DC, or ABj likewife the Arch 
 defcribed from A B will be equal to the right Line 
 B b. Therefore whereas D R is two fifth Parts of 
 the Arch A C, and by confequence B b five of thofe 
 Fifths $ B b will be five times as long as the right 
 Line E which is the Half of the right Line D R, 
 and the fifth Part of the Arch A C, or the right 
 Line B b. Which was to be demonftratcd. 
 
 Therefore if in the Side B C, be taken a part B /, 
 equal to E that part B i being five times repeated, 
 will end in b. 
 
 A compendious Exposition. 
 
 If 2 right Lines whatfoever have a mean Propor- 
 tional between them, which is equal to the Side A B, 
 it will be as the Arch A C to one of them ; fo reci- 
 
 procally 
 
Book II. ROSES. 61 
 
 procally the other of them will be to two fifth Parts 
 of the fame Arch AC; as in Example, becaufe the 
 Side A B is a mean Proportional between the Arch 
 A C, and two Fifths of the fame ; and the fame mean 
 Proportional between the Diagonal Line A C, and 
 the Half of it D I. The right Angle under the Arch 
 A C, and two fifth Parts of the fame, will be equal 
 to the right Angle under B D, and the Half of it D I. 
 Therefore as the Arch A C is to his Diagonal A C, 
 or B D, fo will the Half Diagonal D I be to D R. 
 And as the Half of the Arch AC, to the Half of 
 the Diagonal, fo is the Half Diagonal to D R. 
 
 C O N S E C T. 
 
 From hence appears the Magnitude of the eighth 
 Part ofthewhole Perimeter. For if to the rightLine 
 DR, be added R z, being a fourth Part ofD R, the 
 whole Line D z, will be five of thofe ten Parts, that 
 is half the Arch A C. For when as D R is two Fifths 
 that is, four Tenths, D z will be five Tenths. * * 
 
 PROP. VII. 
 
 r ¥~ , HE fame Lines continuing, draw the right 
 -A. Line D F. I fay D F is a mean Proportional 
 between the whole Arch A C, and his Half 
 Draw the right Line R r parallel to the Side B C, 
 cutting the Diagonal D B in r, and D F in d. Then 
 becaufe D F cuts the Side B C in the Middle in F, 
 the right Line D d will cut the rightLine R r in the 
 middle in d. Let L N be drawn parallel to the fame 
 Side B C, cutting D F in e, and the Side D C in N. 
 Then D N and N L will be equal, and either of 
 
 F 3 them 
 
61 GEOMETRICAL Book II. 
 
 them equal to the Semidiagonal D I, and N L, will 
 be divided in the Middle in e. 
 
 Therefore becaufe by the compendious Expofition 
 of the precedent Prop. Dz,DN,D R,are continual 
 Proportionals: If with the Radius D 2 be defcribed 
 an Arch zf, it will cut the right I -ine N Line; like- 
 wife if with the Radius D N be defcribed an Arch of 
 a Circle N I, it will cut the right Line R r in d. 
 
 Then defcribe the Quadrantal Arch N I O, which 
 (as it is (hewn) w ill pafs through d. Then becaufe 
 D R is the Radius of a Circle, whofe fourth Part of 
 the Perimeter is equal to the Side DC; the right 
 Line D d or D N will be the Radius of a Circle, 
 whofe fourth Part of the Perimeter is equal to the 
 right Line D F. But the right Line D N is the 
 Radius of a Circle, whofe fourth Part of the Peri- 
 meter is the Arch it felf NIO. 
 
 Therefore the right I. ine DF, and the Arch NIO 
 are equal : And the Arch NIO is a mean Proporti- 
 onal between the Arch AC and his Half, there- 
 fore alfo the right Line D F is a mean Proportio- 
 nal. &c. which was to be demonftrated. 
 
 Cor. If from a Point z be drawn aright Line 2 c 
 parallel to the Side B C, cutting the Diagonal D B 
 in c,; D c will be equal to D F. For D C will be a 
 mean Proportional between D z and the double of 
 it. Likewife it is made appear, that D z is equal 
 to half the Arch A C, 
 
 C O N S E C T. I. 
 
 The Arch A C, that is the right Line B b, is a 
 mean Proportional between the Side D C, and the 
 Quintuple of (< viz. a Line five times as long as) the. 
 ^ Half 
 

 ; 
 
o 
 
Book II. ROSES. 6 3 
 
 Half Side B F. For feeing that two Fifths of the Side 
 D C, the right Line D R, and five of thofe Fifths of 
 the Side D C are continual Proportionals. If the 
 Proportion be continued, D R, D C, and the mean 
 Proportional between D C, and the Quintuple of 
 the Half Side B F, will be continual Proportionals: 
 Therefore a Quadrat being made or drawn from 
 the right Line B b, is equal to ten Quadrates drawn 
 from the Half Side B F. From whence alfo it fol- 
 lows, that a Quadrantal Arch being defcribed from 
 the Arch A C, being extended in a right Line, is 
 Quintuple to the Half Side B F. 
 
 C O N s E C T. II. 
 
 The fame Lines remaining, to the Side A D let 
 there be added in a diredt LineDg, equal to the right 
 Line D that is equal to the Arch C L ; moreover 
 cut the whole Line A g in the Middle in h ; then 
 from the Centre 1 o, with the Radius h A, or h g, de- 
 fcribe the Arch of a Circle AY, cutting the Side DC 
 in Y: Then D Y will be a mean Proportional be- 
 tween the Side D C and D z ; and a right Line Y Q 
 being drawn parallel to the Side B C, cutting the 
 Diagonal D B in Q, Y Q_ will be the Side of a Qua- 
 drant (from Archimedes's Demonftration) equal to 
 the Sedtor D C L, or an eighth Part of the whole 
 Circle defcribed from the Semidiameter D A, 
 
 C O N S E C T. III. 
 
 I 
 
 From hence it follows, that the Arch A C is equal to 
 the Compound of the Side BC and a F angent of 30 Deg. 
 
 For becaufe D C is a mean Proportional between 
 the Arch A C and D R • if to D R and D C there be 
 
 F 4 taken 
 
6 4 GEOMETRICAL Book II. 
 
 taken a third Porportional, it will be equal to the 
 Side BC, together with the Tangent of 30 Deg. 
 
 But the Diagram muft be renewed (or drawn anew.) 
 Therefore let ABCDbe a Quadrate, and let it be 
 divided into four equal Parts by the right Lines E F, 
 G H v alfo let it be divided into four equal Parts by 
 the Diagonal Lines A C, B D, concurring in the 
 Centre L Alfo draw the Quadrantal Arches A C, 
 B D cutting E F and G H, in K and X. 
 
 Produce the Lines B C, G H, and let there be taken 
 the right Lines CL, fi'-M, either of them being equal 
 to half the Side, and let B M be joined. Then B M 
 being taken as a Semidiameter, with it, from the 
 Centre B, deferibe an Arch of a Circle, cutting B C, 
 being produced in i. Then becaufe B L being fqua- 
 red, contains as much as nine Quadrates (or Squares) 
 made from the Half Side, and LM H C one of them 
 Quadrates : Alfo the right Line B M, that is, B / 
 being iquared (to wit, a Quadrate being drawn with 
 four Lines, each of them being the Length of B i ) 
 that Quadrate contains as much as ten Quadrates 
 made from the Half Side C H 3 and therefore B i 
 is equal to the Arch AC, or B D. 
 
 Draw A K, and produce it to the Side B C in P ; 
 A P will be a Secant of 30 Deg. and BPa Tangent 
 of 30 Deg. and A P the double of B P. To the Side 
 B C add C k equal to B P, and the whole Line B K 
 will be compofed of the Side B C, and a Tangent 
 of 3 o Deg. B P or C /. 
 
 It remains therefore to be demonftrated, that i . 
 and k meet in the lame Point. 
 
 Let A i be joined, cutting D C in R ; and with 
 the Radius D R, deferibe a Quadrantal Arch R r, 
 cutting the Diagonal Line B D in T. 
 
 From 
 
Book II. ROSES. 65 
 
 From the Centre T with the Radius T D, de- 
 fcribe an Arch of a Circle cutting B C in S, and 
 produce S T to the Side A D in S; then areD R, 
 DT,TS Equals. 
 
 Then becaufe A B is a mean Proportional both be- 
 tween AP and EK, and between B i and D R, like- 
 wife between the Diagonal B D and its Half Cl; 
 it will be as B i or the Arch AC) toB D, fe> C I 
 to D R or S T. And likewife as B D to A P, fb re- 
 ciprocally E K to D R or S. T. Wherefore if there 
 be taken in the Side BC, a certain right Line equal to 
 E K,fuppofe the right Line By, and from thence to 
 the Side A D be drawn y t parallel to A P, it will be 
 as B i to y £, fo B y to S T. Wherefore the right 
 Line y t will pafs by T, and y T will be equal to 
 S T, which is abfurd. Then B S is equal to E K, 
 and S s equal to A P. Therefore becaufe A P is 
 equal toDK, DK will be parallel to S s. Then 
 
 as 
 
66 GEOMETRICAL Book II. 
 
 as B k to D k 9 fo is B S to S T : Therefore B i and 
 B k are Equals, which was propofed. 
 
 CoNSECT IV. 
 
 From hence it follows, that the right Line H R 
 (which is the Difference whereby two Fifths of the 
 Arch A C exceeds the Half Side D H ) is equal to 
 F K the verfed Line of 30 Degrees. 
 
 For if in the Side AB, there be taken A V equ*il 
 to E K, and with the Radius A V be defcribed an 
 Arch of a Circle, cutting A P in Y, the Arch V Y 
 will be equal to the Half Side B F or BY, and A Y 
 equal to A V j and therefore B Y will pafs to x y and 
 the fame B Y will be a Tangent of 30 Degrees in the 
 Circle, whofe Semidiameter is A V or E K. Then 
 produce V K to the Diagonal A C in z, V Z will 
 be equal to E K, wherefore a Perpendicular being 
 dropt down from the Point S, it will pafs through z 
 
 and 
 
Book II. ROSES. 67 
 
 and x ; and B x being produced to the Side D C in 
 d^Cd will be equal to C / : But S x is equal to half 
 the Side, and therefore z X equal to K z. 
 
 In the Side C D, let there be taken the right Line 
 C a equal to D R, and let a b be drawn parallel to 
 the Side B C, cutting £ x produced in b , and let V b 
 be joined. Then becaufe as B i is (that is B C more; 
 the Tangent C /) to the Secant B d, fo is B S to 
 D R, that is to S b $ and as B C to the Tangent 
 C d, fo V ,£ to S x ; if in S x produced, be taken 
 z b equal to S x, zb will be a Tangent of 30 De- 
 grees in the Arch delcribed from V z . 
 
 Then join V £, it will be equal to the Side RC, 
 and becaufe zb is a Tangent of 30 Degrees in the 
 Circle, whole Radius is V z, and C d a Tangent of 
 30 Degrees in the Circle, whofe Radius is A B ; 
 B d and V b will be Parallels ; and join V b equal, 
 to the Side B C, and B C more C d will be equal 
 to the right Line B /. 
 
 Then becaufe it is as B C more C d (that is B /) 
 to B d, fo V z to D R j and as the lame B i to 
 V #, lo V ^ to D R, and S b will be equal to D R: 
 For thele two Analogies can no way be conftituted 
 in any other Point of the right Line &x. 
 
 Then the Equals D H being taken from D R, 
 
 and £ b from S £, there remains HR, S z, both 
 
 Equals ; but S z is equal to F K : Wherefore HR, 
 
 F K are Equals. 
 
 * 
 
 Cor. Join R r, it will pafs by x. 
 
 Alio from hence it follows (V 2;, being produced 
 to D C in e) that the Right Line R e is double to 
 the Difference between G B the Half Side, and the 
 greater Segment A B divided in extream and mean 
 Proportion : For (by the firft Cor. of this Prop.') the 
 
 Fourth 
 
6 8 GEOMETRICAL Book II. 
 
 Fourth Part of the Side A B, or DC, is equal to the 
 right Line F K, together with the Difference be- 
 tween the Half Side, and the greater Segment of the 
 Side : Wherefore half the Side, that is, a e, is equal 
 to the double 6f F K, and to the double of the 
 Difference between the greater Segment of the Side 
 and the Half Side : But a R is double to F K; 
 wherefore R e is double to the Difference between 
 the greater Segment of the Side and the Half Side. 
 
 PROP. VIII. 
 
 Gain defcribe a Quadrate ABC D,alfo defcribe 
 ^ a Quadrantal Arch BD $ and to the Side BC, 
 let there be added C k equal to a Tangent of 30 De- 
 grees, and let the right Line A k be drawn ; more- 
 over in the Side A B, let there be taken A a equal to 
 
 the 
 
Book II. ROSES. 6p 
 
 the right Line A c being made equal to the greatei 
 Segment of the Side A B ; (divided in extream and 
 mean Proportion) then draw a b parallel to the 
 Side B C, cutting A k in b. 
 
 I fay, the right Line a b is the Subtenfe of two 
 fifth Parts of the Quadrantal Arch k m, defcribed 
 with the Semidiameter B h , and thole two Fifths 
 equal to the Side A B. 
 
 For becaufe A a is the greater Segment of the Side 
 B C, being divided in extream and mean Proportion, 
 it is alfo the Side of a Decagon in a Circle, whofe 
 Semidiameter is A B (by the 4 Prop, of the 14 Elem* 
 of Euclid ) and fubtends the tenth Part of the whole 
 Perimeter, that is, a fifth Part of the Half Perime- 
 ter, that is, two fifth Parts of the Arch B D. 
 
 Therefore fince it hath been {hewn, that the right 
 Line B k is equal to the Arch B D, it will be, as AB 
 to A a, fo B &, (that is, the Arch B D) to a b. 
 
 Wherefore a b is the greater Segment of the right 
 Line B k divided in extream and mean Proportion. 
 
 Apply to the Arch k r,i the right Line k p equal 
 to a b then the Arch kp will be two Fifths of the 
 Quadrantal Arch k w, which is the firfl. But the 
 Arch k fti ( by the 4 Corol . of Prop. 7.) is equal to 
 five Half Sides of a Quadrate from AB ; wherefore 
 the Arch k p is equal to the Side A B, which re- 
 mained to be demonftrated. 
 
 C o N S E C T. I. 
 
 If to the Side B C there be added the right Line 
 C l equal to A and from the Centre B, with the 
 Interval B /, be defcribed a Quadrantal Arch / 
 alfo join B p being produced to / n in 0 : The Chord 
 
 Line 
 
7 o GEOMETRICAL Book II. 
 
 Line / o being drawn, it will be equal to the Arch 
 k p, or the Side A B. For (by the 5 Prop, of the 
 13 Elem .) the Side B C will be the greater Segment 
 of the whole B l: Wherefore the Side B C fubtends 
 two Fifths of the whole Arch / n. Therefore fince 
 the Angle k B p is two Fifths of the Angle k B m 
 like wife / 0 will be two Fifths of the Arch / n : 
 Wherefore the right Line / 0, which fubtends the 
 Arch l 0 , is equal to the Side B C, and the lame 
 equal to the Arch k p. 
 
 C O N S E C T. II. 
 
 From hence it is manifeft, that both the Qua- 
 drate it felf A BCD, and all the right Lines from 
 the Centre A to the Side B C (if the Work be pro- 
 duced) are cut from a b (if the Work be produced) 
 
 in 
 
Book I!. ROSES. 71 
 
 in extream and mean Proportion : For the Qua- 
 drate A BCD is divided in the fame Proportion 
 whereby thie Side A B is in a j likewife from the 
 fame a b are divided all the right Lines drawn from 
 A to B C (when the Work is produced) in the 
 fame Proportion whereby A B was divided in a : 
 Wherefore (by the 2 Prop, of the 14 Elem,) they 
 are divided in extream and mean Proportion. 
 
 Cor. From hence a manifeft and brief Method 
 appears of finding proportional Segments of right 
 Lines of what Kind foever. As for Example • If 
 the greater Segment of a Secant of 30 Degrees be 
 fought, draw the Line A K, and let it be pro- 
 duced to the Side B C in G, which is the Secant 
 of 30 Degrees ; it will cut the right Line a b in d , 
 and A d is the greater Segment, and the Refidue is 
 the leffer Segment : Or elfe the Se&ion of a Tan- 
 gent of 30 Degrees being fought, which as afore- 
 faid is half the Secant, and the Side of a Cube in- 
 fcrib’d in a Circle, whofe Diameter is A B. 
 
 Take the Half of A d, that is a d , it will be the 
 greater Segment, and the remaining Part of the 
 Tangent will be the leffer Segment. Likewife, if 
 A F be to be cut according to the fame Proportion, 
 draw A F, it will cut a b in £, and A e will be the 
 greater Segment, and e F the leffer Segment • and 
 the Half of A e the greater Segment of the Half 
 of A F- 
 
 PROP, 
 
71 
 
 GEOMETRICAL Book II. 
 
 PROP. IX. 
 
 rv 'h ... . ; i‘ l • L-? 
 
 ^to cut an Angle given, into any Proportion given. 
 
 L E T the Angle given be B A C, and let the Pro- 
 portion given be as A B to A D. Let the 
 Arch A D E be defcribed, and cut the Arch B C 
 in the Middle in F. Wherefore A F being drawn, 
 it will alfo cut D E in the Middle (fuppofe) in G. 
 Then the Chords B F, D G being drawn, they will 
 be between themfelves as the right Lines A B, A D 
 
 Apply the two Chords D G, G E, to the Arch 
 B F in I and K, fo that the Chords B I, I K may 
 be Equals to the Chords D G, G E, and either to 
 other ; alio draw the Lines A I, A K,of which A K 
 cuts the Arch DE in L, and AI cuts the fame Arch 
 in M. 
 
Book II. ROSES. 7? 
 
 I fay the whole Arch BC being thus divided in 
 K, that the Arch BC (or the Angle given B AC) 
 is to the Arch BK (or to the Angle BAK) as the 
 right Line A B to A D. 
 
 For the Chord BK is to the Chord DL, as AS 
 to AD. Alio as the two Chords BI, IK, to the two 
 Chords BM, ML, fo is AB to AD. But the two 
 Chords BI, IK, are by Conftru&don equal to the two 
 Chords DG, GE. Wherefore the two Chords DG, 
 GE, are to the two Chords DM, DL, as AB to AB. 
 But as two Chords DG, GE, to two Chords DM, 
 D L, fo is the whole Arch D E, to the Arch D L j 
 which I thus make. appear. 
 
 If the Arches B I and I K, be cut in the Middle in 
 u and b ; likewife the Arches D M, M L, being cut 
 in the Middle in c and d ; and the Chords of the Bi- 
 fegments in the Arch BK being drawn ; likewife the 
 Chords of the Bifegments in the Arch DL • alfo thefe 
 will be as A B to AD, and they will always be lo 
 if the Bifegments of the Bifegments proceed infinitely! 
 Alfo the fame is true in the Bife&ions of the Arches 
 KC and LE. But the Chords of the Bifegments 
 infinite in Number, are equal to the Arch itfelf 5 
 for if all the Chords were lefs than all their Arches 
 there might yet a Bife&ion proceed 5 which is con-! 
 trary to the Suppofition. 
 
 Therefore, the Arch DE is to the Arch DL, as the 
 Arch BC to the Arch BK, as the given Line A B, 
 to the given Line AD, and lb alio the given An^le 
 BAC to BAK. Therefore the Angle given BAC 
 is cut in K, in the Proportion given of AB to AD. 
 Which was to be done. 
 
7 4 
 
 GEOMETRICAL Book II. 
 
 prop. x. 
 
 Of Sines, Subtenfes, and other Lines in the Qua- 
 drant of a Circle. 
 
 T H E Tangent of an Arch of 22 Degrees and l 
 is equal to the Excefs whereby the Diagonal 
 of a Quadrate exceeds the Side of the fame. 
 
 Let ABCD be a Quadrate, and in it a Quadran- 
 tal Arch infcribed BD, cutting the Diagonal AC in 
 N Then AN is equal to the Side AB. I fey NC 
 is equal to the Tangent of an Arch of 22 Deg. 7. 
 
 Defcribe the Quadrantal Arch AC, cutting the 
 Diagonal DB in M : Alfo draw MN, it will be pa. 
 rallel to the Side BC, and MN being produced tc 
 both the Sides AB, DC, in L and O. Then DO wil 
 be equal to the Sine of an Arch of 45 Degrees, or tc 
 
Book li. ROSES. 
 
 75 
 
 the half Diagonal D I. Let the Quadrate A B C D 
 be cut by the right Lines EF, GH, interfe&ing at L 
 four Ways. Then is DO a mean Proportional be- 
 tween the whole Side D,C, and his half DH. Where- 
 fore as DC to DO, fo is DO to DH j and like 
 wife the Difference CO to the Difference OH. 
 
 Likewife C O and N O are equal, becaufe the 
 Angles at C and N are half right Angles. Where 
 fore NC is in Power double to CO; alfo CO is in 
 Power double to OH (being double in Power fm- 
 mfies that a Quadrate whofe Side is CO contains 
 as much again as a Quadrate whole Sidels OH ■') 
 for when the Side DC is in Power double to DO 
 and DC, D O, O H continual Proportionals ; C C* 
 * n ^ ver dou ble to OH; therefore alfo HO 
 OC, NC are continual Proportionals ; becaufe NC 
 is double in Power to CO. 
 
 Draw the Chord DN ; then the Angle ODN will be 
 an Angle of 22 Deg. f Alfo the Chord DN cuts the 
 right Line GH in R, and HR will be equal to HO 
 Likewife becaufe RH, NO, NC are continual 
 Proportionals in the Proportion of the Side D C to 
 
 Pi?? n Cll ,° rd DN bei ng produced to the Side 
 BC in P, will cut the Part CP equal to the right 
 
 i! nC N r » f be , m S e 9 ual to the -Excels of the Diagonal 
 above the Side. Wherefore a Tangent of an Arch 
 of 22 Deg. f, ,s equal to the Excefs of the Diagonal, 
 which was to be demonfirated. 
 
 Cor. From hence it follows, that the right Line BP 
 is equal to the double of CO. For if from the Centre 
 C, with the Interval CD, an Arch be drawn, cuttin- 
 CA in a N will be double to IN, and A « equal 
 
 likiJr 5 t t ere (° re ’. wh L en as a C > and BC be equal, 
 tr> R ^ tiac ,s s tbe double of CO will be equal 
 
 G 2 PROP, 
 
j6 
 
 GEOMETRICAL Book II. 
 
 PROP. XI. 
 
 A Tangent of an Arch of 30 Degrees, together 
 with a Tangent of an Arch of 22 Deg. % 
 are equal to the Side of a Quadrate BC. 
 
 Divide the Arch M C in the Middle m b. Then 
 either of the Arches C b, b M will be an Arch of 22 
 Dee. i , and D b will pafs by N ; alio MK is a third 
 Part, that is, two Sixths of the Arch CM. 'I hen when 
 asMK is a third Fart ofMC, and M b an half, the 
 Arch M K will be double to the Arch K. b. 
 
 Then the Angle BDK is a fixth Part of the Angle 
 CDM, that is, a twelfth Part of the right Angle. 
 Produce D b to the Side B C in P. , 
 
 Then becaufe the Angle AKD is two Thirds, that 
 is eight Twelfths of one right Angle, and the Ang e 
 KDP one Twelfth ; if AK be produced until it meet 
 D b being produced, which will happen in P. 
 
Book II. ROSES. 77 
 
 wherefbever it cuts in D b being produced, it will 
 make with it an Angle equal to leven Twelfths of a 
 right Angle, becaufe the Angle KAB is equal to 
 eight Twelfths, and the Angle KDP to one Twelfth ; 
 for the remaining Angle KPD will be feven Twelfths 
 ofone right Angle; for when the Angle C PD is nine, 
 and the Angle which the Tangent of 30 Deg. makes 
 with his Secant is eight, the remaining Angle will be 
 a Complement to two right Angles, that is, to the 
 three Angles BP A, APD, CPD. Therefore when as 
 the Angle C P D is nine, and the Angle DAK eight 
 Twelfths, the other Angle APD will be feven, and 
 all the three Angles together, will be twenty- four 
 T welfths of one right Angle, that is equal to two right 
 Angles, that is to the three Angles of the Triangle 
 APD. Wherefore the Tangent of 30 Deg. Qc. 
 which was to be demonflrated. 
 
 A N I M A D VERS. 
 
 fhis Prop, hath longfince been confuted by fables 
 of Sines , Secants and fangents , calculated / from 
 federal Geometricians by Dr. Wallis. 
 
 Whether Mr. Hobbs or he is to be credited , is left 
 to the Reader s Confederation. 
 
 From hence it follows, that BM, MN, NC, CP, 
 KS, are equal between themfelves. For whereas 
 every one of them is double to the right Line HO or 
 19., they will be equal among themfelves. Befides 
 BP is double to GS ; for it is manifeft from this, that 
 A B is the double of A G. From whence it appears 
 again, that BP is a Tangent of 30 Deg. for AS, 
 which is manifeftly equal to the Tangent of 30 Deg. 
 is the double of GS. 
 
 G 3 
 
 PROP. 
 
7.8 GEOMETRICAL Bookll. 
 
 p r o P. XII. 
 
 A right Line which cuts the Safe AK of an Equi- 
 lateral friangle^ from any Vertical Point in the 
 Middle , is Sefquialter of the tangent of an Arch 
 of 30 Degrees. 
 
 L ET there be taken inAB, a Tangent of an Arch 
 of 30 Deg. AT. Join DT cutting AK in V, 
 and the right lane E K in X, and draw AX. Then 
 the Triangle AT X will be Equilateral, and both the 
 Angles AVD, and KVD are right Angles $ like- 
 wife the Sides AX, XD will be equal. And 1 X is 
 the double of V X. Therefore D V is the 1 riple of 
 VX, that is, Sefquialter Quiz, once and an half) of 
 DX, that is, a Tangent of 30 Deg. which was to be 
 demonftrated. The fame Propofition is demonftrated 
 
 by 
 
Book II. ROSES. 79 
 
 by Euclid , in his 1 8 Prop . of the 14 Elem. but becaufe 
 it is too long to rehearfe in this Place, let it be read 
 by him that doubts of the Truth of it. 
 
 Corol. Therefore the right Line D V or EK, is 
 triple the Difference between the Side DC and the 
 half Diagonal Dior DOj for CO is manifefted 
 to be equal to half the Tangent A T. 
 
 PROP. XIII. 
 
 *fhe Difference between the greater and leffer Seg- 
 ment of a right Line , divided in extreme and mean 
 Proportion , is double the Difference between the 
 fame right Line , and a right Line whofe Power is 
 to it , as 5 to 4. 
 
 L E T the given 
 right Line 
 be AB, to which 
 let there be put 
 at right Angles 
 BF, half of the 
 given Line A B • 
 then draw A F, 
 the Power where- 
 of is to the Pow- 
 er of the given 
 Line, as 5 to 4, 
 and to the Power 
 of BF, as*5 to 1. 
 
 NOfE , That 
 the Word Power 
 ydl! be often 
 
So GEOMETRICAL Book II. 
 
 uf ed ; therefore, by the Power of a Line you are to un- 
 derhand a Quadrate or Square made from the Line 
 mentioned, each Side whereof is equal to the men- 
 tioned Line} as if it had been faid, a Qaudrate or 
 Square drawn from AF (that is, a Quadrate, each 
 of its Sides being equal to AF) is to the Quadrate 
 drawn from the given lane A B, as 5 to 4, and to 
 the Quadrate drawn from B F, as 5 to 1 . 
 
 With the Interval AF, defcribe an Arch of a 
 Circle, cutting A B being produced in Z. Then the 
 Square from the right Line A z , is to the Square 
 made from the given Line A B, as 5 to 4, and to 
 the Square from BF, as 5 to 1. 
 
 I fay, the Difference between the greater and the 
 leffer Segment of the given Line A B, divided in ex- 
 treme and mean Proportion, is the double of B z. 
 
 Divide AB in the Middle in G 3 then AG being 
 taken away from the whole Line A z , the remaining 
 part G £ is the greater Segment of the given Line A B 
 d vidcd in extreme and mean Proportion (by the 
 1 Prop, of the 1 3 Elcm. of Euclid. J From the Point 
 A in the Line A B, take A a equal to Gz ± then be- 
 caufe both AG, GB, and A <2, G z are equal, G 
 B ,2, 41'e likewile equal; and becaufe A a is the 
 greater Segment, the leffer Segment will be a B. 
 
 Therefore is G £ the greater Segment, being great- 
 er than tfB the leffer Segment. The two right 
 Lines B z, G a are equal between themfelves, that 
 is, the double of B z. 
 
 Therefore, the Difference between the greater 
 and leffer Segment is double, &c. Which was to 
 be demonftrated. 
 
 PROP, 
 
Book II. 
 
 ROSES. 
 
 81 
 
 PROP. XIV. 
 
 If the Secant of an Arch of 30 Degrees he cut in 
 extreme and mean Proportion , the greater Segment 
 will he equal to the Semidiagonal of a Quadrate 
 made from the Semidiameter . 
 
 D Efcribe a Quadrate from AB (to wit) ABCD, 
 and divide it into 4 Parts by the right Lines 
 EF, GH, allb divide it into 4 Parts by the Diagonal 
 Lines AC ? BD, all concurring in the Centre of the 
 Quadrate at I; let there be defcribed twoQuadran- 
 tal Arches AC, BD, cutting the Diagonals in M and 
 N, and the right 
 Line EF in K u 
 draw A K, and 
 produce it to 
 the Side BC in 2 ; 
 
 P. Then A Pis' * 
 the Secant of the 
 Arch BK, which 
 is an Arch of 30 ^ 
 
 Degrees 3 likewile 
 BP is a Tangent G 
 of 30 Deg. and 
 the half of the Se- 
 cant AP. 
 
 By the Points 
 M and N, draw 
 the right Line 
 
 LO 
 
8z GEOMETRICAL Book II. 
 
 - 
 
 LO equal and parallel to the Side B C. Wherefore 
 AL, or DO, is equal to the Semidiagonal A I. 
 
 I fay, AL is the greater Segment of the Secant AP 
 divided in extreme and mean Proportion. 
 
 With the In- 
 terval A P de- 
 fcribe an Arch 
 of a Circle Py, 
 cutting A B pro- 
 duced in y. 
 
 From the 
 Point y draw 
 y x parallel to * 
 the Side BC, and 
 equal to half 
 the Secant AP, 
 that is, equal 
 to the Tangent 
 BP. Then A* 
 being drawn, is 
 in Power Quin- 
 tuple to (or five 
 times as much 
 as ) the right 
 
 Line yx. 
 
 With the Interval Ax defcribe an Arch of a 
 Circle xu, cutting AB produced in u. Wherefore 
 (by Elem. 13. Prop. 1.) yx (that is, BP, being taken 
 away from the whole Line A u , the remaining Part 
 will be the greater Segment of the right Line A P 
 (or A z ) being divided in extreme and mean Pro- 
 portion. Likewife yx or BP being taken from A u, 
 the remaining Part AL will be equal to the Semi- 
 diagonal A L 
 
 For 
 
Book II. ROSES. 83 
 
 For it is manifeft by Prop. 10. that the right Line 
 BP is double to the right Line CO or BL. There- 
 fore <uL is equal to y# or BP ; and the remaining 
 Part AL the greater Segment of the Secant Ap or 
 the right Line Ay divided in extreme and mean 
 Proportion. Which was to be demonftrated. 
 
 Corol. From hence it follows, that the half Diago- 
 nal A I is the greater Segment of half the Secant A P, 
 that is, of a Tangent of 30 Degrees, that is, of a 
 Side of a Cube inlcribed in a Circle, whofe Dia- 
 meter is A B. 
 
 PROP, XV. 
 
 A Bigreffion concerning the' Bif cord between the Com- 
 putation of Lines , of Superficies, and of Numbers 
 in the Bemonftrations of Geometricians. 
 
 B Y the 5th Definition of the 5 Elens, of Euclid , 
 Magnitudes are faid to have Reafon or Pro- 
 portion to one another, which being multiplied, may 
 exceed one another. 
 
 From which Definition it is manifeft, that Lines, 
 Superficies, and Solids, can have no Proportion 
 among themfelves 3 for being multiplied together in 
 themfelves, they can never mutually exceed. 
 
 If, notwithftanding,foraLinetherebeufedafmall 
 
 right Angle, there may truly fometimes a Proportion 
 be found between a Superficies and that right Angle, 
 tho it be very fmall • to wit, when two Quadrates 
 are to each other as a Quadrate Number to a Qua- 
 drate 
 
84 GEOMETRICAL Book II. 
 
 drate Number. And one is the Meafure of the 
 other, becaufe they may be compared. 
 
 But the Quadrates which are between themfelves 
 as Quadrate Numbers, are much fewer than thofe 
 which are not as Quadrate Numbers, altho 5 both 
 are innumerable. 
 
 Therefore the Dodtrine concerning the Quantity 
 of Lines is a Science fubfiftent by itfelf, and diftindt 
 from the Science of Superficies, and this diftindt 
 from the Science of Solids. 
 
 Moreover, becaufe all Menfuration begins from a 
 Point, and a Point cannot be confidered as Figurative, 
 how can a Point in a Quadrate Angle any otherwiie 
 be confidered, than as a Quantum common to the 
 Quadrate and Side of it? Por to confider it as Fi- 
 gurative, and not Figurative, is abfurd. 
 
 Befides, how can a Point, which is in the Centre of 
 a Circle, be accounted for nothing, when as it is di- 
 vifible? For in how many Parts foever any Sedtor is di- 
 vided, in fo many Parts likewife is the Centre divided. 
 
 Perhaps, fome may fay, without the known Quan- 
 tity of Figures, very few of Theorems hereafter con- 
 cerning the Proportions of Lines are demonftrable, 
 befides Mechanick Menfuration. But they err, Firft, 
 Becaufe the Proportions of Lines between themfelves, 
 and the Conftrudtion of Figures, and all their Affe- 
 ctions of Motion, are deliver d by Euclid , without 
 the known Quantity of a Quadrate, or of any other 
 Figure, or the Proportion of Figure to Figure. 
 Neither hath any one attempted to demonftrate the 
 Length of a Line by the Magnitude of a Quadrate, 
 before Archimedes ; neither after him (that I know) 
 befides Eutocnts , before Copernicus 3 truly neither 
 hath he demonftrated accurately. 
 
 Neither 
 
Book II. ROSES. 85 
 
 Neither do I fay thefe things, becaufe T would ad- 
 mit Mechanick Operations for lawful Bemonftrations. 
 But in every Geometrical Queftion, I think it much 
 more Prudence, before mechanically Meafuring the 
 Magnitude fought, as much as may be done with 
 Truth, to attempt the neareft Way, and afterwards 
 inquire into the Caufe of that nearnefs, which being 
 found out, will deted the Truth or Falfhocd 3 than 
 rafhly believing of uncertain Reafoning, or the 
 Reckoning of thofe, or the Authority of others, 
 who pronounce unknown things, efpecially, not 
 only where the Certainty of one, but of many 
 Theorems is deftroyed. A diiigent Meafurer de- 
 clares more credible things by Meafuring, than he 
 that realons from falfe Principles ; and will defer- 
 vingly fcorn the Algebraifts, that is, the Arithme- 
 ticians, difputing againft Meafure, and faying. 
 That the Side of a Quadrate, and the Root of a 
 Number are the fame. 
 
 Alfo no Man will think thofe ftudious of Truth, 
 who feeing a Conclufion contrary to their own 
 Knowledge, although fupported with very pro- 
 bable Arguments, are content to refill the foie De- 
 monftration, and negled the Verity of the thing 
 (which fometimes proceeds from the Debility of 
 their own Wit, or from the Omillion of fome Pro- 
 portion, which the Bemonftrator fuppofed to be 
 well known to Geometricians, efpecially the chiefeft 
 of them) for thefe Kind of Men are not fecking 
 the Truth, but Vidory. 
 
 What I have faid concerning the Incongruity of 
 Lines and Superficies, will appear clearly in the fol- 
 lowing Problem, being propoied to Algebraifts. 
 
 PROP, 
 
u 
 
 GEOMb i RICAL Book II. 
 
 PROP. XVI. 
 
 J^Efcribe the Quadrate A BCD, and divide it 
 
 (both) with the right Lines EF, GH, and 
 with the Diagonals AC, BD, interfering in I 
 four Ways. 
 
 In FC being produced, take FK equal to the half 
 of MF, and join MK 3 then with the Radius M F, 
 
 defcribe an Arch F d 
 
 it will cut in d« 
 
 5 
 
 fo 
 
 that MF, M d are 
 
 equal. And MK cuts 
 the right Line Hd in 
 <2, and the Diagonal 
 IDinL AlfoMK 
 will pafs by H. 
 
 Draw E / paral- 
 lel to ID cutting 
 MK in/. Moreover 
 with the Radii M E, 
 MI, def ribe the 
 
 Arches Etf, I £, cutting MK in e and c. 
 
 It is manifeft from this ConftriuStion ; Firft, That 
 Mtf, #H, HK, are Equals between themielves. 
 
 Secondly, (Becaufe M I is double to M E) M b 
 is double to M f 
 
 Thirdly, (Becaufe ME is the double of E a) Mf 
 is the double of fa ; and (becaufe M I is double to 
 IH) H b is double to b a ; and then it will be as FI a 
 to H£, fb Ma to M /, as 3 to 2, or 9 to 6. 
 
 Fourthly, It is manifeft, that as well Kd as b a if 
 to ae , as 3 to 1, and to He as 3 to 2, and fo like- 
 wile will be to ae}> to wit, as 3 to 2, or 9 to 6, 
 
 and 
 
Book II. ROSES. %f 
 
 and therefore C d being joined, is parallel to I D ; 
 therefore d H, H£ are equal. 
 
 Fifthly, It is manifeft, that 'Mjf, fb are equal, 
 and b K, da are equal. 
 
 Then M d is four times two, whereof MK is 
 thrice three. 
 
 Wherefore MK is to Mi or MF, as 9 to 8. 
 
 Then becaufe MK is Quintuple in Power to FK 
 or M b , if F K be taken away from the right Line 
 MK, the remaining Part ^K (by Elem. 13. Prop, j.) 
 will be the greater Segment of MK, or bd of Mt£ 
 being divided in extreme and mean Proportion. 
 
 But b K is equal to da. Wherefore da is the 
 greater Segment of Mi divided in extreme and 
 mean Proportion. 
 
 Then whereas MK is 9, whereof Mi is 8, and 
 M a 3, da will be 5. And fb the whole Mi, his 
 greater Segment da , his lefler Segment M a , will be 
 in Proportion of the Numbers 8, 5, and 3 • which at 
 the End of the Second Proportion of this Book, I 
 have fhewed to be falfe ; and it is again!! Euclid , wiho 
 hath demonftrated, that if the whole Line be ratio- 
 nal, both the Segments of that Line be irrational. 
 Unlefs this Demonftration be confuted, there is no 
 reafon that the arguing from the Powers of Lines be 
 heard any more. No Line drawn obliquely between 
 Parallels ought to be referred to the Proportion of 
 pure Lines ^ becaufe they may be confidered either 
 as Triangles, or as fmall oblique angled Parallelo- 
 grams, whole Longitude is not determined. For 
 the Longitude of a Figure is nothing Purely befides 
 that which is called Altitude. 
 
 PROP, 
 
88 GEOMETRICAL Book II. 
 
 PROP. XVII. 
 
 ft he Side of an Icofahedron is equal to the third Part 
 of the greatefi Semicircle in its own Sphere . 
 
 D Efcribc the Quadrate A BCD, and divide it in 
 the Middle by the right Line EF parallel to the 
 Sides AB, DC ; alfo deferibe a Quadrant ABD, whole 
 Arch will cut the right Line EF in K. Join A K 
 
 H 
 
 being produced to BC in P. Then is BP a Tangent 
 of 30 Degrees, to which add P b equal to the Side 
 BC, the whole Line B h will be compounded of the 
 Side, and a Tangent of 30 Deg. Likewife draw the 
 
 Diagonal 
 
Book II. ROSES. 
 
 85) 
 
 ^gonalA C ’ which the ri § ht Line E F will cut in 
 the Middle in /. 
 
 Triangle^’ **** AKD wil1 be an Equilateral 
 
 In the Side A D take A c a 3d Part of it and 
 join cK, and produce it to the Side Be in <r • and 
 F c will be a 3d Part of F P, and therefore Be will 
 
 of ‘he T ofehio Wh01 ' Li " B ” ^ 3 - 
 
 In the right Lines K A, K D, let there be takert 
 K K. c, either of them equal to a 4th Part of the 
 Diagonal AC, or the Half of A /, and join a e 
 Then the Triangle K«e will be Equilateral : and 
 its Bafe « e will be cut by the right Line K E in the 
 Middle, and at right Angles. 
 
 Of the right Line B P, which is a Tangent of 30 
 Degrees, and of 2 right Lines, whereof either is 
 equal to * c make the T riangle a c y j and e y 
 will be equal. r 9 
 
 By the 3 Points *,C, y , deferibe a Circle, whole 
 Centre is G, and Semidiameter GforGe 
 And becaufe EF or AB, the Diameter of a 
 Sphere, is to B P, that is, to y in Pow<=r as a 
 
 t ° 1 ’ b <;. the Side of a Cube inferibed in a 
 
 Sphere, whofe Diameter is E F. And becaufe (bv 
 the 14th Proportion of this ) the 4th Part of the 
 Diagonal, is the greater Segment of the Side of a 
 Cube, divided in extream and mean Proportion • 
 it will be (by the 8 Prop, of the 13 £/m.) that the 
 right Line « £ is the Side of a Pentagon in the Cir- 
 a.S yi and that Side of a Pentagon one of the 
 
 he f! atS T ^ oints of a Dodecahedron inferibed in 
 he fame Sphere with the Icofahedron. 
 
 Compleat the Pentagon aCyJ't. 
 
 H Draw 
 
9 o GEOMETRICAL Book II. 
 
 Draw the right Line A F cutting the Arch A K C 
 \ n f- and the Power of Bf will be (as is well known 
 to Geometricians) a 5th Part of the Power of the 
 Side A B, or Quintuple in Power to the Diametei 
 
 of the Sphere. , _ TU 
 
 Defcribe aloof off tvith the Radius I H, 
 
 being 
 
 eaual to Bf, the Circle H L, in which the Side of 
 an -Equilateral Pentagon is HL. Then H L will 
 be the Side of an Icofahedron m the fame Sphere, 
 by the i 6 Prop, of the 1 3 Elem. 
 
 Then (by the 5 Prop ■ of the 14 Elem. of Clayl- 
 ms Edition) the right Line H Lis the Side o an 
 Equilateral Triangle infcribed in the fame i e. 
 Infcribe in the Circle WffyJ' «, the Equilne- 
 an^le * r Then a. f will be the Side of 
 hedron in the Sphere, whoie Diameter is B L , 
 
9 \ 
 
 Book II. ROSES. 
 
 rn l\‘ S . be ^ 1 % that the right Line « f is equal 
 
 c n | d P u art of the S « micircle » whole Semidiameter 
 is BP that is, to the Arch BK, which is a ad 
 Part of a Quadrant of a Circle defcribed from A B 
 as a Semidiameter. 
 
 With the Semidiameter Be, deferibean Arch e9 
 m which takers equal to the Arch y Then draw’ 
 
 t Vu' i be equal tbat is , “ the Side of 
 
 an Icolahedron in a Sphere wholb Diameter is E F 
 
 Then with the Radius B8, deferibe an Arch of a 
 Circle cutting B P, it will give a Side of an Icofa- 
 3 , , r °p ’ rr Vvblcb Slde ’ J f !t be the fame B r, will be 
 ct P „ P f the ri S ht Llne B h that is, a 3 d Part 
 o the Arch A C, or the Semicircle above the Dia- 
 meter ER Asis manifefted by Confett. 3. ofProfi.n. 
 
 Likewife a Reafon is to be given wherefore *> or 
 ? > ought to be equal to the 3 d Part of the Arch 
 A C, that is, to the Arch C K. Which very Reafon 
 ™ ybe tBore eafi ly rendered by looking upon the 
 Solid Icofahedron itfelf But becaufe it cannot be 
 expounded , n a Plain, the neareft Way is to de~ 
 lenbe 4 Triangles, whereof 20 make the Superficies 
 of an Icofahedron, and in that Pofition wherein 
 Claims difpofed them, at the End of the 1 6th Prop 
 
 M NO, 3 N OQ,' qo R h ° fe TrianSkS arC D M N > 
 
 thp ^ n f t |! liS i) P ® Ur l l £t tbe B °! e of the Sphere be D, 
 then the Points D, M, N, O, Q, R, will be in the 
 
 Conuve Superfiaes °f the Sphere ; and therefore 
 
 fame § Ph, S ^^x, N °’ ° ^’ will not be in the 
 lame Plain with the Points D and R, which are in 
 
 1 v£!. ain tbe D, ameter of the Sphere. 
 
 from fh^P T S C S ' d u ° f 3n Icolahedr on proceeds 
 from the Pole D to the Pole R, by 5 equal right 
 
 H - 
 
 Lines. 
 
91 GEOMETRICAL Book II. 
 
 Lines, to wit, from D to M, from M to N, from 
 N to O, from O to Q, from Q^o R. And firft m 
 the Motion from D to M, it is moved forward to- 
 wards R. Again, from M to N, it is not moved 
 forward towards R. Thirdly, from N to O, it is 
 moved forward, as much as from D to M. Fourth- 
 ly from O to Q^it is not moved forward towards R. 
 Fifthly from Q_ it is moved forward to the very 
 Joint R. Therefore by 5 equal right Lines, which 
 are the Sides of an Icofahedron, is the Motion made 
 
 from Pole to Pole ; but by reafon of the Digrelfon 
 to the neareft Circles, which divide the Superficies 
 of the Sphere in 5 Parts, the Motion is made by 
 the 5 Sides of an Icofahedron. 
 
 But in the Circumference of a Semicircle, the 
 Motion is made from Pole to Pole by 3 Arches, 
 
 every 
 
Book II. R o S E S. 9 5 
 
 every one of them being equal to the Arch C K or 
 BK. The Motion therefore being taken away 
 which is made by thofe two Sides of an Icofahedron 
 which move not forward, the Ways by the 3 Sides 
 of an Icofahedron, and by 3 Arches of a Semicircle 
 every one of which being equal to the Arch C K or 
 at leaft, by the Chords of thofe Arches, will be equal. 
 But B 6 will be found greater than the Chord B K. 
 Wherefore the Arches every one being equal to B K 
 are equal to 3 Sides of an Icofahedron, and one to 
 one. Which was to be demonftrated. 
 
 This kind of Demonftration will be condemned (I 
 know furely) by Algebraifts, and, perhaps, by others, 
 who do not admit in Geometry, Arguments to be 
 taken from Motion, neither from Meafure. But to 
 this, and many other Geometrical Propofitions 
 they can find out no other Principles from whence 
 they can truly derive a certain Conclufion. For 
 Numbers (as I have often demonftrated) are unapt 
 for this Thing, neither can they proceed rightly 
 from Superficies to Longitudes, nor contrarywife, 
 becaufe that Superficies and Lengths are divers kinds 
 of Quantity. 
 
 H 3 
 
 PROP. 
 
94 
 
 GEOMETRICAL Book II. 
 
 PROP. XVIII. 
 
 A Circle being given , to find a Quadrat tqiidl to it. 
 
 L E T the Circle given, whofe Quadrat is D A C, 
 be infcribed within a Quadrat ABCD ; and 
 BLC the 8th Part of the Circle. Cut the Quadrat 
 ABCD with the Diagonals AC, B D ; alfo with 
 the right Lines E F, I all interfering in I, and 
 
 dividing the Quadrat into 4 Parts both Ways ; and 
 draw D F cutting the Arch CL in P, and by the 
 Point P draw Y Q, cutting the Diagonal BD in Q^ 
 then D Y, Y Q^will be equal. 
 
 With the Radius D F, defcribe an Arch Fc cut- 
 ting the Diagonal BDin r, fothatDF, D c may 
 be Equals 
 
Book II. ROSES. 95 
 
 I fay, the Quadrate from YQor DY is equal to an 
 8th Part of the Superficies of the Circle DCL. 
 
 From the Point c draw c z parallel to Y Q, cut- 
 ting D C in Z. Then is z c (by the 6th Prop, of 
 this) half the Arch AC, and therefore equal to 
 the Arch C L. 
 
 In the Arch C L, take the Arch L V equal to 
 C P, and join D V, cutting Y P in X ; and the 3 
 Lines CYP, will be wholly within the Sector 
 D C L 5 but the 3 Lines P QL are all without the 
 fame Sector. 
 
 Likewife both the 3 Lines together are ( as be- 
 fore I have fhewn, and now will fhew) equal to the 
 Sector A P V. For becaufe the right Line B C is cut 
 in the Middle at F, and the Bafes of the Triangles 
 D C B, D Y Q are parallel ; alio the Bafis Y Q is 
 cut in the Middle in P, and the Triangles DYP, 
 D P Qare equal. 
 
 Now DPL more P QL more CP Y, are equal 
 to DVL, or BCP (becaufe DPL more P QL 
 is equal to D Y P). For D C V more D V P is equal 
 to D CP, or DVL. 
 
 Wherefore DPL more PQ_L more CYP is 
 equal to DCV more DVP. 
 
 Then both the Equals DPL, DCY, being taken 
 away, there remains P QL more C Y Q equal to the 
 Sedtor DVP. And hitherto our Adversaries con- 
 fent or agree. This alfo they muft condefcend to 
 (for it is manifeft) that if CYP, PQL, are equal 
 between themfeives, the T riangle D Y Q, and the 
 Sedlor DCL are likewife equal. 
 
 And I had thought it demonftrated before from 
 this, becaufe the Triangle whofe Vertex is D, and 
 Bafis parallel to the Side BC, no Triangle can be 
 
 H 4 made 
 
9 6 GEOMETRICAL Book II. 
 
 made equal to the Se&or DCL, whofe Bafe pafT- 
 eth not by P j but the Force of the Demonftraticn 
 they have not clearly underftood. Go to then, 
 we will produce, if it be poflible, a more clear De- 
 monftration. 
 
 With the Radius DI, which is a mean Propor- 
 tional between the Side D C, and the half Side D k , 
 defcribe an Arch of a Circle cutting DF in b, and 
 D V in /, and D C in b j and by the Point b draw 
 e q, cutting DC in e, and the Diagonal BD in 7, 
 and D V in 0, Then the Sector D C P will be dou- 
 
 ble of the Sedor D b b, as alfo of the 4 Lines CP b b , 
 likewife the Triangle DX Y double of the Triangle 
 Doe i it is likewife double of the 4 Lines Y X 0 e. 
 Wherefore the remaining Seftor D VP is double to 
 the 3 Lines CYP, and likewife double to the 4 Lines 
 VP b i. And D b i double to the 3 Lines b b ^ 
 
Book II. ROSES. 97 
 
 Then when as the Sedlor D V P is double to the 
 three Lines CYP, and the lame equal to two three 
 Lines, CYP, PQ.L ; then CYP, PQL will be 
 equal between themfelves. 
 
 Which was to be demonftrated. 
 
 Therefore a Quadrate is found (to wit, a Qua- 
 drate from Y Q) equal to the 8th Part of a Circle, 
 to wit, to the Secftor DCL j and fo is effected the 
 fquaring of a Circle, nor now the firft Time, but 
 many Years ago fufficiently demonftrated by divers 
 Methods. 
 
 From this Demonftration likewile I have deduced 
 the duplicating of a Cube, fhewing that the 4 right 
 Lines CB, zc^ cq , and I£, are continual Propor- 
 tionals. But neither have the Algcbraifts under- 
 ftood this j for the SaviU an Profejfor objecfts, that 
 the right Lines CB, Y Q, zc are not continual 
 Proportionals. For if they were, likewile BD, 
 D Q, D-c, would be Proportionals. Let it be fup- 
 pofed as he faith, divide the Line BC in 5 Parts, 
 the Square from the whole Line is 25, and the 
 Square (or Quadrate) from B D (whereas it is the 
 Double of it, that is, of the former Quadrate) 50 j 
 the Quadrate from D (^40, the Quadrate from 
 32 ; and therefore the Quadrate from zc , 16 of 
 thofe 25 of the whole Quadrate A BCD. But the 
 Quadrate from Zc is if of the Quadrate ABCD. 
 But i£ and if are not equal. Therefore CB, YQ, 
 Z c are not Proportionals. 
 
 But Arguments of this Kind ( for the Caufe de- 
 clared at the 1 5th Propofition) are the mere Spirit 
 of Delufion in Algebraifts, in applying thole Num- 
 bers to Quantities, which have not the Proportion 
 of Number to Number. 
 
 If 
 
^8 GEOMETRICAL Book IE 
 
 If BC be divided in 5 Parts, the Quadrate from 
 DF will be 6 £, to wit, the 4th Part of 25. Where- 
 fore D c will be 3 1 and the Quadrate from B D 
 will be 8 Times as much as the Quadrate from 
 B F, or 50. 
 
 But the rrlean Proportional between 8 and 5, will 
 be the Side (I call it the Side, not the Root) of the 
 Number 40. Then not only 50, 40, 32, but alio 50, 
 40, 3 1 i will be continual Proportionals. Therefore 
 what the Caufe of this Difcord ftiould be (fince it is 
 plainly demonftrated, that both zc is equal to Half 
 the Arch A C, and that the Quadrate from Y Q is 
 equal to theSe&orCL) unlels that Lines drawn (that 
 is divifible according to Breadth) cannot be compared 
 
 A 
 
 with pure Lines, that is, without Breadth. But the 
 right Lines CB, zc^eq^k I, are continual Propor- 
 tionals, as I fhall make appear moft clearly in the 
 following Dernonftration. Con- 
 
Book II. 
 
 ROSES 
 
 99 
 
 C O N S E C T, I. 
 
 i 
 
 The Arch CL is lefs than four fifths of the Ra- 
 dius DC; for all the verfed Sines which are pofli- 
 ble to be drawn in the Quadrant DA C, being taken 
 together, are equal to the Area or Content of the 
 Quadrant, and by confequence to the Quadrat D Y 
 QM, or to four fifths of the Quadrate A BCD. 
 And for becaufe thofe verfed Sines are all bounded 
 within the Arch, the 2 Sides B C, C D of the Qua- 
 drat ABCD, ought to be to the Halfs of the Arch 
 AC, in Proportion double of CD to DY ; but it 
 is not fb. For altho’ you fhould divide Lines, or 
 other continued Quantity infinitely, notwithftand- 
 ing you will never attain to nothing ; becaufe con- 
 tinual Quantity is always divifible into Bivifibles. 
 Then thofe Sines, whereof the whole Number of 
 them fill the Area of a Quadrant, will have every 
 one their Breadth, and will always be finite in Num- 
 ber. Alfd thofe verfed Sines are Parallels between 
 themfelves, whereof the greateft D C is the Radius 
 of a Circle, whofe Bound (or End) at C is a fmall 
 Arch. Alfo D C is the Side of a Quadrat ABCD, 
 and therefore (becaufe it hath Breadth) it will be a 
 right Angle, and therefore more than the Radius 
 DC. And part of it will be without the Circle. 
 The Proportion of the reft of the verfed Sines is the 
 fame ; and for becaufe according to Breadth, they 
 may be divided as often as it is poffible to divide, 
 they will never proceed to an Indivifible. 
 
 Therefore the Aggregate of the Parts compaffed 
 about of the right Angles being without the Qua- 
 drant, is more than the Aggregate of the Parts of 
 
 the 
 
loo GEOMETRICAL Book II. 
 
 the Quadrant itfelf AC. Alfo the Arch AC is 
 fomeimall matter lefs than four fifths of the Dia- 
 meter, and the Arch CL, or the right Line 2 c 
 lefs than four fifths of the Side DC. 
 
 PROP. XIX. 
 
 Between a right Line given, and the Half of it ; io 
 find two mean Proportionals. 
 
 L E T the given Line A B, and the Half of it BC, 
 be difpofed at right Angles; join AC, and 
 let AC be cut in the Middle in D ; from the Centre 
 D with the Radius DA, defcribe a Semicircle, and 
 divide it in the Middle in E. Draw A E, C E, and 
 in CB being produced, put BF equal to CE. In 
 AB being produced, take 2 5 ths of the right Line 
 AB, which let be B /, and between AB and Bi I 
 
 find i 
 
lOI 
 
 Book II. ROSES. 
 
 find a mean Proportional B k ; Equal to it in A B 
 produc’d, put BG (which, as above, is demonftrat- 
 ed, is equal to 2 5ths of the Quadrantal Arch de- 
 fcribed from A B) and draw F G, GC. Moreover, 
 divide C F in the Middle in /, from the Centre / with 
 the Radius / F, defcribe a Semicircle by F and G, 
 which will pafs by C ; which I fhew thus. 
 
 The right Line AC, whofe Quadrate is Quintuple 
 of the Quadrate BC, is a mean Proportional between 
 the Quadrantal Arch defcribed from A B and its 
 Half, as is fhewn at Prop. 6. Therefore CE is equal 
 to half the Quadrantal Arch defcribed from AB, for 
 becaufe the Angle by E is a right Angle, CE is a 
 mean between both the right and circular Lines CE, 
 E A, and either of them. Wherefore B F is equal to 
 half the Quadrantal Arch defcribed from AB. And 
 BG is a 3d continual Proportional to BF, and the 
 whole Line AB by Prop.7. Then becaufe the Qua- 
 drantal Arch from AB, and the right Line A B, and 
 2 5th Parts of the Quadrantal Arch from A B, are 
 continual Proportionals ; it will be, as 2 5ths of the 
 Quadrantal Arch from A B (or the right Line B £) 
 to the Half AB, fo AB to the Half of the Qua- 
 drantal Arch from AB. Therefore it is as A B to 
 BF, foBGtoBC. 
 
 In the Arch ABC, apply from the Point A, a right 
 Line AH equal to BC; then AB, CH will be 
 equal, and therefore the Arches HE, EB, or the 
 Angies HCE, ECF equal, and the right Lines H F, 
 B F, likewife C F, A F are equal, and D E produc’d, 
 divides the Angle A FC in the Middle. 
 
 Alio the Triangles ABF, GBC arealike, where- 
 fore the Angles BGC, BAF are equal, alfo the Angles 
 BCG, BF A equal. And (for becaufe the Angle CH A 
 
 is 
 
102 GEOMETRICAL Book II. 
 
 is a right Angle, and CH equal to AB) the Tri- 
 angle ABF, F B G are alike and equal. Therefore 
 both BF, GC, alfo CH (or AB) and FG, are 
 equal between themfelves. 
 
 And A G being divided in the Middle in b, from 
 the Centre h , with the Radius b A, a Semicircle 
 being defcribed, it will pafs by F. 
 
 Then is FGC H right Angled, and the Triangles 
 ABF, FBG alike. Therefore it is as AB to FB, 
 fo F B to B G, and fo B G to BC. Therefore B F, 
 BG are found to be two mean Proportionals be- 
 tween the given Line AB, and the Half of it BC* 
 Which was to be done. j 
 
 C O R. 
 
 It is manifeft from the Precedings, that the greater 
 of the 2 Means is an 8 th Part, and the leffer Mean 
 
 2 5th 
 
Book II. ROSES. 
 
 2 5th Parts of the Quadrantal Arch delcribed from 
 AB. Alfo A B (or CH) is equal toFG$ and 
 BG is equal to that Part of the right Line CH, 
 which is cut off by A B, being computed from the 
 Point C, and the Conftru&ibn of the Problem de- 
 monftrated to the Byes of all Men. 
 
 PROP. XX. 
 
 Of the Centre of Gravity of a Quadrant of a Circle . 
 
 T H E Centre of Gravity (that is ? the Centre of 
 Weightinefi or Heavinefs) of a Quadrant of 
 a Circle is in a right Line from its Centre, dividing 
 the Arch in the Middle, and diftant from the Centre 
 of the Circle, fo much as is the mean Proportional 
 between the Semidiameter, and 2 5th Parts of it. 
 
 Deforibe the Quadrate A B C D, and in it a Qua- 
 drant A D G, and draw the Diagonals A C, B D, 
 whereof B D will cut the Arch A C in the Middle 
 in L. 
 
 Between the Semidiameter DC, and 2 5th Parts 
 of it, find a mean Proportional DR, and with the 
 Radius DR, deferibe an Arch of a Quadrant RS, 
 cutting the Diagonal B D in 2:. I fay 2; is the Cen- 
 tre of Gravity of the Quadrant D AC. 
 
 Cut the Quadrate A BCD into 4 Parts by the 
 right Lines E F, G H, cutting themfelves mutually 
 and at right Angies in I. Join DF, cutting the Arch 
 A C in P 3 and by P draw Y Q^ parallel to BC, cut- 
 ting the Diagonal B D in Q, and E F in V ; alfo 
 compleat the Quadrate D YQjYI, whofe Side QjVt 
 cuts the Arch A C in N. 
 
 Then as the Quadrate from DF, to the Quadrate 
 
 from 
 
io 4 GEOMETRICAL BooklE 
 
 from D P or D C, fo is the Quadrate from D C to 
 the Quadrate from I) Y. And the Quadrate A B C D 
 is 5, whereof the Quadrate D Y QM is 4. 
 
 Join Y M, and it divides D Q in the Middle ; 
 alfo draw the right Lines P k, NO, that parallel to 
 the Side AB, this parallel to the Side BC, they 
 will cut each other mutually, and at right Angles, 
 in the Middle of the right Line D Q 
 
 Therefore becaufe the Quadrate A BCD is to 
 the Quadrate D YQM as 5 to 4, the Quadrate of 
 DQ will be 8, whereof the Quadrate from YQis 
 4, and the Quadrate from DC 5, and the Quadrate 
 from the half D Q2. 
 
 Then the Quadrate from the Half of DQ, is 2 5ths 
 of the Quadrate A B C D ; therefore the Half of D Q 
 is a mean Proportional between the Semidiameter DC, 
 and 2 5ths of it, and therefore equal to D ,2. Like- 
 wife 2 is both the Centre of Magnitude, and alfo 
 the Centre of Gravity of the Quadrate D Y QM ; 
 and the Point 1 the Centre of Magnitude and of 
 Gravity of the Quadrate A BCD. 
 
 Alfo it is {hewn at the 18 Prop, of this, that the 
 Quadrate DYQM, and the Quadrant DAC between 
 themfelves are equal. And that the 3 Lines C YP, 
 PQL, AMN, NQL between themfelves are equal. 
 Therefore if from the Quadrate D Y QM be taken 
 two three Lines equal to P QL, N QL, and to 
 the fame Quadrate be added two three Lines CYP, , 
 AMN at equal Diftances from the Diameters of even 1 
 Height, the 3 Lines CYP, AMN will be equi- 
 ponderant (or weigh alike) but the Diftances Y P, 
 M N, P Q, QN are equal and alike diftant from the 
 Diameters of equal Height N O, P k ; likewife the 
 Points Y, Q, M, D are equally diftant from the 
 ^ ' Centre 
 
Book H. ROSE S. 
 
 i°5 
 
 Centre of Gravity of the whole 2 ; therefore z is 
 the Centre of Gravity of the Quadrant D A C 
 which was to be demonftrated. 
 
 C o 
 
 N S E C T. 
 
 The Centre of Gravity of a Semicircle, is a mean 
 Proportional between two Fifths and one Fifth of 
 the Arch AC. For it is ihewn ( Pro p. 5.) that the 
 right Line, which is a mean between the Semi- 
 diameter, and two Fifths of it, is equal to two Fifths 
 of the Arch A C ; therefore if 2 O be produced 
 to p, fo that zO, Op be equal, then draw D p. 
 p will be the Centre of Gravity of a Quadrant 
 equal toDAC' And becaufe the Point O divides 
 z p in the Middle, the Point O will be the Centre 
 of Gravity of the double of the Quadrant D A C 
 that is, of a Semicirle defcribed with the Radius 
 -UC ; alio D p produced, cuts the Arch C / equal 
 to the Arch L C. Then when as the Quadrant 
 D A C refteth in and the Double of tlie Sedor 
 C / relteth in p, the whole Semicircle will reft in 
 • But z O is a mean Proportional between D z 
 and , ts Half, that is, between two Fifths, and one 
 •J;™ • 5 ^ e -^ r(dl ^ Q becaufe the Triangle 
 
 ?? f ? A S n § ht a "S Ied and equicrural ; {viz. it hath 
 two Sides equal.) 
 
 I 
 
 PROP. 
 
t0 6 GEOMETRICAL Book II. 
 uop. xxi. 
 
 ‘the Centre of Gravity of the Circular Line and 
 Strait Line , ALCA is in the Diagonal BD, di- 
 ftant front the Point B, fo much as is the Length 
 of a tangent of 3 ° Degrees. 
 
 F ind a Tangent of 30 Deg BX, equal *- 0 
 which, from the Point B in the Diagonal BD 
 take B T. I lay that the Point T is the Centre 01 
 Gravity of the two - Lines A LC A. 
 
 For becaufe the Quadrat from A B, is to the Qua- 
 drat from BX, or BT, as 3 to 1, and the Quadrat 
 from the Semidiagonal B I is half of the Quadrat 
 from A B,the Quadrat from B I will be to the Qua- 
 drat from BT, as i to 1, as 3 to 2. 
 
 And the three Lines A B C L A, to the two Lines 
 ALCA, as 2 to 3 : Which I thus Ihew. 
 
 The Gnomon YBM, is a fifth Part of the Qua- 
 drat DYQM, that is a fifth Part of the Quadrant 
 t) aC: Wherefore alfo the three Lines A B C L A is 
 a fifth Part, or two tenth Parts of the Quadrat 
 ABC D ; and the Triangle ABC is the Half; or 
 five Tenths of the Quadrat A B C D : But the Qua- 
 drant D AC is four Fifths, or eight Tenths of the 
 Quadrat A B C D ; therefore the two Lines ALCA 
 is three Tenths of the Quadrat A B C D. 
 
 Then is the Proportion of the three Lines to tne 
 two Lines, the fame which 2 is to 3, that is, recipro- 
 cal of the Proportion, as well of the Magnitude. 
 ALCA, ABCLA, as alfo of the Quadrats BI, BI 
 Join AF cutting the Diagonal B D in a. Then be 
 cauie A B is double to B F, and the Angle^Bl 
 
Book II. ROSES. 
 
 307 
 
 divided in the Middle by the right Line B a, A a 
 will be the double of a F : Therefore the Point a 
 is the Centre of Gravity of the whole Triangle 
 A jB C. 
 
 Cut a T in the Middle in b, and take a c the Tri- 
 ple of T b, and let a be the Centre of the Baliance. 
 It will be therefore as the three Lines A BCL A 
 to the two Lines ALC A, that is, as 2 to 5 fo 
 reciprocally c a, to a T, to wit, as 3 to 2. 
 
 Therefore is T the Centre of Gravity of the two 
 Lines A LC A; and the Point c, the Centre of 
 Gravity of the three Lines A BCL A : Which was 
 to be demonftrated. 
 
 •A M S EA 
 
 / 
 
 
 The End of the Second Book. 
 
 I 2 
 
 SOME 
 
 ; 
 
o L 
 
SOME 
 
 PRINCIPLES 
 
 AND 
 
 PROBLEMS 
 
 GEOMETRY, 
 
 Thought formerly Defperate 5 
 
 now 
 
 Briefly Explained and T>emonftrated. 
 
 The Third Book. 
 
 Written in Latin 
 
 By THOMAS HOBBES : 
 
 And made Englijh 
 
 By FEN. MAJSDET. 
 
 LONDON: 
 
 Printed in the Year M.DCC. XXVII, 
 
Cl VI A. 
 
 Principles md Problems 
 
 ' i 
 '‘I ‘*>W i. _ T **- 
 
 I N 
 
 geometry,^. 
 
 rV 
 
 __ 
 
 C H A P. ■ I. 
 
 .... , JL • • • ‘ A J i 1 
 
 0/ the Subject, Principles , W Method of the 
 Mathematicks . 
 
 * - ;• Wt- V\V C5‘ V ' ' '■ /C C 
 
 E that would apply himfelf to the Study 
 
 of the Mathematicks, muft, in the firft 
 places underftand what the Matter is con- 
 
 ..... cerning which Mathematicians treat, and 
 
 what it is that is enquired into about that Matter. 
 
 We are therefore to know, that the Matter which 
 they treat of, is every thing that hath Magnitude, 
 that is to fay, " 1 L ^ " Ua 
 
 asked. 
 
 J J ^ - 
 
 s to fay, all about which the Queftion may be 
 How great is it? So that the fubjed Matter of 
 Mathematical Sciences, are the Lengths ^Superficies and 
 \ Thicknefs , or Profundity of Bodies: For every Body 
 hath three, and no more Dimenfions, and thefe differ- 
 ing in Kind. From the Motion of a Body, a thing long 
 is produced, which is called a Line j then from the 
 Motion of the Line fprings a Superficies and from 
 the Motion of the Superfici e^fomewhat thick or grofs , 
 
 which 
 
Book III. ‘Principles > 8cc. i i j 
 
 which is alfo called a Solid and Mathematical Body 
 There is no farther Progrefs , for which way fbever 
 a folid Body is moved, a folid Magnitude will be 
 defcribed. They therefore, who among the Kinds 
 of Magnitudes rank Surdofolids, Quadratoqua- 
 drats, 0V. do idly- for thefe are not Figures, 
 but only Numbers, that is, not Magnitude, but 
 Magnitudes. 
 
 Motion, Time, Force, Weight, belong likewife to 
 the Mathematicks j for one of thefe may be laid to be 
 bigger or lefs than another : Nay in fo far the Ma* 
 thematicks treat of Qualities, as one may be called 
 more or lefs fuch than another. For of whatfbever 
 may be adverbially faid, more or lefs , that hath a 
 Foundation in fome Magnitude. The Companion 
 of Motions, Magnitudes, Times, amongft 
 themfelves, pertains alfo to the Mathematicks ; for 
 by reafon of the various Inequalities of Things, one 
 Inequality is more or lefs than another. Inequality 
 therefore hath its own Magnitude, and that is it 
 which is ufually called Proportion or Reafon. Num- 
 ber belongs to the Mathematicks alfo $ for to what 
 Science a Body, or Motion, or Time, or a Line 
 belongs in the Singular j all thefe belong to the 
 fame in the Plural. For when One is nothing elfe 
 but a thing number d 9 Number can be nothing, 
 other than things number d. 
 
 Now to ask how big a thing infinite is, would be 
 an abfurd Queftion, and not at all to be anflver’d. 
 For neither can a finite Magnitude be divided into 
 Parts infinite in Number, nor can an infinite Num- 
 ber be given : when Mathematicians frequently fay ? 
 Let a Line be drawn out in infinitum , they are not 
 fo to be underftood, as if they thought that would 
 
 I 4 be 
 
i i i ‘Principles and Problems Book III. 
 
 be done, but that they leave it to be produced at 
 the R eader’s pleafure. 
 
 The thing that is enquired of thofe Subje&s of the 
 Mathematicks, is in a manner nothing elle, than how 
 great the propoled Magnitude is, as it is compared 
 with another Magnitude of the fame Kind ; or that I 
 may fay it in a W ofd, the Proportion is enquired. It 
 is therefore necelfary to one that ftudies Mathema- 
 ticksjexa&ly to underftand thefe,and all other Terms 
 which Mathematicians ufe j that is to fay, to know 
 the true aild clear Definitions of the fame. 
 
 Quantity (which the Antients no where defined, 
 and who, for want of a Definition thereof, rail* ly dif- 
 putiiig about the Nature of an Infinite, have erre d) I 
 thus define : Quantity is a Magnitude every way de- 
 termined either by Expofition, fo that to one asking, 
 How great is it ? It may be anfwered, As great as you 
 fee • or by Comparifon, fo that it may be anfwered, 
 It is one or more Feet great, or any other fuch way. 
 But above all things, he ought to have an accurate 
 Definition of Proportion, or Mathematical Reafon j 
 andbefides, to know how to Add, Subtrad:, Multi- 
 ply, and Divide Proportions, and the Varieties of 
 Comparifons ^ of which, though moft of them be 
 taught by Euclid , and his Interpreters, yet I fhall 
 reduce almoft all of them to the next Chapter. 
 
 The Matter being known, and what we enquire into 
 concerning it, the next thing is to take the Principles : 
 The Principles of what? the Principles of the Science, 
 or of knowing that we feek for. Now nothing but 
 Truths can be known. The Principles then, of the 
 Mathematicks, are the firft T ruths , which we are not 
 taught, but know them by the Light of Nature, fo 
 foon as we hear them fpoken, They are therefore 
 
 called 
 
Book III. in GEOMETRY. 
 
 called the firft Propofitions, becaufe of them the fir ft 
 of all Syllogifms confift. The Subject then of a Sci- 
 ence, or any part of it, is not to be called a Principle; 
 as a Point of Geometry, or an Unity of Arithme- 
 tick (which as a certain Mafter taught publicklv) 
 tho the firft Book or Eticlid s Elements begins with 
 a Point, and the feventh Book with an Unity. 
 
 The Principles of Mathematicks, or firft Propofi- 
 tions, are Definitions; but that thefe may be ufeful, 
 they muft not only be true, but likewife accurate, left 
 that through fo great Variety of Significations’ we 
 either proceed with Uncertainty, or by continual 
 Diftindtions being beat out of the Path, lofe the 
 Truth by jangling. For as if a Hare fhould ftarc 
 amidft a Company of MaftifFs, the Dogs let loofe 
 would fall upon one another, whilft the Hare is run- 
 ning for it ; fo likewife in a Theatre of thole who 
 difpute for Vidtory, Truth flies away undiicemed. 
 And for thisCaufe (I mean, to remove Diftindtions) 
 wife Men have introduc’d Definitions. 
 
 But to define well, that is, lb to circumfcribe with 
 "Words the fLhing in Hand, and that clearly, and 
 with as much Bevity as can be, that no Ambiguity 
 be left, is a very difficult Task, and not fo much a 
 Work of Art, as of the natural Intclledt. Howe- 
 ver we are eafed of moft part of this Difficulty bv 
 the Induftry of the antient Geometricians ; lave that 
 Euclid’s Definition of a Line needs fome Corredlion, 
 or at leaft a found Interpretation. It is true indeed, 
 that in comparing of Longitudes, there is no re- 
 Ipedl at all to be had to Latitude ; but yet in the 
 Conftrudtion of Figures, Latitude is necefTary ; for 
 without Latitude, it is impoffible to deferibe a Fi- 
 gure; nor can it be known in what part of a broad 
 
 Line 
 
1 14 ‘Principles and Problems Book III. 
 
 Line that invifible Line is to be underftood. The 
 reft, tho’ all of them be not moft exadf, yet be- 
 cause they are true Propolitions, they fpoil not his 
 Demonftrations. 
 
 Now Definitions are of two Kinds, of which the 
 one barely explains the Nature of the Thing, and 
 the other the Caufe or Manner of producing it. But 
 thefe Definitions are moft ufeful to the advancing of 
 Knowledge, which contain the Caufes of, and Manner 
 of producing the Thing defined. For that Saying of 
 Arifiotles is true, tfo know is to know by the Caufe : 
 The reft, which only declare the Effence of the 
 Thing defined, are generally lefs fruitful 3 for no- 
 thing follows from them, that was not before con- 
 tain’d in them 3 nor matters it, whether their Pro- 
 perties be called Definitions, or the Definitions, 
 Properties. 
 
 Petitions are ufually reckoned among Principles 
 alfo 3 but thefe are not the Principles of knowing, 
 but of conftrucfting, that is, of the Defcription of 
 Figures, wherein nothing is required, but that they 
 be pofiible. 
 
 All Prbpofitions Iikewife, whofe Truth is obvious 
 to the Light of Nature, are to be held for Princi- 
 ples. Now there no Doubt to be made of the Truth 
 of a lawful Definition, becaufe they have their Truth 
 from the Confent and Will of Men, who, at their 
 Pleafure, give Names to Things explained. But 
 there are fome Propofitions, which tho’ they depend 
 on Definitions, and may by them be demonftrated 3 
 yet are fo perfpicuous, that even without a Demon- 
 ftration they can force an Affent. Thefe are called 
 Axioms 3 but care is to be had, that they be not held 
 for true upon the bare Authority of Mafters. There 
 
 is 
 
Book III. in GEOMETRY. 115 
 
 is great Difference between the Authority even of 
 the beft Mafter, and the Light of Nature : For 
 there is nothing fo abfurd, but that feme Mafter (in 
 Defence of lome Error) hath fome time written 
 Things as abfurd. 
 
 The Manner like wile of adding, fubtradfing, 
 multiplying, and dividing of Coflick Numbers (that 
 is. Things number’d) is to be foreknown ; which 
 is eafy, and taught by , Vieta in his Ifagoge^ but 
 moft clearly and briefly by Ougbtred , in the Key of 
 Arithmetick ; from whence I have transferred as 
 much as I thought fit into the third Chapter. The 
 Nature and Manner of finding out Square and Cu- 
 bick Numbers, with their Roots, is alio to be learnt. 
 This is handled in the Fourth Chapter. 
 
 The Nature of an Angle is like wife to be known ; 
 and therefore I have explained the Properties there- 
 of in the Fifth Chapter. 
 
 To conclude, if heufe any thing that is well de- 
 monftrated. by any Author for a Principle, he will 
 do well • for every thing known Hands inftead of a 
 Principle, for a farther Progrefs in Knowledge. 
 
 Some perhaps may fay, that 1 place all Geometry 
 in frcecognita^ or Things to be foreknown ; but they 
 are miftaken : For he is a Geometrician (as I Paid 
 before) who knows how to determine a Quantity by 
 Comparifon,, amongft Quantities of the lame Kind. 
 But thole Things which I would have foreknown, 
 are only Properties that attend Geometry, and Steps 
 whereby we are initiated in the Myfteries of fub- 
 limer Geometry, and whereof we Hand in need not 
 only for Figures and Mechanick Works, but alfb 
 for the Knowledge of all natural Caufes, as being 
 all contained in Motion. 
 
 The 
 
1 1 6 ‘Principles and Problems Book III. 
 
 The Principles and Theorems, which are well de- 
 monftrated in the Books of the ancient Geometrici- 
 ans, being well underftood (for I write to thofo who de- 
 fire to improve the Science, the Thanks for the Work 
 being due to the firft Inventors ♦ and no Man can re- 
 jeft what is already done, but who is very learned in 
 his own Conceit) the next Thing is, (not to demon- 
 ftrate, but) to feek out the Caufos of Proportion, 
 that one Magnitude hath to another: For no Man 
 can demonftrate what he knoweth not, nor know but 
 what he hath found out, nor (unlefs very foldom) 
 find out what he hath not fought after. There is no 
 certain Art of Invention by Sagacity, trying, fup- 
 pofing, deducing Confequences from Things fup- 
 pofed : Men often attain either to the Caufes of the 
 Matter fought after, or to fomething impoflible. And 
 by this Sagacity (not by Art) all the ancient Geome- 
 tricians performed what they did, as manifeftly ap- 
 pears, inthatmoft of their Theorems begin with the 
 Note of Suppofition [//]. But (one may fay) how fhall 
 one chance to take that fuppofed, which he Hands 
 moft in need of? There is no certain Method whereby 
 the moft convenient Suppofitions may be chofon ; and 
 therefore we muft try and experiment by meafuring 
 with a Compals, and other Inftruments, how near 
 we can come mechanically to the Truth of the Thing 
 fought. For Inftance, in feeking the Length of the 
 Circumference of a Circle (all Men knowing, by the 
 Light of Nature, that any Arch is bigger than its 
 Chord) it is an eafy matter to come very near the 
 true Length. The remaining Work is no more but 
 to excogitate the poflible Caufes of fo great a Propin- 
 quity, wherein if they fucceed, they have the Princi- 
 ple of a Demonftration. For indeed, in all Doubts, 
 
 Nature 
 
Book III. in GEOMETRY. 117 
 
 Nature it felf fets us upon various Suppofitions. But 
 moft Geometricians enquire nothing now adays, but 
 if it be Euclid's ; tho 5 it be well known that the moft 
 noble Theorems were invented, and the moft beauti- 
 ful Fabricks of the whole World built long before the 
 Time of Euclid. Now all our modern Geometricians, 
 almoft, imagine to find not only the Proportions of 
 Quantities, but alfb of Motions, that is, the Caufes 
 of all natural EffeCls, in the foie Extraction of the 
 Numeral Roots, and that a falfe one too j and the 
 Reafon is, becaufe they are fb taught. 
 
 The laft thing in Geometry is, the efficient Caufes 
 of the thing fought being found, to demonftrate from 
 them as it were backwards : I fay, from the efficient 
 Caufes, becaufe the Caufes of Numerical Properties 
 are, for moft part, nothing but arbitrary Impofitions 
 of Numeral Names; and Arithmetical Operations 
 are the Demonftrations of the Works themfelves. 
 
 CHAP. II. 
 
 Of Reafon or Proportion. 
 
 *• T^Eafon is the Relation of a Magnitude to a 
 I\ Magnitude of the fame kind, according to 
 Quantity ; as two Magnitudes, 2 and 5 being pro- 
 poled ; the Reafon of the firft to the fecond is the 
 Relation of the Quantity of 2 to 5, that is, the 
 Quantity of the Magnitude 2, as it is compared with 
 the Magnitude 5 : For the Inequality of two Mag- 
 nitudes may be more or lefs than the Inequality of 
 two other things unequal. 
 
 2. Therefore, for fhewing Proportion or Reafon, 
 it is neceflary that two abfolute Magnitudes be ex- 
 
 pofed. 
 
1 1 8 ‘Principles and Problems Book III. 
 
 pofed, of which the former is ufually called the An- 
 tecedent, and the fecond the Confequent. 
 
 So that neither both Quantities together, as 2 and 
 5, nor one alone, as 2 or 5 ; neither 4 (therefore our 
 Algebraifts do err, who fay that Fradion and Rea- 
 fon are the fame thing) neither the Difference betwixt 
 2 and 5, but the Quantity of Inequality that is be- 
 twixt two Magnitudes, is that which is called Rea- 
 fon or Proportion : For though both 3 and 2, and 
 9 and 8, differ only by a Unity, yet their Inequality 
 is not the fame. Yet if there be three Magnitudes, 
 the leaft, the mean, and the greateft ; the Reafon or 
 Proportion of the leaft to the mean, is greater thari 
 the Reafon of the fame to the greateft. But on the 
 contrary, the Reafon or Proportion of the greateft to 
 the mean, is lefs than the Reafon or Proportion of the 
 lame to the leaft. We might fay that fencer was 
 bigger, compared to Ulyjfes, than to Ajax ; and Ajax 
 was left, compared with UlyJJes , than with fencer ; 
 the Reafon of this is manifeft ; for by how much a 
 leffer Magnitude comes nearer a greater Magnitude, 
 by fo much it hath a greater Reafon or Proportion to 
 it; and by how much more a greater Magnitude 
 exceeds a left, by fb much it hath alfo to it a greater 
 Reafon cr Proportion. 
 
 3. Now the Proportions or Reafbns of left to 
 greater, and of greater to left, are different Kinds of 
 Quantity ; whereof the one is the Proportion of De- 
 fed, the other of Excels; of which the former, by 
 how much the more it is multiplied, by fb much the 
 Defed becomes always the left ; therefore it can ne- 
 ver exceed or equal the Proportion of Excels. 
 
 4. Reafons or Proportions are the fame, or alike, 
 or equal, in Numbers, when they are of the fame 
 
 Denomi- 
 
Book III. in GEOMETRY. it 9 
 
 Denomination, or may be reduced to the fame De- 
 nomination • as if both the firft of the fecond, and 
 third of the fourth be 4 or ~ : But where the Pro- 
 portions are not expreflible by Numbers, both the 
 firft of the fecond, and third of the fourth, are 
 correfpondent Sides of like Triangles, and are called 
 Proportional, as having Portion for Portion, by the 
 Greeks 'Avdhcyov^ as the fame thing laid again. 
 
 5. Continual Proportionals are fuch, as, being pro- 
 portional, have a middle Quantity common, as i, 2, 4. 
 
 6. The Root of a Square Number is that Number, 
 which, being multiplied by it felt* produceth feme 
 Number : Now the Number produced is called a 
 Square or Quadrat, and the Root is always an ali- 
 quot Part of its own Square • as, if the Root be 8, 
 the Square will be 64, and the Root the eighth Part 
 of it. 
 
 7. It is then manifeft, that the Root of a Square 
 Number is not the Side of a Square Figure, though 
 feme Algebraifts deny this, being now alhamed to 
 confels the Truth. 
 
 8. If there be three Magnitudes continually pro- 
 portional, and from the fame Antecedent three other 
 continual Proportionals j the Proportional of the laft 
 to the laft, will be the double of the Proportion of 
 the fecond to the fecond • as is manifeft in thefe 
 Numbers, 2, 4, 8, and 2, 6, 18 $ the Proportion of 
 18 to 8, is the double of the Proportion of 6 to 4. 
 For as 18 is to 8, fo is 9 to 4; but 9 to 4 hath a 
 double Proportion of 6 to 4. Elem. 14. Prop . 28. 
 
 9. Two Proportions are compounded or added to- 
 gether, when both Antecedents and Confequents be- 
 ing multiplied among themfeives, a Proportion is 
 made. For Example, if the Proportion of 2 to 3 be 
 to be added to the Proportion of 4 to 5 ^ let the An- 
 tecedents 
 
i zo Principles and Problems Book III. 
 
 tecedents 2 and 4 be multiplied one by another, and 
 they make 8 * then let the Confequents 3 and 5 be 
 multiplied one by another, which make 1 5. This 
 being done, the Proportion compounded of the 
 Proportions of 2 to 3, and 4 to 5, is the Proportion 
 of 8 to 15, according to Eucl. Elem. 6 . Def. 5. 
 
 10. Otherwife, if it be thus ; as 4 is to 5, lo 3 to 
 another which fhall be 3 3, the Proportion of 2 to 
 3 l (that is, both being multiplied by 4) will be of 
 8 to 15, that which is compounded of the Proporti- 
 ons of 2 to 3, and 4 to 5. 
 
 11. The Subtradtion of a Proportion from a Pro- 
 portion, is done by dividing the Quantities of the 
 whole Proportion to be fubtradted by the Antece- 
 dent, and the Conlequent of the whole to be fub- 
 tradted by both. For Example, Let the whole Pro- 
 portion be of 8 to 15, the fubtradted of 4 to 5 ; I 
 divide 8 and 15 by 4, the Quotients are 2 and 3 jj 
 Again, I divide the Conlequent of the whole Propor- 
 tion 15 to be fubtradted by both the Quantities, to 
 wit, 4 and 5, the Quotients will be 3 i and 3, which 
 are in the Proportion of 12 to 15, placing them in 
 order, 8, 12, 15. If from the Proportion of 8 to 15, 
 the Proportion of 12 to 15, that is, of 4 to 5, be 
 taken $ it is manifeft that there remains the Pro- 
 portion of 8 to 12, that is, of 2 to 3. 
 
 12. Otherwife, if it be thus ; As the Confequent 
 of the Subtrahend, which is 5, is to its Antecedent 
 4^ fo the Confequent of the whole, which is 15, is 
 to the other 12 ; placing in order 8, 12, 15, it is 
 plain that the Proportion of 12 to 15, that is, of 4 
 to 5, being fubtradted from the whole Proportion 
 of 8 to 1 5, there remains the Proportion of 8 to 
 12, that is, of 2 to 3. 
 
Ill 
 
 Book III. in GEOMETRY. 
 
 13. Proportion is multiplied by laying, As the 
 Antecedent is to the Conlequent, lb the Conlequent 
 to a third, and lo the third to a fourth, &c. 
 
 14. A Proportion is divided by Number, when 
 between the Antecedent and Conlequent, one or 
 more mean Proptionals are interpoled. 
 
 15. If there be never lo many Magnitudes of the 
 lame Kind, the Proportion of the firft to the laft, is 
 compounded of the Proportions of the firft to the le- 
 cond, of the fecond to the third, and fo forth unto 
 the laft ; as in any Numbers, 4, 9, 10, 3, the Pro- 
 portion of 4 to 3, is compounded of the Proportions 
 of 4 to 9, and 9 to 10, and 10 to 3, which is de- 
 monftrated by many. 
 
 1 6. But there is another Kind of compared Quanti- 
 ty, which is alfo called Proportion, not Simply, but 
 Arithmetical Proportion. Now four Quantities are 
 faid to be in Arithmetical Proportion, when the one 
 conftantly exceeds the other by equal Differences. 
 Concerning which Proportion, I fhall only lay, If a 
 Line were cut into as many equal Parts , as it is intelhgi - 
 ble it may be divided into , the Geometrical Proportional 
 Me an between every next two ofthefe Parts^willbethe 
 fame with the Arithmetical Mean. If (for Example) 
 a given Line Ihould be divided into the Parts 2 and 
 16, the Arithmetical Proportional Mean will be 7 * 
 fo then 2, 9, 16 are continual Arithmetical Propor- 
 tionals. Now the Geometrical Proportional Mean is 
 lefs than 6 ; but if the lame Line had been divided 
 into iooooo equal Parts, and both a Geometrical 
 and Arithmetical Mean taken betwixt the Whole and 
 its Parts 99999, how fmall a Difference would be be- 
 tween thole two Geometrical and Arithmetical Pro- 
 portions ? Therefore by how much lels the Difference 
 
 K be- 
 
122- ‘Principles and ‘Problems Book III. 
 
 betwixt the Extremes is, by fo muchlefs is always the 
 Difference between the Geometrical and Arithmeti- 
 cal Mean. Wherefore if fuch Means were every where 
 interpofed, all their Difference would vanifh. This 
 is a moft iiieful. Proportion for finding out the Pro- 
 portions that ftrait Lines have to crooked. 
 
 CHAP. HI. 
 
 Of Algebraical Operations. 
 
 i. r T^HE Algebraifts inftead of Things, that is, in- 
 .1 ftead of Magnitudes, put Letters as the Spe- 
 cies of Things, which in this manner we are taught to 
 Add, Subtrad, Multiply and Divide. If the Species 
 be folitary, as A and B, they are added by interpo- 
 fing the Sign +, which fignifies more. A and B 
 then being added, the Sum is A + B ; but if B be 
 to be fubtra&ed from A, they interpole the Sign — , 
 which fignifies lefs, or a Muld. Therefore A — B 
 is that which remains of the Magnitude A, after that 
 
 B is taken from it : Multiplying by a Number, is w'hen 
 the Number Multiplier is prefix’d, as 2 A, 3 A, &. 
 The Multiplication of A by itfclf is performed by 
 writing A A or A q ; a Line cannot be multiplied 
 by itfelf ; and this is the firft thing wherein the Alge- 
 braifts of this Age differ from Diophantus and the An- 
 cients, who contained themfelves within the Bounds 
 of Arithmetick. If A be to be multiplied intoB, they 
 
 write AB, if it be written fi , A is underftood to b< 
 
 dividec 
 
Book III. in GEOMETRY. 123 
 
 divided by B, which fignifies the Quotient, as if A be 
 
 A. 
 
 10, and B 23 that Mark -- is 5 : This does indeed, 
 
 very well in Numbers- but if A A be a fquare Fi- 
 gure, and B the Divifor a Line 3 then that Quotient 
 A A 
 
 ~B ^ 1CWS ^° W 0 ^ te11 t ^ ie ^ ine ® is in t ^ ie ^ uare Fi- 
 
 gure A A, which is againft the Nature of true Al- 
 gebra, and likewife abfurd 3 • for Planes or Flats 
 are not compofed of aggregated Lines. 
 
 But if the Species be affected with like Marks or 
 Signs, as + A + A, they being added together make 
 2 A. For this is a true Rule of Oughtred , and the 
 Algebraifts 3 in Addition the Species are to be added, 
 and a common Mark to be affixed. If therefore — A 
 be added to — A, the Sum will be — 2 A 3 but if 
 the Marks be different, then this is the Rule 3 let Spe- 
 cies be fubtraded from Species, and the contrary 
 Mark prefixed to what remains 3 fo that + A and 
 — A added together make o, that is nothing. 
 
 3. But if two affeded Species be to be added to 
 two affeded Species, this is the Rule: If they be like 
 Signs, let the Species be added, referving the Signs. 
 So A + B, and A+ B added together make 2 A 
 + 2 B. But if the Marks be different, as in thefe 
 + A — B, and — A +B, the Rule is 3 let the like 
 affeded be reduced under their proper Signs, and 
 the Sum will be + A+ B — A — B, that is 
 nothing. 
 
 4. In Subtraction, if the Species befolitary, and 
 each have the Sign +, as + A + B, and + B be to 
 be fubtraded from +A, the Remains is the Species 
 from which the Subtradion is made, with the Sign + 
 together with the Species fubtraded having the con- 
 
 K 2 trary 
 
1 14 ‘Principles and Problems Book III. 
 
 trary Sign. For B taken from A leaves A — B ; but 
 if both the Signs be — , the Difference of the Species 
 with the Sign -his to be taken for the remaining Part, 
 
 as if 5 be to be fubtradfed from — 7, the Remains 
 
 will be + 2. For it is manifeft, that — 5 is more 
 than — 7 by two Unities. And that it is fo, is eafy 
 to be underllood by any that will take the Pains to 
 confider it. For what is it elfe, to take the Defedt 
 from a given Magnitude, but to add to it the defici- 
 ent Magnitude which it wanted ? If therefore — A 
 be to be taken from A, it is manifeft that there re- 
 mains 2 A ; for by the very taking away of the De- 
 fe<ft, the Acceffion of that which was wanting is ad- 
 ded, in the fame manner as he that remits a Debt, 
 beftows upon his Debtor as much as he hath remit- 
 ted. If feveral Species affecftcd with Signs be to be 
 multiplied by one or more Species affedted, this is the 
 Rule : If the Signs be alike, let the Sign + be pre- 
 fixed to the multiplied Species ; but if different, the 
 Sign — j as in this Example, 
 
 A — B 
 A-C 
 
 A + B 
 A + C 
 
 — CA + BC 
 +AA— A B 
 
 + CA + BC 
 A A + AB 
 
 AA+CA+AB+BC +AA— CA-AB+BC 
 
 Which is true indeed in Numbers ; but if they be 
 Lines, falfe. For nothing can be multiplied but by 
 a Number. 
 
 As many Species as youpleafe, andhowloever af- 
 fe&ed, are underftood to be divided, when they place 
 
 the 
 
Book irr. in GEOMETRY. J2 ? 
 
 the Divifor under the Dividend, as is the 
 
 Quotient of A ~PB divided by B — C, that likewile 
 is true in Numbers ; but if A +B be a Superficie, and 
 B — C a Line, it is abfurd, as we obferved before. 
 What remains is no more but Labour in feekin°- out 
 a commodious Suppofition, from whence they may be- 
 gin, which is not Art, but groping and chance. Now 
 when they have found out any Proportion by chang- 
 ing, converting, multiplying, dividing, compound- 
 ing, and elevating, and deprefling it by a Scale of con- 
 tinual Proportionals, they tofs the fame, until that at 
 length unawares, and by hap-hazard, they ftumble 
 upon fome Equation that may fatisfy an Arithmetical 
 Problem : But it is impoflible by this Method to an- 
 lwer a Geometrical Queftion. Algebra is, indeed 
 one of the Rules of Arithmetic!:, to the attaining the 
 Theory whereof, two or three Days at moft are re- 
 quired, though to the Promptitude of Working, per- 
 haps the Pradice of three Months is necefiary. 
 
 CHAP. IV. 
 
 Of Square Figures , and Square Numbers. 
 
 r. \ Square Figure, and a fquare Number, when 
 XX. they have fo many fquare and equal Parts, 
 are one and the lame thing ; for a whole Figure, 
 and the Number of its Parts are the lame. 
 
 K 3 
 
 2. If 
 
ii6 Principles and Problems Book III. 
 
 2. If fquare Figures be defcribed from crcfcent Ra- 
 dii (or Radius’s increafing) as Numbers in natural 
 Order from anUnity, they differ among themfelves as 
 odd Numbers of the fame Order. For the leaf; Figure 
 is one, to this the next odd Number (which is 3.) 
 being added, makes the next fquare Figure' (4.) 
 To thisagain, the next odd Number (5) being added, 
 makes the third Square (9) and fo on. Therefore 
 theExceffes ofthefaid Squares increafe as odd Num- 
 bers from an Unity, to wit, 1, 3, 5, 7, &e. 
 
 3. If there be three fquare Numbers next to one 
 another in order, and a Unity be taken from the 
 middle, the Number that is left is the mean Propor- 
 tional betwixt the Extremes. For Example, 1 00,8 1 ,6^ 
 are fquare Numbers next to one another in order. 
 Ret the Unity be taken from 81, there remains 80, 
 which is a mean Proportional betwixt 100 and 64. 
 In like manner 49, 36, 25, are fquare Numbers next 
 to one another in order. Take an Unity from the 
 middle 3 6, there remains 3 5 the mean Proportional 
 betwixt 49 and 25. So in thefe 9, 4, 1 j if from the 
 middle 4 one be taken, there remains 3 the mean be- 
 twixt the Squares 9 and 1. Laftly, 400, 361, 324, 
 are Squares next to one another in order : If from 
 the middle 361 an Unity be taken, there remains 
 360, the Mean betwixt the Extremes 400 and 324, 
 and fo in all. 
 
 4. Between two next fquare Figures, if to the leaf! 
 Square its Root be added, the whole will be the mean 
 Proportional. For if to the Square 1, its Root (which 
 is 1 ) be added, the whole 2 is the mean Propor- 
 tional betwixt 1 and4. Inlike manner, ifto the Square 
 4 its Root be added, the whole 6 is the mean Propor- 
 tional 
 
Book III. in GEOMETRY. 127 
 
 tional betwixt 4 and 9 ; and lo in all, as is manifeft 
 by Indu&ion. 
 
 5. If from a Square its Root be taken, there 
 will remain the Mean betwixt it and the next Idler 
 Square. Let 100 be a fquare Figure, if its Root 
 10 be taken from it, there will remain ninety, the 
 Proportional Mean betwixt 100 and 81, the next 
 lelfer Square. In like manner if from the Square 9, 
 its Root 3 be taken, the remaining 6 will be the 
 Mean betwixt 9 and the next inferior Square 4, 
 and fo always. 
 
 6. But if the Squares be not next to one ano- 
 ther, but that one interpoles, the lels Square with 
 two Roots, is the mean Proportional betwixt thofe 
 two Squares. For if to the Square 1 its two 
 Roots, that is 2 be added, there will be 3, the 
 Proportional^ Mean betwixt 1 and 9. Likewile if 
 to the Square 4 its two Roots, that is 4, be added* 
 there will be 8, the Mean betwixt 4 and 16, and fo 
 perpetually. The fame Mean will be had, if from 
 the greater Square, its two Roots be fubtradled; 
 For if from 16, its two Roots be taken, there will 
 remain 8, the Mean betwixt 16 and 4. 
 
 Iflaftly, Two Squares have other two between 
 them, and to the leis Square, the Triple of its Root 
 be added, you will have the Mean betwixt the 
 two extreme Squares. For if 3 be added to the 
 Square 1, there will be 4, the Mean betwixt 1 
 and 16; or if to the Square 4, the Triple of its 
 Root, to wit, 6, be added, there will be 10, the 
 Proportional Mean betwixt 4 and 25 ; or if from 
 25 the Triple of its Root, to wit, 15, be taken, 
 there will remain 10, the Mean betwixt 25 and 
 
 K 4 4* 
 
iz8 ‘Principles and Problems Book III. 
 
 4, and fb always. From which follows a general 
 Rule, that the mean Proportional betwixt two 
 Squares, is fb many Roots of the leffer added to the 
 leffer ; or what remains after fb many Roots are ta- 
 ken from the greater, as the greater Square is diftant 
 Places in the natural Order of Numbers from the 
 leffer. 
 
 7. Hence it is manifeft, that the Root of a 
 Square Number is not (as Algebraifts imagine) 
 the Side of a fquare Figure, nor at all a Line, 
 becaufe a Line adds nothing to a Flat, or 
 Plane. 
 
 8. If two Squares are next to one another, 
 both taken together, are greater than the double 
 Mean, by a Unity,, or one fquare Particle. For 
 1 and 4 are next to one another, both together 
 make 5, the double of the Mean is 4, the Diffe- 
 rence is 1. In like manner 100 and 121 are next, 
 both together are 221, the Mean is no, the 
 double of the Mean 220, the Difference 1, and fb 
 in all. 
 
 9. If Squares admit of one between them, the 
 Extremes taken together, are greater than the double 
 Mean by four • for 1 and 9 admit of one betwixt 
 them, both together make 10, the double Mean is 
 6, . the Difference 4. Likewife 100 and 144 receive 
 one between them ; the Extremes put together are 
 244, the Mean is 120, the double Mean 240, the 
 Difference 4, and fb every where. 
 
 10. If between two Squares, other two inter- 
 pofe, the Extremes together are greater than the 
 double Mean by the Number Nine. For 1 and 16 
 have two betwixt them, added together they make 
 
 *7 > 
 
Book III. in GEOMETRY. 129 
 
 17, the double Mean is 8, the Difference 9. Like- 
 wife 100 and 49 have two betwixt them, put to- 
 gether they make 149, the Mean is 70, the double 
 of that 140, the Difference 9. In like manner, 
 three being interpofed, the Extremes exceed the 
 double Mean by fixteen, and fo forth by fquare 
 Numbers throughout the whole Series of Squares. 
 
 1 1 . Betwixt two Cubick N umbers, two mean Pro- 
 portional Numbers intervene : Now they are found 
 out after this manner ; if the Cubes be next to one 
 another, let the lefs Cube be divided by its Root, 
 then let the Quotient be multiplied by a Number 
 that exceeds the Root by an Unity, the Produdt will 
 be the lefs of the Means. For Example, Take the 
 next Cubes 1 and 8 ; let the leffer Cube 1 be divided 
 by its Root 1, the Quotient will be 1, then multi- 
 ply 1, by one more 1, that is 2. This Number 2 is 
 the lefs of the Means betwixt 1 and 8 ; by the fame 
 Method the greater Mean is found out, 2 being di- 
 vided by 1, the Quotient is 2 j then 2 being mul- 
 tiplied by one more 1, makes the greater Mean 4. 
 Laftly, 4 being divided by 1, the Quotient is 4, and 
 4 being multiplied by one more 1, it makes 8 $ or let 
 the lefs Cube be 64, its Root 4, the next great- 
 er Cube 125 ; 64 being divided by 4, the Quotient 
 is 16; 16 being multiplied by 4 more 1, makes 
 80, the fir ft of the Means. Again, 80 being divi- 
 ded by 4, the Quotient is 20 ; which multiplied 
 by 4 more 1, makes 100 the greater Mean. In 
 the laft Place, if 100 be divided by 4, the Quo- 
 tient will be 25, which being multiplied by 4 more 
 I, makes 125. 
 
 12. If 
 
130 ‘Principles and ‘Problems Book Ilf. 
 
 - 12I If two Cubick Numbers have one Cubick 
 Number^betwixt them, let the lefs Cube be di- 
 vided, hy its Root, and let the Quotient be multi- 
 plied by the Root more 2, the Produft will be 
 the leffer Mean ; let the Cubes be 1, 27: 1 being 
 divided by 1, the Quotient is 1, which multipli- 
 ed by 1 more 2, makes 3 the leffer Mean. Again, 
 
 3 divided by 1 is 3, then 3 multipled by 1 more 
 2,. makes 9 the greater Mean. Laftly, 9 divided 
 by 1 is 9, which being multiplied by 1 more 2, is 
 
 13. In like manner, if the Cubes be 8 and 64, 8 
 being divided by its Root 2, the Quotient is 4, and 
 
 4 multiplied by 2 more 2, makes 16, the fir ft .Mean ; 
 and 16 being divided by 2, makes 8, which multi- 
 plied hy 2 more 2, makes 32 the fecond Mean. 
 To conclude, 32 being divided by 2, the Quotient 
 is 16, which multiplied by 2 more 2 makes 64. 
 In like manner, if the Cubes be 27 and 125 ; 27 be- 
 ing divided by 3, the Quotient is 9, which multi- 
 plied by 3 more 2 makes 45 the firft Mean. Again, 
 45 being divided by 3, the Quotient is 15, which 
 multiplied by 3 more 2 is ^5, and divided by 
 3, the Quotient is 25 ; and this multiplied by 3 
 more 2 makes 125, and fo always in Cubes that in- 
 termit one. 
 
 IftwoCubick^Numbers intermit two Cubick Num- 
 bers, the left being divided by its Root, the Quoti- 
 ent multiplied by the Root more 3, will give the 
 firft Mean. Let 27. and 216 be two Cubes inter- 
 mitting two Cubes ,(to wit 64 and 125) 27 being 
 divided by its Root, the" Quotient is 9, which being 
 multiplied by the Root 3 more 3, makes 54 the 
 
Book III. in GEOMETRY. 131 
 
 firft Mean ; 54 divided by 3 makes 18, which being 
 multiplied by 3 more 3 is 108. Laftly, 108 being 
 divided by 3, the Quotient will be 3 6, which mul- 
 tiplied by 3 more 3, makes 216. 
 
 If two Cubick Numbers intermit three Cubick 
 Numbers, and the lefs Cube be divided by its Root, 
 and the Quotient multiplied by that Root more 4, 
 the Produd will be the firft of the Means. And fo 
 through all the Cubes of Numbers that are in a 
 natural Order, adding 5, 6, 7, c 3 c. you may find the 
 firft Means betwixt any two Cubes whatfoever. 
 
 C H A P. 
 
\%i ‘Principles and Problems Book III. 
 
 CHAR V. 
 
 Of ANGLES. 
 
 Strait or right Line Angle, is the Digreflion. 
 
 of two ftrait (or right Lines) from the 
 lame Point in the Circumference. 
 
 2. The Quantity of a ftrait Line Angle, is the 
 Quantity of the Proportion that the intercepted Cir- 
 cumference hath to the Perimeter of the Circle. 
 
 3- The Quantity then of an Angle is not a Line, 
 neither a Superficie, nor yet a lolid Body, but the 
 Quantity of the Proportion which the Revolution of 
 the Radius hath to another Revolution of the fame. 
 
 4. The Chord of an Angle,: 
 Circumference or Arch, is a 
 ftrait Line that joins toge- 
 ther any two Points of the Cir- j 
 cumference ; as if there be 
 ftrait Lines from the fame 
 Point A B, A b, AC, and the 
 Angle ABD a right Angle $ 
 but the Arch B b C. T hen the 
 ftrait Line BC is the Chord 
 A or fubtenfe of the Arch BC, 
 
 and the ftrait Line B £, the 
 Chord of the Arch B b . 
 
 5. The right Sine of the Circumference is a ftrait 
 Line drawn from one Term of the Circumference, 3 
 to the Radius which pafleth thro* the other Term 
 perpendicularly $ as Be or ab, is the right Sine of 
 the Arch B^. & The 
 
Book III. in GEOMETRY. 
 
 6. The Secant of an Arch or Circumference, is a 
 ftrait Line drawn from the Centre by one Term of 
 the Arch to the ftrait Line which is drawn perpen- 
 dicularly in the other Term to the Radius j as AD 
 is the Secant of the Arch B C. 
 
 7. The Tangent of an Arch, is that Part which is 
 intercepted of the right Line which is drawn perpen- 
 dicularly from the Beginning of the Arch j as B D is 
 the Tangent of the Arch BC, or of the Angle ABC. 
 
 8. Hence it is manifeft, that the right Sine of any 
 Arch, is the half of the Chord of the double Arch, or 
 that the Chord of the Arch, is equal to the double 
 right Sine of the half of the Arch, as if the Arch Bb 
 be the half of the Arch B C, a b doubled will be 
 equal to the Chord BC. 
 
 9. A leffer Arch hath a lefs Proportion to its Chord, 
 than a greater Arch hath to its Chord. For the Pro- 
 portion of the Arch BC to its Chord B C, is the fame 
 with that of B the half of the Arch B C to the 
 half Chord B e. But the Chord of the half Arch BC 
 is the ftrait Line B £, which is greater than Be. 
 Therefore (by Chap. 1. Num. 2.) the Proportion 
 of the lefter Arch Bb to its Chord B £, is lefs than 
 that of the greater Arch BC to its Chord BC. 
 
 10. Hence it follows, that if the Arch Bb were 
 bifefted, and the Bifegmentof it again bifefted, and 
 fo on as often as can be done ; we would at length 
 come to a Segment of an Arch, the Excels of which 
 upon its own Chord fhould be lefs than any given 
 Longitude, and by confequence, lefs than theBreadth 
 of any Line having never fo little Latitude. 
 
 11. in like manner, if a ftrait Line werebifedt- 
 ed, and the Bifegments thereof again bife&ed as often 
 as might be, we would come at length to a Segment 
 
 lefs 
 
1 34 Principles and ‘Problems Book III. 
 
 left than any affignable Quantity, and therefore lefs 
 than the Breadth of a lane having never fo little 
 Latitude. 
 
 1 2. Now that univerlal Saying, Every Arch is great- 
 er than its Chord , would hold universally true, if there! 
 could be given an Arch of a Circle, and a ftrait 
 Line, that had no Breadth ; but neither is there a Line 
 without Latitude, nor if there were, or could be but 
 fuppofed, could it enter into a Demonftration, that 
 handles the Proportion of Superficies and Solids ; be- 
 caule they are different kinds of Quantity, as appears 
 by the fifth Definition, El. 5. of Eucl. Neither (as I 
 conceive) would Dr. IVallis have thought otherwife 
 but that that Definition is not found among the Prin- 
 ciples of Algebra in Oughtred’s Key of Arithmetick. 
 
 13. In any Triangle, if the 
 D Angle be cut in two equal 
 
 Parts, and the ftrait Line 
 that cuts the Angle, cut alio 
 the Bafe - the Segments of the 
 the Bafe will in Proportion oi 
 the Sides. This is clearly de- 
 monftrated in Eucl. Elem. 6. 
 Prop. 3. 
 
 14. There is another Kind 
 of a plain Angle, which they 
 call the Angle of Contadh, fuch 
 
 as the Angle that the ftrait Line BD makes with 
 the Arch BC in the Point B. Which Euclid (or hit 
 Interpreter dCheoii) and many that have followed 
 him, have affirmed to be greater indeed, than an) 
 acute Angle ; and the Angle which AB makes with 
 the fame Arch BC, bigger than any acute Angle 
 yet lefs than a rights which is manifeftly abfurd. Foi 
 
 then. 
 
Book III. in GEOMETRY. 135 
 
 then, as Dr. Wallis will have it, the Angle of Con- 
 tad: had been of the fame Kind with the Angle 
 that is made by the Revolution of the Radius , for 
 every Angle that is not obtule, is either right, or 
 lefs than a right Angle, by fo much as any Angle 
 is acute. 
 
 They were in the firft Place miftaken, in that they 
 thought the Space betwixt the ft'rait Line BD, and 
 the Arch BC to-be an Angle, which (by Numb 3. 
 of this Chapter) is falfe. They were alfo miflaken, 
 in that they took thofe Angles for Quantite^of the 
 fame Kind, which is falfe. For an Angle of Contad, 
 howfoever multiplied, can never equal nor exceed an 
 Angle made by the Circulation of the Radius. 
 
 15. If it were true, that the Angle made by the 
 right Line AB, with the Arch BC, were lefs than a 
 right Angle 3 it might come to pafs that an acute 
 Angle increasing uniformly, would at length become 
 greater than a right, which yet could never equal it : 
 The Ground of which Abfurdities is, that they 
 thought no Line to have any Latitude, nor a Sine 
 any Quantity at all. 
 
 1 6. That an Angle in the Centre is the double of 
 an Angle in the Circumference (if they infift upon 
 equal Arches) is demonftrated by Eucl. El. 3. Prop. 20. 
 But it is to be obferv’d, that to the Effence of an 
 Angle in the Circumference, it is not required that 
 both the Sides fhould terminate in the Circum- 
 ference * for D BC is an Angle in the Circum- 
 ference, though the Point D be not in the Cir* 
 cumference. 
 
 CHAP, 
 
 t 
 
t 36 ‘Principles and ‘Problems Book III. 
 
 CHAP. VI. 
 
 Of the Proportion of the Perimeter to the Radius of 
 the Circle . 
 
 1. A Quadrantal Arch is equal to the Radius, 
 together with the Tangent of 30 Degrees. 
 
 Let the Radius be AB, the half Radius BE, or 
 AF, draw the Line then EF, the Re&angle ABEF, 
 will be the half of the Square from the Radius AB. 
 By the Radius AB let the Arch BG be delcribed, 
 cutting EF in G : The Line then AG being drawn, 
 and produced to B E produced in H, B H will be 
 the Tangent of 30 Degrees, and BG the third Part 
 of the Quadrantal Arch defcribed by AB ; then from 
 the Point G to the Side AB, let the Perpendicular 
 GI be drawn. In BA produced, take AN the 
 double of AI, lo that IN be the Triple of I A ; then 
 draw the Secant NG, ^cutting AF in T, and pro- 
 duce it to BH in q. 
 
 1 lay, that the ftrait Line Eq is equal to the 
 Arch B G. 
 
 Upon AT defcribe a Quadrantal Arch ST, be- 
 caule then the Triangles NIG, NAT are alike ; 
 AT will be two Thirds of the ftrait Line G I, the 
 half Radius ; that is, the third Part of the whole 
 Radius. 
 
 Becaule therefore AT is the third Part of the Ra- 
 dius, the Arch S T will be equal to the Arch B G. 
 Let B G be cut in two in the Middle in /, and A T 
 inM, and let the Perpendicular i h be drawn to AB, 
 
 and 
 
Book III. in GEOMETRY. 
 
 and AM will be the 6th Part of the Radius AB, 
 and the Arch YM equal to the Arch B i ; and as 
 N b is to N A, fb (hall b i be to A M : Draw then 
 and produce N M, it will paft thro' / • now let it 
 be produced to BH in Q. Let b i be produced to 
 the Concourfe with M q in t , and b t will be the 
 Double of b *, and equal to the Chord of the Arch 
 BG. For the right Sine of the Half of any Arch, 
 is the Half of the Chord of the double Arch. 
 Again, let the Arch Bi be bife&ed in e , and AM 
 in L ; and draw the Perpendicular ^to AB, and 
 e a will be the Half Chord of the Arch B /, and the 
 4th Part of the 2 Chords of the Arches B i and / G. 
 Now let A L alfo be the 4th Part of the ftrait Line 
 A T ; draw then and produce the Line N L, and 
 it will paft by e . Now let it be produced to B H 
 in P. Therefore the Arch KL is equal to the Arch 
 Be. In like manner, if M T be divided in X, 
 draw the Line NX, and produce it, it will cut the 
 Arch B G in 0, lo that 0 G is the 4th Part of the 
 Arch BG, and the Quadrantal Arch VX will be 
 equal to the Arch B 0 $ and fo perpetually. The 
 ftrait Lines then, which are drawn from the Point 
 N, will divide both the ftrait Lines AT, B <7, and 
 the Arch BG, into the fame Proportions. 
 
 If then the Bife&ion were continued as much as 
 may be, we would at length come to the Arch of a 
 Circle, left than the Latitude of the Line or Side 
 AB. Alfo according to Chap. 5. Numb. 9. the Pro- 
 portion of an Arch to the Chord always decreafeth. 
 Wherefore the Difference of the Arch to its own 
 Chord decreafeth alfo, until it become left than any 
 affignable Difference. 
 
 L 
 
 Let 
 
138 ‘Principles and ‘Problems Book III. 
 
 Let us then fuppofe the Latitude of the Radius 
 A B to be equal to the leaft Segment, or lefs than it 
 (for by bife&ing, it can never be reduced to no- 
 thing) the laft and leaft Segment will be in the ftrait 
 Line BH to and fo continually all the equal Seg- 
 ments of the Arch BG will lie extended toward the 
 ftrait Line B q ; for Hq is greater than bt , which 
 is equal to the Chord of the Arch B G. Let a e be 
 produced till it meet with N q in u, and a u will be 
 equal to 4 Chords of the Arch B <?, but lefs than the 
 whole B q. In like manner, if the Arch a e be bi- 
 fe&ed, and the right Sine of it drawn, that produced 
 till it meet the ftrait Line N q, it will be equal to 
 8 Chords of the Arch ae, but lefs than the ftrait 
 l ine B q ; and fo always, until one come to the laft 
 and leaft Segment, whole Longitude is equal to the 
 Latitude of the Radius AB. 
 
 Therefore, if the leaft Latitude be allowed to the 
 Side AB, the Arch BG, and ftrait Line B q will 
 be equal ; but if all Latitude be denied to a Line, 
 not only the leaft Part of an Arch will be greater 
 than its" Chord ; but alfo all its Farts to all the 
 others, fo that no Arch can be equal to a ftrait 
 Line. Neither can there be any Line drawn, nor 
 Sine, nor Arch, nor Figure. For a Line without 
 Latitude is nothing, and the Half of it as much. 
 
 Therefore B q is equal to the Arch B G, that is, 
 to the third Part of the Quadrantal Arch delcribed 
 by the Radius A B. 
 
 Now becaufe the Triangles GTF, EG q, are 
 Rectangular and alike, and T F the third Part of 
 the Half Radius A F ; E q will be the third Part of 
 EH (the Excefs of the Tangent BH above the 
 Half Radius BE). The Triple then of £ q (that 
 
 is. 
 
Pafte this on Margin ofP- 139, to f ° ld U P- 
 
 
Book IIL in GEOMETRY. i ^ 9 
 
 is, the Quadrantal Arch defcribed by the Radius 
 A B) is equal to the Triple of the Semiradius to- 
 gether with the Triple of £ q ; that is, to the Ra- 
 dius A B, together with the Tangent of 30 Degrees 
 that is, together with BH, which was to be de- 
 monftrated. 
 
 2. If B q were produced until it were tripled and 
 A F until it were doubled, it is manifeft that a ftrait 
 Line drawn from the Point N by the End of the 
 doubled AF, would cut off, in the ftrait Line BH 
 being produced, a ftrait Line equal to one, com- 
 pofed of the Radius and Tangent of 30 Decrees • 
 but that a ftrait Line drawn from the Point N* 
 palling through AD produced, fbould alfo propor- 
 tionally divide what remains of the whole Arch by 
 BG, is impoflible j becaufe if by the Point G the 
 Tangent were drawn to the Point G ; the ftrait 
 Line G A produced until it were equal to the ftrait 
 Line B N, would do the fame as BN • and therefore 
 ftrait Lines from the Point N, thro’ any Part of 
 the Arch, would not do it. 
 
 SCHOLIUM. 
 
 3. Becaufe I have lhewed in Book 2. (of Geometri- 
 ■al Roles') that a Quadrantal Arch is equal to a right 
 Line, whofe Square is ten times as great as a Square 
 rom the half Radius ; and have now made out that 
 he lame Arch is equal to a ftrait Line, made up 
 “the Radius and Tangent of 30 Degrees- In this 
 ’lace I will, from moft clear Principles, demonftrate 
 hat a Square from that ftrait Line compofed of 
 he Radius, and Tangent of 30 Degrees, is alfo the 
 Jecuple of a Square fipm the Semiradius. 
 
 v L 2 Delcribc 
 
140 Principles and Problems Book III. 
 
 Delcribe a Square ABCD, and let it be divided 
 4 Ways, as well by the right Lines E F, G H, as 
 the Diagonals AC, BD, concurring in the Centre 
 of the Square at m 3 and defcribe the Arch A C 
 cutting E F in 03 and let it be produced^ to jEe 
 
 Side BC in a. B a is then the Tangent of 30 De- 
 grees 3 let the Side B G be produced, and in it be- 
 
 Ed 
 
 C 
 
 & 
 
 S 
 
 
 Hi 
 
 c 
 
 l 
 
 
 
 j 
 
 f 
 
 / / / / 
 
 
 
 e. 
 
 
 w 
 
 
 
 H 
 
 
 i' 
 
 
 
 
 
 
 
 M 
 
 k 
 
 
 
 
 
 
 M 1 X 
 
 z 
 
 ing produced, let 
 Cl equal to the half 
 Side BE, and CL 
 equal to the Tan- 
 gent Ba , betaken. 
 Then defcribe a 
 Square from the 
 whole BL, which 
 is BLZM, which 
 is to be demon- 
 ftrated to be equal 
 to ten Squares from 
 BE or CL 
 The Square from 
 B E is B E m G, the 
 
 Square from the Tangent B a, is B abS, to which 
 let CLYH£ be equal. 
 
 Defcribe alfo a Square from B I, which fhall be 
 B I eg, and let CD, E F be produced to the Side 
 M Z cutting g c in K and k. Let a b aho be pro- 
 duced to the Side MZ in X, and SH to LZ in Y. 
 
 Underhand now, that the 4 Sides of the Square 
 ABCD, are each divided into 12 equal Parts. So it 
 will be that the Square ABCD does contain 144 
 equal little Squares. 
 
 Becaufe the Side BC is 12, the Side BI is 18, the 
 
 Square of the former is 144, of this 324, the Square 
 ^ from 
 
Book III. in GEOMETRY. 141 
 
 from 19 is 361, the Square from 20 is 400 ; and 
 thefe fquare Numbers are in Order next to one ano- 
 ther. Wherefore (according to Chap. 4. Numb. 3.) 
 an Unity being taken from the mean Square 361, 
 the remaining 360 is the mean Proportional betwixt 
 400 and 324, that is, the Root of the fquare Num- 
 ber from 400 multiplied into 324. 
 
 Again, Becaufe BC is 12, the half Side BE is 6, 
 which fquared is 36. Now the Square from 7 is 49, 
 the Square from 8 is 64 ; and theie Squares are next 
 to one another in Order. Wherefore (according. to 
 the lame Number of the faid 4th Chapter ) take gn 
 Unity from the mean Square 49, there will remain 
 48, the mean Proportional betwixt 64 and 3 6 5 or 
 the Root of the Number 64 multiplied into 36. 
 Now 48 is the Square of the Tangent of 30 De- 
 grees in the Square 144, becaufe then 360 is the 
 Decuple of the Square from the half Side BE, it is 
 to the Square 144, as 10 to 4 - To the Tangent of 
 30 Degrees, in the Square 360, will be to the 
 Square of the Tangent dlfo, as 10 to 4, but the 
 Tangent of 30 Degrees, in the Square 360, to wit, 
 the 3d Part of it, is 120, which Number is to 48, 
 the Square of the Tangent in the Square 144, as 10 
 to 4. Therefore 360 is the Square from the right 
 Line compofed of the Radius and Tangent of 30 
 Degrees, which right compounded Line is BL, 
 which was to be demonftrated. 
 
 4. In the right Line I c, take I e equal to the ftrait 
 Line Cl, which being joined to BE, the Arch of 
 the Circle delcribed by the Radius Be will pals by 
 L • for B I hath the Power of 9 Squares, whereof 
 I e hath the Power of 1 3 therefore Be hath the 
 
 L 3 Power 
 
\^z ‘Principles and ‘Problems Book III. 
 
 Power of i o half Sides, the Arch therefore defcribed 
 by it will pals by L. 
 
 5. We mull not here pafs over a Difficulty, which 
 in the 23d Chapter of the Book of Principles , I did 
 indeed mark, but thought fit to leave the Solution 
 of it to the Readers : The Solution of this will much 
 conduce to the illuftrating and facilitating of Geo- 
 metry. This is the Difficulty : The Square CLYH 
 
 (which is the Square 
 of the Tangent of 30 
 Degrees, and equal 
 by Conftrudtion to 
 B a h S ) is to the 
 _ Square C\eh^ as 4 
 to 3, as is commonly 
 known : Becaule 
 
 therefore the Gno- 
 mon I L Z c M ought 
 to be equal to the 
 Triple of the Gno- 
 mon ILYtfZ?H,and 
 fo that Gnomon be- 
 ing tripled and 
 placed upon this, ought to agree with it, but it 
 agreeth not, for it is lels. Hence it may feem con- 
 clufive, that the Side of the Square, which is the 
 Decuple of the Square from the half Side, is greater 
 than the Square from B L 3 but why ? becaufe it 
 is kfs by fo much as the double little Square m 
 or c Z, or e Y. 
 
 For fuppofe the right Line BC to be divided into 
 12 equal Parts, the whole Square A BCD will be 
 144 of thole Parts j the Square CIYH 48, the 
 Square alio AX 48, and the Square DZ 48, all 
 
 thefe 
 
Book III. in GEOMETRY. 143 
 
 thefe make 288. Now feeing the Square from the 
 half Side C I is 36, the whole Square (which is 10 
 times as much) ought to be 360 ; 288 being then 
 taken from the Remainder 72 ought to be 
 equal to the 2 equal Rectangles X D and D Y, and 
 either of them ought to be 36, but it is not fo. 
 The Square FDK.& is indeed equal to the Square 
 from BE or Cl (that is 36) from which if the Rect- 
 angle F l be taken, and the ReCtangie X K added 
 (feeing that is greater than this by the Quantity of 
 the Square k X) the remaining ReCtangle XD will 
 be lels than the Square FK, that is, than 36, as 
 much cis the Scjusvc or V'/Zj ^ for the fnrne 
 Caufe the Redtangle D Y is alfo lefs than 36 by the 
 Quantity of the Square cYorc Z. Therefore the 
 whole Square BLZM is lefs than the Decuple of 
 the Square from the half Side, which is 36, by the 
 Quantity of the double little Square c Z. 
 
 What can be anfwered to thefe Things ? nothing 
 truly, if there be any Ground to find fault with the 
 former Demonftration ; but it is wholly Arithmeti- 
 cal, wherein it is not fo eafy to be miftaken. 
 
 1 will therefore firft fhew the Objection to be of 
 no Force 5 then, whence it is that that Difference of 
 the double little Square cz doth arife. 
 
 That it is of no Force, is manifeft from hence 
 that by the fame Argument it may be proved (it be- 
 ing fuppofed, that the right Line B C is divided into 
 12 equal Parts) that the Square of it alfo is lefs than 
 i 44 j by the Quantity of the fame double Square 
 » ' F ° r leein S Square C m is 36, the right 
 Angle CabH will be lefs than it by the Quantity 
 of the Square b m or c Z. For from the Square C m 
 on the one Side, the Complement E b is taken, on 
 
 L 4 the 
 
144 ‘Principles and Problems Book Ilf. 
 
 the other b h ; but E b is greater than b h by the 
 Square bm. Therefore the Redangle Cabi \ -is 
 leis than 3 6, by the fame little Square cZ ; for the 
 fame Reafon the Redangle A b is lefs than 36, by 
 the fame little Square c z 5 there remains for com* 
 pleating the whole Square, the Square b D, which 
 is the Half of the Square from the Tangent B a. 
 For join a s cutting the Diagonal in p , it will be as 
 S a, or B b to B a 9 fo B a to b p 9 that is, to H b; 
 a b is then the mean Proportional betwixt B b and 
 b H 5 therefore the Square b D is the Half of the 
 
 Square B a b S, that 
 
 
 Ed 
 
 is 24 • therefore the 
 whole Square 144 is 
 equal to 48 + 24+ 
 36 + 36, lefs the 
 double Square c Z, 
 which is abfurd. For 
 it is not to be doubt- 
 ed, but that the 
 Square A B C D is 
 144. Therefore this 
 Objedion is of no 
 Force, and has this 
 v y * Fault befides, that 
 
 the Square from BL 
 is/ by it made lefs than 360, not by the Double, 
 but the Quadruple of the Square c Z. 
 
 The Caufe of this Difagreement is therefore to 
 be enquired into, and it can be no other, but the 
 reckoning of Lines without Latitude in the Propor- 
 tions of Superficies ; whence it follows neceffarily, 
 that that which is contained within the 4 Sides B L, 
 LZ, ZM, MB, is leis than i o Squares from B E, 
 
 by 
 
 
 
 
 
 /f / / 
 
 
 
 c - 
 
 V 
 
 
 
 H 
 
 J? 
 
 
 
 \ 
 
 J? 
 
 
 N 
 
 K \ 
 
 — n l 
 
Book III. in GEOMETRY. 145 
 
 by the Double of c Z ; but that the Sides thcmfelves 
 are equal to the Quadruple of c Z. Therefore that 
 the Square from B L is the Decuple of the Square 
 from the half Side, Bands good, and fhall remain lo ; 
 for it cannot be proved by any Argument, that the 
 Square from B L is lefs than the decupled Square 
 from BE, but that by the lame it will be proved, 
 that the Square from BC is lefs than 4 Squares of 
 the fame BE. We have then, by the Solution of 
 this Difficulty, removed one, and perhaps the great- 
 eft Hindrance to Geometry, to wit, the Computa- 
 tion of Superficies by Lines without Latitude. 
 
 CHAP. 
 
146 ‘Principles and ‘Problems Book IIF. 
 
 (Jj J _.\ • ! t > «•* • -• .• '• >i •>*' * . ’O . • * 
 
 CHAP. VII. 
 
 Of mean E roper tionals. 
 
 i.YTOW to find a mean Proportional betwixt 
 1 J 2 right Lines given, is taught by Euclid's 
 
 El 6. Prop. 13. 
 
 -» 1 «■ •< 
 
 2l To find out 2 mean Proportionals betwixt a 
 right Line given and its Half- let DC be the 
 right Line given, and its Half C G s let them be 
 
 F 
 
 difpofed at a right Angle in C ; then make a Square 
 D A B C, and let it be cut as well by the Diagonals 
 AC, BD, as by the right Lines gf qp, meeting 
 4 Ways in H. 
 
 Take 
 
Book III. in GEOMETRY. 147 
 
 Take the mean Proportional betwixt BC and 
 its Half C G ; now that is equal to the Semidiago- 
 nal HC, by the Radius HC deforibe the Arch 
 C m , cutting q p produced to in, and let the Arch 
 C 771 be cut into 3 equal Parts in k and /, and draw 
 H H / cutting DC in n and 0. Now let C* 
 equal to C n be taken • draw xa parallel and equal 
 to C p j cutting PI C in e, Ce then is a Square. In 
 like Manner by the Point 0, draw a Parallel to the 
 Side B C, cutting H C in b, and by b a right Line 
 be cutting H p in c , and then both eb and H b 
 will be Squares $ for they are Rcdtangular upon 
 the Diagonal of the Square H C. And the Squares 
 Ce, eb, b H are then continual Proportionals, and 
 therefore their Sides alfo (which, are equal to the 
 right Lines C n, n o’, op, each to each) will be 
 continual Proportionals. Now p C, the Height of 
 them all together, is equal to CG or Dpi then 
 DC, D n, Do, Dp, are continual Proportionals. 
 Now becaufe as Cp is to np, fo np to op • it will 
 be alfo, that as C p more C p (that is, the whole 
 DC) is to C p more n p ( that is, D n ) fo D n 
 to Do, and fo D0 to Dp. Then are D 11 and 
 D 0 two mean Proportionals betwixt D C and 
 D p, the Half of the fame D C, which was to be 
 found out. 
 
 3. If then in the right Line C B, there be put 
 C E equal to D n, and C F equal to D 0 be put in 
 D C produced, and F G be joined, and the right 
 Lines DE, EF drawn, the Triangles DEC, EC F, 
 CFG will be Triangles of equal Angles, that is, 
 the Arches fr, tv, zy ( having equal Shanks ) 
 will alfo be equal. 
 
 4. How 
 
By the Radius I B defcribe the Arch B e , cutting 
 IK produced to e - and let it be cut into ,5 equal 
 Parts by the ftrait Lines la, I b, lc 9 Id cutting 
 
 B K 
 
 ‘Principles and Problems Book III. 
 
 4. How to find 4 mean Proportionals betwixt a 
 right Line given, and its Half 
 
 Let the right Line given (in the 2d Figure) be 
 DC, and the Half of it CG * make a Square D C 
 BA, the 4th Part of which is the Square IB. So 
 then the ftrait Line I B is the mean Proportional 
 betwixt the whole BC and its Half CG. 
 
Book III. in GEOMETRY. 149 
 
 B K in E, k, /, m, and let B n equal to B E be taken. 
 The Lines nf, E f being then drawn, the one pa- 
 rallel to B E," and the other to B n, they will meet 
 in the diagonal Line IB at /, and B/ will be^a 
 Square. In the fame Manner, if B n be lengthen’d 
 to o, that it be equal to B£ ; right Lines drawn by 0 
 and k parallel to the oppofite Sides, will meet in 
 the Diagonal IB atg, and therefore fg will be a 
 Square, and fo of the reft. Let then upon I B be 
 made 5 Squares in continual Proportion B f, fg, 
 gh, h i, i I, whofe Sides are equal to the right 
 Lines BE, E k, k l, Im, k I, theft therefore will 
 be likewile continual Proportionals ; putting then 
 together, as KB more KB, that is, the whole CB, 
 is to C K more KE; fo will C K more K E be to 
 C K more K k, and fo C K more K k to C K more 
 K /, tic. Then are CB, C £,' C k. Cl, Cm, 
 C K, 6 right Lines continuoufly proportional; 
 amongft which C E, C k. Cl, C m intervene. 
 Therefore 4 mean Proportionals are found out be- 
 twixt the right Line given and its Half, which was 
 to be done. 
 
 5. Coniectary. If in the right Line D C pro- 
 duced, C F equal to C k be put ; and C m equal to 
 C l be put in B C produced, and in C D there be 
 put CH equal to C m : And laftly, if H K, H M, 
 M F, F E, E D be joined, there will be made 5 
 equiangled Triangles, DEC, C E F, F M C, 
 MHC, HKC. 
 
 6. How to find 2 mean Proportionals between 
 2 ftrait Lines whatfoever given. 
 
 Let 
 
150 ‘Principles and ‘Problems Book III. 
 
 Let (in this 3 Fig.) B C the greater, and C D 
 the lelfer, be given. Let them be difpoled at right 
 Angles in C, and let B C be equal to A C, let A D, 
 A B be joined, and C c equal to the right Line B C 
 be taken, and A c joined. 
 
 H 
 
 From the Point D, draw down D E, making right 
 Angles at D, and cutting A c in E : From the Cen- 
 tre E at the Diftance E D, defcribe an Arch of a Cir- 
 cle D /, ending in A c at/; let the Arch D/be cut 
 into 3 Parts in d and e, and the Lines Ei, E c being 
 drawn, produce them to T>c in a and b. According 
 then to the 2 Art. of this Chap, cb y b a D, will be 
 continual Proportionals ; and (by El. 6. Prop. 1.) the 
 Triangles A CD, ADc, wi/1 be as the Bales CD, 
 D c ; and the Triangles ACc,EDc alike ; and the 
 Triangle AC c, will be to the Triangle E be in the 
 double Proportion of C c to D c ; therefore C c, C b y 
 Ca y CD, will be continual Proportionals. 
 
 7. Con- 
 
Book III. in GEOMETRY. 
 
 151 
 
 7. Confe&ary. If then in the right Line C B, 
 there be put C G equal to Cb , and in A C produced 
 CH equal to Ca be put j HD being drawn, and 
 AG, GH joined^ the Triangles ACG, GCH, 
 HCD, will have equal Angles. 
 
 8. In like manner, if CD were put in the Side 
 CB, and the Arch D/ cut in five Parts, between 
 A C and C D, four mean Proportionals would 
 be found • but if C D were put in A C produced, 
 and the Angle DEf cut in feven Parts, fix mean 
 Proportionals would be found betwixt the fame 
 Extreams. 
 
 9. The natural Caufe of this Truth, from whence 
 arileth a moft evident Demonftration, is this ; Sup- 
 pofe^that the right Lines AC, BC had been length- 
 ned ’till they had been doubled, and the right Lines, 
 which join their Ends, had been drawn, thefe would 
 be fo many diagonal Lines, and would make 3 like 
 equal Triangles. From whence it will follow, that 
 thofe four Sides would have been continual Propor- 
 tionals upon the Account of Equality, of which A C 
 is the 1 ft, C c the 4th. Therefore that 2 Means 
 may be found between A C and C D (becaufe C D 
 is left than C c by the Quantity of D c) feeing the 
 Difference of D c arifes from 3 Mul<fts or Diminu- 
 tions of the Side B C, to wit, c b a D, it is 
 neceflary that the Difference D c be divided into 3 
 Parts, c b aT> continual Proportionals. And 
 becaufe the Angle C A B is half a right Angle, the 
 Angle alio D E c (from which Point E the Divifion 
 of the right Line Dr into 3 Parts, being continual 
 
 Proportionals, 
 
152 'Principles and Problems Book III. 
 
 Proportionals, taketh its Rife) ought likewife to be 
 half a right Angle. 
 
 Now, feeing that that Divifion into 3, is done 
 by a Trifedtion of a half right Angle, and no other- 
 wife ; and in taking 4 Means by a Quinquifedtion 
 *or fivefold Cutting, and no otherwife , alfo in taking 
 6 proportional Means by Septife&ion, or a feven- 
 fold Cutting, and no otherwife ; and fo forward : 
 no other Method will ferve to find out Means that 
 are even in Number, as 2, 4, 6, 8, £ 3 c. except the 
 Trifedtion, Quinquifedtion, Septifedtion, &c. of a 
 half right Angle. 
 
 H 
 
 CHAP- 
 
Book III. in GEOMETRY. 
 
 M3 
 
 CHAP VIII. 
 
 Of the "Proportion of a Square, to the Quadrant 
 of a Circle infer ibed in it. 
 
 i. T E T the Square ABCD be divided, not only 
 !_j by the right Lines, EF, GH, but alfo by 
 the Diagonals A C, B D meeting fourfbldly at I, in 
 the Centre of the Square, and cutting the Arches AC, 
 BD in L and M ; and deferibe from the Centres C 
 and D, the Quadrantal Arches A C, B D mutually 
 cutting one another in the right Line E F at K. 
 
 I fay, that the Square ABCD is to the Qua- 
 drant D A C, as 5 to 4. 
 
 Though I have 
 demonftrated this 
 in another Place, 
 yet for the fake of 
 thole, who cannot 
 fo eafily go along 
 in their Thoughts, 
 with long and dif- 
 ficult Demonftra- 
 tions, I (hall in 
 this Place demon- 
 ftrate the fame by 
 a fhorter and eafier 
 Method. By the Radius D H, deferibe the Arch 
 of a Quadrant H F, cutting the Arch BA C in m 
 and n, and the Diagonal BD in j and r j the Qua- 
 drant then DHF, is the fourth Part of the Qua- 
 
 M drant 
 
154 ‘Principles and Problems Book III. 
 
 drant DC A, and the Space CLF^H, is three 
 Fourths of the fame. Now the fame Space C LF j H, 
 is three Fourths of the Quadrant B A r C $ taking 
 then that common Space CLF s H from both the Qau- 
 drants, there will remain on the one Fart the Qua- 
 drant DHF, on the other Part the Triline or 3 Lines 
 DArC lefs than the Biline or two Lines m 0, more 
 than the two three Lines C H A F that is, the 
 fame three Lines D A r C, and the Quadrant DHF 
 equal to one another ; and therefore the Space 
 B A F s H C, will be equal to the Quadrant B A r C. 
 For that Space confifts of the Triple of the Square 
 DI, and the fourth Part of the three Lines ALCB, 
 that is, of three Fourths of the Quadrant, and four 
 Fourths of the Triline ALCB. Seeing then the 
 three Lines ALCB, are equal to the fourth Part of a 
 Quadrant ; the whole Square A BCD will be to 
 the whole Quadrant D AC, as 5 to 4, which was 
 to be demonftrated. 
 
 2. Hence it is manifeft, that the Biline or two 
 Lines AICLA, is to the Triline or three Lines 
 ALCB, as 3 to 2 j for feeing the whole Square is 
 five, the Triangle ABC will be of which the 
 Triline ALCB is 1 ; wherefore the Biline AICLA 
 is 1 Therefore the Proportion of the faid Biline 
 to the aforefaid Triline, is as 3 to 2. 
 
 3. It is alfo plain, that the Biline m n is equal to 
 the two Trlines C H AF m. 
 
 4. It is alfo clear, that if a right Line were drawn 
 parallel to the Side B C by the Point P, in which the 
 ftrait Line DE cuts the Arch CL, and ends in 
 the Side DC at Y, and in the Diagonal B D at Q 
 The Square from Y Q would be equal to the Qua- 
 drant 
 
Book III. in GEOMETRY. 155 
 
 drant D AC, fee- 
 ing the Square 
 A B C D to the 
 Square from Y 
 is as 5 to 4 ; and 
 the Trilines C YP, 
 
 PQL equal. 
 
 5. It is alio 
 manifest, that the 
 half of the Square 
 ABCD, to wit, 
 the Redangle AE, 
 the Figure AKF 
 being taken away, 
 is the fifth Part of the Square ABCD: For that 
 which is contained within’ the three Lines ALK, 
 KB, and BA, together wuth the Triline B E K, is 
 equal to the Inline ALCB, that is, the fifth Part 
 of the Square ABCD 3 becaufe BEK, C E K are 
 equal, the fame alfo is true of the Redangle D E : 
 Whence it follows, that the two Figures F K D, 
 FKA taken together, are equal to three Fifths of 
 the Square ABCD, that is, to three Fourths of 
 the Quadrant DAC 
 
 Now feeing the Knowledge of thefe Things is of 
 it felf of no great ufe, I might have paffed them 
 over • but that, as I had begun to handle Cyclo- 
 metry, I was willing alfo to perfed it. IJkewife 
 becaufe to the Fulnefs of Cyclometry, it belongs to 
 know alfo the Quantities of its Parts, that is, of 
 its Segments or Angles 3 we mud now treat of 
 thefe. 
 
 6. Deforibe again (in the focond Figure) a Square 
 A BCD, divided as in the firft Figure. 
 
 M 21 
 
 To 
 
1 5 <5 ‘Principles and 'Problems Book III. 
 
 To the Arch C P, let an equal L V be taken, and 
 to the Arch LK an equal C k ; becaufe then LK is 
 the third Part of the Arch LC; LK will be io, K k 
 io, and kC 10, of which LC is 30. Again, becaufe 
 CP, LV are made equal, the Arch CL being 
 divided by the ftrait Line D e in two in the Middle, 
 the Arch PV will be alfb divided in two in the Mid- 
 dle, in e. Both then the Arch C e, and the Arch L e, 
 are 15 apiece : Wherefore the Arch V as well 
 as the Arch P e is 2 f , and the Arch P V 5, and 
 the Arch K V 7 i , the Arch L V then, as well as 
 the Arch PC, is 1 7 if, therefore the Arch C V or 
 KP is 12* From the ftrait Line ED, take E a 
 equal to the half Side E C, and the Remainder D & 
 (according to the 13 E.Prop. 1.) will be the greater 
 Segment of the Side DC, or of the ftrait Line 
 D P divided in extream and mean Proportion. At 
 the Diftance D <2, defcribe an Arch a b cutting the 
 Arch D B in b ; and let D b be produced to the 
 Arch C L in a? ; the Arch C *v then will be 1 2, of 
 which CL is 30. For (by El. 14. Prop. 9.) the 
 ftrait Line D b fiibtends the tenth Part of the whole 
 Perimeter, that is a fifth Part of the Semiperimeter, 
 that is, two Fifths of the Arch DB. The Arch 
 then T> b is two Fifths of the Arch BD j and be- 
 caufe the Angle in the Centre, to wit, D A b is 
 the double of the Angle in the Circumference, to 
 wit, of the Angle CDo), the Arch C v will be two 
 Fifths of the Arch CL: Therefore feeing CL is 30, 
 C <v will be 12 ; now becaufe the Arches L V, CP 
 are equal, CV will be 12 f, of which LC is 30: 
 Wherefore as well L V as C P fhall be 17 i and P e 
 or e V 2 1 , and P V or K e 5. 
 
 If 
 
Book III. m GEOMETRY. 
 
 »57 
 
 If we fuppofe the Arch CL to be 45-, yet the 
 Proportion of the Angles will be found to be the fame 
 For LK, Kk, 
 k C, will be each 
 of them 15, and 
 L e 5 e C each 
 22 and K e 
 7 f 5 L P 18 
 wherefore C P 
 will be 26 and 
 LV as much, and 
 C V 18 4. Now 
 18 i more 26 % 
 make 45 ; but the 
 fame is the Pro- 
 portion of 10 to 
 5, as of 15 to 7 y. It Hands then good, that the 
 Angle EDCj to the half right Angle LDC, is as 
 17 \ to 30, and to L P as 17 = to 12 r, or as 7 to 
 Si and to L K (which is 10.) as 7 to 4. We have 
 then the Quantity of the Angle EDC, that was 
 not known before. 
 
 M 3 
 
 CHAP. 
 
5 3 Principles and Problems Book III. 
 
 C H A P. IX. 
 
 Of Solids and their Superficies. 
 
 A Cube to a Cylinder infcribed in it, is as 5 
 to 4. 
 
 1. Let there be a Cube, whole Bale is the 
 Square ABCDj within this Square let the Circle 
 G E H F be infcribed. According then to the fore- 
 going Chap, the Square ABCD is to the Circle 
 GBHF, as 5 to 4. Upon all the Points of the 
 Compafs, both of the Square A B C D, and Circle 
 G E H F, fuppofe Perpendicular Lines ere&ed, the 
 Height of every one of which is equal to the 
 Height, of EF. So a Cube 
 Will be defcribed, toge- 
 ther with a Cylinder in- 
 icribed in it, arid the Bale 
 of the Cube will be to the 
 Bafe of the Cylinder, as 
 5 to 4. Now the Plane 
 that cuts the Cylinder and 
 Cube Parallel- ways to the 
 Bale, will every where 
 make a Section of the 
 Square, to the Section of the inlcribed Circle, as 
 5 to 4. Therefore as the Square ABCD to the 
 Circle GEHF (that is, as 5 to 4) fo (according 
 to EL 6. Prop. 1.) all the Sections of the Cube to- 
 gether (that is, the Cube) will be (to all the Secti- 
 ons of the Cylinder together, that is, to the Cy- 
 linder) as 5 to 4. 
 
 Superficie of a Cube, is to the Superficie 
 
 of 
 
Book III. in GEOMETRY. 159 
 
 of a Cylinder inlcribed in it, as 5 to 4 : For the 
 Superficie of a Cube is equal to fix Squares from the 
 Side ot Height of the Cube. Now Archimedes , in 
 his Book of a Sphere and Cylinder , hath demon- 
 ftrated, that the Superficie of a Cylinder without 
 the Bales, is equal to the four greateft Circles in a 
 Sphere, and with the Bafes, to fix Circles. The 
 whole Superficies then of a Cylinder, is equal to the fix 
 greateft Circles in a Sphere : Therefore the Superficie 
 of a Cube, to the whole Superficie of a Cylinder, 
 is as fix Squares to fix greateft Circles, that is, as 
 one Square to one Circle inlcribed in it, that is, 
 as 5 to 4. 
 
 3. A Cube to a Sphere infer ibed in it, is as 15 
 to 8 : For a Cube to a Cylinder (as is already 
 Ihewed) is as 5 to 4. Now a Cylinder to a Sphere 
 (as Archimedes hath demonftrated in the firft Book 
 of a Sphere and Cylinder) is as 3 to 2 : Therefore 
 the Proportion of a Cube to a Sphere, is made up 
 of the Proportions of 5 to 4, and 3 to 2; be it 
 then as 3 to 2, fo 4 to another- now that lhall be 
 2 f. The Proportion then of 5 to 2 f, is made 
 up of the Proportions of a Cube to a Cylinder, 
 and of a Cylinder to a Sphere : Therefore a Cube 
 to a Sphere is as 5 to 2 f, that ig, (both being mul- 
 tiplied by three) as 15 to 8. 
 
 4. A Sphere is equal to the half Cylinder in which 
 it is inlcribed, together with the half of the Cone 
 which is inlcribed in the fame Cylinder : For feeing 
 it is demonftrated by Archimedes , that a Cylinder 
 is to a Sphere inferibed in it, as 3 to 2, and to the 
 inferibed Cone, as 3 to 1, a Cylinder, Sphere, and 
 Cone will be as 3, 2 and 1. But 2 (that is, a 
 Sphere) is equal to the Half of the Aggregate of 
 
 M 4 3 and 
 
1 60 'Principles and Problems Book III. 
 
 3 and 1 3 (that is, to the hnlf of the Cylinder, and 
 half of the Cone put together) as is propofed. 
 
 5. Archimedes alfo, in his fecond Book of the 
 Sphere and Cylinder ,- Prep. 1. fhews how a Sphere 
 equal to a Cylinder may be found, to wit, that 
 a Cylinder be taken, which to the Cylinder pro- 
 pofed is as 3 to 2, that is, as the propofed Cylin- 
 der to a Sphere; then that two mean Proportionals 
 be found betwixt the Height of the taken Cylinder, 
 and the Diameter of the propofed Sphere : But to 
 find out two mean Proportionals betwixt two right 
 Lines, was not then found out. How that is to be 
 done, I have demonfirated in the feventh Chapter 
 of this Treatife. 
 
 6. The Superficie of a Sphere is equal to the four 
 greateft Circles in the fame Sphere. 
 
 For if the Arch of the Semicircle E F be turned 
 round upon the Axis E F, it will be the Superficies 
 of the Sphere. From the 
 Point F, let theftrait Line 
 F I be how you will drawn 
 to the Circumference, and 
 from the Point I, draw the 
 right Line I K parallel to 
 the Side BC, cutting the 
 Circumference in K, and 
 the Axis in L ; the Angles 
 at L will be right ; and as 
 F L to L I, fo will L I be 
 to LE, and as F I is to F L, fo will E X be to EL, 
 becaufe of the like Triangles FLI, ELI. Now 
 v/hiift the Semicircle turns upon the Axis E F, let 
 the Circle I K be deferibed from the Point I in the 
 Superficie of the Sphere, and it will fo fall out in 
 
 wbatfo- 
 
Book III. in GEOMETRY. 161 
 
 whatfoever Point of the Circumference E F the Point 
 I be placed ; but as F E is to L E, fo is the Circle 
 upon E F to the Circle upon LE ; for Circles are in 
 the double Proportion of Rays or Radius’s. 
 
 Becaufe then all the Perimeters IK make the Super- 
 fine of the Sphere, and all the right Lines I K make 
 the Circle GEHF ; the Superficie of the Sphere will 
 be to the Circle GEHF in the double Proportion of 
 the Axis E F to the Radius of the Circle GEHF the 
 half of the lame EF : But the Circle delcribed on the 
 Axis EF, is alio to the Circle GEHF in the double 
 Proportion of the Axis EF, to the Semidiameter of 
 the Circle GEHF ; therefore the Superficie of the 
 Sphere is equal to the Circle, whofe Semidiameter is 
 the Axis E F. Now the Circle, whofe Radius is the 
 Axis of the Sphere, is the Quadruple of the Circle 
 GEHF : wherefore the Superficie of the Sphere is 
 alfo the Quadruple of the fame. 
 
 7. Hence it follows, that the Superficie of any 
 Segment, or Portion of a Sphere, is to the Superficie 
 of the remaining Portion, as the Portion of the 
 Axis cut off by I K is to the remaining Portion ; to 
 wit, the Superficie of the Portion FIK, is to the Su- 
 perficie of the Portion EIK, as the Portion of the 
 Axis FL, to the remaining Portion EL. 
 
 8. Laftly, Becaule Archimedes hath demonftrated 
 that the Convex Superficie of a Cylinder is theQua- 
 druple of the Circle, which is the Bafe of theCylinder 
 it follows, that it is the fame thing, as to the Quanti- 
 ty of a Cylindcrical Superficie, whether it be made by 
 the Revolution of the Perimeter of the Bafis about 
 the Axis, or by the Motion of the Bafe by the Sides 
 pf the Cylinder, or by the circular Motion of the 
 Side. For all thefe beget equal Quantities of Super- 
 ficie i 
 
1 6 1 ‘Principles and 'Problems Book III. 
 
 fide ; neither is it hard to be underftood, even with- 
 out a Demonftration, that there is no Difference in 
 making a Superficie, whether the Cirde that is the 
 Bafe of the Cylinder be carried ftrait by the Side, 
 or whether in every Point of the Side Perimeters 
 be feverally defcribed, or whether from the whole 
 Side one Perimeter be defciibed. Now the Sup r- 
 ficie of a Cylinder (that is, the thin Coat of the 
 Cylinder ftript off, its Skin being diiplayed and ex- 
 tended in a Plane) becomes a Triangle, of which 
 one Side is the Diameter of the Bale, the other a 
 ftrait Line, whofe Square is equal to ten Squares 
 from the Diameter of its own Bafis. 
 
 CHAP. X. 
 
 Of a new Method of treating of Solids, and their Su- 
 perficies, by the efficient Caufes. 
 
 LEMMA. 
 
 A N Agent working uniformly in a given Time, 
 for perfecting a defigned Work, if in fome 
 Parts of that Time he work, and in fome others 
 reft ; the Work that is done, will be to the Work 
 left undone, as the Time wherein he worketh, to 
 the Time wherein he refteth. For Example, If a 
 given Piece of Land may, by continued Labour, be 
 •wholly ploughed in three Days; then, if it be only 
 ploughed two Days, and one Day fpent idly ; that 
 which fhall be ploughed, will be to that which re- 
 mains unploughed, as 2 to i ; and what was defigned 
 to be ploughed, will be to what is ploughed, as 3 
 to 2, and to then nploughed,as 3 to x. In like mam 
 
 ner, 
 
Book III. in GEOMETRY. 1 6$ 
 
 ner, how much an Agent, by continued and uniform 
 Labour can do in any time, a double Agent, by the 
 like Labour, can do as much, if one half of the Time 
 be ipent in Work, and the other half idly. 
 
 Moreover, how much the Agent lofeth in Time 
 of working, lb much it muft be thought he relied ; 
 ; and thefe are manifeft by the Light of Nature. 
 
 i. 1 he Convex Superficie of a ftrait Cylinder is 
 equal to the Superficie of a ftrait Cone, having the 
 lame Bale with the Cylinder, but double the Height. 
 
 Here (Fig. i.) take the Cylinder A BCD, fuch 
 as is propofed, whofe Bafe is the Circle BC. Now 
 A BCD is a Square- let the Square A B C D be di- 
 vided in two in the Middle by the right Line G H, 
 parallel to the Side AB, and length- 
 en GHtoK; let then G K be the 
 Double of GH. Join BK, KC, 
 and B H, C H, then the Square 
 A BCD, and the Triangle BKC 
 have the fame Bale, but the Height 
 of the Triangle BKC, is double 
 the Height of the Square A B C D. 
 Suppofe the right Angle DG to 
 turn round about the Axis GH, and 
 by that Circumvolution from the 
 Rectangle D G a right Cylinder will 
 be defcribed j but from the Triangles CHG, CKG, 
 two right Cones, and from the Side DC, the Super- 
 ficie of the Cylinder, and from C H, C K, the Sides 
 of the Cones, two conical Superficies, and from the 
 Point C, the Circumference of the Bafe. Thele 
 Things luppofed, we are to demonftrate, that the 
 conical Superficie of the Cone BKC, is equal to the 
 whole Superficie of the Cylinder A B C D. 
 
 Draw 
 
1 6\ ‘Principles and Problems Book HI. 
 
 Draw the right Line a b as you pleafe, but parallel 
 to the Side BC, cutting the ftrait Lines BH, CHin 
 a and b. The Proportion then of B C to a b^ will 
 be the fame with the Proportion of H a to HBj 
 and becaufe a b is placed any where, the fame will 
 the Proportion be every where; and becaufe the right 
 Line B C by an uniform Motion, and to right Angles 
 by the Sides BA, CD, will compleat both the Cy- 
 linder A B C D, and its whole Superficie ; but the 
 Motion failing according to the Proportions ofTimes, 
 they will defcribe the Triangle BHC; the right Line 
 BC refts as much from working, as it worketh. The 
 “Superficie BHC will then be equal to the two Super- 
 fi cies BAHjCDH, that are not made (according 
 to the foregoing Lemma.) Now becaufe the Diame- 
 ters are in the fame Proportion one to another, as the 
 Perimeters ; the Perimeters alfo will be deficient in 
 the fame Proportion with the Times. The Perime- 
 ters therefore of the Circles, which conftitute the Su- 
 perficie BHK, that is, the Superficie of the Cone it 
 felf B H C, is equal to the Half of the Superficie of 
 the whole Cylinder. But the Triangle B K C is the 
 double of the Triangle BHC, and the conical Su- 
 perficie of the Cone B K C, is the double of the co- 
 nical Superficie of the Cone BHC : Therefore the 
 Superficie of the Cone BKC, is equal to the Convex 
 Superficie of the Cylinder A B C D, which is the 
 Thing propofed. 
 
 2. The Convex Superficie of a Cylinder, is the 
 Quadruple of the Bafe of the fame Cylinder ; for fee* 
 ing the convex Superficie of a Cylinder is made by 
 the Perimeter of the Circle, which is the Bafe of the 
 Cylinder working uniformly; if a Perimeter, which 
 is the double of the Perimeter of the Bafe, fhould 
 
 work 
 
Book III. in GEOMETRY. 165 
 
 work one half of the fame time, and reft another hal£ 
 it will perform as much (according to the foregoing 
 Lemma.) But the Perimeter of the Circle, whofe 
 Radius is H G, is the double of the Perimeter of the 
 Bafe BC ; and the Circle from H G is made by the 
 Radius HG equal to B C, by a circular Motion, that 
 is, the Radius B C ceafing as much as it operateth, 
 to wit, failing in operating according to the Propor- 
 tion of the Times wherein it doth 
 operate. Therefore the Circle, 
 whofe Radius is H G (which is the 
 Quadruple of the Circle that is the 
 Bafe of the Cylinder) is equal to 
 the Convex Superficie of the Cy- 
 linder ABCD. And this is a fhort, 
 clear, and natural Demonftration, 
 drawn from the efficient Caufe, 
 but fuch as Archimedes could not 
 make ufe of ; it being the Opinion, 
 in thofe Times, that a Line ought 
 always to be confider’d without any Breadth, that 4s, 
 as nothing ; and therefore that a Superficie could no 
 ways be deferibed by the Motion of a Line ; fo that 
 becaufe of the Prejudices of others, Archimedes was 
 forced to make ufe of a long Demonftration, lead- 
 ing ad impojfibile , which no Man’s Wit but his own 
 could have overcome. 
 
 3 . A right Cylinder is the Triple of a right Cone 
 inferibed in it : For a Cylinder is made bv the uni- 
 form Motion of the Circle BC (I fay, of the Circle, 
 not ofthe Perimeter) which is the common Bafe both 
 of the Cylinder ABCD, and the Cone BHC. Now 
 Circles have a double Proportion of that which their 
 Diameters have : Therefore the Circle BC, to the 
 
 Circle 
 
1 66 ‘Principles and \ Problems Book ILL 
 
 Circle a b 9 is in a double Proportion of the Diameter 
 BC to the Diameter a £,and io every where. Now 
 the Circle, whilft it maketh the Cone BHC, lofeth 
 every where of its Magnitude in Working, the double 
 Proportion of the Diameter B C to the Diameter a b. 
 But (as has been fhewed, Chap. 2. Art 9.) when mean 
 Proportionals, as well Geometrical as Arithmetical, 
 are taken every where, all together, they are the 
 fame 3 therefore the Circle B C lofeth of its Mag- 
 nitude, whilft it maketh the Cone BHC two Thirds 
 of its own intire Magnitude : Now how much of its 
 Magnitude it lofeth, lb much it cealeth from work. 
 
 It will then make 
 the third Part of 
 that which the fame 
 entire Circle B C, 
 w'ould have made ; 
 that is, a third Part 
 of the whole Cylin- 
 der : Therefore a 
 Cone is the third Part 
 of a Cylinder, that 
 is, a Cylinder is the 
 Triple of a Cone in- 
 fcribed in it. 
 
 H 
 
 F 
 
 4. It may be fhew r ed, by the fame Method, that 
 the Ipiral Space, vvhich Iprings from the firft Revolu- 
 tion of a Circle, is the third Part of the lame Circle. 
 
 From the Centre A, by the Radius A B, let a Cir- 
 cle be defcribed, and eight Ways cut by the ftrait 
 Lines AC, AD, AE, &c. Divide likewife the Se- 
 midiameter eight Ways, of which let one Eighth A a 
 be marked in the ftrait Line AC, then two Eighths 
 in the Radius AD at b 3 three in the Radius AE 
 
 at 
 
Book III. in GEOMETRY. 1 67 
 
 at c, four in the Radius AF at d, five in the Ra- 
 dius A G at e , fix in the Radius A H at /, feven in 
 the Radius A H at g. Now you muft fuppofe that 
 every eighth Part is divided, as the whole Radius, 
 in as many equal Parts, as it can be fuppofed it can 
 be divided into ; fo that of what Parts A B is 8, 
 A^isi, A b 2, Ac 3, A 4, A c 5, A /6, A g 7 * 
 then you muft iuppofe a fpiral Line, (whofe Begin- 
 ning is A, and End B) drawn through all the Points, 
 <2, £, c, d , c, /, g, B ; and this will be the fpiral Line 
 defcribed by Archimedes. We muft then, fhew, that 
 the Space concluded by this fpiral Line, and the Se- 
 midiameter A B, is the third Part of the Circle 
 BCD 
 
 Becaufe the Circle BCD, is made by theCircum- 
 dudtion of the entire Radius A B, and the fpiral 
 Space by the fame Radius, but failing, that is, reft- 
 ing every where according to the double Proportion 
 of the Times (for all the Circles through a , c, d y 
 &c. have a double Proportion of the ftrait lanes 
 A a, A £, A c, &c. the Space which is left without 
 the fpiral Line unmade, will be the double of that 
 which is made by the fpiral Conversion 3 and there- 
 fore both the Spaces, made and unmade together, 
 are the Triple of the fpiral Space. 
 
 5. Now it is manifeft,that fince the fpiral Line A, 
 ci h , c, dj e^fig, B, is continually deficient in the 
 lame Proportion with the Times, it is equal to the 
 half Perimeter of the Circle BCD. 
 
 6. That a Solid a lfo, whofe Bafe is the fpiral Space, 
 is the third Part of a Cylinder whofe Bafe is the 
 Circle BCD. — 
 
 7. Alfo that a Square circumfcribed by the Circle 
 BCD— is to the fpiral Space as 15 to 4. Now (as 
 
 we 
 
1 68 ‘Principles and Problems Book III. 
 
 vie demonft rated before) a Square is to a Circle in- 
 fcribed in it, as alfo a Cube from the Radius, to the 
 Cylinder, as 5 to 4. The Proportion then of a 
 Square to the fpiral Space, is compounded of the 
 Proportions of 5 to 4, and 3 to 1 5 fo that if it be, 
 that as 3 to 1, fo 4 to another, which fhall be i: 
 TheProportion of a Square to the fpiral Space will be 
 made up of the Proportions 5 to 4, and 4 to j • the 
 Square then is to the fpiral Space, as 5 to that 
 is (both being multiplied by 3) as 15 to 4. 
 
 8. It likewife follows from the fame, that if the 
 Circle BCD— be the Bale of a Cylinder, whole 
 
 Bafe is equal to the 
 Diameter of the lame 
 Circle, and that there 
 be a Solid , whole 
 Balls is indeed the 
 fpiral Space, but the 
 Height equal to the 
 Height of the Cy- 
 linder the Sphere 
 wherein the greateft 
 Circle is BCD, will 
 be the Double of the 
 faid Solid. Now we 
 have (hewed before, that a Square into wfiich the 
 Circle BCD is infcribed, as alio a Cube from the 
 Diameter, is to a Cy linder whofe Bafis is the Circle 
 BCD, as 5 to 4, and to a Sphere, as 15 to 8, and 
 to the fpiral Space, as 15 to 4 ; for the Proportion 
 of 15 to 4 being lubtrafted from the Proportion of 
 15 to 8, there remains T |? 9 that is, l. The Solid 
 then whofe Bafis is the fpiral Space, is the half of 
 the Sphere wherein BCD is the greateft Circle. 
 
 9. By 
 
Book III. wGEOMETRY. \ 6 9 
 
 9 - By the fame Method it will be found th=»t 
 any Redangle is to the Parabole inferred in it as 
 3 to 2. 5 
 
 Defcribe then any right Angle ABCD, in whofe 
 Side DC take any where D a, to which in the Side 
 A B, take an equal A d, cutting the Diagonal B D 
 in b, and let that be lo every where ; then a s da 
 every where to d c, fo let d c be to db, and draw the 
 crooked Line Be cD through all the Points c h that 
 this crooked Line is Parabolical, and the Figure 
 AB cc D the half of a Parabole, all Mathematicians 
 do agree. It is then to be (hewed, that the Redangle 
 ABCD is to the half Parabole AB cc D, as 3 to 2 
 
 For defcribe the Redangle ABCD from the right 
 Line AB uniformly, and parallelways moved to the 
 oppofite Side DC. Now 
 the Semiparabole ABccD b 
 is deferibed by the right 
 Line C B moved towards 
 
 v- ■' uiuvi-u Luwarus •• X ™- 
 
 the oppofite Side A B, dill J 
 
 lofing in the double Pro- j 
 portion of the time A B J j 
 to the time D a ; for the ,j 
 Proportion of da to dc ‘ 
 is every where the double *’!' 
 of the Proportion of d a to £ 
 d b ■ and therefore (accord- 
 
 J Ӥ to C.* 1 ' 2 ' Art - 9 -) the da together, are to all 
 the dc, m a double Arithmetical Proportion of all the 
 da to all the db. So then that which is made uniformly 
 wh ? leA ? (that is, the whole Redangle 
 ABCD; to that which is made by the Motion of the 
 Side DC towards AB failing or lofing (that is, reft- 
 
 N ing 
 
170 Principles and ‘Problems Book III. 
 
 ins from work) according to thefubduplicated Arith- 
 metical Proportion of Time, will be that which is left 
 unmade, to wit, C D c c B one Part of the whole 
 Redangle ABCD, of which that which is perfed- 
 ed to wit, the half Parabole AB cc D, is two. 
 
 And fo the Redangle ABCD, is 3 of the lame 
 Parts: The Redangle ABCD is therefore to the 
 Semiparabole infcribed in it, as 3 to 2, and the dou- 
 ble Redangle to the whole Parabole infcribed in it, 
 as 3 to 2. 
 
 10. We may alfo ufe the fame Method in finding 
 out the Proportion ofa right Angle to the Parabolafter 
 (which is called a Cubical Parabole) for if it be every 
 where that as da to dc , and dc to db , fo db to a 
 fourth de (fo that there be four continual Propor- 
 tionals da , dc, db , 
 d e ) the crooked Line j 
 of that Parabolafter 
 will pals through all 
 the Points e j and 
 the Proportion of th* 
 Redangle ABC! 
 will be to the half 0 
 that Parabolafter, a 
 4 to 3 j and fo yoi 
 may proceed to th 
 fecond, third, 
 Parabolafter, who! 
 Proportions to the Redangle in which they are infer; 
 bed (as alfo of Cylinders to the Cone and Conoide 
 infcribed in them) are reduced into the T able of Chaj 
 17. of my Book De Corpove. Such is then the natun 
 Aptitude of the Lemma prefixed to this Chapter, t 
 the Demonftrating of the greateft Problems of pui 
 
 Geometry 
 
 B C 
 
Book III. in GEOMETRY. \ 7 ' v 
 
 Geometry, that he who knows by what Proportion 
 of Proportions (for feeing Proportion is a Quantity, 
 there will be Proportions of Proportions, as well as of 
 other Quantities) every compared Figure is framed 
 he cannot be ignorant of their Proportions one to 
 another. But what (may fome body fay) is that 
 pure Geometry ? Is that Geometry true which is 
 impure? Let him term it mix’d, notimpure. But how 
 can a thing pure, mixed to a thing pure, become 
 impure? But neither muft that be faid; but that 
 then it is mixed, when it is applied to Matter. For 
 it is indeed manifeft, that if it be not applied to 
 Matter, it is ufelefs, and no more but mere hard 
 W ords. I call that pure Geometry, with which no- 
 thing of Arithmetick is blended, but what may alfb 
 agree to a continual Quantity, which is the proper 
 and adequate Subjed of Geometry. Therefore the 
 chief things that corrupt Geometry, are the Surdity of 
 Numbers , Longitude without Latitude , and Latitude 
 without 1’hicknefs , and the lately introduced Do- 
 drine, that Fradion is Proportion, as if the half 
 were the Proportion of the half to the whole. 
 
 CHAP 
 
172 ‘Principles and ‘Problems Book III. 
 
 CHAP. XI. 
 
 Of demonstration. 
 
 I N the firft Place it may be doubted, whether De- 
 monftration goes before, or comes after Know- 
 ledge. Moft fay, that it goes before ; for a Demon- 
 ftration is either' fcientifick or frivolous : It is there- 
 fore the efficient Caufe of Knowledge, and for that 
 reafon goes before its Effedt. It muft then be grant- 
 ed, that the Mailer’s Demonftration goes before the 
 Knowledge of the Scholar ; but no more. _ But if 
 any Conclufion be demonftratyd by any Man, he muft 
 firft have known the Truth of it, before he could de- 
 monftrate it either to himfelf or others. For no 
 Man can demonftrate, that which he knows not whe- 
 ther itbefalfeortrue: It is then manifeft that Know- 
 ledge in its own nature goes before Demonftration. 
 
 In the next Place it may be asked (fince Geome- 
 try is very ufeful and ornamental to Mankind) who are 
 chiefly the Men to whom we are indebted for fo great 
 a Good, the Demonftrators, or not Demonftrators. It 
 is certain that long before Euclid , many large and 
 artful Fabricks were built, the Tower of Babylon, the 
 Pyramids of Egypt, the wonderful W alls and Gardens 
 in' Babylon, the Palace of Perfia, and others ; alfo a 
 Sun-dial was firft ihewed at Lacedemon. Thefe Works 
 without doubt required Knowledge ; yet the Author. 1 
 of them demonftrated nothing, but by natural Lo- 
 gick forefaw the Reality of their future Works, tho 
 others afterward were curious to do it, with tha 
 
 purpofe 
 
Book III. in GEOMETRY. 
 
 purpofe, that they might excite the Minds of many 
 to the Invention of* ufeful Things. The Proportions 
 aJlo of the five regular Bodies (as you may read in 
 the old Greek Epigram) were found out indeed by 
 Pythagoras , tho 5 Plato demonftrated them, and af- 
 ter him Euclid EL 3. Plato's Demonftration is not 
 extant ; fo that we owe thofe good Inventions to fe- 
 veral Inventors, whofe Names (except a few) are now 
 loft, and not to Demonftrators. How then is De- 
 monftration ufelefs? Not indeed to thofe who are 
 taught. Now to be taught I think not very laudable, 
 though to teach, provided it be rightly done, and 
 without Hire, is honourable. But doth not a Demon- 
 ftration oflbme ample Science, hitherto unknown, de- 
 ferve great Thanks? Yes, but he that (hall do that, 
 is to be reckoned among the Inventors of profitable 
 things. And therefore we are indebted to Euclid , 
 who firft of all taught the World the Method of De- 
 monftrating, that is, of found Reafoning. All agree 
 that moft ample Thanks are due to thofe who firft ad- 
 vifed Men to aftociate, and to unite together under 
 the Obedience of one Supreme Power. The next I 
 think is due to them, who fhall perfuade Men that 
 they (hould not violate the Compacts and Agree- 
 ments that once they have made. 
 
 Thirdly, Some may ask what a Demonftration is : 
 Moft Men, and not without good Ground ufe to call a 
 Demonftration, an evident Probation of the Truth, in 
 any dubious Queftion. Everyone thinks that he has 
 enough of Demonftration, when his Mind does fully 
 acquiefce to thofe things that are alledged for Pro- 
 bation. But the ancient Philofophers nuried up in 
 perpetual Difputations, as often as there was anv 
 Debate concerning the Companion of Motions and 
 
 N 3 Mag- 
 
1 74 ‘Principles and Problems Book III. 
 
 Magnitudes, for the mod part they defcrib’d Figures, 
 and put before the Eyes of their Difcipl§s, as if fhew’d 
 by them to the Sight (that is ibmewhat more as they 
 thought than prov’d) call’d their Arguments 
 (in Engllfb ) Demonftrations. But now when any 
 Man is a little too vehement and pofitive in avert- 
 ing, he affirms that he has demonftrated. 
 
 Now a Demonftration is, when the Truth of the 
 Conclufion hath its firft Foundation, in thole things 
 which are already known to them, to whom he that 
 proves them, fpeaks. Nov/ thefe Foundations are 
 Definitions, and befides Axioms, which are not indeed 
 demonftrated, yet ought to be true and well under- 
 flood 5 for from Truths nothing but what is true 
 can be inferred, nor can any thing though never fo 
 true be known, that is not underftood. 
 
 Fourthly, It may be demanded what the Ufe of 
 Definitions and Axioms is : This is the plain Ufe of 
 Definitions, that he to whom a Demonftration is 
 made, may underftand what certain an univerfal Sig- 
 nification of the Word the Demonftrant would have 
 taken ; for the Signification that changeth, deceiveth. 
 Therefore a Definition conduceth to the Underftand- 
 ing, and without Underftanding there is neither any 
 Demonftration, nor in him thatlearneth can there be 
 any Knowledge, but from thence unfeafonable and 
 obicure Diftindions are introduced ; fo that of what is 
 faid, nothing maketh Impreffion on the Imagination. 
 There are indeed, fome Truths which can no Ways 
 be known by Man, but nothing is demonftrated 
 that cannot be underftood by him. 
 
 The Ufe of Axioms confifts in this, that they abbre- 
 viate the too long Series of Demonftrations, to wit, 
 by removing unnecefiary Demonftrations. For the 
 
 Truth 
 
Book III. in GEOMETRY. 17 5 
 
 Truth of Axioms ought to appear fooner, and more 
 clearly than the Means themfelves, by which they 
 are proved. 
 
 In every Demonftration, the Caufe of theConclu- 
 fion ought to be in the Antecedents, by Virtue of 
 which it is inferred • that is, in things before de- 
 monftrated, or known by the Light of Nature. There- 
 fore where the Words cohere and hang together, 
 there will be a Demonftration • for tho 5 the Caufe 
 of thefubjecft Matter be not known, yet the Conclu- 
 fion will retain the Truth of the Principles from 
 whence it is derived. Now Demonftrations of this 
 kind are eafy, tho they little advance Knowledge. 
 
 That is thechiefeft of all Demonftrations, which 
 is drawn from the Production of the fubject Matter, 
 according to the Order of Nature ; and 10 thefe are 
 the moft ufeful Definitions for Knowledge, wherein 
 the Generation of the fubjedl Matter is explained ; 
 tnat is, by what Motion, what Concourfe of Mo- 
 tions, what Proportions of Motions and Times, all 
 Spaces and Magnitudes are determined. 
 
 The next Demonftration to this, is, when (from 
 the Negation of Truth) fomething impoflible is in- 
 ferred. Now this kind of Demonftration has its 
 Force from this, that from a Truth nothing but 
 Truth can be deduced. 
 
17 6 ‘Principles and ‘Problems Book III. 
 
 CHAP. XII. ' 
 
 Of FALLACIES. 
 
 LT^Hat a Man who underftands his Words, that 
 JL is, his own Speech, and fets upon the Proof of 
 it from true Principles (1 mean) natural Definitions, 
 proceeding (lowly in Mathematical Matters, fhould 
 be long and often deceived, and being admonifhed, 
 lhould perfift in his Error, is a thing almoft impof- 
 fible. Speaking properly, there can be no Error in 
 the Intellect 3 for to err in the Intellect, is the fame 
 as not to underftand. The right Ufe of the T ongue. 
 Feet, and Hands, is not demonftrated by a Mafter, 
 but by Exercife we learn it , and though all our 
 Words almoft change their Signification, according 
 to the Variety of the things whereof we (peak or 
 write 3 yet we ufe them at home, in the Fields, and 
 in the Market, without any hurt, becaufe it is 
 enough for Civil Society, if one underftand aright 
 what another fays. 
 
 It is otherwife in Philofophy, where nothing but 
 Truth is fought after 3 but efpecially if Glory attend 
 Invention. It is one thing to walk, another thing to 
 walk upon a Rope 3 the one is eafy, and without great 
 Damage, tho* a Man fhould trip 3 fo that Negligence 
 is there pardonable : It is not lo to a Rope-dancer. 
 Truth walks upon a very fmall Thread, without that 
 metaphorical Latitude of common Converfation 3 
 and efpecially Mathematical Truth, which unlefs it 
 be poifed by the Weight of Definitions and Axioms, 
 it tumbles headlong, to the Laughter of Spe&ators, 
 
 The 
 
Book III. in GEOMETRY. 177 
 
 The chief and moft frequent Caufe then of Fallacies 
 in the Mathematicks, is, that they build their Realon- 
 ing upon Definitions not underftood, orfilfe or ambi- 
 guous ones, from which no T ruth can be deduced. The 
 Greek Philofophers were of the fame Mind alfo, when 
 to this Science they gave the Name Mathematicks , 
 from the Greek Word that fignifies tounderftand ; for 
 commonly when any one, ipeaking to another, doubt- 
 ed if he was underftood or not, he asked, * mMvm , 
 Bo you under ft and? To which it wasanfwered, 
 or * ucL^dtveo $ that is, / do, or do not under ft and. So 
 natural it was to denominate the Mathematicks from 
 theUnderftanding. Another Caufe of Fallacies is, not 
 to know what Motion is, and its Properties 3 that is, 
 to be ignorant of the immediate natural Caufe of all 
 things. Thofe known Fallacies reckoned up by Arif* 
 totle , by which a Child can hardly be deceived, 1 
 purpofely pafs over, as the greateft Bane of Geo- 
 metry. In the firft Place, I condemn a Line without 
 Latitude , a thing unconceivable. Secondly, T he Side 
 ofafquare Figure, fuppofed for the Root ofa Number. 
 Thirdly, The nature of Proportion not underftood. 
 Fourthly, All confideration of infinitum , whether Geo- 
 metrical or Arithmetical. And here I fhould have made 
 an end, had not there been one who affirms, That he 
 can alfo demonftrate thofe things, which are neither 
 intelligible, explicable, nor conceivable • which is in- 
 deed to fay, that all Sciences are not worth a Rufh. 
 
 Nothing (fays he) is more obvious in Nature than 
 continual Quantity , and local Motion. Now the fe either 
 are , or are not divisible in infinitum : Can this Disjun- 
 ctive be denied ? Can it be faid , that they neither are , 
 nor are not thus divifible ? Let any choofe what Mem- 
 ber of this he pleafes ; jhall he remove the Difficulties 
 
 that 
 
178 Principles and ‘Problems Book III. 
 
 that are in it ? Or Jhall he anfwer the Objections' 
 that can be made to the contrary ? 
 
 Though thefe be Queftions, yet in this Place they 
 have the Force of Negations ; let him then object his 
 Difficulties. I fhall explain the Matter itfelf , to wit, 
 that which Mathematical Writers underftand, when 
 they fay, that Quantity is divifible in infinitum. They 
 do not fay nor underftand, that a finite Quantity (fup- 
 pofe a Line) is divifible into Parts that are infinite 
 in Number ; but that a Line never fo little, is of its 
 own nature capable of Divifion. Neither by Divifion 
 do they underftand a material Separation, that is, a 
 Separation of one Part from another 3 but that in all 
 continual Quantity, a Quantity lefs, and that affign* 
 able, is ft ill to be fuppofed ; and that theSignification of 
 thisDivifion is nothing elfe but the uni imitted Confide- 
 ration of a Part, in a Whole never fo little. For of 
 whatfoever it may be {aid, it is a Whole, it may very- 
 well befaid, that there are Parts in it ; but that a mor- 
 tal Man can divide any thing eternally, or if he could 
 do it, that yet the Parts fhould not be of a finite Num- 
 ber, it is impoffible. Now what is faid of a Line, 
 ought likewife to be underftood of Motion, Time, 
 and every thing elfe that is divifible, except Number. 
 Let an Obje&ion now be brought againft this (I 
 confefs I do not remember that ever I read any) that 
 we may fee, if it may not eafily be underftood, 
 whether it be ftrong or weak. Moreover, 
 
 Jfuppofe (fays he) that moft know that famous Ar- 
 gument of Zeno, which is called Achilles, and that 
 how that great Difputer , whilfi that , by apparent 
 Impojfibilties and Ab fur dities on each Side , he demon- 
 fir ated local Motion to be impoffible , he was confuted 
 by one of the Hearers , who rofe and walked through 
 the School. Of 
 
Book III. «/ GEOMETRY. i yy 
 
 Of which Words this is the Sum, That Zeno was, 
 indeed, confuted by the Walker, b t that his Sentence 
 was rightly inferred by a Demonftration leading to 
 an Ablurdity. WTat Zeno s Argument is, that the 
 Reader may judge of it, I will (hew. Seeing the leafh 
 Space that is, cannot be paffed over by Motion, but 
 that the half of that Space is firft to be palled over, 
 and that the leaft Space again has alio its half, and 
 fo perpetually ; Zeno concludes (fubfuming that no 
 Space can be paffed over in an Inftant) that it re- 
 quires an infinite Time to pafs over the leaft Space 
 whatfoever. Now it is manifeft, that if a Space 
 cannot be paffed over without an eternal Motion, it 
 cannot at all be paffed over. 
 
 The Stcicks exemplified this Argument taken from 
 Zeno theirMafter, m Achilles fwiftofFoot, and a How- 
 paced Tortoife ; and they laid (putting Achilles and 
 the Tortoife into the Race) that Achilles cpuld never 
 overtake it, if it were but the leaft Diftance before 
 him. But why ? becaufe whilft Achilles ran over that 
 half Diftance, the Tortoife alfo advanced a little. I 
 fhould be afhamed to repeat thofe Trifles, did I not 
 perceive, that they who leaft ought to be, may yet be 
 deceived by fuch childifh Fallacies : But what is to be 
 anfwered, I fnall anfwer, Firft, That it is no wonder, 
 if he that is unwilling, never overtake a flow Runner 
 that is got before him. For fuch is. the nature of Mo- 
 tion, that he who always will, oris forced to flacken 
 his Swiftnefs in proportion to the Space that is left, 
 will never be able to pafs over even thefmalleft Space 
 whatlbever. Secondly, What is made by any whole, 
 and the half of it, and the half of that half, and fo 
 on, will be left than two ; they are fo far from making 
 an infinite Time. By that Argument of Zeno's then, 
 
i8o Principles and ‘Problems Book III. 
 
 it cannot be inferred, that Iwift Achilles could not in 
 running overtake the Tortoile ; but only this, that if 
 he pleaied he would not. Therefore "the Example 
 of Zenos Sophifm does not prove, that there can 
 be a Demonftration of a thing that is not intelligible. 
 H< oes on ; 
 
 The pure and mofi fimple Mathematicks are the moft 
 eafy , evident , and comprehenfible of all human Sciences . 
 Tet in Geometry and Arithmetic &, how many Propofitions 
 are there firmly demonfirated , which neverthelefs are 
 inexplicable , unconceive able , and incomprehenfible ? 
 I Jh all give a few Infiances , &c. 
 
 Is not this abfurd ? Can he be faid to demonftrate 
 a Truth, that does not make it apparent and intel- 
 ligible ? How can 1 know whether it be true or 
 falfe, if I cannot imagine it in my Mind ? But let us 
 read the Examples, which he lays he will fubjoin. 
 
 1 . That the leaf Space imaginable may be equal to ano- 
 ther Space upon the fame Bafe , and of the fame Height , 
 whofe Sides are drawn out in infinitum. 
 
 That a Parallelogram andTriangles, indeed, of the 
 fame height, upon the lame Bafe, are all equal to one 
 another, is intelligible to every one. What he adds, 
 whofe Sides are drawn out in infinitum , is abfurd ; as a 
 Bench is not called a Bench before it be perfected, 
 to neither is a Space before it be finifhed. 
 
 For it is downright impoffible and abfurd ; nor 
 from Torricellius does it follow, that a Finite is equal 
 to an Infinite. 
 
 2. One Infinite may be greater than another Infinite. 
 Who faid fo ? He is filent ; for it is abfurd. 
 
 3. That all the circular Angles of Cont abb are equal. 
 What is a circular Angle of Contact ? The Expref- 
 fion itfelf is unintelligible. 
 
 4 c fhe 
 
Book III. In GEOMETRY. 181 
 
 4. ST 3 he eternal Appropinquation of two Lines , but the 
 Conconrje impoffible, to wit , Afymptotes. TheCaufe 
 of this is before explain’d in the Sophifm of Zeno. 
 
 5. be Affections of fur {land irrational Tenantries 3 all 
 thefe are fo demonftrated , as that they cannot he denied , 
 and many others , which neverthelefs are inexplicable , in- 
 cotnprehenfible , dfod inconceivable . But what are thele 
 furd and irrational Quantities ? That there are Ibme 
 continual Quantities, which have not the Proportion 
 of Number to Number, and are called incommenfu- 
 rable, there is no Mathematician but knows ; and of 
 thefe alfo that are commenfurable one with another, 
 many are irrational ; becaufe though they be com- 
 menfurable one with another, yet becaule they are 
 not commenfurable to any Quantity taken at Plea- 
 fur e, they are faid to be irrational. 
 
 6 . In Numbers can the Affections of aUnity and 'Ternary 
 be fully comprehended , exprejfed and explained , that in 
 Nature there floould be one Quantity , and no other , which 
 with its infinite Powers afc ending, and Roots defending^ 
 all fioould be equal among them] elves , or rather one and 
 the fame ? The Words themfelves are pretty obfcure ; 
 but I think this is it that he would have faid : Seeing 
 the Iquare Power of a N umber, and a fquar e N umber, 
 alfo the fecond Power and cubical Number, &c. is the 
 lame thing to Arithmeticians ; and that that Number 
 is called the Root of a fquare Number, which being 
 multiplied by itfelf makes any Number, and the Root 
 of a cubickN umber the fame with thatNumber,which 
 being multiplied by itfelf, and again by the Product, 
 maketh any one, and that one multiplied by itfelf ne- 
 ver fo often, makes no more but one, itismanifeftthat 
 all the Products and their Roots are the fame Unities. 
 Now though this Secret be only proper to an Unity 1 
 
 yet 
 
 t 
 
182 Principles and Problems Book III. 
 
 yet it is not hard to be underftood, becaufe nothing 
 can be multiplied by one. Laftly, 
 
 He adds 5 Can fo much as any one of the infinite 
 potential Roots of the Number 3 be explained, thought , 
 or comprehended ? 
 
 It cannot, I know ; and why? Neither that, nor 
 that of many other Numbers, can, I know, and 
 can briefly explain, and that is, becaufe they have 
 no Root. 
 
 CHAP. XIII. 
 
 Of an INFINITE. 
 
 T O the Word Infinitum fomethingis underftood, 
 as Work, Time, Space, f '3c. If it fignify a 
 Work, then, in its Latin Acceptation, it fignifies a 
 Work that is indeed begun, but not as yet brought 
 to the End that the Artift defigned in his Mind • and 
 fo in that Senfe a thing infinite is the fame as a thing 
 unfiniflied or imperfect, but what may be finifhed • 
 luch as a Houfe begun, but not perfected. If any 
 then fhould lay of an Artift, that he had in his Mind 
 to perfed an infinite Work, he would fpeak abfurdly, 
 as if he fhould fay, that it was in his Mind to do that 
 which he never intended : For no Man thinks of do- 
 ing more than what he can accomplifh • fb that no 
 Man can judge of a Work that is not his own, with- 
 out confulting the Artift, whether it be finifhed and 
 perfed, or infinite, that is, unfinifhed and imperfed. 
 
 An 
 
Book III. in GEOMETRY. 185 
 
 An Infinite , if Space be nnderftood, fignifies a Space 
 greater than can be equalled by the greatelt Number 
 of Meafures, as of Feet, Paces, Miles, or even of 
 the Diameters of the Earth, or of the Orb of the 
 fixed Stars ; that is to fay, which cannot be in- 
 cluded within Bounds. In like manner, an infinite 
 Time is that which no Number of Hoars, or Days 
 can equal. 
 
 Therefore of an Infinite , according to this Senle, it 
 cannot be laid, that one is greater than another : 
 Draw then the finite right Line A B, and fuppofe it 
 
 produced beyond p j* ^ B by E in 
 
 infinitum. Therefore both the right Line BE — 
 and ABE — are infinite in Length. Now A B E — 
 is greater than BE — by the whole length of A B, 
 be it fo. Let A B be divided in C, and put A D 
 equal to AC ; and let it be fuppoled lengthen’d ftrait- 
 ways by F in infinitum. A D F will be then greater 
 than D F by the finite Longitude C D, that is, by 
 the Quantity of A B. 
 
 Wherefore CBE — and CAD — are not un- 
 equal ; therefore there is fome certain Point, that 
 is a mean in the infinite Line — DB, fo that the 
 Centre of an infinite Sphere will be the Point C, and 
 (becaufe the Points A and B are taken at Plea- 
 fure) in every Point of an infinite Sphere, will the 
 Centre of the Sphere be ^ and lo the Semidiame- 
 ters of an infinite Sphere from any Centre, whe- 
 ther A, B, or C, are not unequal one to another 
 Therefore one infinite Line is nqt greater than 
 another. 
 
 By 
 
 t 
 
1 84 ‘Principles and Problems Book III. 
 
 By the fame Reafon, if the Line AB be put for 
 an infinite Time, it may be proved that two Eternals 
 cannot be unequal. 
 
 Now the Caufe why a Finite may be confidered in 
 an Infinite, cdnfifts in this, that neither the fubjecft 
 Body is in the Thinker, nor Space, the Image of 
 the Body, in the Thing thought of, but only in the 
 Memory : And in the Memory and Senfe there can 
 be no Infinite. 
 
 But Mathematicians ufe often the Word infinite for 
 indefinite. Now indefinite is the fame as never fo great , 
 and fometimes it is taken for infinitely little , provided 
 it be not nothing. Sometimes alfb infinite goes for as 
 much as ispoffible. But nothing is properly infinite , 
 unlefs it exceed all arguable Number of given Mea- 
 fures. But it is {aid to have been demonft rated by 
 Y orricellius , that a certain acute hyperbolical Solid 
 is, even in this Senfe of infinite , equal to a certain 
 Cylinder, whofe Bafe hath a Diameter equal to the 
 half Bafe of the Hyperbole, but a Height equal to 
 the tranfverfe Axis of the fame Hyperbole. I have 
 often and attentively read the Demonftration of this 
 Problem, and never found any Sophifm in it. Yet 
 I found that the Diftance, which ^orricellius fuppo- 
 fes infinite, is meant of an indefinite Diftance ; nor 
 could it be otherwife underftood by himfelf, who 
 in very many Demonftrations ufeth the Cavallerian 
 Principle of Indivifibles • which Indivifibles of Ca- 
 vallenus are fuch, that their Aggregate may be equal 
 to any given Magnitude. So that fb abfurd a Pro- 
 pofition as this, an Infinite is equal to a Finite , 
 ought not to be alcribed to T 'orricellius • for there 
 . can be no Solid fb fmall, which, being infinite, doth 
 not exceed every finite Solid, as is manifeft by the 
 
 Light 
 
Book III. in GEOMETRY. 185 
 
 Light of Nature; that is, the Abfurdity of Arith- 
 meticians Reafbning about an Infinite, and mea- 
 furing Superficies and Solids by Lines without La- 
 titude ; who obferving no Difference betwixt Arith- 
 metick and Geometry, took the Root of a Number 
 (which is part of its fquare Number) for the fame 
 thing with the Side of a fquare Figure, though they 
 confefs that the Side is no Part of its ow r n Square. 
 So much of Mathematicks. I now expedl what 
 the Algebraifls will fay to the contrary. 
 
 The End of the Third Book. 
 
 G 
 
 ME A- 
 

 
MEASURING 
 
 O F 
 
 Gla zing. 
 Painting, 
 Plaistering, 
 Masonry, 
 
 Joinery, 
 
 Carpenters, 
 
 and 
 
 Bricklayers 
 
 WORKS, 
 
 B Y 
 
 Vulgar Arithmet ick, 
 
 WITHOUT 
 
 Reducing the Integers into the leaf! 
 Denomination. 
 
 The Fourth Book. 
 
 By VEN. MANDEY. 
 
 LONDON : 
 
 Printed in the Year M.DCC. XXVII. 
 
MEASURING 
 
 O F 
 
 Carpenters Work. 
 
 CHAP. I. 
 
 M E N S U R AT I 0 N, 
 
 R Meafuring, is a Science, whereby we 
 are certified” (in fuperficial MeaJtiire) 
 either how many Inches, or how many 
 
 Feet and Inches, or how many Yards, 
 
 Feet and Inches, or how many Squares, Feet, and 
 Inches, there is in any Dimenfion, whether it be of 
 Glafs, Board, Stone-paying, Plaifiering, Painting, 
 Tyling, Carpenters or joiners Work, Sc. And fo 
 confequently it certifies us of the Content of many 
 Dimenfions, by adding the Products of the feveral 
 Dimenfions together. 
 
 It doth the fame in folid Meafure ; there is only this 
 Difference between fuperficial and folid Menfuration ; 
 to wit, in fuperficial Meafure there is only two 
 Sums in a Dimenfion, viz. Length and Breadth, to 
 multiplied one by the other : But in folid Mea- 
 fure 
 
Book IV. Meafuring , See. 189 
 
 fure there are three Sums in a Dimenfion ; to wit. 
 Length, Breadth, and Thicknefs, to be multiplied 
 each by the other ; and this kind of folid Menfii- 
 ration ierveth for the meafuring of Timber, Stone, 
 Digging, Bricklayers Work, for it is commonly re- 
 duced to a Thicknefs, and all manner of folid Bo- 
 dies wharioever. 
 
 The Inftruments that are ufed in taking of the 
 Lengths and Heights (or Breadths) in meafuring 
 of the following Works, are a ten Foot Rod, and 
 a five Foot Rod, and a two Foot Ruler, and fome« 
 times a Line. 
 
 I (hall begin with the Meafuring of Carpenters 
 Work, it being moft eafy to learn, and lb proceed 
 through the other Trades, leaving tne Meafuring 
 of Bricklayers Work until the laft, it being in oft 
 difficult. 
 
 CHAP. IL 
 
 H E Flooring, Roofing, Partitions, and Ceil 
 
 x ing Joifts, being Carpenters Work, is gene- 
 rally meafured by the Square (as it is vulgarly faid) 
 or more properly is reduced into Squares. 
 
 Which Square c'onfifts of (or contains) too fu- 
 perficial Feet, being the Product of a Square Su- 
 perficies multiplied in it felf, being ten Feet in 
 Length, and ten Feet in Breadth. 
 
 Note^ That a Superficies is that which hath only 
 Length and Breadth, as a Square inclofed with four 
 Lines on Paper, is called a Superficies (as appears 
 tnoreat large in the precedent Treatifes ofGeometry.) 
 
 O 3 
 
 And 
 
190 MEASURING BookIV. 
 
 And from thence the meafuring of Flooring, 
 Roofing, Partitioning, Ceiling-Joifting, Boarding, 
 Plaiftering, Painting, Sc. is called fuperficial Mea- 
 fure, becaufe their Thicknefifes are not confidered, 
 nor taken Notice of in Meafurement. 
 
 For generally Boards, Glafs, Painting, Plaifter- 
 ing, are feverally for the moft part of one Thick- 
 nels. 
 
 And although fome Floors, and Roofs, and Par- 
 titions, do require to be made ftronger than other 
 fome, by reafon of the Greatnefs of the Building, 
 and by this means the Timbers are larger and 
 thicker ; yet there is no refpedt had to (nor cogni- 
 zance taken of) the Largenefs and Thicknefs of the 
 Timbers, in the Meafurement 3 but an Allowance 
 is made in the Price, by adding fo much per Square 
 more than is ufually given for ordinary Work. 
 
 Thus 12 Inches in Length, and 12 in Breadth, 
 is called a fuperficial Foot, of Roofing, Flooring, 
 Boarding, Partitioning, Glazing, Painting, Plaifter- 
 ing, Sc. 
 
 Note , When a Carpenter’s Bill of Meafurement is 
 made, there is fet down, 
 
 For fo many Squares of Roofing (at what Price 
 they agree upon per Square) fo much Money. 
 
 Likewife for fo many Squares of Flooring and 
 Boarding, at fo much per Square, fo much Money. . 
 
 Alfo for fo many Squares of Partitioning, at fo 
 much per Square, fo much Money. 
 
 And for fo many Squares of Ceiling- Joifts, Sc. 
 
 The Windows they fet down either at fo much 
 per Light, or fo much per Window.’ 
 
 The Door-cafes at fo much a-piece, either with 
 or without Doors. 
 
 The 
 
Book IV. of Carpenters Worh 191 
 
 The Mantletrees and Tarfels at fo much a-piece. 
 The Lintolling, Guttering, Cornice, and Win- 
 dow Eoards, at io much per Foot. 
 
 Stairs at fo much a Pair, or fo much per Step, 
 
 CHAP. III. 
 
 I T is fuppofed that he that intends to learn the 
 Science of Meafuring is an Arithmetician. 
 
 For although feveral Propofitions may be an- 
 fwer’d, fome Geometrically, and fome on the Line 
 of Numbers (other wile called Gunters Line) and 
 fome on the Lines of Superficies and Solids on a 
 Se&or ; yet without Arithmetick it is impoflible to 
 meafure exactly all kinds of Plains and Bodies. 
 
 Therefore to the Arithmetician, I fay, multiply 
 the Length of any Dimenfion (that is, either Square 
 or Oblong) by the Breadth thereof, and the Pro- 
 dud: is the fuperficial Area or Content. 
 
 I fhall begin with a Dimenfion of Inches, and fb 
 proceed gradually to Feet, and from Feet, to Feet 
 and Inches (or Parts.) 
 
 Note , That two Sides of a Superficies being mea- 
 fured and exprefs’d by Arithmetical Figures (one 
 Sum being the Length, the other the Breadth) is 
 called a Dimenfion. 
 
 Example in Inches. 
 
 Suppofe a Piece of Board or Glafs, or flat Stone, 
 or Painting, or Plaiftering, or any thing, be it what 
 it will, that is to be meafured fuperficially, (vulgar- 
 
 o 4 ly 
 
tpi MEASURING Book IV. 
 
 ly called fiat Meafure) be n Inches in Length, 
 and 9 Inches in Breadth (which two Sums 1 1 and 
 9, are called a Dimenfion) and you would know 
 the Area or Content thereof 3 
 
 In. 
 
 Set down your Dimenfion thus-^J 
 
 Then multiply the Length by the 
 Breadth. 
 
 And the Product is — 99 Inches. 
 
 Which 99 Inches is half a Foot and 27 Inches, 
 or very little more than 4 of a Foot. 
 
 So that the Produd of 1 1 Inches, multiplied by 
 9 Inches, which is 99 Inches, contains 8 fuch In- 
 ches, whereof 12 make a fuperficial Foot, that is 
 to fay, it contains 8 Inches, whereof each Inch is 
 one Inch in Breadth, and 12 Inches long : For 8 
 times 12 is 965 which is | Parts of a Foot ; then 
 the 3 Inches which are remaining, (for the Pro- 
 dud: was 99 Inches, take 66 from 69, and there re- 
 mains 3 • thefe three Inches, I fay) are 7, or, as 
 I call them, 3 Parts of fuch an Inch as aforefaid ; 
 namely an Inch in Breadth, and 12 in Length. 
 
 For, by the way, you muft note. 
 
 That in a fuperficial Foot, there is 
 
 contained 144 Inches, which is the 12 Length. 
 
 Produd of 12 Inches multiplied by 12 Breadth. 
 
 1 2, thus — ■ — — 
 
 144 Produd. 
 
 Or 
 

 iz 
 
 tH 
 
 Book IV. of Carpenters Work. 193 
 
 Or as you may lee in the 
 Geometrical Figure hereun- li 
 
 to annex’d. 
 
 By this Geometrical Fi- 
 gure, you may plainly per- 
 ceive, that there is in a ^2 
 Square Superficies, being 12 
 Inches on each Side, 144 
 Superficial Inches ^ for eve- ^7 j 
 ry little Square in the Ge- tz 
 
 ometrical Figure reprefents 
 
 an Inch : So confequently in half a Foot there is 
 72 Inches, being produc’d from an Oblong, being 
 12 Inches in Length, and 6 in Breadth. 
 
 Which Oblong you may perceive, or imagine in 
 this fquare Geometrical Figure, by taking the whole 
 Length one Way, and half the Length, or 6 of the 
 little Squares, the other way: Therefore I fhall not 
 need here to defcribe the Figure of an Oblong. 
 
 Alfo half a fuperficial Foot, which is 72 Inches, 
 is produced from an Oblong very near a Square, be- 
 ing 9 Inches in Length, and 8 Inches in Breadth, 
 for 9 times 8 is 72. 
 
 Likewife a Quarter of a fuperficial Foot contains 
 36 Inches, which is one half of 72 Inches, or the 
 Product of 6 Inches, multiplied by 6 Inches, or 9 
 by 4. 
 
 I Jh all now proceed to Dimenfions in Feet . 
 
 Suppofe there is a Timber Floor (or a Stone Pave- 
 ment, or a Plaifter’d Ceiling, or a Piece of Wain- 
 fcot, or any other Superficies) that is 22 Feet in 
 Length, and 18 Feet in Breadth, and you would 
 
 know 
 
1 94 MEASURING Book IV. 
 
 know in Carpenters Work, how many Squares there 
 is contained therein. 
 
 Feet. 
 
 _ _ S 22 
 
 1 18 
 
 Set down your Dimenfion thus 
 
 176 
 
 22 
 
 Which being multiplied, the Produ< 5 t is 396 
 
 fuperficial Feet. 
 
 Now to bring thefe Feet into Squares, you mud 
 divide 396, the Content in Feet by 100 (the Con- 
 tent of a Square whole Side is 10 Feet) and the 
 Quotient is 3 Squares, and 96 Feet remaining. 
 
 (96 Feet. 
 
 See the Example g uares> 
 
 Here you fee by the Example, that there is 4 
 Squares of Flooring, wanting 4 Feet ; or 3 Squares 
 and three Quarters of a Square, and 21 Feet. 
 
 If the Dimenfion had been Stone Pavement of 
 Mafonry, then there had needed no Divifion, for 
 they work by the Foot ; lb there had been 396 Feet 
 of Pavement with broad Stone, at lo much per Foot. 
 
 But if it had been Plaiftering, or Painting, or 
 Wainfcoting, or Hangings, or fuch like , then you 
 muft have brought the Feet into Yards, which is 
 done by dividing 396 Feet by 9 Feet, which is the 
 Number of Feet contained in a fuperficial Yard (or 
 Quadrate) being 3 Feet in Length, and 3 Feet in 
 Breadth. See 
 
Book IV. of Carpenters Work. ipj 
 
 Seethe Example — (44 Yards. 
 
 ■■99 
 
 Thus having divided 396 Feet by 9 Feet (or one 
 Yard) you have 44 Yards in the Quotient, which 
 is the Number of Yards contained in a Superficies 
 of Plaiftering, or Painting, or Wainfcot, or Hang- 
 ings, or fuch like, being 22 Feet in Length, and 
 18 Feet in Breadth. 
 
 Another Example in Feet. 
 
 Suppofe a Floor or Partition be 146 Feet in 
 Length, and 97 Feet in Breadth. 
 
 Feet. 
 
 Set down the Dimenfion thus 146 
 
 97 
 
 1022 
 
 1314 
 
 Being multiplied, theProdud is — —14162 Feet, 
 
 Which being divided by 100, produceth 14 1 
 Squares, and 62 Feet : Or you need not divide it, 
 but read it thus. One hundred forty, one hundred 
 and fixty two Feet, which is 141 Squares and an 
 half, and 12 Feet. 
 
 If you were to bring the fame Dimenfion into 
 Yards, then you muft divide the Produd of the 
 Multiplication, being 14162, by 9 Feet (or one Yard) 
 as you did in the foregoing Example. 
 
 (5 Feet. 
 
 See this Example—-; — (1573 Yards. 
 
 999-9 
 
 So 
 
1 9 6 MEASURING Book IV. 
 
 So you have in the Quotient 1573 Yards and 5 
 Feet remaining, which is half a Yard and half a 
 Foot Thus much may ferve for meafuring of Di- 
 menfions of whole Feet. 
 
 1 jhall now proceed to Dimenfions in Feet , and Inches , 
 which is fomethtng more difficult. 
 
 Suppofe a Floor, or Partition, or any other fu- 
 perficial thing, as Plaiftering, or Painting, &c. be 
 2 1 Feet, and 6 Inches in Length, and 15 Feet, and 
 6 Inches in Breadth ; and it is required to know how 
 many fuperficial Feet there is contained therein, and 
 confequentiy how many Squares, or Yards. 
 
 Feet In. 
 
 Set down your Dimenfion thus — - • 
 
 105 
 
 21 
 
 Then multiply the Feet, and the Pro-? 
 
 dudt is 
 
 Then for the 6 Inches in Length, and 6 in 
 Breadth, multiply them Diagonal or Crofs-wife into 
 the Feet, faying 15 times 6 Inches, (or 6 times 15 
 Inches) is 7 Feet and 6 Inches : Or thus more 
 briefly, (6 Inches being half a Foot) fay, The half 
 of 15 Feet is 7 Feet and 6 Inches, which you muft 
 add to the former Produft 315. This being done, 
 in the next place you muft multiply the 6 Inches 
 in the Breadth into the 21 Feet in the Length ; fay- 
 ing, The half of 21 Feet is 10 Feet and 6 Inches, 
 which you muft likew'ife add to the reft. 
 
 Then 
 
Book IV. of Carpenters Work. 1 97 
 
 Then laft of all, you muft multiply the Inches of 
 the Breadth into the Inches of the Length, faying, 
 6 times 6 is 36, which 36 is accounted but for 3 In- 
 ches, the Realon whereof, in the next following 
 Page but one, fhall be (hewn. 
 
 For Note, How many times 12 you have in the 
 Produd of the Inches, being multiplied in them- 
 ielves, fo many Inches you muft add to the former 
 Work, as in this Example, you have 3 times 12 in 
 36 : As you may fee in the following Example. 
 
 Whence Note, That it is ufual to begin the Mul- 
 tiplication towards the left Hand firft, namely, the 
 Feet into the Feet, and afterwards the Inches into 
 the Feet, &c. 
 
 Example of Operation. 
 
 Feet In. 
 
 The Length, 2i\/ 6 
 
 The Breadth, 1 $/\6 
 
 The Feet multiplied - 
 
 The Produd of the Feet - — — 
 The half of 15 Feet, or the Produd 
 of 6 Inches, being multiplied by 15 
 
 Feet — — — 
 
 The half of 21 Feet, or the Produd 
 of 21 Feet, being multiplied by 6 Inches 
 The Produd of 6 Inches multiplied 
 by 6 Inches, which produces 36, or 
 3 times 12, which, as you read be- 
 fore, is 3 Inches — — 
 
 . 10 S 
 
 21 
 
 315 
 
 , 1 
 
 ^ 7 
 
 6 
 
 } IO 
 
 6 
 
 J 
 
 3 
 
 333 
 
 — — 
 
 3 
 
 Then 
 
ip8 
 
 MEASURING Book IV. 
 
 Then add your Sums together, beginning at the 
 Inches on the right Hand, faying, 3 and 6 is 9, and 
 6 is 15 Inches, which is 1 Foot and 3 Inches; then 
 fet down 3 under the Column of Inches, and carry 
 1 to the Feet, and fay, 1 that I carry and 7 makes 
 8, and 5 makes 1 3 ; then fet down 3 in the place 
 of Units, under Feet, and carry one to the next 
 place, faying, 1 that I carry and 1 is 2, and 1 
 makes 3, which I fet down on the left Side of the 
 former 3 (to wit) under the place of Tens : Then 
 I proceed in the next place to 3, which being in 
 the place of Hundreds is 300, and fet that 3 down 
 on the left Side of the two other. 
 
 So the whole Product of 21 Feet and 6 Inches, be- 
 ing multiplied by 15 Feet and 6 Inches, is 333 Feet 
 and 3 Inches, as you have it above in the Example : 
 
 
 F 2 
 
 2d: <T 
 
 s 
 
 s 
 
 
 F , x 
 
 I think it convenient 
 here to add a Geometri- 
 cal Figure for the better 
 underftanding of what 
 hath been faid. 
 
 By this Geometrical 
 Figure, you may per- 
 ceive the Truth of what 
 hath been faid. 
 
 ^ Firft, Here you fee that the whole Squares or 
 Feet are 21 in Length, and 15 in Height or Breadth, 
 which produce 315 Feet. 
 
 Secondly, In the upper Part of the Figure, you 
 fee 21 half Squares or half Feet, which is, the 6 In- 
 ches which you multiplied by the 21 Feet, the Pro- 
 dud whereof was 10 Feet and an Half; fo in the 
 Figure, accounting two half Squares (or half Feet) 
 for one whole one, you will find 10 Feet and an 
 Half, as before by Arithmetick. Thirdly, 
 
Book IV. of Carpenters Work. 1 9 9 
 
 Thirdly, On the right Side of the Figure, you lee 
 1 5 Oblongs or half Feet, which are the fix Inches 
 that you multiplied by the 15 Feet, which did pro- 
 duce 7 Feet and an half 3 fo in the Figure, account- 
 ing two halfs for one whole, you will have 7 Feet 
 and an Half, as you had before by Arithmetick. 
 
 Fourthly, In the uppermoft Angle, on the right 
 Side of the Figure, you lee a little Square, which is 
 6 Inches in Length, and 6 Inches in Breadth : This 
 is the 6 Inches that you multiplied by 6 Inches, 
 which produced 36 Inches, for which there was let 
 down but 3 Inches : The Realon whereof is this ; 
 
 Thefe 3 Inches are the f of 12 Inches, fo is 36 
 the ? of 144 Inches, of which I have declared be- 
 fore, that there is 144 Iquare fuperficial Inches in a 
 fuperficial Foot. 
 
 Therefore for Conveniency and Brevity in adding 
 feveral Dimenfions together, the odd Inches being 
 multiplied in themfelves, 144 Inches arc accounted 
 as 12 Inches and 12 of 144 are accounted as one 
 Inch. 
 
 Now if you will give your felf the Trouble to tell 
 the Squares in the Geometrical Figure, which Squares 
 represent Feet, and the half Squares or Oblongs 
 which reprelent half Feet, and add them together, 
 you will find that they make 333 Feet befides the 
 little Square in the upper Angle of the Figure on 
 the right Side, which reprelents 3 Inches or - of 
 a Foot , fo you will have 333 Feet and 3 Inches, 
 as was produced by Arithmetick before. 
 
 By what hath been declared, I think it eafy for 
 a mean Capacity to underhand how to c? ft up the 
 foregoing Dimenfion, and bring it into I eet. 
 
 Now 
 
200 
 
 MEASURING Book IV. 
 
 Now to know how many Squares of Carpenters 
 Work there are in this Dimenfion, whofe Producft is 
 333 Feet and 3 Inches ; you muft either divide 333 
 by ioo, or elfe cut off the two laft Figures with a 
 Stroke thus 3 1 3 3 and read it thus, 3 Squares, and 
 3 3 Feet and 3 Inches. 
 
 So in 14162 Feet, cut off the two laft Figures 
 thus ; 141162, and then you read it 141 Squares 
 and 62 Feet. 
 
 So in otherNumbers, confifting of three Figures 
 or more, cut off the two laft Figures to the right 
 Fland, and the remaining Figures will be Squares. 
 
 But if you would know how many Yards there is 
 in 21 Feet and 6 Inches in Length, and 15 Feet 
 and 6 Inches in Breadth ; you muft, as before is 
 taught, divide the Product of the Dimenfion, being 
 333 Feet, by 9, being the Number of Feet in a 
 fquare Yard fuperficiaL 
 
 As in Example, 
 
 So your Quotient is 37 Yards; the odd 3 Inches 
 muft be added after you have divided. 
 
 tfake another Example . 
 
 & 
 
 mCn Yards. 
 
 Suppofe you have a Floor or Partition, that is 
 35 Feet and 10 Inches in Breadth, and 46 Feet 9 
 Inches in Length. ^Set down your Di~ 
 menfion as before, and multiply the / 
 
 whole Feet 46, by the whole Feet 35; 4 \/ °9 
 
 and the Work will ftand thus, 35 / \ 10 
 
 Secondly, Multiply the Inches into 230 
 the F eet crols- ways, laying 9 times 35 138 
 
Book IV. of Carpenters (Vork. i o i 
 
 ^ which muft be let apart by it leli^ as you 
 may lee in the Example in this Page. 
 
 Thirdly, You muft multiply the io Inches by the 
 4 6 Feet, which produces 460 5 this you muft add to 
 the 3 15, which before you fet apart, and they make 
 775 Inches, which you muft divide by 12 to 
 bring the Inches into Feet, which being divided 
 the Quotient is 64 Feet, and 7 Inches remaining,' 
 which 64 you muft add to the Multiplication of 
 the Feet, and the 7 to the Inches. 
 
 See the Example of the Inches mul- 46X/09 
 tiplied into the Feet. 3 10 
 
 The Produdl of the 10 Inches be- 
 ing multiplied into 46 Feet is 460 Inches 
 
 The Product of 9 Inches multi- 
 plied by 3 s Feet is 315 Inches 
 
 Which being added, is 775 Inches 
 
 The 775 Inches being divided by 
 12, produces, as in the Margin, 64 Feet 
 and 7 Inches, which muft be added to 
 the Multiplication of the Feet in the preceding Pao-e 
 and then the Work will ftand thus: 46\^^p 
 
 35 /\io 
 
 Fourthly, You muft multiply the 230 
 10 Inches belonging to the Breadth, 138 
 by the 9 Inches belonging to the 64 n 
 Length, and the Produdl will be 90 
 Inches of 144 Inches ; or 7 Inches and 
 an Half of 12 Inches, for if you di- 
 
 P 
 
 *K7 
 
 i pep: 
 
 vide 
 
201 
 
 MEASURING Book IV* 
 
 vide 90 by 12, you will have 7 in the Quotient, 
 and 6 remaining, which 7 in the Quotient is 7 In- 
 ches, and mull be added to the Place of Inches ; 
 and the 6 that remains over and above the Quoti- 
 tient, being the i of 12, is half an Inch, which is 
 feldom accounted in Carpenters Work : But al- 
 though it is not ufually done, yet if you are mind- 
 ed to^be exadt in Meafuring, you may fet down thefe 
 odd Parts of Inches by themfelves, and at laft add 
 them all together, and for every 12 of thefe Parts 
 you may add 1 Inch, and for 144 of them you muft 
 add 1 Foot, as you read before. 
 
 Then adding alL your Sums together, you find 
 that 46 Feet 9 Inches, multiplied by 55 Feet 10 
 Inches, is 1675 Feet 2 Inches. 
 
 Which if you are to bring into Squares, you cut 
 off the two Figures next the right Hand thus 16 [ 7 y 
 and then you read it 16 Squares and 75 Feet (or 16 
 Squares and 4 of a Square.) 
 
 An Example of the whole Work. 
 
 Feet In. 
 
 The Length, 
 The Breadth, 
 
 The Product of 46 by 5, 
 The Product of 46 by 3, 
 
 The Produdt of all the Inches multi 
 
 230 
 
 138 
 
 plied by all the Feet, 
 
 64 07 
 
 The Produdt of the) 10 Inches mul- 
 
 tiplied by 9, 
 
 The whole Frodudt 
 
 16 1 75 02 
 
 Alfo 
 
Book IV. of Carpenters Work. 203 
 
 Alfb by the way, take Notice that Girders Ends, 
 and Ends of Breft-Sommers, and Places, and fuch 
 like, muft be remember’d and let down in your 
 Bill of Mealurement. 
 
 Alio you muft allow in your Meafure, for the 
 Ends of the Joifts that lie in the Walls ; if it be 
 Flooring unboarded that you are to meafure, then 
 you ought to allow 9 Inches into each Wall that 
 way the Joift Ends are laid. 
 
 But if you meafure Flooring and Boarding, then 
 if you allow 6 Inches, it is reafbnable, becaufe 
 though the Timbers go into the Walls 9 Inches, yet 
 the Boarding goeth but home to the Wall. 
 
 And as you add thefe Things, you muft alfo re- 
 member to dedu< 3 : the Stairs and Chimneys where 
 the Workman findeth Materials, elfe not. 
 
 Thus much will ferve as to the Meafuring of 
 Flooring, Partitioning, and fuch like ; In the next 
 Place I fhall treat 
 
 Of the Meafuring of Hoofs . 
 
 Suppole a Building to be 30 Feet in Breadth, 
 from the Outfide of one Wall to the Outfide of the 
 ether, and 65 Feet in Length, from out to out of 
 the Walls, and you would know how many Squares 
 of Roofing there are in the Roof of this Building. 
 
 If the Roof be true Pitch, you need notmeaiure 
 the Length of the Rafter, but take this for a gene- 
 ral Rule : 
 
 That the Length of the Rafter is J of rhe Breadth 
 of the Building, therefore the Breadth being 30 
 Feet, the Length of the R_after will be 22 Feet and 
 6 Inches, whi.ch is } of 30) which being doubled 
 
 P 2 (F<fr 
 
io 4 MEASURING Book IV. 
 
 (for fo it muft be for both Sides) makes 45 Feet : 
 Then multiply 45 by the Length of the Building 
 being 65, and the Product will be 29 | 25, which 
 is 29 Squares and a Quarter. 
 
 But the ufual way, which is fomething briefer, 
 is thus: Multiply the Length 65 by the Breadth 
 30, and that produces 1950 Feet for the flat of the 
 Building ; then add half the Number of Feet in the 
 flat (to wit) 975 to 1950, and the Product is 2925 
 as before. 
 
 You may add the half thus, laying, The half of 
 19 is 9, then becaufe twice 9 is but 18, and there 
 remains 1 , you muft carry that one to the next Fi- 
 gure being 5, and for that i, which remained of 
 the 19 after it was halfed, you muft add 10 to 5, 
 which makes it 15. 
 
 Then fay. The half of 15 is 7 and 1 remaining, 
 add that 1 to the Cypher, and it makes 10. 
 
 Then lay. The half of 10 is 5. 
 
 See both the Examples . 
 
 The Length of the Rafter on one Side 
 of the Roof 
 
 The Length of the Rafter on the 
 other Side 
 
 The Length of bothRafters being added 
 The Length of the Building to be mul- 
 plied by 45 
 
 The Product 
 
 1 
 
 Feet 
 
 In, 
 
 22 
 
 6 
 
 22 
 
 6 
 
 45 
 
 O 
 
 65 
 
 O 
 
 225 
 
 270 
 
 2925 
 
 O 
 
 An 
 
Book IV. of Carpenters Work. 205 
 
 An Example of the other way , called Menfuration 
 on the Flat. 
 
 .. ^ n r ^. v ; \ r 
 
 Feet 
 
 In. 
 
 The Length, 
 
 65 
 
 0 
 
 The Breadth, 
 
 30 
 
 0 
 
 The Product of the Length by the 
 
 
 
 Breadth on the Flat, 
 
 1950 
 
 0 
 
 The half Flat to be added to the Flat, 
 
 975 
 
 0 
 
 The Producft of the Flat and half 
 
 2925 
 
 0 
 
 Note , When you meafure Roofing this laft way, 
 being called Meafuring on the Fiat, you muft write 
 over your Dimensions, Flat and Half or i i to figni- 
 fy, that after you have multiplied the Length by the 
 Breadth, you muft add half that Product to it ielf. 
 
 .Note* If the Building that you are to meafure be 
 broader at one End than it is at the other, you muft 
 meafure the Breadth in the Middle, both for the 
 Flooring and the Roofing; or elfe meafure each 
 End, and add the Sums together, and take half the 
 Producft : As, fuppofe a Building to be 14 Feet 
 wide at one End, and 12 Feet wide at the other, 
 they being added, make 26, the half whereof is 13 
 Feet for the Width of the Building. 
 
 How to meafure a Gable-end. 
 
 Suppofe a Timber Building be 40 Feet in Breadth, 
 and you have meafured all the Carcafs of the Build- 
 ing, exsept the Gable-end, or Ends ; 
 
 f 
 
 To 
 
MEASURING Book IV. 
 
 To meafure this Gable-end, multiply half the 
 Bale by the whole Perpendicular ; or half the Per- 
 pendicular by the whole Bale, gives you the Area, 
 or Content of it. 
 
 I call the Plate, whereon the Rafters ftand, the 
 Bafe. 
 
 The Bafe being 40 Feet, the Perpendicular is 2 z 
 Feet and 4 Inches. 
 
 See the Example, 
 
 The half of the Bafe, 
 
 The whole Perpendicular, 
 
 The Frodudt of 2 multiplied by 20, 
 The Product of 20 by 20, 
 
 The Product of the 4 Inches multi- 
 plied by the 20 Feet, 
 
 The total Product, 
 
 &be other Example . 
 
 The whole Bale, 
 
 The half of the Perpendicular, 
 
 The Prcduft of 40 by 1, 
 
 The Product of 40 by 10, 
 
 The Frodud of the 2 In. by the 4oFeet, 
 
 The total Produ< 3 :, 
 
 Feet 
 
 In. 
 
 20 
 
 0 
 
 22 
 
 4 
 
 40 
 
 0 
 
 400 
 
 0 
 
 6 
 
 8 
 
 446 
 
 8 
 
 Feet. 
 
 In. 
 
 40 
 
 0 
 
 ii 
 
 2 
 
 40 
 
 0 
 
 40 
 
 0 
 
 6 
 
 8 
 
 446 
 
 8 
 
 Thus you fee both ways agree, and the Gable- 
 end contains 4 Squares and 40 Feet 8 Inches. If 
 
Book IV. of Carpenters Worn. 2 07 
 
 If there be two Gable-ends equal, or broth alike 
 in the Building, then you tnuft fet down the total 
 Produd twice. 
 
 If there be four Gable-ends, and each of them of 
 one Bignels, then you muft fet down the total Pro- 
 dud four times. 
 
 Note., If the Building be fqitare Qviz .) hath four 
 right Angles, and the Breadth 40 Feet $ and the 
 Roof true Pitch $ 
 
 The Length of the Rafter, and alfo of the Hip 
 Rafter, and the Angles which they make ; alio the 
 Length of the Diagonal Line, and of the Perpen- 
 dicular, are as in the following Table. 
 
 As you may fee by the Figure A B C D hereunto 
 annexed. 
 
 The Breadth of the Building, 
 
 The Length of the Rafter, 
 
 The Length of the Hip Rafter, 
 
 The Length of the Diagonal Line, 
 
 The Length of the Perpendicular, 
 
 Rafter Angles 
 Hip Angles 
 
 Explanation of the Fig. ABC D. 
 
 A B. The Breadth of the Building. 
 
 AP. The half Breadth. 
 
 AE, or EB, The Length of the Rafter. 
 RP The Perpendicular. 
 
 ^4 
 
 Feet In. 
 
 40 00 
 
 30 00 
 
 36 00 
 
 56 06 1 
 
 22 04 i 
 
 Deg. Min. 
 48 10 
 
 41 50 
 
 3 $ 22 
 51 38 
 
ao8 MEASURING Book IV. 
 
 AF, or FB, The Length of the Hip Rafter. 
 
 AGH V The. Apgle that the Hip makes at 
 Top, and K i , the Meafure of it. 
 
 HAG, The Angle that the Hip makes at the 
 Foot, and LI the Meafure of it. 
 
 P A E, The Angle that the Rafter makes at Foot, 
 and P a the Meafure of it. 
 
 AEP, The Angle that the Rafter makes at Top, 
 and the Meafure of it. 
 
 LIP, The Quadrant of a Circle divided into 90 
 Degrees, for the meafuring of the Angles. 
 
 A C, The Diagonal. 
 
 LM, The Line over which the Rafters muft be 
 placed, that the upper Ends of the Hip Rafters are 
 fixed to. 
 
 The Length of the Hip Rafter is found, by 
 taking the Length of the Rafter A E, and Getting it 
 on the Perpendicular from P to F, draw F A the 
 Hip Rafter. ' 
 
 If you have a Roof of any other Width, you may 
 defcribe it from this Figure, or you may find the 
 Length of the Hip Rafters, and the Length of the 
 Diagonal, and likewife of the Perpendicular by the 
 Rule of Three, and the foregoing Table. 
 
 The Angles are always equal to thofe in the Ta- 
 ble, let the Width be what it will, provided the 
 Building be right angled, and the Roof the ufual 
 true Pitch (that is, the Rafters Length to be \ of 
 the Breadth of the Building. 
 
 S to Meafure a Hipt Roof. 
 
 Suppofe a Building to be 60 Feet in Length, and 
 40 Feet in Breadth, ftanding alone, being Hipt from 
 
 each 
 
tne unas uemg a j.iiaugic, wm^n 
 
 fure, as you did the Gable-end. 
 
 Multiply 40 Feet being the Bafe, by 15 Feet be- 
 ing one half of the Rafter (or Perpendicular as you 
 may call it, although it lean from an upright) and 
 the Product is 600. 
 
 The 
 
p 
 
 Place this between Page 208 and 209. 
 
 • -V* V14V JL» 
 
 i-'uimmg. 
 
 tfo Me a fun a Hipt Roof. 
 
 Suppofe a Building to be 60 Feet in Length, and 
 40 Feet in Breadth, (landing alone, being Hipt from 
 
 cacli 
 
Book IV. of Carpenters Work. 2 op 
 
 each Angle, the Angles being right, and the Roof 
 true Pitch, and it is required to know how many 
 Squares of Roofing there is in it. 
 
 Multiply the whole Length, being 60 Feet, by 
 30 Feet, the Length of the Rafter, for one Side of 
 the Roof. 
 
 Or add 30 and 30 together, they make 60 for both 
 Sides of the Roof, then multiply 60 by 60, the Pro- 
 duct is 3600, or 36 Squares in that Hipt Roof. 
 
 Which I thus de monfir ate. 
 
 The Length of the Roof on the Ridge, from Flip 
 to Hip is 20 Feet, and the Length of the Roof at 
 the Plate is 60 Feet. 
 
 Add thefe two Sums together, and they make 80 
 Feet, whereof take half, that is 40 Feet, which is the 
 mean Length of the Roof, if you meafure in the 
 Middle between the Top and Bottom; then multi- 
 ply this 40 Feet by 60 Feet, being the Length of 
 both Rafters on each Side the Roof, and the Pro- 
 duct is 2400, which is the Meafure of the two Sides 
 of the Roof 
 
 Proceed we, in the next Place , to the two Ends ; 
 
 Being each of them 40 Feet in Length, and the 
 Rafter of each End 30 Feet in Length ; each of 
 the Ends being a Triangle, which you mult mea- 
 fure, as you did the Gable-end. 
 
 Multiply 40 Feet being the Bafe, by 15 Feet be- 
 ing one half of the Rafter (or Perpendicular as you 
 may call it, although it lean from an upright) and 
 the Product is 600. 
 
 The 
 
2io MEASURING Book IV. 
 
 The other End of the Roof being the fame as 
 this, add 600 to 660 it makes 1200, which being 
 added to 2400, the Produd of the two Sides, make 
 3600, or 36 Squares as before, which was to be de- 
 monftrated. 
 
 Suppofe you were to meafure the Roofing of a cor- 
 ner Houfe, being 60 Feet long, and 40 Feet broad, 
 and joining to other Houles each Way, then there will 
 be but one Hip, and one Sleeper, yet the Dimenfions 
 will be the fame, and the lame Number of Squares in 
 it, as in the former that was Hipt, provided the 
 Building be Square, and the Roof true pitch. 
 
 See the Examples . 
 
 
 - 
 
 
 Feet 
 
 In. 
 
 The whole Length of the Roo£ 
 
 60 
 
 00 
 
 The Length of both Rafters, 
 
 60 
 
 00 
 
 The Produft or Number of the Squares 
 
 3600 
 
 00 
 
 Sthe Demon/} ration. 
 
 
 Feet 
 
 In. 
 
 The Length of the Roof on the Top is 
 To which add the Length of the Roof 
 
 20 
 
 00 
 
 at the Plate 
 
 60 
 
 00 
 
 The Sum is 
 
 80 
 
 00 
 
 The Length of both Rafters 
 
 Being multiplied by 40, the Mean* 
 between 60 and 20 5 
 
 60 
 
 00 
 
 40 
 
 00 
 
 The Product is 
 
 Which is the Content of the two Sides. 
 
 2400 
 
 00 
 
 
 Now 
 
Book IV. of Carpenters Work. 1 1 1 
 
 Now for the two Ends. 
 
 Feet In. 
 
 The Length of the Bafe, 40 00 
 
 Multiplied by the half Length of the Rafter, 15 00 
 
 200 00 
 40 
 
 Produces 600 00 
 
 To which add the Meafurement of the a 6qo oq 
 
 other End, ^ 
 
 TheSumis the Content of both the Ends, 1200 oa 
 To which add the Meafure of both Sides, 2400 00 
 
 The Sum is the Area or Content of ^3600 00 
 
 the whole Roof, as before, $ ~ 
 
 How to find the Length of the Hip for a Roof that 
 is 20 Feet in Breadth , by the Rj'le of L hree . 
 
 You fee by the foregoing Figure ABCD, that a 
 Roof being 40 Feet in Width or Breadth, gives a 
 Hip Rafter that is 36 Feet in Length 3 therefore 
 ftate your Queftion thus : If 40 give $6, what will 
 20 give? 
 
 36 * 
 
 20 7 * 0 /' ^ 
 
 — — ( iS 
 
 A00\ 
 720 ' 
 
 * 
 
 Multiply and Divide, and you 
 will find 18 
 
 After thefame Manner you may find any ofthereft. 
 You muft remember to add the R-afters Feet, and 
 
 Eves Boards, in the Bill of Meafurement. 
 
 CHAP. 
 
212 
 
 MEASURING Book IV, 
 
 CHAP. IV. 
 
 Meafuring of Glaziers Work. 
 
 T HE Meafuring of Glaziers Work is the fame 
 as of Carpenters Work ; only Carpenters Work 
 is not meafured any nearer than two whole Inches. 
 
 But Glaziers Work isfometimes meafured to f of 
 an Inch ; and the beft and readieft way to meafure 
 Glazing, is to take the Dimenfions with a Hiding 
 Ruler, fuch as Glaziers generally ufe, which Ruler 
 is divided Decimally, a Foot into ioo Parts, fo 
 that a whole Foot contains ioo Parts or Divifions; 
 sf of a Foot contains 75 Parts ^ ~ of a Foot contains 
 50 Parts,, and i of a Foot contains 25 Parts. 
 
 Thofe that defire to meafure the Decimal Way, 
 I refer them to a Treatife put forth by my Friend 
 ho. Hammond (Entituled, A new and exalt Way of 
 j\ lenfiiration ) which I writ for him when I was Ap- 
 prentice, and indeed it is a very .ready Way of 
 Meafuring, provided our two Foot Rules were di- 
 vided into 20 equal Parts, or a Foot Rule into 10 
 Parts (or, as I may call them, Inches) and every 
 one of thofe Parts divided into 1 o equal Parts. 
 
 But to come to the Thing intended (to wit) to 
 meafure Glazing by vulgar Arithmetick, and with 
 a two Foot Ruler divided into 24 Parts, and each 
 of thofe into 8 Parts, as they are commonly made 
 and ufed. 
 
 Suppofe you have a Pane of Glafs which is 5 Feet 
 8 Inches and an half in Length, and 5 Feet 7 In- 
 ches and 4 in Breadth, and you would know the 
 Area or Content thereof : 
 
 Set 
 
Book IV. of Glaziers Work. 2 1 3 
 
 Feet In. Parts. 
 
 Set down your Dimenfions thus, ^ ^ ^ ^ 
 
 And read it, 5 Feet 8 Inches and 6 • 
 
 Parts of 12, which is theT Inch ; by 
 5 Feet 7 Inches, and 3 Parts of 1 2, which is the \ Inch. 
 
 Then multiply the Feet and Inches one by the 
 other, as you are taught in the foregoing Pages, fay- 
 ing, 5 times 5 is 25 Feet 3 which being let down un- 
 der the Feet, multiply crols-ways 5 by 8, which is 
 40 Inches, or 3 Feet and 4 Inches. This being 
 let down, 
 
 In the next Place, multiply 7 by 5, which is 35 
 Inches, or 2 Feet and 11 Inches. This being let 
 down, 
 
 In the next place multiply the 7 Inches by the 8, 
 it is 56 Inches, of which 144 make a Superficial Foot, 
 for which you mu ft fet down 4 Inches, becaule you 
 can have but 4 times 12 out of 56, and there re- 
 mains 8. Therefore I fet down 4 Inches and 8 
 Parts, of which Parts 12 make an Inch, 12 of which 
 Inches are accounted a fuperficial Foot. 
 
 In the next place multiply the Parts of Inches 
 crols-ways 3 firft into the Feet, faying, 6 times 5 make 
 30, which is 2 Inches and 6 Parts. This being let 
 down, 
 
 Say 5 times 3 is 1 5, or 1 Inch and 3 Parts, which 
 you muft put alio under the reft. 
 
 Then multiply the Parts of Inches crols-ways into 
 the Inches, faying, 6 times 7 is 42, which is 3 Parts 
 and an half, for which I let down 3 under the Place 
 of Parts of Inches, and 6 againft it towards the right 
 Hand, which fignifies i Part of a Part (for 12 of 
 thele Parts make one fuch Part, whereof there are 
 
 12 
 
214 MEASURING Book IY. 
 
 12 in an Inch) Line Meafure. This being done, 
 multiply 3 by 8 (which is 3 Parts by 8 Inches) 
 that makes 24 Parts, for which you muft fet down 
 2 under the Place of Parts of Inches. 
 
 Note, That as in multiplying the Inches by the 
 Inches, you account every 12 of the Product for x 
 Inch, fo in multiplying the Parts of Inches into the 
 Inches, you muft account every 12 of the Product 
 for 1 Part ; as in multiplying 3 Parts by 8 Inches 
 produceth 24, which is twice 12, therefore I fet 
 down 2 under the place of Parts. 
 
 Laftly, you muft multiply the Parts of Inches by 
 the Parts (if you will meafure fo near ) faying, 3 
 times 6 is 18, which is 1 Part and a half of a twelfth 
 Part of an Inch Line Meafure ; fee the Dimenfion 
 caft up. 
 
 Example. 
 
 X 11V. x J * J X XO 
 
 TheProduft of6 Parts by 7 Inch, is 
 
 The total Product is 
 
 Feet In. 
 
 Parts. 
 
 05 
 
 8 
 
 6 
 
 os 
 
 7 
 
 3 
 
 *S 
 
 O 
 
 0 
 
 03 
 
 4 
 
 0 
 
 02 
 
 11 
 
 0 
 
 00 
 
 4 
 
 8 
 
 00 
 
 2 
 
 6 
 
 00 
 
 1 
 
 3 Pa. 
 
 00 
 
 0 
 
 3 6 
 
 ; 00 
 
 0 
 
 2 0 
 
 00 
 
 0 
 
 0 li 
 
 31 
 
 11 
 
 io 7 I 
 
 Thus 
 
Book IV. of Glaziers Work 2 1 5 
 
 Thus you lee that the Area or Content of 5 Feet 8 
 Inches and an Half (or 6 Parts of 12) in Length, be- 
 ing multiplied by 5 Feet 7 Inches and 4 (or 3 Parts 
 of 12) in Breadth, is 31 Feet, 11 Inches and 10 Parts 
 of an Inch, accounted to have 12 Parts, and 7 and £■ 
 of one of thofe Parts $ but as for the laft 7 Parts and 
 4, they are not worth the fetting down nor taking 
 notice of, the Value of thofe Parts being lo Imall. 
 
 The Truth of this Area or Content may be 
 proved three leveral Ways. 
 
 Firft Decimally, Second Geometrically, Thirdly 
 by Vulgar Arithmetick. 
 
 Firft Decimally thus, the Decimal of 8 Inches and 
 l is 71, and the Decimal of 7 Inches and i is 60 
 and fome fmall matter more : Therefore 
 in Decimals, I fet down the Dimenfion 
 thus j 
 
 And having multiplied it, I find the 34260 
 Producft to be 319760, then cut off the 2855 
 
 2 Figures next to the Left Hand with a ~"~ 
 
 Stroak, as you fee in the Example, and 3 1 1971 
 they represent 31 Feet. 
 
 Then the Remainder is 2760 of ioooo, and by 
 cutting of 2 Figures more thus 97I60 it is 97 of 100 
 and fomething more, it is near 98 Parts of 100, 
 92 of which Parts, is the Decimal of n Inches, and 
 the 6 Parts which remain are of an Inch ; fo 
 that by this you fee the Decimal Way of Meafuring 
 and this Vulgar doth agree. . 
 
 Secondly, you may prove it Geometrically, if you 
 defer ibe a Geometrical Figure (after the lame Man- 
 ner that the Figure is delcribed inP. 189 of Carpen- 
 ters Work , being drawn according to a Scale f) 5 Feet 
 
 8 Inches 
 
 F. P, 
 
 5 57* 
 
 I 560 
 
21 6 MEASURING Book IV. 
 
 8 Inches and i in Length, and 5 Feet 7 Inches and 
 i in Breadth ; and then divide the Feet and Inches, 
 and Parts ; and you will find it to agree with the 
 Area or Content before produced. 
 
 Thirdly, it may alfo be proved by Vulgar Arith- 
 metick thus. 
 
 Reduce the Feet and Inches into the leaft Deno- 
 mination (to wit) Quarters of Inches, and then mul- 
 tiply them one by another. 
 
 To do this, multiply 68 Inches (which are the 
 Inches in 5 Feet and 8 Inches) by 4, and that 
 brings them into Quarters of Inches Line Meafure, 
 then add 2 to the Produd, for the half Inch in 
 the Length. 
 
 68 
 
 4 
 
 272 
 
 2 
 
 Thus in the Length you have 274 Quarters of Inches. 
 
 The Breadth being 5 Feet 7 Inches and -f, multi- 
 ply 67 (the Inches in 5 Feet and 7 Inches Line 
 Meafure) by 4, being the Quarters in an Inch Line 
 Meafure, and to the Frodudt add i, which is the i 
 Inch in the Breadth. 
 
 67 
 
 4 
 
 268 ' 
 
 1 
 
 And in the Breadth is contained 269 Quarters of Inch. 
 
 Thus 
 
Book IV. Of Glaziers Work. 1 1 7 
 
 Thus having brought theLength and Breadth into 
 the leaft Denomination (to wit. Quarters of Inches) 
 I let down my Dimenfion thus, and multiply it : 
 
 " O 1 ' r j 
 
 274 
 
 269 
 
 2466 
 
 i6 44 
 
 548 
 
 The Product is 73706 Quarters. 
 
 Of which Quarters there are 1 6 in an Inch fuperfi- 
 cial Meafiire (to wit, 4 in Length, and 4 in Breadth) 
 and of which Inches there are 144 in a Foot fuper- 
 ficial Meafure (to wit, 12 in Length, and 12 in 
 Breadth.) Therefore I multiply 144, the Inches in 
 a fuperficial Foot, by 16, the Quarters in a fuper« 
 ficial Inch, and 
 
 144 
 
 16 
 
 864 
 
 144 
 
 In a luperficial Foot there is 2304 Quarters. 
 
 Then divide 73706, the fuperficial Quarters of 
 Inches in the whole Dimenfion, by 2304, the fuper- 
 ficial Quarters of Inches in a Foot. 
 
 Ql 
 
 228 
 
21 8 MEASURING Book IV, 
 
 228 
 
 2 > 
 
 7 17 06 ( 31 Feet. 
 
 & 
 
 And the Quotient is 31 Feet, and the Remainder 
 is 2282 Quarters, which are not a Foot. 
 
 Therefore divide 2282 by 192 (which is the Num- 
 ber of Quarters contained in 12 Inches in Length, 
 and 1 Inch in Breadth, which 192 is the Produdt of 
 12 by 16.) 
 
 12 
 
 16 
 
 72 
 12 
 
 192 Quar. 
 
 And the Quotient is 11 Inches, and 170 the Re- 
 mainder, which is 170 Parts, or Quarters of an Inch, 
 in Breadth, and 12 Inches in Length 5 which Inch, 
 as I told you before, contains 192 Quarters. 
 
 Wherefore divide this 170 by 16 (to wit, the 
 Quarters that are contained in an Inch in Length, 
 and an Inch in Breadth.) 
 
 1 
 
 i 70 (10 Parts. 
 
 i 
 
 And the Quotient is 10, and the Remainder 10. 
 
 Thus you lee the Produdt of 5 Feet, 8 Inches, and 
 ) in Length, being multiplied by 5 Feet, 7 Inches, 
 
 and 
 
 17 
 
 2 fi 1 Inches. 
 
 i & Ztit v 
 
 *9 
 
Book IV. Of Glaziers Work . 2 1 9 
 
 and £ in Breadth, is 31 Feet, 11 Inches, 10 Parts, 
 and t| of a Part, which of a Part, is in Value 
 the feme that 7 and £ is, of a Part containing 12, 
 as you found it by the firft Way of working in Vul- 
 gar Arithmetick. 
 
 Another Example. 
 
 Suppofe a Pane of Glafe be 5 Feet, 3 Inches and £ 
 long, and 2 Feet, 4 Inches and ~ broad, and you 
 defire to know the Content thereof. 
 
 Feet. In. Parts. 
 
 Set down your Dimenfion thus ^ 3 6 
 
 J 246 
 
 Then multiply the Feet and Inches one into ano- 
 ther, faying, 5 times 2 is 10 Feet ; fet that down 
 under the Feet ; then multiply crofswife, feying, 3 
 times 2 is 6 Inches ; fet that under the Inches. 
 
 Then fay, 4 times 5 is 20 Inches, or 1 Foot and 8 
 Inches ; fet the 1 under the Feet, and the 8 under 
 the Inches. 
 
 Then fay, 4 times 3 is 12, which is 1 Inch. 
 
 Next multiply the Parts crofsways into the Feet, 
 faying, 6 times 2 is 12, which is 1 Inch, which muft 
 be fet under the Place of Inches. 
 
 Then multiply the 6 Parts by 5 Feet, it makes 30 
 Parts, which is 2 Inches, and 6 Parts ^ which being 
 let down, 
 
 Multiply the Parts into the Inches, feying, 6 times 
 3 is 18, which is 1 Part and an half j for which I 
 put 1 under the Place of Parts, and 6 againft it 
 towards the R ight Hand, which fignifies i or T $ of 
 a Part. 
 
 o.* 
 
 Then 
 
220 MEASURING Book IV. 
 
 Then multiply the 6 Parts by the 4 Inches, it 
 makes 24, which is two Parts ; therefore add 2 un- 
 der the Place of Parts, 
 
 Laftly, multiply the Parts by the Parts, faying, 
 6 times 6 is 36, which is T l of a Part j fet this 3 
 under the 6 on the Right Hand. 
 
 Then adding them all together, the Content is 12 
 Feet, 6 Inches, 9 Parts, and tI of a Part ; which 7 l 
 you need not fet down as you read before, they 
 iignifying very little ; then it will be 12 Feet, 6 
 Inches, and 9 Parts, or \ of an Inch. 
 
 See the whole Work, 
 
 Feet. In. Parts. 
 
 The Length 05 3 6 
 
 The Breadth 02 4 6 
 
 The Produd of 5 Feet by 2 Feet is 10 o o 
 The Produd of 3 Inches by 2 Feet is 00 6 o 
 
 The Produd of 4 Inches by 5 Feet is 01 8 o 
 
 The Product of4lnchesby 3 Inchesis 00 1 o 
 
 The Product of 6 Parts by 2 Feet is 00 1 o 
 
 The Produd of 6 Parts by 5 Feet is 00 2 6 Pa. 
 
 The Product of 6 Parts by 3 Inches is 00 o 16 
 The Produd of 6 Parts by 4 Inches is 00 o 2 
 The Produd of 6 Parts by 6 Parts is 00 o 03 
 
 The Total is 12 6 9 9 
 
 Note, That in meafuring of Glazing, many times 
 in a Building there are feveral Window-frames of one 
 bignefs, and in one Window-frame there are feveral 
 Panes of Glafs of one Bignefs or Dimenfion j as in a 
 
 fix 
 
22 1 
 
 Book IV. Of Glaziers Work. 
 
 fix Light Window there are 6 Panes of Glafs, 3 of 
 them of one Dimenfion, and 3 of another ; fo in a 
 four Light Window there are 4 Panes of Glafs, 2 
 of one bignefs, and 2 of another. 
 
 In a fix Light Window you need meafure but one 
 Pane of a Sort, and fet down each Dimenfion j and 
 becaufe there are 3 of a Sort of one bignefs, fet down 
 3 againft your Dimenfion, which fignifies 3 times 
 that Dimenfion. 
 
 If it be a four Light Window, and have 2 Panes 
 of Glafs of one bignefs, and two of another, then 
 fet down 2 againft your Dimenfion, 
 
 E$atnplf. - 
 
 If you were to meafure a fix Light Window, and 
 the upper Panes were each of them 2 Feet 6 Inches 
 in height (or length) and 1 Foot 2 Inches in Breadth ; 
 and the lower Panes 6 Feet 4 Inches in Length, 
 and 1 Foot 2 Inches in Breadth ; you may make 
 but two Dimenfions of thefe 6 Panes of Glafs. 
 
 Feet. In. Parts. 
 
 Setting them down thus 2 6 o ? (3) 
 
 1 2 o S 
 
 Feet. In. Parts. 
 640 
 120 
 
 The 3 in the Circle thus (3) fignifies 3 times the 
 Dimenfion that it Hands againft ; therefore when 
 you have caft up the Dimenfion, the Produdt thereof 
 
 0.3 
 
 you 
 
an MEASURING Book IV. 
 
 you muft multiply by 3, and fet that down for the 
 3 Panes. 
 
 Example. 
 
 Feet In. Parts. 
 
 2 6 o ? (3) 
 
 i 2 o> 
 
 2 
 
 0 0 
 
 The Produft of one Pane is 2 
 
 6 
 
 4 
 
 I 
 
 II 0 
 
 Which multiply by 3, becaufe £ 
 there are 3 Panes of that bignefs * 
 
 3 
 
 And the Product is 8 
 
 9 0 
 
 Being the Content of the 3 Panes. 
 
 Then the other 3 Panes, being each of them 6 Feet 
 4 Inches in Length, and 1 Foot 2 Inches in Breadth, 
 
 Feet In. Parts. 
 
 Set down thus 6 4 0^(3) 
 
 And multiply 1 200 
 
 The Produft of 6 Feet by one, is 6 00 
 
 The Product of 2 Inches by 6 Feet is 1 00 
 
 The Product of4 Inches by 1 Foot is o 40 
 
 TheProdu6tof2lnchesby4lnchesiso o 8 
 
 The Total Product or Content is 7 48 
 
 Which multiply by 3 
 
 Saying 
 
Book IV. Of Glaziers Work. 223 
 
 Feet In. Pare 
 
 748 
 
 3 
 
 m rwniiB mm 1 iw mmmmmmrnm 
 
 Saying 3 times 8 is 24 Parts or 2 \ 
 
 Inches, which carry to the Inches, and ( o 
 
 fet a Cypher under the Parts, thus \ 
 
 Then fay 3 times 4 Inches is 12 
 Inches, and 2 that I carry is 14 Inches, 
 that is, 1 Foot, and 2 In. fet down the 
 2 In. and carry the 1 Foot to the Feet 
 
 Then fay 3 times 7 is 2 1 Feet, and 1 > 
 that I carry is 22 Feet j fet that down S 22 00 
 
 Add them, and the Product is 22 2 o 
 
 For the Area, or Content of the 3 Panes. 
 
 Feet In. Parts. 
 
 The Content of the firft 3 Panes is 08 9 o 
 
 The Content of the laft 3 Panes is 22 2 o 
 
 - ■ 1 ■■ .1 li n n... .1 m -T . 
 
 Being added together is 30 11 o 
 
 Which is the Content of all the fix Panes in the 
 fix-light Window. 
 
 Then fo many fix-light Windows of the fame 
 bignefs as you have in the Building to meafure, mul- 
 tiply the Number of them by 30 Feet and 1 1 Inches, 
 and the Produft is the Content of them all. 
 
 To meafure a four-light Window. 
 
 A four- light Window having 2 Panes of one big- 
 nefs, and 2 Panes of another ; you may fet down 
 but 1 Pane of each bignefs, and (2) againft it thus. 
 
 Suppofe the upper Panes to be 2 Feet 1 Inch in 
 length, and 1 Foot 8 Inches and l in breadth, a- piece; 
 
 Q. 4 And 
 
214 ME ASURING Book IV. 
 
 . And the lower Panes to be 4 Feet in length, and 
 1 Foot 8 Inches and | in breadth a-piece. 
 
 AJVglll W i L.1I LVVW 
 
 And fet them down thus 
 
 The Content of i Pane is 
 The other Pane being thefame,add 3 
 
 Panes is 
 
 Secondly, Set 
 lower Panes thus 3 
 
 The Content of one Pane is 
 
 O J J 
 
 multiply it by 2 5 or elfe add 
 The Content of both Panes is 
 To which add the Content of 
 the upper Panes 
 
 And theContent of the4Pane$ 
 in the 4 Light Window is 
 
 Feet In. Parts. 
 2 1 o>( 
 1 8 6 S 
 
 2 0 
 
 O 
 
 V 4 
 
 1 
 
 O 
 
 0 
 
 8 
 
 1 
 
 6 
 
 3 7 
 
 2 
 
 <137 
 
 2 
 
 1 7 2 
 
 4 
 
 nenfion of the t 
 
 Feet In. Parts. 
 
 4 ° 
 
 °>i 
 
 1 8 
 
 6 5 
 
 4 0 
 
 O 
 
 2 8 
 
 Q 
 
 2 
 
 
 6 10 
 
 % 
 
 O 
 
 j 6 10 
 
 O 
 
 13 s 
 
 0 
 
 \ 1 2 
 
 4 
 
 \ 20 IO 
 
 4 . 
 
 Hot 
 
Book IV. Of Glaziers Work. 225 
 
 ' How to fet down Dimenfions in your Pocket-Book. 
 
 Note , Before you begin to fet down your Dimen- 
 fions, it is convenient to divide the Breadth of your 
 Page or Leaf into fo many feveral Columns as you 
 think convenient, with Lines drawn of Ink ; '1 he 
 Leaves of your Pocket-Book being of the Breadth of 
 this Book, you may divide a Leaf into four Parts 
 or Columns. 
 
 You muff likewife before you let down any Dimen- 
 fions, exprefs the Workmafter, and the Workmens 
 Names, ahb the Place where,, and the Day of the 
 Month, and Date of the Year that you meafiire. 
 Likewife if the Work that you are to meafure, be 
 Glazed with fquare Glafs, you muff write Squares 
 above your Dimenfions, and over thofe Dimenfions 
 which are glazed with Quarries, you muff write 
 Quarries ; that when you come to make the Bill of 
 Meafurement, you may exprefs them feverally, be- 
 caufe they are of feveral Prices. 
 
 In the next Page I will fet down all the Dimen- 
 fions which you have been taught to caft up in 
 Glazing, and fome others, with the Product to 
 each Dimenfion, 
 
 Glazing 
 
MEASURING Book IV. 
 
 Glazing done by A. B. for C. D. in Long-Acre, 
 and meafured the 22 d of January, 1727. 
 
 Quarries. 
 
 Produdls. 
 
 Squares. 
 
 Produdfe. 
 
 F. I. P. 
 
 F. I. P. 
 
 F. I.P. 
 
 F. I. P. 
 
 5 8 6? 
 
 S 7 35 
 
 31 II 10 
 
 4 3 
 
 1 2 oi 
 
 04 11 06 
 
 5 3 6? 
 246^ 
 
 12 06 09 
 
 2 0 0? 
 
 I 6 o> 
 
 03 00 CO 
 
 2 6 0K3) 
 j 2 0^ 
 
 < _ 
 
 1 08 09 00 
 
 3 0 0? 
 
 2 3 
 
 06 09 00 
 
 <5 4 0K3) 
 12 0^ 
 
 22 02 OO 
 
 609? 
 jo §5 
 
 3 ° 05 03 
 
 Z I 0?(2) 
 
 I 8 6> 
 
 07 02 04 
 
 1 2 .07(2) 
 
 3 0 oi 
 
 07 00 00 
 
 4 0 °?G 0 
 x 8 65 
 
 13 08 00 
 
 • 
 
 52 01 09 
 
 
 96 03 n 
 
 
 
 Explanation of the Columns . 
 
 In the firft Column towards the Left Hand, are the 
 Dimenfions (which you have been taught to caft 
 up) of Glazing done with Quarries. 
 
 In the fecond Column you have the Product of 
 each Dimenfion juft againft it. 
 
 In 
 
Book IV. Of Glaziers Work. zzy 
 
 In the third Column you have 5 Dimenfions of 
 Glazing done with Square Glals. 
 
 In the laft Column you have the Product of each 
 Dimenfion juft againft it ; and at the Bottom you 
 have the total Sum of all the Products in that Co- 
 lumn, being 52 01 09. 
 
 Likewife at the Bottom of the lecond Column you 
 have the total Sum of all the Produbls of the Dimen- 
 fions done with Quarries, which is 96 Feet 3 Inches 
 and 1 1 Parts. As for the odd Parts, you may leave 
 them out when you make your Bill of Meafiirement. 
 
 When you are meafuring, and letting down the 
 Dimenfions in your Book, whether it be of Glazing, 
 or any other Trade, you muft leave every other Co- 
 lumn vacant or empty, that lo having let down all 
 your Dimenfions in your Book (which you gene- 
 rally do before you caft up any) when you caft 
 them Up (which muft be in another Book or Sheet 
 of Paper) you may enter the Produbl of each Di- 
 menfion juft againft it, as you lee in the Page be- 
 fore. 
 
 If there be another to meafure againft you, you 
 muft, after you have done letting down all your Di- 
 menfions, compare your Dimenfions together, to fee 
 if the Dimenfions in both your Books agree, 
 
 tfhe Reafon why you fet down the Produff of each 
 Dimenfion juft againft it in the next Column to- 
 wards the Right Hand , /j. 
 
 If you meafure againft another Meafurer, and there 
 Ihould be a Miftake in either of your callings up of 
 the Dimenfions (as it often happens thro 5 Security or 
 
 Negli- 
 
228 MEASURING Book IV. 
 
 Negligence) then one by reading over the Dimen- 
 fions in his Book, with the Produft to each Dimen- 
 fion as he goes on ; and the other looking in his 
 own Book, the Miftake will loon be found, which 
 muft be rectified between you. 
 
 Therefore, to be certain in cafting up your Di- 
 men lions, you ought to caft them up twice, if not 
 three times ; (to wit) after you have caft them all 
 over once, begin and caft them over again, and lee 
 whether it agrees with your firft cafting up s if not, 
 then caft them up again. 
 
 When you make your Bill of Mealurement, you 
 jnuft fet your Name to it at the lower End of the Bill. 
 
 An Example of a Bill. 
 
 Glaziers Work done by A.B , for C.D, in Long- 
 Acre , and mealured the 22d of January ^ 1727. 
 
 /• s. d. 
 
 For 96 Feet, and 3 Inches of Glazing 1 
 
 done with Quarries, at 3 d. per >2 00 1 V 
 Foot 
 
 For 52 Feet, and 1 Inch of 
 with Squares, at 7 d per F001 
 
 The Sum is 3 10 5? 
 
 Meafnred the Bay and Tear 
 above-written by V. M. 
 
 You may fometimes happen to meet with Panes 
 of Glals of various Forms • as fometimes the Top of 
 a Pane of Glals is concluded with a Semicircle, fome- 
 times 
 
 Glazing 
 
 1 
 
 1 10 4 
 
Book IV. Of Glaziers Work. 229 
 
 times with a Scheme or Segment of a Circle, fome- 
 times Triangular, and fometimes Ovalar, and fome- 
 times an Oval in Glafs, or a round Pane. 
 
 For the Meafuring of each of thefe, and fuch 
 like, I refer you to the Meafuring of Planes in the 
 Fifth Book, which doth fucceed this Treadle. 
 
 Thus much may fuffice the ingenious Fra6ticer 3 
 as to the Meafuring of Glazing. 
 
 CHAP. V. 
 
 T HE Meafuring of Joiners Work is the fame 
 that is taught before, only there is this Diffe- 
 rence ; Carpenters Work is brought into Squares, 
 and Glaziers into Feet • and Joiners Work mull be 
 brought into Yards. 
 
 Which to do, after you have caft up all your 
 Dimenfions, and brought them into Feet, divide 
 the whole Produdt by 9, and that gives you the 
 Number of Yards. 
 
 You muft divide by 9, becaufe 3 Feet in Length, 
 and 3 in Breadth, contain 9 Feet, which is a fuper- 
 ficial Yard. t 
 
 I will fuppofe fame Dimenfions to be caft up, 
 whole Contents or Products are thefe following. 
 
 For it would be ncedlcfs to fpend time to teach 
 the calling up of Dimenfions in Feet and Inches 
 again, having fpoken largely and plainly thereto in 
 the precedent Pages. 
 
 Wainicot 
 
MEASURING BookIV. 
 
 Wainlcot. 
 
 Feet. Inches, 
 no 08 
 304 07 
 065 06 
 
 934 02 
 041 00 
 
 073 01 
 
 *s as 
 
 ** 29 (169 Yards, and 8 Feet. 
 999 
 
 1529 00 The Total ; which divide by 9 to bring 
 
 the Feet into Yards, as you fee above, 
 
 and in 1529 Feet, you find 169 Yards, 
 and 8 Feet; 
 
 Which muft be let down in the Bill of Mealure- 
 ment at lo much per Yard. 
 
 Then, for Cornices, and Bale, and Sub-bafe, 
 Joiners do them by the Foot, Line Meafure, or 
 Running Meafure, as it is called, if they do no other 
 Work with it : But if they do other Work joining 
 with it, they meafure it all together by the Yard. 
 
 Likewife Architrave and Frieze they do by the 
 Foot Line Me%lure. 
 
 The Chimney-Pieces at fo much a-piece, and 
 lometimes meafure them in to the other Work ; de- 
 ducting nothing for the Vacancy of the Chimney. 
 
 CHAR 
 
*3 i 
 
 Book IV* Of ‘Painting . 
 
 CHAP. vr. 
 
 P Ainters Work is meafured like Joiners Work, 
 and brought into Yards, only there is this Dif- 
 ference : 
 
 When you meafure Painting upon Cornices, or 
 Moldings of any Sort, you muft have a Line, and 
 girt the round Moldings, and bend the Line into 
 the hollow Moldings. Thus when you are to mea~ 
 fore a Room, you begin with your Line at the Top 
 of the Cornice, where you may fatten it with afmall 
 Nail, and girting the round Moldings, and bending 
 the Line into the Hollows and Angles, bring the 
 Line down to the Bottom of the Painting, then 
 meafore the Length of the Line with your Rule, and 
 let down that Sum. 
 
 Then begin in one Angle ofthe Room, and meafure 
 the Length of that Side whereof you took the Height, 
 and fet down the Length under the Height which be- 
 fore you let down, and that is one Dimenfion. 
 
 And if the oppofite Side of the Room be like unto 
 that which you have already meafured and fet down, 
 and the Room fquare, you may either fet down that 
 Dimenfion twice, or elfefet a Figure of (2) enclofed 
 againft the Dimenfion, which fignifies that Dimenfion 
 is 2 times : Alio you muft take the Length of the 
 other 2 Sides, and fet them down by the fame per- 
 pendicular Height 3 or you may take the Length of 
 each Side of the Room, and add them all together, 
 and fo make but one Dimenfion of all the Sides by 
 the perpendicular Height. 
 
 Then after you have fet down all your Dimenfions, 
 and brought them into Feet and Inches by catting 
 
 them 
 
2jz MEASURING Book IV. 
 
 them up, you muft divide the whole Product by 9, 
 and that brings the Feet into Yards, as you read be- 
 fore : For you muft not think to learn to meafure any 
 one of theie Trades only, but you muft begin at the 
 beginning of Menfuration (to wit) of Carpenters 
 W ork, and fo proceed gradually ; and be fure to be 
 perfect in calling up one Dimenfion before you en- 
 ter upon another. 
 
 Thus proceeding with Care and Diligence, until 
 you have attained to the meafuring of Glazing, you 
 may with eale afterwards meafure all the reft of the 
 Works that are fuperficial. 
 
 Indeed, the meafuring of Bricklayers Work is 
 fomething more difficult, becaufe it is reduced to a 
 thicknefs, and fo becomes folid Meafure. 
 
 As for the Painting of Windows, they are gene- 
 rally fet down at fo much per Light, and Cafements 
 at lo much a-piece. 
 
 The next Work to be meafured is Plaiftering, of 
 which I need lay no more than this, that it muft be 
 brought into Yards , and where a Sommer or Girder 
 lies below the Ceiling, it is ufually dedufted where 
 the Workman finds Materials^ but not elle. 
 
 Allb in rendring where Materials are found by the 
 Workman, lometimes 4 is deducted for the Quarters j 
 but not where Workmanffiip only is found, becaufe 
 it might be rendred as foon if there were no Quarters. 
 
 The meafuring of Mafon’s Work (to wit, fiat 
 Paving, and fuch like) is meafured the lame Way 
 as all the reft. There is no need of Divifion, for 
 Mafon’s Work by the Foot. 
 
 I come now to treat of the Menfuration of 
 Bricklayers Work, 
 
 CHAP. 
 
Book IV. of Bricklayers Work. 233 
 
 CHAP. VII. 
 
 B Rickwork, as you read before, muft be reduced 
 to a certain or ufual Thicknefs, in cafting of 
 it up, for which Reafon it becoihes folid Meafure. 
 
 Which ufual Thicknefs is 14 Inches, or, as it is 
 generally called, one Brick and half 
 
 By one Brick and half, is meant the Length of 
 a Brick, and half the Length ; or the Length of 
 one Brick, together with the Breadth of another. 
 
 For the Breadth of two Bricks, with a Joint of 
 Mortar between them, i(s anfwerable to the Length 
 of a Brick, the Length whereof ought to be, and ge- 
 nerally is 9 Inches, and the Breadth 4 Inches and 
 which being added together with a Joint of Mortar 
 makes 14 Inches, the aforefaid ufual Thicknefs. 
 
 And as Carpenters Work is brought into Squares, 
 Bricklayers Work muft be brought into Rods, which 
 Rod contains 272 Feet and 3 Inches, and is produc’d 
 from 1 6 Feet and 6 Inches in Length, being multi- 
 plied by 16 Feet and 6 Inches in Height or Breadth j 
 or a Quadrate whofe Side is a Pole or Perch. 
 
 When you come to meafure the Brickwork of a 
 Building, you will find the Walls thereof to be of 
 feveral Thicknefles ; fome thicker than a Brick and 
 half, and fome thinner, as 1 Brick Walls, or Walls 
 9 Inches in Thicknefs. 
 
 Which feveral Thicknefles muft all be brought, 
 or reduced to the ufual Thicknefs of x Brick and 
 half, which to do, obferve the following Rules. 
 
 1. Set down each Thicknefs in a Column by it 
 felf, (to wit) fet down your Dimenfions that are 
 
 R 1 Brick 
 
* 34 ME ASURIH& Book IV; 
 
 i Brick (or 9 Inches) in Thicknefs, in a Column 
 by themfelves. - - ' ] > 
 
 Alfb thofe Dimenfions that are 1 Brick and \ in 
 T hicknels, muft be fet in a Column by themfelves. 
 
 Likewife thofe of 2 Bricks ihThickhefs by them- 
 Pelves, and thofe of 2 Bricks aiid in a Column by. 
 themfelves. / . ‘ 
 
 And thofeof 3 Bricks in a Column by themfelves. 
 
 . So likevyife if your Walls be thicker than 3 Bricks, 
 (in Length) fet each Thicknefs by it felf 
 ‘ ^ fetting the Dimenfions by themfelyes, is meant' 
 to iet.them in/a Pageor Column ato.rvc pyggtfy, 
 and not to interfiux plmehfibns*Qf Teveral ' T h;ck 7 
 neffes in one Page or Column. . 
 
 Notwithftanding, fome Meafurers. fet 'down their' 
 Dimenfions as. they Like them of fevcral Thickneflcs 
 in one Column ; but Ido not think it fo convenient 
 for a Learner, 'as to ftt them down federally. 
 
 2.' After you- have taken your Dimenfions and 
 caft them up, and by adding the feveral Frodu&s 
 together (of one Thicknefs) have produced the 
 total Produdb, then to reduce it to 1 Brick and 
 wf mu)| 
 
 If the Dimenfions that you have caft up, be 1 
 Brick’s Length in Thicknefs, that is the Breadth of 
 2 Bricks j then you muft multiply your total . Pro- , 
 duift by two, which is- the Thicknefs of the Wall in 
 the ' lekft Denomination (to wit) two 4 Inches, or 
 the Breadth of 2 Bricks, and divide that Product 
 by 3, (becaufe there is the Breadth of 3 Bricks in 
 a Brick and f) and the Quotient is the Sum of the 
 1. Brickwork, reduced to 1 Brick and \ Brickwork. 
 
 Example, 
 
Book IV. of Bricklayers Work. 235 
 
 Example *■ 
 
 Suppofe you hdve caft up feveral Dimenficns of 
 1 Brick’s Length in Thicknels, and the Products 
 (or Contents) of each Dimenfion being added, to- 
 gether, produceth a total Produd of 4976 Feet of 
 Brick-work, being one Brick thick. 
 
 To reduce this to 1 Brick and f thick, multiply 
 4976 by 2 (to wit, the Breadth of 2 Bricks, and 
 that brings it into the leaf* Thicknefs or Denomi- 
 nation-, namely, 4 Inch-work, or Brick-work i a 
 Brick thick.) 
 
 4976 
 
 So the Produd of 4 Inch-work is 
 
 9952 
 
 Which divide by 3 (becaule there are the 
 Breadth of 3 Bricks in 1 Brick. and j) and in the 
 Quotient you have 3317 Feet, of 1 Brick and £ 
 Work, and i remaining, 
 
 it 
 
 See the Divifion fBJ CiJ ‘ 7 
 
 The 1 that remains is i Foot of 4 Inch-work, or 
 4 Inches of j Brick and £ Work being reduced. 
 
 For if you divide 12 (which are the Inches in that 
 Foot) by 3 (the Number of 4 Inches or f Bricks 
 ih a Brick and i ) the Quotient will be 4j which 
 is 4 Inches or f of a Foot. 
 
 So likewife your Dimeniions that are thicker than 
 1 Bri ck and f ) as iuppofe 2 Bricks in Length, which 
 
236 MEASURING Book IV. 
 
 is called 2 Brick- work, which 2 Brick-work con- 
 tains the Breadth of 4. Bricks, you muft in reducing 
 2 Brickwork to 1 Brick and \ work, firft multiply 
 the Sum by 4, and divide the Product by 3. 
 
 Example. 
 
 If you have 5845 Feet of 2 Brickwork, multiply 
 it by 4. 
 
 See the Multiplication 5845 
 
 4 
 
 The Produdt is 23380 Feet 
 
 Of 4 Inch Work, which you muft divide by 3. 
 
 I 
 
 See the Divifion, £$'$'#0(7793 
 
 And the Quotient is 7793 Feet of 1 Brick and 
 \ Work, and 1 remaining, which is 1 Foot of 4 
 Inch Work, which being reduced, as you arc taught 
 in the foregoing Page, is 4 Inches (or \ of a Foot) 
 of 1 Brick and ~ Work. Thus in 5845 Feet of 
 Brickwork being 2 Bricks thick (or the Walls being 
 the Length of 2 Bricks in Thicknefs) you have, 
 when it is reduced, 7793 Feet, and 4 Inches of 1 
 Brick and \ Work, (the ufual Thicknels.) 
 
 If your Dimenfions be 2 Bricks and \ in Thick- 
 nefs, you muft multiply the total Product by 5, and 
 divide the Product of that Multiplication by 3. 
 
 If your Dimenfions be 3 Bricks in Thicknefs, you 
 may fave the Labour of dividing and multiplying, 
 
 becauie 
 
Book IV. of Bricklayers Work. 2 37 
 
 becaufe a Wall of 3 Bricks thick is juft fo much more 
 as a Wall of 1 Brick and £ thick * therefore if you 
 fet down the Sum of your Product in 3 Bricks twice, 
 it will be the fame as if you fhould multiply by 6, 
 and divide the Produdt by 3, for this Reafon, that 
 6 is as much more as 3. 
 
 In meafuring of Brickwork, you muft fet down 
 your Dimenfions with the Number of their Thick- 
 nefles over them, or adjoining to them, namely, 
 over Dimenfions of 1 Brick in Thicknefs fet 1 B. fig- 
 nifying 1 Brick ; and over Dimenfions that are 1 
 Brick and £ in Thicknefs, write £ B, and over Di- 
 menfions 2 Bricks in Thicknefs, w'rite 2 B, Sc. 
 
 And caft up your Dimenfions as you are taught in 
 meafuring of Carpenters Work and Glaziers Work. 
 
 But in Brickwork as well as in Carpenters Work, 
 it is ufiial to meafure no nearer than whole Inches, 
 that is to fay, f Inches and 4 of Inches, nor £ of 
 Inches are not fbt down. 
 
 For which Parts of Inches an Allowance is made 
 when you come to divide your Number of Feet to 
 bring them into Rods. 
 
 fo bring or reduce Feet into Rods . 
 
 After you have reduced all your feveral Thick- 
 neffes, and brought them to the ufual Thicknefs of 
 a Brick and £, you muft divide your whole Product 
 by 272, (being the Number of Feet contained in a 
 Rod) to bring or reduce the Feet into Rods. 
 
 Taking no notice of the 3 Inches belonging to the 
 272 Feet, being both contained in a Rod. 
 
 For you read before, that in a Rod is contained 
 272 Feet, and 3 Inches, which 3 Inches is allowed to 
 
 R 3 make 
 
a ? 8 MEASURING Book IV. 
 
 make good for the Parts of Inches^; which is not 
 let down when you fet down the pimenfions. 
 
 But if you are minded to be exact and curious 
 in meafuring, you may fet down the odd Parts of 
 Inches, and caft them up as you were taught in 
 meafuring of Glazing, and divide the tptaj Produd 
 by 272 Feet and 3 Inches, but this is feldom or 
 never done in Brick-work. 
 
 Folded between Page 240, 241, you have de- 
 ligned the Ground-plat of a Building j which toge- 
 ther with the Inftrudions that follow, will much 
 aflift you in making an Eftimate for Building from 
 a Defign given. 
 
 It will alfo inftrud you how to take your Dimen- 
 fions, and to fet them down in your Book, better 
 than by a Multitude of Words.* 
 
 ST 'he Explanation of the Defign. ■ 
 
 This Defign is the Ground-plat of a Building, 
 being 25 Feet in the Front, and 40 Feet in the 
 Flank or Depth. 
 
 The Front and Rear-Front Walls are 2 Bricks 
 and - thick. 
 
 The Flank Walls are 2 Bricks thick, as you may 
 perceive by the Scale annexed to the Defign. 
 
 You may fuppofe this Defign to be the Ground 
 Floor, having no Cellar beneath it. 
 
 And the Story to be 1 1 Feet and 6 Inches in Height 
 from the Foundation to the Top of the next Floor. 
 
 The Windows are 4 Feet in Breadth, and 6 Feet 
 6 Inches in Height or Length. 
 
 The Door-cafes in Front and Rear are 4 Feet 
 wide, and 9 Feet in (Length or) Height. 
 
 The 
 
Book IV. of Bricklayers Work. i j p 
 
 The Chimneys, you fee, ftand back to back, the 
 Jaums being 2 Bricks thick. 
 
 The Enclofure of the Stairs, and the Partition 
 next the Entr^r, are of Timber, > . r . 
 
 What hath been faid isftTficieiTt for Explanation'. 
 
 I {hall, in the next platfe*, Iriftruft you how to 
 meafure the Brickwork of this Story in the Ground- 
 plat, which being well underftobd, you may with 
 eafe meafure all the reft of 'the . Stories over it. 
 
 I {hall begin with the Front. - 
 
 And becaufe all Walls are (or faould be) made 
 with a Bafis, therefore we will. fupp ole the Front 
 Wall and Rear Front- Wall to 'be 3 Bricks thick, 6 
 Inches high, which is 2 Courfes of Bricks j- for 3 
 Inches is generally allow’d for a Courfe (or the 
 Thicknefs of a Brick laid in Mortar. 
 
 So the firft Dimenfion will be 25 Feet in Length, 
 by 6 Inches in Height, 3 Bricks thick. 
 
 The Thickneflfes mnft be always fet over the Di- 
 menfions (as you will fee in the next Page.) 
 
 And becaufe the Front and Rear are both of one 
 Length, you need not fet down the Dimenfion 
 twice, but fet a Figure 2 enclofed in a Circle thus 
 (2) which fignifies the Dimenfion to be 2 times. 
 
 I think it will be beft for Inftru&ion, to caft up the 
 Dimenfions as they are taken, (or fet down) but after 
 you underftand the W ay, you rAuft-let downfall your 
 Dimenfions firft, and caft them up . afterwards. 
 
 I fhall alfo fet the feveral Thickneffes of the Di- 
 menfions (of this Story of the Ground-plat) in one 
 Column, becaufe there will be but a few Dimenfions 
 in this Story, and not above a Dlrhenfion or two or 
 one Thicknefs,. and-it is not worth while to fet down 
 a. Dimenfion op two in a Column by it felf 
 
 R 4 Example* 
 
24 ° 
 
 MEASURING Book IV. 
 
 Example . 
 
 The Length of the Front is 
 The Height of the Brickwork is 
 2 Courfes, or 
 
 The Product is 
 
 The Rear Front being the fame Di- 
 menfion, and being joined in one 
 Dimenfion by the Figure (2) you 
 
 muft add 
 
 Which being added, the Content of > 
 the 3 Brickwork is 5 
 
 And becaufe 3 Bricks is the Double p 
 
 3 B. 
 Feet. In 
 
 25 
 
 of 1 | B. add 
 
 
 00 
 
 00 q 6 
 12 o$ 
 
 12 06 
 
 00 
 
 2 s 00 
 
 ( 2 ) 
 
 And the Content of this firft Dimen- 
 fion being reduced into 1 f B. is 
 
 2j dly. I proceed to the remaining Part 
 of the Front and Rear, being in 
 Height 
 
 In Length 
 
 Being multiplied 
 
 The Content of the Front is 
 The fame being added for the Rear qq 
 
 The Content of 2 i B. work inFront > 
 
 and Rear is 3 $ 5 ° 00 
 
 Which to reduce, you muft multi- > 5 
 
 ply by 3 — — 
 
 The Prod, of 4 Inch, or i B. work is 2750 
 
 Which 
 
 SS 
 
 22 
 
 2*7 < no 
 

#%*, 0 <b iuo]^ 
 
 B Ztw CTl Pacj . Q.Sr 
 
 'Bcerc 
 
 » 
 
Book IV- of Bricklayers Work. 241 
 
 Which muft be divided by 3, to reduce it to 
 1 i B. in Thicknefs. zz 
 
 *7*0(916 
 
 S'?? 
 
 And the Quotient is 916 and 2 remaining, which 
 is 916 Feet of 1 i B. and 2 Feet of h B. which is 8 
 Inches of 1 i B. being reduced. 
 
 Having done with the Fronts, we proceed to take 
 the Dimenfions of the Flank-Walls. 
 
 And becaufe we meafured the Fronts from Out 
 to Out, as we call it in brief, (to wit, from the Out- 
 fide of one Flank-wall, to the Outfide of the other) 
 we muftmeafure the Length of the Flank-walls with- 
 in (to wit, from the Infide of the Front-wall, to the 
 Infide of the Rear Front-wall) fo the Thicknefs of the 
 two Front- walls, which is 3 Feet and 10 Inches, be- 
 ing taken out of the 40 Feet in Depth, 
 
 f B. 
 
 * Feet. Inches 
 
 The Length of one Flank-wall is 36 02? (2) 
 
 The Height of the Bafe is 00 o 6 $ 
 
 And becaufe both Walls are alike, or 
 equal in Length and Height, you 
 muft put (2) againft the Dimenfi- 
 on, and it ferves for both Walls 
 TheCont.in 2 i B. work of both Walls is 36 02 
 
 Being multiplied by J> 
 
 TheProdudt of i B. work is 180 10 
 
 Being divided by 3 to reduce to \ B. 
 
 It is 60 Feet, 3 Inches and i of an Inch of 1 i- B. 
 Work. The 
 
 18 or 
 18 or 
 
24 * ME ASURING Book IV, 
 
 2 B. 
 
 The next Dimenfion of the Flank-wall is 36 02 >(2) 
 
 11 00 5 
 
 36 
 
 Being multiplied 361 10 
 
 TheCont.in 2 B. work of one Wall is 397 10 
 To which add the other Wall 397, 10 
 
 - 
 
 TheCont in 2 B. work of both Walls is 795 08 
 Which multiply by 04 
 
 The Con.of both Walls in l B. work 3182 08 
 
 Being reduced to 1 ? B. the Content 
 of both Walls are 1060 Feet and 
 10 Inches. 
 
 H 81(1060 
 
 Although I caft up and reduce each Dimenfion 
 as I proceed, yet after you underhand the Way, you 
 mull reduce all your Dimenfions of oneThicknels at 
 one time. * 
 
 Having caft up and reduced both the Fronts, and 
 both the Flank-walls, the next Dimenfions to be 
 taken are the Chimneys : Which are ufually agreed 
 for, at fo much per Fire-place (or Chimney). Alfo 
 lbmetimcs they arc done by the Rod, at the Rate 
 of the other Work, and then they mult be mea- 
 sured. 
 
 Theft two Chimneys in the Dcfign, ftanding Back 
 to Back, are 5 Feet a-piece between the Jaums. 
 
 The Jaums are 2 Bricks thick. 
 
 The Wall between the Chimneys is 14 Inches 
 thick ; to which is added 9 Inches, for a falling 
 Back to each Chimney. • There- 
 
Book IV. of Bricklayers Work. 143 
 
 Therefore we will take the Wall firft by it felf, 
 which is 9 Feet io Inches in Length, and n Feet 
 6 Inches, the aforefaid Height of the Story, and 
 fet it down. 
 
 1 i B. 
 
 The Heighth ri 06 
 
 The Length 09 10 
 
 99 
 
 4 06 
 
 9 02 
 
 OS 
 
 TheCont. of theWall between the Chim. 113 01 
 
 Altho 3 heretofore I have put Feet or F. over the 
 place ofFeet ; and In. or I. over the Place of Inches. 
 
 Suppofing now that you know one Place from ano- 
 ther, I lhall defift doing it, as in the laft Dimenfion ; 
 and ouly fet the Thickneifes over the Dimenfions. 
 
 In the next place, fuppofethe Mantle- tree to lie 5 
 Feet high, and the falling back to begin at 3 Feet high 
 The Mean between 5 and 3 is 4, which is the 
 Height of the 9 Inch Work that is added to each 
 Chimney for falling back, and the Length is 5 Feet. 
 
 1 B. (2) 
 
 The Dimenfion of the falling back >05 oo 7 
 of both Chimneys /04 00 j 
 
 
 20 
 
 00 
 
 
 20 
 
 00 
 
 The Content in 1 B, 
 
 40 
 
 00 
 
 Multi- 
 
244 MEASURING 
 
 Multiply it by 
 
 The Produdt is 
 
 Book IV. 
 
 1 B. 
 
 40 OO 
 
 2 
 
 80 
 
 2 
 
 Which divide by 3 $0 (26 
 
 Having reduced it, you find 26 Feet and 8 In- 
 ches of 1 i B. work in both fallings back. 
 
 The 4 Jaums being 3 Feet 6 Inches deep apiece, 
 above the falling back ; being added together, make 
 14 Feet of 2 B work, which muft be multiplied by 
 the Height of the Story. 
 
 For although the Jaums in the next Story, be 
 but 1 i B. and from the Mantle-tree in this Story 
 it be wrought but a Brick and half, yet the Wings 
 being added, makes two Bricks. 
 
 2 B. 
 
 The Breadth of the 4 Jaums being? 
 
 added together is 5 1 ^ 00 
 
 The Height of the Story is 1 1 06 
 
 *4 
 
 147 
 
 The Cont. of the 4 Jaums in 2 B. work is 161 00 
 
 Being multiplied by 4 
 
 The Cont. of the 4 Jaums in J B. work is 644 
 
 12 
 
 Which divide by 3 in the Margin ^#4(214 
 
 * 3ZZ 
 
 And 
 
Book IV. of Bricklayers Work. 24.J 
 
 And the Content of the Jaums being reduced to 
 1 i B. is 214 Feet, and f of a Foot, or 8 Inches. 
 
 The 2 Breads of the Chimneys are next to be 
 mealured, which are 5 Feet apiece between the 
 Jaums ; and altho 5 they are but 9 Inches thick a- 
 piece, yet confidering the crofs Withs, and the Peers 
 that are wrought from the falling back till the Wings 
 gather to them, they are ufually meafured at the 
 fame Thicknefs that the Jaums are, and at the 
 Height of the Story, abating nothing for the Va- 
 cancy between the Floor and the Mantle-tree. 
 
 2 B. 
 
 The Length of the Bread between £ ^ ^7 (2) 
 the Jaums is 
 
 The Height of the Story is 
 
 The Dimenfion being (2) add 
 
 The Content of the 2 B. work in 
 both Breads is 
 
 Which multiply by 
 
 1 
 
 05 
 
 00 c, 
 
 II 
 
 06$ 
 
 ojr 
 
 
 52 
 
 06 
 
 57 
 
 06 
 
 57 
 
 06 
 
 115 
 
 00 
 
 4 
 
 
 O 
 
 so 
 
 rt- 
 
 00 
 
 HI 
 
 Which divide by 3 4^0(153 
 
 S'*? 
 
 Being reduced, the Content is 153 Feet and f of 
 a Foot (or 4 Inches) of 1 £ B. 
 
 Thus having taken all the Dimenfions in one 
 Story, and cad them Pp 3 The 
 
*4 6 MEASURING Book IV. 
 
 The next Thing to be done,, is to take the De- 
 ductions of the Windows and Doors, in the Fronts. 
 
 Note, That the Deductions (to wit, Windows and 
 Doors, and all Vacancies whatfoever that you mea- 
 gre) muft be fet in Columns by themfelves,; with 
 this Note Dea. (fignifying Deductions) over them 
 together with the Thicknefs of the -Wall wherein the 
 Windows or Doors Hand that you are to deduct ■ 
 
 , Example. 
 
 . Led. 2~B. 
 
 The Height or Length ofa Window is 06 06 ; (V) 
 
 The Breadth is . • D+ 00 £ 
 
 Arid becaufeihere are 4 Windows in both Fronts, 
 and ah of one Bignefs; therefore put (4) againft ' 
 the Dimen lion, fignifying that that Dimenfion is'4 
 times (or that there are 4 Windows of that Bignefs.) 
 
 _ £ ■ Ded 2 \ B. 
 
 The Height of a Window 
 The Breadth 
 
 Being^ multiplied 
 
 of 2 i b. work in i 
 Which multiply by 
 
 orb 
 
 Which multiply by 
 
 To bring it into the leaft Denomi- ? 
 nation, and it is j* ^ 2o 
 
 Which divide by 3 ; and 
 Being reduced to 1 l B. there 
 is 173 Feet and 4 Inches, 
 
 06 
 
 06 
 
 04 
 
 00 
 
 24 
 
 2 
 
 • 
 
 00 
 
 as 26 
 
 00 
 
 4 
 
 - V T 
 
 is 104 
 "5 
 
 
 
 ** r.J 
 
 HT 
 
 1 
 
 (l 73 
 
 k- There 
 
BooklV. of Bricklayers Work. 247 
 
 There are alfo in Front and Rear 2 Door-cales 
 to be deduded (dedud Signifies. to take out) of the 
 fame Thicknefs of 2 i B. 
 
 Bed. 2 i 
 
 ^he Height of a Door is * J 09 00? (2) 
 
 The Breadth . 04 00 £ 
 
 ,1 ‘ 00 ~ 
 
 Bdng, multiplied 36 
 
 " 2 
 
 i , 
 
 a 1 . 
 
 The FrpduCt of 2! I B. work in both? 
 
 Doors is ~rr S ” 
 
 .;T Which multiply by 5 
 
 Being brought into i B. work is 360 
 
 W^ieh divide by 1 3, 'as in the Margin 3^0 (120 
 
 10 • 3 i o t 'Q >5 
 
 being reduced to FlB. there is 120 Feet, as in the 
 Margin. f:; ’ co ;• o . . /! lo ’ *• ' 
 
 [ ): Note^ Tho - 1 reduce the Dedudions here £ai In- 
 ftrudion $ hereafter you need not reduce them, but 
 take the Produd>of the Deductions of one Thick- 
 nefs out of th^ P^odud'of Dimenfions of thpfame 
 Thicknels. _ 
 
 In the following Page I will fet down all the Dir 
 mentions as the^ were taken, with the Produd of 
 each Dimention in a Column juft againft it. 
 
 And although I faid befqre, that you might di- 
 vide a Page or Leaf of your Measuring Book -into 
 4 Parts or Columns , yet in Meafuring of Brick- 
 layers Work, it Will be necetifary to divide a Page 
 only in two Parts, (as you fee in the next following 
 Page) that fb you may have room to fet a Name to 
 each Dimenfion for Diftindion-fake. 
 
 Dimen - 
 
248 MEASURING Book IV. 
 
 Limenfions of the Defign. 
 
 3 B. 
 
 Bafis of the Front and Rear °° £ ^ 
 
 Front and Rear 
 
 2 i B. 
 
 C25 00 >(*) 
 
 c 11 00 S 
 
 2 i B. 
 
 Bafis of both the Flank 536 027(2) 
 
 Walls l 06J 
 
 2 B. 
 
 Both the Flank Walls 4 02 7 (2) 
 
 ( IX OQ J 
 
 iB. 
 
 The Wall between thejn 06 
 Chimney ^09 10 
 
 1 B - 9 
 
 The falling back of bothjoj; 00^(2) 
 Chimneys £04 oo( 
 
 2 B. 
 
 
 The 4 Jaums 
 
 J *4 
 
 1 11 o6j 
 
 2 B. 
 
 The Forepart or Breafts of c 1 1 06 7(2) 
 
 co< 00 S 
 
 both Chimneys t 05 00 
 
 Produffs. 
 
 SB. 
 
 25 00 
 
 2i B. 
 55 ° 00 
 
 2 IB. 
 36 02 
 
 2 B. 
 795 08 
 
 1 iB. 
 
 112 01 
 
 1 B. 
 040 00 
 
 2 B. 
 
 161 OQ 
 
 2 B. 
 
 115 00 
 
 Having fet down all the Dimenfions with their 
 Products, in the next place we mud fet down the 
 Deductions of the Windows and Doors with their 
 Products. 
 
 Deductions 
 
Book IV. of Brie 
 
 klayers Work. 
 
 2 49t 
 
 f 
 
 Deductions in the Defgn. 
 
 Ded. 
 
 2 l B. 
 
 
 Products « 
 
 The 4 Windows 
 
 06 06 > 
 04 OO J 
 
 ( 4 ) 
 
 2 f B. 
 
 1 04 0 
 
 The 2 Doors 
 
 2 i B. 
 
 09 OO £ 
 04 00S 
 
 0) 
 
 2|B. 
 072 0 
 
 The next Work is to add the Products of eacfc* 
 feveral Thicknefs into one Sum. 
 
 ProdhCts of feveral Zhicknejfes, 
 
 2 IB. 
 
 2 B. 
 
 if B. 
 
 iB. 
 
 SS 0 00 
 
 7 9 S 08 
 
 113 1 
 
 40 0 
 
 36 02 
 
 161 00 
 
 
 l " - ,r " 
 
 586 02 
 
 115 00 
 
 
 
 
 1071 OO 
 
 
 
 The feveral Products of each Thicknefs being: 
 added, 
 
 In the firft Column of the left Hand, you have 
 25 Feet of 3 B. 
 
 In the fecond Column you have 586. 2. of 2 ~ B, 
 
 In the third Column you have 1071. 8. of 2B. 
 
 In the fourth Column you have 113. 1. of 1 
 
 In the fifth Column you have 40 Feet of 1 B. 
 
 The next Work muft be to take the ProduCts of 
 the Deductions out of the Products of the Di« 
 menfions. 
 
2jo MEASURING Book IV. 
 
 Products of Ded. in 2 ~ B. 
 
 104 o 
 72 o 
 
 The total Product of Ded. in 2 ~ B. is 176 o 
 
 Which 176 Feet of 2 \ B. v/ork being contained 
 in the Windows and Doors, muft be fubtradted 
 from 586 Feet and 2 Inches, being the total Product 
 of all the Dimenfions that are 2 £ B. in Thicknefs. 
 
 For this Reafon. 
 
 Becaufe when we meafured the Front and Rear, 
 we meafured the whole Length and Breadth over 
 the Windows and Doors, allowing no Abatement 
 for them. 
 
 Notc 9 That whatfbever Windows or Doors, or 
 other Vacancies you mealure over when you take 
 the Dimenfions, you muft remember to ded udl them 
 out of the total Product of the Dimenfions of the 
 fame Thicknefs wherein they are fituate. 
 
 Example, 
 
 The Doors and Windows being in 2 ? B. work ; ? 
 
 I fet down the total Product of all the Dimenfions / f86 02 
 of that Thicknefs, which is j 
 
 The total Produft of all the Dedu&lons of that ? I7 ^ co 
 
 Thicknefs which are to be lubtra&ed is 5 _Z 
 
 The Pvemainder is 410 02 
 
 So likewife if there had been Deductions in the other 
 Thicknefies, you muft have lubtrafted them before you be- 
 girt to reduce your feveral Thicknefies to the ufual Thick- 
 nefs of 1 i B. 
 
 But feeing we have no other Ded. in this Defign, to fub. 
 traft out of any other Thicknefs, the next Work will be to 
 reduce each Thicknefs to the ufual Thicknefs of 1 ^ B. 
 
 Beginning with the greateft Thicknefs fto wit 3 B.) we 
 find 2f Feet, which, as you r^ad before, becaufe It is juft 
 as thick again as 1 \ B. you need not multiply it by 6, and 
 ‘ I divide 
 
Book I V. of Bricklayers Work. z 5 1 
 
 •iB.Work 2oyo 10 
 
 (*8 3 
 
 The 2 remaining in Dlvifion ) * 0^ 
 is 8 Inches, and the 8 Inches of >42$# ( 1428 
 tB. is 2 Inches of ~ B. which ^ ^ ^ ^ ^ 
 added is 10 Inches, fo the 2 B. work being 
 reduced is 
 
 The next Thicknefs being 1 ~ B. needs ni 
 reducing, becaufe it is the ufual Thicknefs 
 the Produft whereof is 
 
 The next Thicknefs to be reduced is 
 i B. the total Produa whereof is 
 Which multiply by 
 
 And the Produa of ~ B. work is 
 
 22 Which divide by 3, and the Quotient is 
 
 /.(T 26 Feet and 8 Inches, being reduced ; 
 
 ' The Produa of all the ThicknefTes 
 %% being reduced is 
 
 S 2 
 
 1 B. 
 
 40 
 
 2 
 
 80 
 
 duced. 
 coyo o 
 
 divide it by 3, but add juft fo much more to the) Produfts re- 
 Dimenfion as it is (to wit 2y to 2yjand fo youj 
 will have yoFeet, which muft be let down in a ‘ 
 
 Column with 1 7 B. over it. 
 
 The next Thicknefs to be reduced is 2 — B. 
 the total Produa whereof is ("the Deduftions 2 be- 
 ing taken out) ^410 Feet and 2 Inches, which 
 multiply by y, and divide the Produa by 3. 
 
 Example * 
 
 21B. 
 
 41° 2 AH 
 
 f 20y0 
 
 
 
 The j that remains in Divifton is 4 Inches, 
 and the 10 Inch. in the \ B. work is 5 Inches of 0685 7 
 
 1 {B. work, which being added is 7 Inches 
 fo the 2 ~ B. work being reduced is 683. 7 
 which muft be fet in the Column. 
 
 The next Thicknefs to be reduced is 2 B. 
 
 2 B, the total Produa whereof is 1071 I 
 
 Which multiply by , 
 
 And the 4 Inch (or J. B ) work is 4286 8 
 
 1428 
 
 0113 
 
 10 
 
 0026 8 
 
 2302 
 
 Haying 
 
z ;5 2 MEASURING Book IV. 
 
 Having reduced all the Thickneffes to i 4 B. in 
 Thickneis, you find the total Product to be 2302 
 Feet and 2 Inches. 
 
 Which 2302 muft be divided by 272, the 
 Number of Feet contained in a Rod, as you read 
 before. 
 
 Example. 
 
 126 
 
 2 ^ 0^(8 
 27 2 - 
 
 Being, divided, the Quotient is 8, and the Re- 
 mainder is i 26, which is 8 Rods and 126 Feet and 
 2 Inches (if you add the two Inches that belonged 
 
 to the 2302 Feet which you divided.) 
 
 Note, The whole Rod containing 272 Feet, the 
 half Rod contains 136 Feet, and the Quarter of a 
 Rod contains 68 Feet ; for twice 136 is 272, and 
 
 four times 68 is the fame. . c 
 
 Then take 63 Feet, 'or { of a Rod, out ot 
 the 126 Feet that remain, by fubtradting 68. 
 
 And. the Remainder, is 5.8 Feet. 
 
 So there is 8 Rods and \ and 58 Fept.of 1 \ B, 
 work in one Story of the Defign. 
 
 It is convenient, before I proceed farther, to add 
 fomething more concerning meafuring of Chimneys. 
 
 Firft Ncte ^ ‘ If ydii are to meafure a Chimney 
 Handing alone or. by it felf^ without any Party- wall 
 being adjoyned. 
 
Book IV. of Bricklayers Work. 155 
 
 Then girt it about for the Length, and the 
 Height of the Story is the Breadth. 
 
 TheThicknefs mull be the fame that your Jaums 
 be, provided the Chimney be wrought upright from 
 the Mantle-tree to the Ceiling* not deducting any 
 thing for the Vacancy between the Floor (or Hearth) 
 and the Mantle-tree, becaufe of the Gatherings of 
 the Breaft and Wings to make room for the Hearth 
 in the next Story. 
 
 Secondly Note, If the Chimney-back be a Party- 
 wall, and the Wall be meafiired by it folf, then 
 you muft meafure the Depth of the 2 Jaums, and 
 the Length of the Breaft, which being added toge- 
 ther, will be your Length, and the Height of the 
 Story your Breadth, at the fame Thicknefs your 
 Jaums be. 
 
 Thirdly Note, When Chimneys are agreed for at 
 a Rate by the Fire-place or Chimney-building, and 
 the Back of the Chimneys ftaiidagainft a Wall (that 
 you meafure befides) you muft deducft 4 Inches in 
 Thicknefs out of the Menfuration of the Wall 
 againft the Chimneys^ which 4 Inches is allowed for 
 the Backs of the Chimneys. 
 
 Fourthly Note, When you meafure Shafts of 
 Chimneys, girt them with a Line round about the 
 leaft Place of them for the Length, and their Height 
 lhall be your Breadth; and if they be 4 Inch-work, 
 then you muft fet down your Thicknefs at 1 B.work: 
 But if they be wrought 9 Inches thick (asfometimes 
 they are when they ftand high and alone above the 
 Roof) then you muft account your Thicknefs 1 4 
 B. in Confideration of Withs and Pargetting, and 
 Trouble in Scaffolding,, 
 
254 
 
 MEASURING Book IV. 
 
 3° 
 
 02 
 
 18 
 
 03 
 
 240 
 
 
 303 
 
 
 7 
 
 06 
 
 550 
 
 06 i 
 
 Obfer-ve thefe following Rules , which will much help 
 you in cafting up your Dimenfeons of any Kind , and 
 fa-ve you the Labour of Divifion , to know how 
 many Feet there is in any Number of Inches that 
 you multiply into Feet. 
 
 Let the firft Example be 
 
 Firft, Having multiplied the whole 
 Feet, in the next place you multiply the 
 Inches into the Feet, laying 2 times 18 
 is 36 Inches; or thus, 18 Inches being 
 a Foot and an half, fay, 2 Feet and 2 
 half Feet are 3 Feet, which being let 
 dcwn : 
 
 In the next Place you come to multiply 3 into 
 3 o, which you need not multiply ; for 3 being one 
 fourth Fart of 12, fay. The one ^ of 30 Feet is 7 
 Feet and 6 Inches ; which is a quicker Way than to 
 multiply 30 by 3, and divide the Produdt by 12. 
 
 Let the fecond Example be 
 
 The Feet being multiplied and let 
 down, next you are to multiply 4 by 
 42 ; but 4 being •§• Part of 12, fay, 
 the one Third of 4 is 1, and the one 
 Third of 12 is 4, which is ( being 
 added) 14 Feet. 
 
 In the next Place you come to mul- 
 tiply 5 25, which you may fay is 25 
 half Feet lacking 25 Inches, that is, 1 0 Feet and 
 5 Inches. 
 
 25 
 
 04 
 
 42 
 
 05 
 
 50 
 
 
 IOO 
 
 
 *4 
 
 
 10 
 
 5 
 
 
 1 
 
 1074 
 
 6 
 
 Or 
 
Book IV. of Bricklayers Work. iff 
 
 Or thus, 24 Inches being 2 Feet, fay 5 times 24 
 Inches is 10 Feet, then add the 5 Inches, it makes 
 10 Feet and 5 Inches. Laftly, 5 by 4 is 20, which 
 is 1 Inch and 8 Parts, which 8 Parts is not account- 
 ed, nor fet down, (as you have read before.) 
 
 Let the third Example be 
 
 ICO 
 
 94 
 
 10 
 
 The Feet being multiplied, you pro- 
 ceed to multiply 6 by 100, but 6 being 
 the half of 12, fay the half of 100 is 50 
 Feet, which being writ down, you pro- 
 ceed to multiply 7 into 94, then fay 
 the half of 94 is 47 Feet, which being 
 fet down, there is ilili remaining 94 
 Inches, which you may ask your lelf 
 how many times 12 you can have in 
 94? Your Anfwer w.ill be 7 times, and iq Inches re- 
 maining, therefore you muft fet down 7 Feet and 
 10 Inches. Laftly, you proceed to multiply the 
 Inches by the Inches, faying 7 times 6 is 42, which 
 3 Inches and \ ; fo the total Produdt is 9505 
 
 400 
 
 900 
 
 So 
 
 47 
 
 7 
 
 9505 
 
 is 
 
 230 
 
 31Q 
 
 Feet and 1 Inch, as you fee in the Margin above. 
 
 Let the fourth Example be 
 
 The Feet being multiplied and fet 
 down, you proceed to multiply 310 by 
 8 ; but inftead of multiplying, take the 
 ■§ of 310, becaufe 8 is of 12 ; faying 
 thus, the -f of 3 is 2, but the f of 1 you 
 cannot take in Integers, therefore you 
 add a Cypher next to the Figure 2, 
 and proceed to take the -| of 10, which 
 is 6 Feet and 8 Inches ^ fo you find 
 
 S 4 
 
 2300 
 
 690 
 
 206 
 
 115 
 
 57 
 
 71679 8 
 
 the 
 
5 5 6 MEASURING Book IV. 
 
 the f of 310 Feet to be 206 Feet and 8 Inches ; 
 which being fet down. 
 
 The next is 230 Feet to be multiplied by 9 In- 
 ches j which 9 Inches is ■£ of 12 Inches, therefore 
 you muft take | of 230 Feet, faying, the half of 
 230 is 1 15, and ~ of 115 is 57 Feet and 6 Inches, 
 which being added to 115 make 172 Feet and 6 
 Inches, as you fee by calling up the Dimenfion, 
 
 Let the fifth Example be 
 
 The Feet being multiplied and fet 
 down, inflead of multiplying 10 by 94, 
 you mull divide 10 into 2 Parts (to 
 wit) 6 and 4, 6 being -J, and 4 being 
 •j of 12, and then fay, the \ of 94 is 
 47, and the i of 94 is 31 Feet 4 In- 
 ches, which being fet down, you pro- 
 ceed to know the Product of 48 Feet 
 multiplied by 1 1 Inches, which to do 
 fay thus, It is 48 Feet lacking 48 Inches, or 4 
 Feet, therefore I fet down 44 Feet ^ behold the 
 Example or Dimenfion call up in the Margin 
 above. 
 
 Thus I have Ihewn you from 1 Inch to 12 In- 
 ches (beginning at 2 Inches, and ending at 1 1 In^ 
 ches) how to work them into Feet, to fave Multi- 
 plication and Divifion * which way, after you are 
 expert in it, will fave you fome Trouble and Time 
 in calling up many Dimenfions. 
 
 94 
 
 11 
 
 48 
 
 10 
 
 752 
 
 
 376 
 
 
 47 
 
 
 31 
 
 4 
 
 44 
 
 9 
 
 4635 
 
 I 
 
Book IV. of Bricklayer s Work. 2 57 
 
 Some things to be obferved in meafurlng of 
 Brickwork . 
 
 Sometimes Walis are wrought 2 Inches thicker 
 than any of thefe Thicknelfes before fpoken of. 
 
 Which 2 Inches ferve for a Water-Table to 
 the Wall, which is ufually let off, about 2 Feet 
 above the Ground;; therefore you may mealure the 
 Wall at the fame Thicknefs that it is above the 
 Water-Table, and add the 2 Inch-work to it thus : 
 
 Suppofe a Wall 20 Feet in Length, and 2 B. thick 
 above the Water-Table ; 
 
 After you have taken the Dimenfion of the Wall 
 from the Bottom to the Height that you are to take 
 it, at 2 B. thick, then add 20 Feet iiiLengtbby the 
 Height of the 2 Inch-work (to wit) from the Bot- 
 tom to the' fet ting off, or Water-Table; which be- 
 ing halfed is fo much 4 Inch-work, and then re- 
 duce it to 1 ^ B. work. 
 
 Secondly Note, That all kinds of ornamental 
 Work in Brick, are generally done at a Rate by the 
 Foot, except there be an Allowance of a Sum of 
 Money over and above the Rate by the Rod, or 
 except a good Rate is allowed by the Rod, and all 
 the ornamental Work to be meafured into the 
 Rodwork. 
 
 By ornamental Work you are to underftand 
 ftrait or circular Arches over Windows or Doors, 
 Facio’s with or without Moldings, Architrave round 
 the Windows, or rubbed Returns, Friezes and Cor- 
 nices, Ruftick Quines, &c. In fine, all kind of 
 Work that is hewed with an Ax, or rubbed upon 
 
 a 
 
a) 8 MEASURING Book IV. 
 
 a Stone, is ornamental Work, and ought to be paid 
 for befides the Rod- Work. 
 
 Thirdly Note, When you meafure Arches, either 
 ftrait or circular, you muft meafure them in the 
 Middle (that is to fay) if a ftrait Arch be 1 2 In- 
 ches in Height or Depth, you muft meafure the 
 Length of it in the Middle of the 12 Inches, which 
 Length will be longer than if you meafure it on 
 the under Side next the Head of the Window, by 
 fo much as one Side of the fpringing of the Arch is 
 skewed back from the Upright of the Jaum. 
 
 Alfo in circular Arches, obferve that the upper 
 Part of the Arch is more in Length (being girt 
 about) than the under Part, becaule it is a Seg- 
 ment of a greater Circle, cut off by the fame right 
 Line that the leffer is, therefore it muft be girt in 
 the Middle. 
 
 Fourthly Note, That Facio’s, Architrave Friezes, 
 Cornices, Bafe Moldings and Plinths, are meafured 
 by the Foot, Line-meafure (or Running-meafure, 
 as it is commonly called.) 
 
 Fifthly Note, You muft take notice of the Jaums 
 of Windows and Doors being fplaved on the Infide 
 of Buildings, at the Rate by the Foot Running- 
 meafure. 
 
 Sixthly Note, Meafuring of Tyling is the fame 
 as meafuring of Roofing, which is taught at the 
 beginning of this Tra 6 t in meafuring of Carpenters 
 Work ; only there is this Difference, you muft mea- 
 fure the Length of all the Vallies, and add fb many 
 Feet as they are in Length, to the Number of Feet 
 in the Meafurement of the Roof ; you muft alfo 
 add the Eyes and Barge-courfes, Tyling being done 
 by the Square like Roofing. 
 
 You 
 
Book IV. of Bricklayers Work. 259 
 
 You may alfo meafure Tyling by the Flat and half 
 of the Building, after the fame Manner that you are 
 fhewed to meafure Roofing (in meafuring of Carpen- 
 ters Work) adding the Barge-courfes and Eves; alio 
 there you may fee how to meafure Hipt Roofs. 
 
 Seventhly Note, That Brick-paving is done by 
 the Yard, and is brought into Yards by dividing the 
 Produd of the Feet by 9, as you have read in mea- 
 furing of Painting, Plaiftering, 
 
 Eighthly Note, The meafuring of Gable-ends in 
 Brickwork, is the fame as in the meafuring of 
 Carpenters Work, wherein you are taught to mea- 
 fure them. 
 
 Ninthly, Remember in the meafuring of Walls 
 joining to each other anglewife, that you take the 
 Length of one Wall to the Outfide of the Angle, 
 and the Length of the other to the Infide of the 
 Angle. 
 
 Tenthly, If you fhould have a Gable-end, to 
 jjieafure, and you know the Width of the Houfe, 
 or the Length of the Bafe Line of the Gable-end, 
 and you defire to know the Length of the Perpen- 
 dicular, you may find it by the Proportions of the 
 Table that is in the meafuring of Carpenter’s Work, 
 and the Rule of Three. 
 
 Or briefer thus, Suppofe the Width of the Gable- 
 end to be 20 Feet, and you would know the Length 
 of the Perpendicular ; take the Length of the Rafter 
 which is 15 Feet, being •£ of the Width (for if the 
 Roof be true Pitch, of the Breadth is the Length 
 of the Rafter) to which add half the Length of the 
 Rafter Quiz.') 7 Feet and the Produd is 22 Feet 
 and f 3 the half whereof (to wit) 1 1 Feet and 3 
 Inches is the Perpendicular. 
 
 Although 
 
160 MEASURING Book IV. 
 
 Although this is a Way commonly ufed, yet it is 
 a fmall Matter more than the Length of the Per- 
 pendicular, as by the former Rule will appear. 
 
 Thus much may fufiice a diligent and pra&ical 
 Reader concerning Meafuring, and of which you 
 may fee more in the Appendix. 
 
 As to the Meafuring of Vaults and Arches, the 
 following Treatifes, intituled, tfhc Meafuring of 
 Plains and Bodies , will inftrudt you, as alfo the 
 Appendix. 
 
 And becaufe Brickwork is done at feveral Rates 
 per Rod, and many times there happens a Number 
 of Feet to remain, after you have divided your 
 whole Product by 272.25 (to bring your Number 
 of Feet into Rods) it will be convenient to fhew, 
 how to find the Price of 1 Foot of Brickwork, or 
 of any Number of Feet, at any Rate per Rod. 
 
 Having the Price per Rod given , and the Number 
 
 of Feet that you defire to know what they come to , 
 'work thus: 
 
 Firfi:, Reduce the Price of a Rod into Shillings, 
 by multiplying the Pounds by 20 (the Number or 
 Shillings in one Pound) and if the Price of a Rod 
 be Pounds and Shillings, then add the Shillings, to 
 the Product of the Pounds, being multiplied by 
 20, fo you will have the Price of a Rod in Shillings: 
 Next multiply the Number of Shillings by 12 (the 
 Pence in one Shilling) and that brings the Price of 
 a Rod into Pence. Then work according to the 
 Rule of Three, faying, 
 
 Exampk 
 
Book IV. of Bricklayers Work. 26 1 
 
 Example 1. 
 
 If I have 5 1. for a Rod of Brickwork, what mufl: 
 
 I have for 34 Feet, according to that Rate? 
 
 State the £>ueftion thus. 
 
 If a Rod, which is 272 Feet and •§■, or 272 
 give 5 1 what will 34 Feet give ? 
 
 1. s. 
 
 Multiply 5 00 
 
 by 20 
 
 The Product is loo Shilling* 
 
 Which multiply by 12 Pence 
 
 And the Product is 1200 Pence 
 
 Which multiply by 34 Feet 
 
 4800 
 
 3600 
 
 And the Produd is 40800 
 
 . Which divide by 272.25, adding 4 Cyphers to 
 the Product or Dividend. 
 
 - * 7 ; 
 
 2 % $0 
 
 7f'000 d. 
 
 40^000000 (149. T J-f 
 
 ;$:*£**** *5 
 a 
 
 J 27 £22 
 
 ‘ ^7 7 
 
 2 
 
 And the Quotient is 149 d. and t Is of a Penny, 
 being more than 3 q* for the 34 Feet, at $1. per- 
 Rod. Which 
 
i6z MEASURING Book IV. 
 
 Which 149 Pence divide by 12, and the Quo- 
 tient « i*j. and T J or 5 Pence 3 Farthings, the 
 Rate of the 34 Feet defired. 
 
 ' * (5 s. d. q. 
 
 *49 (12 5 3 
 
 * zz 
 
 l 
 
 Example 2. 
 
 If I have j ) l. 5 for a Rod, what comes 20 Feet 
 to at that Rate ? 
 
 State the ^htejlion thus , and it faves you the La- 
 bour of multiplying by 20. 
 
 F. S. 
 
 If 272.25. give 105 what will 20 Feet sive> 
 Multiply by 12 Pence S 
 
 210 
 
 105 
 
 The Product is 1260 theRateofa Rod in Pence 
 which multiply by 20 the Number of Feet, And 
 
 the Produd is 25200: To this Produd add 4 
 
 Cyphers, and divide by 272.25. 
 
 6 s ?%00 
 
 2}200000 ($ z 56 
 
 272 
 
 The Quotient is 92 d. and of a Penny, which 
 
 92 a. being divided by izd. gives you 7 s. 8 d. and 
 
 the 
 
Book IV. of Bricklayers Work. 
 
 the Fra&ion remaining namely ~r~5 is fomewhat 
 above 2 Farthings. 
 
 So at s l- 5 s - P er R °d, 20 Feet comes to 7 s. 8 d. 
 
 2 q. 
 
 Example 3 . 
 
 If I have 5 /. 55. for one Rod of Brickwork, 
 what muft I have for i Foot ? 
 
 /. 
 
 Multiply 
 
 By 
 
 °5 
 
 20 Shillings 
 
 The Product is 
 Add the 
 
 100 
 
 5 Shillings 
 
 It is 
 
 Which multiply by 
 
 105 Shillings 
 1 2 Pence 
 
 210 
 
 105 
 
 The Price of a Rod is 1260 Pence 
 
 Which fliould be multiplied by the Third Num- 
 ber in the Queftion (to wit) 1 Foot ; but becaufe 
 multiplying any Number by 1 doth not encreafe it. 
 Therefore divide 1260 c?. by 272.25 Feet. 
 
 220 
 
 5 
 
 *7 i 0000 d* 
 
 i 2600000 ( T |~ of a Penny, being more 
 2722$$* than 2 q. and not 3 q. 
 
 27 222 
 27 2 
 
 And the Quotient is 4 d. T f§- of a Penny. 
 
 Then to know the Value of the Fra&ion -£§■, 
 
 Multiply 
 
264 MEASURING Book IV. 
 
 Multiply the Numerator 62 by 4 (the Number 
 
 of Farthings in one Penny) the Produd is 248, 
 which divide by the Denominator 100, and the 
 Quotient is 2 Farthings, and the Remainder of 
 a Farthing; which Numerator 44, is Ids than half 
 of the Denominator xoo, and therefore Ids than 
 half a Farthing. 
 
 Example. 
 
 f 1 * k i • ■ u ^ 
 
 The.Numerator 62 of the Fra&ion 
 
 Multiply by 4 Farthings 
 
 TheProdudt is 2483 which divide 
 
 by the Denominator ioo. 
 
 And the Quotient is 2 Far- £^48 q. 
 things, and fomething more. j l00 ( 2 Too 
 
 Thus the Rate of 1 Foot of Brickwork after the 
 Rate of 5 /. 5 j. per Rod, is found to be Four- 
 pence half-penny half Farthing • which was re- 
 quired. 
 
 The fame Rule holds in Tyling v or Carpenters 
 Work, only changing the Number 272. 25. (the 
 Feet in a Rod) and inftead of it ufe 100 (the 
 Number of Feet in a Square.) 
 
 The End of the Fourth Book, 
 
MEASURING 
 
 O F 
 
 Superficial PLAINS,' 
 
 B Y 
 
 Vulgar ARITHMETIC^ 
 
 without reducing the Integers into the 
 leaft Denomination. 
 
 Together. With 
 
 Directions for Meafiuring 
 
 l *’ *, » * ^ ^ * ' ' • •. V*" * j .7 . ' 1 ' > ‘ 
 
 L A N D. 
 
 ' ■ — - - ■ - ■■■ "— * 
 
 £ The FIFTH BOOK. 
 
 By VEN. MANDET. 
 
 LONDON: 
 
 Printed in the Year M.DCC. XXVII. 
 
M E AS U RING 
 
 O F 
 
 Superficial P L A I N S. 
 
 book v. 
 
 CHAP. I. 
 
 By meaf tiring of Plains, is meant the finding the 
 Area or Content of Superficial Figures , fuch as are 
 Quadrates, 'Triangles, Circles , &c. and as a 
 Line is made by the Motion of a Point, fo like - 
 •wife a Superficie is made by the Motion of a Line, 
 
 N meafuring of Plains, the firft Figure to 
 begin with (as being moft eafy) is a 
 Quadrate or Square, which is a Figure 
 inclofed with four equal Sides, making 
 
 four right Angles. . 
 
 The next is an Oblong or long Square, which is a 
 
 Figure having four right Angles, and four Sides, but 
 
 not equal or of one Length. Two of which Sides be- 
 ing oppofite, are of one Length and Parallel ; and^t ^ 
 
Book V. Of Superficial ‘Plains. 267 
 
 other two Sides being of another Length, are op- 
 pofite and parallel. 
 
 The meaiuring of thefe two kinds of Plains, you 
 have been already lhewn in the preceding Traci of 
 Meafuring (to wit, by multiplying the Length by 
 the Breadth.) 
 
 PROP. I. Fig. I. 
 
 To meafure ( or find the Content of) a Rhombus^ 
 being a Figure like a Quarry of Glafs, having four 
 equal Sides ; and two Pair of equal Angles. 
 
 S Uppofe the Rhombus ABCD in Fig. I. one Side 
 w hereof is 5 Feet, and 4 Inches ; take this for 
 your Length. 
 
 Then for the Breadth take the neareft Difiance be- 
 tween any two of the Sides, as fuppofe the Prick 
 Line EF being 5 Feet, multiply thefe two, and the 
 Produd is 26 Feet, 8 Inches, which is the Area or 
 Content of the Rhombus ABCD. 
 
 Otherwife thus , 
 
 Draw the Diagonal Line AC, which divides the 
 Rhombus into two Triangles 3 next draw the Per® 
 pendicular Line DB. 
 
 Then take the Diagonal Line AC for your Length, 
 and the half Length of the Perpendicular for your 
 Breadth, and multiply them, and the Produd is 
 the Content, 
 
268 MEASURING BookV. 
 
 See the Examples cajl up both Ways. 
 
 Feet In. 
 
 ThcLength of one SideAB (the other three > 
 being each the fame Length) is 5 5 4 
 
 The neareft Diftance (or EF) is 5 o 
 
 25 
 
 1 8 
 
 Being call up the firft Way, the Content is 26 8 
 
 cfhe Second Way. 
 
 F. In. P. 
 
 The Length of the Diagonal Line AC is 9 2 o 
 
 The half Length of the Perpendicular is 2 10 1 1 
 
 18 4 
 
 7 6 
 1 8 
 
 8 3 
 
 1 
 
 The fame Content as before 26 8 o 
 
 The multiplying of the Parts of Inches into the 
 Feet and Inches, you are taught in the foregoing 
 Trait of Meafuring of Glaziers Work. 
 
 PROP. 
 
Book V. Of Superficial Tlains. 2 69 
 
 PROP. II. Fig. 11. 
 
 Fo meafure ( or find the Content of) a 'Trapezium, or 
 four-fided Figure, two fides whereof are parallel. 
 
 ( 1 
 
 S Uppofe the Trapezium ABCD in Fig. II. whole 
 longeft Side A B is 9 Feet, 4 Inches ; , the Side 
 parallel to it is DC, being 7 Feet, 2 Inches and 1| of 
 an Inch ; the Breadth being taken the neareft Way, is 
 the prickLine whofeLength is 6 Feet, and ^ of an Inch. 
 
 Add the two parallel Sides together, whereof take 
 half the Sum for your Length ; then take the neareft 
 Diftance between thole two Sides that are neareft 
 together for your Breadth, and multiply them, the 
 Product whereof is the Content. 
 
 • 1 
 
 Example. 
 
 The longeft Side AB, is 
 The fhorteft parallel Side DC, is 
 
 Thofe two being added, is 16 7 4 
 
 The mean Length being one half, is 8 3 8 
 
 Thisbeing mult by the neareft Diftance 6 o 3 
 
 48 Sec, 
 1 6 
 
 4 
 
 a 9 
 a 
 
 50 o 11 
 
 F. Ia P. 
 9 4<S 
 7 a 10 
 
 The Content required is 
 T 3 
 
 The 
 
27 ° MEASURING Book V. 
 
 The 1 1 beyond the Place of Parts of Inches is A 
 Parts of t,: Part of an Inch, or T \\ Parts of an Inch, 
 which is not confiderable, except the Superficies that 
 you meafure be fome rich or coftly thing. 
 
 Note, That I account, or fuppofe the Inch to be 
 divided into 12 Parts, for two Reafons : The firft 
 is. The more Parts it is divided into, the nearer to 
 the Truth you may take the Length and Breadth of 
 any Superficies or Solid, and likewife proceed the 
 nearer to the true Content. 
 
 The fecond Reafon is, Becaufe the Parts of Inches 
 thus divided, havethe fame Proportion to the Inches, 
 that the Inches have to the Feet, and by that means 
 thefe Parts may be multiplied into the Feet and 
 Inches ; which according as the Inch is divided on our. 
 common Rulers into 8 Parts, thofe Parts cannot be 
 multiplied into the Feet and Inches, but the Length, 
 and likewife the Breadth of any Superficies muft be 
 reduced into the leaft Denomination, and then mul- 
 tiplied each by other, which cauieth a great deal of 
 Multiplication, and alfo ofDivifion, beforeyou arrive 
 at the Superficial Area or Content, as you may fee 
 in the preceding Trad at the beginning of Mea- 
 furing of Glaziers Work, from Page 216, to Page 
 219, where 1 have proved this W ay of multiplying 
 ' the Parts of Inches into the Feet and Inches to be 
 true, both by Decimal Arithmetick, and alfo by 
 reducing the Length and Breadth into the leaf!: De- 
 nomination (to wit, Quarters of Inches.) 
 
 And indeed (as Mr. Ou^btved faith in his Cirelesof 
 Proportion) the dividing of the Inches on the Rulers, 
 into Halfs, and Quarters, andHalf-quarters, is moft in- 
 artificial ■, and I fuppole was done before Decimal 
 Arithmetick was brought to light. Which if the Foot 
 
 were 
 
BookV. Of Superficial ‘Plains. 271 
 
 were divided into io Parts (which may as well be 
 called Inches, as now it is divided into 1 2 Parts) and 
 each of thofe Parts into 1 o Parts, and the Length and 
 Breadth of Plains or Solids taken therewith, and fet 
 down Decimally, and caft up by Decimal Arithme- 
 tick ; the Science of Meafuring would be much 
 eafier than now it is, and lave a great deal of 
 Labour. 
 
 Note , That a four-fided Superficies which hath 
 none of the Sides parallel, likewife every plain 
 Figure of more Sides than four, being propoled to 
 be meafured, muft with Diagonal Lines be divided 
 into Triangles. 
 
 And Note, That every fuch Figure containeth fo 
 many Triangles as it hath Sides, abating two out of 
 the Number (to wit, if the Figure to be divided into 
 Triangles have fix Sides, it will contain four Triangles, 
 if feven Sides, five Triangles) ; then thefe Triangles 
 are to be meafured leverally, and their Contents being 
 added, gives the Content of the irregular Figure. 
 
 PROP. III. Fig. III. 
 
 *10 meafure any triangle. 
 
 M ultiply the longeft Side (being ufually called 
 the Bafe) by half the Perpendicular (which 
 is a Line let fall Perpendicular from the Angle, 
 oppofite to the Bafe) and the Produd is the Con- 
 tent of the Triangle. 
 
 Suppofe a Triangle ABC in Fig. III. whole Bale 
 AC is 7 Feet, the Side CB 6 Feet, and the Side BA 
 5 Feet, and the Perpendicular, which is the prick 
 
 T 4 Line, 
 
272 MEASURING Book V. 
 
 Line 3Feet,2lnches,thehalfwhereofis2Feet, ilnch i 
 which being multiplied by the Bafe Line 7 Feet, makes 
 14 Feet, 7 Inches, for the Content of that Triangle. 
 
 Example . 
 
 The Length of the Bale is 
 
 Half the Length of the Perpendicular is 
 
 F. I. 
 7 0 
 
 2 1 
 
 The Content of the Triangle is 
 
 14 7 
 
 Otherwife thus: 
 
 
 Multiply the whole Perpendicular by half the Bale. 
 
 Example. 
 
 The whole Perpendicular is 
 
 Being multiplied by half the Bafe 
 
 F. I. 
 
 4 ^ 
 3 6 
 
 
 12 6 
 2 1 
 
 The Content (as before) is 
 
 14 7 
 
 I 
 
 PROP. IV. Fig. IV. 
 
 the Side of an Equilateral 'triangle being given ^ to 
 find the Perpendicular Arithmetically . 
 
 L E T the Side of the Triangle given be A C in 
 Fig. IV. whole Length is 8 Feet. 
 
 Firft fquare the Side, then fquare half the Bafe, 
 
 and 
 
Book V. Of Superficial ‘Plains. 275 
 
 and fubtrad that from the Square of the Side, and 
 the fquare Root of the Remainder, is the Length 
 
 of the Perpendicular. 
 
 Example . 
 
 F. I. 
 
 The given Side is A C 8 o 
 
 Which to fquare, mnltiply by 8 o 
 
 The Square of the Side A C is 64 o 
 
 The whole Bafe is 8, the half whereof is 40 
 Which to fquare, multiply by 40 
 
 The Square of half the Bafe is 160 
 
 Which fubtrad from 64 o 
 
 The Square of the given Side p 16 o 
 
 and the Remainder is 3 48 o 
 
 The Square Root whereof is 6 Feet, 1 1 Inches, 
 and T | Parts of an Inch Fere, being the Length of 
 the Perpendicular required. 
 
 It is fuppofed that the Reader hereof can extrad 
 Square and Cube Roots ; if he cannot, he may 
 learn it in Mr. Wingates Arithmetick, in which 
 Book they are plainly and briefly demonftrated ; or 
 he may work by Logarithms. 
 
 PROP- 
 
% 7 4 MEASURING Book V. 
 
 PROP. V. Fig. IV. 
 
 the Perpendicular and Side of an Equilateral ’triangle 
 being given , to find the Content of that triangle . 
 
 I N the former Diagram, Fig. IV. the Perpendi- 
 cular 6 Feet, n Inches T f Fere , and the Side 
 £ Feet being given, the Content is required. 
 
 Multiply the whole of either, by half the other, 
 and the Product is the Content required. 
 
 Example. 
 
 F. I. P. 
 
 The whole Bafe or Side is 800 
 
 The half of the Perpendicular is 3 5 7 ^ ere 
 
 24 
 
 3 4 
 4 8 
 
 The Content of the Triangle is 2788 Fere 
 
 If you multiply the whole Perpendicular 6 Feet, 
 11 Inches, by half the Side 4 Feet, it produceth the 
 lame as before (to wit) 27 Feet, 8 Inches, T f of an 
 Inch almoft, for Fere fignifies almoft or near. 
 
 to find the Content of the fame Equilateral triangle , 
 without knowing the Length of the Perpendicular. 
 
 Make it as 30 to the Square of one Side, fo 1 3 to a 
 fourth Number, which will be the Area requir’d. 
 Example. 
 
 6 4 2 * F. 
 
 _JJL % (27 fi the fame as before. 
 
 i)z 220 
 
 64. 
 
 PROP. 
 
Book V. Of Superficial Thins. 
 
 2 75 
 
 PROP. VI. Fig. V. 
 
 • * 
 
 <fhe Perpendicular and Bafe of a right angled Tri- 
 angle being given, the Content of that triangle 
 is required. 
 
 I N Fig. V. the Perpendicular FG, being 12 Feet* 
 and the Bafe CE 2 5 Feet, is given, and the 
 Content required. 
 
 Multiply the whole of either by half the other, as 
 12 Feet by 12 Feet, 6 Inches, or 25 Feet by 6, the 
 Product is 1 50 Feet, the Superficial Content required. 
 
 Or multiply the whole by the whole, the Produd 
 is 300, whereof take half, it is 150 as before. 
 
 Or thus , without the Perpendicular. 
 
 Multiply* the two Sides 15 by 20, the Produd is 
 300, whereof take half for your Demand • or mul- 
 tiply half of one Side by the whole of the other Side, 
 the Produd is 150, as before. 
 
 Thefe Rules hold good in thofe right-angled Tri- 
 angles, whofe longeft Side (or Bafe) being fquared, 
 the Produd is as much as the Square of both the 
 other Sides, as in Fig. V. the Square of CE is 625, 
 the Square of CF is 225, and the Square of F E 
 400 ; which two laft Squares being added, make 625, 
 the Square of the Bafe ^ folikewife, in a right-angled. 
 Triangle, whofe Sides are 6 Feet, 8 Feet, and 10 
 Feet, it holds the fame Proportion $ for the Square 
 of 10, which is 100, is as much as the Square of 6, 
 which is 36, and the Square of 8, which is 64, 
 being added together. 
 
 PROP. 
 
z 76 MEASURING Book V. 
 
 PROP. VII. Fig. VI. 
 
 c fhe Perpendicular and Bafe of an Ifofceles 'triangle 
 given^ to find the Area or Superficial Content . 
 
 I N Fig. VI. add all the three Sides together, where- 
 of take half, then take the Difference of each Side 
 from that half, and multiply thofe three Differences 
 one by another, the Produdt whereof multiply by the 
 half Sum of the three Sides, and the Square Root 
 of that Product is the Superficial Content. 
 
 Example . 
 
 The 3 Sides being added, make 56, the half 
 whereof is 28, then the Difference of the Side A B 
 (being 20) from 28 is 8 j the like Difference is be- 
 tween the Side B C and 28, namely 8 ; and the 
 Difference between the Side AC (being 16) and 28, 
 is 12. Now thefe 3 Differences being multiplied 
 one by another, namely, 8 by 8 is 64, and 64 by 
 .12, makes 7 68, which being multiplied by 28, the 
 half Sum of the 3 Sides, the Product is 21504, 
 whereof the Square Root is 146 Feet, 7 Inches, 
 
 8 Parts, and T ! of a Part, or 146 Feet, 7 Inches, 
 and of an Inch. 
 
 Which Rule is general for all right-lined Tri- 
 angles whatfoever. 
 
 PROP. 
 
Book V* Of Superficial Plains. 
 
 277 
 
 PROP. VIII. Fig. VII. 
 
 to find the Perpendicular of any ‘triangle Arithme- 
 tically , the Sides being given. 
 
 I N Fit VII. let the Sides given be ED 6 Feet, 
 "DF 8 Feet, and FE 10 Feet, it is required to 
 
 find the Perpendicular ? t n c 
 
 Square the 3 Sides feverally, then add the Square of 
 the Bafe Cor longeft Side) to the Square of one of 
 the other two Sides, as to the Square of ED, 10. 
 whence fubtrad the Square of the remaining Side, and 
 from the Remainder take half, which divide by the 
 Bafe, the Quotient thereof being fquared, deduct 
 or fubtra&it from the Square of the (horteft Side, 
 and the Square Root of the Remainder is the Length 
 of the Perpendicular. 
 
 Example. 
 
 The Square of ED is 36, the Square of DF is 64, 
 and the Square of FE is 100 ; then the Squareof the 
 Bafe 100 being added to the Square of ED 36, makes 
 1 3 6, from whence the Square of D F being graft- 
 ed, which is 64, the Remainder is 7 a, whereof the halt 
 is 3 6, which being divided by the Bafe 10, produceth 
 3 Feet i ; or 3 Feet, 7 Inches, 2 Parts, and , sofa Part, 
 for the Idler Segment of theBafe EG; the Square of 
 which Segment is 12 Feet |y, or 12 Feet, 11 nc 
 and 6 Parts, and being fubtraded from the Square of 
 ED 3 6, the Remainder is 23 Feetrf, 01-23 Feetand lS 
 of an Inch +, whofe Square Root is 4 Feet, 9 Inches, 
 
278 ME ASURING Book V. 
 
 and 7 Parts +, is the Length of the Perpendicular 
 DG required. But this being very troublefcme, I 
 will give you a Table in the Appendix , which will 
 Ihew the Length of feveral Perpendiculars. 
 
 PROP. IX. Fig. VIII. 
 
 *fbe Side of a Pentagon being given, to find the Area 
 or Content . 
 
 I N Fig. VIII. let the given Side be io, it is re- 
 quired to find the Area of that Pentagon. 
 Defcribe an Ilblceles Triangle on any Side of the 
 Pentagon, whofe Top is the Centre of the Pentagon, 
 as ADB, whofe Perpendicular being found is 6 Feet, 
 i o Inches, and 5 Parts + of an Inch, divided into 12 
 Parts, which being multiplied in half the Perimetry 
 25, produces 171 Feet, 8 Inches, and 5 Parts, for 
 the Area required. 
 
 This Rule is general in all kind of regular Poly- 
 gons, of how many Sides foever, as well for their 
 Superficial Content, as finding their Perpendicular. 
 
 But becaufe it is troublefcme finding the Centre 
 Geometrically, I will here Ihew how to find it by 
 Arithmetick. 
 
 The Side of a Pentagon being given, to find the Ra- 
 dius of a Circle infer ibed within that Pentagon- 
 
 The Rule is this. As 182 to 125, fo is the Side of 
 the Pentagon, to the Semidiameter of a Circle in- 
 feribed in it. 
 
 Example. 
 
 The Side given is 10 Feet, what is the neareft 
 Diftance from the Centre of the Pentagon to one of 
 the Sides? 
 
 Multiply 
 
gook V. Of SttpcirjicMil tains* 
 
 Multiply andDivide according to theRule ofThree, 
 and you will find 6 Feet, i o Inches, and j Parts + . 
 
 PROP. X. Fig. IX. 
 
 q’he Diameter of a Circle being given, to find the 
 Circumference thereof Arithmetically. 
 
 T H E Proportion of the Diameter to the Circum- 
 ference was firft found out by Archimedes, who 
 brought the Length of the Perimeterofa Circle within 
 the Limits of Numbers, very little differing from the 
 Truth, demonftrating the lame to be lefs than three 
 Diameters and a feventh Part, but greater than three 
 Diameters and x 7 \ Parts of the Diameter, fo that fup- 
 pofing the Radius to confift of iooooooo equal Parts, 
 the Arch of a Quadrant will be between 15714285 
 and 1 5704225 of the fame Parts. But fince Ludovicus 
 Van Cullen hath come yet nearer to theT ruth, and pro- 
 nounced from true Principles that the Arch of a Qua- 
 drant (putting as before iooooooo for the Radius) 
 differs not one wholeUnitefrom theN umb. 15707963. _ 
 But for our purpofe we fhall make choice either of 
 the Proportion in whole Numbers by Archimedes, 
 which is as 7 to 22, or of the Proportion which Me- 
 tius hath given, which is as 1 13 to 355, fo is the 
 Diameter to the Periphery or Circumference. 
 
 Let the Diameter given be 14 Feet, in Fig. IX. 
 it is required to find the Circumference thereof 
 Multiply the Diameter given (being 14) by. 22, 
 the Prod iid is 308, which divided by 7, bringeth 44 
 Feet for the Circumference required ; or multiply the 
 Diameter by 3 9 the Produd is 44, as before ; but this 
 makes the Circumference a fmall Matter too great, 
 
 as 
 
aBo MEASURING BookV. 
 
 as you may fee if you multiply 14 by 355, the Pro- 
 duct is 4970, which being divided by 1 13, the Quo- 
 tient is 43 and ^ the Remainder, which is m 
 Parts of an Unite (whether it be a Foot, or an 
 Inch, or any other Meafure) divided into 113 
 
 Parts. 
 
 If the Circumference be given, and the Diame- 
 ter required, it appeareth by this Rule, that the 
 Circumference 44 being multipily’d by 7, and the 
 Produd divided by 22, brings 14 the Diameter. 
 
 PROP. XI. Fig. IX. 
 
 tfhe Diameter and Circumference of a Circle being 
 given^ to find the Area or Superficial Content 
 thereof Arithmetically divers ways . 
 
 E T the Diameter of the Circle be 14, as in 
 
 JLj Fig. IX. and the Circumfererence thereof 44. 
 It is required to find the Superficial Content. 
 
 Multiply half the Diameter by half the Circum- 
 ference, it fhews the Content of any Circle. 
 
 Thus the half Circumference is 22, which being 
 multiplied by the Semidiameter 7, the Produdt is 
 154, the Superficial Content required. 
 
 Or multiply the whole Circumference 44, by the 
 Semidiameter 7, the Produdt will be 308, whereof 
 take half, which is 154, as before. 
 
 Or multiply the Square of the Diameter 196 by 
 11, the Product will be 21563 which divide by 
 14, brings 154, as before. 
 
 PROP* 
 
Book V. Of Superficial Plains. 281 
 
 PROP. XII. Fig. IX. 
 
 Having o'nly the Diameter of a Circle given , to find 
 the Content . 
 
 S AY, as 7 is to 22, lo is the Square of the Semi- 
 diameter to the Content of the Circle. 
 
 Let the Diameter given be 14, as in Fig. IX. 
 Then fay by the Rule of Three, if 7 give 22, 
 what will 49 give ? (which 49 is the Square of the 
 Semidiameter •) multiply the third Number 49, by 
 thefecond Number 22, the Product is 1078, which 
 divide by the fir ft Number 7, and you have 154 
 for the Content, as before. 
 
 PROP. XIII. Fig. IX. 
 
 Having the Circumference given , to find the Content . 
 
 T HE Rule is at 88 (which is 4 times 22) is to 
 7, fo is the Square of the Circumference to 
 the Content. 
 
 Example. 
 
 Let the Circumference be 44, the Square thereof 
 is 1936 ^ then fay by the Rule of Three, 
 
 If 88 give 7, what will 193^6 give? 
 
 Multiply 1936 by 7 
 
 The Product is 1 3552 
 
 U 
 
 Which 
 
i8 z ME A S U RING Book V. 
 
 Which divide by the firft Number 88, and the 
 Quotient is 1 54 Feet. 
 
 4 ? 
 
 * 2 f -MO 54 Feet, the Content of the- Circle. 
 
 y 
 
 %$ .... 
 
 HU3: 
 
 PROP. XIV. % IX. 
 
 tfhe Content of a Circle being given , to find the 
 
 Diameter. 
 
 1 
 
 THHE Rule is as 22 to (4 times 7, which is) 28, fi> 
 1 is the Content to the Square of the Diameter. 
 
 Example. 
 
 Let the Content given be 154* and it is required 
 to find the Diameter. 
 
 If 22 give 28, what will 154 give ? Multiply and 
 Divide, and you will find 196, the Square Root 
 whereof is 14, the Diameter required. 
 
 - ■: 93I2£T.U: > £& 'io ■ .i ’ j'Ai - 
 
 PROP. XV. Fig. IX. 
 
 fbe Content of a Circle being given , to find the 
 Circumference. 
 
 T H E Rule is, as 7 is to (4 times 22, which is) 
 88, fb is tne Content to the Square of the 
 Circumference. 
 
 x:; >* 
 
 Example. 
 
Book V- Of Superficial ‘Plains . 
 
 
 Example. 
 
 Let the Content given be 154, and the Circum- 
 ference required to be found. 
 
 Then lay, by the Rule of Three, 
 
 If 7 give 88, what will 154 give? Multiply and 
 divide, and you will find 1936, the Root whereof 
 being 44, is the Circumference required. 
 
 PROP. XVI. Fig. IX. 
 
 fhe Content of a Circle being given , to find the Side 
 of a Square , the Content of which Square fhall be 
 equal to the Content of the Circle . 
 
 E Xtra& the Square Root of the Content given, and 
 that Root is the Side of the Square required. 
 
 Example. 
 
 Let the Content given be 154, the Square Root 
 whereof is 12 Feet, 4 Inches t h Parts of an Inch, 
 which is the Side of a Square, whofe Content is 
 equal to the Content of the Circle given. 
 
284 MEASURING Book V. 
 
 PROF. XVII. 
 
 The Diameter of any Circle being given, to find the 
 Side of a Square , the Area of which Square (hall 
 be equal to the Area of the Circle of the given 
 Diameter . 
 
 T HIS Propofition may be fblved feveral Ways, 
 as firft according to Archimedes his Propor- 
 tion in whole Numbers, thus : 
 
 As 7 is to 22, io is the Square of the Semidiame- 
 ter to the Content required. 
 
 Let the Diameter given be 7 Feet, the Semidia- 
 meter will be 3 Feet, 6 Inches, the Square whereof 
 is 12 Feet, 3 Inches; which being multiplied by 22 
 Feet, the Product is 269 Feet, and 6 Inches, which 
 divided by 7, makes 38 Feet, 6 Inches for the Con- 
 tent of the Circle, which Content is a fmall Matter 
 more than the true Content. 
 
 Secondly, According to Metius his Proportion, 
 thus : • 
 
 As 11 3 to 355, fo is the Square of the Semi- 
 diameter to the Content required. 
 
 Example. 
 
 T he Square of the Semidiameter aforefaid is 1 2Feet, 
 3lnches, which being multiplied by 3 5 jFeet, produces 
 4348 Feet, 9 Inches; which Product being divided 
 
 by 
 
Book V. Of Superficial ‘Plains. 285 
 
 by xi 3 Feet, makes 38 Feet, 5 Inches, 9 Parts, and 
 T | of a Part, which Content is lefs than the Content 
 produced from Archimedes ’ s Proportion, by more than 
 t 4 Parts of a Superficial Foot, and the greater the 
 Diameter is, the more the Difference will be. 
 
 Or thus. As 452 (which is 4 times 1 1 3) is to 
 355 : So is the Square of the Diameter to the Con- 
 tent required. 
 
 Thirdly, It may be folved by Decimal Arithmetick, 
 thus ; as 1 Foot to Feet 3.1416, fo is the Square of 
 the Semidiameter to the Content required. 
 
 Example. 
 
 Feet 3.1416 being multiplied by Feet 12.25, P r °- 
 ducethFeet 38.484600, which friould be divided by 
 the firft N umber 1 ; but becaufe 1 doth neither increale 
 any Number being multiplied by it, nor diminilh any 
 Number being divided by it, therefore we find the 
 Content to be Feet 38.484600, which being reduced 
 to Feet, Inches, and Parts, is 38 Feet, 5 Inches, 9 
 Parts, 9 Seconds, and 4 Thirds. 
 
 Fourthly, It may be folved Decimally thus : 
 
 As Feet 1 1 4.. 59 1 5492^0 360 Fect,fois the Square 
 of the Semidiameter to the Content required. 
 
 Example. 
 
 360 Feet being multiplied by Feet 12.25, pro- 
 duceth Feet 4410.00 which being divided by Feet 
 114. 5915492, produceth Feet 38.48451, which 
 being reduced, is 38 Feet, 5 Inches, 9 Parts, 
 9 Seconds, and 2 Thirds ; which is a fmall Matter 
 left than the former Content. 
 
 U 3 
 
 But 
 
286 MEASURING Book V. 
 
 But peradventure the Pleader hereof may not un- 
 derftand Decimal Arithmetick, I will therefore ex- 
 hibit one Way more by Vulgar Arithmetick, for 
 the folving of this Proportion ; which is this : 
 
 Multiply the Diameter given, by ioFeet, 7 Inches, 
 7 Parts, 5 Seconds ; and divide the Product by 12 
 Feet, it produces the Side of a Square, which being 
 multiplied by itfelf, is the Content of the Circle 
 required. 
 
 Which I look upon to be as quick a Way as by 
 Decimals, efpecially if the Diameter given confifls 
 of Feet, Inches, Halfs, Quarters, &c. and the So- 
 lution be required in the fame, and cometh within 
 a Trifle as near the Truth. 
 
 And here I fhall a little infift upon the multiplying 
 the Fractional Parts of Inches into one another, for 
 two Reafbns. 
 
 Firft, As being a Way very ufeful, and faves a great 
 deal of Labour (to wit) of redicing the Integers 
 into the leaft Denomination (or Fractional Parts) it 
 being the ufual Way fo to do, among thofe who un- 
 derftand not Decimal Arithmetick. 
 
 Secondly, Becaule hitherto I have had occafion 
 only to multiply Twelfths of Inches into Feet and 
 Inches ; and having occafion now to multiply other 
 Twelfths into thole Twelfths, and other T welfths 
 into thefe Twelfths, &c. which I name thus, 
 
 Feet) Inches , Parts , Seconds , $ birds , Fourth s, &c. 
 
 And here Note, that as a Foot contains 12 Inches 
 Line mealure, fo an Inch is fuppofed (or accompted) 
 to contain 12 Parts Line meafure (although upon the 
 Rulers the Inches are divided but into 8 Parts, or Half- 
 
 Quarters) 
 
Book V. Of Superficial Plains. 28 7 
 
 Quarters) and each Part to contain 12 Seconds, and 
 each Second to contain 12 Thirds, and each Third 
 to contain 12 Fourths, &?£. And fo the Proportion 
 might be continued, if there were occafion, to 
 Fifths, Sixths, 
 
 But according to fuperficial Meafure, a Foot 
 contains 144 Inches, and an Inch 144 Parts, and a 
 Part 144 Seconds, &c. 
 
 Aifo obferve, that each Place of Twelfths is di- 
 ftinguiflied by having a Letter put over it ^ thus, 
 over the Feet write F, over Inches I, over Parts P, 
 over Seconds S, over Thirds T, over Fourths 
 Fo, fcfo as beneath. 
 
 And for the better underffanding how to multi- 
 ply the Fradlions or Twelfths into one another, ob- 
 ferve the Rules in the following Table. 
 
 1. Note, That Inches mult, into F. every 12 of the Product is 
 and any Numberlefs than iz is Inches 
 
 2. In. mult, into (or by) In. every t z of the Prod, is ' 
 and any Number of the Prod, lefs than iz are Parts 
 
 3. Parts mult, into Feet, every iz of the Prod. is 
 and any Number under 12 are Parts. 
 
 4. Parts multiplied into Indies, every 12. of the Prod, is 
 and any Number of the Prod, under 12 are Seconds 
 
 ?. Parts multiplied by Parts, every \z of the Prod, is 
 and any Number lefs than 1 2 are Thirds 
 
 6. Seconds multiplied into Fe. every 12 of the Prod, is 
 and any Number lefs than 1 2 are Seconds 
 
 7. Sec. mult, into Inches, every 12 of the Prod, is 
 and any Numberlefs than 12 are Thirds. 
 
 8. Sec. mult, into Parts, every 12 of the Prod, is 
 and any Number lefs than 1 2 are Fourths. 
 
 9. Sec. multiplied into Sec. every 12 of the Prod, is 
 and any Number lefs than 12 are Fifths, 
 
 jc. Thiids multiplied into Feet, every 12 of the Prod, is 
 and any Number lefs than 1 2 are Thirds. 
 
 ji. Thirds multiplied into Inches, every xz of the Prod, is 
 and any Number lefsthan 12 axe Fourths. 
 
 12.. Thirds multiplied into Parts, every 12 of the Prod, is 
 and any Number lefs than 1 2 are Fifths. 
 
 I?. Fourrhs multiplied into Feet, every 12 of the Prod is 
 and any Number left than iz are Foiuths 
 
 F.I.P.S.T.F- 
 
 0 1 
 
 .0 1 
 
 o o 1 
 O O C I 
 O O I 
 000; 
 
 00061 
 
 u 4 
 
 Farther 
 
288 MEASURING Book V. 
 
 Farther in Fradtional Parts you need not proceed. 
 
 Here Note, if the greateft Denomination of the 
 Dimenfion to be caft up be Inches, and the Con- 
 tent be required in fuperficial Inches ; then Parts mul- 
 tiplied into Inches, every 12 of theProduft is one 
 Inch , and Seconds multiplied into Parts, every 12 
 is one Part, &c. 
 
 Explanation of the fable in 3 or 4 Examples. 
 
 Firft, Suppofe you have 6 Parts to be multiplied 
 into 7 Feet 3 
 
 Say 6 times 7 is 42 3 then ask yourfelf how many 
 times 12 is 42? and the Anfwer is 3 times, and 6 
 remaining 3 therefore according to the 3d Rule of the 
 preceding Table, fet 3 in the Place of Inches, 
 noted with the Letter I, and the 6 that remains, 
 being left than 12, you muft write in the Place 
 of Parts, noted above the Dimenfion with the Let- 
 ter P. 
 
 Secondly, Suppofe 8 Thirds to be multiplied into 
 8 Inches, fay 8 times 8 is 64, which contains 5 
 twelves, and 4 remaining 3 therefore, according to 
 the nth Rule, fet 5 in the Place of Thirds, noted 
 by T 3 and the 4 that remains, fet in the Place or 
 Column of Fourths, noted with Fo. 
 
 Thirdly, Suppofe you have 10 Seconds to be 
 multiplied by n Parts, fay 10 times n is no, 
 which contains 9 times 12, and 2 remaining 3 then, 
 according to the 8th Rule of the foregoing Table, 
 let down 9 in the Place of Thirds, noted on the 
 Top with T, and the 2 that remains fet in the Place 
 of Fourths, noted by Fo, 
 
 Thefe 
 
Book V* Of Superficial ‘Plains. 289 
 
 Theft being premised, I come now to folve the 
 foregoing P yop ■ (to wit} the Diameter of a Circle 
 being given, whofe Length is 7 Feet, it is required 
 to find a Square, whofe Content (hall be equal to 
 that Circle. 
 
 The Proportion is, as 12 Feet is to 10 Feet 7 In- 
 ches, 7 Parts, 5 Seconds ; fo is the Diameter to the 
 Side’of a Square, or the Square of the Diameter to 
 the Content required. 
 
 The Rule thus; Multiply the Diameter by 10 
 Feet, 7 Inches, 7 Parts, 5 Seconds, and divide the 
 Product by 12 Feet. The Quotient, together with 
 the Remains, if there happen to be any, is the Side 
 of a Square, which being multiplied in itfelf, is equal 
 to the Conteht required. 
 
 Example. 
 
 F. 
 
 The Length of the Diam. given is 7 
 Which multiply by iol 
 
 I. 
 
 1 0 
 
 1 7 
 
 p. 
 
 0 
 
 7 
 
 s. 
 
 0 
 
 s 
 
 T.Fo- 
 oi 0 
 
 The Prod, of xo F. by 7 F. ss 70 
 The Prod, of 7 Inches by 7 F. is 4 
 The Prod, of 7 Parts by 7 F. is 0 
 The Prod, of 5 Sec. by 7 F. is 0 
 
 I 
 
 4 
 
 0 
 
 1 
 
 2 
 
 IX 
 
 The Total Produft is 74! y[ 3|n 
 
 Which Product, according to the foregoing 
 Rule, muft be divided by 12. Therefore divide 
 the 74 Feet, by 12 Feet, thus. 
 
 Inches 
 
MEASURING Book V. 
 
 apo 
 
 * (2 Inches. 
 
 *4(6 Feet. 
 
 i x 
 
 And the Quotient is 6 Feet, and 2 remains, which 
 is 2 Inches, fo the Side of the Square equal to the 
 Circle, is 6 Feet, 2 Inches, 5 Parts, 3 Seconds, 11 
 Thirds ; for inftead of dividing the 5 Inches, 3 
 Parts, 11 Seconds (that belonged to the 74 Feet 
 in the Total Produft) by 12, only do thus ; Re- 
 move them one Place forwarder towards the Right 
 Hand, it is the fame as if you had divided them by 
 12 ; fb then the 5 Inches will ftand iij the Place of 
 Parts, and be 5 Parts, and the 3 Parts will be 3 
 Seconds, and the 11 Seconds will be 11 Thirds; 
 which obferve as a general Rule in any Number to 
 be divided after this manner. 
 
 When you have divided the Feet by 12, if there 
 happen to be no Inches remaining, then fet down 
 your Feet under the Place of Feet, and put a Cypher 
 in the Place of Inches, and then fet your other 
 Fractions next ; as in this laft Example, if there 
 had been no Remainder befides the Quotient, then 
 you muft have put o in the Place of Inches, and 
 then the 5 Inches which belonged to the 74 Feet 
 will be changed into 5 Parts, as if they had been 
 divided by 12. And likewife the 3 Parts will be 
 turned into 3 Seconds, Sc. 
 
 So 
 
Book V. Of Superficial Thins. 291 
 
 So that 74 Feet, 5 Inch. 3 Farts, 1 1 
 Seconds, being divided by 12 
 Which is the Side of a Square 5 ) 
 
 which multiply by 
 The Prod, of 6 Feet by 6 Feet, is 
 The Prod, of 2 Inch, by 6 Feet, is 
 The fame again is 
 The Prod, of 2 Inch, by 2 Inch, is 
 The Prod, of 5 Parts by 6 Feet, is 
 The fame again is 
 The Prod, of 5 Parts by 2 Inch, is 
 The fame again is 
 The Prod, of 5 Parts by 5 Parts, is 
 The Prod, of 3 Sec by 6 Feet, is 
 The lame again is 
 The Prod, of 3 Sec. by 2 Inch, is 
 The fame again is 
 The Prod, of 3 Sec. by 5 Parts, is 
 The fame again is 
 The Prod, of 11 Thirds by 6 F. is 
 The fame again is 
 The Prod, of n Thirds by 2 In. is 
 The fame again is 
 
 Farther you need not proceed, 
 the Value of the Produds being 
 inconfiderate. 
 
 F. I.P. S. T.Fo. 
 
 ",i tj*i 
 
 f>As\ 3 !“ 
 
 36 
 
 1 
 
 1 
 
 o 
 
 216 
 2 6 
 00 10 
 10 
 2 
 
 1 6 
 1 
 
 The Total Produd is 
 
 10 
 
 lot 
 
 38 59] 10] 71 2 
 
 Which is the Content required, and agrees with 
 the Decimal way within Parts of an Inch. 
 
 Although I have fet the Produd of each Multipli- 
 cation one under another, for Demonftration ; yet 
 
 never- 
 
292, MEASURING Book V. 
 
 neverthelels when you are perfect in calling up, you 
 may let the Produ&s of 4 or 5 Multiplications in 
 one level Line. 
 
 Let a fecond Example be^ 
 
 A Diameter given, whole Length is 42 Feet, 
 and the Content of that Circle is required. 
 
 According to Archimedes his Proportion in-whole 
 Numbers, working as is ftiewn before, the Content 
 will be 1 3 86 Feet. 
 
 And according to Metius , in whole Numbers the 
 Content will be found to be 1385 Feet and Aj of a 
 Foot, which reduced, is 1385 Feet, 5 Inches, 3 
 Parts, 8 Seconds. 
 
 Let us try what the Content will be our Way. 
 
 The Diameter i$ 
 Which multiply by 
 
 F. 
 
 42 
 
 10 
 
 I. P.S.T.F. 
 
 o 
 
 7 
 
 o oj o] o | 
 
 75 
 
 The Prod, of 10 F. by 42 F. is 420 
 The Prod, of 7 Inch, by 42 Feet, is 24 6 
 The Prod, of 7 Parts by 42 Feet, is o 24 
 The Prod, of 5 Sec. by 42 Feet, is | 
 
 6 
 
 i7i6. 
 
 The Total Product is 446) 7|iij6| I j 
 
 Which 446 Feet, divide by 12. 
 * 
 
 Inches. 
 
 44^(37 Feet. 
 
 1 
 
 The 
 
Book V. Of Superficial Thins. 295 
 
 The Quotient is 3 7 Feet and 2 Inches, to which add 
 the 7 Parts 1 1 Seconds 6 Thirds, by changing their 
 Places one Degree forwarder toward the Right Hand, 
 
 F 
 
 For the Side of the Square 3 7 
 
 Which to fquare, multiply by 37 
 
 The Prod, of 7 Feet by 3 7 is 259 
 
 The Prod, of 30 F. by 37 F. is 111 
 
 The Prod, of 2 Inch, by 37 Feet is 6 
 
 The fame again is 6 
 
 The Prod, of 2 Inch, by 2 Inch, is 
 
 The Prod, of 7 Parts by 37 Feet, is 
 
 The fame again is 
 
 The Prod, of 7 Parts by 2 Inches, is 
 
 The fame again is 
 
 The Prod, of 7 Parts by 7 Parts, is 
 
 The Prod, of 1 1 Sec. by 3 7 Feet, is 
 
 The fame again is 
 
 The Prod, of 1 1 Sec. by 2 Inch, is 
 
 The fame again is 
 
 The Prod, of 1 1 Sec. by 7 Parts, is 
 
 The fame again is 
 
 The Prod, of 11 Sec. by 11 Sec. is 
 
 The Prod, of 6 Thir. by 37 Feet, is 
 
 The fame again is 
 
 The Prod, of 6 Thir. by 2 Inches, is 
 
 The fame again is 
 
 The Prod, of 6 Thir. by 7 Parts, is 
 
 The fame again is 
 
 2 
 
 2 
 
 21 
 
 21 
 
 P.S. T.I 
 
 7 o. 
 
 7 1 
 
 11 
 
 6 
 
 0 
 
 71 
 
 11 
 
 6 
 
 0 
 
 4 
 
 
 
 
 7 
 
 
 
 
 7 
 
 
 
 
 1 
 
 2 
 
 
 
 1 
 
 2 
 
 
 
 
 4 
 
 1 
 
 
 33 
 
 11 
 
 
 
 33 
 
 11 
 
 
 
 
 1 
 
 10 
 
 
 
 1 
 
 IC 
 
 
 
 
 6 
 
 S 
 
 
 
 6 
 
 s 
 
 
 
 
 10 
 
 
 18 
 
 6 
 
 
 
 18 
 
 6 
 
 
 1 
 
 
 1 
 
 
 
 
 1 
 
 3 
 
 
 
 
 3 
 
 The Content or Total Prod, is 13S5I 5I 8j o| 1 
 
 Which 
 
*94 MEASURING BookY. 
 
 Which Content is four Parts more than the Con- 
 tent, ^according to Metrus\ Proportion, but near 
 four Parts leis than the Content according to Ay~ 
 ehimedess proportion. 
 
 And by the way, note, That a Circle of i4Feet Dia- 
 contains 4 times as much as a Circle that is 7 F. 
 Diameter. The Rule is thus, the Diameter 7 is con- 
 tained in the Diameter 14 two times / now the Square 
 of 2 is 4, therefore a Circle whofe Diameter is 14, 
 contains a Circle whofe Diameter is 7 four times. 
 
 Tike wife theDiameter in the laftExample being 42, 
 
 contains the Diameter 7 fix times ; for the Square of 
 6 is 3,6, therefore a Circle whofe Diameter is 42, 
 contains 36 Circles of 7 Feet Diameter apiece. 
 
 For if you multiply 38 Feet, 5 Inches, 9 Parts, 10 
 Seconds, + (being the Content of a Circle whofe 
 Diameter is 7) by 36, you will find it produce the 
 Content of a Circle whofe Diameter is 42. 
 
 Example. 
 
 F. I. P. S.T. 
 
 ACircIewh0feDia.is7F.theC0n.is 38I <1 0I106 
 
 Whic^ multiply by 36 1 
 
 228! O I ol OjO 
 1 14 
 
 , i$A 7 l 3 Q[i 8 'o 
 
 A Cir. whofe Dia. is 42 F . contains 13 8 5 j s\ 7] 6jo 
 
 Let the tfhird Example he 
 A Diameter whofe Length is 38 Feet, 6 Inches, 
 and an half and half a quarter, and it is required to 
 find the Content of that Circle. 
 
 The half Inch is 6 Parts, but the half quarter is \ 
 Part of an Inch, which muft bereduced to Twelfths, 
 
 thus j 
 
Book V. Of Superficial Thins. 295 
 
 thus 5 Multiply the Numerator of the Fradion (to 
 be reduced) by 12, and divide the Produd by the 
 Denominator of the lame, and the Quotient is the 
 Numerator of a Fradion that hath 12 for his Deno- 
 minator; and if any Number remain befides the 
 Quotient, multiply that Remainder by 12, and 
 divide the Produd by 8, fo long until you have no 
 Remainder befides the Quotient. 
 
 Example. 
 
 « of an Inch is to be reduced ; multiply the Nu- 
 merator i by your new Denominator I2 3 it is ItiJ 
 12 • divide it by the Denominator 8, the Quotient 
 is 1 and 4 remains ; the 1 in the Quotient is 1 Part, 
 which add to the f Inch or 6 Parts, and then it 
 makes 7 Parts: Then multiply the Remainder 4 
 by 12, the Produd is 48 ; which divide by the 
 Denominator 8, and the Quotient is 6 and no Re- 
 mainder, which is 6 Seconds. So \ of an Inch being 
 reduced to I2ths, is. 1 Far. 6 Sec. F. I. P. S.T 
 
 Now I fet down the Diameter thus 38 
 
 Which 1 multiply by 
 
 10 
 
 6 7 
 7I 7 
 
 380I 
 
 
 
 
 
 5 
 
 3 
 
 6 
 
 
 
 22 
 
 2 
 
 4 
 
 I 
 
 
 
 5 
 
 10 
 
 
 
 
 22 
 
 2 
 
 4 
 
 
 
 
 3 
 
 6 
 
 
 
 
 5 
 
 
 
 
 
 15 
 
 TO 
 
 
 
 
 
 3 
 
 6 
 
 
 
 
 2 
 
 6 
 
 The Produd is 
 
 4°9 
 
 1 x 1 1 x 1 1 3I0 
 
 Which 
 
2 96 MEASURING Booky. 
 
 Which 409 divide by 12. 
 
 F. I. P. S. T. 
 
 *#(* In. The Side of the Sq. is 34I 1 n nj 3 
 #09^(34 F. Which multiply by 34I 1 1 1 1 1 1 3 
 
 * zz ■ 
 
 The Content required is 
 
 136 
 
 31 
 
 2 
 
 
 
 1022 
 
 IO 
 
 I 
 
 
 
 2 
 
 IO 
 
 
 II 
 
 IO 
 
 
 
 31 
 
 2 
 
 II 
 
 IO 
 
 
 
 
 
 10 
 
 
 
 
 
 31 
 
 2 
 
 II 
 
 
 
 
 31 
 
 2 
 
 II 
 
 10 
 
 
 
 
 8 
 
 6 
 
 
 
 
 
 8 
 
 6 
 
 
 0671 3 1 ill- 8 1 610 
 
 This laft Example could not be folved fo near the 
 Truth by vulgar Arithmetick any other way, without 
 reducing the 38 Feet, 6 Inches, and half Inch into 
 half quarters of Inches, which is a long way about. 
 
 PROP. XVIII. 
 
 S the Superficial Content of a Circle being given , to 
 find the Diameter. 
 
 L E T the Content given (being found accord- 
 ing to Archimedes's Proportion) be 346 Feet 
 6 Inches ; and the Diameter required. 
 
 The Proportion is as 1 1 is to 14, fo is the Content 
 given, to the Square of the Diameter required. 
 
 The 
 
Book V. of Superficial 'Plains. i p/ 
 
 The Rule thus, Multiply the Content given by 
 14, the Product whereof divide by 11, and the 
 Square Root of the Quotient is the Length of the 
 Diameter. 
 
 Example. 
 
 Multiply the Content given, being 346 Feet 6 
 Inches, by 14, the Product is 4851, which being 
 divided by n, the Quotient is 441 for the Square 
 of the Diameter, the Roof whereof is 21, the 
 Length of the Diameter requir’d. 
 
 Or thus, and more exact, As 355 to 452, fo is 
 the Content given, to the Square of the Dia- 
 meteten 
 
 Example. 
 
 Multiply the Content given, being 346 Feet 6 
 Inches, by 452, the Produ6t is 156618, which be- 
 ing d : vided by 355 produces 441 Feet and 
 of a Foot, which, when reduced, is 441 Feet 2 
 Inches 1 Part, 6 Seconds, the Square Root where^ 
 of is 21 Feet 1 Inch 5 Parts 6 Seconds, the Dia- 
 meter required. 
 
 To thofe who underftand Decimals, the Propor- 
 tion for folving the 18th Proportion is this, 
 
 As 1. 00000 is td 1.27324, fo is the Cdntent given 
 to the Square of the Diameter. 
 
 Therefore if you multiply 1.27324 by the given 
 Content, the Square Root of that Product will be 
 the Diameter required. 
 
25>S MEASURING Bookv. 
 
 PROP. XIX. 
 
 if he Diameter of a Circle being given ; to find the 
 Side of a Square which may be infcritfd within 
 that Circle. 
 
 T H E Rule is this : Square the Diameter, then 
 take one half of the Product, then find the 
 Root of that Product, it will be the Side of the 
 Square defired. 
 
 Example. 
 
 Let the Diameter given be 28 Feet 3 Inches, 
 the Square whereof is 798 Feet 9 Parts, the Half 
 whereof is 399 Feet 4 Parts and 6 Seconds , the 
 Square Root whereof is 19 Feet 11 Inches 8 Parts 
 5 Seconds +, for the Length of the Side of the 
 Square required. 
 
 Decimally thus. As 1. 000000 is to .70710 6, fo is 
 the Diameter to the Side required. 
 
 Therefore if you multiply the faid .707106 by the 
 Diameter given, the Product will be the Side of the 
 infcrib’d Square requir’d • for, as you read before, 
 if you fhould divide the Product by 1. 000000, it 
 w ill ftill be the fame. 
 
 PROP. XX. Fig. X. 
 
 i The Diameter and Arch-Line of a Semi-circle being 
 given ; to find the Area thereof. 
 
 L ET ABC in Fig. X . be a Semi-circle given, 
 whofe Diameter is A C, and the Arch-Line 
 ABC, it is required to find the Area of that 
 Semi-circle. Multi'* 
 
Book V* of Superficial ‘Plains. 299 
 
 Multiply half the Arch-Line n by the Semidia- 
 meter 7, the Produdt will be 77 for the Area 
 required. 
 
 PROP. XXI. Eg. X. 
 
 *fhe Semi diameter and Arch-Line of a Sett, or of a 
 Circle being given ; to find the Area . 
 
 I N Fig. X. let BCD be the Sector of a Circle* 
 whole Semidiameter is DC, orDB, and the 
 Arch-Line B C, and it is required to find the 
 Area. 
 
 Multiply the Semidiameter 7, by half the Arch- 
 Line BC 5 Feet 6 Inches, the Product is 38 Feet 
 6 Inches for the Area required. 
 
 PROP. XXII. Fig. X. 
 
 Any Segment or fart of a Circle being given , to find 
 the Content . 
 
 I N Fig. X. let AF BE be the Segment of a Cir- 
 cle, the Content whereof is required. 
 
 Firft find out the Centre of the Arch-Line A F B, 
 (by the 21ft Prop, of the 20th Page of Geo- 
 metry) then draw the Lines D A and D B, and 
 caft up the whole Figure A F B D, as in the laft 
 Prop, which will be 38 Feet 6 Inches, then find the 
 fuperfidal Content of the Triangle A BD, which is 
 24 Feet 5 Inches 9 Parts, and dedud: the fame out 
 ot the whole Content 38 Feet 6 Inches ; the Re- 
 mains is 14 Feet and 3 Parts, for the Content of 
 the given Segment, which was required. 
 
 By 
 
 X 2 
 
3oo MEASURING Book V. 
 
 By this Rule (obferved with Difcretion) may all 
 manner of Segments or Parts of a Circle, whether 
 greater or lefs than a Semi-circle, be eafily mca- 
 iured. 
 
 PROP. XXIII. Fig. X. 
 
 2 o find the Diameter of a Circle , by having one Part 
 of the Diameter given , alfo having the Length 
 of the Chord crojfing the Diameter in the given 
 Part. 
 
 I N Fig. X. let F E be the Part of the Diameter 
 given, alfo let AB be the given Chord which 
 cuts off rhe given Part of the Diameter ; it is re- 
 quired to find the whole Diameter Arithmeti- 
 cally. 
 
 F E the Part of the Diameter given, is 2 Feet 
 and - \ of an Inch, AB the Chord interfering it is 
 9 Feet 10 Inches of an Inch. 
 
 The R ule is thus : Square one half of the Chord, 
 and divide the Produdl by the Part of the Diame- 
 ter given, which Quotient being added to the laid 
 Part, is the Length of the Diameter required. 
 
 Example. 
 
 The whole Chord AB is 9 Feet 10 Inches 10 
 Parts, the Half whereof is 4 Feet 1 1 Inches 5 Parts, 
 the Square of which is 24 Feet 6 Inches 2 Parts and 
 4 3ds, which being divided by 2 Feet 7 Parts the 
 Part of the Diameter given) the Quotient is 1 1 Feet 
 11 Inches 8 Parts, to which 2 Feet 7 Parts being 
 added, make 14 Feet for the Length of the Dia- 
 meter required. 
 
 PROP. 
 
Book V. of Superficial ‘Plains. 301 
 
 PROP. XXIV. Fig. X. 
 
 Any Segment of a Circle being given , whofe Chord - 
 Line doth not exceed the Chord of the Quadrant 
 of the fame Circle 3 to find the Content wi. bout 
 finding the Diameter , and without deferring any 
 more of the Circumference. 
 
 A Lthough, by the precedent Prop. XXII. any 
 Segment of a Circle may be meafured • yet 
 becaufe the finding of the Center, and meaiur'ng 
 of the Sedtor, and then meafuring the Triangle and 
 fubtradting it from the Sedtor, requires a great 
 deal of 1 rouble and Time 3 I will here fhew how 
 to do it with 1 els Trouble, and very near the Truth* 
 in Imall Segments. 
 
 Let the Segment given be the fame (whofe Con- 
 tent was found by 22 Prop. ) in Fig. X.- whofe Chord- 
 Line A B is 9 Feet 10 Inches and 9 Parts, and the 
 Part of the Diameter F E (cut off by the Chord 
 Line, is 2 Feet 7 Parts and 6 Seconds 3 the Content 
 of which Segment A F BE is required. 
 
 Note, If the verfed Sine, viz. the Line F E be 
 not given, you muft divide the Chord A B in the 
 Middle, and from that middle Point, as at E, raile 
 a Perpendicular to the Arch-Line. 
 
 Then take the whole Length of the Chord A B, 
 and two Thirds of the Length of the Line F E, 
 to which two Thirds add 7 Parts, then multiply 
 thofe two Lengths, and the Product gives you the 
 Content. 
 
 X 3 
 
 Example, 
 
MEASURING Book V. 
 
 Example. 
 
 The verfed Sine FE is 2 Feet 7 Parts 6 Seconds, 
 two third Parts whereof is 1 Foot 4 Inches 5 Parts, 
 to which 7 Parts being added, makes it 1 Foot 5 In- 
 ches 5 which multiply 3 d by the Chord A B, (name- 
 ly) 9 Feet 10 Inches 9 Parts, produces 14 Feet and 
 2 Parts for the Content, which is 1 Part lefs than 
 the Content found by the 22 Prop. 
 
 Another Example . 
 
 I nFig'XI. the Semi-diameter is 14 Feet, and the 
 Content of the Quadrant A B C D will be found by 
 the former Rules, according to Metiuss Proportion, 
 to he 153 Feet 11 Inches 1 Part 6 Seconds. 
 
 Now it is required to find the Content of the 
 Segment A BCE. 
 
 By fubtrading the Content of the T riangle A CD, 
 which is 98 Feet 4 Parts and 5 Seconds, from the 
 Content of the Quadrant which is as above, there 
 remains 55 Feet 10 Inches 9 Parts for the Content 
 of the Segment A BCE. 
 
 Let us fee what the other Way will produce. 
 
 The Length or Part of the Diameter BE is 4 
 Feet 1 Inch 2 Parts, two third Parts whereof is 2 
 Feet 8 Inches 9 Parts 4 Seconds, to which add 7 
 Parts, and it makes 2 Feet 9 Inches 4 Parts 4 Se- 
 conds 3 which being multiplied by the Chord-Line 
 AC, whole Length is 19 Feet 9 Inches 7 Parts, 
 the Produd is 54 Feet 11 Inches 6 Parts for the 
 Content of the Segment 
 
Book V. of Superficial Plains. 303 
 
 A third Example. 
 
 Let the Content of the two little Segments, A B, 
 B C, in Fig. XI. be required. 
 
 The Line F G is i Foot, two third Parts where- 
 of is 8 Inches, to which 7 Parts being added, makes 
 it 8 Inches 7 Parts, which being multiplied by 10 
 Feet 10 Inches, the Length of the Chord-Line A B 
 produces 7 Feet 8 Inches 1 1 Parts, for the Content 
 of the Segment A F B G, the other Segment B C 
 being equal to this, add 7 Feet 8 Inches 1 1 Parts to 
 the Content of the firft, and the Product is 15 Feet, 
 
 5 Inches 10 Parts for the Content of both the little 
 Segments. 
 
 The Triangle ABCE, whofe Content is 40 Feet, 
 
 6 Inches 5 Parts, being deducted from the Content 
 of the Segment ABCE, which is 55 Feet 10 Inches 
 9 Parts, there remains 15 Feet 4 Inches 4 Parts, 
 for the Content of the two little Segments 3 which 
 is half an Inch lefs than was found before. 
 
 PROP. XXV. Fig. XII 
 
 An irregular Flat or Figure being given 3 to find the 
 Area or Content thereof. 
 
 L ET abedefghi in Fig. XII. be an irre- 
 gular Plat, whofe fuperficial Content is re- 
 quired. 
 
 Reduce the fame into as many Trapezia’s as it 
 will contain, at firft the Trapezium a b c d 3 Se- 
 condly, a d e f 3 Thirdly, a f g, i, and there re- 
 mains the Triangle, i g h. In which three Tra- 
 pezia’s, draw the Diagonals b d, d £, and f i 5 which 
 
 X 4 (hall 
 
5 o4 MEASURING Book V. 
 
 Ihail be as common Bales to each Triangle on either 
 Side 3 on which Bales let fall the Perpendiculars 
 from the ieveral Angles at a, c, e, and g • then in 
 every 1 rapezium take the Length of the Bafe by 
 itlelf, and the Length of the two Perpendiculars 
 thereon falling, joined together, in one Number by 
 itlelf, then multiply one half of the one in the 
 whole of the other, and the Frodud is the Area of 
 that Trapezium, which relerve by itlelf, and work- 
 ing in like lort with the reft ; and laftly, the Tri- 
 angle i, g, h : Colled all their Products together, 
 which fhall fhew the Content required. 
 
 PROP. XXVI. 
 
 tfo find the Superficial Content of any Oval. 
 
 L~T~ Ake both the Diameters, and multiply them 
 i one into another; then extrad the Square Root 
 of that Produd, and that Root will be the Diame- 
 ter of a Circle, whole Content will be equal to the 
 Content of the Oval. For the Area of a Circle is 
 to the Area of an Ellipfis, as the Square of the 
 Diameter of that Circle is to the redangled Figure 
 of the tranfverfe and conjugate Diameters of the 
 Ellipfis by the 6th oi Archimedes s Conoides and 
 Spheroides, 
 
 ♦ 
 
 Example. 
 
 Let Fig. XIII. be an Oval, whofe longeft Dia- 
 meter is io Feet, and the fhorteft Diameter is 6 
 Feet, and the Content required. 
 
 Multiply 
 
Book V. of Superficial ^Plains. 30^ 
 
 Multiply the Length ioby the Breadth 6 ? the Pro- 
 dud; is 6 o, the fquare Root whereof is 7 Feet 8 In- 
 ches 1 1 Parts 5 Seconds, which is the Length of the 
 Diameter, the Content of whofe Circle is equal to the 
 Content of the Oval • then multiply the Diameter 
 by 10 Feet 7 Inches 7 Parts 5 Seconds (as you were 
 fhewn at the 5th Example of Prop. XVII. of this) 
 and the Produd is 82 Feet 4 Inches 6 Parts 3 Se- 
 conds, which divide by 12, it produces 6 Feet 10 
 Inches 4 Parts 6 Seconds 3 Thirds, which being 
 multiplied by it felf Qviz. fquared) the Produd is 
 47 Feet 1 Inch 5 Parts 1 1 Seconds, the Content of 
 the Oval required. 
 
 See the whole Work. 
 The Length of the Oval 
 The Breadth of the Oval 
 
 Being multiplied, the Produd is 
 The Square Root whereof is 
 
 F. 
 
 I. P. 
 
 S. 
 
 T. 
 
 10 
 
 b 
 
 0 
 
 1 0 
 
 °l 
 
 6 
 
 1° 
 
 ! 0 
 
 1 0 
 
 °[ 
 
 6o|o 
 
 1 0 
 
 1 ol o| 
 
 7 
 
 8 
 
 i" 
 
 S 
 
 1 
 
 10 
 
 7 
 
 1 7 
 
 S 
 
 1 
 
 70 
 
 4 
 
 
 
 
 6 
 
 8 
 
 8 
 
 5 
 
 5 
 
 4 
 
 1 
 
 6 
 
 6 
 
 
 
 9 
 
 2 
 
 3 
 
 4 
 
 
 4 
 
 1 
 
 2 
 
 11 
 
 
 
 4 
 
 8 
 
 2 
 
 
 
 4 
 
 2 
 
 4 
 
 
 
 2 
 
 11 
 
 
 The Produd is 
 
 8*141 6 I 3l *1 
 
 Which 
 
;o 6 MEASURING 
 
 Which Produdt divide by 12. 
 
 (1 Inch. And it is 
 
 2 (o Which multiply by 
 (6 Feet 
 
 The Content of the Oval is 
 
 Book V. 
 
 F. I.P.S.T. 
 
 6|io|4 6 \ i \ 
 6;io! 4 6/ 3I 
 
 36 
 
 8 
 
 4 
 
 1 
 
 4 
 
 s 
 
 2 
 
 3 
 
 4 
 
 2 
 
 5 
 
 2 
 
 3 
 
 4 
 
 2 
 
 
 
 3 
 
 5 
 
 3 
 
 
 
 3 
 
 5 
 
 2 
 
 
 
 
 1 
 
 6 
 
 
 1 
 
 
 1 
 
 6 I 
 
 
 
 
 
 A 
 
 47l ililn|3l 
 
 In Decimals thus : Multiply the Length of the 
 Oval by the Breadth, and divide the Produdl by 
 I.27324 Foot, it gives the Content. 
 
 PROP. XXVII. Fig. XIV. 
 
 tfo find the Superficial Content of a Cylinder , the 
 Diameter being given. 
 
 XTOfr, That a Cylinder is a folid Body, which 
 i. >1 may well be reprefented by a Roll either of 
 Timber or Stone, fuch as are ufed in Gardens for 
 the rolling of Walks. 
 
 .The Proportion is. As 7 to 22, or As 113 to 
 355 j So is the Diameter and Length of the Side 
 multiplied one by the other, 1*0 the fuperficial Con- 
 tent of the Outfide of the Cylinder, belides the two 
 Bafes or Ends, 
 
 The 
 
Book V- of Superficial ^Plains. 3 07 
 
 The Rule thus : Multiply the Diameter by the 
 Length or Side ; then multiply that Product by 
 355? the Product whereof divide by 113, and it 
 gives you the Content of the round Superficies. 
 
 Let Fig. XlV. be a Cylinder, whole Diameter is 
 4 Feet, and the Side or Length 10 Feet, and the 
 Content required. 
 
 Firft, Multiply the Diameter 4 by the Length 
 10, the Produdt is 40 * which multiply by 3 55, and 
 the Product is 14200 3 which divide by 113, and 
 the Quotient 125 Feet, and 75 remaining, which 
 is ,tt of a Foot; and being reduced, is 125 F. 7 I. 
 
 1 1 P. 6 S. for the Content required. 
 
 And if you had wrought after the Proportion of 
 
 7 to 22, the Content would have been 125 F. 
 
 8 1 . 7 P. fere , which is a fmail matter more than 
 the former. 
 
 If you have the Circumference and Length of a 
 Cylinder given, and the Content required ; 
 
 Multiply the Circumference by the Length, and 
 the Product is the fuperficial Content, 
 
 \ Example. 
 
 Let the Circumference and Length of the Cylin- 
 der, Fig. XIV be given, and the Content re- 
 quired. 
 
 Multiply the Circumference 12 F. 6 I. 9 P. 7. S. 
 by the Length 10 F. and the Product is 125 F. 7 L 
 j 1 P. for the Content required. 
 
 If you are minded to add the fuperficial Content 
 of the two Bafes or Ends of the Cylinder, you will 
 find the Content of them as you are Ihewn before 
 in meafuring of Circles, v 
 
 PROP, 
 
308 MEASURING BookV. 
 
 prop, xxviii. rig . xv . 
 
 find the Superficial Content of a Cone. 
 
 A Cone is a Body which hath a Circle for its 
 Bale, from whence it dinrivfhcth equally 
 (like a round Spire of a Steeple) till it end in a 
 Point. 
 
 Let Tig. XV. be a Cone, the Circumference of 
 whole Bale is 24 F. and the Length of the Side 10 F. 
 and the Content required. 
 
 If you require the Content of the whole Cone, 
 that is to fay, the Content of the Bale, as well as 
 the Content of the Outfide , then mulriply the whole 
 Side by half the Compafs or Periphery of the Bale; 
 and to the Product add the Content of the Plain of 
 the Bale. 
 
 Example. 
 
 The whole Circumference of the Bafe being 24 F. 
 the half thereof is 12 F. which being multiplied by 
 the Length 10 F. produces 120 F. for the Content 
 of the Outfide, befide the Bale: Then by the 13th 
 Prop, of this, find the Content of the Circle whole 
 Circumference is 24, and add it to the former Con- 
 tent, and the Produft is the fuperficial Content of 
 the Cone, together with its Bale. 
 
 Example. 
 
 F. I.P.S. 
 
 TheCont. oftheCone without the Bale is i2o'olo 
 The Content of the Bafe is 45 i9i9 
 
 1 te srv* 1 iteCone ’ tos " her 
 
 PROP. 
 
Book V. of Superficial Thins. 309 
 
 PROP. XXIX. Eg. XVI. 
 
 Zo find the Superficial Content of a Pyramid . 
 
 A S a Cone hath a round Bafe, and d'minilheth 
 equally till it end in a Point j fo a Pyramid 
 hath an angular Bafe of 3, 4, 5, 6, or any Number 
 of Sides, and diminilheth to a Point at the Top. 
 
 Let Fig. XVI. be a Pyramid, whofe Bale is a 
 Square, whofe Side is 5 Feet, and the Length from 
 the Bafe to the Top 10 Feet, and the fuperficial 
 Content required. 
 
 Add the four Sides of the Bafe toge ther, of which 
 Product take half 5 then multiply that half by the 
 Side or Length. 
 
 Example. 
 
 One Side being 5 Feet, the four Sides added 
 make 20 Feet, the Half whereof is 10 Feet ; which 
 multiplied by 10 Feet the Length, produces ico 
 Feet for the fuperficial Content, befides the Bafe : 
 If you are minded to add the Content of the Bale 
 to it, which is 5 times 5 Feet, or 25 Feet, then the 
 fuperficial Content of the Pyramid, together with 
 the Bafe, will be 125 Feet. 
 
 Likewife, if the Bafe be Triangular, you mull 
 add the three Sides of the Bafe together, and take 
 half of the Produft, which multiplied by the Length, 
 gives you the Content : T hen if you would add the 
 Content of the Bafe to the Content of the Outfide, 
 meafure it as you are Ihewn before in meafuring of 
 Triangles, 
 
 Alfo 
 
 * 
 
310 MEASURING BookV. 
 
 Pr ^ ir i lf r th i? ? afe c .^ e a Pentagon, take half the 
 Produd of the five Sides of the Bafe (being added 
 together) and multiply it by the Length, and the 
 Produd gives you the Content : If you are minded 
 to add the Bafe find the Content, as you are fhewn 
 at Prop. IX. of this. 
 
 PROP. XXX. 
 
 find the Superficial Content of a Globe or Sphere. 
 
 Multiply the Diameter or Axis by theCircum- 
 .]-> 1 ference, and the Produd will be the fuper- 
 ficial Content of the round Body or Globe. 
 
 Example. 
 
 Let the Diameter be 14 Feet, the Circumference 
 will be 44 fere, which being multiplied one by the 
 other, the Produd is 616 Feet for the Content of 
 the Superficies of the Globe. 
 
 Or find the Content of the Circle that hath the 
 fame Diameter as the Globe, and multiply that 
 Content by 4, and the Produd is the fuperficial 
 Content required. 
 
 Example. 
 
 A Circle whofe Diameter is 14, the Content is 
 154, which multiplied by 4, the Produd is 616, 
 as before. ’ 
 
 PROP. 
 
Book V. of Superficial ‘Plains. 3 1 1 
 
 PROP. XXXI. Fig. XVII. 
 
 tfo find the Superficial Content of a Fragment , or Part 
 of a Globe. 
 
 T H E Superficie of any Fragment, or Portion of 
 a Globe, is to the Superficie of the remaining 
 Portion, as the Portion of the Axis or Diameter cut 
 ofl^ is to the remaining Portion of the feme. 
 
 Example. 
 
 Let there be in Tig. XVIJ. a Portion of a Globe 
 given ABC, whole Part (or Segment) of the Axis 
 (or Diameter) is 5 Feet, and the whole Diameter 
 14 Feet, and the fuperficial Content of the Circular 
 Part is required. 
 
 Say, by the Rule cf Three, If 14 give 616, what 
 will 5 give? Multiply 616 by 5, the Product is 
 3080 3 which divide by 14, and you have 220 Feet 
 for the fuperficial Content of the Segment, or Part 
 of the Globe. 
 
 I fijall now conclude the Meafuring of Plains, with 
 fotne Directions whereby to meafure Land. 
 
 Wherein Note, That 5 l Yards, or 16 7 Feet in 
 Length, is called (according to Statute) a Pole or 
 Perch j and one Perch in Breadth, and 40 in Length, 
 is called a Rood ; and 4 Perches in Breadth, and 40 
 in Length, is one Acre : So that an Acre contains 
 160 fquare Perches, half an Acre 80 Perches, and 
 a quarter of an Acre, commonly called a Rood, 40 
 Perches, In 
 
MEASURING Book V. 
 
 In meafuring of Land, it is ufual to take the 
 Lengths and Breadths with a Chain, the Length of 
 which Chain fome make to contain 4 Poles, (or 
 Perches) as Mr. Gunter and others 5 and fome make 
 it to contain 2 Poles, as Mr. Kathborn , and others. 
 tfhe Chain which I would advife is thus : 
 
 To contain 2 Perches (or 33 Feet) in Length, 
 each of which Perches, divide into 12 equal Parts, or 
 (as 1 may call them) Links, and each Link into it 
 equal Parts, with little Wyers or Notches crofs-wife. 
 
 Thus the Length of a Perch will be divided into 
 1 44 Parts, and the whole Chain, confiding of 2 Per- 
 ches, will contain 288 Parts 3 in the Middle of which 
 Chain it will be convenient to have a pretty large 
 Ring, that fo you may diftinguifh where one Perch 
 ends, and the other begins : Alfo midway between 
 that large Ring and one end of the Chain hang a 
 Curtain-Ring 3 and likewife hang another Curtain* 
 Ring midway between the other end and the large 
 Ring: So the whole Chain will be divided into four 
 equal Parts or half Perches by the three Rings. 
 
 The Chain being thus divided, you have every 
 whole Pole equal to 12 Links, or 144 Parts; every \ 
 of a Pole equal to 9 Links, or 108 Parts ; every half 
 Pole equal to 6 Links, or 72 Parts : And laftly, eve- 
 ry quarter of a Pole equal to 3 Links, or 36 Parts. 
 
 The Chain being divided after this manner, you 
 have only three Denominations to fet down, name- 
 ly, Chains , Links , Parts , if you will meafure fo 
 near the Truth ; but in mod Menfurations you need 
 proceed no farther than to Chains and Links, and 
 then you have but two Denominations to fet down. 
 
 Alio the Manner of calling up the Dimenfions, is 
 the fame as is fhewn all along before, only altering 
 
 the 
 
Book V. Of LA NT). 315 
 
 the two firft Denominations, to wit, whereas you did 
 fet over the Dimenfions F. I. P. lignifying Feet, 
 Inches, Parts , fo in meafuring of Land, you muft 
 fet over the Dimenfions C. L. P. fignifying Chains, 
 Links, Parts: For as a Foot contains 12 Inches, 
 and an Inch 12 Parts, Line-Meafure (or in Length) 
 fo likewife a Pole or Chain contains 12 Links, and 
 one Link 12 Parts. 
 
 You muft remember to double the Number of 
 your Chains when you fet them down, becaufe a 
 Chain contains 2 Poles or Perches. 
 
 Suppofe in Fig. V. CFEB to be a Piece of Land 
 lying in form of a long Square, and the Length FE 
 being meafured with the Chain, to contain 8 Chains, 
 6 Links, and 6 Parts ; becauie the dou- C.L.P 
 
 ble of 8 is 1 6, I let it down thus 16 
 
 Alfo the Breadth EB fet thus, containing 
 
 Which I multiply as if they were Feet, 
 Inches, and Parts 
 
 
 6 
 
 6 i 
 
 ! 
 
 ing 14 
 
 9 
 
 3! 
 
 
 64 
 
 4 
 
 6 
 
 
 167 
 
 7 
 
 1 
 
 6 
 
 1 12 
 
 4 
 
 4 
 
 
 5 
 
 
 
 !i 
 
 And the Product: in fquare Perches, is 244(410] i| 
 
 — — >• 
 
 Which 244 Chains or Perches you muft divide 
 by 160 to bring them into Acres, as you read above 
 that an Acre contains 160 fquare Poles or ro 
 Perches. ( 8 4 
 
 So you have 1 the Quotient, which is 1 ■ 
 
 Acre, arid the 84 that remains being Per- 
 ches, divide by 40, the Number of Perches in a Rood, 
 and you have the Quotient 2 Roods and (4 
 4 Perches remaining. £4(2 
 
 Thus the Content of that Piece of Land 40 
 
 y is 
 
3 14 MEASURING Book V. 
 
 is found to be i Acre, 2 Roods, 4 Perches, and 
 the 4 Links which remain in the Multiplication is 
 one third Part of a Perch. 
 
 Again , 
 
 Suppofe F/jr. VII. to be a Field inclofed with 
 three Sides, like a Triangle EDF, and you require 
 to know the Content thereof ; 
 
 Let the Side ED be 120 Chains or Perches, 
 the Side DF 160 Chains or Perches, and the Side 
 FE 200 Perches, and the Perpendicular DG 95 
 Perches. 
 
 As you were fhewn in meafuring of Triangles, take 
 half the Bafe or Side EF, which is 100 Perches, and 
 multiply it by the Perpendicular DG 95 Perches. 
 
 Example. 
 
 The Length of half the Bale EF*, is 
 Which multiplied by the Perpendicular 
 
 CL. P. 
 ioojo 
 95 '° 
 
 500 
 
 900 
 
 The Produd is, in fquare Perches 9500I0I0I 
 
 ■Which being divided by 160, produces 59 Acres, 
 (6 1 Rood and an half; (for if you divide 
 
 * f 0(0 60 by 40, it is one and a half) which 
 
 00(59 is the Content of that Triangular Piece 
 *600 0 f Land. 
 
 16 
 
 Example. 
 
Book V. Of LAND. 315 
 
 Example. 
 
 A fquare Piece df Land being in Breadth 120 
 Chains, 9 Links, 6 Parts ; and in Length 125 Chains^ 
 6 Links, and 2 Parts, and the Content is required. 
 
 C. L. P; 
 
 The double of 120 is 240, fet thus 240 
 
 9 
 
 1 6 I 
 
 The double of 125 is 250, fet it thus 250 
 
 6 
 
 1 A 
 
 12000 
 
 4 
 
 6 
 
 480 
 
 125 
 
 1 
 
 120 
 
 
 3 
 
 187 
 
 so 0 
 
 
 > 
 
 The Product in Perches is 60321) 7 1 1 o [ 
 
 Which divide by 160, and the Quotient is 377 
 Acres, and 1 remaining, which is 1 Perch * to which 
 add the 7 Links that are in the Pfo- 
 dudt of the Multiplication, and the * * 
 whole Content will be 377 Acres, izzzQt 
 1 Perch, and 7 Links, which is half 377 
 
 a Perch, and 1 Link. 1^000 
 
 The 1 o Parts being but of a fmall *66 
 
 Value, are feldom fet down. * 
 
 Forafmuch as in fome Countries they allow 1 8 Feet tb 
 the Foie or Perch , and in other Countries moire , as 
 20 or 24 ; allow 160 of their Perches to an 
 
 Acre : It will he convenient here to Jhew how to re- 
 duce one kind of Meafure into, another: 
 
 Y i Example 
 
MEASURING Book V. 
 
 316 
 
 Example, 
 
 Suppofe it were required to reduce 4 Acres, 3 
 Roods, and 4 Perches of Statute Meafure (that is, 
 16 Feet and ail half to the Pole) into cuftomary 
 Meafure of 18 Feet to the Pole. 
 
 Firft, reduce your given Quantity, 4 Acres, 3 
 Roods, and 4 Perches, into the leaft Denomination, 
 to wit. Perches ; by multiplying the 4 Acres, by 1 60 
 (the Number of Perches in one Acre) and the Produdt 
 is 640 Perches in the 4 Acres: Then multiply the 3 
 R oods by 40 (the Number of Perches* in one Rood) 
 and the Produdt is 120 Perches in the 3 Roods. 
 Laftly, add the 640 Perches, and the 1 20 Perches, alfb 
 the 4 Perches together, and the total Product is 764 
 Perches, being contained in the given Number, 
 •viz. 4 Acres, 3 Roods, and 4 Perches. 
 
 Secondly, fquare the two Poles or Perches, that 
 is to fay, multiplyeach initfelf: Asfirft, multiply the 
 Pole of 18 Feet by 18 Feet, and the Produdt is 324 
 Feet: Then multiply the Pole of 16 Feet and an 
 half, by 16 Feet and an half^ and the Produdt is 272 
 Feet and 3 Inches. 
 
 Thirdly, fay by the Rule of Three; As 324 
 (which is the Square of the 18 Feet Pole) is to 272 
 Feet and 3 Inches (which is the Square of the 16 Feet 
 and an half Pole) fo is 764 Perches (the Number 
 given) to a fourth Number required. 
 
 Firft, multiply 764 by 272 Feet and 3 Inches, 
 and the Produdt is 207999, which divide by the 
 firft Number in the Queftion, namely, 324, and 
 the Quotient is 641 Perches, Hi of a Perch, which 
 being abbreviated, is || of a Perch. 
 
 Fourthly, 
 
Book V- Of LAND. 317 
 
 Fourthly, to reduce the Perches into Acres, divide 
 them by 160; fo 641 Perches being divided by 160, 
 you find 4 Acres and 1 Perch remaining. 
 
 Thus you lee, that 4 Acres, 3 Roods, and 4 Perches 
 of Statute Meafure, of 16 Feet and a half to the 
 Pole, being reduc’d, is 4 Acres, 1 Perch, and 5 1 of 
 a Perch cuftomary Meafure of 1 8 Feet to the Pole. 
 
 An Example of the whole IVorh 
 
 The Quantity given is 4 Acres, 3 Roods, 4 Perches 
 of Statute Meafure, to be reduced to Cuftomary 
 Meafure of 18 Feet to the Pole or Perch. 
 
 Firft reduce the Quantity given into the leaft 
 Denomination, namely. Perches. 
 
 Firft reduce the 4 Acres. 
 
 The Perches in 1 Acre is 160 ; being multiplied. 
 
 The Perches in 4 Acres is 640 
 
 •r:nivyr; 
 
 Secondly, Reduce the 3 Roods. 
 
 The Perches in 1 Rood is 40 ; being multiplied, 
 
 '■ ' '• i '* •*- •; J ‘ <■; -*■ J } ! ' tT* ' ' ' *- 
 
 The Pdrches in 3 Roods is 126 
 
 • . - -v, ^ 
 
 The third Number 4 being the leaft Denomina- 
 tion, namely,' Perches, needs no reducing. 
 
 Now add the Perches together, 640 
 ; 120 
 
 4 
 
 AndyoufifidtheQuan.givencontains 764 Perches. 
 
5 « S MEASURING BookV. 
 
 The Quantity given being brought into Perches, 
 the enfuing Work will be to find the Number of 
 Superficial Feet that each Pole contains. 
 
 And firft the Pole of 
 Being fquared, or multiplied by 
 
 Contains, or the Produdt is 
 
 Secondly, The Pole of 
 Being fquared, or multiplied by 
 
 Contains, or the Produdfc is 
 
 Thirdly, by the Rule of Three fay. As 324 Feet 
 is to 272 Feet 3 Inched, fo is 764 Perches to a 
 Fourth Number required. F. I. 
 
 Multiply the Second Number 272I3I 
 By the Third 7 64 1 I 
 
 1088 
 
 1632 
 
 1904 
 
 191 
 
 The Product is 207999 1 
 
 Which 
 
 18 Feet. 
 18 
 
 144 
 
 18 
 
 324 Feet. 
 
 F. I 
 16 
 1 616 
 
 96 
 168 
 S3 
 
 272(3] 
 
0 
 
Book V. Of LA N^D. 319 
 
 Which divide by the firft Number 324, and the 
 Quotient is 641 Perches and j£j Parts or a Perch, 
 which abbreviated is Parts of a 
 Perch, which is almoft a Perch. (3 
 
 Fourthly, Having now reduced 6(1 
 the Quantity of given Perches 
 (namely, 764 of Statute Meafure) *0?#w(64i 
 to 641 Perches, and -}£ of a Perch S'* £ 4 # 
 
 of Cuftomary Meafure of 18 Feet 
 to the Pole or Perch 3 the next Work ? 
 is to bring thefe Perches into Acres : 
 
 Which to do, divide them by 160, and the Quotient 
 (as you fee in the Margin) is 4 Acres and 1 Perch ; 
 to which add the i* of a Perch, and 
 then you have 4 Acres 1 Perch and 
 of a Perch of Cuftomary Mea- 
 sure (of 18 Feet to the Pole) con- 
 tained in 4 Acres, 3 Roods, and 4 
 Perches of Statute Meaftire, of 1 6 Feet and an half 
 to the Pole, 
 
 (1 Perch 
 (4 Acres. 
 
 A Second Example . 
 
 Suppofe, on the contrary, that 4 Acres, 1 Perch, 
 and of a Perch of cuftomary Meafure of 1 8 Feet 
 to the Pole, were required to be reduced into 
 Statute Meafure of 16 Feet and an half to the Pole. 
 
 As in the laft Example, Firft, reduce the given 
 Quantity into the leaft Denomination, to wit. Feet , 
 for in this Example, the leaft Denomination is Parts 
 of a Perch, which are Feet. 
 
 Therefore firft multiply the 4 Acres by 160, and 
 the Produdt is 640, to which add the 1 Perch given* 
 and it is 641 Perches. 
 
 Y4 
 
 Secondly, 
 
320 MEASURING BookV. 
 
 Secondly, to bring thefe Perches into Feet, you 
 muft multiply them by the Number of Feet contained 
 in a Perch ; which as you found before in the laft 
 Example, by multiplying 18 by 18, the Product 
 was 324: So then you muft multiply 641 Perches 
 by 324 Feet, and the Product will be 207684 Feet: 
 To which you muft add the Fraction whole 
 Value you muft find thus: 
 
 Firft multiply the Numerator 35 by the Number of 
 Feet in a Perch, to wit, 3 24, and the Produd is 1 1 340, 
 which divided by the Denominator 36, and the 
 Quotient is 315 Feet, which is the Value of the 
 Fradioivls, and muft be added to the 207684 Feet 
 and then the whole Number of Feet will be 207999 . 
 Thus you have the Quantity given reduced into 
 the. leaft Denomination, namely. Feet, 
 
 And becaufe the Statute Pole 1 6 Feet and an half is 
 a Fraction, and being Iquared, it produces a Fradion, 
 and muft be the firft Number in the Queftion, accord- 
 ing to the Rule of Three, and by confequence, the 
 Divilorof the Divifion, which cannot be in Vulgar 
 Arithmetick, to divide by an Integer, mix'd with a 
 Fraction, altho 3 it may be done Decimally: We 
 muft therefore reduce the Feet into Inches, by mul- 
 tiplying 207999 by 12, the Produd is 2495988 
 Inches ; likewile multiply 272 Feet by 12, and the 
 Produd is 3264, to which add the 3 inches that be- 
 long to the 272 Feet, and it is 3267 Inches. 
 
 Then divide 2495988 by 3267, the Number of In- 
 ches in a Statute Perch, and the Quotient is 764 Per- 
 ches of Statute Mealure, which divide by 160, and 
 you have 4 Acres in the Quotient, and 1 24 Perches 
 remaining j which 124 divide by 40, the Perches 
 in a Rood, and you have 3 Roods and 4 Perches. 
 
Book V. Of LA N'T). 3 2 1 
 
 So that 4 Acres, i Perch, and i 5 6 of a Perch of 
 Cuftomary Meafure, being reduced to Statute Mea- 
 fure, contains 4 Acres, 3 Roods, and 4 Perches, 
 which was required to be done. 
 
 After the fame Method may any Cuftomary 
 Quantity whatfoever be reduced. 
 
 I fhall not need to mention any more Examples 
 of Land that is circular, or irregular Plats, having 
 already fhewn how to find their Contents. 
 
 Note, When you meafure the Length or Breadth 
 of a Field with your Chain, you muff meafure in a 
 ftrait Line as near as you can, otherwife you will 
 make the Length longer than it is. 
 
 I fhall conclude the meafuring of fuperficial Plains 
 with this Advice : 
 
 If you intend to learn any thing that is herein con- 
 tained, you muft not only read, but likewifeput Pen 
 to Paper, and caft up the Examples, and alfo otherEx- 
 amples of your own propofmg, and with a little 
 Ufe, mix’d with Diligence, you will foon attain to 
 your Defire; and you will find this Way of multi- 
 plying Fra&ions into Integers very pleafant and 
 brief, in refpeft of the other Way in Vulgar Arith- 
 metick, whereby the Integers muft be reduced into 
 the leaf! Denomination, which is 4 times more 
 Labour than this Way. 
 
 rfhe End of the Fifth Book. 
 
 / 
 
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 , 
 
MEASURING 
 
 O F 
 
 SOLIDS. 
 
 3"bB SIXTH BOOK. 
 
 By VEN MAN BET. 
 
 T" ' 
 
 LONDON: 
 
 Printed in the Yea* M.DCC. XXVII. 
 
M EASURIN G 
 
 O F 
 
 SOLIDS. 
 
 r\ / 
 
 Y Meafuring of Solids, is meant the Mea- 
 furing of Stone, Timber, digging of 
 Earth, and all folid Magnitude what- 
 foever. 
 
 And as from the Motion of a Line is made a 
 Superficie ; fo likewife from the Motion of a Super- 
 ficie, is made a Solid or Mathematical Body. 
 
 And as a Superficie confifts of Length and 
 Breadth, fo a Solid or Mathematical Body confifts 
 of Length, Breadth, and Depth (or Thicknefs.) 
 
 There is no farther Progrefs ; for which Way 
 foever a folid Body is moved, a folid Magnitude 
 will be thereby deicribed. 
 
 In meafuring of Solids, t ftihli begin with the 
 Cube, it being moft eafy to mealure, 
 
 A Cube, 
 
Book VI. Of SOLIDS. 315 
 
 A Cube is a Rectangular or Square Solid, that 
 hath an equal Length, Breadth, and Depth, and 
 is comprehended under fix equal Squares, and may 
 well be reprefented by a Dye, which is itfelfa little 
 
 Cube. 
 
 And as in meafuring of Plains, you only multiply 
 the Breadth by the Length, for the gaining of the 
 Content ; 
 
 So in meafuring of Solids, you mult firft multiply 
 the Breadth by the Length, for the gaining the Su- 
 perficial Content ; and then multiply that iuperficial 
 Content by the Depth or Thicknefs of the folid 
 Body, and the Produd is the folid Content. 
 
 Of Timber or Stone being fquare and 
 equal- fide d. 
 
 PROP. I. Fig. I- 
 
 Fig. I. being a Cube 12 Inches in Length, and 12 
 Inches in Breadth, and 12 Inches in Depth-, c what 
 is the folid Content thereof ? 
 
 F IRST multiply 12 Inches the Length, by 12 
 Inches the Breadth, and the Produd is 144 
 Inches for the fuperficial Content ; which mu tiphed 
 by 12 Inches the Depth, produces 1728 Inches lor 
 the/olid Content of that Cube, 
 
 Example. 
 
 t 
 
$16 MEASURING 
 
 Example . 
 
 The Length 
 The Breadth 
 
 The fuperficial Content is 
 Which multiply by the Depth 
 
 Book Vi. 
 
 Inches , 
 
 12 
 
 12 
 
 24 
 
 12 
 
 *44 
 
 12 
 
 288 
 
 x 44 
 
 And the fblid Content required, is 1728 
 
 Where by the Way, you may take Notice, That 
 as a fuperficial Foot contains 144 fuperficial Inches, 
 fo likewife a cubical or folid Foot contains 1728 
 cubical Inches, being produced as above. 
 
 So that by a Foot of Timber or Stone, or any 
 other fblid Material, in whatfoever kind of Solid it 
 be found, is underftood a Cube containing 1728 
 cubical Inches, and confequently half a Foot Solid 
 contains 864 cubick Inches, and a quarter of a Foot 
 
 contains 432 cubick Inches. 
 
 - \ 
 
 PROP. II. 
 
 If the Side of a Cube of timber or Stone he 4 Feeh 
 5 Inches , what is the Content of that Cube ? 
 
 F IRST find the Content of the Side, by multiplying 
 4 Feet, 5 Inches by itfelf, that is, by 4 Feet, 5 In- 
 ches^ 
 
 t 
 
Book VI. Of SOLIDS. 327 
 
 ches, it produces 19 Feet, 6 Inches, 1 Part for the 
 fuperficial Content of the Side j which Product 
 multiplied by the faid 4 Feet, 5 Inches, produces 
 86 Feet, 1 Inch, 10 Parts, and 5 Seconds, for the 
 Content required. 
 
 Example, 
 
 F. I. P.S.T. 
 
 The Side given is 
 Which multiply by itfelf 
 
 1621 
 1 8 
 1 8 
 
 The Superficial Content of the Side is 19 1 61 i| 
 Which multiply by the Depth or Side 41 5] | 
 
 76 2 65 
 
 2 
 
 7 11 4 
 
 The folid Content required, is 
 
 86[ ilidjjl 
 
 PROP, 
 
MEASURING 
 
 Book VI. 
 
 PROP. III. Fig. II. 
 
 Suppofe in Fig. II- ABCD to be a ftrait Piece of 
 timber , terminated at both Ends by two equal long 
 Squares ( which may be called the BafesJ the 
 Length of which Piece BD is 3 Feet , 4 Inches , 
 and the broader Side of the Bafe AB 1 Foot , 8 
 Inches , and the narrower Side BC 1 Foot , 1 Inch , 
 the Content of this Piece be required $ 
 
 • % 1 
 
 F IRST find the fuperficial Content of the Bafe 
 or End, by multiplying the broader Side 
 1 Foot, 8 Inches, by the narrower Side 1 Foot, 
 r Inch, the Product is 1 Foot, 9 Inches, 8 Parts 
 for the fuperficial Content of the End^ which r 
 Foot, 9 Inches, 8 Parts, being multiplied by the 
 Length 3 Feet, 4 Inches, produces 6 Feet, 2 Parts, 
 8 Seconds, for the folid Content of that Piece of 
 Timber. 
 
 ✓ 
 
 i ■ 
 
 Example, 
 
Book VI. Of SO LIT>S. 
 
 Example . 
 
 The broader Side of the Bafe AB, is 
 The narrower Side BC, is 
 
 Being multiplied. 
 
 The fupeficial Content of the Bafe is , 
 Which multiply by the Length AE 
 
 The folid Content required, is 6!oj2!8 
 
 PROP. IV. 
 
 Suppofe a Piece of ‘Timber or Stone , whofe Length is 
 3 Feet 4 Inches , be terminated at both Ends by two 
 equal Quadrats ( or Squares ) the Side whereof 
 is 1 Foot 7 Inches $ what is the Content of fuch 
 Piece ? 
 
 I R S T find the Content of one End or Bafe, 
 by multiplying 1 Foot 7 Inches, by 1 Foot 
 7 Inches, it produces 2 Feet 6 Inches, and 1 Part 
 for the Content ; then multiply the Content of the 
 End by the Length 3 Feet 4 Inches, and it pro- 
 duces 8 Feet, 4 Inches, 3 Parts, and 4 Seconds, 
 for the Iblid Content. 
 
 7j An 
 
 1*9 
 
 F.I. 
 8 
 1 
 
 RS.T 
 
 *\*\ 
 
 I1I8 
 
 t | 8 
 
 3 4 
 
 34 
 
 3 
 
3 3 o MEASURING Book VI. 
 
 An Advertisement. 
 
 The ufual Way among many Men to meafure 
 Timber, whole Sidesare unequal (although it be a 
 falfe Way) is this : They ufually add the Breadth 
 and Depth together, and take the half for the Side 
 of a mean Square 3 and after they have found the 
 Content of that mean Square, as they call it (falfe 
 Square as I term it) they multiply it into (or by) 
 the Length, and the Product they conclude is the 
 Content of the Piece. 
 
 Let us compare this falle Way with the true Way. 
 
 Suppole the Piece of Timber in the preceding 
 Prop. III. whole Sides are unequal, to wit, one Side 
 of the Bafe (or End) A B, is 1 Foot 8 Inches, and 
 the other Side of the fame Bafe (or End) BC, is 
 1 Foot 1 Inch. 
 
 Example of the falfe Way. 
 
 Firft they add the two Sides together, namely, 
 
 1 Foot 8 Inches, and 1 Foot 1 Inch, which make 
 
 2 Feet 9 Inches, the half whereof is 1 Foot 4 Inches 
 and 6 Parts, which they multiply by itfelf, namely, 
 1 Foot 4 Inches and 6 Parts, and the Produd is 1 
 Foot 1 o Inches 8 Parts and 3 Seconds, which they 
 alfo multiply by the Length 3 Feet 4 Inches, and the 
 Produd is 6 Feet, 3 Inches, 7 Parts, and 6 Seconds, 
 for the Content, which is more than the true Con- 
 tent by above a quarter of a Foot, and the more un- 
 equal the two Sides are, the more falle this Way of 
 Meafuring gives the Content. 
 
 Of 
 
Book VI. Of SOLIDS. 3 31 
 
 Of Timber or Stone, confiding of 3, 5, or 
 more Sides. 
 
 PROP. V. Fig. III. 
 
 Admit Fig. III. to be a ftrait Piece of Fmber or 
 Stone , whofe Length A B is 30 Feet^ to be ter - 
 minated at both Ends with two equilateral Fr i angle s y 
 the Side whereof GF is 1 z Feet 3 the Content 
 whereof is required . 
 
 Fake this as a general Rule for regular Polygons. 
 
 F IRST meafure all the Sides round about, 
 or girt them with a Line 3 then take half that 
 Length for one Sum or Breadth 3 then find the 
 Center H of the Bafe, which you may do by find- 
 ing the Middle of the Bafe Line, as alfo of one 
 Side 3 and from the Angles oppofite to the Bale 
 Line and that Side, draw two Lines to the Middle 
 of them, and where thofe Lines interfeft, there is 
 the Center of the Triangle, as in the Figure at H 3 
 then take the neareft Diftance from the Center 
 to one of the Sides for the other Sum or Depth 3 
 then multiply that Breadth and Depth one by the 
 other, and the Produdt is the fuperficial Content of 
 the End or Bafe of the Piece, which Content mul- 
 tiplied by the Length of the Piece, produces the 
 folid Content. 
 
 7 , z 
 
 Example . 
 
W MEASURING Book VI. 
 
 Example . 
 
 The Girt of the 3 Sides is 36 Feet, the half 
 whereof is 18 Feet, which multiply by the neareft 
 Space from the Center to one Side, which is 3 Feet, 
 5 Inches, 6 Parts, 10 Seconds, and the Product is 
 62 Feet, 4 Inches, and 3 Parts, for the Content 
 of the Bafe or End of the Piece, which multi- 
 plied by the Length of the Piece 30 Feet, produces 
 1870 Feet, 7 Inches, and 6 Parts for the Content 
 of the Piece. 
 
 $ he Operation . 
 
 FI.P.S.T. 
 
 The 3 Sides being added, is 36 Feet,? ^ 
 the half whereof is * 
 
 Which multiply by the neareft? 
 Space from the Center, to wit, 5 $ 
 
 0 
 
 5 
 
 © 
 
 6 
 
 O O 
 
 IO O 
 
 s %' s . 
 
 1 
 
 Thefuperficial Content of the Bafe is 621 
 Which multiply by the Length 30I 
 
 4 
 
 3 ] 
 
 
 1860, 
 
 IOJ 
 
 .’1 
 
 11 
 
 ! 
 
 The folid Content of the whole? jg I 
 
 Piece is 5 ' 1 
 
 7 
 
 ‘1 
 
 
 Otberjuife 
 
m 
 
 Book VI. Of SO LIT) S. 
 
 Otherwife thus : 
 
 Find the Perpendicular of the Triangle by the 
 4th Prop, of meafuring of Plains ; then multiply 
 the Perpendicular by half the Bafe, the Product 
 whereof being multiplied into the Length, gives the 
 folid Content. 
 
 Example, 
 
 The Perpendicular is 
 
 Which multiply by half the Bafe 
 
 ; ' CjL l) vd 
 
 The Content of the End is 
 Which multiply by the Length 
 
 > ■ ■ tl. , adj X't 
 
 The Content (as before) is 
 
 F.T P.S, 
 
 "it 
 
 60 
 
 2 
 
 43 
 
 62I4I3 
 30 1 > 
 
 i860 7(6 
 10 
 
 870(7161 
 
MEASURING Book VI. 
 
 114 
 
 PROP. VI. Fig. IV. 
 
 Admit Fig. IV. were a ft r ait Piece of 'Timber^ ter- 
 minated at both Ends with two equal Pentagons , 
 whofe Side AB is 12 Inches , and the near eft Di- 
 ftance from the Center C, to the Middle of the Side 
 at D is 8 Inches , and the Length E F, 1 5 Feet, 
 and the Content required. 
 
 Example. 
 
 r pHE 5 Sides added 
 JL the half whereof is 
 Which multiply by CD 
 
 F.I.R 
 
 o|8 
 
 % 
 
 The Content of the Bafe is 
 Which multiply by the Length 
 
 10 
 
 The folid Content is 
 
 2Sl 
 
335 
 
 Book VI. Of SO LIT) S. 
 
 Of fquared 'timber or Stone , being bigger at one End 
 than at the other , in which Form fome timber trees 
 grow ; and being fell'd , fo hewn> and brought 
 
 to a Square . 
 
 PROP. VII. % V. 
 
 Admit Fig. V. ft? a fir ait Piece of timber or Stone , 
 
 terminated at both Ends with two unequal Squares ; 
 the Side of the greater Square ( or End ) A B, is 
 I Foot 8 Inches , the Side of the leffer Square 
 (or Upper-end J EF, is i Foot 3 Inches , 
 
 Length A E 20 ; what is the Content ? 
 
 the Rule is this : 
 
 i.YTMND the Content of the lower End (or 
 JL greater Square) ABC. 2. Find the Con- 
 tent of the upper End Cor leffer Square) EGDF. 
 3. Multiply one Content by the other. 4. From 
 that Produd extrad the Square Root. 5. Add 
 this Square Root, and the whole Content of both 
 the Bafes into one Sum. 6. Multiply this Sum by^ 
 one Third Part of the Altitude (or Length) of 
 the Piece, and the Produd will be the folid Con- 
 tent required. 
 
 Z 4 
 
 Which 
 
33 6 MEASURING Book VI. 
 
 Which Rule is demonftrated by Mr. Oughtred , for 
 meafuring a Segment of a Pyramid, in Cb. 19. Prob. 
 21. of his Claris Mathematics. 
 
 Example . 
 
 Pirft, Find the fuperficial Content of the lower 
 End ABC, by multiplying 1 Foot 8 Inches, by 1 
 Foot 8 Inches, it produces 2 Feet, 9 Inches, 4 Parts, 
 for the fuperficial Content of the lower End. 
 
 Secondly, Find the Content of the upper End (or 
 lefifer Square) EGDF, by multiplying 1 Foot 3 
 Inches, by 1 Foot 3 Inches, the Product whereof is 
 1 Foot, 6 Inches, and 9 Parts, for the fuperficial 
 Coutent of the letter Square (or upper End.) 
 
 Thirdly, Multiply one Content by the other, to 
 wit, 1 Foot, 6 Inches, and 9 Parts, by 2 Feet, 
 9 Inches, and 4 Parts, and the Produ# is 4 Feet, 
 4 Inches, and 1 Part. 
 
 Fourthly, Extra# the Square Root of 4 Feet, 4 
 Inches, and 1 Part, which Root is 2 Feet and 1 
 Inch, 
 
 Fifthly, Add this Square Root and the Content 
 of both the Bafes together, all three in one Sum, 
 and the Product is 6 Feet, 5 Inches, and 1 Part. 
 
 Sixthly, Multiply the laft Sum by a Third Part 
 of the Length, namely, 6 Feet 8 Inches, and the 
 Produ# will be 42 Feet, 9 Inches, 10 Parts, and 
 8 Seconds, for the fblid Content required. 
 
Book VI. Of SOLIDS. 337; 
 
 £be Operation . 
 The Side AB is 
 
 Which fquare or multiply by itfelf 
 
 F.I. P. S 
 
 8 1 °|°f 
 81 olo| 
 
 8 
 8 
 
 5i 4 
 
 The fuperficial Cont. of the lower End is 2I9I 4I 
 
 The Side of the lefTer Square, or upper ? r 
 End, is # ^ 
 
 Which fquare or multiply by itfelf 1 
 
 13 9 
 '3 
 
 The fuperficial Cont. of the upper End is i)6| 9 
 
 Which multiply by the Cont. of the lower 2 9I 4 
 The Product of one End multiplied by ? 
 the other is 
 
 The Square Root whereof is 
 
 4 1 
 
 21 
 
 To which add the Cont. of the lower End 219 
 Alfo add the Content of the upper End i|6 
 
 And the Produd is 6|5| 
 
 Which multiply by 1 Third of the Leng. 6|8 
 
 The folid Content required is 
 
 S6 
 
 2 
 
 4 
 
 3 4 
 6 6 
 
 42 9I10I8 
 
 An 
 
MEASURING Book VI. 
 
 53 8 
 
 An Advertifement. 
 
 The ufual Way which a great many Men ufe in 
 me a faring fuch a Piece of 'Timber or Stone , is this , 
 which is a falfe Way , and finds a Content lefs than 
 the true Content . 
 
 They take the Sides of the Square about the 
 Middle of the Piece, and this Square of the Middle 
 of the Piece they fuppofe to be a mean Square be- 
 tween the greater End and theleffer End j then they 
 multiply the Content of this Square by the Length 
 of the Piece. 
 
 Let us compare this falfe Way with the true Way, 
 and fee what Difference there will be between the 
 Content of the Piece found by the foregoing 7th 
 Prop, and the Content of the fame Piece found ac- 
 cording to the ufual (but falfe) Way. 
 
 Example, 
 
 The Side of the Square taken in the Middle of 
 the Piece is 1 Foot, 5 Inches, and 6 Parts, which 
 multiplied by itfelf produces 2 Feet, 1 Inch, 6 
 Parts, and 3 Seconds, for the Content of the mean 
 Square, as they account it ; which Content they 
 multiply by the Length of the Piece 20 Feet, and 
 it produces 42 Feet, 6 Inches, and 5 Parts, for the 
 folid Content which is above a quarter of a folid 
 Foot lefs than the true Content found before ; and 
 themoi^e theTree (or Solid whatfoever) diminifheth, 
 the more the Content, meafured this falfe Way, 
 will want of the true Content. 
 
 Example * 
 
Book VI. Of SOLIDS. 339 
 
 Example. 
 
 Suppofe a Piece of timber, being the fame Length 
 20 Feet were i Fo.ot 8 Inches Square , at the lower 
 for greater) End, and i Foot Square at the upper 
 forkjfer) End, and the Content required. 
 
 Flrft, We will find it the true Way. 
 
 The Content of the greater End 
 (whole Side is i F. 8 1.) is 
 The Content of the lefler End (whofe ? 
 
 F. I. P. S. 
 
 \ ' V, J 
 
 Side is x F.) is > 
 
 Which being multiplied produces 
 
 4 ! 0 
 
 2| 9I 4I 
 
 The Square Root whereof is x 
 
 To which add the Content of the? 2 
 greater End j 
 
 Alfo add the Content of the lefler End 1 
 
 8 
 
 
 9 
 
 4 
 
 0 
 
 0 
 
 The Product is 5I 5I 4 
 
 Which mult, by 4 Part of the Length 6 1 8 
 
 30 
 
 1 
 
 3 
 
 4 
 
 
 2 
 
 6 
 
 2 
 
 8 
 
 3 
 
 4 
 
 
 
 
 2 
 
 
 
 The true folid Content is 
 
 36I 3I 6| 8| 
 
 Now 
 
540 . ME ASURING Book VL 
 
 Now let us fee what the ufual (but falfe) Way 
 will produce. 
 
 r F.I.P. 
 
 The Side of the Square taken in the Middle is i 4I0I 
 Which multiply by itfelf, namely/ 1 4I0I 
 
 
 > 
 
 W 
 
 4 * 
 
 The Content of the mean Square is 
 
 I 94 
 
 Which multiply by the Length 
 
 2o 
 
 
 20 1 
 
 15 618 
 
 The Content (the falfe Way) is 
 
 3 * 16 , 8| 
 
 Which Content is left than the true Content by 
 almoft 3 of a folid Foot, which in meafuring of a 
 great Quantity of Taper-grown Timber would be 
 confiderable. 
 
 Of Pyramids. 
 
 A Pyramid is a Solid, comprehended under plain 
 Surfaces, and from a Triangular, Quadrangular, or 
 any Multangular Bale, diminifheth equally left and 
 left, till it finifh or end in a Point at Top 5 fuch are 
 the Spires of lorne Steeples. 
 
 Note, If a Pyramid be cut into two Segments or 
 Parts by a Plain parallel to the Baft, one of thoft 
 Segments or Parts will be a Pyramid (to wit, that 
 which diminifheth to a Point) and the other Seg- 
 ment or Part will have two unequal Bafes, fuch as 
 is Fig . V. The manner of finding its Content is 
 ihewn at Prop . VII, 
 
 PROP. 
 
Book VI. Qf SOLIDS. 
 
 34 1 
 
 PROP. VIII. Fig. VI. 
 
 Admit Fig- VI. were <? Pyramid, whofe Bafe is an 
 Equilateral 'triangle, and one Side of the Bafe 
 AB is i Foot, and the Perpendicular of the Bafe 
 C D xo Inches, 4 Parts, 6 Seconds, and the Length 
 of the Pyramid RS 8 Feet ; what is the Content of 
 this Pyramid ? 
 
 the Rule for finding the Content, is this : 
 
 M ultiply the Content of the Bafe by one Third 
 Part of the Height of the Pyramid, and the 
 Product is the folid Content. (By Height is to be 
 utiderftood a Perpendicular Line, that falleth n orn 
 the Top of the Pyramid, to the Middle or Center or 
 the Bafe : You are not to underftand by Height, 
 the Length of the Slope-line on the Outfide RS, 
 for that is longer than the Altitude or Height.) And 
 before we proceed any farther, it will be convenient to 
 ihew how to find the Height of any Pyramid. 
 
 How to find the Height of any Pyramid. 
 
 Firft, Square the Semidiameter of the Bafe. 
 Secondly, Square the Length of the Pyramid, 
 
 which is the Slope-line RS. 
 
 Thirdly, Subtract the lelTer Square out of the 
 
 b Fourthly, Find the Square Root of the Remain- 
 der, and that is the Height required. 
 
 Example. 
 
342. MEASURING Book VI. 
 
 Example. 
 
 Firft, The Semidiameter of the Bale EF is 3 In- 
 ches, 5 Parts, 7 Seconds, which being fquared is 
 12 Inches (or 1 one of thole Inches whereof 5 12 
 make a fuperficial Foot.) 
 
 Secondly, The Length of the Pyramid R S is 8 
 Feet, which being fquared is 64 Feet. 
 
 Thirdly, The lelfer Square, 1 Inch being taken 
 out of the greater Square 64 Feet, there remains 63 
 Feet and 1 1 Inches. 
 
 Fourthly, The Root of 63 Feet and n Inches 
 being extracted, is 7 Feet, 1 1 Inches, 1 x Parts • 
 which is the Height of the Pyramid required. 
 
 And this Rule is general for the finding the Height 
 of any Pyramid, whether Triangular, Quadrangular 
 or Multangular, always remembring to find the 
 Center of the Bale, that lo you may have the Semi- 
 diameter thereof ; which Centers are found by draw- 
 ing Lines from the Angles, to the Middle of the 
 Sides oppofite to them. 
 
 The Operation. 
 
 _ I. P. S. T 
 
 The Semidiameter of the Bafe> [ I } 
 
 ^ being EF, is $ 7 | 
 
 The Square of the Semidiameter is 
 
 9 
 
 2 
 
 I 
 
 1 
 
 1 
 
 3 
 
 2 
 
 11 
 
 1 
 
 3 
 
 2 
 
 11 
 
 
 1 
 
 9 
 
 
 
 1 
 
 9 
 
 
 ii 0! 
 
 : o| 
 
 o\ 
 
 io 7 
 
 The 
 
Book VI. Of SOLIDS. 545 
 
 F. I. P. S. 
 
 The Length of the Pyramid RS is 8 
 Which fquare or multiply by itfelf 8 
 
 o| o o 
 
 of the Semidiameter 
 
 And there remains 
 
 The Square Root whei 
 the Height) is 
 
 64! 0 
 
 E 1 1 
 
 oj 0 
 
 1 
 
 63I11I 0] ot 
 
 l 7]“ 
 
 1 1 1 jn + 
 
 , k , 
 
 Thus having found the Height of the Pyramid, 
 we will proceed in the next Place, to find the Con- 
 tent of the Bale. 
 
 F. 1 . P. S. 
 
 The Perpendicular C D is ojio 
 
 Which multiply by half the Bale AC I 6 
 
 6 \ 
 
 The' Content of the Bale is 
 Which multiply by * of the Height 
 
 o 
 
 2 
 
 51 2 3 
 
 
 + 
 
 4 
 
 I 
 
 11 
 
 6 
 
 1 
 
 ' 2 
 
 4 
 
 7 
 
 
 1 
 
 
 4 
 
 IO 
 
 2 
 
 2 
 
 The lolid Content of the Pyramid, is i| i| 9I1 if 6 
 
 PROP, 
 
 Os t 
 
344 MEASURING Book VI. 
 
 PROP. IX. Fig. vii. 
 
 Admit Fig. VII. were a Pyramid, whofe Length AE 
 « l S Feet, and the Bafe of it ABC D a Square, 
 whofe Side AB, or CD, is 2 Feet 6 Inches 5 what 
 is the Content ? 
 
 F IRST find the Altitude or Height, which is 
 the prick Line EF, by the Rule in the fore- 
 going Prop. VIII. then multiply one Third Part of 
 the Heighth by the Content of the Bale A BCD 
 and the Product is the Content required. 
 
 fthe Operation. 
 
 F. I. P. S. T 
 
 The Side of the Square of the Bale } 
 
 AB, or CD, is £ 2 
 
 Which multiply by itfelf 
 
 The Content of the Bale is 
 
 3 | 
 
 I 
 
 31 o 
 
 24 
 
 2 
 
 9 
 
 3 
 
 
 I 
 
 6 
 
 2 
 
 1 
 
 6 
 
 s 
 
 4 
 
 6 f 
 
 2 
 
 1 
 
 
 
 31 
 
 
 
 The folid Content of the Pyram. Is 31 1 i| 8( 6j 7 j 
 
34J 
 
 Book VI. Of SO L I <DS. 
 
 How to find the third Part of any Number. 
 
 Suppofe you were to find the one Third Part of 
 the Height of the Pyramid aforefaid. 
 
 F. I. P.S.T. 
 
 Set down your Number thus 14I x 1 l 4 l 71 ol 
 
 Then fay, The f Part of I 4, Feet is 4] 8 j 
 
 The l Part of 11 Inches, is j 3 8 
 
 The £ Part of 4 Parts, is I 1 4 
 
 The j Part of 7 Seconds, is I 2 4 
 
 Being added, the Produdl is 4I11I9I6I4I 
 
 Which is the Third Part required. 
 
 After the lame manner you may find the Third 
 Part of any other Number ; you may like wile find 
 the Fourth Part or Fifth Part of any Number after 
 the fame manner. 
 
 PROP. X. Fig. VIII. 
 
 Admit Fig. VIII. were a Pyramid, whofe Length 
 AH is 1 5 Feet , and the Bafe thereof a Pentagon , 
 whofe Side AB ^ 2 Feet, 6 Inches, and the Con- 
 tent required . 
 
 F IRST find the Content of the Bale ABCDE 
 (by the Rule delivered at the 9th Prop, of 
 Book 5. Of Meafuring of Plains J which I will here 
 repeat. 
 
 I4I7I0 
 
 8 
 
 
 
 1 
 
 4 
 
 -T 
 
 
 2 
 
 4 
 
34<5 MEASURING Book VI. 
 
 tfhe Rule is this : 
 
 As 182 is to 125: So is the Side of the Pentagori 
 given, to the Semidiameter of a Circle infcribed 
 within that Pentagon. 
 
 Then fay, If 182 give 125, what will (the Side 
 of the Pentagon) 2 Feet 6 Inches give ? 
 
 F. I. 
 
 Multiply the Second Number 125 o 
 
 By the Third Number 2 6 
 
 250 
 
 62 
 
 And the Product is 
 
 3i2|6| 
 
 Which divide by the firft Number 182, and the 
 Quotient is 1 Foot, and 130 remains, which multi- 
 ply by 12, and the Product is 1560, 
 which divide by 182, and the 03 ° 
 
 Quotient is 8 Inches, and 104 re- ( l ^ 00t ’ 
 
 mains , which multiply by 12, and i 
 the Produdl is 1248, which divide 
 by 182; and the Quotient is 6 
 Parts, and 1 56 remaining ♦ which 
 multiply by 12, and the Product 
 is 1872, which divide by 182, 
 and the Quotient is 10 Seconds, 
 and 52 remains $ which multiply 
 by 12, and the Product is 624, 
 which divide by 1 82, and the Quo- ✓ 
 tient is 3 Thirds, as you may fee (8 Inches, 
 
 in the Margin; There is yet a rtf* 
 
 1560 
 
 Remainder, 
 
Book VI. Of SO LIT) S> 34 7 
 
 Remainder, but the Value of it is 
 not T3o T <5 Part of an Inch 3 and in 
 many Cafes you will not need to 
 proceed any farther than to Parts 
 Of Inches. 
 
 In the next Place, there is thef 
 6 Inches that belong to the 312 
 Feet, to be divided by 182 3 but 
 becaufe the Divifor is greater 
 than the Dividend, multiply the 
 6 Inches by 144, and the Produd 
 is 864 Seconds^ then divide the 
 
 .«* 
 
 104 
 
 12 
 
 208 
 
 104 
 
 1248 
 
 (15 * 
 
 Parts* 
 
 156 
 
 12 
 
 V C l t 
 
 
 864 by 182, and the Quotient is 
 4 Seconds, which you muft add to 
 the 1 o Seconds, &nd they make 1 
 Part and 2 Seconds. 
 
 So is there found the Length of 
 the Semidiameter of the Circle 
 inlcribed within the Pentagon, 
 
 (whofe Side was given) namely, 
 
 1 Foot, 8 Inches, 7 Parts, 2 Se- 
 conds, and 3 Thirds, which is the 
 neareft Diftance from the Center of the Pentagon, 
 to either of the Sides. 
 
 The next Work will be to find the Content of 
 the Bafe of the Pyramid. 
 
 312 
 
 1872 
 
 0(5 
 
 i % (10 Sec 
 
 Firft, 
 
 Aa 2 
 
r 34S MEASURING Book VL 
 
 Firft, add the 5 Sides togther, and > 
 they make 1 2 Feet and 6 Inches )> 
 the half whereof is i 
 
 Which multiply by the Semidia-? 
 meter found, which is 5 
 
 And the Content of the Bale 
 ABCDE, is 
 
 Which multiply by a Third Part 
 
 And the Solid Content of the 
 Pyramid, is 
 
 F. 
 
 I. 
 
 p. 
 
 s. 
 
 T. 
 
 6 
 
 3 
 
 0 
 
 0 
 
 0 
 
 1 
 
 8 
 
 7 
 
 2 
 
 3 
 
 6 
 
 i 3 
 
 ! 
 
 9 
 
 6 
 
 4 
 
 2 
 
 
 
 6 
 
 ! 
 
 1 3 
 
 i| 
 
 I 
 
 
 10 
 
 ! 8 
 
 8 
 
 II 
 
 0 
 
 ; 5 
 
 1 
 
 
 
 
 So 
 
 
 4 
 
 1 7 | 
 
 
 3 
 
 4 
 
 
 
 
 
 3 
 
 4 
 
 
 
 S 1 
 
 1 7 
 
 8 
 
 j 
 
 [ 
 
 I fhall not inftance any more Examples of Py- 
 ramids ; for if the Bafe confift of 6 or more Sides, 
 the Content of the Bafe may be found by the Rules 
 delivered in Book 5. and then by multiplying the 
 Content of the Bafe into one Third Part of the 
 Height of the Pyramid, the Product is the folid 
 Content. 
 
 Of 
 
34 9 
 
 Book VI. Of SO LIZ)S. 
 
 ’ ■ ' if \ ' ! 
 
 X . »uk » ,'ij j>.\ ' iy " 
 
 Of round Solids , y&c# tfn? Cylinders 9 Cones $ 
 Spheres , Columns , &c. 
 
 • . > r , * 
 
 PROP. XL JRg. IX. 
 
 Admit Fig. IX. were a Cylinder (by Cylinder is 
 meant fuch a like Solid as is ufed in Gardens for the 
 Rolling of Walks ) whofe Length A B is 6 Feet 9 
 and the Diameter B C 1 Foot and 4 Inches , and 
 the Content required . 
 
 ’The Rule for Me a faring fuch a Solid , is this : 
 
 M ultiply the Content of the Bafe BC 5 or AD^ 
 by the Length A B, and the Produdt is the 
 folid Content. 
 
 Firft let us find the Content of the Bale, the 
 Rule is this : As 452 (that is four times 113) is to 
 355 3 So is the Square of the Diameter to the Con- 
 tent required. 
 
 The Diameter is 1 F. 4 I. the ; 
 
 > 
 
 F. I. 
 
 P. 
 
 S. 
 
 Square of it is 1 
 
 f 1 
 
 9 
 
 4 
 
 0 
 
 Which multiply by 
 
 355 
 
 
 r 
 
 
 tv C) 
 
 355 
 
 266 
 
 US 
 
 3 
 
 4 ! 
 
 1 
 
 And the Product is 
 
 .631! 1 
 
 I 4 J 
 
 • 
 
 A a 3 
 
 
 Which 
 
35 ° 
 
 ( r 79 
 * 
 
 179 
 
 12 
 
 358 
 
 179 
 
 2148 
 
 (34° 
 
 »t#8 
 
 4f2 
 
 Foot. 
 
 4080 
 
 MEASURING Book VI. 
 
 Which divide by the firft Num" 
 bcr 452, and the Quotient is 1 
 Foot, as you fee in the Margin, 
 and 179 remains ; then continuing 
 multiplying the Remainders by 12, 
 and dividing the Produces by 452, 
 tilP you come to the Denomina- 
 tion of Thirds, as you fee in the 
 Margin, at laft your Quotient is 
 9 Parts, and 12 remaining, which 
 12 is inconfiderate in Value; but 
 (4 Inches, for the 1 Inch and 4 Parts that 
 belong to the 631 Feet; and for 
 the 12 that remain above your 
 laft Divifion, you may add 6 
 Thirds ; for ii you fhould con- 
 : 1 tinue multiplying and dividing 
 the Remains, you would produce 
 6 Thirds. 
 
 {12 
 
 408*0 (9 Parts. 
 
 Then the Content of the Bale is 
 found to be 
 Which muft be multiplied by the 7 . 
 Length AB 5 
 
 F. I. P. S. T 
 
 I- 
 
 41 ‘ 
 
 And the Producft is 8 1 4I 61 3I 
 
 Which is the folid Content of the Cylinder 
 
 Of 
 
35 * 
 
 Book VI. Of SOLID S. 
 
 h j \ -*7' f r j* 
 
 Of a Cone . 
 
 A Cone is a Solid, which hath a Circle for its Bafe, 
 from whence it grows equally lefs and lefs (like a 
 Sugar-Loaf) till it finifh or end in a Point at Top. 
 
 PROP. XII. Eg, X. 
 
 J.'JvJjO t JO » 
 
 Admit Fig. X. were a Cone , whofe Length AC & 
 1 5 , and the Diameter of the Bafe A B i Foot 
 
 4 Inches what is the Content thereof ? 
 
 \ - I 
 
 T H E Rule for finding the folid Content of a Cone, 
 is the fame as for the finding the folid Content 
 of a Pyramid, to wit, by multiplying the Content of 
 the Bafe by one Third Part of the Height. 
 
 Firft find the Height of the Cone (according to 
 the Rule prefer ibed in Prop. 8. of this Book) for 
 the Height of a Cone is found by the fame Rule 
 that the Height of a Pyramid is found. 
 
 F. I.P. 
 
 8|o 
 8 o 
 
 i. The Semidiam. of the Bafe A B, is 
 Which fquare (or multiply by itfelf ) 
 
 The Square of the Semidiameter is 0I5I4I 
 
 A a 4 
 
 2. The 
 
352 MEASURING Book VI. 
 
 F. I. P. S. T. 
 
 2. The Length of the Cone> 
 
 AC is 3 15 
 
 Which fquare (or multiply by > 
 itfelf) S 15 
 
 The Produd is 
 3. Out of the Square of the 
 Length AC, fubtrad the Square 
 of half the Diam. AB, to wit^ 
 And the Remainder is 
 
 75| 
 15 1 
 
 225 
 
 
 
 
 i 
 
 5 
 
 4 
 
 
 2241 61 8| 
 
 The Square Root whereof is 14I11I 9 I 11 fere 
 Which is the Height of the 
 Cone , of which Height we muft 
 take one Third Part, which is 
 
 
 11 
 
 11 
 
 Having found the Height, and taken a Third 
 Part thereof, the next Work will be to find the 
 Content of the Bale. 
 
 Then lay , by the preceding Rule in Prop. XI. 
 As 452 is to 355 : So is the Square of the Diameter 
 to the Content of ,the Bale. 
 
 • • ' _ ; /i, _ 
 
 F. 
 
 I 
 
 P 
 
 The Square of the Diameter is 
 
 1 
 
 9 
 
 4| 
 
 Which multiply b.y 
 
 355 
 
 
 1 
 
 
 355 
 
 Il8 
 
 4 
 
 
 266 
 
 3 
 
 
 And the Product is 
 
 631 
 
 i I! 
 
 4l 
 
 Which 
 
Book VI. Of S 0 L 1 2) S. 
 
 F. 
 
 Which divide by 452, and you? 
 will find the Cont. of the Bale ^ 
 to be (as in Prop. XI.) ^ 
 
 Which multip. by one Third Part? 
 the Height of the Cone * 
 
 35? 
 
 I. P. S. T. 
 
 I 
 
 4 
 
 9 
 
 0 
 
 6 
 
 4 
 
 11 
 
 11 
 
 3 
 
 B- 
 
 4 
 
 11 
 
 II 
 
 3 
 
 s 
 
 1 
 
 4 
 
 8 
 
 3 
 
 
 
 3 
 
 8 
 
 8 
 
 3 
 
 
 3 
 
 3 
 
 8 
 
 2 
 
 
 
 
 1 
 
 8 
 
 ;i 
 
 1 
 
 
 2 
 
 2 
 
 Thefolid Content of the Cone is 6|nl 8j 2I 8| 
 
 Note, If a Cone be cut into two Segments (or 
 Parts) by a Plain parallel to the Bafe ; one of thofe 
 Parts will be a Cone, and the other Part will have 
 two unequal Circles for the Bales : As in Fig. XI. 
 the upper Part is a Cone, and the lower Part is a 
 Segment of a Cone, which hath two unequal 
 Circles for the Bafes, 
 
 PROP. XIII. Fig. IX. 
 
 A Segment of a Cone may well be reprefented by a free 
 that grows taper or dimini [hing. 
 
 A Dmit the Segment in Fig. XI. were a round 
 Taper Piece of Timber, whole Length AB 
 is 1 5 Feet, and the Diameter of the lower Bafe AC 
 1 Foot 4 Inches s and the Diameter of the upper 
 End, or Bafe DB, is 1 Foot ; what is the Content 
 of this Piece ? 
 
 The 
 
354 MEASURING Book VI, 
 
 The Length A B being 15 Feet, and the Diame- 
 ter AC being 1 Foot 4 Inches 5 the Length, and 
 likewife the Diameter is the fame as was the Cone 
 in the laft Propofition. 
 
 For finding the Content of this Piece of Timber 
 for Segment of a Cone) you muft have refpecft to 
 the Rule delivered in the preceding 7th Prop, of this 
 Book, for the fame Rule that ferveth for fquare 
 Timber, whofe Bafes or Ends are unequal, ferveth 
 likewife for round T imber, or Bodies whole Bales 
 or Circles at the Ends are unequal. 
 
 Example. 
 
 The Content of the lower Bafe AC, is, by the 
 laft foregoing Propofition found to be 1 F. 4 I. 9 P. 
 o S. 6 T. 
 
 Now we muft find the Content of the upper Bale, 
 or Ieffer End BD, whofe Diameter is 1 Foot. 
 
 And becaufe the Square of the Diameter is but 1 
 Foot, the Content of the Bafe cannot be fb much 
 as 1 Foot ; therefore we muft ftate the Queftion in 
 Inches, faying thus : 
 
 If 4J 2 give 355 j what will the Diameter, whofe 
 Square is 1 2 Inches, give ? 
 
 For, if you remember, the Rule is, As 452 is 
 to 355 : $° IS the Square of the Diameter, to the 
 Content of the Circle of the Bafe (or End of the 
 Piece.) 
 
 Therefore 
 
Book VI. Of SOLIDS. 355 
 
 Inches. 
 
 Therefore multiply the fecond Number 3 55 
 By the third Number in the Queftion, to wit 
 
 12 
 
 And the Produft is 
 
 710 
 3 55 
 
 4260 
 
 Which divide by the firft Number 452, and the 
 Quotient is 9 Inches, and of an Inch 3 which, 
 when reduced, as is (hewn before, 
 by multiplying the Remainders by ^^0 ( 9 Inches. 
 12, and dividing the Produ&s by 4^ 
 
 452, you will find 9 Inches, 5 Parts 
 
 1 Second, and 2 Thirds, for the Content of the 
 
 upper End or Bale of the Piece. 
 
 Having now the Content ofboth the Bafes, the next 
 Work will be to multiply them one by the other. 
 
 Bafe BD, is 
 
 Which multiply by the Co 
 tent of the greater Bafe AC 
 
 F. I. 1 
 
 3 < 
 
 X 1 - 
 
 r. 
 
 Fo. 
 
 ; 0 
 
 9 
 
 s 
 
 I 
 
 2 
 
 0 
 
 s- 
 
 4 
 
 9 
 
 0 
 
 6 
 
 0 
 
 
 9 
 
 6 
 
 9 
 
 2 ) 
 
 1 9 
 
 
 3 
 
 1 
 
 8 
 
 4 
 
 1 
 
 . - 
 
 
 5 
 
 3 
 
 9 
 
 8 
 
 
 
 
 1 
 
 4 
 
 6 
 
 on j 
 
 
 
 
 
 
 1 1 
 [ 2 
 
 n 
 
 I 
 
 1 
 
 10 
 
 9 
 
 2 ! 
 
 Bafes multiplied, is 
 
 The Square Root whereof is 3 I. 7 P. 6 S 
 
 Which 
 
35 6 MEASURING Book VL 
 
 Which Square Root, and the Content of both 
 the Bales, muft be added together. 
 
 The Square Ropt is 
 The Content of the greater Bafe is 
 The Content of the leffer Bafe is 
 
 find the Produd of the Addition is 
 
 Which fhould be multiplied by 
 | Part of the Height, namely, 4 F. 
 11 I. 11 P. 3 S. 8 T. but becaufe 
 f Part of the Height, and ? Part 
 of the Length differ but by 8 Se- 
 conds, you may multiply it by T 
 Part of the Length, namely, 5 Fe. 
 and the Produd is 
 
 Which is the folid Content of the Segment of 
 the Cone. 
 
 \ L . - 
 
 Of a- Globe or Sphere, . . 
 
 A Globe or Sphere is a perfed round Body, con- 
 tained under one Circular Plain, the middle Point 
 whereof is called the Center, from whence all flrait 
 Lines drawn to the Outfide are of equal Length, 
 and are called Semidiameters, or rather Semiaxes. 
 
 F.I.P. S.T. 
 
 0 
 
 1 
 
 0 
 
 3 
 
 4 
 9 
 
 7 
 
 9 
 
 S 
 
 6 
 
 0 
 
 1 
 
 °i 
 
 6 
 
 A 
 
 2 
 
 i 5 
 
 h 
 
 
 io| 
 
 2 1 
 
 ■ 1 
 
 1 
 
 3 
 
 9 
 
 2 
 
 H 
 
 3 
 
 \ 
 
 4 
 
 12I5I0! 2|4l 
 
 PROP. 
 
Book VI. Of SO LIVS. 35 7 
 
 PROP. XIV. Fig. XII. 
 
 Let Fig. XII. be a Sphere, whofe Axis AB is 8 
 Feet j -what is the folid Content thereof ? 
 
 if he Rule is this : 
 
 A S 21 is to ii (or. As 42 to 22): So is the 
 Cube of the Diameter or Axis to the folid 
 Content required. 
 
 F. 
 
 The Axis is 8, and the Square of it 64 
 
 Which multiplied by the Axis 8 
 
 The Cube of the Axis is 5 * 2 
 
 Then fay by the Rule of Three : If 21 give 1 1, 
 what will (the Cube of the Diameter) 5* 2 g ive ? 
 
 Multiply 512 by 11, and 
 the Produd is 5632 (as you fee 
 in the Margin) which divide by 
 thefirftNumber in theQueftion, 
 to wit, 21, and the Quotient is 
 268 Feet, and f of a Foot, 
 which being reduced, is 2 Inc. 
 
 3 Parts, and 5 Seconds. 
 
 So that the folid Content of 
 the Sphere, whole Axis is 8 
 Feet, is 268 Feet, 2 Inches, 3 
 Parts, and 5 Seconds, 
 
 512 
 
 11 
 
 512 
 
 512 
 
 5632 
 
 f t 
 
 .6% Feet. 
 
 pi i i 
 22 
 
 PROP. 
 
35§ MEASURING Book VI. 
 
 PROP* XV. 
 
 A Sphere is given, whofe [olid Content is 268 Felt, 
 2 Inches , 3 Parts , 5 Seconds * *j fife 
 
 Length of the Axis ? 
 
 £ the Rule is this : 
 
 A S 22 is to 42 : So is the folid Content given, 
 to wit, 268 Feet, 2 Inches, 3 Parts, 5 Seconds, 
 to the Cube of the Axis, whofe Length is required. 
 
 State the Queftion thus: If 22 give 42, what 
 will 268 F. 2 I. 3 P. 5 S. give ? 
 
 Multiply the third Number by 
 the fecond (as in the Margin) 
 and the Product is 11264 Feet 
 fere • which divide by the firft 
 Number in the Queftion 22, and 
 the Quotient is 512 (which is 
 the Cube of the Axis) the Cube 
 Root whereof is 8 Feet, which 
 is the Length of the Axis re- 
 quired. 
 
 The briefeft Way to extradl 
 Square and Cube Roots, is by a 
 Table of Logarithms. 
 
 F. I. P. S. 
 
 268| 2 
 42! 
 
 I’M 
 
 536 \ 
 
 J| 
 
 I0727|IO 
 
 6 i 1 
 
 1 
 
 17I6I 
 
 1 1263I1 1 [ 1 1 '6[ 
 
 (512 
 
 2Z 
 
 PROP, 
 
Book VI. Of SO Lift S. 
 
 359 
 
 PROP. XVI. Fig. XII. 
 
 Admit in Fig. XII. CED were a Segment ( or Por- 
 tion J of a Sphere, whofe Segment of the Axis EF 
 is 2 Feet (the whole Axis AB being 8 Feet ) and 
 the Chord (or Subtenfe ) of the Segment CD, is 
 6 Feet, ii Inches, and 2 Parts-, what is thefolid 
 Content of this Portion of the Sphere? 
 
 •the Kale is this : 
 
 F Irft, increafe the Altitude of the other Segment 
 (not given) by half the Axis. 
 
 Then fay by the Rule of Three : As the Altitude 
 of the other Segment not given, is to the Altitude 
 (or Height) of the given Segment (or Portion) 
 So is the Altitude of the other Segment increafed 
 by half the Axis, to a fourth Proportional. 
 
 Secondly, Square half the Chord (or Subtenfe) 
 of the given Segment, and multiply the Square of 
 that half Chord by the fourth Proportional found 
 by the firft Work (or Rule.) 
 
 Then lay by the Rule of Three: As 21 is to 22 
 So is the Produdl of the Square of half the Chord 
 of the Segment given, multiplied by the fourth 
 Proportional (found as above) to the folid Content 
 of the Segment given. 
 
 Example \ 
 
3<So MEASURING Book VI. 
 
 Example. 
 
 The Segment not given is CDG, whofe Altitude 
 FG is 6 Feet, which mull be increafed by half the 
 Axis, namely, 4 Feet, and then it is 10 Feet. 
 
 Then fay. As (the Altitude of the other Segment 
 not given, which is) 6 Feet, is to (the Altitude of 
 the given Segment) 2 Feet : So (is the Altitude of 
 the other Segment not given, being increafed by half 
 the Axis, to wit, 6 Feet more 4 Feet, that is) 10 
 Feet, to a fourth Proportional required. 
 
 Let us find this fourth Proportional. 
 
 Firft, Multiply 10 F. by 2 F. and the Product is 
 20 F. which divide by 6 F. and the Quotient is 3 F. 
 ? Parts of a Foot, that is, 3 F. 4 I. which is the 
 fourth Proportional. 
 
 ^ In the next Place, Square half the given Chord 
 CF ; the whole Chord CD is 6 F. 1 1 I. 2 P. the 
 half whereof is 3 F. 5 I. 7 P. fere, the Square 
 whereof is 12 F. o I. 1 P. 2 S. + which muft be 
 multiplied by the fourth Proportional found, 
 namely, 3 F. 4 1 . and the Product is 40 F. o L 
 3 P. 10 S. 8 T. 
 
 Then fay by the Rule of Three ; As 21 is to 22 : 
 So is the Square of half the given Chord, multiplied 
 by the fourth Proportional found (as before) 40 F. 
 o I. 3 P. 10 S. 8 T. to the Content required. 
 
 Then multiply the third Number by the fecond, 
 and the Produdtis 880 F. 7 I. 1 P. 6 S. 8 T. which 
 divide by the firft Number 21, and the Quotient is 
 41 F. and kt of a Foot, which (when reduced) is 
 
 41 
 
Book VI. Of SOLIDS. 161 
 
 4 i F. i t I. fere, which is the fblid Content of the 
 Segment required. 
 
 For note, That a Sphere is equal to two Cones, 
 having the Height and the Diameter of their Bafe 
 the fame with the Axis of the Sphere : Or, which 
 is all one, a Sphere is two Third Parts of a Cy- 
 linder, having the Height and the Diameter of the 
 Bale the fame with the Axis of the Sphere. 
 
 But in cafe you have only the Segment of a 
 Sphere given, and not the Axis of the w hole Sphere, 
 then you muft find the Axis. 
 
 PROP. XVII. Fig. XII. 
 
 *Fbe Segment or Portion of a Sphere being given, and 
 the Axis required $ 
 
 %*be Rule is this : 
 
 S Quare one half of the Chord of the Segment, and 
 the Produdfc divide by the Altitude of the given 
 Segment ; then to the Quotient add the Altitude of 
 the given Segment, and the Product is the Length 
 of the Axis required. 
 
 • . ~ • f • 
 
 C- ■ , . ' i 
 
 Example . 
 
 • i ■ * ir ■ lii: . ; :v f*j 'j [ /r- ■ , ) 
 
 Admit in Fig. XII. the Segment given to be C E D, 
 and the Axis AB required to be found. 
 
 The given Chord CD is 6 F. 11 1 . 2 P. the half 
 whereof is 3 F. 5 I. 7 P. w'hich being fquared, is 
 12 F. which divide (by the Altitude £F) 2 F. and 
 
 B b the 
 
362 ME ASURING Book VI. 
 
 the Quotient is 6 F. To which add the Altitude of 
 the Segment (EF) 2 Feet, and the Product is 8 F. 
 which is the Length of the Axis of the whole Sphere, 
 whereof C E D is a Segment. 
 
 - 
 
 P R O P. XVIII. 
 
 fhe Axis of a Sphere being given ; to find the 
 fuperficial Content : 
 
 One Ride is this : 
 
 A S 7 is to 22 : So is the Square of the Diameter, 
 J\. to the fuperficial Content required. 
 
 Example. 
 
 ' T \ . . f , > 
 
 . / j >. 
 
 Let the Diameter or Axis given be 8 Feet, which 
 being fquared is 64 Feet. ■ 
 
 Then fay by the Rule of Three : If 7 give 22, 
 
 what will 64 give ? . 
 
 Multiply 64 by 22, and the Product is 1408; which 
 divide by 7, and the Quotient is 20 1 Feet and j of a 
 Foot ; which reduced, is 201 F. 1 1 . 8 P. 7 S. fere , 
 which is the fuperficial Content of the Sphere. 
 
 On the contrary, If the fuperficial Content of a 
 Sphere be given, and the Axis required, 
 
 . ,i\ 5iVj s '0 1 ’-i \ isj i \ \ * * ' * K \ . .. ) i %'• *ii • *' 
 
 1 ‘he Rule is this : 
 
 As 22 to 7: So is the fuperficial Content given, 
 
 to the Square of the Axis. 
 
 d . Example, 
 
Book VI. Of S O L 1 2 ) S. 363 
 
 Example. 
 
 >,h ! , 
 
 The Content given is 201 F. 1 1 . 8 P. 7 S. what 
 is the Axis ? 
 
 Multiply the Content given by 7, and* the Produft is 
 1408, which divide by 22, and the Quotient is 64, 
 which is the Square of the Diameter 3 the Square 
 Root whereof is 8, the Axis or Diameter required. 
 
 PROP. XIX. Fig. XIII. 
 
 A Portion or Segment of a Sphere being given to find 
 the Content of the Convex Superficies . 
 
 A Dmit in Fig. XIII. BAC to be a Portion of a 
 Sphere, whole Altitude is A E 4 Feet, and 
 whofe Bafe or Chord is B C 8 Feet, and the Con- 
 tent of the Convex Superficies is required. 
 
 Draw the ftrait Line AB from the Pole A, to 
 the Bafe in B ; then, I fay, the Content of a 
 Circle, whofe Radius (or Semidiameter) is AB is 
 equal to the Content of the Convex Superficies of 
 the Portion BAC. 
 
 Let us try. 
 
 The Semidiameter AB is 5 F. 7 I. 10 P. 6 S. 
 now the Content of a Circle delcribed from that 
 Semi diameter (being found by Prop. XVII. of Mea~ 
 firing of Plains J will be 100 F. 6 I. 6 P. which is 
 the Content of the convex Superficies of the Portion 
 BAC, which was required. 
 
 Bb 2 
 
 Which 
 
ME ASURING Book VL 
 
 Which may be proved by the laft foregoing Propo- 
 rtion ; for the Content Superficial of a Sphere, whofe 
 Diamt ter is 8 F. was found to be 201 F. il. 8P. +: 
 Now this Portion B AC is half the Sphere of the 
 fame Axis ; therefore double the Content of it, which 
 was found to be too F. 6 I. 6 P. and the Produdt is 
 201 F. and 1 1 , for the Content of the whole ; which 
 is the fame that was found by the preceding Propofi- 
 tion, within (or want) 8 P. which want of the 8 Parts 
 is ca idl'd by the Length of the Line AB, it being a 
 lmall Matter more than 5 F. 7 I. 10 P. 6 S. 
 
 The Truth of this laft Propofition, that is to fay. 
 That the Circle defcribed by the Radius AD, or 
 AB, is equal to the convex Superficies of the Por- 
 tion of a Sphere B A C, is demonftrable from the 
 Companion of Motion, thus: 
 
 Let the Plain A E B D be underftood to make a 
 Revolution about tbe Axis AE ; and it is manifeft, 
 that by the ftrait Line AD, a Circle may be defcri- 
 bed ; and by the Arch AB, the Superficies of a Por- 
 tion of a Sphere; and laftly, by theSubtenfe AB, the 
 
 Superficiesofa right Cone will be defcribed. Now fee- 
 ing both the ftrait Line A B, and the Arch AB, make 
 one and the fame Revolution, and both of them have 
 the fame extreme Points A and B ; the Caufe why the 
 Spherical Superficies which is made by the Arch, is 
 greater than the conical Superficies which is made by 
 the Subtenfe, is. That AB the Arch is greater than 
 AB the Subtenie ; and the Caufe why it is greater, 
 confifts in this ! That although they be both drawn 
 from A to b? y et the Subtenie is drawn ftrait, but 
 the Arch angularly, namely, according to that Angle 
 which the Arch makes with the Subtenie, which Angle 
 
 is equal to the Angle DAB (for an Angle of Con- 
 x tingence 
 
Book VI, Of SOLIDS. 3 63 
 
 tingence adds pothing to an Angle of a Segment.} 
 Wnerefore the Magnitude of the Angle DAB is the 
 Caufe why the Superficies of the Portion delcrked 
 by the Arch A B is greater than the Superficies of 
 the mht Cone deferibed by the Subtenfe AB. 
 
 A aain, The Caufe why the Circledefcribed by the 
 Tangent AD, is greater than the Superficies of the 
 mlit Cone deferibed by the Subtenle AB (notwith- 
 standing that the Tangent and Subtenfe are equal, and 
 both moved round in the fame time) is this: That AD 
 ftands at right Angles to the Axis, but AB obliquely ; 
 which Obliquity confilts in the fame Angle DAB. 
 Seeing therefore the Quantity of the Angle DAB is 
 .that which makes the Lxcefsboth of the Superficies of 
 the Portion, and of the Circle made by the Radius 
 A D, above the Superficies of the right Cone deferi- 
 bed by the Subtenfe AB ; it follows, That both the 
 Superficies of the Fortion, and that of the Circle do 
 equally exceed the Superficies of the Cone. W here- 
 fore the Circle made by AD or AB, and the Sphe- 
 rical Superficies made by the Arch AB, are equal 
 to one another, which was to be proved. 
 
 Of the Proportions between a Cube , a Priftn, and a 
 Pyramid^ a Cylinder , Sphere , and Cone , whofe 
 Altitudes and Bafes are as follow. 
 
 F. I. P.S.T. 
 
 A Cube , whole Side or 
 1 F. 4 I. the Content is 
 
 Bale is 
 
 24540 
 
 B b 3 
 
 A Priftn 
 
MEASURING Book VI. 
 
 . I. P. s. T. Fo 
 
 3 \66 
 
 A Prifm (having its Bafe 
 and Altitude feverally equal 
 to the Bale of the aforefaid 
 Cube, or of any other Rectan- 
 gular Parallelepipedon is the 
 half of it, and the Content is 
 A Square Pyramid , having 
 its Height and Bafe feverally 
 equal to the Bale of the afore- 
 laid Cube, is \ Part of it, and 
 its Content is 
 
 A f r i angular Pyramid , 
 whofe Height is as aforefaid, 
 and each Side of the Bale as 
 afore (that is, i F. 4 I.) the 
 Solid Content is 
 
 A Cylinder , having the 
 fame Height and Diameter 
 with the Cube aforefaid, the^i 10 
 Solid Content of fuch a Cylin- 
 der is 
 
 A Sphere ^ whofe Axis is') 
 as aforefaid, its Solid Con-Ci 
 tent is $ 
 
 A Cone , whole Altitude") 
 and Diameter of its Bafe are f 
 feverally equal to the Side of^o 
 the Cube aforelaid, its Solid V 
 Content is J 
 
 8 
 
 9 5 9 4 
 
 10 9 4 8 
 
 7 5 4 8 4 
 
 Hence it appears that a Cuhe is double the Prifm , 
 and three times as much as the Square Pyramid of 
 equal Bale and Altitude. 
 
 The 
 
Book VI. Of S O L IT) S. 3 67 
 
 The 'Triangular Pyramid is fomething more than 
 
 I of the Cube. 
 
 The Cylinder, whole Diameter and Height is fe- 
 verally equal to the Height of the Cube, is in pro- 
 portion to it as 11 to 14 ; or the Cylinder contains 
 
 II of thole Parts whereof the Cube contains 14 
 The Globe or Sphere , whole Axis is equal to the 
 
 Height of a Cube, contains 1 1 fuch Parts, whereof 
 the Cube contains 21 ; or the Sphere is in propor- 
 tion to the Cube, as 11 to 21. 
 
 The Cone, whofe Diameter and Altitude are fe- 
 verally equal to the Altitude and Diameter of the 
 Cylinder, is in proportion to the Cylinder as 1 to 3. 
 
 Whence it appears, that the Sphere is f of the 
 Cylinder, and the Cone I of the Sphere. 
 
 Of GAUGING. 
 
 G Auging is comprehended in the Meafuring of 
 Solids, and there is only this Difference between 
 Gauging (or meafuring of Veffels) and meafuring 
 of other Solids : The Content of the latter is given 
 in Feet, Inches, Parts, Sc. but the Content of the 
 former is given in Gallons, Quarts, Pints, Sc. 
 
 But before you can give the Content in Gallons, Sc. 
 you mull firft find the Iblid Content in Cubick Inches, 
 Farts, Sc. and having firft; found the Content in 
 Inches and Parts, you muft afterwards reduce them 
 into Gallons, Quarts, and Pints, Sc. 
 
 There are feveral Methods Ihewn for the finding 
 the Content of thefe irregular Solids ; but the Me- 
 thod Ihewn by Mr. Ought red, is generally efteemed 
 for one of the belt ; and therefore I fhall make ufe 
 of his Method. 
 
 Bb 
 
 4 
 
 Moft 
 
3 68 Me afuring of Solids. Book VI. 
 
 Moft Liquid Veffels, fuch as are Pipes, Hogfheads, 
 Barrels, Kilderkins, Firkins, are made in form 
 of a Spheroides, having the two Ends equally cut 
 off 3 and accordingly may be meafured thus : 
 
 Meafure the two Diameters of the Veffel in Inches , 
 the one at the Bung-hole , the other at the Head , and aljo 
 the length within ; and by the Diameters founds find out 
 the Area or Contents of the Circles : 'Then add together 
 two third Parts of the Content of the greater Circle , and 
 one third Part of the Content of the leffer Circle. 
 Lajtly , Multiply the Producl of thefe two Sums added 
 together , by the Length of the Veffel ; fo Jhall you have 
 the Content of the Veffel in Cubic k Inches , 
 
 Of which 231 make a Wine Gallon; and 272, 
 and 3 Parts, make an Ale or Beer Gallon, according 
 to Mr. Ought red , who would have a Gallon to con- 
 fift of a Number of Cubick Inches, the Square 
 Root whereof is Palms 5 i. 
 
 That 231 Cubick Inches make a Wine Gallon, 
 is the Opinion of moft Men ; but the Quantity of 
 the Ale or Beer Gallon, is net as yet fully agreed 
 on : Formerly the Ale Gallon hath been accounted 
 to contain 288 Cubick Inches, and \ or 9 Parts ; 
 but fince the Excife, it is computed tp contain but 
 282 Cubick Inches, 
 
 <■ V 
 
 PROP, 
 
Book VI. GAUGING. 
 
 369 
 
 PROP. XX. 
 
 Admit a Vejfel to have the Diameter at the Bung 3 2 
 Inches , and at the Head 18 Inches , and the Length 40 
 Inches , what is the Content of this Vejfel in Gallon's, 
 and Parts of a Gallon ? 
 
 T Hefe Dimenfions being given in Inches, and 
 the Content required in Inches, you muft have 
 relpeft to the Note at Page 288. Book 5. 
 
 We muft, firft of all, find the Content of the 
 two Circles belonging to the two Diameters given, 
 by the fecond Rule delivered at the 17th Propofi- 
 tion of the Fifth Book. 
 
 J Inches. 
 
 The Square of the Diameter 32, is 1024 
 
 Which according to the Rule, multiply by 3 55 
 
 And the Product is 
 
 5120 
 
 5120 
 
 3072 
 
 363520 
 
 Which divide by 452 (accord- 
 ing as the aforefaid Rule directs) 
 and the Quotient is (as in the 
 Margin ) 804 Inches, and #£ 
 Parts of an Inch ; which reduced, 
 is 804 I 2 P. 11 S. 8 T. for the 
 
 (1 
 
 12 I . 
 (804 
 
 4 
 
 Content 
 
370 Meafuring of Solids. Book VI. 
 
 Content of the greater Circle, to wit, the Circle at 
 the Bung, whole Diameter was 32 Inches. 
 
 In the next Place, the Square of the Dia-> 
 meter 18 Inches, is 5 3 2 4 
 
 Which multiply by ^ 
 
 And the Produdt is 
 
 1 620 
 1620 
 972 
 
 115020 
 
 Which divide by 452 (as in the Margin) and the 
 Quotient is 254 Inches, and -Jyf of an Inch ; which 
 
 reduced, is 254 I. 5 P. 
 7 S. 6 T. and is the Con- 
 tent of the leflfer Circle at 
 the Head, whofe Diameter 
 was 18 Inches. 
 
 The next Work is, to 
 take the two third Parts of 
 the Content of the greater 
 Circle, and one third Part of the Content of the 
 lefler Circle, and add them together. 
 
 O 
 
 20(1 
 
 2462(2 
 
 *i?020 ( 254 lnches. 
 4Z222 
 
 4 
 
 The 
 
Book VI. QAUGING. 
 
 1 . 
 
 The Content of the greater ^ g 
 Circle, is j 
 
 The Content of the lefler £ 
 Circle, is J **" 
 
 P. 
 
 2 
 
 5 
 
 S. 
 
 11 
 
 7 
 
 37 
 
 T. 
 
 8 
 
 6 
 
 Two third Parts of the greater; ^ 6 
 Circle, is ^ 
 
 And one third Part of the lef- ? g 
 fer Circle, is 5 
 
 I 
 
 *i 
 
 II 
 
 i“ 
 
 9 - 
 
 6 
 
 Which being added, the Pro-? J 
 
 dud is J i I I i 
 
 Which multiply by the Length 4° ' 1 ' 
 
 24800 
 
 36 
 
 1 33 
 
 1 8 
 
 1 H 
 
 I O 4- 
 
 
 The Content of the VefleH g 
 
 in Cubick Inches, is 5 + 
 
 6 
 
 2 
 
 
 The next Work will be, to find how many Gal- 
 lons is contained in 24839 Cubick Inches. 
 
 And if you would know the Content in Wine 
 Meafure, divide the Number of Inches by 23 1, and 
 the Quotient gives you the Content in Wine Gallons. 
 
 But if you would know the Content in Ale Mea- 
 fure, then divide the Number of Cubick Inches by 
 28 2, and the Quotient gives you the Content . Or if 
 you would know the Content in Ale Meafure, accord- 
 ing as the Gallon was accounted (before the Excife) 
 to contain 28S Cubick Inches, and \ (or 9 Parts.) 
 After you have reduced the Inches into Wine Mea- 
 fure, then multiply the Content in Gallons (and 
 Parts, if there be any) by 4, and divide the Product 
 
572 Meafaring of Solids. Book Vf. 
 
 by 5, and the Quotient gives you the Content re- 
 quired ; for 231 being multiplied by 5, produces 
 1155 I. which being divided by 4, the Quotient is 
 2 88 I. and l (or 9 P.) for 231 bears the lame Pro- 
 portion to 288 Inches and 9 Parts, that 4 does to 5. 
 
 Example. 
 
 (1 
 
 * 7(22 
 
 2 #%%# (107 Gallons. 
 
 il 
 
 Suppofe you would know how many Gallons of 
 Wine Meafure is contained in 24839 Cubick Inches; 
 
 divide theNumber given by 
 231 (as in the Margin) 
 and the Quotient is 107 
 Gallons, and i\ T Parts of 
 a Gallon ; which reduced, 
 is 107 Gallons and an half, 
 and about $ of a Pint. 
 Then if you would know how many Ale Gallons it 
 contains, accounting 288 1 . 9 P. to the Gallon, fay, 
 
 As 5 is to 4 ; fo is the 
 Content in Wine Gallons, 
 to the Content in Ale Gal- 
 lons. 
 
 Multiply 107 Gallons 
 and an half, by 4, and the 
 Product is 430, which di- 
 vide by 5, and the Quoti- 
 ent is 86 Gallons, and near 
 
 ? a Quarter of a Pint. 
 
 4^0 ( 8<7 Gal. 4 t Pint. 
 
 Or if you would know how many Ale Gallons 
 (of £88 I. 9 P. to the Gallon) there is in 24839 I. 
 
 6 P ? without reducing it firft to Wine Mealure, 
 
 do thus : 
 
 43 ° 
 
 Multiply 
 
Book VI. G AUGING. 
 
 Multiply the Content L ? ‘ 
 given inches, by iz (to *4*39 « 
 
 bring them into Parts) and 
 
 the Produd is 298068, to ' g 
 
 which add the 6 Parts that 49*78 
 
 are in the Content befides *4^39 
 the Inches, and the Product 
 
 is 298074 P- Cas in the 
 
 “XW the Inch* 
 
 of the Gallon into Parts, 
 
 by multiplying 288 by 12, 
 
 and the Produd is 345° i 
 to which add the 9 Parts, 
 and the Produd is 34*5 
 Parts ; laftly, divide 298074 
 
 by 3465, and the Quotient 
 is 86 Gallons, and ot 
 a Gallon; which 84 Parts 
 being reduced is 7 Inches, 
 which is not a quarter or a 
 Pint, for a quarter ot a 
 Pint contains 9 Inches. 
 
 By the fame Method you 
 may find the Content (ot 
 
 298068 
 
 6 
 
 298074 
 
 L P. 
 288 9 
 12 o 
 
 576 
 
 288 
 
 3456 
 
 9 
 
 3465 
 
 (8 
 
 nay find the Content (ot (4 
 
 my Number of Inches and 0 r # (8<f Gallons. 
 
 Parts <dven) in AleMeafure, 3 f * 
 
 t®theGahoa 
 
 Vow the finding the two third Parts of t be 
 
’74 Meafimng of Solids. Book VI. 
 
 As 1728 I. is to 904 I. 9 P. 4 S. 5 T. 2 Fo fo 
 is the Square of the Diameter at the Bun=> to two 
 third Parts of the Content of the Circle thereof 
 
 Again: As 1728 I. ts 10452 I. 4P. 8 S. 2 T. 7 F 
 fo is the Square of the Diameter at the Head to 
 one third Part of the Content ofthe Circle thereof 
 
 Having thele proportional Numbers, the Opera- 
 tion will he after this manner. 
 
 Firft, Square each Diameter. 
 
 Secondly, Multiply the Square of the Diameter 
 at the Bung by 904 1 . 9 P. 4 S. 5 T. 2 Fo. 
 
 Thirdly, Multiply the Square of the Diameter at 
 the Head by 452 1 , 4 P. 8 S. 2 T. 7 Fo. 
 
 Fourthly, Add thefe two Products together. 
 
 Fifthly, Divide the Produd of the Addition bv 
 1728. 3 
 
 Sixthly, Multiply the Quotient by the Length 
 of the Veflel, and the Produd is the Content in 
 Cubick Inches. 
 
 Example. 
 
 I. 
 
 I. The Diameter at the 7 
 
 Bung is 32 Inches, the >1024 
 Square whereof is J* 
 
 II. Which multiply by 904 
 
 P. 
 
 0 
 
 9 
 
 S. T. 
 
 0 0 
 
 4 5 
 
 Fo. 
 
 0 
 
 2 
 
 4096 
 
 92160 
 
 768 
 
 341 
 
 4 | I 
 426 1 8 
 
 ! 170! 
 
 8 
 
 The Produd is 9264.95/ 6j 0 | ioj 8/ 
 
 III. The 
 
Book VI. GAUGING. 375 
 
 I. P. S. T. Fo. 
 
 III. The Diameter 
 
 at the? 
 
 
 
 
 
 Head is 18 Inches. 
 Square whereof is 
 
 the 5*324 
 
 oj 
 
 4l 
 
 0 
 
 8 
 
 0 
 
 0 
 
 Which multiply by 
 
 452 
 
 2 
 
 7 
 
 648 
 
 2l6| 
 
 54 
 
 l 
 
 1 
 
 
 1620 
 
 
 
 189 
 
 
 • : o'f 'Ult' 
 
 I29608j 
 
 I 1 
 
 ! 1 
 
 
 ! 
 
 h 
 
 The Produft is 
 
 146574 
 
 1 s 
 
 1 9 
 
 9 
 
 0 
 
 IV. To which add 
 former Product 
 
 the ^ 926495 
 
 1 6 
 
 1 0 
 
 10 
 
 8 
 
 And the Sum is 1073069I n|iol 7I 8 
 
 V. Divide this laft Sum by 
 1 7285 and the Quotient is 620 1. 
 and V7II Parts of an Inch, which 
 being; reduced, is 620 I. 11 P. 
 10 S. 5 T. 
 
 VI. Multiply this laft Sum by 
 the Length of the Veflel 40 I. 
 and the Produdt is (as in the 
 Margin) 24839 I. 6 P. 8 S. 8 T. 
 which is the folid Content in 
 Cubick Inches and Parts, and 
 the fame that was found the 
 firft Way. 
 
 362(0 I. 
 10 7 306(9 (620 
 
 i 7 
 
 *7 2 % 
 
 I. P. S.T. 
 62o|ii|ioJ$ 
 
 40I 1 ) 
 
 24800*33! 4I 
 
 3 6] 8 1 1 6 | 8 
 
 24839! 61 8 8 
 
 For 
 
For meafuring or gauging of Brewers Tuns or 
 Velfels, whether they be fquare or round, or any 
 other Form, work thus: 
 
 Firft find the Content in Cubick Inches, by the 
 preceding Rules of meafuring fuch Bodies ; then di- 
 vide that Content by 282 (the Number of Inches 
 agreed upon for a Gallon to contain fince the Excifc) 
 and the Quotient is the Content in Gallons. Laftly, 
 to bring the Gallons into Barrels, divide the Num- 
 ber of them by 36 (the Gallons in one Barrel) and 
 the Quotient gives you the Content of the Veffei 
 or Tun, in Barrels. 
 
 t'. • 
 
 THE 
 
Place this at the End of the Sixth Book. 
 
'W i 
 
 , * 1 
 
 
 THE 
 
 
THE 
 
 APPENDIX. 
 
 Treating in the FIRST PLACE 
 
 O F T H E 
 
 MEASURING 
 
 O F 
 
 CHIMNEYS 
 
 REFORMED. 
 
 By VEN. MANDET. 
 
 LONDON: 
 
 Printed in the Year M.DCC. XXVII- 
 
a 
 
 
 V. I 
 
 i 
 
 r > tT 
 
 c J: 
 
 1 v 
 
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 O 
 
 ■ 
 
 ' / 
 
 * 1 a ~ ./%/ ' ■ 
 
 
 
appendix. 
 
 ng in the fir ft place of the Meaftt- 
 ring of Chimneys Reformed. 
 
 HE Brick-work of Chimneys being a 
 1 great part of it not to be leen, hath 
 occafioned much Error in the Meafu- 
 
 ring thereof ; and, it may be, not 
 
 one Meafurer in ten is fenfible how the Funnels are 
 wrought, and in what manner they are carried up . 
 I thought it might be a Service to the Pub >ck to 
 correct this Error that fo neither the Gentleman, 
 nor he that pays for the Work, nor the Workman 
 that doth it, may receive any Injury., The uiuai 
 way of girting round the Jaumbs and Brealt o u 
 Chimney, for the Length to be multiplied by the 
 Height of the Story, is erroneous, and more in one 
 Story than in another, as will appear hereafter. 
 Some Tears ago, it was my Hap to meafure the 
 Bricklayers Work of an Houfe, wherein Mr. Leo- 
 nard Sowersby , the late notorious Meaiurer, was 
 concerned againft me. We had oftentimes meaiured 
 together before that time, and alfo fince in levera 
 5 C c 2 P laccs ’ 
 
$80 MEASURING 
 
 Places : But here we happen’d to difagree about 
 meafuring the Chimneys very confiderably, which 
 gave me occafion to delineate the Chimneys on Pa- 
 per, and to Ihew him wherein he was miftaken : The 
 Chimneys were a double Stack placed Diagonal ways 
 againft the Party Wall, as you may lee in Fig. i. 
 But before we proceed any farther, it will be con- 
 venient that you underftand fome Symbols which I 
 fhall have occafion to ufe in demonftrating the Men- 
 furation of the following Particulars in this Appen- 
 dix, only to abbreviate the Work. 
 
 Signification . 
 
 Equal to. 
 
 More. 
 
 Lels. 
 
 Multiplied by 
 Deduct. 
 
 Triangle. 
 
 Square. 
 
 Oblong. 
 
 The Sum. 
 
 The Difference. 
 
 One Brick-work. 
 
 One Brick and half work. 
 
 Feet. Inches. 
 
 Proceed we now to the Mealure of thefe two 
 Chimneys in the Shop or Parlour Story * for in the 
 Cellar there were Piers wrought, and Arches turn’d, 
 if my Memory fail not. 
 
 And fir ft, Mr. Sowersbys way was thus : He girt* 
 them round from the Wall, which Length was 16/. 
 2. /. and tnultiplied that by the Height of the Story, 
 
 which 
 
 Symbols. 
 
 +' 
 
 x 
 
 dd 
 
 A 
 
 □ 
 
 CD 
 
 z 
 
 • X 
 
 i B 
 ItB 
 /. i. 
 
of Chimneys Reformed. 381 
 
 which'was 10 /. Si. at i Brick in Thicknefs, and 
 would allow no more. 
 
 feet in. 
 
 16 2 
 
 10 8 
 
 160 o 
 1 8 
 
 10 8 
 
 1 
 
 172 5 of 1 Brick- work, which being reduced, is 
 
 1 14 f. nt . of one Brick and 
 
 I told him, that he had not allowed the tP«e 
 Meafure of the Work by a great deal, and that I 
 did conceive the trueft and readieft way, was to 
 meafure them as a Solid, and dedud the Vacan- 
 cies $ which accordingly I did in the following man- 
 ner ; The two Chimneys made the half of a Qua- 
 drate, each fide being 8 /. i /. which Quadrate was 
 divided in two in the middle by the Party Wall, as 
 you may perceive by Fig. i. Therefore I multiply 
 8/. i /. by 8/ i /. it makes 65/. 4 i. halfwhere- 
 of is 32 /. 8 i. which being multiplied by the Height 
 of the Story, viz. by 10 /. 8 i. is = 348 /. 5 /. the 
 Iblid Content : In the next place we mull dedud 
 the Vacancies ; the Chimneys were both alike in 
 Magnitude, and each Chimney was 4/ wide be- 
 tween Jaumb and Jaumb, and 1 /. 8 i. deep, from 
 Breaft to Back, and the Mantle-trees lay 4/. high 
 from the Floor. The neareft way, as I conceive, is 
 to dedud the whole as a Vacancy, and afterwards 
 
 C c 3 to 
 
382 MEASURING 
 
 to add the Breafts, Wings, and Wieths : The Vacancy 
 of one Chimney is 4/. xi/.8 8 L x 10/. 8 /. 
 
 = 71/. 1 /. which being doubled for both Chim- 
 neys is 142 /. 2 /. which deduced from the Solidity, 
 leaved Cubick feet, 206/. 3 /. which muft be re- 
 duced to 1 Brick and half in thicknefs, by laying. 
 As 14 to 12, fo 20 6f to 17 6 f 6 u of 1 Brick and 
 half Work. Now becaufe we have deduced all the 
 Breadth and Depth of the Chimneys, by the Height 
 of the Story, and there being the Breafts, and 
 Wings, and Wieths, included in this Deduction, we 
 muft add the Quantity of them to the 176/. 6 i. 
 
 The mean Length of one Wing, and both the four 
 Inches of the Breaft, and the croft Wieth, is 8 f 
 5 i. which multiplied by 6 /. 8 i. the Height of the 
 IVfontle-tree to the Top of the Story, is =56/ 1 /. 
 of i Brick- work, which reduced is 18 /. 8 i. of one 
 Brick and half work in one Chimney, the double 
 of which for both Chimneys, is 37 /. 4 /. which 
 added to 17 6 f. 6 i. makes 213/. 10 /. of one Brick 
 and half work, the true Content of thele two Chim- 
 neys in the Parlour Story : Then we muft add the 
 Pargetting of thefe two Funnels in this Story, (for 
 fince we mealure no more Brick-work, than is real- 
 ly done in the Chimneys, we ought in Juftice to mea- 
 fiire, and allow for the Pargetting ; by reafon the 
 Chimneys are more troublelome in working than 
 ftrait Walls ; and yet when they are meafured 
 by the Rod, the Bricklayer hath no greater Price 
 by the Rod for them, than he hath for working the 
 foait Walls : The mean Girt of the four Sides of 
 one Funnel, is 5 /. 8 /. which multiplied by 6 f 8 L 
 — 37 / 9 which doubled for both Funnels, is 75 f 
 
 6J . 
 
of Chimneys Reformed. $85 
 
 6 i. which divided by 9 is 8 Yards + 3 /. + 6 i. 
 of Pargetting, at 4 d. per Yard, is 2 s. 9 d. 
 
 feet. in. 
 
 So that the true Meafure of thefe 1 of ii 
 
 two Chimneys, is 2 , 13 10 \Brick- 
 
 And Mr. Sower shy made the \work. 
 
 Content to be but 1 14 1 1 ) 
 
 The Difference 9 8 1 1 
 
 A very coiifiderable Quantity of Work, for the 
 Workmen to lofe in one Story, befides the Parget- 
 ting of the Funnels, and fo proportionally in every 
 
 Story. 
 
 * 
 
 E XAMPL-E II. 
 
 Siippofe the Dining-Room Story to he 11 fhigh * wid 
 the Chimneys the fame Girt as in the Parlour 1 See 
 Fig. 1, and the fame Bignefs within. - 
 
 feet, in 
 
 a»rlj • . . . ■ 
 
 Firft, Mr. Sowershys Way was* 
 
 
 16 
 
 161 19 ? ..Vi 
 
 177 10 of 1 Brick in Thicknefi, being =5= t© 
 1 1 8 6 of 1 Brick % W ork. 
 
3 84 MEASURING 
 
 0 ~i * . £ “ ; - s . i ? A 
 
 Now my way. 
 
 
 feet. 
 
 ifl. 
 
 The Solidity 
 
 8 
 
 1 
 
 vi 
 
 8 
 
 1 
 
 ;bhS< 01 • • 
 
 ; ' .> 1 
 
 
 i ) 4 jii.i *!--'rr 
 
 >. 707/ / 
 
 (5 4 
 
 8 
 
 V 1 1 v £ r 
 
 - 'i Oj ji-y : . 
 
 8 
 
 ir 8 0 
 
 65 
 
 4 
 
 
 * 32 
 
 8 
 
 - M f“T i *>).,! ; V 70-f* : 
 
 which v a 
 
 > ' Vfif i'V -f: 
 
 0 
 
 TJ v J HI V 0 * V • j J 00 * > i V V > i 1 ; . ; . 
 
 32 
 
 
 
 327 
 
 4 
 
 The Solid Content 
 
 359 
 
 4 
 
 The Solid Content of the two Chimneys in the 
 Dining-Room Story, is $S 9 f- 4 i. of Cubicle Feet; 
 now we muft dedu<ft the two Funnels of the Par- 
 lour Story, and the two Funnels pf this Story: One 
 Funnel of the Parlour Story, is ifi zi. x i f z i. — 
 i /. 4 i. x 1 1 f. the Height of the Story is = 14/. 8 i. 
 the Double whereof, for both the Parlour Funnels, 
 is 29 f. 4 i. Then fuppofing the Chimneys in the 
 Dining-room Story to be the fame Breadth between 
 the Jaumbs, and the fame Depth as in the Parlour 
 Story, we will deduct the Breadth by the Depth, 
 the whole Height of the Story, and add the Wings, 
 and Breafts, and Wierhs afterwards: Well then 4/ 
 x 1 / 8 z. — 6 f. 8 i. x i 1 /. the Height of the Story, 
 U s= 7 3 /. 4 i. for one Chimney, the double whereof 
 
of Chimneys Reformed. 
 
 for both Chimneys, is 146/. 8 t. which added to 
 29/. 4 /. the other Deduction makes 176 /. which 
 muft be dedq&ed from the Solidity, and there re-p 
 mains Cubick Feet 183 and 4 1. Say then, if 14 give 
 12, what will 183 give? Multiply and divide ac- 
 cording to the Rule of Proportion, and you will 
 have 156/. to t. of one Brick and half Work, to 
 which we muft add the Wings, and Breafts, and 
 Wieths : The mean Length of both Wings, and of 
 the half Brick-work of the infide and outlideBreaft, 
 and crofs Wieth, is 10 /. 5 /. for one Chimney, the 
 double whereof, is 20/. 10 i. for both Chimneys, 
 which multiplied by 7 /. the Height of them, from 
 the Mantle-tree to the Top of the Story, produces 
 145 fioi. of half Brick-work, in the Breafts, Sc. of 
 both Chimneys, which reduced, is 48 f 7 /. of one 
 Brick and half Work, to be added to 156 f. 10 i. 
 which makes 205 f. $ u of 1 Brick and half Work 
 in the Dining-Room Story, being 8 6 f 11 /. more 
 than Mr. Sower sbys way : next we muft meafure the 
 Pargetting of the four Funnels: The four Sides of 
 one of the Parlour Funnels, is = 4 f 8 t. (for we fup- 
 pofe the Funnels to be 14 /. □) which multiplied 
 by 1 1 f the Height of the Story, is = 5 1 f 4 /'. the 
 Double whereof is 102/. 8 i. for both Funnels : And 
 for the Pargetting of the other two Funnels, from the 
 Mantle-tree, to the Top of the Story, the mean 
 Length of the 4 Sides, is 5/. 8 /. x 7 f the Height from 
 the Mantle-tree to the Top of the Story, is = 39 f 
 .8 i. for one, the Double whereof, is 79 f 4/. for 
 both Funnels, which added to 102 f. 8 /. the Par- 
 getting of the Parlour Funnels makes 182, which di- 
 vided by 9, produces 20 Yards and 2 f. which at 4 d t 
 per Yard, comes to 6 s. + 9 A* 
 
 ' ' EX A Mr 
 
386 MEASURING 
 
 EXAMPLE III. 
 
 Of Meafuring Chimneys that ftand at right Angles, 
 adjoining to a Party or other Wjall, as Fig. 2. 
 
 Suppofe Fig. 2. to be the firft Story : The ufual 
 way is to girt the Chimney about, for the Length, 
 and multiply that Length by the Height of the Story, 
 allowing the Work to be of the fameThicknefi as the 
 Jaumbs, which in fome Cafes it comes pretty near 
 the Truth, and in others it makes lefs than tne true 
 Quantity by a great deal : The Jaumbs of this Chim- 
 ney being 3 f from the Wall, and the Chimney 4 f 
 within, and the Jaumbs 9 in Thicknefs, the Girt is 
 11 f 6 i. x by 10 /. the Height of the Story, is =5 
 1 1 5 / of one Brick- work, which reduced is = 7 6 f. 
 8 i. of one half Brick-work. Let’s try it by mea- 
 furing each Particular 3 the two Jaumbs being ad- 
 ded together, make 6 f. x 10 f =60 f the Back, 
 is 4 f x 10 f =40/. of two Brick-work, which be- 
 ing doubled is = 80 f of one Brick-work : The 
 outfide Brealt is 4 f x 6/. (that is from the Vacan- 
 cy under the Mantle-tree, to the Top of the Story, 
 fuppofing the Mantle-tree to lie 4 f. high from the 
 Floor,) is = 24/. of half Brick-work, = 12/ of 
 one Brick-work: Then the infide Breaft, and Wing, 
 and Wieth, the mean Length, is 3 f 9 i. x 6 f = 22/ 
 6 i. of half Brick-work, = 1 1 /. 3 /. of one Brick- 
 work. All thefe being added together, makes 163 /. 
 3/. of 1 Brick- work, being = 108 f 10 i. of one 
 half Brick-work, out of which we muft deduft the 
 falling back for the Funnel, being 1 /. 2 i. x 7/. = Sf 
 2 i. of one half Brick- work, and there remains 
 
 100 f 
 
of Chimneys Reformed. 387 
 
 100 f. 8 i. one and a half Brick-work, being 24/. 
 more than the ufual way. But if the Jaurrft s had 
 been one Brick and a half inThicknefs, the uiual way 
 of Meafuring would have wanted but 5 /. 8 /. of 
 the true Meafure or Quantity ; the Pargetting muft 
 be added, which is about 4 Yards, at 4 d. per Yard, 
 comes to 1 s. 4 d. 
 
 EXAMPLE IV. 
 
 Jo meafure Fig. 2. in tie fecond Story. 
 
 Suppofmg it to be 11 /. high to the Top of the 
 Floor, and the Girt the fame, as in the firft: Story : 
 
 1 x /. 6 i. x x 1 f. = 1 26 / of one Brick- work =84 f. 
 4 t. of one and a half Brick-work, according to the 
 ufual way of meafuring : We will try it the true 
 way, the two Jaumbs = 6 f. x 11 f..— 66 f. of one 
 Brick-work. The Back 4 / x 1 if. = 44 f. of two 
 Brick-work — 88/ of one Brick- work ; the outer 
 Breaft, 4/. x 7 /. = 28 /. of half Brick- work = 14 f. 
 of one Brick-work ; the inner Breaft, and Wing, 
 and Wieth, is \f. 9 i. the mean Length, which x 7/. 
 the Height from the Mantle-tree to the Top of the 
 Story, is = 2.6 f 3 i. of half Brick-work = 13 / 1 i. 
 of one Brick-work, all being added, makes 18 1 f. 
 1 i. of one Briek-work, which reduced, makes 12 of 
 8 i. of one and a half Brick-work, out of which we 
 muft dedudt the Funnel of the firft Story, and the 
 Funnel of this Story : The Funnel of the firft Story, is 
 if 2 i. x nf. = 12 f. 10 i. of oneand a half Brick- 
 work, (for we fuppofe the Funnels to be 14 i. □ 
 within) the Funnel of this Story, is 1 / zi. x 7/ = 8/ 
 % i. of one and a half Brick-work, which added to 
 
3 8 S MEASURING 
 
 the 12 f. io i. makes 21 f to be deduced from 120 f. 
 $ i. fo there remains 99 / 8 /. of one and a half 
 Brick-work, being 15/4 *. more than the ufual way. 
 Then alfo the Pargetting of thefe two Funnels is to 
 be meafur’d ; the four Sides of the firft Story Fun- 
 nel being added together, is = ±f 8 x 1 1 /. 51 f 
 
 4 i. the Sides of the Funnel in tnis Story being added, 
 the mean Length is 5 / 4 /. x 7 f 5= 37 /. 4 /. both 
 being added, produce 88/ 8 /. which isnine Yards; 
 and 7 / of Pargetting, at 4 d . per Yard, comes to 
 3 s - '} d- 
 
 EXAMPLE V. 
 
 To tneafure Fig. 2, in the third Story. 
 
 Suppofing the Story to be 9 f 6 /. to the Top of the 
 Floor, and the Girt the fame as in the two other 
 Stories, 1 1 f. 6 /. x 9 /. 6 /. = 109 / 3 i. of one Brick- 
 work, = 72 f 10 /. of one and a half Brick- work ac- 
 cording to the ufual way. Now for the true way, the 
 t*wo Jaumbs added make 6/ x 9 / 6 i. = 57 f. of one 
 Brick-work. The Back 4/ x 9 / 6 /. = 38 / of half 
 Brick-work = 19 / of one Brick-work ; the two 
 Wieths of the Funnels, is 2/ 4 /. x 9 f 6 /. = 22/ 
 2 i. of half Brick-work = 11/ 1 i. of one Brick- 
 work ; theoutfide Breall 4/ x 5/ 6 /. = 22/ of 
 half Brick-work, 11 of one Brick ; the two Wings 
 and infide Breaft, and Wieth, the mean Length, is 
 5 / x 5 / 6 i. = 27 /. 6 i. of half Brick-work = 13/ 
 9 z. of one Brick-work ; all thefe being added, make 
 in f. 10 of one Brick-work = 74/7 /. of one 
 and a half Brick-work, which exceeds the ufual way, 
 but if. 9 i. befides the Pargetting : For which, the 
 four Sides of one Funnel, is 4/ Si. x 9/ 6u = 44/ 
 
 4/, 
 
of Chimneys Reformed. 
 
 4 i. the other Funnel being the fame, is = 44/ 4 /. 
 and the Funnel in this Story, the mean Length of the 
 four Sides, is 8/2/. x 5 /. 6/.= 44/] ni. all three 
 being added, make 133/! 7;. = 14 Yards and 7 f 
 at 4^. /tfrYard, comes to 4 j. 11 d, which added to 
 the Price of the Excels of the Brick-work, is 5 s. 6 d. 
 being too much for a Workman to lofe in one Story. 
 
 EXAMPLE VI. 
 
 Of Meafuring Angle Chimneys , or Chimneys ftanding 
 in an Angle, 
 
 We will fuppofe Fig. 3. to be the Chimney in the 
 Cellar, or Ground Story, and to be 9 }, high from 
 the Foundation to the Top of the Floor. The ufual 
 way of Meafuring this Chimney, to take the 
 Length of the Forefide, or Bread, and to multiply it 
 by the Height of the Story, and to allow the Work 
 to be half Brick thinner than the Breadth of the 
 Jaumbs ; nay fome will allow the Work to be but 
 one Brick thick, let the Jaumbs be of never fc much 
 in Thicknefs : Well then the Length of the Forefide of 
 Fig. 3. is 7/] x 9/ = 6 3/. of one and a half Brick- 
 work, becaufethe Jaumbs are two Bricks thick : We 
 will try it the true way ; fuppofmg the Walls A and 
 B that are at right Angles to be old Walls, or if 
 new, to bemeafured before by themfelves: But be- 
 fore we proceed to the true way of Meafuring thefe 
 kind of Chimneys, it will be convenient to infert a 
 Rule, whereby we may know by Meafuring the 
 Forefide, what the Length of the two fide Walls* A 
 and B are, which are included by the Forefide, which 
 is from the 47 Propofition of the firft Book of Eu- 
 clid j and 5 tis thus, Square the Length of the Forefide, 
 
 and 
 
 1 
 
39° MEASURING 
 
 and take J part of the Product for the fuperficial Con- 
 tent : This Rule is of good ufe for the dedu&ing of 
 any Angle Chimney out of Cieling, or Paving, or 
 Flooring, &c. Therefore note it well, for it is of 
 great ule. Now to come to the Bufinefs, the Length 
 of the Forefide is 7/. which fquared,or x 7/ = 49/. 
 a fourth Part of it is 1 2 /. 3 i. for the fuperficial 
 Content, which multiplied by the Height of the Sto- 
 ry, is 9 f. produces Cubick Feet 1 1 o f 3 /. then we 
 will dedudfc the V acancy of the whole Length, Breadth, 
 and Height, including the Breads, and Wings, and 
 add them afterwards, well then 4/ x 1 f 6 i. = 6/. 
 x 9 /. = 54/. which deducted from 1 1 o f 3 i. leaves 
 Cubick Feet 56/. 3 i. which muft be reduced to one 
 and a half Brick-work, by making it as 14 to 12, fo 
 56/. 3 1. to 48 f 3 u of one and a half Brick- v/ork. 
 In the next place, we muft dedudfc the Funnel from 
 the Mantle-tree upwards, which muft ftand in the 
 Triangle, let it be 1 fi 2 i. x 1 f = 1 / 2 x 5/.== 
 5 /. 1 o /. of one and a half Brick-work, which de- 
 duct from 48 f 3 i. and there remains 42 /. 5 i. of 
 one and a half Brick-work : In the next place we muft 
 add the Breads, and Wings, andWieths: The out- 
 fide Bread, is 4 / x 9/^36/ of half Brick- work 
 = 12 f of one and a half Brick-work : The infide 
 Breaft, and Wing, and Wieth, is $ f x 5./. (the 
 ‘Height from the Tarfels to the Top of the Floor) 
 makes 25/ of half Brick-work = 8 f 4 i. of one 
 and a half Brick-work s tnefe two being added, make 
 2 4 /. of one and a half Brick-work, which ad- 
 ded to the 48 /. 3 /. we had before, makes 68 f. 7 i. 
 of one Brick and a half Work, for the true Content 
 of this Chimney, being 5 f 7 /. more than the ufual 
 w^y makes it, the Pargett'ng is like wife to be mea- 
 fuAcd, and allowed for. L X- 
 
 % 
 
of Chimneys Reformed. 391 
 
 example vii. 
 
 Suppofing the lame Chimney in Fig. 3. to be 4 f. 
 wide, and the Jaumbs but 14 /. thick on the Fore- 
 fide in the fecond Story, which Story we will for 
 Example luppofe to be 1 of high 3 the Length of the 
 Forefide will be 4 f + 1 f. 2 /. + j f 2 /. = 6f. 4 /. 
 which multiplied by 10 f. makes 63 f. 4 /. of one 
 Brick-work, if you allow the Work to be half a Brick 
 thinner than the Jaumbs, as moft Meafurers gene- 
 rally do, which reduced, makes 42 /. 2 i. of one 
 and a half Brick- work. But to find the true Quan- 
 tity of the Chimney in this Story, we work thus ; 
 the Length of the Forefide is 6f 4 i. which, when 
 Iquared, makes 40/1 i. 4 f. a fourth Part of it is 
 10 /. (rejecting the 1 i. and 4 ^.) for the Area of the 
 Bale, which multiplied by 10 f the Height of the 
 Story produces iooCubickFeet: Then we muft de- 
 duct the Vacancy of the Funnel in' the firft Story, 
 being 1 / 2 /. x 1 /. x 1 o f = 1 1 /. 8 /. Alfo we muft 
 deduct the Vacancy in thi$ Story, being 4/ x 1 f 6 i. 
 — 6 /. x 10 /. == 60 f which added to the other De- 
 duction xi/ 8 z. makes 77 / 8 which taken from 
 100/. leaves Cubick Feet 22 f 4 i. which muft be 
 reduced to one and a half Brick-work, by making 
 it as 14 to 12, lo 22 / 4/. to 19 f of one and a 
 half Brick- work, to which we muft add the Breafts, 
 and Wieths, and Wings, the mean Length of which, 
 is 9/. x 6 f = 54/ of half Brick-work = 18/. of 
 one and a half Brick-w'ork, which added to the 19 f 
 of one and a half Brick-work, makes 37/ of one 
 and a half Brick-work ; to which the Pargetting 
 muft be added of both Funnels. 
 
 E X AM- 
 
$ 9 z MEASURING 
 
 EXAMPLE VIII. Fig. 4. 
 
 Suppofing Fig. 4. to be a double Stack of Chiffi* 
 neys in the Ground Story {landing fingly by them* 
 felves, and the Height oi the Story, to be 10/. the 
 Chimneys to be each 4 / wide, and 1 8 i. deep* 
 * and the Jaumbs one Brick thick : To meafiire thefe 
 Chimneys the ufiial Way, is to girt them about, or 
 to take the Length of one Jaumb, and theForefide, 
 and double them for the Length, and multiply it by 
 the Height of the Story ; the Length of one Jaumb 
 is 5/ and the Length of the Forefide is 5 / 6 i. which 
 being added, makes 10/. 6/. the Double whereof is 
 21 /. for the Girt, or Length, which multiplied by 1 of 
 the Height of the Story is = 210 f. of one Brick- 
 work, which reduced, is = 140/ of one and a half 
 Brick-work, according to the uftial way : Let’s try 
 it the true way; the two Jaumbs added, are 10/. 
 which xio/. = i oof. of one Brick-work = 66/8 i. 
 of one and a half Brick- work : The Back 4/ * 10/ 
 = 4 of of two and a half Brick-work = 66 / 8 i. of 
 one and a half Brick-work * One outfide Bread be- 
 ing 4/ in Length, the other being the fame, and ad- 
 ded makes 8 / of half Brick- work ; the mean Length 
 of the two infide Breads, and Wings, and Wieths, 
 is 6/ 9 /. which add to the 8/ and it makes 14/ 
 9 i. which x by 6/ the Height from the Tarfels to 
 the Top of the Story, makes 88/6 of half Brick- 
 work, which when reduced, produces 59 / of one 
 and a half Brick-work, which three Contents being 
 added, makes 192/4 i. of one and a half Brick- 
 work, out of which we mud dedudl the Funnel, 
 from the Tarfels to the Top of the Story, which is 
 1 / x 6/ — 6 f. of one and a half Brick- work, be- 
 caufe the Funnel, is 14/, by 12/. Well then dedu<d6/ 
 
 from 
 
of Chimneys Reformed. 39 ^ 
 
 from 192 f 4 i. and there remains 186/4 i. of one 
 md a half Brick- work, for the true Content of thefe 
 :wo Chimneys in thefe Story ; to which mull be ad- 
 ded the Pargetting of the Funnel, which Content 
 found the true Way, is 4 6 f 4 i. of one and a half 
 Brick-work more than the .ufual Way, befides the 
 Allowance to be added for the Pargetting. 
 
 An Advertisement. 
 
 Becaule the Pargetting of the Funnels is a Work 
 differing both in Quality and Price from the Brick- 
 work, it will be convenient to find out a Method, 
 that it may be reduced so its Equivalency in Brick- 
 work, which may be done thus ; one Yard of Par- 
 getting comes to 4 d. and one Foot of one and a 
 (half Brick- work, at 5/. per Rod, comes to 4^. 
 1 rvi, therefore we may allow one Yard of Par- 
 igetting againft one Foot of Brickwork, without any 
 great difference, it being but fomewhat more than 
 a Farthing, and the Pargetting of a Funnel 1 f in 
 Height is about half a Yard , therefore, when we 
 make Deductions for the Funnels, if we allow but 
 one half of the Deduction, the other half will re- 
 compence for the Pargetting, and fo we need not 
 trouble ourfelves to meafure the Pargetting any 
 more hereafter in what follows, but ufe this Method 
 of taking but one half of the Deduction. 
 
 EXAMPLE IX. Fig. XIV. 
 
 Let’s try the fame Stack of Chimneys in the fecond 
 Story, which we fuppofe to be 12/ high: And 
 firft by the ufual way, we will imagine the Girt to 
 be the fame as in the Ground Storv, VfZ. 21 /. x 1 2./. 
 = 252 f. of one Brick-work, which reduced is = 
 
 D d 168 f. 
 
394 MEASURING 
 
 1 68 f of one Brick and a half Work for the Content 
 of the two Chimneys, according to the ufual Way 
 of Meafuring ? We will try it our Way, the two 
 Jaums added, make io /. x 12 /. = 120/. of one 
 Brick-work= 80/. of one Brick and a half Work :| 
 The Back is 4/. x 12/. = 48 f of two Brick and 
 a half Work = 80 /. or one Brick and a half Work : 
 The two outfide Breafts, and infide Breafts, and 
 Wings, and Weiths being added together, make 
 14/ 9 /. x 8 /. (for we fuppofe the Mantle-tree to 
 lie 4 f. high from the Floor) produces 1 18/. of half " 
 Brick-work, which reduced is 39 /. of one and a 
 half Brick-work, which three Contents being added ■ 
 together, make 199 /. of one and a half Brick-work, 
 out of which we muft deduct the Funnel in each 
 Story - the Funnel of the Ground Story, is if x if 
 2 /. x 12/. = 127. of one Brick and a halfWork, 
 and abating one half of the Deduction for the Par- 
 getting, there remains 6 f of one Brick and a half 
 Work, to be deducted for the Funnel of the Ground 
 Floor in this Story ; alfb we muft deduCt the Fun- 
 nel of this Story from the Mantle-tree, which is 
 1 f 2 L x 1 f x 8 /. = 8 f of one Brick and a half 
 Work, which allowing one half for the Pargetting 
 of the fame, there remains 4/. of one and a half 
 Brick- work ; thefe two Deductions being added, 
 make 10/. to be taken out of 199 /. and then there ' 
 remains 189 /. of one Brick and a half Work, the I 
 true Content of thefe two Chimneys in this Story, I 
 which is 21 / of one Brick and a half Work morel 
 than the Contents, by the ufual Way of Meafuring. 
 
 EXAMPLE X. Fig. 5. 
 
 The Invention of placing the Funnels without or 
 beyond the Jaums of the Chimneys, as in this fifth 
 
 Figure, 
 
of Chimneys Reformed. 395 
 
 Figure, hath been but of late Years ; and indeed 
 where the Breadth of the Room is but narrow, it 
 is a good Contrivance, by Realon the Chimneys do 
 not come fo far into the Room as they would do, 
 if the Funnels were placed behind them, there be- 
 ing the Breadth of the Funnels, or 1 6 t. laved in the 
 Breadth, which is compenfated in the Length : Now 
 to meafure this Chimney, fuppofing the Wall it 
 Hands againft, to be either an old Wall, or elfe to 
 have been meafured before by itfelf- the ufual Way 
 is to girt the Chimney and Funnel about for the 
 Length, and to multiply that Length by the Height 
 of the Story, which we will fuppofe to be io / the 
 Girt then is 12/ 2 i. x 10/ == 121 /. 8 /. of one 
 Brick- work = 81 /. it. of one Brick and a half 
 Work. Let’s try it our Way, each Jaum is 2/. from 
 the Wall, that is 19 /. the Depth of the Chimney, 
 and 5 /. for the Breadth of the Back, which two 
 Jaums being added, make 4/ x 10/ = 40 f of one 
 Brickwork=26/. 8 i. of one Brick and a half Work: 
 The Back is 7 f x 10/. =70/. of half Brick-work 
 =2 3/ 4/. of one Brick and a half Work; the Front 
 of the Funnels, is 3 f x 10/. = 30/. of Brick-work 
 16 i. thick, which muft be reduced to Brick-work of 
 14 i. the ufual Thicknefs, by making it as 14 to 16, 
 fo 30/ to 34/ 3 /. of one Brick and a half Work ; 
 the outfide and infideBreafts,and Wing, and Wieth, 
 are 7/. 10 i. X 6/. (that is from the Tarfels to the 
 Top of the Stpry) =47/. of half Brick-work — 1 $f 
 8 /. of one Brick and a halfWork, all thefe being 
 added together, make 99 f. 1 1 /. of one Brick and 
 a halfWork for the true Content, it being 18 /.of 
 one and a half Brick-work, more than the Content 
 found according to the ufual Way of Meafuring. 
 
 D d 2 A fA- 
 
39* 
 
 A TABLE of Brickwork from 20 s, to 50 s. 
 per Rod. 
 
 d. q. C- 
 
 •The 
 
The Table of Brickwork continued. 
 From 4 /. ioj. to 6 /. io s. per R.ocL 
 
 1. s. 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 4 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 y 
 
 r 
 
 10 
 
 11 
 
 12 
 *3 
 *4 
 iy 
 
 1(5 
 
 17 
 
 18 
 
 19 
 
 oo 
 
 per Rod, i Foot is- 
 
 3 ' 
 
 4 
 
 l. 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 1 3 
 
 14 
 iy 
 1 6 
 
 17 
 
 18 
 , i 9 
 6 co 
 6 i 
 6 2 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 6 
 
 3 
 
 4 
 
 6 
 
 7 
 
 8 
 
 9 
 
 io 
 
 d. 
 
 3 
 
 4 
 4 
 4 
 
 - 4 
 ’ 4 
 
 * 4 
 " 4 
 
 - 4 
 ' 4 
 
 - 4 
 ” 4 
 “ 4 
 
 - 4 
 
 - 4 
 ““ 4 
 
 - 4 
 - 4 
 
 - 4 
 ~ 4 
 “* 4 
 
 4 
 
 - 4 
 4 
 
 - y 
 
 * y 
 
 * y 
 ~ r 
 
 - y 
 
 - y 
 
 - y 
 
 - y 
 y 
 y 
 y 
 
 q- 
 
 3 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 0 
 
 1 
 I 
 I 
 I 
 I 
 
 1 
 
 2 
 2 
 2 
 2 
 2 
 
 c. 
 
 87 
 
 °y 
 
 2 3 
 
 40 
 
 y* 
 
 76 
 
 93 
 
 11 
 
 28 
 
 46 
 
 64 
 
 81 
 
 99 
 
 17 
 
 34 
 
 l z 
 
 6 9 
 
 87 
 
 3 °y 
 
 3 
 
 39 
 
 57 
 
 74 
 
 9 * 
 
 10 
 
 28 
 
 46 
 
 63 
 
 81 
 
 99 
 
 1 17 
 34 
 y* 
 
 69 
 
 87 
 
 oy 
 
 22 
 
 4 ° 
 
 ys 
 
 7 y 
 
 397 
 
 D d 3 
 
 
39 8 
 
 A fABLE of Tyling, from 2 s. 6 d. 40 s. 
 per Square . 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 2 
 
 3 
 3 
 3 
 3 
 3 
 3 
 5 
 3 
 
 3 
 
 4 
 4 
 
 y 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 *3 
 
 *4 
 
 16 
 
 17 
 
 18 
 
 1 9 
 
 d. 
 
 6 per Squ. 1 Foot is* 
 
 8 — 
 
 9 
 
 10 
 
 1 1 
 
 o 
 
 2 
 
 3 
 
 4 
 
 6 
 
 8 
 
 10 
 
 o 
 
 6 
 
 c 
 
 6 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o — 
 o — 
 o — 
 o — 
 o — 
 o — 
 
 d. 
 
 o 
 o 
 o 
 o 
 o 
 o 
 
 O I 
 O I 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 o 
 
 0 
 
 1 
 I 
 I 
 I 
 I 
 I 
 I 
 
 1 
 
 2 
 2 
 2 
 
 c. 
 
 20 
 
 2 4 
 
 28 
 
 I 32 
 I 36 
 I 40 
 
 44 
 
 48 
 
 I f2 
 
 1 t 
 
 I 60 
 
 I 
 I 
 I 
 I 
 
 1 9Z 
 
 2 
 2 
 2 
 2 
 3 
 
 64 
 
 68 
 
 76 
 
 84 
 
 1 6 
 40 
 64 
 88 
 36 
 
 3 84 
 
 0 32 
 
 0 
 
 1 
 
 1 
 
 2 
 2 
 
 80 
 28 
 76 
 
 2 4 
 7Z 
 3 20 
 ; 68 
 o 1 6 
 
 0 64 
 
 1 12 
 
 n>e 
 
The Table of Tyling continued. 
 
 s. 
 
 d. 
 
 d. 1 
 
 q- 
 
 c. 
 
 20 
 
 0 per Sq. 
 
 i Foot is 2 
 
 1 
 
 60 
 
 21 
 
 O - 
 
 2 
 
 2 
 
 08 
 
 22 
 
 0 — 
 
 - 2 
 
 2 
 
 
 
 o — • 
 
 2 
 
 3 
 
 04 
 
 ^4 
 
 o 
 
 " ‘ -rr 2 
 
 3 
 
 *2 
 
 2f 
 
 0 
 
 3 
 
 0 
 
 O 
 
 26 
 
 O ■ 
 
 . 3 
 
 0 
 
 48 
 
 27 
 
 0 
 
 . 3 
 
 0 
 
 9 <> 
 
 28 
 
 0 
 
 — — 3 
 
 1 
 
 44 
 
 29 
 
 0 — - 
 
 3 
 
 1 
 
 92 
 
 30 
 
 0 
 
 3 
 
 2 
 
 40 
 
 3 1 
 
 o 
 
 - - — — 3 
 
 2 
 
 88 
 
 32 
 
 0 
 
 - — 3 
 
 3 
 
 36 
 
 33 
 
 0 — 
 
 - • 3 
 
 3 
 
 84 
 
 34 
 
 0 
 
 4 
 
 0 
 
 32 
 
 3 r 
 
 0 
 
 — 4 
 
 0 
 
 80 
 
 3 6 
 
 0 > 
 
 4 
 
 1 
 
 28 
 
 37 
 
 0 
 
 4 
 
 1 
 
 76 
 
 38 
 
 o 
 
 - — 4 
 
 2 
 
 24 
 
 39 
 
 0 
 
 4 
 
 2 
 
 72 
 
 40 
 
 0 * — 
 
 4 
 
 3 
 
 20 
 
 d a 4 
 
 ^Tableo/ 
 fendiculars for 
 Gable-ends . 
 
 Bafe. Perpendicular. 
 
 F. I. 
 
 F. 
 
 p. p. 
 
 10 0 
 
 1 
 
 7 6 
 
 11 0 
 
 6 
 
 0 7 
 
 12 0 
 
 6 
 
 8 7 
 
 13 0 
 
 7 
 
 4 9 
 
 14 0 
 
 7 
 
 10 2 
 
 i$ 0 
 
 8 
 
 4 9 
 
 1 6 0 
 
 8 
 
 11 6 
 
 17 0 
 
 9 
 
 6 4 
 
 18 0 
 
 10 
 
 0 1 
 
 19 0 
 
 10 
 
 7 8 
 
 20 0 
 
 11 
 
 2 r 
 
 21 0 
 
 n 
 
 9 3 
 
 22 0 
 
 12 
 
 3 9 
 
 23 0 
 
 12 
 
 10 7 
 
 24 0 
 
 J 3 
 
 5 4 
 
 0 
 
 14 
 
 0 2 
 
 26 0 
 
 *4 
 
 6 9 
 
 27 0 
 
 
 1 6 
 
 28 0 
 
 
 8 3 
 
 29 0 
 
 16 
 
 3 0 
 
 3° 0 
 
 16 
 
 9 8 
 
 31 0 
 
 17 
 
 4 * 
 
 32 0 
 
 17 
 
 u 3 
 
 33 0 
 
 18 
 
 6 0 
 
 34 0 
 
 19 
 
 0 7 
 
 3£ 0 
 
 19 
 
 7 4 
 
 36 0 
 
 20 
 
 2 0 
 
 37 0 
 
 20 
 
 8 9 
 
 38 0 
 
 21 
 
 3 6 
 
 39 0 
 
 21 
 
 10 2 
 
 40 0 
 
 22 
 
 4 3 
 
 10 0 
 
 or 
 
 7 6 
 
 9 0 
 
 or 
 
 0 r 
 
 8 0 
 
 04 
 
 r 8 
 
 7 0 
 
 03 
 
 II I 
 
 6 0 
 
 03 
 
 4 \ 
 
 1 * ° 
 
 02 
 
 9 6 
 
 O F 
 
O F 
 
 MEASURING 
 
 Superficies and Solids. 
 
 Concerning a Line of Sines and Cycloids. 
 
 '* r\ i \ 1 “ r - • ) C- 
 
 DEFINITION I. 
 
 Line of Sines, is of right Sines, bowing 
 the Termination oT the ordinate Appla- 
 cates to the Arch of the Quadrant. Or it is, 
 a crooked Line defcribed from the ex- 
 tream Point of the Semidiameter, attending by an 
 equal Motion through the bowed Arch of the Qua- 
 drant, approaching to the oppofite Point, accord- 
 ing to the Ratio of the verfed Sines of the Arch 
 paffed over, each Definition fhall be explained in 
 the firft Propofition. 
 
 DEFINITION II. 
 
 A Cycloid is the bounding or terminating of the 
 compounded, from the Sines, and from the fuperior 
 Arches, by the ordinate Applicates to the Diame- 
 I * ter 
 
Superficies and Solids. 4° l 
 
 ter of the fame Circle : Or it is a crooked Line de- 
 fcribed from a moveable Point, by a mixt Motion 
 from equal Motions of the Orb, and of the Center: 
 Both {hall be explained in Prop. 23 . 
 
 DEFINITION III. Fig. 1 . 
 
 A Figure is faid to proceed by its Elements, in 
 which Lines are taken or accepted parallel to the 
 Bafe, if the Bafe is a Line, or parallel Superficies, 
 if the Bafe is a Superficie. 
 
 We will explain it : Let there be a Rectangle 
 MD take how many Ibever Parallels to. the Bale 
 M N as PO 8c. the ReCtangle M D is faid to 
 proceed or increafe by MN, PO, and the other 
 Elements of the fame Kind : In like manner the Cy- 
 linder M D proceeds or increafes by the Circular 
 Planes MN, PO, 8c. Likewife the Triang. MAN 
 by the Lines M N, P V : The Cone M N A by the 
 Circle MN, PV, 8c. 
 
 definition IV. 
 
 A Figure is Ifoparallel, which proceeds by Ele- 
 ments equal to the Bafe, as a Rectangle, a Cylin- 
 der, a Priim, 8c. 
 
 DEFINITION V. 
 
 Homogene Figures are thole which proceed by 
 proportional Elements (for Example) the Rectangle 
 MD, and the Cylinder M D, are Homogene Fi- 
 gures, becaufe as the Line M N to P O, fo the 
 Plane MN to the Plane PO. 
 
 DEFI- 
 
402 MEASURING 
 
 definition vi. 
 
 The Center of Gravity is that from which Quan- 
 tity being hung down by a Perpendicular, ’tis Equi- 
 ponderate, or in Equilibrium. 
 
 DEFINITION VII. 
 
 A BallancS is a Line, to the Extreams of which 
 Weights being annexed, and being hung perpendi- 
 cular from fome Point of the fame Line, they make 
 Equilibrium, or weigh equal. 
 
 DEFINITION. VIII. 
 
 Moments are the Endeavour of Weights hanging 
 compared to one another. 
 
 POSITION I. Fig. I. 
 
 Homogene Figures, of the fame Kind of Altitude 
 and Bale, are equal ; being of the lame Altitude, 
 they are as their Bafes ; being of the fame Bafe, they 
 are as their Altitudes ; being of divers Bales and Al- 
 titudes, they are in Compofition, or they are com- 
 pounded of their Bales and Altitudes ; Let there 
 be two Homogene Figures of the fame Kind, BAC, 
 BCD, of the fame Bafe and Altitude; they are 
 equal; For as AC is equal to CD, fo E I is equal to 
 I K, and lo of the reft, if there were never lo many 
 Lines drawn parallel to the Bafes AC, CD; there- 
 fore they are equal. Let them be of the fame Al- 
 titude, and not of the fame Bafe, as AB C, A BD, 
 
 then 
 
Superficies and Solids. 405 
 
 then they are as their Bafes AD, AC ; for as AD 
 to AC % EK to El, and fo of the reft. There- 
 fore ABD is to ABC as AD to AC, that is, as 2 
 to 1 Let them be of the fame Bale, but not ol 
 the fame Altitude, as ABC, AEC, then they are 
 as their Altitudes AB, AE; for as AB to AE fo 
 GI to GI, and fo of the reft ; therefore ABC is 
 to AEC as AB to AE, that is, as 3 to 2. Laftly, 
 let them be of divers Bafes and Altitudes, as ABD, 
 AEC then they are in Competition, or they are 
 compounded of the Bafes, and of the Altitudes, 
 that is, as the Reftangles under BA, AD, and un- 
 der EA, AC i 2 o wit , AEC is to ABC, as the 
 Rectangle under EAC to the Redang. under BAC, 
 viz 2 S' 6, 25 : : 5 • 12, 5- Eut EAC is to BAD 
 as the’Redtangle under BAC to the Redangle un- 
 der BAD. Therefore EAC is to BAD, as the 
 Redangle under EAC to the Redangle under 
 BAD, that is in a compounded Ratio, from the 
 Ratio’s of AE to AB, and of AC to AD: If the 
 Bafe is a Plane, the Figures will be as Ifoparallels 
 under the Bafes and Altitudes : Alfo the fame man- 
 ner of Demonftration may be applied to other Ho- 
 mogene Figures. 
 
 POSITION II. Fig. 2. 
 
 A Quadrant is to the Triangle, in Composition, 
 or compounded from the Ratio of the Semidiame- 
 ter to the Altitude, and from the Ratio of the Arch 
 of the Quadrant to the Bafe of the Triangle ; Let 
 the Quadrant be ADC, and any Triangle ACH, 
 I fay, ’tis Homogene to the Quadrant ; for as AG 
 to CH, fo AB to BF : Therefore as AC to AB, 
 
4°4 MEASU RT N G 
 
 fo CH to BF ; but as AC to AB, fo the Arch 
 CD to the Arch BE; therefore the Figures proceed 
 by proportional Elements, by Defin. 3 therefore 
 they are Homogene by Definition 5. therefore thev 
 are compounded of the Bafes and Altitudes bv 
 Pofition 1 and the Bafe of the Quadrant is’ the 
 Arch of the fame, but the Altitude is the Radius 
 or Semidiameter of the fame. 
 
 COROLLARY I. Fig. 2. 
 
 Hence affirming the Bafe of the Triangle CG 
 equal to the Arch CD, and the Altitude CA • the’ 
 Triangle ACG, will be equal to the Quadrant 
 ACD : Afluming the Bafe lelTer as CK, or greater 
 as CH : ACK will be to ACD, as CK to CG • 
 and ACH to ACD, as CH to CG; laftly BCK 
 to ACH, will be as the Redangle under BCK m 
 the Redangle under ACG. 5 
 
 COROLLARY II. 
 
 Hence the Redangle under the Radius AC and 
 me Half of CG, is equal to the Quadrant ACD • 
 therefore the Redangle under AC, and the Doul 
 . e of CG, is equal to the Circle , but under four 
 times CG is double of the Circle : Hence laftlv 
 any Sedor, fuppofe ACM, is to a Triangle, fup DO fe 
 ACK, as the Arch CM to CK. * 
 
 POSITION III. fig. 3. 
 
 A Cylinder is fefquialter, or once and a half of an 
 Hemilphere of the lame Altitude, and Bafe: Let a 
 
 Quadrant 
 
Superficies and Solids . 405 
 
 Quadrant be ALB, a Redtangle AL, a Triangle 
 BML, all turned about BL 3 there will be gene- 
 rated an Hemifphere from the Quadrant ABL3 a 
 Cylinder from the Re&angle AL; a Cone from the 
 Triangle BML 3 thefe being granted, that begat 
 from the Triline AML, is Homogene to the Cone, 
 begat from BLM 3 For fince Circles are as the 
 Squafes of their Radii, let any one be EG, the 
 Square of GE or BD is equal to the Squares of 
 GD and GB or GF 3 therefore that begat from 
 GE equals thofe begat from GD and GF or DE, 
 therefore that begat from GF, is equal to that 
 begat from DE, therefore that begat from LM is 
 to the begat from FG, as to that begat from DE : 
 Therefore the begat from the Triline AML, and 
 from the Triangle BML are Homogenes by Defi- 
 nition 5. and they are of the fame Bafe, viz. of the 
 begat from LM, and of the Altitude BL, therefore 
 they are equal by Pofition 1. Then fince that be- 
 gat from the Triangle BML being a Cone, is one 
 third of the Cylinder begat from the Rediangle AL, 
 that begat from the Triline AML will be like- 
 wife one third of the fame Cylinder 3 therefore that 
 begat from ALB the Quadrant, viz. the Hemi- 
 , fphere, will be equal to two Thirds of the fame Cy- 
 linder, therefore the Cylinder is Sefquialter, or one 
 and a half of the Hemifphere. 
 
 Suppofe the Radius AB = to 5 /. the Altitude 
 AM equal, or = to 5/. the Diameter = 10 f. Then 
 for the Solidity of the Hemifphere, 10 x 10 = 100 x 
 10 = 1000, the Cube of the Diameter 1000 x 11 = 
 1 1000, which divided by 21, gives in the Quotient 
 524 fere for the Solidity of the whole Sphere, the 
 Half whereof 262 f is the Solidity of the Hemi- 
 fphere. For 
 
4© 6 MEASURING 
 
 For the Cylinder whole Diameter is io / and its 
 Altitude 5/ 
 
 The Area of the Bale -= *78 /. 54 x $f the Al- 
 titude = 392 f 70 for the Solidity of it. 
 
 The Cone, the Area of its Bale 78 f 54 x by f 
 of the Altitude 5 f. which is x, 7 fere makes for 
 the Solidity of the Cone 130 /. 9. 
 
 POSITION IV. Fig. 3. 
 
 The Superficie of the Hemifphere is equal to the 
 Superficie of the Cylinder, of the fame Bafe and 
 Height, the Bafes being taken away, viz. the He- 
 mifphere is Homogene to the Cone : For as that 
 begat from LM to that begat from GF, fb that be- 
 gat from the Arch LA, to that begat from the Arch 
 GP, to wit, as the Square of LM to the Square of 
 GF. But the common Altitude is BL, therefore the 
 Figures are as their Bafes $ but the Hemifphere is 
 double of the Cone by Pofition 3. Therefore the 
 Bafe of the Cone begat from LM, is half of the 
 Superficie of the Hemifphere, begat from the Arch 
 LA : And the Superficie of the Cylinder begat from 
 AM, is double of the Circle begat from LM, viz. 
 equal to the Redtangle under AM, and the Qua- 
 druple of the Arch of the Quadrant by Coroll 2. of 
 Pofition 2. therefore the Superficie of the Cylinder 
 is equal to the Superficie of the Hemiiphere : Hence 
 the Superficie of the Sphere is quadruple of the 
 Area of the greater Circle. 
 
 Example, Diameter 1 o f the Periphery is 31/. 
 417 ; the Area of the Circle, 78 /. 543. 
 
 The Area 78, 543 x 2 = 157 f 086 the Super- 
 ficie of the Hemifphere. 
 
 The 
 
Superficies and Solids. 407 
 
 The Periphery 31, 41 7x5 = 157 / 085, the Su- 
 perficie of the Cylinder. 
 
 Another Way. The Hemifphere is Homogene to 
 the Cone, therefore the Superficie of it multiplied by 
 one Third of the Height of BL, produces the Solid: 
 But the Cylinder, forafmuch as it proceeds by Cy- 
 lindric Superficies, ’tis Homogene to the Triangle; 
 therefore the Superficie begat from MA, drawn 
 into half of ML or BL, produces the Solid ; and 
 the Ratio of the Produ&s, or of the Solids is I, 
 the other Ratio is \ : Therefore the other which 
 is of Superficies is |, therefore the Superficies are 
 equal. 
 
 POSITION V. Fig. 3- 
 
 A Sphere being cut by parallel Planes, the Seg- 
 ments of the fpherical Superficie are as the Seg- 
 ments of the Diameter, cut by the Planes at right 
 Angles. Example, Let the Plane be EK ; I fay 
 the Superficie begat from the Arch AD is to that 
 begat from the Arch DL, as BG to GL, to wit 
 that begat from ABM is equal to the Hemifphere, 
 by Pofition 3. And that begat from ABE pro- 
 ceeding by Cylindric Superficies, of which the firft 
 is begat from AE, is Homogene to that begat from 
 ABD proceeding by Sphericals, of which the firft 
 is that begat from AD: And fince that begat from 
 GF is equal to that begat from DE, and fince that 
 begat from BFG is to that begat from BED, as 
 the begat of the Bale from FG to that begat from 
 DE : the begat from BFG, will be equal to that 
 begat from BDE: But that begat from BFG is 
 equal to that, begat from the Triiiue AED ; and 
 
40 8 MEASURING 
 
 by taking away the common begat by the Triline 
 VED, the Remainers will be equal, viz. the begat 
 from BVD, and AEV, therefore the begat from 
 BAD is equal to that begat from BAE, but thele 
 are Homogenes of the fame Altitude AB, and 
 therefore of the lame Bale, by Pofition i. And the 
 Bales of the begotten from AE, and from the Arch 
 AD : But that begat from AE, is to that begat 
 from EM, as A E to EM, or as BG toGL; there- 
 fore that begat from the Arch AD, to that begat 
 from the Arch DL, as BG to GL ; the lame being 
 alfumed, it may be demonftrated in any other Point 3 
 Therefore the Segments of the Superficie of the 
 Hemilphere, are as the Segments of the Semi- 
 diameter for the fame Planes ; therefore the Seg- 
 ment of the Superficie of the Sphere, are as the 
 Segments of the Diameter. 
 
 POSITION VI. Fig. 4. 
 
 If as AD the Radius of the Quadrant ADK, 
 or the whole Sine, to DI the right Sine of the An- 
 gle BAL, fo DI to DC ; AD will be to AI, as 
 A I to AC, of which alfo AD, the Difference is the 
 fame DC, which is the verfed Sine of the Angle 
 DBI, the double of DAI in the Quadrant DBM, 
 under the Radius BD fupduple, or half of the 
 former AD ; to wit, the fubtenle DH, and all 
 the other will be cut in the Middle by the Periphery 
 DMA : Thefe Things appear from the Elements. 
 
 POST 
 
Superficies and Solids. 409 
 
 POSITION VII* Fig. 5. 
 
 If there be any Fig. fuppofe the Semicircle A HM, 
 which let be turned about AC ; that begat from 
 AMH is to that begat from the Reftangle A L, as 
 the Ifoparallel under the Bafe AMH, and from the 
 Altitude A M, to the Parallelipiped under the- Bale 
 AL and the Altitude AM, to wit, that begat 
 from A M is to that begat from* D I, as the Square 
 of A M to the Square of B'li, left by the Square of 
 B D, that is to the Redtang. under the Bafe D I, 
 and the Altitude AM. The fame may be demon- 
 ftrated by affuming any other. Therefore that be- 
 gat from AMH is to that begat from A L, as the 
 Ifoparallel under the Bafe AMH, and the Altitude 
 AM, to the Parallelipiped of thedame Altitude un- 
 der the Bale AL. If weaffume any other Fig. (for 
 Example) the Triang. AMH, or any other Triline, 
 the fame may be demonftrated generally. 
 
 POSITION VIII. Fig, 6. 
 
 Homogene Figures of the fame Altitude have the 
 Center of Gravity equidiftant from the Bafe. Let 
 CAB, EAB be homogene Figures, and let D A be 
 the Diftance of the Centre of Gravity of the Fig. 
 CAB from the Bafe AC ; it will be the fame of 
 the Center of the Fig. EAB from the Bale AE. For 
 if to the Perpendicular D F you hang down CAB, 
 it will equiponderate. Therefore the Moments of 
 the Trapezium ACFD, and of the Triang. DFB, 
 are equal ; but they are compounded of the Quanti- 
 ties and of the Diftances by exchanging. And fmee 
 
 E e the 
 
4 io MEASURING 
 
 the Trapezium ADGE is to DGB the Triang. as 
 ADFC is to DFB, the Center of Gravity of D F B 
 will be in OL; and it will be, as the Trapezium 
 ADFC to DFB, fo OD toDI ; the Center of 
 the Trapezium will be in I H. Therefore alfo the 
 Center of GDB in OM, and of the Trapezium 
 ADGE in IK. Therefore of the whole Fig. ABE 
 the Center of Gravity will be in DG. The fame 
 maybedemonftrated in any other Homogenes, either 
 Solids or Planes. I {hew thefe in hafte, and briefly, 
 which, in Cavalerius his 5th Exercife , at Propofition 
 the 9th, and others that have writ concerning Sta- 
 tic ks in Geometrical Kigour^ may be found demonftra- 
 ted more at large. Therefore fince thefe things are 
 fuppofed, it will fuffice to {hew them briefly. 
 
 POSITION IX. Fig. 7. 
 
 If two Figures (for Example) E A, BAC, do 
 weigh on the common Axis BA, the Moments 
 will be as the Solids begat from the fame, being 
 turned about BA ; becaufe the Moments are com- 
 pounded from the Ratio of the Quantities, and from 
 the Ratio of the Diftances of the Center of each Fi- 
 gure from the common Axis BA. Therefore the 
 Moment of K G is to the Moment of G I, in the 
 compounded Ratio of the whole to the whole, and 
 of the i to the i ; that is, in duplicate of G K to 
 GI ; that is, as the Square of GK to the Square of 
 G I ; that is, as that begat from G K, to that be- 
 gat from G I. The fame may be {hewn by aflfuming 
 any other. Therefore the Moment of the whole 
 EA, is to the Moment of B A C, as that begat from 
 I A to that begat from BAC. Thefe things I fhew 
 
 briefly , 
 
Superficies and Solids. 4 1 1 
 
 briefly ; they may befeen demonftrated more at large 
 in the aforefaid Caualerius , and in Y orricellius , con- 
 cerning the meafuring of the Parabola, Lem. 31. 
 
 COROLLARY. 
 
 Hence from a given Ratio of the Figures, and of 
 the Diftances from each Center (for Example) of 
 GF, O H, the Ratio of the Solids may be known, 
 or of the Generators or Begetters from the fame Fi- 
 gures ; and back again from the Ratio given of the 
 Solids and of the Diftances, the Ratio may be had 
 of the Figures j and from the Ratio given of the So- 
 lids and of the Figures, the Ratio of the Diftances 
 is had. 
 
 P R O P. I. Fig. 8. 
 
 An homogene Figure of a Figure of Sines, the 
 Bafe of which is quadruple of the Axis, equals the 
 Superficie of the Hemifphere under the Radius, equal 
 to the Bafe of the Fig. of Sines. That this Propor- 
 tion may be underftood, fomewhat of ConftrudHon 
 muft be ufed. Let BA E be at right Angles, and let 
 A E be quadruple of AB ; let any Quadrant be 
 APL ; divide the Arch of it AL in the middle in S, 
 the Sine SZ. Divide AB in the middle in D, and 
 let it be as the whole Sine A P to S Z, fo A E to 
 the Ordinate applied DH. In the fame manner 
 may be found the other Ordinates applied from the 
 Arch AL and the right Line AB proportionally 
 divided. Laftly, through the Extreams of the Ap- 
 plieds fuppofe the Curve Line B H E to be drawn : 
 Let A Q be equal to A B, and let it be as A E to 
 
 E e 2 D B. 
 
D B, fo AE to DK ; and do the fame in the other 
 Applieds, the Fig. ABK.Q will be homogene of 
 the Fig. ABHE. - Laftly, let it be as AC to AB, 
 as Radius, or the Semidiameter to the Arch of the 
 Quadrant ; let AP be equal to AC, and let DI 
 be equal to SZ; and in like manner apply the other 
 Sines ; BT P will be the Line of Sines ; and call 
 ABF the Fig. of Sines, AB the Axfo, AP the Ba- 
 fis, A P L the generating Quadrant, CBT the right 
 Segment, YTP the verfed Segment, CT, 1 DI 
 Ordinates applied ; T T parallel to the Axis, A C T P 
 the right Trapezium of the Fig. ABT T the verfed 
 Trapezium; that begat from the Fig. about BA 
 the right Solid ; that about AP the verfed Solid, and 
 it will appear that AE, the Quadruple of A B, is 
 equal to the Periphery of the Circle under the Ra- 
 dius AC or AP : Alfo it appears, that the Fig. 
 A B E is homogene of the Fig. of Sines : Laftly, 
 turn about P A in the Plane of the Quadrant, the 
 Point A will defcribe the Arch A S L, equal to the 
 right Line AB, which if it be turn'd about L P, 
 that A will defcribe the Periphery under the Radius 
 P A ; fo S will defcribe his under the Radius SZ, 
 and each other Point his under the Radius which is 
 the Sine terminated at the generating or begetting 
 Point : and at length the whole Arch ASL, the Su- 
 perficie of the Hemifphere. Thefe things being 
 done, the Fropofition may be eafily demonftrated. 
 
 For the Fig. ABHE is homogene of the Fig. of 
 Sines ABTP, by Befin. 5. the Bafe A E is qua- 
 druple of the Axis A B ; and it is as A P to S Z, 
 fo AE to DH ; AE is equal to the Periphery under 
 the Radius AP of the Begat, viz. from A. There- 
 fore DH is equal to the Periphery under the 
 
 Radius 
 
Superficies and Solids. 4 1 3 
 
 Radius SZ of that Begat, viz. from S; there- 
 fore the Fig. ABE, and the Superficie of the Hemi- 
 fphere begat from the Arch A S L, are homogene 
 Figures by Defin. 5. They are of the lame Bafe, 
 for AE is equal of the Periphery of the Begat from 
 A ; alfo of the fame Altitude, for the Arch AL is 
 equal to the right Line AB : Therefore the Figures 
 are equal by Pofition x. 
 
 PROP. II. 
 
 The Fig. ABHE is to ABKQ, as AE to AC ; 
 likewife to the Fig. of Sines as AE to AP, that is, 
 as the Periphery to the P.adius j becaufe, fince they 
 are homogene by Conftrudion, and of the fame Al- 
 titude AB, they arc as the Bales by Pofition the ift. 
 Therefore as A E to AQand AB. The fame may 
 be demonflrated concerning all other Homogenes. 
 
 PROP. III. 
 
 Any Fig- homogene of the Fig. of Sines ABTP. 
 and of the fame Altitude AB, is equal to the 
 Redang. under its Bafe and AC or AP : For fince 
 ABHE is equal to two Circles under the Radius 
 AC or AP, by Pofition 4. it will equal the Redang. 
 A F by Corol. 2. of Pofition 2. Therefore fince 
 ABHE is to ABTP as AE to AP, by Prop. 2. it 
 will be as the Redang. A F to the Redang. A L. 
 Therefore ABTP is equal to the Redang. under 
 the Bafe AP, and AC. In like manner it may be 
 fhewn, that ABKQ, is equal to the Redangle 
 AM , and fo of the reft. 
 
 E e 3 
 
 COROL 
 
414 
 
 MEASURING 
 
 COROL I. 
 
 Hence the Fig. of Sines is equal to the Quadrate 
 of its Bale, or to the Radius of the generating Qua- 
 drant, fo A B T P is equal to the Quadrate A L, 
 for A L is the Quadrate, becaufe A P is equal to 
 AC. 
 
 COROL. II. 
 
 The Triline BMP is equal to the Difference of 
 the Square of the Radius, to wit, of A L, and of 
 the Semicircle under the fame Radius 5 for A M is 
 equal to the Semicircle by Corol. 2. of Pofition 2. 
 and AB T P is equal to the Square of the Radius. 
 
 COROL. III. 
 
 Hence the aforefaid Triline is equal to the Re6l- 
 angle E M under the Radius, and the Difference of 
 the fame Radius, and the Arch of the Quadrant. 
 
 COROL. IV. 
 
 The fame Triline is equal to twice the Segment 
 of the Circle under the Radius A P, contained by 
 the Arch of the Quadrant and the Subtenfe. 
 
 COROL. V. 
 
 The Segment contained by the right Line BP, 
 and the curve Line BTP, that is the Line of Sines, 
 is equal to the Triline ACL. 
 
 COROL 
 
Superficies and Solids. 
 
 415 
 
 C O R O L. VI. 
 
 As A O the Square of the Arch of the Quadrant 
 to AM, or the Semicircle • fo AM to AL the Square 
 of the Radius: Hence the Semicircle is a mean Pro- 
 portional between the Square of the Radius, and the 
 Arch of the Quadrant. 
 
 C O R O L. VII. 
 
 The Triline C B P equals the Triline ACL: 
 Likewife BCT equals the Triline PLT: Alfo 
 BCY equals the Se&ion under the Arch Y P, and 
 the right Line YP. 
 
 S C H O L. 
 
 We muft imagine in our Mind the Arch ASL un- 
 bent from Curvity, and defle&ed into the right Line 
 A B , likewife each Periphery ordinately applyed at 
 right Angles to the Arch ASL (for fo the Radius 
 falls in with the Arch) being defle&ed alfbinto right 
 Lines equal to AB, ordinately applyed at right An- 
 gles : So the whole Superficie of the Hemifphere 
 from the Arch ASL begat in the Plane Fig. goes 
 ABHE away, and the Quantity remains the fame, 
 but the Round only and the Curve are changed into 
 right Lines and Planes. 
 
 PROP. IV. Fig. 9. 
 
 If the Fig. of Sines be turned about the Axis, 
 that begat or generated is fubduple of the Cylinder 
 
 E e 4 of 
 
4 i 6 MEASURING 
 
 of the fame Bafe and Altitude : Let the Fig. of 
 Sines be ABC, the Quadrant ADC, the Arches 
 DG, IC are equal : And let the Sine EG be tran- 
 flated into OR, and KI into LV j OB will be 
 equal to A L, alfo to the Arch D G or I E : And 
 fince the Square of AC or A I is equal to the 
 Squares of KI, and I C, or E G, that is the Squares 
 LV and OR ; the Difference of the Squares L V, 
 LM, will be equal to the Square of O R, and the 
 Circles are as the Squares of the Radius’s j there- 
 fore that begat from L V, with that begat from 
 OR, is equal to that begat from VM : In like man- 
 ner that begat from RS, is equal to that begat from 
 L V or Z j therefore that begat from ABC, and 
 from the Triline BSC are homogene ; therefore 
 they are equal fince the Bafes are equal, and are in 
 the fame Altitude • therefore either of them is fub~ 
 duple to that begat from AS, to wit, of the Cylin- 
 der of the fame Bafe and Altitude. 
 
 C O R O I, I. 
 
 Hence the Semiparabola under the Axis AB, and 
 the Bafe A C, is indeed greater than the Fig. ABC, 
 yet being revolved about the Axis, begets a Solid 
 equal to that begat from the Fig. ABC : But if AB 
 be divided in the middlein N, and NP be drawn, 
 the aforefaid Semiparabola will cut the Line of Sines 
 in PI ; and the Applicates or Applyeds above NP in 
 the Parabola are greater, but beneath leffer than in 
 the Line of Sines. 
 
 COROL 
 
4 *7 
 
 Superficies and Solids. 
 
 C O R O L. II. 
 
 That begat from any Fig. homogene of the Fig, 
 of Sines, is fubduple of the Cylinder of the fame 
 Altitnde and Bafe, for they are Homogenes begat 
 of homogene Figures. 
 
 C O R O L. III. Fig. 8. 
 
 That begat from the Quadrant ABQ, to that 
 begat from the Fig. ABKQ, is as 4 to 3, becaule 
 that begat from the Quadrant, to wit, the Hemi- 
 Iphere, is to the Cylinder of the fame Bafe and Al- 
 titude, as 4 to 6. 
 
 C O R O L. IV. Fig. 8. 
 
 That begat from the generating Quadrant ALP 
 or ACP, is to that begat from the Fig. A BP, is as 
 half of the Radius CA to two Thirds of AB, equal 
 to the Arch of the Quadrant ASL , that is accord- 
 ing to Archimedes ' s Cyclometry, as 28 to 3 : But to 
 that begat from the Triline CBP, as one Third of 
 AC, to the Difference of two Thirds of AC and 
 two Thirds of AB, that is as 28 to 5. 
 
 C O R O L. V. 
 
 If there be a mean Froportional between AB, 
 AP, the Parallelipiped under the Square of it, and 
 of the Altitude A B, will be equal to that begat 
 from the Fig. ABTP ; for it is fubduple of the 
 lame Cylinder, fince the Bafe of it is equal to the 
 Semicircle. 
 
 C 
 
MEASURING 
 
 4t 8 
 
 C O R O L. VI. 
 
 That Begat from homogene Figures A B T P of 
 the fame Height or Altitude, are as the Square of the 
 Bafes of the Figures, (for Example) that begat from 
 ABTP is to that begat from ABKQ, as the 
 Square of AP to the Square AQ. 
 
 C O R O L. VII. 
 
 That begat from the Lunula contained by the 
 Arch of the Quadrant BVQ, and the Curve Line 
 BKQ, is fubduple of the begat from the Trian- 
 gle A BQ: For let that begat from the Re&angle 
 AO be 6, that begat from the Triangle ABQwill 
 be 2 j from the Fig. ABKQ, 3 ; from the Qua- 
 drant A BVQ, 4 ■ therefore that begat by the Lu- 
 nula will be 1 : Therefore the Subdupleof that be- 
 gat from the Triangle ABQ, and equal to the be- 
 gat from the Segment contained by the Curve Line 
 BKQ, and the right Line BQ: Hence as the Su- 
 perficie begat from the right Line BQ, will divide 
 in the middle that begat from the Quadrant, <viz. 
 the Hemifphere , fo the Superficie begat from the 
 Curve Line BKQ, will divide in the middle the Be- 
 gat from the Segment of the Quadrant contained 
 by the Arch BVQ, and the right Line BQ 
 
 C O R O L. VIII. Fig. 10. 
 
 If you let the Quadrant be AT MB, and the 
 Fig. homogene of the Fig. of Sines ATNB, 
 the Reftangle AG, A T divided in the middle in 
 
 Cs 
 
Superficies and Solids . 41 
 
 C , CF Parallel to AB, the right Line begat from 
 the four Segments CDEOF are equal: For let that 
 begat from CF be 16, that begat from CD will be 
 4 * from CE, 8 ; from CO will be 12 ; therefore from 
 DE, 4> from EO,4; from OF, 4: And therefore 
 equal. 
 
 C O R O L. IX. 
 
 If you draw two Lines RP, LH Parallels to EF, 
 and equidiftant from the lame, fince that begat from 
 RS and IH are equal, for AL, RT, RS are 
 equal, likewife thofe begat from LK, MP will be 
 equal to the begat from R M and K H : And by 
 fubtrafting the Equals begat from RS and IH, the 
 Remainers will be equal, to wit, the begat from SM, 
 KI : The fame may be demonftrated from any 
 other being alTumed : Hence from the Cone begat 
 from ATB being fubtrafted from the Hemifphere, 
 the Remainer will be divided in the middle by the 
 Plane CF. 
 
 C O R O L. X. Fig. 9. 
 
 If AL, OB are alfumed equal, the Cylinder un- 
 der theBafe from the Circle from AC, and from the 
 Altitude AL, is equal to the begat from the right 
 Segment OBR, and from the Trapezium ALVC, 
 to wit, the bespat from the Triline C V M is equal 
 to that begat from the Segment OBR. 
 
 C O R O L. XI. Fig. 9, 
 
 Hence that begat from LVRO is to the Rem- 
 nant as N L to L A : Hence may be had a mean 
 
 Fruf- 
 
420 MEASURING 
 
 Fruftum in any given Ratio to the whole 3 if the fef- 
 quitertia Ratio be required, let NL be to N A, as 
 $ to 4, the begat from LVRO will be to the be- 
 gat from the Fig. ABC, as 3 to 4, that is NL to 
 NA. 
 
 C O R O L. XII. 
 
 Laftly, hence that begat from OBR, and CMV 
 are equal, likewife that begat from ALVC, and 
 from BR S , alio the begat from RTY and from 
 LV0T are equal, 
 
 PROP. V. 
 
 Any right Segment of a Fig. of Sines is to the 
 whole Fig. as the verfed Sine of the Arch of it, to 
 which the Altitude of the Segment is equal to the 
 whole Sine : For Jet any. Segment be NBP whole 
 Altitude NB is equal to the Arch DH, and there- 
 fore F1F equal to N P : I lay the Segment NBP is 
 to the whole Fig. ABC, as the verfed Sine of DFI 
 to the whole Sine D A : For lince the Fig. ABC is 
 homogene to the Superficie of the Hemilphere, this 
 will be divided in the Ratio of the Segments of the 
 Radii, by Pofition 5. (for Example) If you revolve 
 the Arch DC about DA, the Superficie begat by 
 the Arch DH, is to that begat from the Arch D C, 
 as the Segment NBP to the Fig. ABC : But that 
 begat from the Arch D H is to that begat from the 
 Arch DC, as DF to DA by Pofition 5. Therefore 
 the Segment NBP is to A B C, as D F the verled 
 Sine of the Arch DH, equal of the Altitude of the 
 
Superficies and Solids. 421 
 
 Segment NB to D A the whole Sine. The fame 
 may be demonftrated in any other Segment. 
 
 C O R O L. I 
 
 Hence may be given the Segment which is to the 
 whole Fig. in any given Ratio, (for Example) let 
 the Ratio given be of DE to DA, let OR be ap- 
 plied equal to EG, the Segment OBR will be to 
 the whole ABC, as DE to DA : Hence if a Seg- 
 ment be required which (hall be one quarter of the 
 whole, let DF (for Example) be one quarter of 
 DA, let N B be applyed equal to F G ; N B P wih 
 be to ABC, as 1 104: Hence if A T be half of 
 AB, T B 8 will be one Third of ABC ; and there- 
 fore*!’ 9 will divide the Fig. of Sines equally. 
 
 C O R O i; II. 
 
 Hence the right Segment is to the whole Fig. as 
 the Applved to the Triline of the Fig. conjoined 
 equally diftant from the Bafe of the Triline, alio 
 the Bale of the Segment from the Bafe of the Fig. 
 OA will be equal from the Bafe of the Triline 
 SM; I lay the Segment OBR is to the whole 
 ABC, as the Applyed to the I nline V M to the 
 Bafe B S : For let the Sine E G or Z I be equal to 
 OR, and therefore OB equal to the Arch DG or 
 CT, the Segment OBR is to the whole ABC, as 
 DE or ZC, to DA or AC: But LV is equal to 
 AZ or AE ; therefore VM is equal to ZC; there- 
 fore O B R is to A B C, as VM to BS : The fame 
 may be demonftrated concerning any other Segment. 
 
 PROP. 
 
 t 
 
MEASURING 
 
 422 
 
 PROP. VI. Fig. 11. 
 
 The Ratio may be defined of the verfed Sine to 
 the whole Fig. of Sines : Let the Fig. of Sines be 
 API, the Quadrant AMT, the Triline annex’d 
 PVT, let AK be equal to A T, and therefore the 
 Quadrate AN ; let any verfed Segment be B R T, 
 draw LR indefinitely, let LR be equal to IF ; laft- 
 ly, through the Point X, in which BR the Axis of 
 the Segment will cut KN the fide of the Quadrate 
 AN, draw AXS from S, let fall SC parallel to the 
 Axis RB ; fince the whole Fig. APT is to the right 
 Segment LPR, as AT to A C, it will be to the 
 Remnant, to wit, to the verfed Segment BRT as 
 AI to Cl ; therefore the Ratio fought is found. 
 
 C O R O L. I. 
 
 Hence the whole Fig. is to each Segment ART, 
 LPR, as AT toCT. 
 
 C O R O L. II. 
 
 Hence the whole Fig. is to the verfed Segment, as 
 the whole Sine to the Bafe of the Segment lefs by 
 the Compound, whereby the whole Sine exceeds the 
 right Sine of the Arch equal to the Axis of the Seg- 
 ment, and from the Line which is to the Excels, 
 whereby the aforefaid Arch exceeds the Radius, as 
 the Sine of the Complement of the aforefaid Arch 
 to the Radiu's. 
 
 PROP. 
 
Superficies and Solids . 4M 
 
 PROP. VII. Fig. 12. 
 
 If a Fi". of Sines be turned about a Parallel to 
 the Axis ere&ed in the Extream of the Bafe, that 
 begat from the Fig. will be to the Cylinder of the 
 lame height under the Circle of the Bale, to thai 
 begat from the Fig. of the Bafe, as the Compound 
 from the Radius, and the Excefs whereby the Radius 
 exceeds half the Axis to the whole Axis : For let the 
 Fig. of Sines be AifC infilling perpendicularly to 
 the Plane of the Quadrate AE, and let the Solid 
 proceed by the Squares AE, N I, fs’c. under the 
 Bafe AE, and the Altitude AB, and therefore ho- 
 mogene to that begat from the Fig. ABC turned 
 about the Axis AB, it will make the Ifoparal- 
 lel under the Bafe ABC, and the Altitude A D, 
 to wit, proceeding by the equal Planes ABC, 
 D K E : Let A X be equal to A C, Iikewife A N 
 equal to B N, and X V equal to X N ; the Ifopa- 
 rallel aforefaid will be to the Parallelipiped A F, to 
 wit, under the Quadrate AE, and the Altitude AB, 
 as ABC to the Re&angle A L • for they are ho- 
 mogene Ifoparallels, therefore if they lhall be of 
 the fame Altitude, they are as their Bafes by Pofi- 
 tion 1 ; therefore as AX to AB, the aforelaia 
 homogene Solid proceeding by the Squares AE, 
 NI, is to the Parallelipiped AF, as AN to AB, 
 by Prop. 4. therefore it is to the aforefaid Ifoparal- 
 lel, as AN to AX ; therefore to the Difference of 
 each, as AN to N X : And by taking away from 
 the Parallelipiped A F the Solid proceeding by the 
 Squares B F, I H, to wit, under the Applicates to 
 the Triline KFE ; and by taking from the Relidue, 
 
424 MEASURING 
 
 the aforefaid Homogene proceeding by the Squares 
 AE, NI, under the Applicates of the Fig. the Re- 
 fidue will proceed by Supplements parallel to the 
 Gnomons IG, IS : For as from the Square NH 
 1H being taken away, and from the Refidue NI be- 
 ing taken away, there will remain the Supplement 
 of the Gnomon IG, IS : So it may be done in any 
 other Square aflumed j and fince IG is equal to IS, 
 and that from any other Quadrate alfumed, the Ex- 
 cels whereby the Refidue of the Parallelipiped by 
 taking away the Solid, proceeding by the Squares 
 BF, 1H, will exceed the Ifoparallel under the Bafe 
 ABC, and the Altitude AD, equal to the Excels 
 whereby the aforefaid Ifoparallel exceeds the afore- 
 faid Homogene proceeding by the Squares AE, 
 NI : Therefore this Homogene it folf is to the 
 aforefaid Ifoparallel as AN to AX, and to this the 
 Difference of both being added, as AN to A V ; 
 laftly to the Solid, proceeding by the Squares B F, 
 IH, as AN to VB. 
 
 But fince the Solid proceeding by the Squares BF, 
 IH, is homogene to that begat from the Triline 
 KFE, or BLC turned about LC ; this begat will 
 be to the Cylinder of the fame Bale and Altitude, 
 asBV to BA; therefore that begat from BAC, 
 turned about CL, will be to the fame Cylinder, as 
 A V to A B ; therefore as the Compound from the 
 Radius AX and XV, to wit, the Excels whereby 
 the Radius AX exceeds AN the half of the Axis 
 AB, for XV is equal to XN. 
 
 COROL 
 
4*5 
 
 Superficies and Solids . 
 
 CORO I.. I. 
 
 v » - ;■''✓» nt oj .-j.'.'i'.- -.»'J i y k 
 
 ~ Thefe may eafify be reduced to Calculation ac- 
 cording to the Cyclometry of Archimedes , for A N 
 will be to AX as n to 14 $ and to A V as. n to 
 I 5Q and to V B as 1 1 to 5 * Henceif the Cylinder 
 begat from AL turned about CL be 22, that begat 
 from the Triline BCL will be $, and from the 
 Fig. AB 17 * and the Parallelipiped AF, to th£ Ifo- 
 parallel iunddr A B C and A D as 22 to 14. 
 
 C O R O lim IL 
 
 If between NGj NP, there be a mean Propor-^ 
 t tonal, and alio between the Sides of other Rect- 
 angles Parallels of NM: The inlid Proceeding by 
 the. Squares 'under the aforefaid mean Proportionals, 
 is equal to the Ifbparallel under ABC and AD: 
 But if from the Applicates ordinately applied to 
 BA, by the aforefaid mean Proportionals, the Fig. 
 turned about AB, that begat will be to the Cylinder 
 of the fame Bafe and Altitude as AX to AB, that 
 is, as 14 to 22. 
 
 PRO P. VIIL Fig, 13* 
 
 If the Fig. of Sines is CAE, and the other com- 
 mon Axis EAB, and let the folid Ifoparallel under 
 E A B, and the Altitude AT be equal to A E : And 
 laftly, the Solid be cut by the Plane BIE, the 
 lower Fruftum RIQB is to the whole Solid, as the 
 Radius CE to the Axis CA, but the upper Fruf- 
 tym, as the Difference of the Axis, and of the Ra~ 
 
4 16 MEASURING 
 
 dius to the Axis , and to the other Fruftum as the 
 aforelaid Difference to the Radius : We will de- 
 monftrate it ; let IT be equal to the Radius E C, 
 and let the Fig. of Sines be T O I, and the accom- 
 panying Triline OAI, thefuperior Fruftum E ABI 
 will proceed by right Segments the Parallels BAE, 
 NDM, &c. And BAE is to NDM, as AO to DF, 
 by Coroll. 2. of Prop. 5. The fame may be demonftra- 
 ted concerning any otherSegment • therefore the Fruf- 
 tum E ABI is Homogene to the Triline AOI , there- 
 fore as the Triline AOI to the Reftangle AT, lo the 
 Fruftum E ABI to the Iloparallel under the fame Bale 
 E AB, and the Altitude AI : But the Recftangle AT 
 is to the Triline A O I as the Axis A I to the Diffe- 
 rence of the Axis AI or AC, and of the Radius CE : 
 Therefore the aforelaid Ifoparallel is to the afore- 
 faid Fruftum in the lame Ratio, therefore to the 
 Fruftum RIQB as the Axis to the Radius ; there- 
 fore the Fruftum R I QB, to the Fruftum E A B I, 
 as the Radius to the aforelaid Difference. 
 
 C O R O L. L 
 
 Hence the Fruftum RIQB, lb far as it proceeds 
 by right Trapeziums, parallels to the Plane RIQ, 
 ^tis Homogene of the Fig. of Sines TO I ; to wit 3 
 as TI to * F, fo RlQtt) ZMNV. 
 
 CORO L. II. Fig. 9* 
 
 Hence any right Trapezium is to the whole Fig^ 
 as the Applicates of the Fig. equally diftant from 
 the Bale of the Fig. and of the Bale of the tight 
 Segment joined to the Trapezium, diftant from the 
 
 V ertex. 
 
Superficies and Solids . 427 
 
 Vertex of the Fig. to the Radius; (for Example) 
 let the right Segment be OBR, let L V be applied 
 equally diftant from AC, as OR from the Vertex 
 B, the Trapezium AORM will be to the whole 
 ABC, as LV to AC; becaufe fince the whole 
 ABC is to OBR, as LM to VM, it will to the 
 Remnant, <viz. AORC, asLMtoLV. 
 
 COROL. III. Fig. 13. 
 
 Hence as the Re&angle T A will be divided by 
 the Curve OIF I, fo the aforefaid Ifoparallel by 
 the Plane BIE ; therefore according to Archimedes $ 
 Ratio, the upper Fruftum is to the lower, as 4 to 
 7 ; but to the whole as 4 to 1 1 ; but the lower to 
 the whole, as 7 to 11. 
 
 PROP. IX. 
 
 The Center of Gravity of the whole Fig. of 
 Sines, or of the double, with the common Axis, 
 will divide the Axis fo, that the Segment towards 
 the Vertex of the Fig. is to the whole Axis, as 
 the Radius to the Arch of the Quadrant : Let the 
 Fig. of Sines be CAE, with its lociate C AB, the 
 common Axis AC, fo divided in K, that the Seg- 
 ment K A be made to the whole C A, as the Ra- 
 dius to the Arch of the Quadrant ; K will be the 
 Center of Gravity of the Fig. E AB ; that it will 
 be in the Axis AC, appears, fince AC divides in 
 the Middle all the Applicates EB, HG, &c. which 
 is in K, fhall be demonftrated : Let fall K X paral- 
 lel to A I divided in the Middle in S, and draw 
 S d' parallel to AC; draw I SC ; thefe are pa- 
 rallel, in I is the Center of Gravity of the Fruf- 
 
 Ff 2 turn 
 
4*8. MEASURING 
 
 turn RIQB, becaufe it will pafs by the Center of 
 Gravity of all the Re&angles of the Parallels E R : 
 Alfo in SC will be the Center of Gravity of E A B I 
 becaufe it pafles through the Center of Gravity of 
 all the rebbangled Parallels HA. For 'Example, let 
 the Center of the Fruftum EABI be . in P, and let 
 the Point Y be in which JS J cuts KX, and draw 
 PY 9 ; fince the Triangles SPY, cM Y are Propor- 
 tional, it will be as PY to 6 Y, fo S Y to Y ^ that 
 is, A K to KCj but P Y is to Y L as the Fruftum 
 R'lQR to the Fruftum EABI f for the Diftances 
 are as the Weights by exchanging : Therefore AK* 
 is to KC as the lower Fruftum to the upper, there- 
 fore to AC as to the whole j therefore as the Ra- 
 dius to the Arch of the Quadrant. Now this hath 
 been demonftrated in general ; concerning each Ifo- 
 parallel you may confult Torricelli us with Cavelier ins 
 at Exercife the 5th, 'Proportion the 17th. 
 
 C O R O L. I. 
 
 Hence the Center of Gravity of the Ifqparallel 
 will be Y ; becaufe KX will be drawn through the 
 Centers of all the parallel Planes BAE, RIQ, there- 
 fore it will be in KX: Moreover, will paft 
 through the Center of Gravity of all the parallel 
 Planes of BQ; therefore it will be in S therefore 
 in Y. : 
 
 C O R O L. II. 
 
 Hence the Center of Gravity of the Fig. of Sines 
 CAE, is in the right Line K H, as appears : Like- 
 wile ,of the Fig. CAB in the right Line KG, to 
 
 wit. 
 
Superficies and Solids. 4 ip 
 
 wit, the Center of both equally diftant from EB, 
 or from the Vertex A, fince the whole EAB is Ho- 
 mogene of each afunder. 
 
 . , . ' • - 
 
 ! jT" iL I v> t • i «• / - V. ’ - > 4* ' M ‘.J I * • •. 
 
 PROP. X. Fig. 14. 
 
 The Center of Gravity of the Fig. of Sines is dif- 
 tant from the Axis one quarter of the Axis, and 
 from the Bafe, by the Excefs it felf, whereby the 
 Axis exceeds the Bafe : Let the Line of Sines be 
 AMC, with the Redlang. CY , the Moments of 
 both Figures librate in CM, they are as the Solids 
 begat from the Figures turned about CM, by Po- 
 fition the 9th. And that begat from CY is double 
 of that begat from ACM by Prop. 4. but the Ra- 
 tio of Solids begat from* Figures is compounded 
 from the Ratio of the Figures, and from the 
 Ratio of the Diftances of the Center of both 
 from the common Axis by Pofition 9 ; therefore 
 the Moment of the Re&angle CY is to the Mo- 
 ment of the Fig. CMA, in the Compofition from 
 the Ratio CM to CO, or its equal MD, \yhich 
 is the Ratio of the Figures, and from the Ratio of 
 EQ, or ST, the half EX ; for this is the Diffance 
 of the Center Q_from the Axis CM, to the Diftance 
 of the Center of the Fig. CMA from the fame Axis 
 CM; that is, the Moment of the Reftangle is to 
 the Moment of the Fig. as the Redbing. under CM, 
 CT to the Subduple Redtangle, under ohe of the 
 Sides equal to CO, and- under the other, which may 
 eafily be found by drawing OR, alfo CR, PVS; 
 for CP is equal to CQ, and EV is one quarter of 
 MC ; for fince EP, C() are equal, as CO to CE, 
 the half of CM, lo CT, the half of CO to CS, the 
 • : ■; O ■ F f 3 ' half 
 
4^0 MEASURING 
 
 half of CE ; therefore CS or EV is half of CE ; 
 therefore one quarter of CM ; therefore CS is the 
 other Side of the Re&ang. fought, therefore Yis the 
 Diftance of the Center of Gravity of the Fig. CMA 
 from the Axis MC. Laftly^ fince CD is the Ex- 
 cels whereby the Axis CM exceeds the Bate CA, 
 by drawing DF parallel to the Bafe, and afliiming 
 DF equal to CS, the Center of Gravity of the Fig. 
 CMA is in F. 
 
 C O R O L. I. 
 
 Hence fince ME is to DE, as 4 to 1, that is, as 
 the Rectang.CY to the Reftang. under one quarter 
 of CM, and under CA, that is,* as the Semicircle 
 to the Semiquadrant, CA will be to DF as the 
 Square of the Radius to the Semiquadrant : There- 
 fore equal weighing being made, and afiuming the 
 Weight to the Arm CX, with the Perpendicular 
 CM, the Semiquadrant will equiponderate the Fig. 
 CMA, for the Moments are equal, viz. compound- 
 ed from the Ratio of CX to CY, and from the 
 Ratio of CY to CX. 
 
 C O R O L. II. 
 
 Hence is had the Center of the Triline MKA, 
 viz. by drawing EG equal to EQ, and through G 
 draw FGH : It will be as CD to DM, fo FG to 
 GH * H will be the Center fought, and the thing 
 may eafily be reduced to Calculation ; let CM be 
 11, CA will be 7, likewife CO, MD will be 7, 
 therefore CD will be 4, DE will be 1 \ ; EV 2 
 NM, or IK, 2 1 ; LK will be 2 
 
 COROL. 
 
Superficies and Solids. 43 1 
 
 COROL. III. Fig. 13. 
 
 Hence the Center of Gravity of each Fruftum of 
 the Ifoparallel in Prop. 8. may be found $ for fince 
 the upper Fruftum EABI is Homogene to the Tri- 
 line OAI, by Prop. 8. the Center of the Fruftum 
 will be equally diftant from the Bafe EAB, as the 
 Center of the Triline from the Bafe AO : And 
 thefe Diftances are had by Coroll. 2. let it be AD, 
 draw DP parallel to AC $ I fay, the Center of 
 the Fruftum is in P , for fince 3 tis in CS, alfo in 
 DP, it will be in P,' in the Point of the common 
 Seftion. In like manner, the Center 9 of the other 
 Fruftum is had, as well by drawing the right Line 
 P 0 through Y, as alfo by the fame Method, fince 
 this Fruftum is Homogene of the Fig. of Sines TIO^ 
 by Coroll. 1. Prop. 8. 
 
 COROL. IV. 
 
 Hence may be known the Quantity of the Line 
 DP , for fince AC is to AS, as DP to DS ; and 
 fince AC is double of AS, DP will be double of 
 DS : Therefore by putting the Axis AI to be n, 
 DS will be 2 {, and confequently DP will be 5 ^ 
 becaufe ? tis the Double of 2 -f-, or of DS. 
 
 PROP. XL Fig. 14. 
 
 If the Fig. of Sines be turned about the 
 that begat from it is to the Cylinder of the fame- 
 Bafe and Altitude, as the Re&ang, under the Bafe, 
 
 p f 4 and 
 
4 3~ MEASURING 
 
 and the Excels whereby the Axis exceeds the Bafe; 
 to the Redtang. under the Axis and half of the Axis • 
 for let the Fig. be CMA, the Redfang. €Y . turn 
 both Figures about AX, the Solids begat are in 
 Compofition from the Ratio of the Figures, which is 
 CO to CM, and from the Ratio of the Diftances 
 of the Center of both From AX, which is CD to 
 CE by Pofition 9. and Prop. 9. Therefore that 
 begat from CMA is to that begat from CY, as the 
 Rectang. under CO, CD, to the Redlang. under 
 CM, CE. Therefore as the Redtang. under the 
 Bafe, and the Excefs whereby the Axis exceeds 
 the Bafe, to the Redtang. under the Axis, and half 
 the Axis. 
 
 C o R o L. I. 
 
 Hence is had that begat from the Triline MAK » 
 for by taking away that begat from CMA bein» 
 known, from that begat from CY being known’ 
 the Refidue will be the begat from the Triline 
 MKA alfo known ; and that the thing may be re- 
 duced to Calculation, let that begat from CY be 
 121, that begat from CMA will be 56 ; therefore 
 that begat from the Triline MR A will be 65. 
 
 C O R O L. II. 
 
 _ And that we may find that begat from the Fig. 
 CMA turn’d about MK, let that begat from the 
 Fig. CMA turn’d about CA, be an Homogene So- 
 lid under the Square of the Bafe CM, and the 
 Altitude CA be called B : Let the Parallelipiped 
 df the fame Bafe and Altitude be called C : Let the 
 
 Ifoparallel 
 
Superficies unci Solids. 43 ? 
 
 Ifoparallel under the Fig- of the Baft CMA, and 
 the Altitude CM be called D ; then C will be to 
 D.as MC to MD : Add toD the Excefs whereby D 
 exceeds B ; C will be to the Aggregate from D, and 
 the aforelaid Excefs, as that begat from CY to that 
 begat from the Fig. CMA turn’d about MK; 
 And by taking away this begat from that, the Re- 
 fidue will be the begat from the Triline MKA 
 turn’d about MK. Theft may be demonftrated in 
 the fame Manner as is demonftrated at Prop. 7 • but 
 fhorter. That begat from MA is to that begat 
 from MCA turn’d about MK, as the Reefing, 
 under MC, ME to the Re&ang. under MD, MD ; 
 that is, as 121 to 98. 
 
 PROP. XII. Fig. 15- 
 
 If you let any right Segment of a Fig. of Sines 
 be of the fame Altitude with the verfed Segment 
 of the fame, it will be to the whole Fig. as the Baft 
 of the verfed Segment to the Bafe of the whole 
 Fig. For let the Fig. of Sines be ARB, the Qua- 
 drant ABD ; the right Segment ILR, the verfed 
 Segment of the fame Height CNB : I fay, ILR 
 is to ARB, as CB to AB ; for fince MN and 
 AC are equal, alfo AC and TC ; FS will be 
 equal to IL ; becaufe IA is equal to MR, there- 
 fore IR will be equal to the Arch DS ; therefore 
 the Segment IRL is to the whole Fig. ARB, as 
 TD equal to VF, or CB to the whole AC, by 
 Prop. 
 
 COROL. 
 
434 MEASURING 
 
 C O R O L I. 
 
 Hence the right Trapezium AILB is to the 
 whole ARB, as the right Sine of the Arch equal 
 of Altitude to the Trapezium, ’viz . AI to the 
 whole Sine AB. 
 
 C O R O L. II. 
 
 Hence the whole Fig. ARB is to the Segment 
 MRN, as AB to SX ; but to the Segment 1 LR, 
 as AB to TD : Likewife to the Trapezium AMNB, 
 As AB to TS ; and to the Trapezium AILB, as 
 AB to AT. ’ 
 
 PROP. XIII. Fig. 1 6. 
 
 The Solid under the Altitude of the Bale of the 
 Fig. of Sines proceeding by right Segments under 
 the Altitudes ordinately applied, parallel to the 
 Axis, is Homogene to the Triang. Let ABC be 
 the Fig. of Sines, for Example, theBafe 5 let alfo the 
 Fig. of Sines be ABD under the right Angle BAD, 
 and under the ordinate Applicates Parallels BA, 
 OH; let the right Segments, as OHG be equal 
 to EAF : I fay, this Solid under the Bale ABC, 
 and the Altitude AD proceeding by the aforefaid 
 right Segments, of which the Vertexes are termi- 
 nated at the right Line AD, is Homogene to the 
 Triang. (for Example) ASD. For ABC is to AEF, 
 or HOG, as AD to HD, or AS to HK, by Prop. 
 12. The fame may be fhewn by affuming any 
 other right Segment. Therefore the whole Solid 
 ABCD is Homogene to the Triangle ASD. 
 
 COROL. 
 
Superficies and Solids. 435 
 
 C O R O L. I. 
 
 Hence the aforefaid Solid ABCD is half the 
 Ifoparallel under the Bafe ABC, and the Altitude 
 AD: For as the Triangle ASD is to theSquaie 
 A I, fo is the Solid ABCD to the Ifoparallel 
 
 ab’cd. 
 
 C O R O L. II. 
 
 Hence the Solid, which proceeds by Re&angle 
 Parallels under the right Sine, and of the Comple* 
 ment, being applied to AB in the two Figures 
 ABC, ABD is equal to the former Solid. For let 
 there be any Applicate EO, to which another EF 
 approaches at a right Angle OEF ; EO is the 
 right Sine of the Arch equal to EB • and EF trie 
 right Sine of the Arch equal to EA , therefore tis 
 the Sine of the Complement of the firft Arch ; 
 therefore the Reclang. E. F, is under the right Sine, 
 and of the Complement : The fame may be Ihewn 
 by affuming any other right Angle. Therefore that 
 Solid, which proceeds by the Rectangles of this 
 fort is equal to the Solid ABCD ; therefore Sub- 
 duple of the Ifoparallel under the Bafe ABC, and 
 the Altitude AD. 
 
 PROP. XIV. Fig. 17. 
 
 The Solid under the Bafe from the generating 
 Quadrant, and from the Altitude of the Axis of 
 the Fig. of Sines, proceeding by Sectors parallel to 
 
 the Bale, which lhall be in the Ratio of Altitudes, 
 
 is 
 
436 MEASURING 
 
 is Homogene to the Triang. Let the Quadrant be 
 ALC, the Altitude AF, equal to the Arch of the 
 Quadrant CL, and let any Sedov be GID, un- 
 der the Arch ID, equal to GF, fi nee the right 
 Sine of it is GH ; I fay, the aforefaid Solid is Ho- 
 mogene to the Triang. AFC' .r Becaufe the Sedor 
 ACL is to the Sedor GID, as the Arch CL to 
 the Arch DI ; that is, as AF equal to the Arch 
 CL to GF, equal to the Arch DI ; that is, as AF 
 equal to the Arch CL to GF, equal to the Arch 
 DI, that is, as AC to GO; therefore the Solid, 
 and the Triangle proceed by proportional Elements, 
 therefore they are Homogene. 
 
 COROL L 
 
 Hence the aforefaid Solid, proceeding by Parallel 
 Sedors, is one Eighth of the Cylinder under the 
 Bale of the Circle, whole Radius is AC, and un- 
 der the Altitude AF. 
 
 COROL IT. 
 
 Hence, if a Solid happens of the fame Altitude 
 AF, proceeding by Redangle Triangles, under 
 the Bale and Altitude, from the right Sine and 
 the Complement, it will be \ of the Ifoparallel 
 Solid, whofe Bafe is the Fig. of Sines, and the Al- 
 titude the fame of the Fig. of the Safe ; {(or Ex- 
 ample) let the Triang. be GHD under GH the 
 right Sine of the Arch DI, or KL equal to GF, 
 and. under HD, equal to BR, the right Sine of the 
 Arch KC equal io GA. Therefore GHD is under 
 the right Sine, and the Sine Complement : The famo 
 
 may 
 
Superficies and Solid*. 4 37 
 
 may be (hewn in any other alfumed. And fince this 
 Solid proceeding by the aforefaid Triangles, is Sub- 
 duple of the Solid proceeding by Rectangles under 
 the fame right Sine, and the Sine of the Comple- 
 ment; for they are Homogenes, and of the fame 
 Altitude, and every . Rectangle Is double of its 
 Triangle. And fince, the Solid proceeding by the’ 
 aforefaid Redangle is fubduple of the Iloparallet 
 Solid under the Safe of the Fig. of Sines, and the 
 Altitude the fame as the Fig. from the Bafe, byi 
 Coroll. x. of Prop. 13. the aforefaid Solid preced- 
 ing by Triangles will be \ of the aforefaid Ifo- 
 parallel. 
 
 CORO L HI. 
 
 ■ Hence is had the whole Solid A. CL, 1 'iZ- tne 
 Aggregate from the' Precedents by the Sedors, 
 arid from the Precedents by the Triangles, for the 
 Precedents by the Triangles is 4 of the afore- 
 faid: Ifoparallel, which is equal to the Cube un- 
 der the Side AC; Hence the Ratio’s may be re- 
 duced to Calculation, let the Parallelipiped under 
 the Square of the Bafe AC, and the Altitude AF 
 be 14, it will be to the Cube under the Side AC 
 as 14 to 8 ~ ; therefore to tneProcedents by Tri- 
 angles as 14 to 2 tt ; therefore to the Quadrant, 
 or Fourth of the Cylinder, under the Bafe from 
 the Quadrant ALC, and the Altitude AF, as 14 
 to 11 ; therefore to the Procedents by the Sedors, 
 as 14 to 5 rf ; therefore to the Aggregate afore- 
 faid, as 14 to 7 t 
 
 CORO L 
 
 1 
 
438 MEASURING 
 
 C O R O L. IV. 
 
 Hence, if from the aforefaid Aggregate be taken 
 away the Solid proceeding by parallel Squares AI 
 under the Altitude AC, which is equal to | of the 
 Cube, and therefore 5 f| of the Refidue j that 
 which proceeds by Redangles parallel to GE will 
 he 1 which if it be added to the Solid AEC, 
 the Aggregate will be 9 ff. 
 
 C O R O L. V. 
 
 Hence, if there be a Parallelipiped under the 
 Altitfide AC, and the Square of the Bafe AF, it 
 will be to the aforefaid Parallelipiped in the 3d 
 Coroll, as 22 to 14 j fince the Parallelipiped under 
 the Square of AF, and the Altitude AC, is to the 
 Solid, which proceeds by the Squares under AF, 
 BH ; that is, under Arches of the fame Altitude, 
 as 121 to 56, by Prop. 11. Coroll. 1. that is, as 22 to 
 10 tt? if from 10 we take away the Aggregate 
 9 yjj as in Coroll. 4. the Refidue will be f-|, that 
 is, the Solid under the Altitude AC, proceeding by 
 Squares, under the Differences of the Sines, and of 
 the Arches, which is to the Solid, proceeding by 
 the Squares under the Sines, as 1 1, to 98. 
 
 C O R O L. VI. 
 
 The Solid AEC proceeds by Re&angles parallels 
 of AE, under the Arches, and right Sines : So 
 AE is under AL the whole Sine, and AF equal 
 to the Arch of the Quadrant. Alfo BD under BH, 
 
 equal 
 
Superficies and Solids. 439 
 
 equal to the Arch KC, and BK the right Sine of 
 the Arch CK : The fame may be Ihewed in any 
 other affumed. 
 
 COROL VII. 
 
 Hence if a Solid be made of the Altitude AC, 
 proceeding by ReCtangles under the Arches, and 
 under the Compounds from the Arches and the 
 Sines, from which take away a Solid of the fame 
 Altitude, proceeding by Squares, under the Arches* 
 and which we appoint at Coroll. 5. to be to the 
 Parallelipipcd of the fame Bale and Altitude AC* 
 as 1077 to 22 : Without doubt the Solid remain- 
 ing of the lame Altitude AC will proceed by Rect- 
 angles under the Arches, and the Sines 3 therefore 
 5 tis equal to the Solid A£C: Therefore fmee AEC 
 is 7 7-7-3 the aforefaid Solid, proceeding by Rectan- 
 gles under the Arches, and the Compounds from 
 the Sines, will be 17 . to which if we add the 
 
 Solid prQceeding by the Squares of the Sines, like- 
 wife proceeding by the ReCtangles under the Ar- 
 ches and the Sines, the Aggregate will be 31 
 and this is the Solid of the Altitude AC proceed- 
 ing by the Squares under the Compounds from the 
 Arches and the Sines. Allb the Parallelipiped is 
 had under the Altitude AC, and the Square of 
 the Bafe under the compounded from the Arch 
 of the Quadrant, and the Radius ; for it is to the 
 Parallelipiped of the fame Altitude under the 
 Square of the Bale AF, as 324 to 121, or as 
 58 to 22. 
 
 PROP, 
 
 \ 
 
440 MEASURING 
 
 PROP. XV. Fig. 1 8 . 
 
 If the Fig. of Sines be doubled or whole, they 
 ■will refemble two Trilines joined by the Bales, the 
 whole Fig. is Homogene to the Solid begat from 
 the Fig. of Sines turned about the Axis; for ERM 
 will be whole or double of the Fig. of Sines, to 
 which the two TrijinefcRDR, ADR are alike, be- 
 ing joined by the Rales in AR : I lay, the Fig. 
 EARM is Homogene to that begat from the Fig. 
 of Sines (for Example); of EAM turn’d about EA ; 
 to wit, as EM to DT, lo DT to DR ; and ac- 
 cepting DF, DC equal ; as EM to CQ, the right 
 Sine of the Arch equal to CA, fo CQ^ to CX, the 
 vcrfed Sine of double the Angle in the Quadrant, 
 whofe Arch is equal to AD, the Half of AE ; and 
 as EM to FO, the right Sine of the Arch equal to 
 FA, lo FO to FZ, of which alfo FY the Diffe- 
 rence is ZY equal to CX, by Pofition the 6th: 
 But as EM to DR, lo that begat from EM to 
 that begat from DT ; and as EM to FZ, fo that 
 begat from EM to that begat from FO ; and as 
 EM to CX, fo that begat from EM to that begat 
 from CQ, for they are begat as the Squares ; there^ 
 fore the Fig. EARM is Homogene to that begat 
 from EAM turn’d about EA. 
 
 C O R O L. I. 
 
 Hence EARM is fubduple of the Redan. ES, 
 becaule the aforefaid begat is Homogene ; yea, 
 fince the Triline ADR is equal to the Triline EDR, 
 the Rectang. EK will be equal to the Fig. EARM, 
 - > r and 
 
Superficies and Solids. 441 
 
 and fince this is Homogene to the aforefaid begat, 
 from thence alio it follows, that the aforefaid be- 
 gat is fubduple of the Cylinder of the fame Bafe 
 and Altitude. 
 
 C O R O L. It 
 
 Hence the Center of Gravity of the Fig. FARM 
 is had • for fince the Center of ERM is had, fup- 
 pofe 7, and of the Triline AER, fuppofo ^ • draw 
 7^, and fo divide it in H, that H y be to H as 
 the Triline ERA to ERM, that is, according to 
 Archimedes's Ratio, as 7 to 43 H will be the Cen- 
 ter of Gravity of that you feek, as appears from 
 the aforefaid. 
 
 C O R O L. III. 
 
 Hence by ufing Calculation, and by drawing yir^ 
 H 0 ; E tt is to ED, as 4 to 1 1, and to EA, as 4 to 
 22 i and 7 r 9 to 9 D, as 4 to 7 : Therefore let Dt 
 be ii, and Gtt will be 4 j E tt 6 ED 17 4 7 EA 
 34 4 : Therefore E 0 to 9 A, as 10 £ to 24 
 
 C O R O L. IV. 
 
 The Center of Gravity of the begat from the 
 Fig. of Sines EATM turnd about AE is 0, for 
 the aforefaid begat is Homogene of the Fig 
 EARM ; therefore the Center of both will be 
 equally diftant from the Bale. 
 
 G g 
 
 PROP 
 
MEASURING 
 
 44 * 
 
 PROP. XVI. Fig. 19. 
 
 If we turn about the Axis Plane Figures Ho- 
 mogene to the begat from the Fig. of Sines turned 
 about the Axis, the Ratio is had of the begat to 
 the Cylinder of the fame Bafe and Altitude , for 
 let EARM be Homogene to the begat from the 
 Fig. of Sines, as aforefaid, turned about EA $ that 
 begat from it, is to that begat from ES in the Ra- 
 tio compounded from the Ratio of the Fig. EARM 
 to E S, that is, half 5 and from the Ratio of I H to 
 DR : For I fuppofe in H to be the Center of Gra- 
 vity of the Fig. EARM, for the aforefaid Solids 
 are as the Moments of each Fig. vibrating in E A, 
 by Pofition 9 ; and the Moments are in the afore- 
 faid compounded Ratio, as hath been already oft- 
 times repeated. 
 
 C O R O L. I. 
 
 Hence is found the begat from ERM $ now be- 
 caufe we have the begat from the Triline EDR, 
 by Prop. 7 ; therefore equal to this is the begat 
 from the Triline ADR, both being taken away 
 from the begat from EARM being known, the 
 Refidue will be the begat from E R M. This may 
 alfb be demonftrated otherwife ; for the begat from 
 ERM is to the Cylinder of the fame Bafe and Al- 
 titude, as the Ifoparallel under the Bafe ERM, and 
 the Altitude E M, to the Parallelipiped under the 
 Bafe EP, and the Altitude EM, by Pofition 7. 
 
 COROL 
 
Superficies and Solids. 
 
 443 
 
 COR 0 L. II. 
 
 Hence alio is had that begat from MPRj to 
 wit, by taking away the begats from ERM, and 
 from EDR known, from the begat from EP 
 known. 
 
 PROP. XVII. Fig. 12. 
 
 The Center of Gravity is had of that begat from 
 the Triline annex’d to the Fig. of Sines turn’d about 
 the proper Axis. Let the Triline be CB L turned 
 about CL, from it the Center of Gravity of the 
 begat is had from the Centers of the other Solids 
 being known, of which the Axis is common • to 
 wit, we have the Center of the Parailelipiped un- 
 der the Bafe AE, and the Altitude AB, equal to 
 QY in the Scale. Alio of the Solid proceeding by 
 the Squares A E, NI, under the Sines, Homogene 
 to that begat from ABC turn’d about AB ; let it 
 be T. Alfo of the Ifoparallel under the Bafe ABC, 
 and the Altitude AD of the proceeding by the 
 Rectangles A E, N M, for it is Homogene of the 
 Fig. ABC; let it be X. And as the Difference of 
 each to the Solid proceeding by the Squares A H, 
 NI, fo TX to XZ in the Scale annex’d to the 12 
 Fig. Z will be the Center of the aforefaid Diffe- 
 rence ; to wit, of the Solid proceeding by Rectangles 
 parallel to GI ; alfo in Z is the Center of the other 
 equal Solid proceeding by Rectangles parallel to IS ; 
 alfo the Center of the I foparallel Solid is had, under 
 the Triline Bale C LB, and the Altitude CE; for 
 let QY the Scale, be divided in the Middle in 0 , and 
 
 G g 2 a$ 
 
444 MEASURING 
 
 as the forefaid Ifoparallel under the Safe CLB, to 
 the Ifoparallel as above under the Baft ABC, fo X 0 
 to OeA; cP will be the Center of the Ifoparallel under the 
 Bafe CLB ; laftly, let it be as the Solid proceeding 
 by Rectangles parallel IS as above, alio known, of 
 which t-he Center is Z, lo Z ^ to J y, the Center of 
 the Solid proceeding by the Squares parallel BF, 
 I H will be in 7. 
 
 But this is Homogene to the begat from the Tri- 
 line EFK burned about E F ; therefore if QY is the 
 Axis of the forefaid begat, the Center of Gravity of 
 it will be in 7 ; laftly, if as 7O to fb the begat 
 from CAB turned about CL, to the begat from the 
 Triline C L B 3 tt will be the Center of gravity of 
 the begat from CAB : In like manner the Center of 
 Gravity may be had of the begat from the Triline 
 CLB turned about A B, fo T 6 to 0 # • /3 will be the 
 Center fought. 
 
 PROP. XVIII. Fig. 19. 
 
 The Center of Gravity is had of that begat from 
 a Fig. Homogene, to the begat from the Fig. of 
 Sines turned about the Axis ; for let the forefaid Ho- 
 mogene Fig. be EARM turned about EA, we 
 have the Center of the begat from both the Trilines 
 EDR, ADR which is in D; likewife of the be- 
 gat from ERM, to wit, of the Homogene to the 
 Ifoparallel Solid under the Bafe ERM, and the Al- 
 titude EM by Pofition 7. which is in Z, by grant- 
 ing that the Center known of the Fig. ERM is in 
 X ; fo laftly, ED Will be divided in 1 , that ZI is to 
 ID, as the begat from both the Trilines EDR, 
 ADR, is to the begat from ERM; I will be the 
 Center of the begat from the Fig. EARM. 
 
 PROP 
 
445 
 
 Superficies -and Solids. 
 
 PROP. XIX. 
 
 If the forefaid Fig. Homogene to the begat from 
 the Fig. of Sines as aforefaid, be turned about the 
 Bafe, the Ratio is had of the Solid begat to the 
 Cylinder of the fame Bale and Altitude ; let them 
 be the fame as above, and turn the Fig. E ARM 
 about EM 3 let the Center of the Fig. be H, the 
 Solid begat is to the Cylinder in compofition from 
 the Ratio of E D to EA, and from the Ratio of 
 LH to ED, that is, as the Redtangle under 
 ED, LH, to the Rectangle EA, ED, that is as 
 LH to EA ; for the Solids begat are compounded 
 of the Figures, and of the Diftances of the Centers 
 from the common Axis about which the Revolution 
 is made by Pofition 9. 
 
 CORO L. I. 
 
 Hence alio are had other begats, to wit, from 
 EDRM j becaule we have the begat from E R M, 
 by Prop. 11. alfo the begat from the Triline EDR, 
 by Corol. 1. of Prop. 11, therefore the Aggregate 
 from both, therefore the begat from EDRM : Alio 
 we have the begat from the Triline ADR, for by 
 taking away the begat from ADRM from the be- 
 gat from EARM, the refidue will be the begat 
 from ADR; likewife we have the begat from 
 SMRA, for by taking away the begat from 
 EARM from the Cylinder begat from ES, the re- 
 fidue will give the begat from S MR A, from which 
 if we take away the begat from the Triline M P R 
 
 Gg 3 known. 
 
44 6 MEASURING 
 
 known, the refidue will give the begat from 
 SPRA. 
 
 C O R O L. II. 
 
 Alfo we have the begat from E A R M turned 
 about MS, for by taking away the begat from 
 SMRA known, from the Cylinder begat from 
 MA, the refidue will give the begat from EARM. 
 
 PROP. XX. Fig. 19. 
 
 The Center of Gravity is had of the begat from 
 EARM turned about B M, for we have the Cen- 
 ter of the begat from E R M, which is N; alfo of 
 the begat from both Trilines EDR, ADR Ho- 
 mogenes to the Ifoparallel Solid under the Bale 
 EAR, and the Altitude EA ; and fince the Center 
 of the Triline EAR, is known by Corol. II. of 
 Prop. 10. let it be in Q, draw Q^V, the Center of 
 the begat from EAR is in V : So laftly, divide 
 VM in O, that O V be to OM, as the begat from 
 ERM is to the begat from EAR, E will be the 
 Center of Gravity of the begat from EARM, the 
 thing may eafily be reduced to Calculation. 
 
 PROP. XXI. Fig. 20. 
 
 We may find the Ratio of the begat from any 
 Segment of a Fig. of Sines; let the Fig. of Sines be 
 DAS, any Segment AQ/T ; let the Fig. DAFL 
 be Homogene to that begat from the Fig. of Sines, 
 as above ; the begat from AQT, will be to the be- 
 gat from DAS, as the Triline T A X to the whole 
 
 DAFL; 
 
Superficies and Solids. 44 7 
 
 DAFL ; let DC be equal to AT, let alfo the Fig. 
 of Sines be DFY, the Ratio of DFV to DAS 
 being known, which is i, the Ratio alfo will be 
 known of DAS to AL, which is -pf ; therefore from 
 thefe is had the Ratio of D A S to D A F L, which 
 is 4 - AL, therefore the Ratio is known of DAFL 
 to the Triline DCP; therefore alfo of the begat 
 from DAS turned about D A to the begat from 
 the Segment T A Q. 
 
 In another manner, the Ratio being known of 
 the Redangle DK to DG, equal to the Fig. 
 DAFL, alfo to the Trapezium D P I L, which be- 
 ing taken from DK, the relidue are the Trilines 
 DEP, LKI ; therefore the Ratio is had of the 
 Redangle DG to the Triline DCP. 
 
 C O R O L. I. 
 
 Hence by mandating the Segment T AQ_ into 
 DCZ, and by making the Fig. of Sines EMK by 
 producing the Axis FM, and then dividing VN 
 in the Middle, draw RY parallel to LS ; if the two 
 Trilines LKI, DCE librate in YR, it will be 
 equal weight ; For as I K to D E, fo any Parallel 
 of IK to any Parallel ofDE equidiftant from RY ■ 
 but as D C E to L K I, fo the begat from T A Q^or 
 DCZ to the begat from S y d, to wit, as the begat; 
 from DZ is equal to the begat from d. y, fo the be- 
 gat from any Parallel to DZ, is equal to the begat 
 from the Parallel to 7 equidiftant from YR : T here- 
 fore the Moments of the Weights of thofe begat 
 from DCZ, and from S 7 d in YR are equal ; like- 
 wife the begat from C^SZ, to wit, as PI is equal 
 to EH, and any other Parallel to PI, equal to ano- 
 
 G g 4 ther 
 
44 8 MEASURING 
 
 ther Parallel equidiftant from OR, fo the be^at 
 from Cd equal to the begat from SZ; And any 
 other begat from a Parallel to C d' equal to the be- 
 gat from another equidiftant from Y R. 
 
 C O R O L. II. 
 
 Hence is. had the Diftance of the Center of Gra- 
 vity of the Segment TAQ_from the AxisTA, for 
 fince the Ratio is had of the begat from T A Q, to 
 the begat from the Redangle under T A, TO / al- 
 fo the Ratio of the Segment T A to the forefaid 
 Redangie, fince thofe begat are in compofition, or 
 are compounded from the Ratios ofthe Figures, and 
 of the Diftances of the Center of each from the 
 common Axis T A Laftly, fince we have the Di- 
 ftance of the Center of the aforefaid Redangle, 
 which is — TQ, from thence we have the other Di- 
 ftance, viz, of the Center of T A Q/rom the fame 
 TA; which that it •may appear more clear, fee Fig. 
 21 . let the Ratio of thofe begatters be AB, AD, 
 let the Ratio of the Figures be AB, AC, let BC be 
 a Diftance known, to wit, of the Center ofthe Red- 
 angle common from the Axis ; let the Redangles be 
 AE, AG, AB, draw KC, FIL; FG will be the 
 Diftance fought ; and the Redangles are AE, LG, 
 as AB, AD, and in compofition from the Ratios of 
 AB to A C, and of BE to FG. 
 
 COROL. III. Fig. 20. ' 
 
 Hence is had the Diftance of the Center of Gravity 
 of the Trapezium DAQZ from the Axis AD; be- 
 caufe we know thcDiftance of the Center of theRed- 
 
 angle 
 
Superficies and Solids. 445 * 
 
 angle DQ, alio of the Segment TAQj therefore we 
 know the Aggregates from both $ likewife we have 
 the Diftance of the Center of Gravity of the Trape- 
 zium DTQS from the Axis DA, fince we have 
 the Diftances of the whole DAS, and of the Part 
 TAQ, of the other Part alfb may be had. 
 
 C O R O L. IV. 
 
 Hence is had the Diftance of the Center of Gra- 
 vity of the verfed Segment ZQS from the fame DT, 
 to wit, fince we have the Diftances of the whole 
 DTQS, and of the other Part, to wit, of the Seg- 
 ment ZQS, therefore alfb the Diftance of the Cen- 
 ter of the fame ZQS from the Axis ZQ; there- 
 fore we have the begat from the Segment ZQS turn- 
 ed about ZQ, for it is to the Cylinder of the fame 
 Bale and Altitude in the compounded Ratio from 
 ZQS to Z T, and from the Ratio of the Diftance 
 of the Center ZQS from ZQto the half of ZD. 
 
 C O R O L. V. 
 
 Hence from defcribing the Quadrant DST fmce 
 the Perpendicular is had falling on ZS from the 
 Center of Gravity of the Segment Z QS, likewife 
 the Perpendicular, falling on ZS from the Center of 
 Gravity, Z S is had alfo the Perpendicular falling 
 on the fame D S, from the remaining Fig. S QT. 
 
 PROP. XXII. Fig. 22. 
 
 The Homogene Fig. begat from the Fig. of Sines 
 turned about the Axis, is equal to the Circle whofe 
 
 Diameter 
 
45© MEASURING 
 
 Diameter is the Bafe of the Fig. Let the forefaid 
 Homogene Fig. be ASIH, let the Semicircle be 
 AKHj fince this is equal to the Re&angle A I, 
 and therefore the whole Circle equal to the Red- 
 angle A L, to which ASIH is equal, this will be 
 equal to the Circle. 
 
 C O R O L. I. 
 
 Any Parallels being drawn to AS, as C T, F V, 
 XZ: As A S is equal to the Arch A K H, io C R 
 to the Arch ONH; andXM to the Arch N FI ; 
 from X M appears from conftru&ion of the Line of 
 . Sines lMHj for fince FX is the Sine of the Arch 
 KN, •viz. equal to MZ; MX will be equal to 
 NE, to wit, FI is fuppofed equal to the Arch 
 KHj and fince RT is equal to the Arch <u A, 
 CR will be equal to the Arch ^ KH^ the fame 
 may be fhewn of any other affumed. 
 
 C O R O L. II. 
 
 Hence by taking away the Semicircle AKH, the 
 remaining Fig. is equal to the Semicircle ; and the 
 Fig. of Sines ASZH is equal to the Quadrate un- 
 der AH- Laftly the Leaf HISZ,H is equal to 
 the Difference of the forefaid Square, and of the 
 Circle under the Diameter A H. 
 
 C O R O L. III. 
 
 If from the Center of Gravity of the Fig. ASIH 
 you let fall a Perpendicular on AH, fuppofe on D, 
 AD will be to DH, as 3 to 5, to wit, from the 
 
 Center 
 
Superficies and Solids . 451 
 
 Center AIH it will fall on F, likewife from the 
 Center of the Triline API it will fall on B; let 
 FA be 7, A B will be 2 7^ ; fo BF being divided 
 in D, that B D be to D F as A P I to A I F, that is, 
 as 4 to 7, or as 16 to 28 , DF will be 1 there- 
 fore DA 5 -7 ; D H 8 •£, therefore AD to DH, as 
 5 f to f , 8 1, or as 21 to 35, that is as 3 to 5, hence 
 if it be divided into 8 equal Parts, AD will be 3, 
 and D H will be 5 of the fame Parts. 
 
 C O R O L. IV. 
 
 If from the Center of the Fig. of Sines F I H, 
 we let fall a Perpendicular on G I, F G will be \ of 
 the whole F 1 3 and as FH to FG, fo the Square 
 under FH, to the Semiquadrant of the Circle under 
 the Radius F H ; therefore by hanging from F the 
 Fig. FIH in the Perpendicular IF to the Arm of the 
 Beam or Ballance A F, the Semiquadrant in A will 
 make equal Weight, or equal Moment ; for fince 
 the Moments are compounded of the Quantities and 
 of the Diftances, the Moment in G is to the Mo- 
 ment in A, as the Re&angle under F H, F G, to 
 the Rectangle under FG, FHj to wit, in G is the 
 Quantity F H, the Diftance G F ; but in A the 
 Quantity F G, the Diftance FA or F H, therefore 
 both Moments are equal. 
 
 C O R O L. V. 
 
 If HcA be equal to AS, alfo FI equal to FY, and 
 you make Y ft ^ the Line of Sines, the Fig. IH J Y 
 will be Equal and Homogene to the Re&angle LY 3 
 for as H <P is equal to L 8, ib BM is equal to Z 3 the 
 
 fame 
 
452. MEASURING 
 
 lame may be fhewn of any other affumed ; there- 
 fore the forefaid Fig. is Homogene to the Re&angle 
 LY, therefore equal to the fame, fince "tis of equal 
 . Altitude HF, and of equal Bafe H G, therefore the 
 forefaid Fig. is equal to the Circle under the Radius 
 FH, therefore alfo of the Fig. A S I H. 
 
 C O R O L. VI. 
 
 If the forefaid Fig. be hung from F in the Per- 
 pendicular IF, on the End or Arm of the Ballance 
 A F, the Semicircle A K H in A will make equal 
 Weight, for divide HF in the Middle in X, and 
 draw X Z it will go or pafs through the Center 
 of the Fig. therefore • the Moment in X is equal 
 to the Moment in A, fince the Ratio of the Weights 
 is and of the Diftances \ therefore compounded 
 of f, therefore the Moments are equal ; the fame 
 will be done if we hang the Perpendicular L H on 
 the Arm of the Beam or Ballance H y equal to 
 HF. 
 
 C O R O L. VII. 
 
 Let there be the fame Perpendicular LH, and the 
 Arm of the Ballance H y, for the Fig. of Sines F I H 
 being hung, the Square under HF lefs •§■ of the Cir- 
 cle will make Equilibrium, as is manifeft ; but for 
 the other Fig. F Y & H, they will equiponderate \ of 
 the fame Circle, lefs by the Square under HF* 
 
 C O R O L. 
 
Superficies and Solids . 4 53 
 
 C O R O L. VIII. 
 
 But if the Perpendicular be AP, the Arm of the 
 Ballance AE equal to AF, for the Fig. AS IF be- 
 ing hung ; they will make equal Weight £ of the lame 
 Circle, lefs by the Square under A F ; becaufe 3 tis 
 equal to the Fig. H cP Y F already hung to the Per- 
 pendicular LH, and of the fame Pofition ; and for 
 the Fig. of Sines F I H hung to the Perpendicular 
 A P, it will make Equilibrium with £ of the fame 
 Circle, more by the Square under A F ; therefore 
 from the Ratio of the whole Fig. A S I H being 
 hung, they will make Equilibrium £ of the forelaid ■ 
 Circle, becaufe if we add I, lefs by the Square of £ 
 more by the Square, the Sum will be £ $ therefore 
 as 8 to 6, or as 4 to 3, fo E A to AD, for from 
 the Center of the Fig. A SI H the Perpendicular will 
 fall in D ; but AF is equal to AE, therefore AD 
 is to D H, as 3 to 5, as is already (hewn above. 
 
 C O R O L. IX. 
 
 If we fuppofe the Circle whole, the fame Center 
 A, fince the Diftances AF, EF are equal according 
 to Equilibrium, the fame thing will happen in th'e 
 Moment E, therefore £ of the Circle ; therefore thole 
 of the Moment E compound l £ of the Circle, but 
 the oppofite equal Moment compounds x -|of the Cir- 
 cle ; therefore the Diftances are \n the fame Ratio 
 by changing, therefore as 12 to 10, or 6 to 5, fo 
 E A to another (for Example) to AO, in O will 
 fall the Perpendicular from the Center of the Fig. 
 
454 MEASURING. 
 
 And fince A F is equal to AE, let AH be 12 AO 
 will be 5, and OH will be 7. 
 
 C O R O L. X. 
 
 Hence if the Fig. A S 1 H be turned about A S 
 that begat is to the Cylinder of the fame Bale and 
 Altitude, as 3 to 8, for it is compounded from the 
 Ratio of the Fig. ASIH to the Rectangle under 
 A S, AH which is -h, and from the Ratio of A D 
 to AF which is |, therefore ’tis compounded of 4 . 
 
 o 
 
 C O R O L. XI. 
 
 If we fuppofe an whole Circle from the Center 
 A, and the Fig. turned about as above, or before 
 the Ratio of the begat to that of the Cylinder is 
 had ; to wit, being compounded from the Ratio of 
 the Fig. ASIH more by the Semicircle, to the 
 Redangle under AH and A S more by AF ; and 
 from the Ratio of A O to A F. 
 
 C O R O L. XII. 
 
 If from the Fig. ASIH be taken away the Se- 
 micircle, that it remain in Equilibrium, the Remain- 
 ers as ftanding above, the Semicircle ought to be 
 taken away, that is, j to the Moment E ; therefore 
 there remains f for the fame Moment E at Equilibri- 
 um; therefore fince the Ratio of Weights is f, the Ra- 
 tio of Diftances will be f ; therefore if A C he fub- 
 duple of AE, or AF, the Perpendicular will fall in 
 C, from the Center of Gravity of the Fig. ASIH 
 by taking away the Semicircle A KH. 
 
 C O R O L. 
 
Superficies and Solids . 45 5 
 
 C O R O L. XIII. 
 
 If we turn about the Fig. ASIH, by taking 
 away the Semicircle AKH, that begat is to the 
 Cylinder, as r to 8, viz. in Compofition from the 
 Ratio of the Fig. to the Rcftangle under AS, AH 
 which is 4 ; and from the Ratio of AC to AE 
 which is i ; therefore the compounded is T ; Hence 
 that begat from the whole ASIH is triple of the 
 forefaid begat; and the begat from the Semicircle 
 AKH double; hence by affuming the whole Cir- 
 cle that begat from the Fig. is to the Cylinder un- 
 der the fame Bafe and Altitude AS, as 5 to 8, to 
 wit, the begat from the Semicircle is to the afore- 
 faid Cylinder, as 2 to 8 ; therefore that begat from 
 the Circle, as 4 to 8,- that begat from the Refidue, 
 as 1 to 8, therefore that begat from the whole, as 
 5 to 8. 
 
 C O R O L. XIV. 
 
 The Center of the Fig. ASIH is had, for by 
 drawing from the Center of the Fig. AIH known 
 the mht Line to the Center of the Triline A I S 
 known, and that will be the Center of the whole 
 Fig. in which the forefaid right Line will cut the 
 Perpendicular by falling in D, fuppofe ». 
 
 C O R O L. XV. 
 
 Hence is had that begat from the Fig. ASIH 
 turned about AH, for it is to the Cylinder of the 
 
 lams 
 
456 MEASURING 
 
 fame Altitude and Bafe, as the Rectangle under 
 AF, D to the Re&angle under AH, AP, to 
 wit, in the Compofition from the Ratio of the Fig. 
 to the Redangle and of the Diftance D to the 
 Diftance AP. 
 
 Therefore that I may comprife the whole in brief, 
 let BEFC be like to the former, alfo BESA, they 
 will referable BGHC, HGIA, the two Semicir- 
 cles are under the Diameters AB, BC; like wife the 
 Fig. of Sines B F C, B S A ; let S & be equal to ML, 
 and the other in like manner to the other Sines, be- 
 in terminated with the Line A cA E ; laftly let the 
 Fig. of Sines be A IB, the Semicircle BXA, the 
 Parallels OR, ZV, thefe being granted or fup- 
 
 pofed, if about EG we turn BEFC, that begat is 
 to the Cylinder of the fame Bafe and Altitude, as 3 
 to 8 ; if by taking away the Semicircle the Remain- 
 cr of the Fig. be turned about, that begat is to the 
 Cylinder, as 1 to 8 ; if we turn about the Fig. of 
 the Semichord BGIAL, that begat is to the fore- 
 laid Cylinder, as 5 to 8 ; in like manner is had that 
 begat from B F C, which is to the Cylinder of the 
 fame Altitude, as the Square of the Radius to the Se- 
 micircle, alfo the begat from the Triline B F C. 
 
 C O R O L. XVI. 
 
 If we turn the Fig. BESA about the Tan- 
 gent in A, that begat is to the Cylinder of the fame 
 Bafe and Altitude, as 5 to 8, viz. in Compofition of 
 the Figures i, and of the Diftances from the Center 
 4 : If we turn about the whole A EC, that begat 
 will be f of the Cylinder under the fame Bafe and 
 Altitude, for the Ratio of Diftances is of Equality • 
 
 there- 
 
Superficies and Solids. 457 
 
 therefore thofe begat are as the Fig. If we fubtradt 
 the Semicircle, and the remaining Fig. B E S A be 
 turned about, that begat will be to the Cylinder as 
 3 to 8, to wit, compounded from the Ratio of the 
 Figures^, andoftheDiftances l ; if we fubtradl each 
 Semicircle, and turn about the Reiidue of the whole 
 Fig. AEC, which is like to a Lilly, that begat will 
 be to the Cylinder as i to 4, which is the Ratio or 
 Proportion of the Figures, for they are equal of dis- 
 tance ; if we turn about the Fig. of the half Chord, 
 it will be to the Cylinder under the Altitude BG,and 
 of the Bafe begat from A B, as 7 to 8 : If laftly the 
 whole Heart, that begat will be to the Cylinder 
 under the Altitude BG, and of the Bale begat from 
 AC, as 6 to 8, *viz- the begat from the Fig. AEC, 
 is to the aforefaid Cylinder as 4 to 8, that begat 
 from the Lilly, as 2 to 8, therefore that begat from 
 both Semicircles as 2 to 8, therefore that begat from 
 the Heart, that is from the Fig. together and the 
 double Semicircle, is as 6 to 8 ; hence that begat 
 from the Semi-heart turned about BG, is equal to 
 that begat from BCSA about the Tangent A; alfb 
 that begat from the Fig. BCSA about BE, is equal 
 to that begat from the Refidue of the fame Fig. 
 from which the Semicircle is taken away, about the 
 Tangent A. 
 
 C O R O L. XVII. 
 
 If the Revolution be made about AC, fince the Ra- 
 tio of the Figures, and of the Diilances is had from 
 tjie Center, the Ratio of thofe begat is had which is 
 compounded from both, and we have the Center as 
 well of the Fig. AEC, as of the Lilly, as appears : 
 Alfo from the turning abont being made about the 
 
 H h Tangent 
 
458 MEASURING 
 
 Tangent is had that begat from the Fig. BESA; for 
 by taking away from the Cylinder the begat from the 
 lame turn'd about BA, the Remainer will give that 
 begat from the fame Fig. about the Tangent E. 
 Likewife is had that begat from the remaining Fig. 
 BESA from the fubtra&ed Semicircle ; to wit, is 
 had that begat from the Semicircle ALB being 
 turned about the Tangent E, for it is half of the 
 Cylinder under the Altitude BA, and of the Bafe 
 begat from EB, lefs the Sphere under the Diame- 
 ter AB; but by taking away the begat from the 
 Semicircle is had that begat from the Remainer, 
 therefore that begat from the Lilly ; therefore alio 
 the begat from the Triline BSE, likewife that be- 
 gat from the Fig. ESAIG, which is half of the Cy- 
 linder, for fmce the Diftances are equal, thofe begat 
 are as the Figures. 
 
 C O R O L. XVIII. 
 
 If we turn about the Tangent G the Fig. of the 
 half Chord, the begat from it is had; to wit, the 
 begat from the Semicircle ALB is had ; and as the 
 Parallelipiped under the Square of GB, and the Al- 
 titude BA to the Cylinder under the Altitude GB, 
 and of the Bafe from the Circle AXBL, more by 
 , the folid Homogene to the Hemifphere, proceeding 
 by the Squares of the Sines under the Altitude AB, 
 lo that begat from the Redtangle G A turned about 
 G, to that begat from the Semicircle ALB ; there- 
 fore by taking away the begat from BESA turned 
 about BA, from the begat from the Reftangle 
 GA turned about G, the Remainer fince being be- 
 gat from a Semicircle, will give that begat from the 
 
Superficies and Solids. 459 
 
 half Heart about G, hence are had thole begat from 
 the whole Circle AXEL, .from the whole Figure 
 of the Heart, from the Lilly, from the double Lilly 
 by taking away the Circle on both Sides 3 all thefe 
 may be reduc'd to Calculation, fuppofing Archi- 
 medes his Ratio. 
 
 C O R O L. XIX. 
 
 The Center of Gravity of that begat from BESA, 
 turned about BE, may eafily be found • to wit, the 
 Center of that begat from the Triline BSE is in D, 
 but of that begat from BSA 'tis equally diftant 
 from AC, as the Center of the Fig. BSA 3 therefore 
 the Centers of the Begats ABS A and BSE are had 
 in EB 3 therefore is had the Center of the begat 
 from BXA, therefore from the common of the 
 begat from BXAS 3 therefore of the begat BES AX, 
 alfo of the begat from the Refidue BESA, to which 
 the taken away is BLA3 therefore alfo the Center 
 of that begat from the Figure of the Heart. 
 
 C O R O L. XX. 
 
 The Fig. contained by the Curve Line A & E, and 
 by the Curve Line ASE is Homogene to the Semi- 
 circle ALB, and equal to the fame, as appears 3 
 alfo that begat from B S E A X and BE ^ A are 
 equal 3 likewife the Refidue of the Fig. BE^A, by 
 taking away the Semicircle BLA, is equal to the 
 Fig. BESA 3 alfo that begat from both are equal 3 
 likewife that begat from MSA about MA is had, 
 which is Homogene to the Solid of that which pro- 
 ceeds by the Squares of the Arches known 3 alfo 
 
 H h 2 that 
 
4<$o MEASURING 
 
 that begat from M of' A about MA, which is Ho- 
 mogene to the Solid which proceeds by the Squares 
 of the Compounds from the Arches and Sines alfo 
 known by Corol. 7. Prop. 14. But the Solid pro- 
 ceeding by the Squares of the compounded from the 
 Arches and Sines IL, TV, is equal to the pro- 
 ceeded by the Squares M c/', and of the Parallels 
 terminated at the Curve Line A J E , alfo equal to 
 the proceeded by the Squares of IL and QR, 
 Laftly, if from the Parallelipiped under the Square 
 of GB, and the Altitude BM, be taken away twice, 
 the Ifoparallel under the Bafe from the Triline 
 LSB, lefs by the folid proceeding by the Squares 
 under LS, and the other Differences of the Arches 
 and Sines under the Altitude BG, being known by 
 Corol. 5. Prop. 14. the Refidue will give the Solid 
 under the Altitude BM, proceeding by the Squares 
 GB or IL equal to the proceeded by the Squares 
 BE, M c/' under the fame Altitude ; hence is had 
 the whole folid under the Altitude AB, proceeding 
 by the Squares BE, M c0 5 &c. and this is to the Pa- 
 rallelipiped under the Altitude AB, and of the Bafe 
 from the Square of BE, as that begat from the 
 Fig. BE o' A turned about BA to the Cylinder of 
 the fame Height and Bafe. Thefe Things have 
 been fo often demonft rated that it were a Shame to 
 repeat them. 
 
 PROP. XXIII. Fig. 24. 
 
 The Semicycloid terminates the compounded from 
 the Arches and Sines ordinately applied to the 
 Diameter of the Circle ; that thefe Things may be 
 underftood with thofe which are faid and demon- 
 
 ftrated. 
 
Superficies and Solids. 4 6 1 
 
 flrated, {omewhat of new Conftru&ion is to be ufed • 
 let CB be the Diameter of the Semicircle BBC in- 
 filling at Rectangles to CG, and draw how many 
 foever Parallels OT, AI, KX, let LM be equal 
 to the Arch LB ; alfo DF equal to the Arch 
 DLB, OS to the Arch PLB • Lallly, CG equal to 
 the Arch of the Semicircle CDB, and through the 
 extream Points BNFG, fuppofe the Curve drawn ; 
 moreover, fuppofe the Semicircle CDB, fo moved 
 in CG by a right Motion of the Center through 
 AI, and by the Motion of the Orb about A, that 
 each Motion be equal and made together : Further- 
 more affume LM equal to KL ; BE equal to AD ; 
 PQ_ equal to OP ; and the fame may be done in 
 any alfumed, and through the extream Points 
 BMEQC draw the Curve ; Laflly, Let the Curve 
 CRFH be like to the former BNFG, and the Fig. 
 of Sines IYH ; thefe being pofited, I will demon- 
 ftrate the Line BNFG to be the Semicycloid, that 
 is the Curve Line defcribed from the Point B, in 
 the aforefaid Motion of the Semicircle ; by Defini- 
 tion 2. move B by the Motion of the Orb through 
 the Arch BD, it will recede from BC by the Space 
 AD, that is the Sine of the Arch DB ; and toge- 
 ther or at the fame time let the Motion of the Cen- 
 ter move with an equal Motion to the Orb, it will 
 recede from BC by the Space DF, equal to the Arch 
 DB, for equal Spaces fuppofe equal Motions in equal 
 Time, therefore from both Motions together the 
 Point B will recede from BC, by the Space AF$ 
 and fince the Arch BD palling over is in the Line 
 AI, it will be in F : In the fame Manner it may be 
 proved by the Arch BL palling over, that the Point 
 B is in N ^ from the Arch PLB palling over ’tis in 
 
 Hh 5 83 
 
4 6 -z MEASURING 
 
 S ; from the Arch CDB paffing over, ? tis in G ; 
 Laftly, From any other Arch paffing over, to ter- 
 minate the Applicate. 
 
 And that all Ambiguity may be taken away, I 
 ffiall fubjoin iome Definitions of Words; the Curve 
 Line ENG is called a Semicycloid ; the Fig. of 
 half the Cycloid CBFG ; the Axis BC ; the Bafe 
 CG ; the generating or begetting Circle BDC ; the 
 ordinate Applicate AF, the right Segment KBN ; 
 the verfed Segment y FG; the right Trapezium 
 CKNG j the verfed Trapezium CBF y. 
 
 C O R O L. I. 
 
 Any Segment ordinately applyed, intercepted be- 
 tween the Arch of the Semicircle CDB, and the 
 Curve BFG, is equal to the Arch cut of (for Ex- 
 ample) DF is equal to the Arch cut of DB ; LN 
 equal to the Arch LB ; PS equal to the Arch 
 PLB. 
 
 C O R O L. II. 
 
 Hence the Curve Line A^E, in Fig. 23. is 
 an half Cycloid ; and B Ac/'E the Fig. of the Cy- 
 cloid. 
 
 C O R O L. III. 
 
 Hence the Semicycloid above, being rightly deter- 
 mined or limited, as well by the Motion of the Cir- 
 cle or Wheel, or of the Circle on a Plane, fo that 
 the Motion of the Orb, and of the Center are equal, 
 as by the ordinate Applicates to the Diameter of 
 
Superficies and Solids. 4 6^ 
 
 the Circle, compounded from the Sines and the 
 Arches. 
 
 COROL. IV. Fig. 24. 
 
 The Fig. EEC is Homogene to the Semicircle 
 BDC ; for as AD is to AE, fo KL to KM, &c. 
 5 tis called the Terminate of the double of the Sines, 
 and it is an Ellipfis, as appears 3 likewife the Fig. 
 contained by the Curves BCD and BEC is Homo- 
 gene on both Sides. 
 
 COROL. V. 
 
 The Fig. contained by the Curves BEC and 
 BFG is Homogene to the Triline BHG, to wit, 
 EF is equal to FI, for EF is the Difference of the 
 Sine AD, and the Arch DB • alio FI more DA 
 equals the Arch DC, therefore FI is equal to EF, 
 likewife ST more OP equals the Arch PC 3 there- ♦ 
 fore by fubtra&ing from OT, the Arch PC more 
 PO, the Refidue is QS 3 and affuming KB equal 
 to OC, by fubtradfcing from KX, the Arch LB 
 equal to PC, or LN more KL equal to CP, that 
 is, by fubtradting the whole KN, the Refidue NX 
 will be equal to QjS, the fame may be fhewn in any 
 other affumed 3 therefore they are Homogene Fi- 
 gures 3 it may be fhewn otherwife thus, fince OP 
 more ST is equal to the Arch PC, and confequcnt- 
 ly of the right Line PR, and Jet PQ^ be equal to 
 PO, QR will be equal to ST 3 in like manner 
 MN is equal to VX, therefore MV equal to 
 NX 3 therefore the Triline GPIB Homogene of 
 the Fig. contained by the Curves CEB, and CFB, 
 
 H h 4 and 
 
4(54 MEASURING 
 
 and the right Line HB, therefore aifo equal by Po- 
 fiton i. hence alfo equal to the Semicircle CDB, 
 is the Fig. contained by the Curve Lines CDB, 
 
 CEB. 
 
 COROL. VI. 
 
 Hence the Triline GIF is equal to the Triline 
 IFH, alfo the Trapezium CEFG is equal to the 
 Trapezium CFIG. 
 
 C O R O L. VII. 
 
 The Triline DEB is Homogene to the Fig. of 
 Sines • for as DF is equal to 1 Y, fo LN is equal to 
 XZ, viz. to the Arch LB ; 1 lay the fame by any 
 other affumed, therefore the Triline is Homogenc 
 to the Fig. therefore alio equal. 
 
 PROP. XXIV. 
 
 The Fig. of the Cycloid is equal to three gene- 
 rating Circles, or, which is the lame, the Fig. of the 
 half Cycloid is equal to three half Circles j this Pro- 
 portion fome Men have already demonftrated, as 
 tforriceltius in his Appendix concerning the Dimen- 
 fion of a Cycloid ; I fhall not retrude from his In- 
 vention, but after my Manner fhall explain the 
 Thing by a four-fold Demonftration. 
 
 i. The Re&angle Cl is equal to the Circle of 
 the Generator or Begetter by Corol. 2. Pofition 2 3 
 the Triline GIF is equal to the Triline IFH by 
 Corol. 6. of Prop. 23 ; ABE is equal to the Se- 
 micircle by Corol. 5. therefore ABE more Cl, or, 
 
 which 
 
Superficies and Solids . 
 
 which is the fame, the Fig. of the Semicycloid CBG 
 is equal to three Semicircles. 
 
 2. CH is equal to two Circles ; CEB equal to the 
 Circle, therefore the Refidue is equal to the Circle • 
 but the half of the Refidue is the Triline GHB by 
 Corol. 5. therefore equal to the half Circle ; there- 
 fore the other half contained by the Curve Lines 
 BEC, BFG, and the right Line CG, is equal to 
 the half Circle, therefore the whole Fig. CBG is 
 equal to three Semicircles. 
 
 3. The Triline DFB is equal to the Fig. of Sines 
 IYH by Corol. 7. of Prop. 23. therefore to the 
 Square under AB, by Corol. 1. Prop. 3. the Tra- 
 pezium CAFG is equal to the Circle lefs by the Tri- 
 line GIF ; likewife the Triline EFB is equal to the 
 Difference of the Quadrant, and of the Quadrate ; 
 therefore ABF is equal to the Semicircle more 
 by*the forefaid Difference • the Trapezium is equal 
 to the Circle lefs by the fame Difference ; therefore 
 both together, that is, the whole Fig. CBG is equal 
 to three Semicircles. 
 
 4. The Fig. of the half Heart in Fig. 23 is equal 
 to three Semicircles, as hath been fhewn at large in 
 Prop. 22. and this is equal to the Fig. CBG, be- 
 cause Homogene; for both proceed by Applicates 
 compounded from the Arches and Sines, by Corol. 
 2. therefore the Fig. of the half Cycloid is equal to 
 three Semicircles, therefore the whole Fig. of the 
 Cycloid is equal to three generating Circles. 
 
 COROL. I. Fig. 24. 
 
 The Seftion contained by the right Line BG,and 
 the Curve Line BFG, is equal to the Semicircle, 
 becaufe the Triangle CBG is equal to the Circle. 
 
 COROL. 
 
4 66 MEASURING 
 
 C O R O L. II. 
 
 The Triline & FB is equal to the Quadrate un- 
 der AB; becaufe the Triangle A B cP is equal to the 
 Quadrant, and the Fig. ABF equal to the Qua- 
 drant, more by the Quadrate. 
 
 C O R O L. III. 
 
 The Triline LBN more by the Trapezium 
 CPSG is equal to the Redangle KH, becaufe 
 P QC is equal toRBL; QJb C is equal to X H V 3 
 Laftly, The Trapezium CRSG is equal to the Tra- 
 pezium BN VH : The fame may be ftiewn if any 
 other be affumed : And KH is to the Circle as 
 BK to BAj hence the Trapezium DLNF more 
 D P S F is to the Circle, as A K to A B ; hence fhe 
 Cylinder under the Altitude K B, and of the Bafe 
 from the Circle under the Radius AB-, is equal to 
 the Ifoparallel under the Bafe KB and the Altitude 
 A B, for they are compounded from the Ratio of 
 the Bafes which is B A to B K, and of the Alti- 
 tudes, which is B K to B A. 
 
 C O R O L. IV. 
 
 E B is equal to the Quadrant, becaufe the Tri- 
 line EFB is equal to the Difference of the Quadrant 
 and the Quadrate ; alfo J F G is equal to twice the 
 Segment of the Circle contained by the Arch of the 
 Quadrant, and the Subtenfe ED, likewife theCurve 
 Line BFG will divide in the Middle the Triangle 
 BGHj alfo the Triline C F G will contain four 
 
 times 
 
Superficies and Solids. 4 67 
 
 times tne aforefaid Segment, therefore his double 
 of the Triline HFG. 
 
 PROP. XXV. 
 
 The Ratio is had of any given right- Segment of 
 the Fig. of the Cycloid to the whole Fig. Let hi ft 
 of all °the Segment be ABF, it is to CBG, as the 
 Quadrant more by the Square under tiB to^tme^ 
 Semicircles, that is, according to Archimedes's Ra- 
 tio, as 25 to 66. Moreover, let another Segment 
 be’KBN, the Ratio is known of LBN, or XZ & 
 to the Square under AB, by Prop. 6. therefore to 
 the Quadrant; alfo the Ratio will be known of 
 the t Segment KBL to the Quadrant ; for it is 
 equal to the Re&angle under t TV and AB, Ids 
 by theTriang. BKL; therefore the Ratio is had 
 of the Segment KBN to the Quadrant ; therefore 
 to the whole CBG, which contains 6 Quadiants . 
 In the fame manner is had the Ratio of the Seg- 
 ment OBS, becaule CT is had, alfo GTS equal to 
 MNB, which being taken from CT, and the Refi- 
 due being taken from the Whole CBG, the Rcfi- 
 due will be OBS : Or fhorter thus, Since EMNF 
 is equal to FFrS, from the given OI, ABF, and 
 MNB, is had OBS equal to the aforefaid taken to- 
 gether. 
 
 PROP. XXVI. 
 
 The Center of Gravity of the Cycloid may be 
 had ; let the Fig. of half the Cycloid be CBG, by 
 taking away the Semicircle CDB, the Remainer is 
 an Homogene Fig. Flomogene to that begat from 
 
\6 8 MEASURING 
 
 the Fig. of Sines turned about the Axis, lince each 
 Fig. proceeds by Applicates equal to the Arches ; 
 but the Perpendicular falling from the Center of 
 the aforefaid Homogene Fig. will lb divide theBafe 
 that the Segments lhall be in Ratio J, by Corol. 3 . 
 of Prop. 22. Therefore from the Center of the 
 Fig. contained by the Curves BDC, BFG, and 
 the right Line CG, the Perpendicular falling on 
 the Axis BG, it will cut it lo, that the verfed Seg- 
 ment B is to the verled Segment C, as 5 to 3. But 
 if the aforefaid Homogene Fig. refemble a Semi- 
 circle in that Manner which is faid above, the Per- 
 pendicular falling from the Center on the Axis will 
 Fo cut it that the Segments are in Ratio ~, by Co- 
 roll. 9. of Prop. 22. but the Fig. added to the Se- 
 micircle is Homogene of the Fig. CBG, by Corol!. 
 2 . of Prop. 23. Therefore a Perpendicular falling 
 from the Center of jithe Fig. CBG on the Axis, will 
 lb cut it, that the verled Segment B will be to the 
 other verfed C, as 7 to 5. Hence if CB be of 12 
 equal Parts, byaffuming or taking from the Vertex 
 B 7 of them, there will be the Center of the Fig. 
 of the Cycloid. 
 
 COROL. I. 
 
 Hence the Center of Gravity of any right Seg- 
 ment of a Fig. of a Cycloid is had ; for Example, 
 let it be ABF : The Center of Gravity of the 
 Fig. IYH is had by Prop. 10. therefore by draw- 
 ing from this Center the Perpendicular on BA, it 
 will pals through the Center of the Homogene Tri- 
 line DFB, by Pofition 8. In like manner is had 
 the Center of the Quadrant ADB, and by draw- 
 
Superjicies and Solids. 4 69 
 
 ing from it a Perpendicular on AB in the End it 
 will fo divide the Segment of the Axis intercepted 
 by the Perpendiculars, that the verfed Part B will 
 be to the other verfed A, as the Quadrate to the 
 Quadrant, there will be the Center of the Segment 
 BAF. And that we may fhew fomewhat of Calcu- 
 lation, the Center of ABD is eafily had, for fince 
 the Ratio of the Figures, that is, of the Quadrate 
 under AD, and of the Quadrant ADB is tt : Like- 
 wife the Ratio of the Solids of thofe begat being 
 made by the Revolution about AD and let the 
 other of the Diftances be half of AB, (for Example) 
 7, the other Diftance will be 5 this is the Di- 
 ftance of the Center of Gravity of the Quadrant 
 from AD. Moreover, the Center of the Triline 
 DFB will be diftant from A I, £ of the whole DF, 
 which fince it is 22 by luppofing that AB is 14, 
 therefore the Diftance will be of Parts 5 
 
 Let another Segment be KBN, we have the Di- 
 ftance of the Center of Gravity of the Segment 
 XZH from the Axis XZ, by Coroll. 4. of Prop. 
 21. therefore alfo of the Homogene Fig. LNB. 
 Likewife the Diftance of the Center of the Segment 
 KLB from the fame KL will be known ; there- 
 fore if it be made as above, we lhall have the Di- 
 ftance of the Center of the whole KBN from the 
 lame KN. Laftly, Let the Segment be OBS, the 
 Center of ABF will be had, likewife of KBN, 
 therefore alfo of the other Part AKNF, alio of 
 ABE, and of KBM, therefore of EFB, likewife of 
 EMNF, therefore alfo of FITS, likewife of GI, 
 therefore of OAFS, therefore allb of the whole 
 Segment OBS : And that thefe may be reduced to 
 Calculation, fuppofe the Quadratures of the Circle. 
 
 COROf 
 
470 MEASURING 
 
 C O R O L. II. 
 
 Hence alio the Center of Gravity is had of the 
 other Parts of the Fig. of the Cycloid in the Axis 
 BC, for Example, of MNB ; for fmce the Center 
 of the whole Segment KBN is had, alfo of KBM, 
 we {hall have alfo the Center of the other Part 
 MNB, likewife of STG, and of XHV, which 
 are Homogene Figures to the Triline MNB : We 
 have alfo the Center of the Triline CFG, fince we 
 have as well the Center of the whole CBG, as of 
 the Part CEB, alfo of COSG. Alfo of the Fig. 
 contained by the Curve Lines BEC, BFG, and 
 the right Line CG being had ; to wit, from the 
 Center of the whole CBG, and of the Part CEB, 
 therefore alfo the Triline GHB, alfo of CEFG, 
 likewife of. FIHB, alfo of the Section contained by 
 the right Line BG, and the Curve Line BFG, and 
 laftly of any Trapezium. 
 
 PROP. XXVII. 
 
 If we turn about the Fig. of the Cycloid about 
 the Bale, that begat is to the Cylinder of the fame 
 Altitude, and the Bafe of the Circle under the Ra- 
 dius of the Axis of the equal Fig. as 5 to 8. Let 
 the Fig. of half the Cycloid be CBG turned about 
 the Bafe CG, that begat is to the Cylinder begat 
 from the Redlang. CH, as 5 to 8 - to wit, in a 
 compounded Ratio from the Ratio of the Figures 
 and of the Liftances which is H or I, *viz. the 
 Diftance of the Center of the Fig. CBG from CG, 
 is to CA, as 5 to 6, by Prop. 26 : Add that the 
 
Superficies and Solids. 471 
 
 Fig. CBC is Homogene to the Fig. of \ the Heart, 
 and that begat from it to that begat from thefe is 
 Homogene ; and that begat from the Fig. of the 
 ~ Heart to the Cylinder, as 5 to 8, by Coroll 15. 
 of Prop. 22. therefore alfo that begat from CBG 
 to the Cylinder begat from CH, as 5 to 8 ; the fame 
 may be laid concerning that begat from the whole 
 Fig. of the Cycloid to the Cylinder under the Bafe 
 from the Circle begat from CB, and from the Al- 
 titude of double CG. 
 
 C O R O L. IV. 
 
 Hence that begat from the Triline GHB is to 
 that begat from the Fig. CBG, as 3 to 5 ; but to 
 the Cylinder as 3 to 8. 
 
 PROP. XXVIII. 
 
 That begat from the Fig. of half the Cycloid 
 turned about the Bale, by taking away the genera- 
 ting Semicircle, is to the aforefaid Cylinder as 3 to 
 8 • for this is an Homogene of the Homogene Fig. 
 begat from the Fig. of Sines turned about the Axis, 
 therefore thofe begat from both are Homogenes : 
 But that begat from the other is to that Cylinder as 
 3 to 8, by Coroll. 15. of Prop. 22. therefore alfo 
 that begat from the other to its Cylinder, as 3 to 
 8 ; for let the aforefaid Fig. contained by the 
 Curves BDC, BFG, and the right Line CG be 
 turned about CG, that begat is to the Cylinder, 
 compounded from the Ratio of the Diftances, which 
 is g, by Prop. 26. and Coroll. 3. of Prop. 22. arid 
 from the B.atio of Figures, which is i j but from 
 
 thefe 
 
 t 
 
47* MEASURING 
 
 thefe the Compound is J, therefore the aforelaid 
 begat is to the Cylinder as 3 to 8. 
 
 C O R O L. I. 
 
 Hence that begat from the aforelaid Fig. is equal 
 to that begat from the Triline GHB, which is to 
 the Cylinder as 3 to 8, by Coroll, of Prop. 27. 
 
 C O R O L. II. 
 
 Hence that begat from the Semicircle CDB is to 
 that begat from the aforefaid Fig. as 2 to 3, and 
 to the Cylinder as 1 to 4. 
 
 ", ^ : 
 
 C O R O L. III. 
 
 And fince that begat from CEB is double of that 
 begat from CDB, for thole begat are as the Figures j 
 that begat from the Fig. contained by the Curve 
 Lines BEC, BFG, is fubduple of that begat from 
 the Semicircle CDB 3 for that begat from the Fig. 
 contained by the Curve Lines BDC, BEC, is equal 
 to that begat from CDB, therefore as 2 , but that 
 begat from the Fig. contained by the Curve Lines 
 BDC, BFG, and the right Line CG, is as 3 * 
 therefore that begat from the Fig. contained by the 
 Curve Lines BEC, BFG, and the right Line CG, 
 is as 1 ; therefore | of the Cylinder, and i of that 
 begat from CD A.. 
 
 COROL 
 
 1 
 
Superficies and Solids. 473 
 
 C O R O L. IV. 
 
 That begat from the aforefaid Fig. to wit, that 
 contained by the Curve Lines BEC, BFG, is equal 
 to that begat from the Trfiine CGH ; for fince all 
 the Applicates are equal, the Superficies in both 
 Figures of the begat are equal 3 for Example, that 
 begat from QS equal to that begat from RT, that 
 begat from EF equal to that begat from FI, there- 
 fore the Solids begat are equal : Hence that begat 
 from the Triline GHB is triple of that begat from 
 the Triline CGH. 
 
 C O R O L. V. 
 
 That begat from BCH turned about CG is to 
 the Cylinder begat as 7 to 8, for fince that begat 
 from the Triline CGH is to the Cylinder as 1 to 8* 
 that begat from BCH will be to the Cylinder as 7 
 to 8. 
 
 C O R O L. VL 
 
 Hence at laft are had others begat, fuppole from 
 the Triline CFG 3 for fince we have that begat 
 from the Triline CGH, alio that begat from the 
 Triline, GFH, which is to that begat from the 
 Redfcangle under GH, IF, as the Triline to the 
 Redtangie we have that begat from CFG, alfo 
 from the Fig. contained by the Curves CDB, CFH, 
 and by the right Line BH : For fince that begat 
 from the Semicircle CDB is to the Cylinder as 2 to 
 8 , and that begat from BCH as 7 to 8 3 that be- 
 
 I i -gat 
 
474 MEASURING 
 
 gat from the aforefaid Fig to the Cylinder as 5 to 
 8, therefore equal to that begat from CBG, and 
 fince that begat from CEB is as 4, that begat from 
 the Fig. contained by the Curves CEB, CFB, and 
 the right Line BH will be to the Cylinder as 3 to 
 8, therefore equal to that begat from the Fig. con- 
 tained by the Curves BDC, BFG, and the right 
 Line CG. Hence thofe begat from CDA, and 
 from the Triline CFG, will be equal to that begat 
 from the Triline BHF. 
 
 PROP. XXIX. 
 
 That begat from any right Segment of a Cycloid 
 turned about the Bafe may be had ; let ABF be 
 turned about AF, that begat from DFB is equal 
 to that begat from IYH, for they are homogene 
 Figures, therefore the \ of the Cylinder of the fame 
 Bafe and Altitude by Prop. 4. But that begat from 
 the Quadrant ADB is to the Cylinder of the fame 
 Bale and Altitude as 2 to 3, therefore let AD be 
 7, DF 1 1 ; that begat from ABF will be to the 
 Cylinder under the Altitude AF, as 10 £ to 18, or 
 as the Cylinder under the fame Bafe and Altitude 
 Compounded from •§ of AD, and \ of DF, that is, 
 as 61 to io8< 
 
 Let another Segment be KBN, fince we have the 
 Ratio of KBN to the Redtang. under KM, KB, 
 and the Diftance of the Center of Gravity of both 
 from KM j we have the Ratio compounded from 
 the Ratio's of the Diftances, and of the Figures, 
 but this is the Ratio of the Solids begat by Pofi- 
 tion 9 : the fame may be faid of any other Seg- 
 ment $ for Example, of OBS. 
 
 COROL 
 
Superficies and Solids * 475 
 
 C O R O L. 
 
 Hence are had thole begat from ADB, DEB, 
 EFB apart j and that begat from EFB is equal to 
 that begat from FIH , alio that begat from BAF 
 turned about BH, for we have the Diftance of the 
 Center of ABF from BH : Alfo of thofe begat 
 from DEB, EFB turned about BH j alfo from the 
 Trapezium BAFH, and from any other from the 
 general Rule delivered. 
 
 PROP. XXX. 
 
 We have the Center of Gravity of the begat as 
 well from the Fig. of the Cycloid, as from any 
 whole Segment turned about the Bale, for it is in 
 the Point in which the Bafe and the Axis are cut 
 crofs-wife ; to wit, the Center of the Solid begat 
 is in the Axis, is drawn through the Centers of all 
 the Planes ; alfo in the Bafe of the Figures which 
 is alfo drawn through the Centers of all the Circles ; 
 therefore the Center of the begat rnuft needs be in 
 the Concourfe or Meeting: of both. 
 
 R O P. XXXI. 
 
 We have the Ratio of the Solid begat from the 
 Fig. of the half Cycloid turned about the Axis to 
 the Cylinder of the fame Bafe and Altitude ; let all 
 be as above, the half Cycloid, the Fig. CBG turned 
 about the Axis BC ; that begat from it is to the 
 Cylinder of the fame Bafe and Altitude, as the 
 Solid under the Altitude CB, proceeding by the 
 
 I i % Square? 
 
47 6 MEASURING 
 
 Squares CG, OS, AF, KN, &c. to the Parallelipi- 
 ped under the Altitude CB, and of the Bafe from 
 the Quadrate CG; for as the Parallelipiped is homo- 
 gene to the Cylinder, fo the aforefaid Solid, pro- 
 ceeding by the Squares homogene to that begat 
 from CBG : But the Ratio is known of the afore- 
 faid Solid to the Parallelipiped by Coroll. 20. of 
 Prop. 22. therefore from thence is had the Ratio of 
 the aforefaid begat to the Cylinder, which we (hall 
 prove by Calculations beneath. 
 
 C O R O L. I. 
 
 Hence are had many Solids begat, being made 
 by the Revolution about BC, to wit, that begat 
 from the Triline BHG, alfb that begat from the 
 Fig. contained by the Curves BDC, BFG, and 
 the right LineCG, by taking away that begat from 
 BEC the double of that begat from BDC ; like- 
 wife that begat from CAFG ; for if from the Pa- 
 rallelipiped under the Altitude CA, and of the Bafe 
 from the Square of CG be taken away twice, the 
 Ifoparallel under the Triline Bafe IGF, and the 
 Altitude CG, lefs by the Solid under the Altitude 
 IG, proceeding by the Squares under FI, ST, and 
 from other Differences of the Arches and Sines, 
 the Remainer will proceed by the Squares CG, OS, 
 AF, and thefe are known by Coroll. 20. of Prop. 22. 
 And it is as the aforefaid Parallelipiped to the afore- 
 faid Remainer, fo the Cylinder begat from Cl, to 
 another Solid, this it felf will be that begat from 
 CAFG, which is homogene to the aforefaid Re- 
 mainer. 
 
 COROL, 
 
Superficies and Solids . 477 
 
 C O R O L. II. 
 
 Hence is had the Center of Gravity of CBG, 
 to wit, we have already the Diftance of it from the 
 Bafe CG, by Prop. 26. the Ratio of the Begetters 
 is had which is compounded from the Ratio of the 
 Figures which is had, and from the Ratio of Di- 
 ftances of which one is had, to wit, the half CG, 
 therefore the other alio is had, for from two com- 
 pounded Terms being given, alfo from the two 
 other of the compounding Ratio, fince to the third 
 Term of one is had, the fourth Term of the lame, 
 as is fhewn above : Therefore is had the Perpendi- 
 cular falling from the Center of the Fig. on the 
 Bafe CG, alfo the Perpendicular falling from the 
 lame on the Axis CB ^ therefore is had the Con- 
 courle of both, in which is the Center of the Fig. 
 CBG. 
 
 COROT. III. 
 
 Alfo the Center of the Triline HBG is had, 
 which may be demonftrated in the fame Manner, 
 alfo that begat from the fame Triline turned about 
 GH ; for we have the Ratio compounded of the 
 Pittances, and of the Figures, therefore of the So- 
 lids begat : Alfo we have the Center of the Tra* 
 pezium CAFG, to wit, the Perpendicular falling 
 from it on the Axis CB j therefore that begat from 
 the fame Trapezium turned about GI, which be- 
 ing taken from the Cylinder begat from GA, we 
 have that begat from the Triline 1FG, 
 
 I i 3 
 
 PRO P, 
 
478 MEASURING 
 
 PROP. XXXII. 
 
 The Solid begat from the Segment turned about 
 the Axis is had, whofe Bafe will fall in the Center 
 of the generating Circle ; (for Example) let ABF 
 be the begat from it, it is Homogene to the Solid 
 under the Altitude AB, proceeding by the Squares 
 under the Compounds from the Arches and Sines, 
 as appears ; for AF is compounded from the Sine 
 AD, and from the Arch DB; alfo KN from the 
 Sine KL, and from the Arch LB, &c. And the 
 Ratio of this Solid will be known to the Paralleli- 
 piped of the fame Altitude and Bale, therefore alfo 
 of the aforefaid begat from ABF to the Cylinder : 
 I fay, the aforefaid Ratio will be known by Coroll. 
 
 of Prop. 14.. and Coroll. 20. of Prop. 22. 
 
 C O R O L. I. 
 
 Hence is had the Center of Gravity of the Seg- 
 ment ABF, which may alfo be had in another 
 Manner, to wit, from the Center being known of 
 the whole Fig. CBG, and of the Trapezium CAFG ; 
 alfo that begat from BFC is had, viz- the double 
 pf that begat from ABF. 
 
 C O R O L. II. 
 
 That begat from DFB is had, to wit, by taking 
 #way the begat from ADB : And therefore the 
 Center of the fame DFB, alfo that begat from the 
 Fig. contained by the Curves BDC, BFC, and 
 the fame Center of the Fig. Alfp that begat front 
 
Superficies and Solids. 479 
 
 the Triline EFB, by taking away that begat from 
 or by AEB, the double of the begat from ADB, 
 and the fame Center of the Triline * alfo that be- 
 gat from the Lunula contained by the Curves BEC, 
 BFC by taking away the begat from or by BEC $ 
 alfo the Centers of the Figures are had, by the Be- 
 gats being known. 
 
 C O R O L. III. 
 
 And from the Revolution made about HG, is 
 had that begat from the Triline GHB, of which 
 the Center is had ; alfo from the Fig. CBG, by 
 reafon of the fame Ratio ; alfo that begat from the 
 Fig. contained by the Curves BDC, BFG, and the 
 right Line CG ; alfo from the Fig. contained by the 
 Curves BEC, BFG, and the right Line CG ; alfo 
 that begat from the aforefaid Lunula $ alfo from 
 the Triline CGF or BHF, to wit, becaufe the 
 Centers are known. 
 
 PROP. XXXIII. Fig. 16. 
 
 From a Solid being cut which proceeds by Rect- 
 angles under the right Sines and Complements, the 
 Ratio of the Segments is had ; let the aforefaid So- 
 lid be ABDC proceeding by the Segments BAC, 
 OHG, &c. or by ReCtangles parallel to EG, which 
 is under the right Sine and the Complement, as is 
 fhewn in Coroll. 2. of Prop. 13. let it be cut by the 
 Plane OHG, fince it is Homogene ASD, the Seg- 
 ments of the Solid are as the Segments of the Triang. 
 And thefe are as the Squares of the Segments of the 
 Bafe AB, BD, therefore by knowing the whole 
 Solid, the Segments will be known, 
 
 I i 4 
 
 COKQU 
 
480 MEASURING 
 
 C O R O L. I. 
 
 Hence from the Solid being cut by the Plane EG 
 we may know the Segment intercepted by the 
 Planes BAC, OHG, from which, if we take away 
 the Ifoparallel under the Bafe EFA, and the Alti- 
 tude AH, the Refidue will be known, of which the 
 Bafe is EG, and the Altitude EB. 
 
 * COROL. II. Fig. 17. 
 
 Hence the Segments of the Solid AEG cut by 
 the Plane DB will be known ; to wit, the Ifopa- 
 rallels had under the Bafei from the Trapezium 
 ALKB, and the Altitude AG: But the Segment 
 under the Bafe GIDH, and the Altitude GF may 
 be had thus ; the Solid will be known proceeding 
 by the Sedtors parallel to GID towards FE, it is 
 Homogene to the Triangle, therefore the Half of 
 the Ifoparallel under the Bafe from the Sector GID, 
 and the Altitude GF ; the Solid alfo will be known 
 proceeding by Triangles parallel to GHD towards 
 FE, for it is fubduple of the Solid known proceed- 
 ing by Redtangles under the right Sines, and of the 
 Complements, by Coroll. 1. Therefore the Segment 
 will be known, intercepted by the Planes AE, and 
 BK, which being taken from the whole Solid AEC, 
 we have the remaining Segment DBG 
 
 COROL III. 
 
 Hence the Solid Segment under the BafeGHDI, 
 and the Altitude GF, is equal to the Ifoparallel un- 
 der the Bafe KLl^fP, and the Altitude LI. 
 
 COROL, 
 
Superficies and Solids . 4 ^ 1 
 
 COROL IV. Fig. 24. 
 
 If we return to the Fig. GBG, the Solid pro- 
 ceeding or growing up by Redtangles under the 
 Sines and the Arches, that is, under AD, DF, KL, 
 LN, ciV. is Homogene to the aforefaid Solid : Hence 
 if it be cut by the Plane LN, the Ratio of the Seg- 
 ments will be had as above, and by adding the 
 Solid proceeding by the Squares of the Sines AD, 
 KL being known, the Solid is had proceeding by 
 the Redtangles under AF, AD, KN, KL, that is, 
 under the Sines and the Compounds from the Sines 
 and Arches ; alfo the Ratio of the Segments is had 
 by dewing the Plane KN. 
 
 PROP. XXXIV, 
 
 From the Revolution being made about the Axis 
 is had that begat from the Fig. contained by the 
 Cycloid, and the right Line the Subtenfe of the 
 fame : Let CIG be f the Cycloid, let the Fig. con- 
 tained be the Curve BFG, and the right Line BG, 
 the Solid will be had from it begat by being turned 
 about BC, to wit, if from the Solid begat from 
 CBFG, we take away the Cone begat from the 
 Triang. CBG, the Refidue will give that begat from 
 the aforelaid Fig. 
 
 CORO L. I. 
 
 Hence is had the Center of Gravity of the afore- 
 faid Fig. alfo of that begat from F B, to wit, by 
 taking away the begat from A V B being known, 
 
 from 
 
4 8i measuring. 
 
 from that begat from AFB alfo known : Hence is 
 had that begat from »'FG; alfo the Center of 
 Gravity as well of d F B as of d F G. 
 
 C O R O L. II. 
 
 If there be another Line of .-£■ the Cycloid GDB, 
 the common Subtenfe BG, the Center of both to- 
 gether is in AF, fince both the Applicates are pa- 
 rallel to AF, and equidiftant, they are equal there- 
 fore in the Point d 9 as appears. 
 
 C O R O L. III. 
 
 Hence is had that begat from both taken together, 
 for it is to the Cylinder begat from BG in the Ra- 
 tio compounded of the Figures and of theDiftancesj 
 the Diftances are equal, for the Center of both Fi- 
 gures is in o", therefore thofe begat are as the Fi- 
 gures : The Redang. is double of the Fig. therefore 
 that begat from the Eg. is fubduple of that begat 
 from the Redang. 
 
 C O R O L. IV. 
 
 Hence that begat from the aforefaid Fig. is fef- 
 qmalter of the Cone begat from the Triang. BCG, 
 and fubfefquiteiiiim of that begat from the Triangle 
 BGH : Hence that begat from both the Trilines 
 taken together, is equal to that begat from the Fig. 
 And the begats are as the Figures, fince the Center 
 of both is in A 
 
 C O R O L, 
 
Superficies and Solids, 4^3 
 
 C o R O L. v. 
 
 If we turn the aforefaid Fig. about CG, fince 
 that begat from CBG is to the Cylinder as 5 to 8, 
 that is, as 15 to 24; that begat from the afore- 
 faid Fig. will be as 12 to 24, therefore that be- 
 gat from the Triline BCG as 3 : And that begat 
 from the Triline GBF 1 as 9. Hence that begat 
 from 4 the Fig. contained by the Curve BFG, and 
 the right Line BG as 7 ; becaufe that begat from 
 the Triang. GBH is as 16, therefore that begat 
 from the other \ as 5 : Hence you fee the Proceflion 
 by odd Numbers, from the beginning being drawn 
 from the Triline BCG, 3, 5 ■> 7* 9 •’ Laftly, that be- 
 gat from HGB as 21. Moreover, in this the two 
 Trilines CBG, GBH agree with the Parabolicks, 
 becaufe that begat from one is triple of that begat 
 from the other. 
 
 SCHOLIUM. Fig. 13. 
 
 But T come to Calculations ; Let the whole Cy* 
 cloid be BAE, for altho’ above it hath been in- 
 ftead of the Fig. of Sines, now I fuppofe it to be a 
 Cycloid which is turned about BE : As CA begets a 
 Circle, fo alfo all the other ordinate Applicates pa- 
 rallels ; LetCA, the Axis of the Fig. (for Example) 
 be 14, EB the Bafe 44, according to the Archime- 
 dean Ratio ; let the Homogcne Solid arifing or pro- 
 ceeding by the Squares under the Diameters of the 
 Circles begat from the aforefaid ordinate Applicates 
 under the Altitude EB be called the greater Ho- 
 mogcne , let there be another under the Altitude 
 
484 measuring 
 
 CE, proceeding or arifing by the Squares of the Ap- 
 plicates Parallels to CA, call it the lefTer Homo- 
 gene: This is fubodtuple of the greater, as appears 9 
 let the Ifoparallel be under the Bafe BAE, and the 
 Altitude AI cut by the Plane EIB, the letter Homo- 
 gene is equal to the upper Fruftum ; fince it is dou- 
 ble of the half Fruftum, to wit, the Homogene will 
 proceed by Squares, and the half Fruftum by Tri- 
 angles, both under the fame Altitude CE, and eve- 
 ry Square is double of its Triangle. Moreover, let 
 the Center of Gravity of the Fig. BAE be in K, fo 
 that CK be to CA as 5 to 12, the aforefaid Fruftum 
 will be to the Ifoparallel as 5 to 12 • therefore al lb 
 the lefTer Homogene as 5 to 1 2 $ therefore the 
 greater Homogene as 40 to 12, that is, as 10 to 3. 
 Therefore fince the Fig. EAB is equal to three 
 Circles under the Diameter CA, the Circle under 
 the Diameter double of CA will be to the Fig. 
 CAB, as 4 to 3 ; therefore the Cylinder under the 
 Bafe from the aforefaid Circle, and the Altitude 
 AI to the Ifoparallel as 4 to 3, therefore the great- 
 er Homogene to this Cylinder as 1 o to 4, or as 5 
 to 2. But this Cylinder is to that begat from the 
 Re&ang. under BE, CA turned about EB, as CA 
 to EB, that is, as 14 to 44, therefore the greater 
 Homogene will be to that begat from the aforefaid 
 Reftang. as 35 to 44. But the Parallelipiped under 
 the Bafe from the Square of the double of CA, and 
 the Altitude CB, is to the aforefaid begat as 14 to 
 11, that is, as 56 to 44, and to the greater Ekn 
 mogene as 56 to 3 5, that is, as 8 to 5 : Therefore 
 the greater Homogene to that begat from the Fig. 
 EAB as 35 to 27 that is as 14 to xi, therefore 
 the greater Homogene to the Cylinder under the 
 
Superficies and Solids. 
 
 Bale from the Circle of the Diameter CA, and the 
 Altitude AI, will be as 40 to 4, that is, as 10 to 
 1 i therefore that begat from E AB to the Cylinder 
 under the Bafe from the generating Circle, and the 
 Altitude CA, is as 55 to 14. 
 
 But let IY or DF be 11, CB 14, the 
 Parallelepiped under the Bale from the 
 Square IY, or DF, and the Altitude IH 
 will be 847, that drawn into three is 
 The Cube under the SidelH, will be 
 343, which, drawn into three, makes 
 The Solid proceeding or arifing by 
 the Squares of the Arches IY, XZ, 
 under the Altitude IH, will be 392, 
 which, drawn into three, makes 
 
 The Solid Homogene to the Sphere, 
 proceeding by the Squares of the Sines 
 under the Altitude IH, will be 228 •§-, 
 which, drawn into three, makes 
 
 The Solid proceeding by Redbangles 
 under the Arches and Sines, and under 
 the fame Altitude IH, will be 297 
 which, drawn into three, makes 
 The Difference of both Solids 
 This Difference being added to the 
 laft Solid, the Aggregate will be 
 
 This Aggregate being taken from the 
 Solid, proceeding by the Squares of the 
 Arches, the Remainer will be 
 
 And this is the Solid proceeding or 
 arifing by the Squares of the Differences 
 between the Arches and the Sines, 
 
 \ 
 
 1 
 
 l 
 
 2541 
 
 1029 
 
 1176 
 
 686 
 
 892 i 
 
 206 i 
 1099 
 
 77 
 
 It 
 
4 8<5 MEASURING 
 
 If from the Parallelipiped under the ^ 
 Square CG, and the Altitude CA, we 
 take away twice the Ifoparallel under the 
 Bafe from the Triline GIF and the Alti- 
 tude CG, left by the Solid proceeding by ^ 
 the Squares FI, ST, to wit, by the 
 Squares of the Differences, as above, the 
 Remainer will be the Solid, proceeding by 
 the Squares CG, OS, AF, to wit ^ 
 
 Being added to the Solid proceeding by ' 
 the Squares of the Arches ; twice the Solid 
 proceeding by the Re&angles under the 
 Arches and Sines, left by the Solidpro- 
 ceeding by the Squares of the Sines, and 
 the Aggregate proceeding by the Squares 
 AF, KN, to wit, compounded from the 
 Arches, and the Sines will be J 
 
 One being added to the other, and will ? 
 be the Solid proceeding by the Squaresr 
 CG, AF, &c. > 
 
 And the Parallelipiped under the Bafe) 
 from the Quadrate CG, and the Altitude S 
 CB, is S 
 
 Therefore the Parallelipiped is to the 
 Solid proceeding by the Squares CG, AF, 
 £ 3 c. that is the Cylinder begat from the 
 Re&angleCH, to that begat from the Fig. 
 CBG turned about BC, as 484 to 287. 
 
 9779 
 
 2275 
 
 12054 
 
 20328 
 
 Hence if we take away from that begat from the 
 Fig. CBG, that begat from the Triangle CBG, 
 that is j of that begat from CH, the Remainer will 
 give the begat from the Fig. contained by the Curve 
 Line BFG, and the right Line BG, to wit 125 -§ ; 
 
 therefore 
 
 1 
 
7% 
 
Superficies and Solids. 4^7 
 
 therefore that begat from the other ~ Portion 
 116 Therefore that begat from the Triline 
 BCG 45, therefore that begat alfo from the othef 
 
 Triline 197. , „ 
 
 And by taking away twice the Solid proceeding 
 by the Squares of the Sines, from the Solid which 
 proceeds by the Squares of CG, AF, theRemainer 
 will be 10682 ; therefore if that begat fromCDFG 
 is 287, that begat from the Fig. contained by the 
 Curves BDC, BFG, and the right Line CG will 
 be 2<4 4 ; and confequently that begat from the 
 Semicircle CDB 325 therefore that begat from 
 CEB 130 -f ; therefore that begat from the Fig. 
 contained by the Curves BEC, BFG, and the 
 riaht Line CG, is 56 4 ; and that begat from CH 
 turned about CB to that begat about CG, as 847 
 to 539, that is 484 to 308 ; therefore that begat 
 from CBG about BC, to that begat from CBG 
 about CG, as 287 to 192-: That begat from the 
 Triline BCG is had in another Manner, to wit, by 
 taking away that begat from the Fig. contained 
 by the Curves BFG, BPG, half of that begat 
 from CH, from that begat from CBG, that is 242 
 from 287, the Remainer 45 will be that begat from 
 the Triline BCG. 
 
 FINIS. 
 
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 A Flew Tranflation of AZfop's Fables, adorn’d with 
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efeiAt 
 
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 I5c53