ARTS and SCIENCES, Abfolutely Neceflary to be underftood by BUILDERS and WORKMEN in general. viz. THE Builder’s Complete Affiftant; o R, LIBRARY O F . ARITHMETICS Vulgar and Deci- mal, in whole Numbers and Fra&ions. T I. GEOMETRY, Lineal, Superficial, and Solid. III. ARCHITECTURE, Univerfal. IV. MENSURATION. V. PLAIN TRIGONOMETRY. VI. SURVEYING of Land, VII. MECHANICK POWERS. VIII. HYDROSTATICKS. Illustrated by above Thirteen Hundred Examples of Lines > Superficies, Solids , Mouldings , Pedeftals , Columns , Pilafiers, En- tablatures , Pediments , Impofts, Block Cornices , Ruftick Quoins, Frontifpieces , Arcades , Porticos , &c. Proportioned by Modules and Minutes, according to Andrea Palladia ; and by Equal Parts. LIKEWISE Great Varieties of Trufted Roofs , Timber Bridges , Centerings , Arches , Groins , Twifted Rails , Compartments , ObeliJks,Vafes , Pedeftals for Buftos, Sun-Dials, Fonts , £sV. and Methods for raifing heavy Bodies by the Force of Levers, Pulleys , Axes in Peritrochio , Screws , and Wedges ; as alfo Water , by the common Pump , Crane, &c. Wherein the Properties, and Preflure of the Air on Water, EsV. are explained. The whole exemplified by 77 large Quarto Copper-plates. By BATTY LANGLEY. The Fourth Edition. LONDON : Printed for I. and J. TAYLOR, at the Architectural Library, No. 56, Holuorn. A table O F THE A T PLAT E S, AND Pages wherein they are explained. Plates..] Pages wherein explained. L T~)AGES 62, 63, 64, 65, 66. II. Jr 66,6 7*68,69,7®, 71. HI- 773* 74* 75* 7^, 77* 7^- IV. 7(9, 80, 8 x, 82, 83, 84, 85. V. 84., 85, 86, 87. VI. 8 7, 88, 89, 90, 91, 92, 93. VII- 9 1 * 97* 93* 94* 95* 9 6 * VIII. 96, 97, 98, 99, 100, lor. IX. ioi, 102, 103. X. ic'6, 107, 1 14, 1 15, 139. XI. 139, 140. XII. 3 20, I 2 I. XIII. 120, I 2 I* XIV. 129,145. XV. a 10, 124. XVI. 142, 143. XVII. 142, 143. XVIII. 104, 142. XIX. 105, 106, 107. XX. 103, 107, ! 08, 136. XXI. 109, 1 10, hi, XXII. 109, I XI, I 12. XIII. 113, 114. XIV. 113, 114, 1 1 5, 1 16, 137. XXV. 1 17. XXVI. 1 16, 1 17. XXVII. 117. XXVIII. 104, 118, 119, 125, 137. XXIX. I 19, 122, 123. XXX. 122, 123, 124, 141. XXXI. 122, 123, 124. XXXII. 105, 1 1 8, 125, 126^ 127. XXXIII. 125, 126, 128, 129. XXXIV. 105,128. XXXV. 127. XXXVI. 130. XXXVII. 130. XXXVIII. 117, 130, 13 1. XXXIX. 105, 132, 133, 137. XL. 133. Plates.] Pages wherein explained* XLI. Pages 105, 132, 133, 134, 138. XLII 105, in 124, 13 1, i 34 , 135. XLIII. 1 3 1, 136, 14 [, 146. "o 5 ’ H*. XLV. 138, 144. XLVI. 145, 146. XLVII. 138, 144, 146. XLVIII. 146. XLIX. 146, 147. L. 147. LI. 150. LIT. 149, 150, 151. LIII. 149, 151, 1 52, LIV. 132. LV. 152, 153. LVI. 152, 156, 157. LVII. 153, 154. LVIIT. 155. LIX. 154, 156. LX. 155, 157,158. LX1. 1 59. LX 1 I. 159, 160. LXIII. 109, no, i6r, LXIV. 16 1, 162,163. LXV. 164. LXVI. 165, 166. LXVH. 163, 166, 167. LX VIII. hi, 166. LXIX. 168. LXX. 168. LXXI. 168, 169. LXXII. 169, 173, 175. LXXIII. 173, 174, 175, 176, LXXIV. 173, 174, 175, 176. LXXV. 177, 178, 180, i8r, 182, 183, 184. LXXVI. 185, 186, 187, 188, 189, 190, 191, 192. LXXVII. 193, 194, 195, 196, 197, 198, 199, 200. 2 A A B E OF THE CONTENTS. PART I. Lect. Of Arithmetics I. F Numeration. 1 II. Of Addition. 6 IH. Of Subtra&ion. 24 IV. Of Multiplication. 33 V. Of Divifion. 43 VI. Of Reduftion. 47 VII. Of the Golden Rule. 49 VIII. Of Fractions. 5 1 IX. Of Square and Cube Roots. 5 ^ PART II. Of Geometry. Introduction. 61 I. Of Definitions. 62 II. Of Angles. 7 * III. Of Lines* 73 IV. Of Plain Figures. 79 V. Of infcribing Geometrical Fi- gures. 90 VI. Of Proportional Lines, 93 Page Lect* XX. PART III. Of Architecture. [. Of Mouldings. 97 [I. Of making Scales. 102 [II. Of the principal Parts of an Order. io 3 tV. Of the Tufcan Order. 105 V. Of Tufcan Frontil'pieces, t 3 V. 109 VI. Of the Doric Order. 1 1 3 VII. Of the Ionic Order. 119 VIII. Of the Corinthian Order. 125 IX. Of the Compofite Order. 132 X. Queries on the Orders. 136 XI. Of the Grotefque Order. 138 XII. Of the Attic Order. ibid, XIII. Of Wreathed Columns. XIV. Of Flutes and Fillets. XV. Of placing Columns. XVI. Of Ornaments for the Enrich- ments of the Five Orders. 14 2 XVII. Of rufticating Columns. *44 XVIII. Of Block Cornices. ibid . XIX. Of Doors and Windows, 145 Of Pediments. 14® XXI. Of Truffed Partitions. 147 XXII. Of Naked Flooring. 148 XXIII. Of Roofs. 15® XXIV. Of Angle Brackets. 15.0 XXV. Of Niches. ibid . XXVI. Of Timber Bridges. 159 XXVII. Of Brick and Stone Arches. 161 XXVIII. Of Centering to Arches and Groins. * 6 3 XXIX. Of Stair-cafes. 165 XXX. Of Ornaments for Buildings and Gardens. 168 PART IV. Of Mensuration. I. Rules for meafuting Superficies. 169 II. Rules for meafuring Solids. 172 PART V. Plain Trigonometry* I. Of the Solution of Plain Triangles* 177 II. Of Heights and Diftances. 1 79 139 ibid. 141 PART VI. Of Surveying Lands, feV. IM- PART VII. Of Mechanicks, I. Definitions of Matter, Gravity, and Motion. 1 8 5 II. Of the Laws of Nature. 187 III. Of Mechanical Powers in ge- neral. I 9 ° IV. Of the Balance. 19 2 V. Of the Leaver. 19} VI. Of the Pulley. 195 VII. Of the Axis in Peritrochio. 196 VIII. Of the Wedge. ibid. IX. Of the Screw. *97 X. Of the Velocities with which Bodies are raifed. 198 PART VIII. Of Hydrostaticks. 199 THE THE Builder’s Complete Assistant; OR, A Library of Arts and Sciences , <&c. PART L Of Arithmetics Sect. i. Of the fcveral Parts of Arithmetick, and the Notation or Art of expreffing Numbers by Chara&ers, and to read their Values. P JVhat is Arithmetick ? • M. Arithmetick is a Greek Word, and imports an Art or Science, that teaches the Ufe and Properties of Figures, or right Art of numbering. P. What doth right numbering confjl of? M. To denote any given Quantity with proper Characters, and to exprefs them by Words, which is called Notation. P. How many are the Kinds of Notation ? M There are many Kinds of Notation by which Quantity is exprefled, but the jnoft ufual are Literal and Figural. P. What is Literal Notation ? M. The expreffing Numbers by Letters, and is therefore called Literal , and which was anciently made ufe of by the Hebrews or Jews, Chaldeeans , Syrians , Arabians , Perfians , and others of the Eaftern Nations. The Greeks alfo expreffed Numbers by divers of their alphabetical Letters, and initial Capital Letters of fome of their numeral Words, as, n n«7!t, Five ; A CsUx, Ten ; JE Exaloy, an Hundred; X Xbuoy, a Thoufand; M Mvgio», Ten Thoufand. P. Pray what Kind of Letters are ufed now for Notation ? M. Divers of the Roman Capitals ; which Method, ’tis very reafonable to believe, the Latins firil took from the Greeks , as is very evident from the initial Letters of feveral of their numeral Words, as follows, viz . The Capital C, which is the initial Letter of Centum, the Latin Word for an Hundred, is now ufed of itfelf to fignify an Hundred. P. But pray how is half an Hundred exprejfed? M. By the Capital L. P, Pray why is half an Hundred exprejfed by an L ? M. You muft underftand, that the ancient Form of the Capital C was thus written E ; and as it then fignified an Hundred, therefore the Ancients fignified half a Hundred by one half Part of it, as thus L ; which being like unto the Ca* pital L, therefore . Printers take the Liberty to denote half a Hundred by that Letter. P. / * Of NUMERATION. • P. 1 thank you. Sir ; pray proceed. M. I will. The Capital Letter D, which is the initial Letter of Decern (the Latin for Ten), was anciently ufed by the Latins to denote Ten; and one Half thereof, as thus U>, did alfo denote Five. Now as this half Letter hath more of the Likenefs of the Capital V, than of any other Capital* therefore Printers and others have ufed the V (inftead of the half Letter u) for Five ; and to denote Ten, in- ftead of ufing the Capital D, as the Ancients did, they joined together two Vs at their narrow Ends, the one upright, the other downright, in Manner of the Capi- tal Letter X, which now is ufed to denote Ten. Again, as Mille is Latin for a Thoufand, therefore the Ancients ufed the Capi- tal M to denote a Thoufand, as it is now ufed at this Day ; and as the old Cha- radter of the Capital Mwas this (J), whofe right-hand Side being like unto the Capital D, therefore Printers, &c. denote five Hundred by the Capital D. You are alfo to note. That, as this ancient M O) had l'ome Refemblance of the Letter I, placed between two Cs, of which one is turned the wrong Way, as thus CI3, therefore thofe Letters are now ufed by fome to denote a Thoufand, inftead of the Letter M ; and I3 to denote five Hundred, inftead of the Letter D. P. Pray by what Character did the Ancients ufe to denote One ? M. Both Greeks and Latins denoted One by one fingle Stroke, as being the na- tural and moft fimple Character of one fingle Thing ; and therefore One is repre- fented by the Letter I. Now from thefe feveral Characters the following Num- bers are exprefled by the Romans or Latins , viz. I One, II Two, III^Three IV or IIII Four, V Five, VI Six, VII Seven, VIII Eight, IX Nine* X Ten! XI Eleven, XII Twelve, XV Fifteen,- XX Twenty, XXX Thirty, XT Forty, L Fifty, LX Sixty, LXX Seventy, LXXX Eighty, XC Ninety, C a Hundred, CC two Hundred, CCC three Hundred, CCCC four Hundred, D or I3 or Io five Hundred, DC fix Hundred, DCC feven Hundred, M or Cl 3 or clo a Thoufand, I33 five Thoufand, CCI33 ten Thoufand, I333 fifty Thoufand, an HundredThoufand, I3333 five HundredThoufand,CCCCl3330 a Million ; and fo MDCCXXXVIII, or CI3DCCXXXVIII, denotes the Date ox this Year, One Thoufand Seven Hundred and Thirty-eight. P. But pray , Sir , why are Nine and Eleven denoted by the fame Letters ? M. As the I, being fet after the X, thus XI, adds One to it, and makes it Eleven ; fo on the contrary, when the I is fet before the X, thus IX, it leflens its Value One, and therefore fignifies but Nine. For the fame Reafon the I, placed before the V Five, leflens its Value One, and fignifies but Four. The fame is alfo to be cbferved of Forty and Ninety, where the X, being fet before the L Fifty, leflens its Value Ten, and fignifies but Forty ; and being placed before the C, a Hun- dred, leflens its Value Ten, and fignifies but Ninety. And it is further to be ob- ferved that fome ufe I IX to denote Eight, and XXC to denote LXXX, as being more concife. The V and L are never repeated, nor are any of the other Characters repeated more than four Times ; the I repeated four Times, thus IIII, fignifies Four ; but the V is Five, not 1 1 III . So likewife 4 Cs, thus CCCC, fignify four Hundred; but five Hundred is denoted by D, or I3, asaforefaid, and not by CCCCC. Now as by this Method the Notation of Numbers by Letters is very- tedious, the Figural Notation was invented, as being more expedite. P. What is Figural Notation M. The Manner of exprefling Quantities by the Ten Arabick Characters, viz. I2 345 67890, which fignify as follows, viz . 1 one, 2 two, 3 three, 4 four, 5 five, 6 fix, 7 feven, 8 eight, 9 nine, o nought, Cypher, or nothing. P. Pray how long may thefe Characters have been ufed in England ? M. Dr. Wallis in his Treatife of Algebra, Page 12, fays they were introduced about the Year One Thoufand One Hundred and Thirty, which is fix Hundred and eight Years fince. P . How many d fin Cl Parts is Arithmetic h divided into ? M. Three ; two of which are properly called Natural, and the third Artificial. P. What are thofe which you call Natural ? M. The o/ numeration. wlifh ^&ar, and :CV ™ h <“ Ufi u the " K F '> re '' ten Times, and together fignifv oneSundred ‘ T ‘° n" m' U mcreafe the 10 two Hundred, 300 three Hundred, 400 foui- Htmief f &h n o£d. CyPKer ’ as ICOO > >t will iucreafe the .oot^Tim^ I'mIS fourVhltd Ma ^ r, i“° r f ; fi „ e ;r Tho t nd> r°A ee Th ° *** *°°° the 1000 will be made ten, Thoufand a T *! t T* " C S >her - as tlu,s ‘doo, they always increase he T^tT* ^ ° r m ° re C M her *> lue is Increafed fror^° as thus *00000, the Va- ner the Addition of Another S Z\tnr ^ 1 “* fo “ Man- themunto ten Hundred Thoufand, which *Sd“ mL^ 0000 ’ o'"' haS f be T Addition ofthe Cy. ± tr'S i Add, t, on of Cyphers,’ unto o,fc Thoufand »f Unity, by the I Unit 4 0/ NUMERATION. I, Unit io, Ten . ioo, one Hundred, or ten times Ten 1000, one Thoufand, or ten times one Hundred ioooo, ten Thoufand . r , i 00,000, one Hundred Thoufand, or ten times ten Thoufand 1.000. 000, one Million, or ten times one Hundred Thouiand 10.000. 000, ten Million , . 100.000. 000, one Hundred Million, or ten times ten Mi io 1000.000. 000, one Thoufand Million, or ten times one Hundred Millions. P. J perfectly underjiand the Increafe that is made by adding of a Cypher- or Cy- pher J any of thedine Figures ; but how are Numbers to be underload when divers of them are placed together , either with or without Cyphers , 12, or I2 3 , This I will make very eafy to you. juft fa the very fame Manner as is done by the Addition “ Cyphers . 1 Example if to i, I place 2, as thus 12, they together figmfy Twelve, which is no more than the Value of the z, placed in the Cypher’s Ai^ add^d to l O i^ud fo in like Manner 13 fignifies Thirteen, 14 Fourteen, 15 T f een, We. to lit 23 fignifies Twenty-three, 25 Twenty-live, «*. So ’tis plain tl«tthe h rlttf the following « a tr:eTo7oX or Periods by which they are to be numbered or exprelled. t*-, e «! t °l .2 43 O C5 O O O * W CO •"rt CJ o .2 Jh Ctj r u HTU, htu, Thds. Billions of Bil- E. lions. F. «-M o g nd o .2 3 < > Thds. Units. 444» 444> HTU, HTU, Thds. Mill, of Mill. C. D. ° e. .-a o a S 1 1 j I 2 j each reprelenting the number of Times that they are contained in the next greater Meafure. Thus in a Mile, there is contained I 90 ° 8 o Barley-corns Length, or 63360 Inches ; or 5280 Feet; or 7946! Fle- mijh Ells; or 1760 hards; or 1408 Englijh Ells; or 880 Fathom; or 320 btatute Poles ; or 293! Woodland Poles ; or 251 1 Foreft Poles ; or 80 Chain ; or 8 furlongs ; as exhibited in the lowermoif Line of the Table. Again, ad- mit it was required to know what Number of Inches is in a Furlong, life, pro- ceed as follows. f h ft, find out the word Furlong on the right hand Side of the Table, and bringing yo;ir Eye level therefrom, until you come under the Title (or Column or ) Inch, in the fecond Column, there Hands 7920, the Number of Inches con- tained in one Furlong, as required. Likewife under the Title Foot , Hands 660, the Number of Feet in a Furlong ; and fo in like manner, any other Meafure, or its Paits of which ’tis compofed, may mod readily be found by Infpe&ion. ' J* !r * I am ver y much obliged to you for your painful information of Long Meafures, pray be plea fed to infrudl me in like manner , of fuch Square Meafures as are ufed in Bufinefs. M. I will. 1 he fquare Meafures by which Works, fee. are performed and fold, are, the Yard, the Foot, the Square, and the R.od, or Pole. P. What do you mean by the Foot ? You have already infor?ned me, that a Foot is a Length containing 12 Inches, vJhich I already know* . ^ ' s ver y true a F° 0l: ’ n Length is 1 2 Inches as you fay, but a fquare Foot is a fquare Space, each Side thereof equal to 12 Inches ; that is, as well in Length as in Breadth, and contains 144 fquare Inches. C P. Pray / 1 4 Of ADDITION. P. Pray explain ibis to mi in fucb a manner as 1 may rightly uniirjlani it ; for atprefe.nl I cannot comprehend your Meaning. M. I will, ’tis very eafily underftood : Suppofe that the Square ABC u,fig. IX PI. LVII. have each of its Sides equal to one Foot in Length. And each Side divided into 12 equal Parts ; that is, the Inches in a Foot. Then I fay that if from the leveral Divifions of the Inches at the Points 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, it, and 12, in the Sides A B and A C, right Lines be drawn from Side to Side, refpe&ively oppolite, they will form 1 44 little Squares or fquare Inches : For every one thereof will be an Inch Iquare precifely. Hence it is, that a fquare Foot contains 144 fquare Inches. P. Sir , I underjland you perfectly well, and upon the fame Principle I fuppofe that a fquare Yard contains fquare Feet. M. ’Tis true. For if each of the Sides of the Square A B C D, fig . I. P l LVII. contain one Yard, divided into 3 equal Parts 01 Feet, as at the Points 1 , 2, 3, 4, feV, and the Lines 3, 7 ; 4, 8 ; and 1, 5 ; 2,6; be drawn, they will divide the fquare Yard into nine little Squares, each containing one fquare Foot. Therefore ’tis evident, that one fquare Yard contains 9 fquare Feet, as you have before ob- ferved. P. Ifee plainly that it doth , hit what do you mean by the Meafiure which you call a Square ? . . r M. A Square of Work is a Space containing 100 fquare Feet, or it is a iquare Figure whole Sides are each equal to 10 Feet, divided into Feet, as the Square A B C D, fig. IT. PI. LVII. r .. ,, P. I underjland you. Sir , and fee that if from the federal rejpeatve Divijions of Feet , there be right Lines drawn , in the fame Manner as before in the fquare Foot and Yard . , they will generate 100 little Squares , each equal to one fquare Foot. Pi ay wherein is this kind of Mcaf ure uf ed? _ M, In the Menfuration of Flooring, Tyling, Slating, tSc. which you 11 be acquainted with, when you come to learn Menfuration. P. Thank you, Sir, pray be pie afed to proceed. M. I will. The next fquare Meafure is a Rod or Pole, and is a Space contain- ing 2725. fquare Feet. P. Pray Jhew me its Figure. T rm . M. I will. Suppofe each Side of the Square A B C D, fig. III. PL L\^II. to contain 16 Feet 4, divided into 16 Feet and 4 as at the Numbers 1, 2, 3, &c. in the Sides A B and AC. Then I fay, that if the right Lines 1 a, 2 b, 3 c, 4 d, 5 e , feV. be drawn, as before in the preceding fquare Figures, they will generate 256 complete little Squares, each containing one fquare Foot, as in the Scheme. P. Very well , Sir, but I thought that you faid, that a fquare Rod contained 272 fquare Feet , and herein you produce but 256. M. Within the Square of 16 Feet \ A B C D, there are 32 little long Squares, or Oblongs, marked with Dots ; now as each of thefe Oblongs is 6 Inches m Breadth, and one Foot in Length, therefore one of them is equal to but 4 of one of the whole fquare Feet. And confequently the 32 being taken together, are equal to but 16 whole Feet. Now if unto 256 You add 16 The Sum is 272 The Number of Feet in one Rod. And laftly the little Square r, at the corner D, having each of its Sides equal to but 4 a Foot or fix Inches, therefore it contains but p of a Foot ; that is, 36 Inches, which is but V of 144, the Number of fquare Inches (as before proved) in one fquare Foot. Therefore the Sum of the whole Square is equal to 272 $ Feet. Having thus defined unto you thefe feveral fquare Meafures, I will in the next place proceed to feme Examples of the Addition of fuch Quantities. XIV. Of ADDITION. J 5 XIV. Addition offquare Feet. Note, That as the fquare Foot is divided into Quarters, therefore one Quarter contains 36 fquare Inches. Sq. Feet. Qrs. Sq. In. Coiled into one Sum thefe feveral Quantities, viz. O23 3 3 1 Rule. - 729 2 29 For every 36 Inches carry 80 1 1 to the Quarters, for every - 7 1 0 3 S' 4 Quarters carry 1 to the fq. Feet, which add as Integers. 1005 1 H Sum I muft alfo inform you, that the fquare Foot is by fome divided into 1 2 equal Parts, each being 1 2 Inches long, and one Inch in breadth, as abcdefgbiklm, VIII. PI. LVII, Which Parts are called long Inches, of which you’ll fee more at large in crofs Multiplication hereafter. By this manner of dividing the fquare Foot, its Parts are moft readily added together, as following. Example. Sq. Feet. Inches, Colled into one Sum thefe Quantities, viz. 999 Rule, 10 10. For every 12 Inches, carry 7 6. 1 to the Feet, and add them 2 3 as Integers. CO O' O' Sum 1120 2 XV . Addition of fquare Yard Meafure, Example. Yds. Feet, Colled into one Stup thefe Quantities, viz. Rule, For every 9 Feet carry 1 to the Yards, and add them as Integers. Sum 62 2 Colled into one Sum thefe feveral Quantities of Floor- < ing, viz. Sum quare Meafure , as Flooring , See, Sq. Feet. - 10 •95 I2 3 • 75 ' Rule, 70 • 8 3 Add up the Feet as 70 .96, and for every too 10 2 5 the Squares, 100 S° 9 3 396 27 XVII. Addition of fquare Polo Meafure. Rote, 1 hat in Bufinefs the fradional Part or one Quarter of a Foot is generally omitted, and then, 4 The Rod is taken at 272 Feet, The 3 Quarters 204 The Half ^6 The Quarter 68 C 2 TO Of ADDITION. 1 6 Rod. Qr. Ft. To add thefe Quantities together, this is the Rule. For every 68 27 3. 30 Feet carry one to the Quarters^ and for every 4 Quarters carry 1291 38 to the Rods. *b 3 * 2 The Quantities in the Margin, are given to be added into one 8 x 9 Sum. Sum 82 1 11 XVIII. Addition of Land Meafure. Note , That an Acre of Land contains 160 Poles or 4 Roods, and each Rood 4 ° fquare Poles or Perches. Rd. P. 3 - 39 - 2 21. 1 3 Sr 3 - 3 ' 8 - 1. 30 Collett thefe feveral Quan- tities into one Sum, viz. Rule. For every 40 Poles carry x to the Roods, for every 4 Roods carry 1 to the Acres, which add as Integers. Total lie 2 03 A Tallc of fquare Meafures. Sq. Inches 144 Feet 1296 9 Yard s 14400 100 ii| Squares 39204 2725 3 °f S) J f 2 2S Stat 1568160 10890 1210 108-re 40 6272640 43560 484c 43 Ire 160 (Roods 4|Acre 'J'hus have I delivered unto you, all the ufeful fquare Meafures, by which all rpanner of Iuperficia| Works are meafured. I fhall now exhibit them together in this Table, which by Infpettion will fliew their refpettive Quantities, in any of thq Jeffer Meafures, P. Pray Jhew me the Ufeofthis Table. M. I will. Suppofe it was required to know how many fquare Feet were con- tained in one Acre Qf Land, Statute Meafure ; looking in the fecond Column, un- der the Title Feet, and again# the word Acre, lla: ds 43560, the Number of fquare Feet in an Acre of Land, as required ; and fo in like manner any othei Meafure in the Table. P. I thank you, Sir, I underfand it, and fo in like manner an Acre of Land is equal to 6272640 fquare Inches , or 4840 fquare Yards, or 435/0 Squares of 100 Feet \ or 160 fquare Statute Poles’, or 4 Roods. And a Rood is equal to 1568160 Jquare Inches, or to 10890 fquare Feet , or to 1210 fquare lards ; or to 108^5 Squares of 100 Feet ; or to 40 Statute Poles. M. ’Tis very "yvell, I find you have a right Underflanding of its Ufe. I fliall in the next place proceed to inform you of the feveral Weights ufed in this King- dom, from which the feveral Meafures of Capacity were taken. P. I thank you, Sir, but if there ’were any fond Meafures necejfary to. follow the fuperficial or fquare ones now taught me, I Jhould gladiy know them. M. There are folid Meafures which you are to be informed of, as the folid Foot, which contains 1728 folid or cubick Inches; and the folid kard, which contains 27 folid Feet, a Tun of Timber 40 folid Feet, and a Load 50 folio feet. • But Of A D D I T I O N. I? But beiore I can inform you thereof regularly, I muft teach you Multiplication or otherwife you cannot fo readily, or fo well underhand them. P. I ajk pardon for my Forwardnefs. Pray proceed to the account of the Weights you was mentioning. * \-\n L 1 wil1 ' The or5 g’ nal of a11 Weights ufed in this Kingdom was a Grain of Wheat, taken out of the middle of a well-grown £ar, and being well dried ox or them were called and made a Penny Weight , 20 Penny Weights one Ounce *and 12 Ounces one Pound. See the Statutes of 5 1 Hen. 3. 3 1 Ed. 1 . 1 2 Hen. 7. ’ But the Moderns fince the making of thefe Statutes, have divided the aforefaid Penny Weight into 24 equal Parts, which are called Grains , and is the leaft Weight now jn common ufe. 6 P . What do you call this original Weight ? M. It is called Troy Weight, becauie his fuppofed to be the fame that was uiecl by the Trojans. By this Weight OJbrigbt, a Saxon King of England, 200 Years before the. Conqueft, caufed an Ounce Troy of Silver to be divided into twenty pieces, which at that time were called Pence, and at that time an Ounce of Silver was worth but 20 Pence. This value of Silver continued unto the Reign of Hen. VI. who to prevent the enhancing of Money in foreign Parts, valued the Ounce at thirty Pence, and ac- cordingly divided the fame into thirty pieces, each being then a Penny. And the old lennys made in OJbrighf s time went then for Three-pence half-penny each and which continued unto the time of Ed. IV. who valued the Ounce of Silver ^°/7 Pef ] C f’ and divid ? d into 4° P^ces each a Penny, and then the old Penny of OJbrigbt s went for Two-pence. J This continued until the Reign of Hen. VIII. who valued the Ounce of Silver Tv n CC ’ wd,cd was not ^tered until the Reign of Queen Eliz. who valued the old Penny of OJbrigbt at Three-pence ; fo that at that time, all Three-pences coined by Qfieen Eliz. weighed but one Penny Weight, every Six-pence two Penny Woght, and the like proportion in Shillings and other pieces then coined 1 rhislaft Alteration was the caufe of the Ounce Troy of Silver to be valued at 60 Pence, or five Shillings, as it now is at this Time. ' we^hed^ Wei§h£ • JeWeIS ’ Gek1, Silver ’ C ° rn ' Bread > and a11 Liquids are XIX. Addition of Troy Weights. • Thefe Weights are added together l?y the following Rule. for every 24 Grains carry 1 to the Penny Weights, for every 20 Penny Weights carry 1 to the Ounces, and for every 12 Ounces carry 1 to die Pounds'’ E X A M P L E. lb- Oz. Pw. Gr. 22 1 1 20 l 6 9 1 1 17 20 S 3 4 l 6 1 1 7 8 Sum 77 5 2 1 1 But befides thefe common Divifions of the Troy Pound, I find in the Prefent ate of England, for the \ear 1699, that the Grain is fubdivided as following ’mz. 1 Gram is divided into 20 Mites, 1 Mite into 24 Droites, 1 Droite into 20 w"ig“tismade? n0t mt0 2 * Blanks > from which thc Mowing Table of Troy Blanks Of ADDITION. Blanks 24 Periot 480 20 Droite 11520 480 24 Mite 230400 9600 480 20 Grain 5529600 230400 M _0_. -N 00 O 24 Penny Weight 102892000 4608000 230400!. 9600 480 20 Ounce izjPound 1 1*234*704,000 55,296,000 27648ooji 15200 576oj2 4 o Thefe Weights are added together by the following Rule. For every 24 Blanks carry one to the Periots, for every 20 Periots carry 1 to the Droites, for every 24 Droites carry one to the Mites, for every 20 Mites carry j to the Grains, for every 24 Grains carry one to the Penny Weights, for every 20 Penny Weights carry one to the Ounces, and for every 12 Ounces cany 1 to the Pounds. E x AMP L E . 12 20 ^24 20 24 20 24 Pounds Oun. Pwts. Gr. Mites Droit. Per. Blanks To 16 7 9 18 *5 x 7 19 23 Add f 20 ^ 02 5 1 1 7 l 9 *3 16 18 14 16 18 16 15 11 Total 40 0 fo 4 1 1 1 14 2 Now feeing that by this Table a Grain contains two Hundred and thirty Thou- fand, four Hundred Parts, or Blanks, furely the Commodities that have been fold by thefe Weights mull have been of great Value, as that they themfelves mult be real Atoms, or at leaft as fmall as one particle of the fineft kind of band. But this Example I give you more for Curiofity than real Ufe. By Avoirdupoife Weight all kind of heavy Commodities are fold, as Iron, Lead, Brafs, Copper, Grocery Wares, &c. whofe fmalleft part is called a Dram, of whjch i r 6 make one Ounce, 16 Ounces one Pound, and 112 Pounds one Hun- dred Weight, 56 Pounds half a Hundred, and 28 a quarter of a Hundred. P. Pray is the Pound Troy , and Pound Avoirdupoife equal to each other ? M No. The Pound Avoirdupoife is equal to one Pouncj two Ounces and I2 Penny Weights, of Troy Weight, and the Pound Troy is but nearly 13 Ounces 2 Drams and a half of Avoirdupoife ; fo that the Pound Avoirdupoife is about two Ounces 13 Drams and a half Avoirdupoife, greater than the Iroy Pound, which is very near a fixth part of a Pound Avoirdupoife, lefs than a Pound Avoirdupoife. And therefore fix Pound of Bread, which is fold by Troy Weight, is very little heavier than five Pound of Butter or Cheefo, which is told by Avoirdupoife Weight. So that thofe who believe the Pound Troy and Found Avoirdupoife to be equal, are much miftaken ; but, however, though the Pound Troy is lefs than the Pound Avoirdupoife, yet the Ounce Troy is heavier than the Ounce Avoirdupoife, for 292 which are the number of Penny Weights in 14 Ounces 12 Penny Weights, which are equal to one Pound Avoirdupoife, being divided into 16 equal Parts, each Part will be found to be but 18, and five 11 x- teenths, which are the Number of Penny Weights in one Ounce Avoirdupoife, of which the Ounce Troy contains 2©. ^ S Of A D D I T I O N. I 0 r ght Troy ’ is ,o ° ib - the haif Hundred s ° ib - and The following is a Table of Avoirdupoife Weights. Drams 16 Ounce 23 6 16 Pound 7168 448 28 Quarter of a Hundred H 33 6 896 5 6 2 Half a Hundred 28672 1792 1 12 4 2 A Hundred 20 A Ton Weight 57344 ° Oh 1 ^ 1 co 1 0 2240 80 40 XX. Addition of Avoirdupoife Weight, Thefe Weights are added together by the following Rule. For every 16 Drams carry i to the Ounces ; for every 1 6 Ounces carry i to the rounds ; for every 28 Pounds carry 1 to the Quarters j for every 4 Quarters carry x to the Hundreds ; and for every 20 Hundred carry 1 to the Tons. Example. The firft weighed The Second The Third The Fourth The Fifth Anfwer To. H. Qc p. Oz. Dr. 7 Z S 3 2 7 r 3 H 2 Ji 2 J 4 10 1 1 I demand the to- 9 r 9 1 9 7 J S tal Weight of the 27 18 *5 2 -5 12 9 whole. J 7 1 I X *5 67 0 0 5 *3 00 r. Fray vjby is this kind of Weight called Avoirdupoife ? , ' F rom the French, Have your Weight ; that is, you (hall have full Weight and therefore the 12 Pounds over and above 100 are added. * "* ' ™y * s t ^ je ^' ro y Pound divided in any other manner than the preceding ? J . No : but the Troy Ounce is, by Apothecaries, as follows, viz. Firft into S calIed Scru « a Sc,uple mt ° »* — -20 Grains 3 Scruples! 8 Drams -12 Ounces “is equal to- ■ 1 Scruple,^ 1 1 Dram, 1 1 Ounce, 'l Pound, whofe Marks, • or Characters, A 3 1 are ih ■at A 1 uuna, * v jf) “’bought anVfold bTlvoWu^ Weight C °' n P 0Un Tierce C I 4273 and half v. 97 02 { Tertian 1 19404 1 Hogfliead 1 14553 | Pipe 1 29106 I Tun J ,.58212 J Four Veflels contain thefe feveral Quantities, I de- mand the total Sum of the whole. Total Example I. XXI. Addition of Beer Meafure. Ba. K. F. G. 3 1 3 8 ° 3 7 1 2 C o 2 : 16 Rule. For every 9 Gallons carry t to the Firkins ; for every 2 Firkins carry x to the Kil- derkins ; for every 2 Kilder- kins carry 1 to the Barrels, which add as Integei s. Note, That altho’ 4 Firkins of 9 Gallons each, which are equal to 36 Gallons, make 1 Barrel of Beer ; yet a Barrel ot Ale contains but 32 Gallons. Example II. B. Hhs. Gal. 3 1 Four Veflels contain thefe ieveral Quantities, I de- mand the Total. S 3 61 2 7 39 Total 26 1 34 Rule. For every 63 Gallons carry 1 to the Hogflieads ; for every 2 Hogiheads carry 1 to the Butts, and add the Butts as Integers, XXII. Addition of Wine Meafure. j Tu. Pip. Ter. Tier. Run. Gal. Four Veflels con- 5 1 1 1 1 37 tain thefe Quan- 7 0 1 1 0 40 tities, .1 demand 7 1 0 0 1 41 the Total. . 2 1 1 1 1 39 Total 24 I o 33 Rule. For every 4 Qu. carry 1 to the Gal- lons ; for every 42 Gallons carry 1 to the Runlets ; for every 2 Runlets carry 1 to the Tierces ; for every 2 Tierces carry 1 ^the^ 21 for every 2 Pipes, 0/ ADDITION. Tertians, for every 1 and half Tertian, carry 1 to the Pipes carry 1 to the Tons, and add the ’Fons as Integers. XXIII. Addition of Dry Meafure. Note, That 4 Bufhels make one Sack or Comb ; 2 Combs 1 Quarter ; 4 Quar ters one Chaldron of Corn ; 5 Quarters 1 Wey ; 2 Weys 1 Laft. Example I. Chal. Quar. Comb. Bufh. Gall. Collect thefe feveral f 9 1 1 3 7 Quantities into one< 5 0 o 26 Sum, Rule. to the Loads, and add them as Integers. Total 15 087 XXX. Addition of Timber and Planks. Note , That 50 folid Feet make Collett into one Sum thefe leveral Quantities of Timber, •viz. 1 Load. Loads. Feet. 2 45 Rule . 3 42 For every 50 Feet carry r 2 28 to the Loads, and add them 1 37 as Integers. 2 49 Total 14 01 Note , That in the Addition of Planks, 1 Inchin thicknefs, every ’600 Feet is 1 Load ; ot 1 Inch and half thicknefs, 400 Feet ; of 2 Inches thicknefs, 300 Feet; of 3 Inches thicknefs, 200 Feet; and of 4 Inches thicknefs, 150 Feet. XXXI. Addition of folid Yards. Note , That in 1 folid Yard there are 27 folid Feet. ,■ Yards. Feet. { 3 26 Rule. 2 17 For every 2 7 Feet carry 1 to ' 4 25 theYards,andaddthe Yards 5 26 as Integers. Total 17 13 XXXII. Addition of Motley. Note , That l. Hands for Pounds ; s. for Shillings ; d. for Pence ; and qr. for Farthings ; with relpett to Libra , which lignifies a Pound ; Solidus , a Shil- ling ; Denarius , a Penny ; and dfuadrans , a Farthing. Collett into one Sum thefe feveral Sums, viz. 1 . s. d. qr. Rule. I 2 *7 11 3 For every 4 Farthings car- 1° 9 2 ry 1 to the Pence ; for 12 7 8 3 every 1 2 Pence carry 1 to ' 15 19 1 1 2 the Shillings ; for every .123 16 7 3 20 Shillings carry 1 to the Pounds, which add as In- l 7 S 18 1 1 tegers. Total As I have thus gone through the Addition of all that is necelfary, I fliall there- fore conclude this Letture with obferving, 1. That a Load of Earth is one folid Yard. 2. A Hundred Weight of Nails, Iron, Brafs, fcV. is 112 Pound. 3. A Hundred of Deals or Nails, fix Score, or 120. 4. A Bundle of 5 Feet Laths, 100 ; and 4 Feet in Length, 120, which fhoplcf be 1 Inch and half in Breadth, and half an Inch in thicknefs. 5. A Fodder of Lead, is 19 Hundred and a half, or 2184 Pounds Avoirdupoife. 6. A Bale of Paper is ten Reams ; a perfett Ream, 20 Quires, or 500 Sheets ; 1 perfett Quire, 25 Sheets. Dz /• *4 Of ADDITION. A folid or Cubicle Foot of line Gold, weighs' D tto of Standard Gold Ditto of Qujcklilvejr Ditto of Lead lb. ioths. 4 Ditto of line Silver Ditto of Standard Silver ■ — — Ditto of Copper Ditto of Brais Ditto of Call Brafs Ditto of Steel Ditto of Iron Ditto of Tin — Ditto of Marble — Ditto of Glafs — — Ditto of Ala better Dhto of Ivory — — — Ditto of Clay moderately moift Ditto of iandy Gravel of common Moifture Ditto of Sea Water Ditto of River W 7 ater — » — Ditto of Dry Oak 1352 1 180 874 7°7 693 658 562 521 500 490 477 457 196 161 117 X13 1 1 2 96 64 62 57 ®. A circular Foot contains 1 13 fquare Inches, and one feventh of an Inch ; that is, there are fo many fquare Inches in a Circle of one Foot Diameter, which I call a circular Foot, for the fame reafon as a fquare foot, which makes a iquare Figure, is called a fquare Foot. 9. A food or Cube Foot, is 1728 folid Inches, that is, 12 Times 144, the fquare Inches in a fquare Foot. jo. A Cylindrical Foot is 1573 folid Inches, and two fevenths of an Inch ; that is, 12 times 1 1 3 and one feventh, the fquare Inches in a circular Foot. XI. A Cylindrical Foot of Sea Water, is about 50 Pound and a half, and of frefh Water, about 49 Pound and one tenth. L E C T. HI. Of Subtraction. ft Subtra&ion is a Rule for finding the Difference of any two Numbers, by JVJl* taking or drawing the lefler from the greater, whereby the Difference or excels (which is called tile Remainder) will appear. P. Pray what is particularly to be obferved herein ? M. To take care that you always place the lefler Number under the greater, and that the Units, Tens, fsY. of the Subtrahend, be placed under the Units, Tens, Hundreds, &c. of the given Number. P. p r ay which of the two Numbers are the Subtrahend and which the given Number ? . M. The greateft is the given Number, and the lefler the Subtrahend, as this Example makes plain. 1 . Subtraction of Integers. Example I. Place your Numbers as in the Margin, and be- ginning at the right hand, fay, 1 from 7, there remains 6, ■ ■ d 2 from 8, remains 6 Note , if in Subtracting, any want flrould hap- pen, then borrow 10 from the next Place, and for every 10 fa borrowed, carry x to the next Place. From 87 the given Number, take 21 the Subtrahend, rem. 66 the Difference Excefs. or Opetation 2 5 Of SUBTRACTION. Operation. Firft, 3 from 4 remain Example II. 1 ; fecondly, 4 from 2 I cannot, but I bought 7524 Bricks 4 from 1 2 (for borrowing 10, makes and have fold 5643, what are remaining the 2, 1 2) and there remains 8 ; third- _ ly, 1 I borrowed, and 6 is 7, from Anfwer x88t remain. 5 I cannot, but (borrowing 10 as be- fore) 7 from 1 5, reft 8. Laftly, 1 that I borrowed, and 5 is 6, from 7, reft 1 fo the remains is 1881. 1 * P. Pray bozO jhall I know when Subtraction is truly performed? M. All ki'ids of Subtraction are proved, by adding the Subtrahend and Remains together, which will be equal to the given Number, if the Subtra&ion be truly performed. As for example, if 5643, 1 the Subtrahend, be added to 1881, 7524 given Number, the remains, their Sum will be 7$ 24, 5643 Subtrahend. as in the Margin, which being equal to the given Number, the Subtraction 1881 remains, is theiefore truly performed. _ 75-4 Sum of given N° and Subtrahend. Other Examples for Practice. From 547213 From 772543279 take 439197 take 619987654 remains 108016 remains 152555625 Proof 5472 *3 Proof 7725432 79 II. Subtraction of Money. Example I. Example II. j. d. ! 1 • /. *. d. From 19 1 1 3 From 272 19 JO take 17 9 2 take 229 15 9 rem. 2 2 1 rem. 43 4 t Proof 19 1 1 3 Proof 272 in 10 Example III. Example IV. 1 . s. d. P I. s. d q. From 275 5 I 2 From 927 5 7 I take 199 ! 9 3 3 take 832 19 8 3 rem. 75 5 9 3 rem. 94 5 10 2 Proof 275 512 Proof 927 5 71 In thele laft two Examples, at the Farthings you borrow 4, and carry 1 to the Pence, becaufe 4 Farthings make one Penny ; at the Pence you borrow 12 and carry 1 to the Shillings, becaufe 12 Pence make 1 Shilling; and at the Shilling? you borrow 20 from the Pounds and carry 1 to the Pounds, becaufe 20 Shillings make 1 Pound. The Pounds you lubtraCt as Integers. * III. Subtraction of Inches and 10 tbs. Example I. Inch, joths. From 372 09 take 245 09 Example II. Inch, ioths. From 342 5 take 213 9 Example III. Inch. ioths. From 971 2 take 725 9 rem. 127 00 rem. 128 6 rem. 245 3 Proof 372 09 Proof 342 5 Proof 971 2 Here, 26 Of SUBTRACTION. Here, at the ioths, you borrow lofrom the Inches, and carry i to the Inches, becaufe 10 Parts make i Inch. IV. Subtraction of Feet and Inches. Example I. Feet. Inch. From 279 5 take 217 ii rem. 61 6 Example II. Feet. Inch. From 972 3 take 165 7 rem. 806 8 Example III. Feet. Inch. From 999 8 take 777 11 rem. 221 Proof 279 S Proof 972 3 Proof 999 8 Here, at the Inches, you borrow 12 Inches, or 1 Foot, irom the Feet, and x’to the Feet, becaufe 12 Inches make 1 Foot. carry Example I. From 217,9 take 206,5 rem. 011,4 V. Subtraction of Decimals . Example II. From 2754,8 take 1234,9 rem. 1519,9 Example III. From 29,02 take 561,97 rem. 167,05 Proof 729,02 Proof 217,9 Proof 2754,8 Here you fubtraft the whole as Integers. VI. Subtraction of Duodecimals. M ^uode^ and as thefe Examples are of Feet, Inches, and^ParoTyouTre to fbf/rve, that the Inches are each divided mto ,a Parts, the fame as the Feet are divided into 12 Inches. Example I. Example II. Feet. Inch. Parts. Feet. Inch. Parts. From 12 7 3 From 9 2 9 9 take 07 11 11 take 73 n 11 Example. III. Feet. Inch. Parts. From 67 2 9 take 27 10 10 rem- rem. 18 9 10 rem. 39 11 Proof 12 7 3 Proof 92 9 9 Proof 67 2 9 Here at the Parts and the Inches, you borrow 12, and carry 1 to the Inches, and to the Feet, becaufe 12 Parts make 1 Inch, and 12 Inches 1 loot. VII. Subtraction of Yards, Feet and Inches. Example I. Example II. Example III. Yds. Feet. Inch. Yds. Feet. Inch. Yds. Feet. Inch. From 127 2 7 From 72 1 3 From 172 o 5 take 97 2 11 take 43 2 9 take 99 10 rem. 29 8 rem. 28 rem. 7.2 Proof 127 2 7 Proof 72 1 3 Eroof 172 o 5 Here, vou borrow 12 at the Inches, and carry I to the Feet ; and borrow 3 at the Feet, and carry 1 to the Yards ; becaufe 12 inches make 1 Foot, and 3 feet 1 Yard. VIII. 27 Of SUBTRACTION. VIII. Subtraction of Cloth Meafure. Example I. Example II. Example III. Yds. Qurs. Nails. Yds. Qurs. Nails. Yds. Qurs. Nails. Irom 527 i 2 From 270 a 1 From 127 3 2 take 399 3 3 take 21 1 3 2 take 96 3 3 rem. 127 1 3 rem. 58 rem. 30 3 3 Proof 527 1 2 Proof 270 2 1 Proof 127 3 2 Here, at the Nails, and at the Quarters, you borrow 4, and carry 1 to the Quarters and to the Yards, becaufe 4 Nails make 1 Quarter, and 4 Quarters 1 Yd. IX. Subtraction of Ftemijh Meafure. Example I. Example II. Example III. Ells. Inch. Ells. Inch. Ells. Inch. Prom 2794 22 From 37255 18 From 32594 22 take 1372 26 take 27532 20 take 12345 rem. 1421 23 rem. 09722 25 2 3 rem. 20148 26 Proof 2794 22 Proof 37255 18 Proof 32594 22 Here, at the Inches, you borrow 27 and carry 1 to the Ells, becaufe 27 Inches naake one Flemijh Ell. ' X. Subtraction ecaufe 60 Seconds make 1 Minute, and 60 Minutes 1 Hour. XIX. Subtraction of fquare Feet and fquare Inches. Example I. Example II. Example III. Feet. Inch. Feet. Inch. Feet. Inch, rrom 729 19 From 927 075 From tee 139 take 672 141 take 526 133 take 274 141 rem. 56 22 rem. 400 084 rem. 280 142 Proof 729 19 Proof 927 75 Proof 555 139 Mere at the Inches you borrow 144, and carry 1 to the Feet, becaufe that 144 Iquare Inches make 1 fquare Foot. XX. Subtraction of fquare Feet and long Inches. Example I. Example II. Example III. Feet. Inch. Feet. Inch. Feet. Inch, from 127 7 From 271 05 From ce r 04 take 93 1 1 take 136 10 take 449 10 rem. 33 8 rem. 134 7 rem. 105 6 Proof 127 7 Proof 271 5 Proof 555 04 Mere at the Inches you borrow 12 and carry x, becaufe 12 long Inches (which are each 12 Inches long and 1 wide) make 1 fquare Foot, XXI. Subtraction of fquare Tard Meafure . Example I, Example II. Example III. Yds. Feet. Yds. Feet. Yds. Feet. From 73 7 From 92 3 From 27 5 take 5 1 8 take 57 7 take 18 g rem. 21 8 rem. 34 4 rem. 08 6 Proof 73 7 Proof 27 3 Proof 92 j Here at the Feet you borrow 9 and carry 1, becaufe 9 fquare Feet mak« 1 fquare Yard. XXII. Subtraction of Jo lid Yards. Example I. Example II. Example III. From 45 2t From 72 20 From 9-’ 19 take 36 26 take 49 23 take 96 24 rem. 08 22 rem. 22 22 rem. 00 2 z Proof 4 e 2 1 Proof 72 20 Proof 9 / jo : at the Feet you borrow 27 and carry i, ard. becaufe 27 folk! Feet make i E XXIII. 3 ° Of SUBTRACTION. XXIII. Subtraction of Squares, as of Flooring , &c. Example I. Squ. Feet. From 2 5 98 take 15 99 Example II. Example HE rem. 09 99 Squ. Feet. Squ. Feet, From 29 1 1 From 127 86 take 2 1 75 take 97 99 rem. 07 3 6 rem. 29 87 Proof 2Q 1 1 Proof 127 86 00 and carry x, becaufe 100 fquare Feet Roofing, ' Tyling 1c. Proof 25 98 e at the Feet you borr re of Work, as of Floe.— a7 OJ y XXIV. Subtraction of Land Meafures. I. Of fquare Statute Poles. r T? _ __TT T7v A 1\7 1 2 Proof 127 5 5 5 Here at the Grains yon borrow 24, at the Penny Weights 20, and 12 at the Ounces, becaufe 24 Grains make 1 Penny Weight, and 20 Penny Weights I Ounce, and 12 Ounces 1 Pound. XXVII, Subtraction of Apothecaries Weights. Example I. Example II. lb.Oun.Dr.Scr. Gr. lb. Oun. Dr. Scr. Gr. From 1 2 9 4 1 From 127 5 3 i 17 take 9 1 1 7 2 *9 take 99 10 7 2 18 rem. 2 9 4 1 16 rem. 27 b 3 1 19 Proof 1 2 9 4 1 *5 Proof 127 5 3 1 17 Here at the Grains you borrow 20, at the Scruples 3, at the Drams 8, and 12 at the Ounces, becaufe 20 Grains make x Scruple, 3 Scruples x Dram, 3 Drams 1' Ounce, and 12 Ounces x Pound. XXVIII, Subtraction Example I. Hun. Qurs. lb. Oun. Dr. From 27 2 21 13 10 take 21 3 27 13 15 of Avoirdupoife Weights . Example II. Hun. Qurs. lb. Oun. Dr. From 25 1 18 7 11 take 17 3 24 14 T2 rem. 05 2 21 13 11 rem. 07 1 2 1 8 *5 Proof 27 2 21. 13 10 Proof 23 1 18 7 1 1 Here, at the Drams and at the Ounces you barrow 16, at the Pounds 28, and 4 at the Quarters, becaufe 16 Drams make i Ounce, 16 Ounces 1 Pound, 2 3 Pounds 1 Quarter of a Hundred, and 4 Quarters 1 Hundred. XXIX. Subtraction of Beer Meafure. Example I. Example II. From 27 0 0 2 X From 22 57 take iS 1 1 3 3 take 18 b2 rem. 08 0 0 7 2 rem. 03 58 Proof 27 0 0 2 1 Proof 22 57 In Example I. borrow 4 at the Quarts, 9 at the Gallons, 2 at the Firkins and at the Kilderkins, becaufe 4 Quarts make 1 Gallon, 9 Gallons 1 Firkin, 2 Fir- kins 1 Kilderkin, and 2 Kilderkins 1 Barrel. In Example II. at the Gallons borrow 63, becaufe 63 Gallons make 1 Hogf- head. E 2 XXX, 32 Of SUBTRACTION. XXX. Subtraction of Wine Meafures. Example I. Example II. Tuns Pipes Tier. Gall. Tuns Pipes Tier. Gall. From 57 o i 35 From 20 o 1 27 take 52 » 2 40 take 15 i 1 Pi rern. 4 0 * 37 rem. 4 0 2 28 Proof 57 0 1 35 Proof 20 0 1 27 Here at the Gallons you borrow 42, at the Tierces 3, and 2 at the Pipes, becaufe 42 Gallons make 1 Tierce, 3 Tierces 1 Pipe, 2 Pipes 1 Ton. XXXI. Subtraction of Dry Meafure. Example. Quarters Sacks Bulh. Pecks Gall. Quarts. From 50 o 2 2 o 2 take 39 1 3 3 1 3 10 0 ' 2 2 0 3 5° 0 2 2 0 2 Here you borrow 4 at the Quarts, 2 at the Gallons, 4 at the Pecks and Bufhels, and 2 at the Sacks ; becaufe 4 Quarts make 1 Gallon, 2 Gallons 1 Peck, 4 Pecks I Bufhel, 4 Bufhels 1 Sack, and 2 Sacks x Quarter. XXXII. Subtraction of Timber. Example I. Example II. Example. III. Loads From 123 take 1 1 7 Feet 44 49 Loads Feet From 57 '38 take 26 39 Loads Feet From 75 38 take 25 47 rem. 005 45 rem. 30 49 rem. 49 41 Proof 123 44 Proof 57 33 Proof 75 38 Here at the Feet you borrow 50, becauie 1 Load of Timber contains 50 folid Feet. XXXIII. Subtraction of Plank 1 Inch thick. Note, 600 Square Feet at one Inch thick, make 1 Load. Example I. Example II. Example III. Loads Feet Loads Feet Loads Feet From 127 425 From 372 47 2 From 725 500 take 38 599 take 263 5 2 5 take 632 584 rem. 88 426 rem. 108 547 rem. 092 5r6 Proof 127 425 Proof 372 472 Proof 725 500 Here at the Feet you borrow 600, becaufe 600 Feet make 1 Load, as aforefaid. Note , If the Thicknels of Plank be 1 Inch and half thick, then borrow 400 ; if two Inches thick, borrow 300 ; if three Inches thick, borrow 200 ; and lahly if four Inches borrow 1 50, becaufe '1 Inch and |T thicknefs 2 Inches I make one 3 Inches [ Load of 4 Inches J Plank. XXXIV. • Feet at' 33 Of SUBTRACTION. Example I. Loads Bricks From 27 49 1 take 13 499 XXXIV. Subtraction of Bricks . Note, 500 make 1 Load. Example II. Loads Bricks From 14 057 take 21 451 Example III. Loads Bricks From 23 372 take 14 428 rem. 13 492 rem. 01 106 rem. 08 444 Proof 27 49 1 Proof 14 057 Proof 23 37 2 make i Load. Example I. Hund. Bags From 27 19 take 14 24 XXXV . Subtraction of Lime . Example II. Hund. Bags From 22 19 take 17 21 Example III. Hund. Bags From 18 it; take 11 21 rem. 12 20 rem. 04 23 rem. 06 J 9 Proof 27 19 Proof 22 J 9 Proof 18 *5 Here at the Bags you borrow 25 and carry x, becaufe 2 5 Bags (which ought to »e a Bufhel each) make a Load of Lime. Example I. Loads Bufh XXXVI. Subtraction of Sand. Example If. Loads Bufh. From 18 16 From 21 1 1 take 15 *7 take 20 16 rem. 02 J 7 rem. 00 J 3 Proof 18 16 Proof 21 1 1 Example III. Loads Bufh. From 29 12 take 25 13 rem. 03 15 Proof 29 12 XXXVII. Subtraction of Lime* Example. Months Weeks Days Hours Min. Seconds From 11 2 20 20 41 53 take 10 3 26 23 59 59 Proof 00 2 2 1 20 4 1 54 11 2 20 20 4 1 53 . J 7 " uvv , * iiiiiicient ixumDer or Examples of all the va- rious Kinds of Bufineis in general, and which I think are much more copious than has been yet taught by all the Matters that have wrote on Arithmetic T fliall now proceed to Multiplication. ’ LECT. Of Multiplication. P What is Multiplication ? • M : B y Multiplication is meant an Increafe, and therefore to multiply is to increafe from a fmall Number to a greater ; and which being confidered, is no more than the adding of divers Numbers together* For 34 7 7 7 Of MULTIPLICATION. 21 For if 3 times 7 be added together the Sum is 21, as in the Margin : And if 3 be multiplied into 7, the Product is 21 alio. Hence ’tis plain that Mul- tiplication is nothing more than a compendious Manner of adding Numbers together, and therefore may be called fhort Addition. P. Pray , w hat is principally to be objerved herein ? jj/f t Three Numbers or Members, which are called the Multiplicand, the Multiplicator or Multiplier, and the Product. P. Pray , what is the Multiplicand , Multiplier , and Produft ? M. In every Multiplication, there are always two Numbers given to be mul- tiplied into each other, which are called the Multiplicand and the Multiplier, or Multiplicator, either of which being placed uppermoft is called the Multiplicand, and the lower the Multiplier ; as for Example, if 8 be multi- plied into -9, as at A, then 8 is the Multiplicand and 9 the Multiplier ; or if 9 be multiplied into 8, as at B, then 9 is the Multiplicand, and 8 the Multiplier, and the Number 2, ariling by 9 times 8, and by 8 times 9, is called the Product. But however as it is belt to make the greateft Number of the two the Multiplicand,' therefore it is molt ufually done, ob- ferving to place the Units, Tens, &c. of the Multiplier, un- der the Units, Tens, fee. of the Multiplicand. A 8 Multiplicand 9 Multiplier 72 Product B 9 Multiplicand 8 Multiplier 72 Product I. Multiplication of Integers. P. Bovj is Multiplication performed ? ' _ . M. The Multiplication of Integers is performed by the following Rules. R die I. Write down the Multiplicand and Multiplier under each other as aforefaid, and draw a Line under the Multiplier to leparate it fiom the 1 roduct, that arifes from its fir ft Figure. R U L E II. Multiply every Figure of the Multiplier into the Multiplicand, obferving as you proceed to carry one for every Ten, to the next Place, and fet the Remains under it, and the Produces arifmg from thefeveral Figures of -the Multiplier being added together, their Sum is the general Produbt of the whole Multiplication. Rule III. When the Multiplier confifts of many Figures, as in the following Example, the Produbt arifing from each Figure is to be placed by itfelf in fuch manner that the firft or right hand Figure thereof may hand under that Figure of the Multiplicator from which the Laid 1 rodubt aiifes. Thefe will be made Familiar by the following Example. Exam p l e . Multiply 7 2 54, by } 3 49, which place as in the Margin. Begin with 9 the firft Figure of the Multiplier, and thereby multiply ali the Figures in the Multiplicand as follows. Firft fay 9 times 4 is 3b, let down 6 and carry 3 > fot the three Tens ; then fay 9 times 5 is 45, and 3 I carry is 48, fet down 8 and carry 4 ; then 9 times 2 is 18, and 4 I cany is 22, fet down 2 and carry 2 ; then 9 times 7 is 63, and 2 I carry is 65, which being the laft in the Multiplication therefore fet down 65, and that Produbt will be 65286, as at A. Ptoceed in the lame man- ner to multiply the remaining three Figures of the Multiplier, 4, 3, and 7, into the Multiplicand, and their 1 rodubts will bo a s at B, C and D, and which with that of A, being added to- gether, will be 53,309,646, the Produbt requited. Rule IV. When Numbers given have one or more Cyphers at the right Hand, the Mul- tiplication may be performed, without Regard being had to the Cyphers, until the Produbt of the other Figures be found, to which they are then to be a nnexe d. 7 2 54 7349 65286 A 29016 B 2 1 762 C 50778 D 53,309,646 35 In Multiplication it is of very great ufe to know readily the Product of any two of the nine Digits or Figures ; for which Purpofe this Table mull be learnt perfedtly by Heart. Multiplication Table. The Ufe of this Table is eafy. Suppofe the Product of 8 is requir- ed ; look for 8 on the Side and 9 on the Top, and againft thofe Numbers in the Angle of Meeting is 72, the Produft required. So 7 times 9 is 63, and 5 times 12 is 60, as in the Angles of Meeting you will find, and fo of all other Numbers. I 2 3 4 5 6 7 8 9 1 2 2 3 4 6 6 9 8 1 2 10 l S 1 2 18 H 2 1 16 24 OO ^ w fS 24 3 6 4 8 12 16 20 24 28 32 36 48 5 10 l 5 20 2 5 3 ° 35 4 c 45 60 6 12 18 24 30 3 6 42 48 54 72 — — — — — — — — 7 14 2 1 28 3.5 42 49 5 6 6 3 84 8 16 24 32 40 48 5 6 64 72 96 — — — — — — _ - — — 9 18 27 3 6 45 54 63 72 81 108 12 24 36 48 60 7 2 84 96 108 144 Examples. 36 Of MULTIPLICATION. Examples for Practice. Example I. Example II. Example III. Mult. 27960 Mult. 972403 Mult. 7235 Uy 200 By 30007 By 1000 7235000 Prod. 5592,000 Prod. 6806821 ■ i.. . - 1 2917209 — — 29178896821 Prod. In the firft Example I contradfted my Work, by placing the 2 of the Multiplier Tinder the Units of the Multiplicand, which fhould always be done, when the other Figures of the Multiplier to the right Hand are all Cyphers. In the fecond Example I contracted my Work, by omitting the Cyphers in the Multiplier, .d multiplying only by the 7 and the 3. In the third Example, I add three Cyphers to the Multiplicand, becauie one neither multiplies or divides. Multiplication of Integers may be performed without giving any Trouble to the Mind, in carrying on the Tens, according to the Rule I. as follows. E x A m p l e I. Multiply 8342 by 7, as in the Margin. Operation. Firft, 7 times 2 is 14, which fet down ; then 7 times 4 is 28, which fet down, 2 before the 1, and 8 under the 1 ; then 7 times 3 is 21, fet 2 before the 2, and 1 under; then 7 times 8 is 56, fet 5 before the laft 2, and 6 under ; laftly, add the two Numbers 52214, and 618 together, as they ftand, their Sum will be the true Pro- duct required. 8342 7 52214 618 58394 E X A 98254 37 2 9 871436 1285 1 10108 8640 651328 3645 2201 12 7465 m p l e II. Multiply 98254, by 3729, as in the Margin. . The Operation of this Example is the fame as the laft, only it is 4 times repeat- ed ; and when the Produft of any Figure is lefts than 10, place a Cypher in the Place, where if it had made 10, or more than 10, the Figure for 10, or above 10, mull have flood, as you will fee in the Product that arifes by 2, the fecond Fi- gure of the Multiplier. ^ Product of the 9. ^ Product of the 2. | Produft of the 7. | Product of the 3. 366389166 Produtt of the whole. For a Proof of this manner of working, I have fubjoined the fame Example, worked 98254 A 3729 after the common Method, as at A. 884286 196508 687778 294762 Produdft of the 9. Produft of the 2. Prod u£t of the 7. Product of the 3. 366389166 Product of the whole as before. As I have thus explained the Multiplication of Integers, you are to obferve, that therein is this Analogy , viz. As an Unit is to the Multiplier, fo is the Mul- tiplicand to the Product, P. Pray 37 . ... 0/ multiplication. Po Pray explain this, for at prefent I don' t conceive tv bat you mean . A?. I will : by this Example. Suppoiing one Load of Timber colt 50 Shil- lings, how much will iz Loads coft ? 3 l \ P ° ad 5 be multiplied by 50 Shillings, aS in the Margin, the Product 600 Shillings, is the Anfwer : and therefore one Load being confidered as an Unit, bears the farbe Proportion to c© Shil- lings, the Multiplier, as iz Loads, the Multiplicand, do to 6co • Shillings the Product. K' r 'L: Cl ' a - y truc ' Sir . ’ P ra y proceed, ;for you make Multiplication a Pleafure to me. you ’ how 1,1 Cal “ *>“ “r “«** Contragt-ion I. To multiply any given Number ( fuppofe 547 J by XI, Rule. Set down the Multiplicand twice, the lower one bone 7 removed one place, either towards the right or left Hand, as at A and B, whereat A ’tis placed one place towards the left Eland, and at B, one place towaids the right Hand. 12 600 547 B 547 6017 6017 Contraction II. To multiply any given Number (fuppofe J92C) by 12, obfervina'^c^f^ f * ie p ‘&j 1Ies in the Multiplicand, by the Units in the Multiplier, next on the ri? m- U R ° C ^ ec ’ t( ? add that figure of the Multiplicand, which {lands Fx ,mn ^ T , ’ U e ° daa 0t the Fi s ure you multiply by. As for Example, multiply 7925, by 14, as in the Margin. . r J J g InthVrTr ? b 2 °’ fe i d ° Wn °’ and cirr y 2 J then 4 times 2 is deb* Hand of V 7 - l8 h ID 5 3nd 5 . 3t being the next Figure on the right 7925 flandot 2, which you are then multiplying, make 15, fet down 5 jJ and carry r , then 4 times 9 is 36, and 1 I carry is „ • arid * he 1 ™ thC d Ri “1 t « *’■ “ ** fet ll0 "' n 9 » which ftand next on 11ml Olv id r’ \° the Prodna ° f the Figure you multiply by : as for Example, multiply 99725, by 1 15, as in the Margin. r j j r > T * irl l» 5 T times 5 . IS 2 5 > Pet down 5 and carry 2 ; then 5 times 2 is ’ an tair y is 12, and 5 at ^ is 17, fet down 7 and carry 1 - then 5 times 7 is 35, and 1 I carry is 36, and 2 at b is 38, and 5 at a is 43 ; let down 3 and Carry 4 ; then 5 times 9 is 45, and 4 I carry .I 49, and 7 a f f is 56, and 2 at b is 58, fet down 8 and carry q , then 5 times 9 is 45, and 5 I carry is 50, and 9 at d\% 59, and 7 at - ' 18 f ’ fet dovvn 6 f hd cair y 6 i then 6 I carry, and 9 at , is 15, and 9 at d is 24, let down 4 and carry 2 > which being added to 9 at makes n, which fet down, and which makes the Product 1 1468375, as required. ' Contraction IV. To multiply any given Number (fuppofe 725432 ) by ior, 102, 103, 104, faY. Rule. Multiply the Figures in the Multiplicand by the Units of the Multblier r!lrH y0U H Pr0Ceed ’ add that F J gUre ° f y ° l]r Multiplicand that {lands next S ExamSr d ’ ° nC ’ Unt °v the Prodl?aof that Figure you multiply by : as for Example, multiply 725432 by xo 9 , as in the Margin. P / / edeb a 99725 ^5 11468375 F Firft, 3 S 0/ MULTIPLICATION, fe dcba Fir ft, 9 times 2 is 18, fet down 8 and carry 1 ; then 9 times 3 U 725432 27, and x I carry is 28, fet down 8 and carry 2 ; then 9 times 4 is 109 36, and 2 I carry is 38, and 2 at a is 40, fet down o and carry 4; . then 9 times 5 is 45, and 4 1 carry is 49, and 3 at b is 52, letdown 2 79072088 and carry 5 ; then 9 times 2 is 18, and 5 I carry is 23, and 4 at c is 27, fet down 7 and carry 2 ; then 9 times 7 is 63, and 2 I carry is 65, and 5 at i is 70, fet down o and carry 7 ; now 7 I carry, and 2 at e is 9, fet -own 9 ; and becaufe you have nothing to carry to the 7 at/, therefore fet down ", and the Product will be 79072088, the Product required. II. Multiplication of Decimals. M. Multiplication of Decimals, both in placing the Multiplicand and Multi- plier, is the fame as the Multiplication of Integers, only when your Work is completed, you muft obferve, that with the dafti of your Pen you cut off as many places of Decimals in your Product, as there are places of Decimals both in your Multiplicand and Multiplier, and in cafe of want in yourProdudt, prefix Cyphers to the left Hand. . , , ^ , T It is alfo to be obferved, Firft, that it will be convenient to make that Num- ber the Multiplicand, which contains the mod Places, though fometimes it may be lefs in Quantity. Secondly, that if the Multiplicand and Multiplier be both Decimals, that is, both Parts of Integers, the Produft will be a Decimal. Thirdlv, if Multiplicand and Multiplier be mixed, that is, Integers and Decimal Parts of Integers, the Product will be mixed. Laftly, it the Multiplicand t.nd Multiplier be mixed, and the other a Decimal, the Product will be fometimes mixed, and fometimes a Decimal. Example I. Cf Decimals alone. » 743 2 > 7 '3 Example II. Of Integ. and Decimals. 7» 2 345 b 2 5 22296 743 2 52024 Facit 5299016 361725 144690 7 - 345 Facit 9I042125 Example III. Where the Multiplicand is mixed, and Multi- plier a Decimal. 72,4072 >357 5068494 3640350 2172216 2518693594 In Example I. of Decimals alone, the Produft is, 5299016, that is, it is r 2 00016 Parts of an Integer, or 1, divided into 10,000,000 Parts, becaufe the Denominator of every Decimal conti fts of as many Places of Cyphers annexed to r, as there are Places in the Decimal. . In Example II. there being 7 Places of Decimals m the Multiplicand, I tnere- fore have cut off 7 Places of Figures from the Produft, and the Product is 9 Inte- gers, and ,04312 5 Parts of an Integer, divided into 10,000,000 Parts. ^ In Example III. I have alfo cut off 7 Places of Decimals, becaufe there are 4 Places in the Multiplicand, -and 3 in the Multiplier, and the Product is 25 In- tegers, and ,8693594 Parts of an Integer, divided into 10,000,000 Parts. III. Multiplication of Duodecimals, vulgarly called Crofs Multiplication . . As in Decimal Multiplication, the Integer is divided into 10 ; io here it is di- vided into 12 Parts, as a Shilling into 12 Pence, or a Foot into 12 Inches. - In the following Examples I fuppofe the Integers to be Feet ; and the Duodeci- mals Inches. As this kind of Multiplication may be performed, as well by taking the aliquot or even Parts of 12, out of the Multiplicand, as will be immediately fliewn, as by multiplying the Multiplier into the Multiplicand ; before I proceed any farther, you are to obferve, that the aliquot (which are the even) Parts of a Foot are as follow, viz. In -12 there is twice 6, three times 4, four times 3, fix ' times Of M U L T I P L I C A T I O N. 39 times 2, eight times x and f, and 12 times 1 ; and therefore, 6 is a half, 4 is one third, 3 is one quarter, 2 is one fixth, 1 and half one eight, and 1 one twelfth. * In this kind of Multiplication there is a great Variety, as follows. I. To multiply Feet , Inches , and Parts, into Inches , by aliquot Parts. Rule. Place under the Multiplicand, the Number of Times that the aliquot Multiplier can be had, in the Feet, Indies and Parts, obferving to begin at the left Hand, and for every one that remains at the Feet, more than the Times that the aliquot Multiplier can be had in them, to add 12 to the Inches, and fo the like to the Parts, &c. In Example I. 6 being contained twice in 12, I therefore lay the two’s in 20 is 10, the two’s in 8 is 4, and the two’s in 6 is 3 ; fo that the Produd is 10 Feet, 4 Inches, 3 Parts. In Example H. 4 being contained 3 times in 12, therefore I fay the three’s in 16 is 5 times, and 1 remains, let down 5 under the 16 ; then the 1 remaining being a Foot, equal to 12 Inches, I add it to the 8 Inches which makes 20, and then fay, the three’s in 20 is 6 times, fet down 6 under the Inches, and carry the 2 Inches remaining to the Parts, which 2 being equal to 24 Seconds, and added to the 7, makes 31 Seconds, wherein I find three 10 times, and I remains, therefore I fet down 10 under the Seconds, and the 1 being one third of 3, the aliquot Part, is equal to 4 Seconds, and the Produdt to 5 Feet, 6 Indies, 10 Parts, 4 Se- conds. In Example III. 3 Inches being contained 4. Times in 12, I therefore fay the fours in 27 is fix times, fet 6 under 27, and 3 remains, equal to 36, and 11 is 47, which contains 4 I I times, fet 1 1 under Inches, and remain 3, equal to 36, and 9 is 45, which contains 4 11 times ; fet 1 1 under Parts, and the remaining 1, being one Quarter of 4, the aliquot Part is equal to 3 Seconds, and the Product to 6 Feet, 1 1 Inches, 1 1 Parts, 3 Seconds. II. To multiply Feet , Inches , and Parts , into Inches , by multiplying the Multiplier into the Multiplicand. Rule. Firfi, Place a Cypher infiead of an Integer, under the Parts of the Mul- tiplicand, and the Inches of the Multiplier, one place farther to the right Hand. Secondly, multiply the Inches of the Multiplier, into the Parts, Inches, and Feet, of the Multiplicand, as if they were Integers or whole Numbers, carrying i for every 12, and fetting down the fil'd remains, when any, under the Figure you multiply by, £sY. To illuftrate the preceding Rule by aliquot Parts, I have here made ufe ofth« following Examples. Example II. Example III. Feet. Inch. Parts. Feet. Inch. Parts. 16 8 7 27 11 9 o 4 03 Example I. Feet. Inch. Parts., Multiply 20 8 6 By 00 6 Inches Product 10 4 3 Example II. Feet. Inch. Parts. Multiply 16 8 7 By 4 Inches Product 56 10 4 Example III. Feet. Inch. Parts. Multiply 27 11 9 By 3 Inches Product 6 11 11 Example I, Feet. Inch, Parts, 20 8 6 o 6 10 3 9 5 10 11 11 Irs Example I. 6 times 6 is 36, fet down o, and carry 3, then 6 times 8 is 48, F x an tden 7 times 11 is 77, and 3 Multiply 4. carry is 80, fet down 8^and carry 6, which put one Place to the left. Secondly 9 times 5 is 45, fet down 9 and carry 3 ; then 9 times 6 is 54, and 3 I carry is 7 7 ; fet down 9 and carry 4; then 9 times 1 1 is 99, and 4 I carry is 103, fet down 7 and carry 8. Third- y* 11 ^ mes j> is 55, fet down 7 and ^ cairy 4 j then 11 times 6 is 66, and 4 * S / 0 7 ^ et down 10 and can T 5 5 then 1 1 times 1 1 is 121, and t I carry is 126 Thirds? d ° Wn ’ and the Produdt ,s * 3 6 Feet > 1 Inch, t Part, 5 Secondhand ir t/,^T U T r m f ters » ot whether the Feet , Inches , , or be firft multiplied, A f -at their t efpettvve Products are but duly placed. ” " V. To multiply any Number of Feet and Inches into any Number of Feet and Inches. t F,r / Tlldtip y the Fect mto themfelves as Integers. Secondly, inftead of multiplying the Feet into the Inches, take the aliquot Parts of a Foot, as often LJ /a Can , b f ^ OUnd , ln theFeet > that fond diagonally againft them (by Rule I. heieot), and halve them when required. Thirdly, the Inches multiplied in* themfelves, every 12 is an Inch, the Remains are Parts. P In Example I. the Feet being firft multiplied into the 6 8 8 11 l 7 9 9 126 10 7 136 i 1 r ' 3 1 1 Example I. Feet. Inch, Multiply 272 3 By 2 18 6 yj, - t, UHU lilt ivet, proceed to the Feet into the Inches as following : 'u u S 3 Flc ^ es ’ s tde 4 td °f 12, therefore by Rule I. hnd the fours in 2 1 8, faying the 4’s in 2 1 is 5 times, and 1 remains, fet down 5 as at A j and then fay, the 4 ’s in is 4 times, and 2 remains, fet down 4," and the 2 remaining being the half of 4, therefore fet down half one for it, viz. 6 Inches ; then will 54 Feet, 6 Inches, yhich is equal to a quarter Part of 218 Feet, be the Prooudt of 218 Feet, multiplied into 3 Inches. Se- condly, as 6 is contained twice in 12, therefore to mul- W 2 76 Feet into 6 Inches, is no more than to take its Ijalf, or fay, the 2’s in 2 is r, fet down 1 at B, and fay, the 2 s in 7 is thrice, fet down 3 next after the 1, and carrying the 1 to the 2, which makes 12, fay, the 2’s in 1 2 is 6 tiqies, fet down 6, and then the Product of 2 72 Feet, into 6 Inches, will be 1 36 Feet. Thirdly, multiply the 6 Inches into 3 t0 ' InCh> 6FartS * and 'he whole Product i, 59486 Fee* 2 176 272 344 A 54 6 B 136 1 6 59486 7 6 7 Inches and 6 Parts. In Example II. Firft, as 9 Inches is three quarters of 1.?, therefore to multiply 531 Feet into 9 Inches, firft take the half of 531, which is 265 — 6 as at A, and the half of 265 — 6, which is 132 — 9 as at B. .< Secondly, as 2 is the fixth of 12, therefore take the 6’s in 752, which is 125, as at C. Thirdly, the Inches into themfelves, make 1 Inch 6 Parts, and the Whole 6 Part"*^ 6 ^* ^ Pxam ^ e F * s 399^35 Feet, 4 Inches, Example II. F. I. Multiply 752 9 % 53 1 ^ 399835 4 6 Ih 4 2 O/MULTIP LI CATIO N. Example III. F. Multiply 392 325 I. t 1 1 i960 784 1176 A 27 B 196 C 98 D65 4 Eo T I 127786 g It Example IV. F. I. Multiply 524 4 37 2 5 1048 3668 1572 A 124 B 131 C 87 Example F. Multiply 723 By 312 1446" 7 2 3 S 6i 5 , A 256 B 43 C 361 D x 20 8 195270 5 x 8 V. I. 7 8 T 370957 4 8 Example VI. F. I. 2 >79 JO 172 10 518 1813 259 A 86 B 57 4 C 129 6 D 86 4 8 4 44907 10 4 In Example III. Firft, as 1 Inch is the twelfth Fart of 12, therefore to multiply 375 Feet into 1 Inch, take the 12 s in 325 Which are 27 1, as at A., Secondly, fo multiply 392 Feet into 11 Inches, fir ft take the half of 392, which is 196 as at B, whofe half Is 9829 at C ; and which two Produ£ts are equal to 39 ^ hcet multiplied into 9. Now as the remains to 1 1 is 2, which is a fixth Part of 12, therefore by Rule I. take the 6’s in 392, which is 65 Feet 4 Inches. Laftly, the Inches multiplied into themfelves make 1 1 Parts, and the feveral ProduftS added, are 127786 Feet, 5 Inches, and 11 Parts. In Example IV. Firft, as 4 Inches is the third of 12, therefore to multiply 372 Feet into 4 Inches take the 3’s in 372, which are 124 as at A. Secondly, as in 5 there are two aliquot Parts of 12, viz. 3, which is a 4th, and 2 which is a 6th, therefore firft take the 4» s in 524, which are 1 31 as at B, and then the 6’s in <24, which are 87 4. Thirdly, the Inches into them- felves, are 1 Inch 8 Parts, and the whole Prodilft 195270 Feet, 5 Inches, 8 larts. In Example V. Firft, as in 7 Inches there are two aliquot Parts of 12, viz. 6 which is a halt, and 1 which is a 1 2th, therefore to multiply 512 Feet into 7 Inches firft take the halves or 2’s in 512 Feet, which aie 256 as at A, then the 1 2’s that are in 43 as at B. Secondly-, as in 8 there are alfo 2 aliquot Parts of 12, viz. 6 and 2, therefore to multiply 723 Feet into 8 Inches, firft take the halves or 2’s in 723, which are 361 6 as at C, and then the 6’s, which are 120 6 as at E. Thirdly, the Inches into themfelves, are 56, equal to 4 Inches 8 Parts, and the whole Produdt 370957 Feet, 4 Inches, 8 Parts, In Example VI. Firft, as in 10 there are two aliquot Parts of 12, viz. 6 which is half, and 4 which is a third ; therefore to multiply 1 72 Feet into 10 Inches, firft take the halves or 2’s in 172 Feet, which are 86 as at A, and then the 3’s, which are 57 4. Secondly, there being the fame aliquot Parts m the other 10 Inches, therefore firft take the halves or 2 s in 2 59 Feet, which are 1 29 6, as at C, and then the 3 s, which are 86 4, as at D. Thirdly, the Inches 10 into ic equal to 100, are equal to 8 Inches, 4 Parts, and the whole Produft to 44907 Feet, 10 Inches, and 4 Parts. Thus 0/ M U L T I P L I C A T I O N. 4 3 Thus have I given you a Number of Examples in all the Variety of odd Inches That can happen, which being well underftood will make the Menfuration of Superficies and Solids very eafy and delightful to every Capacity. And, in con- fideration that fome Kinds of Works are performed by Yard Meafure, I (hall therefore, before I proceed to Divifion, {hew the Multiplication of Yards and Feet. IV. Multiplication of Yards and Feet, Note, i ft, That Yards multiplied into Yards produce Yards. 2dly, That Yards multiplied into Feet every 3 is a Yard, the remains more than 3 are lono- Feet, a long Foot is one Foot in Breadth, and 3 Feet in Length. 3dly, Feet multiplied into Feet produce Parts, which are fquare Feet, 3 of which make 1 long Foot aforefaid. Operation. Firft, .he Yards being multiplied as Integers, to Feet, multiply 251 Yards into 1 Foot, as 1 is the third Part of 3, the 273 1 Feet in a Yard, therefore take the thirds of 251, which are 251 2 83 2, as at A. Secondly, as 2 is two thirds of 3, therefore to — multiply 272 Feet into 2 Feet, take the thirds, twice in 273, which are 91, and 91 as at B and C. Thirdly, the Feet multi- *3 -» plied into themfelves are two Parts, and the whole Product is a o equal to 68788 Y'ards, 2 Feet, and 2 Parts. R 3 2 The next Thing in Order, to conclude this Le£ture, is to {hew, 68788 2 How to prove Multiplication. Rule. Make that which was your Multiplier your Multiplicand, and then multiplying as ufuaJ, if the Produd be the lame, your Work is true: if not ’tis falfe. * L E C T. V. Of Division. D ivifion is nothing more than a compendious Subtraction ; for as many Times as the Divifor can be fubtraded out of the Dividend, fo many Units is the Quotient. In Divifion there are four principal Parts to be obferved ; iz. 1. The given Number which is to be divided, called the Dividend.' 2. The o-iven Number by which the Dividend is to be divided, called the Divifor. 3. & The Number arifing from the Number of Times that the Divifor is contained in the Dividend, which is called the Quotient. And laftly, a Number that fometimes happens to remain when the Divifion is ended, lefs than the Divifor, which is called the Remains. Divifion in general is per- I formed by this Analogy, viz. ^7 As the Divifor is to r, fo is the Dividend to the Quotient ; which I {hall illuftrate by the following Examples. Example. ’Tis required to divide 997 2 543 by 37 25; firft place the Dividend and Divifor as at D E, leparated by a Crotchet as F. Alfomake another Crotchet as G to fe pa rate the Dividend from the Quotient. Secondly, make a 7 able of Divifors as in the Margin, thus ift, place 3725 and againft it fet 1; 2dly, q 2032 remains, double 3 7 25, as at A7450, and ) F E G Table of Divifors. ' 2 5 ) 997 2 543 2(2666411,1 H 3 7 2 5 1 f 745 ° : •• : : ah cd e A 7450 2 : K 1 ii 75 3 £2322,5 : : : C 14900 4 h 2235 0 : : : L 18625 5 : : : B 22350 6 2 2475^4: : M 26075 7 k 2235 0 : i N 29800 S l 2404,3 : O 33525 9 22350 n 16932 p 14900 againft 44 Of DIVISION. againft it fet 2, fignifying that 7450 is the Divifor 2 Times. Thirdly, add 3725 and 7450 together, which make 1 1175, as at k, againft which let 3* Fourthly, to J 1175? add 3725, which make 14900, as at f, and againft it fet 4* Fifthly, to 14900, add 3725, which ’make 18625, as at l, and againft it fet 5. Proceed, in like Manner, to add the firft and laft together, until you have repeated the Operations. 9 Times, placing the Number of Times againft each. Or other- vvife, multiply the Diviior 3725, by 2, 3, 4, 5, 6, 7, 8, 9, and their Products will be as againft A, K, C, L, B, M, N, O. This being done, the Work is very eafy, and is thus performed. Firft, as 3725 cannot be had in the firft 3 Figures of the Dividend 997 , therefore under the fourth Figure 2, make a Point; then fay, how often 3725 in 9972 : Look in the Table of Divifors for the leis neareft Number to 9972, which is 7450, againft which ftands 2, as at A. Place 2 in the Quotient as at a , and 7450 under 997 2 > as anc l lubtraiSt 7450 from 9972, the Remains is 2522, as the firft 4 figures towards the left. Band at g. Secondly, make a Point under 5 in the Dividend, which bring down and place againft 2522, as thus, 25225 for a new Dividend* Then fay, how often 3725 in 25225 ; look in the Table of Divifors, for theneareft lefs Number, which is 22350, againft Which ftands 6 ; place 6 in the Quotient, as at b, and 22350 under 25225, as at h, and fubtradf 22350 from 25225, the Remains is 2475, as the firft 4 Figures to the left at i. Thirdly, point the next Figure 4, in the Dividend, and bring it down to 2475, as t l lus > 2 47 S 4 > a ! *"> f°Q a lecond new Dividend. Then fay, how often 3725 in 24754; look in the Table of Divifors, and the neareft lefs Number is 22350, againft which ftands 6, as at B ; place 6 in the Quotient, as at c, arid 22350 under 24754, as at £, and fubtraft 22350 from 24754, the Remains is 2404, as the firft 4 figures to the left at /. Fourthly, point the next Figure 3, in the Dividend, and bring it down to 2404, as thus, 24043, as at l , for a third new Divifor. Then fay, how often 3723 in 24043 ; look in the Table of Divifors for the neareft lefs Number, which is 22350 (as before), againft which ftands 6 ; place 6 in the Quotient, and 22350, under 24043, and the Remains is 1693, as the firft 4 figures to the left at ?i. Fifthly, point the next and laft Figure 2 of the Dividend, and bring it down to 1693, as thus, 16932, as at^, for a fourth new Divifor. Then fay, how often 3725 in 16932 ; look in the Table of Divifors for the neareft lefs Number, which is 14900, againft which ftands 4 ; place 4 in the Quotient, as at e, and 14900 under 16932, and fubtradting 14900 from 16932, the Remains is 3032, and which being the laft Remains, is 2032 Parts of 3725, and which together make a Fraftion thus, f^f, which mult be let in the Quotient, next after 26664, as in the Margin. Note , That as many Points as are placed under the Figures of the Dividend, fo many Figures will be in the Quotient. The Value of this Fraction, or any other, in the Parts of the Integer may b£ found as following. Admit the Integers, in this Example, to be Pounds Ster- Iing. Firft, multiply 2032, the Remains, by 20, th« Shillings in a Pound, as at A, and divide the Pro* dwft 40640, by 3725, the former Divifor, as at 13 , and the Quotient 10 are Shillings, and 3390 remains, as at C. Secondly, multiply 3390, tint Remains, by 12, the Pence in a Shilling, as at D, and divide the Produdt 40680, by 3725, the former Divifor as at E, and the Quotient xo are Pence, and 3430 remains. Thirdly, multiply 3430, ^ ^ e " mains, by 4, the Farthings in one Penny, as at F, and divide the Product 13720, by 3725, as before, and the Quotient 3 are Farthings, and 2 545 remain?, which are 2545 Parts of 3725 of a Farthing, the Farthing being divided into 3725 Parts. The Man- A 2032 20 B 3725)40640(10 Shillings 3 7 2 5 C 3390 rem. D 12 E 372 5(4°68o) i oPence 37 2 5 ' 3430 *4 Of DIVISION. 45 37*5) x 37 2 ° (3 Farth. 1 1 X 7 S 2 545 rem. ncr of reducing this and other Fra&ions, into the leaft equivalent Parts, is taught in Le&ure VIII. It this example be well underftood, it is fully fufficient for performing all Varieties of Cafes in whole Numbers, that can happen, and more efpe- _ eially when you have alfo learned the following Contractions in Divifon. I. When theDivifor is 10, 100, 1000, feV. cut from theDivi- A 10) 732 { o dead, the fame Number of Figures to the right Hand as are B 100) 275 j 43 Cyphers in the Divifor, and the Figures remaining to the Left C 1000) 72 i 354 are the Quotient required. So 7320, divided by 10, I cut off the laft figure o, and 732 remaining to the Left, is the Quotient required, as at A. In like manner, 27743, divided by 100, the Quotient is 275/^ . and 72354, divided by 1000, the Quotient is 72 as at B and C, as the Figures cut on to the right Hand, are fo many Parts of the Divilor. And as in every of thefe Cafes, the Divifor is decimally divided, therefore thefe Remains are Decimal Pi actions ; and though I have here let their Denominators under each for Plainnefs bake, yet in Practice they are to be omitted, and the Fractions annexed to the whole Numbers, as following, viz. 10,732, not and 275.43, not 275 To% ; and 72,354, not 7 - vWarj °f which I have already advertiied you in the preceding Lectures. Ii. When your Dividend and Divifor confft of Cy- phers to the right Hand, cut off an equal Number of Cyphers in both, and then proceed as before taught by 63000, cut off three Cyphers in each, and divide Margin. III. If your Divifor have Cyphers annexed, and your Dividend none, cut off as many Figures in your Dividend, as there are Cvphers in your Divi- ior, and then proceed as before. So to divide 73 2 S 479 by 1200, cut off 79, the laft two Figures in the Dividend, and dividing 73254 .hy 12, the Quotient will be 6104, and 6 remains as in the Margin. The 6 remaining, is* to be placed be for* 79, cut from the Dividend, making it 679, and which is the true remains, and the Numerator of the Fraction /iV®, as annexed to the Quotient. To p rove Divifon . Multiply the Quotient by the Divifor, and to the Product add the Remain* when any, and if the Work be true, their Sum will be equal to the Dividehd. Division of Decimals. Divifion of Decimals is performed in every Refpeft as whole Numbers, and for difcovering the true Value of the Quotient, this is the general Rule : Rule. The Places of Decimal Parts in the Divifor and Quotient, being accounted together, nm ft always be equal in Number with thofe in the Dividend ; and therefore as many iigui es as are cut of in the Dividend , fo many ?nufl be cut of ' in the Divifor and Quotient : or thus ; cut of as many Figures in the Quotient, as will make thofe cut of in the Divifor equal to thofe in the Quotient ; always obferving, that if there be not fo many in the Quotient,* to add Cyphers to the left Hand. And alfo , that if your Dividend be an Integer, or have lefscutoff than in the Divifor, to add Cy- phers to the Dividend, till they are equal. This general Rule admits of four Cafes. 63)000} 7735)000 (122 : So to divide 7735000 7735 by 63, as in the 1200)73254179(6104/’° 72 ... 054 43 6 rem. £ Cafe 46 °f Example. 55,635) 4672,56 5 (182 25.635 : DIVISION. 2 10906 205080 58265 51270 4 2 7 ) 7254,271 427- • 2984 • 2989 • 6995 rcm. Cafe 1. When the Places of Decimal Parts in the Divifor and Dividend are eqhal in Number, as in this Example in the Margin, where both Divifor and Dividend are mixed Numbers, then the Quotient will be all whole Numbers. Example II. Divide 7254,271, by 427, as in the Margin. Here the Dividend is a mixed Number, and the Divifor is Integers, and as here are three Decimals in the Dividend, and none in the Divifor, therefore cut off 012, the la ft 3 Figures in the Quotient, and the Quotient will be 17,012. *47 rem. ,0x25) 7500 (60 750 Example III. Divide 75 by ,0125, as in the Margin. Here the Dividend is Integers, and the Divilor a Decimal ; and feeing that 75, the Dividend, confifts but of two Places, I therefore add two Cyphers to it, making it 7500, that thereby both Divifor and Dividend may be made Fractions, ___ and by their being both of equal Number of Places, there- . fore by Cafe 1, the Quotient is Integers. _ CaCe i When there are not fo many Places of Decimal Parts in toe Di\ idem , as there are in the Divifor, then annex Cyphers to the Dividend, to make them equal, and the quotient will be all whole Numbers, as m iy> 00 ,725) 3425,000 (4724 2900 * * ‘ 5250* * 5057 ’. *. 1750 • 1450 * 3000 2900 100 rem, ,725) 3425, 00000 (4724,13 29°° ' * 5250 • 5057 * 175° 145° 3000 2900 575 rem. , Divide 3425, by ,725, as in the Margin. Now- here the Dividend being Integers, and the Divilor a Decimal, to bring out In- tegers in the Quotient, I add 3 Cyphers to 3425, the Dividend, and the Quo- tient is 4724, and 100 re- mains. But if ’tis required to have the Quotient to a greater Exa&nefs, then I add a competent N umber of Cyphers more to the Dividend. In the follow- ing Example, at A, in the Margin, ’tis required to have two Places of Deci- mals, after .the Integral Part of the Quotient, where Of DIVISION. 47 ’ the Quotient is 4724,13 , and 575 remains ; for by adding two Cyphers more to the Dividend, than was required before to make the Diviior and Dividend equal; and cutting off the fame Number of. Places from the Quotient, leave 13 for the fractional Fart required, and 575 remains. In this Manner, by annexing of a greater Number of Cyphers, you may come nearer to the Truth; but in all Cafes like this, where the Divifor is not contained an exact Number of Times in the Dividend, there will always be a Remainder. Cafe 3. When the Number of Places of Decimal Parts in the Dividend exceed thofe in the Divifor, cut off the Excefs of Decimal Parts in the Quotient. As for Exam- ple, divide 71,4038, by 7,54, as in the Margin ; where the Number qf Decimal Parts in the Dividend is 4, and but 2 in the Divifor ; therefore, as the Excefs is 2, cut oft 47, the lalt two Places in the Quotient, 7» 54) 7D4038 (947 . 6786 Cafe 4. If after Divifion is finifhed, there are not fo many Figures in the Quotient, as there ought to be Places of Decimal Parts by the general Rule, then fupply their Defeat by prefixing Cyphers before the Figures produced in the Quotient. As for Example, divide 13973 by 43. Now here the Dividend is a Decimal, and the Divifor 'is Integers, whofe Quotient is 323. But as in the Dividend there are 5 Places, therefore, according to the general Rule, I prefix two Cyphers before the Quotient 325, mak- ing it ,00325, which is the true Quotient required. o rem. 43) > 1 3975 G00325 129 107 85 2I S 2I 5 o rem. Note, When any Decimal Fra&ion, or mixed Number, is to be divided by an Unit, with any Number of Cyphers annexed, remove the Separatrix as many Places towards the left Hand, as there are Cyphers annexed to the Unit ; fo if 57,27, were given to be divided the Quotient will be Now, from the pieceding Examples, it maybe obferved, firft, That when the Dividend is fupei ior to the Divifor, the Quotient is either Integers, or Integers and Decimals : and laftly, That when the Divifor is fuperior to the Dividend, the Quotient is a Decimal, and which in both Cafes holds good in all other Ex - * amples. EEC T. VI. Of Reduction. R Education is nothing more than Multiplication or Divifion, or both, and its the in whole Numbers is for changing Quantity out of one Denomination into another, as greater into lefs by Multiplication, or lefs into greater by DR vifion, > 0 ' G Exampls* ' 4 8 0/ REDUCTION. Example I. In 5287 fuperficial Feet, bow many fuperfeial Inches? 5278 Here, becaufe 1 fuperficial Foot contains 144 fuperficial Inches, 144 therefore multiply 5-78 by 144, and the Product 700032, as in the Margin, is the Anfwer required. 2 1 11 2 2 1 1 1 2 5278 760032 Example IT. In 760032 fuperficial Inches , bow many fuperficial Feet ? 144) 760032 (5278 Here you divide 760032 the Number given by 144, the fquare Inches in a fquare Foot, and the Quotient is 5278. Now thefe two Examples, which are tonverie to each other, illullrate all that can be done in Reductions, and therefore I need only add the following Rules, by which Redudtions in general may be performed. 1152 1152 o rem. Rule 1. To reduce Pounds into Shillings, multiply the Pounds by 20, the Shil- lings in a Pound, the Produd will be Shillings ; and to reduce Shillings into Pounds, divide the Shillings by 20, the Quotient will be Pounds.. Rule 2. To reduce Shillings into Pence, multiply the Shillings by 12, the Pence in a Shilling, the Produd will be Pence ; and to reduce Pence into Shillings, divide the Pence by twelve, the Quotient will be Shillings. Rule 3. To reduce fquare Yards into Feet, multiply the Yards by 9, the fquare Feet in a Yard, and the Produd will be Feet ; and to reduce fquare Feet into Yards', divide the Feet by 9, the Quotient will be Yards. Rule To reduce folid Yards into folid Feet, multiply the Yards by 27, the folid Feet in a lolid Yard, and the Produd will be folid Feet ; and to reduce folid Feet into folid Yards, divide the Feet by' 2 7, and the Quotient will be folid Yards. Rule 5. To reduce fquare Statute Rods into fquare Feet, multiply the Rods by 2 ~ 2 1 ? t hc iquare Feet in a fquare Rod, and the Produd will be lquare Feet ; and to reduce iquare Feet into fquare Rods, divide the Feet by 272^, and the Quotient will be fquare Rods. Rule 6. To reduce Squares of Roofing, Tyling, &c. into fquare Feet, multi- ply the Squares by loo, the fquare Feet in a Square of Work, and the Produd will be iquare feet ; and to reduce fquare Feet into fquare Rods, divide the Feet by 272F, and the Quotient will be iquare Rods. Rule 7. To reduce folid Feet into folid Inches, multiply the Feet by 1728, the Number of folid Inches in one folid Foot, and the Produd will be folid Inches ; and to reduce folid Inches into folid Feet, divide the folid Inches by 1728, and the Quotient will be folid Feet. Rule 8. To reduce Loads of Timber to folid Feet, multiply the Loads by 50, the Number of folid Feet in a Load of Timber, and the Produd will be folid Feet ; and to reduce folid Feet into Loads, divide the iolid Feet by 50, and the Quotient will be Loads. Thele Rules, which are very plain, being underftood, will render the Reafon of all other Kinds of Redudion eafy to the meaneft Capacity ; and as the Re- dudion of Decimals willbe bell underftood when Vulgar Fradions have been explained, 400 : 288: 1 123 1008 The Golden Rule, or Rule of Three. 4g explained, I fhall therefore proceed to the Golden Rule, or Rule of Three in whole Numbers. LECT. VII. The Golden Rule, or Rule of Three. T HIS Rule loi its excellent Ufe is called the Golden Rule , and teaches to find a fourth Number, which fhall have the fame Proportion to one of three N umbers given, as they have to one another, and therefore is alio called the Rule of Proportion. This Rule is Dircdl, Indirect, and Compound. I. The fingle Rule of Three Direct finds a fourth Number in fuch Proportion to the third, as the fecond is to the firft ; or as the fecond is to the firft, fo is the third to the fourth. f Example I. If the Diameter of one Circlehe 7, and its Circumference 22, what is the Circumference of another Circle pohofe Diameter is 14 Feet '? Rule. Firit place your Numbers as in the Margin, fecondly, D. C. D. C. multiply 14 the third Number by 22 the fecond Number, and divide their Produdt 308 by 7 the firft Number, the Quotient 44 is the fourth Number and Anfvver required. Now you muft obfervethat as the firft and third Numbers are always of like Kinds, viz. both Diameters, fo likewifeare the fecond and fourth Numbers of like Kinds, being both Cir- cumferences, of which the firft is always given, and the laft is the Anfwer required. Note , When the fourth Number is thus found, place it next after the third Number, with two Dots of Separation between them as is done at c. The fame Kind of Separation muft be al- io always placed between the firft and fecond Numbers, as at a. Put between the fecond and third, always place four Dots or Points, as at h. Thefe Points of Separation, fo placed, fignify the following Words, viz. the two Points at a thus fignify the Words, is to , the four Points at h thus : fignify the Words, fo is } and the two Points at c thus :, fignify the Word to ; and therefore the four Numbers, 7 : 22 :: 14 : 44, are thus to be read, nnz. as 7 is to 22, fo is 14 to 44. And in lo like Manner, all other Numbers having the fame Analogy. Example II. If the Circumference of a Circle be 2 2, whofe Diameter is ameter of another Circle whofe Cicumference is 44 ? Here the Nature of the Qneftion requires the two firft Numbers to be placed the reverie to thole of the foregoing Example ; for as there the 4th Number required was the Circumference of a Circle, fo here on the contrary the Dia- meter of a Circle is required. Put the Manner of working by multiplying the third Number by the fecond, and dividing by the firft, is the fame here as before, as is leen in the Margin, where the Quotient 14 is the Diameter required. Now as in both thele and all other Examples in the Rule of Three Diredt, the fourth Number is always equal to, 9r more than the fecond ; fo in the Rule of Three Indiredl, the fourth Number is always lefs than the fec.ond : and as the 4th Number in the Diredt Rule is found by multiplying the fe- coud and third Numbers together, and dividing of their Produdt by the firft Num- ber ; fo on the contrary in the Indiredl Rule you multiply the firft and lecond into one another, and divide their Produdt by the third, as following. If. The Rule of Three Indirect, Example. If 20 Men can perform a certain Quantity of Work in 50 Days, how loner a 'Time will 40 Men be employed to perform the fame ? 7 : 2 2 : : 1 4 : 44 a b 21 c 28 28 7) 3° 8 (44 28: 28 28 o rem. 7, what is the Di- Analogy. C. D. C. D. 22 : 7 : : 44 : 14 7 22) 308 (14 22 : 88 ,/ 88 o rem. RuU' 5 ° Men. 20 tfhe Golden Rule, or Rule of Three. Days. Men. Days. 40 25 5 ° 20 4c) 1000 (25 Rule. Multiply 50 the fecond Number by 2® the fir ft, and their Produft 1000, divide by '40 the thirclNumber, and the Quotient 25 is the Anfvver required. ow many III. The Golden Rule Compound . In the Golden Rule Compound, there are five Numbers given to find a fixth in proportion thereto, which Numbers muft be fo placed, as that the three firft may contain a Suppofition, and the two laft a Demand. And that you may place your Numbers truly, always cbferve, that the firft Number be of the fame Denomination with the fourth ; the fecond of the fame DcnominatiorLwith the fifth ; and the third with the fixth required. , Example I. If 20 Bricklayers, in 136 Days, perform 680 Rods of Brick-\yo| 3 t Rods can 12 Bricklayers perform in 28 Days ? ; y . Rule. Firft, ftate your Numbers as in the Mar- gin ; fecondiy, multiply the two firft Numbers to- gether, viz. 136 into 20, whofe ProducSt is 2720, as alio the two laft, 1 2 and 28, whofe Product is 336. Now the Anfvver to this Queftion is found by the Rule of Three Direft, for making 2720 (the Product of the firft two Terms) the firft Num- ber; the third given Number, 680 Rods, your fecond, and 336 (the Product of the two laft), your third Number ; then 228480, the Produft of 680, multiplied into 336, the two firft Numbers, being divided by 2720, the Quotient is 84, as in the Mar- gin at A, which is the fixth Number, and the An- swer required. 2720 2720 680 M. D. 12 28 1 2 33 6 33 6 680 26880 2016 2720) 228480 91760 10880 10880 (84 A ' To prove the Golden Rule. As the four Numbers are Proportionals, that is, the 4th is to the 2d, as the 3d is to the 1 ft ; therefore the Square of the two Means (which are the fecond and third) are always equal to the Square of the two Extremes (which are the firft and laft) : that is to fay, if the Product of the firft and laft Numbers, multi- plied into each other, be equal to the Product of the two middle Numbers multi- plied together, the Work is right, elfe not. So 228480, the Product of 336, multiplied into 680, which are the two Means of the laft Example, as in the Margin at A, is equal to 228480, the Product of 84, mul- tiplied at 2720, the two Extremes of the fame Example, as atB. Hence ’tis plain, that when the given Numbers, in the foregoing three Varieties of the Rule of Three are truly ftated (and which indeed is the only Difficulty in the whole), the Manner of performing the Operations is very eafy. - - 336 2720 A 680 B 84 26880 12880 2016 21760 228480 228,480 6 LECT. Of V ulgaf and D eci mal Fractions. L E C T. VIII. Of Vulgar and Decimal Fractions. I. Notation of Fractions . A Fraflion is a broken Number, fignifying one or more Parts, proportionally of any Thing divided, and therefore is always lefs than Unity. It confxffs of two Numbers fet one over another, with a Line between them, as which fignifies one fourth, or quarter of ah Integer or Unit ; and fo in like manner, \ fignifies one half ; | three fourths, or three quarters ; two thirds ; | one third ; \ three eighths ; | five eighths, feY. The upper Number is called the Numerator, and the lower the Denominator. In all Fractions, as the Numerator is to the Deno- minator, fo is the F raflion itfelf to that Whole, of which it is aFraflibn. Flence ? tis plain, that there may be infinite Fractions of the fame Value one with ano- ther, for there may be infinite Numbers found, which fhall have the fame Pro- portion one to another. So i> fri are each of the fame Value' as -f ; and *, are each of the fame Value with 4 * When the Numerator is lefs than the Denominator, the Fra flic n is lefs than an Unit, and therefore is called a Proper Fraction ; but when the Numerator is either equal to, or greater than its Denominator, the Fraction is called Improper , becaufe ’tis equal to, or greater than an Unit. So 3 . is equal to i, as alfo and ■*, &c. and 4 is equal to 1 and | to x \. Fraflions are fingte or compound : Single Frac- tions are fuch as hav'e but one Numerator, and one Denominator, as | two thirds, ■§• three fifths r f, nine elevenths, five twefths, &c. Compound Fractions are Fraflions of Fractions, and are fuch as confilt of more than one Numerator, and one Denominator, as 4 of of that is to fay, one Farthing, which is 4 of a Penny, which is of a Shilling, which is f s of a Pound Ster- ling. All Fraflions, whofe Numerators and Denominators are proportional to onfe another, are equal to one another, as before cbferved. So 1 is equal to f, and 4 to Lfc. When Integers and Fractions are joined together, as 1 4, or 7 -jtj, or 1 5 J, they are called mixed Numbers. Things commonly exprefTed I by Fraflions, are the Parts of Coin, Weight, Meafure, c sic. So Inches are Frac- tions, in refpeft of Feet, and Feet are Fraflions in refpeft of Yards, Rods, fer. As Addition and Subtraflion of Fraflions cannot well be performed without the Knowledge of the Reduflion, I fhall therefore firft teach you Reduflion. II. Reduction of Vulgar Fractions. By Reduflion you are taught, firft, how to bring Fraflions into their leaft equivalent Parts, and their various Denominators into common Denominators, or into one Denominator. Secondly, to find the Value of any Fraflion in the known Parts of the Integer. And laftly, to reduce whole or mixed Numbers into improper Fraflions, and improper Fraflions into mixed Numbers. I. Fo bring Fractions into their leaft equivalent Parts. Rule. Firft, Divide the Denominator by the Numerator, and the Divifor by the Remainder, if any be : thus continue to divide the la ft Divifor, by the laft Remains, till nothing remain, and the laft Divifor is your greafeft common Mealure ; by which dividing the Numerator and Denominator, and their Quo- tients being placed in a fraflional Manner, will be a new Fraflion equal to the given Fraflion, and in the leaft Parts. Example. Let 41? be a Fraflion given, to be reduced into its leaft Terms. Firft, the Denominator 819, divided by 637, 637) 819 (1 the Numerator, the Remains is 182, as at 637 A. Secondly, the Divilor 637, divided by — 182, the Remains, as at B, the Remains is 91. A 182 rem* Thirdly, the laft Divifor - 82, being divided by the laft Remains 91, as at C, and o re- 1S2) 637 (3 mains; therefore 91, the laft Divifor, is 546 the greateft Common Meafure required. Fourthly, divide 637, the Numerator cf the B 91 rem. given Of Vulgar and Decimal Fractions. given Fradlion, by 91, as at D, and the Quo- tient 7 is a new Numerator. Fifthly, divide 819, the Denominator of the given Fraction, by 91, as at E, and the Quotient 9 is a new Denominator. Laftly, the lad two Quotients, 7 and 9, being placed as at F, will be the new Fraction required ; and equal to the given Fradtion. 5 * 091)182(2 j 82 D 91)637(7 new Numerator 63? o rent. £91)819(9 new Denominator. 819 o rem. J £ new Fraction equal to f f t.. Note, When it happens that your lad Divifor is an Unit, the Fradtion is in its lead Terms already, becaufe 1 neither multiplies nor divides. It is alfo to be observed, that fome Fradtion s may be abbreviated, by halving both your Numerator and your Denominator, as often as you can, and which may always be done, when both Numerator and Denominator end with a. Cypher. II. to reduce fever al Fractions, whofe Denominators are different, into other Fractions having a common Denominator. Ride. Fird, multiply the Denominators into themfelves, and their Produc'd is a new Denominator common to every Fradtion. Secondly, multiply every Nume- rator into each Denominator continually, except its own, which fliall be new Numerators. Example. Let |, be Fractions given to be reduced into other Fractions, which Jhall have one common Denominator. j 7 , Operation. Fird, to find the common Denominator, I fay, the * ^ 71 Denominator 2, into the Denominator 4, is 8; and 8 into the Deno- || 4° nfinator 6, is 48, the new Denominator required, which place under a b c each Fradtion, as at a b «. Secondly, to find the new Numera- tors, I fay, the Numerator 1 into the Denominator 4, is 4 ; and 4. into the Denominator 6, is 24, which I fet over 24 at a. Then the Nu- merator 3, into the Denominator 2, is 6, and 6 into the Denominator 6 is 36, which I place over 48 at b. Thirdly, the Numerator 7, into the Denominator 2 is 10, and 10 into the Denominator 4 is 40, which I place over 48 at c. Then will -|f, and which have one common Denominator, be equal to the given Fractions i, |, as required. III. to find the Value of any vulgar FraBion in the known Parts of the Integer. Rule. Multiply the Numerator of the Fradtion by the known Parts of the next lelfer Denominator, and that Produdt being divided by the Denominator, the Quotient is the Parts of that Denominator required. Example. How many Inches are contained in fdo °f a F°°t» 75 as the next lefler denominative Parts of a Foot are Inches ? I there- 12 fdre multiply 75, the Numerator, by 12, the Inches in a Foot, and the Produdt 900, being divided by 100, the Denominator, the ioo)9]oo( Quotient 9, is the Number of Inches, which are equal to as re- quired. This may alfo be found by the Rule of Three Diredl. For 100 ; 1 2 : : 73 ; 9. Of Vulgar and Decimal Fractions. If the given Fraction AV be Parts of a Yard, and *tis required to know how many Feet and In- ches are equal thereto, multiply the Numerator 75, by 3, the Feet in a Yard, as at A, and the ProduCt 225 being divided by the Denominator 100, the Quotient is 2 Feet, and 25 remains. Now in all Kind of Cafes, when a Remainder happens, multi- ply the Remainder by the Parts of the next lefs Denomination, and divide by 100 as before. So here, as Inches are the next lefs Denomination, therefore the Remainder 25 being multiplied by 12, the Inches in a Foot, and the Product 300, divided by 100 as before, the Quotient is 3 Inches. Thefe two Quotients, 2 Feet and 3 Inches, are the Feet and Inches which are equal to j-gTS of a Yard, as required. After the fame Manner, the Value of of a Pound Sterling, will be found to 1 cf’ir' 2 t0 hnd, after having multiplied the Numerator into 20, t ie Shillings in a Pound, which are the next lefs Denomination, and divided the Product by 480 the Denominator; multiply the Remains by 12, the Pence in a Shilling ; and the Remains of that Produd, after dividing it by 480, multiply by 4, the Farthings in a Penny, the next lefs Denomination, &c. IV. To reduce whole or mixed Numbers into improper Fra ft ions % and improper Fraftions into mixed Numbers. . y our Number be an Integer, and the given Denominator be 12, it is done bymaking an Unit the Denominator, and 12 the Numerator, as thus 12 Secondly, if the given Number be mixed, as t then making 12 the Deno- minator, add 7 to 12, equal to 19, is the Numerator, and the Fraction is thus exprefled *£. Thirdly, to reduce an improper Fraction to a proper Fraction, di- vide the Numerator by the Denominator, the Quotient will be Integers, and the Remains, if any, will be a Numerator to the former Denominator. So 4*, is 4 {h l° r 59 divided by 12, the Quotient is 4, and 1 1 remains. V. To reduce a compound Fra ft ion into a Jingle Fra ft ion. Rule. Multiply all the Numerators one into another, for a new Numerator, and the Denominators, one into another, for a new Denominator, which being placed in a Fradtion, will be the Fraction required. , n ’ s 54V1 that is 11 Pence, which is 1 4 of a Shilling, which is •j-0 of a 1 ound, is that is, it is yet 11 Pence, becaufe the new Denominator 240, is equal to the Pence in a Pound Sterling. III. Addition of Fractions. Before the Addition of Fractions can be welfperformed, you muft firft obferve to reduce every given Fraction to be added, into its lealt Terms, and then the -Work is veryeaiy, as appears by the following Rules. I. To add Fraftions of the fame Denomination, Rule. Add all the Numerators into one Sum, for a new Numerator, keeping the lame Denominator ; and when the new Numerator is greater than the Deno- minator, divide the Numerator by the Denominator, and the Quotient will be the Integers and Parts. A S° ‘f tt5 i d A, T \» given Fraftions to be added, the Sum of the Nu- merators added together, is equal to 32, and the Fraction is ; and as the Nu- merator 32, is greater than 12 the Denominator, therefore divide 32 by 12, and the Quotient is 2 Ar» equal to 2 or 2 |, which is the Sum of the Fractions a3 required. ’ _ lb Fo add Fraftions of divers Denominations. Rule. Fii it, reduce the F raCtions to be added, into one Denomination. Se- condly, add all the Numerators into one Sum. Thirdly, if the Sum of the Nu- merators be greater than the Denominator, divide the Sum of the Numerators by t ie Denominators, as before taught, and the Quotient is the Sum required. But H when A 75 A 3 100) 2I25 (2 Feet 25 rem. 12 Inch, in a Foot. 100) 3I00 (3 Inches* 5*4 Of Vulgar and Decimal Fractions, when the Sum of all the Numerators is lefs than the Denominator, than the 9w» of the Fradtions is the new Numerator required. IV. Subtraction of Fractions. Rule o Firth, reduce the two Fractions into one Denomination. Secondly, fub- traift the Idler Numerator from the greater, and the Difference is the Remain* required. V. Multiplication of Fractions. ^ Before Fradtions can be multiplied, if there be any mixed Numbers, they muff be reduced into improper Fractions, and if any are compound Fractions, they mull be reduced to tingle Fractions ; and then the Fradtions being all reduced t® the lowed Terms, this is the Rule. Firfr, multiply the Numerators into each other, their Procmcv is a new Nu- merator. Secondly, multiply the Denominators into each other, and their Pro duct is a new Denominator. So f, multiplied by f, the Produdt is |f, equal to I . ; and fo in like manner J„ Q,, f, T S T , multiplied into each other, their Produdt is > which reduced into the lead Terms, is., f T %., Now from hence ’tis plain, that the Multiplication of Fradtions is the very fame thing, as to reduce a com- pound F radti on into a fmgle Fradtion, as w r as but now taught in the Reduction of Fradtions. Andfo in the fame Manner, ten thoufand Fradtions placed before, ©ne another in a right Line, may be multiplied into each other. VI. Division of Fractions. Before any proceeding can be made in the Divilion of Fradtions, that ais mixed or compound, and not in their lead Terms, they mud be prepared as be- fore was taught in Multiplication, and then proceed by the following Rule. Rule. Multiply the Denominator of the Diviior , by the Numerator of the Di- vidend, and their Sum is the Numerator of the Quotient; and the Numerator of the Divifor being multiplied into the Denominator of the Dividend, the Produce is the Denominator of the Quotient. Suppofe 4 be to be divided by ®, as in the Margin at A, then 6 the Denominator of the Divifor, multiplied into 3, the Numerator of the Dividend, the Produdt is 18 for the Numera- tor of the Quotient, and ^ the Numerator of the Divifor, multi- plied into 4 the Denominator of the Dividend,, the Product 20 is the Denominator of the Quotient required. So divided by A B fry) in -yes as at B, the Quotient is equal to y. , _ ' , . A general Rule for nil Sorts of compound &i e vifions* I. A hen toere :s a traction in the Divifor or Dividend. Rule , Multiply the Divifor and the Dividend by the Denominator of the Frac- tion, adding the Numerator to that to which it belongs, and their Proaudts be* ing divided as Integers, the Quotient will be the true Quotient required. So 271, divided by 7 the Divifor 7 multiplied by 9 the Denominai 01 of the Fradtion, whofe Produd is 63, being added to 8 the Numerator of the Fradtion, their Sum 7 1 is a new Divifor. And then 271, multiplied by the Denominator 9, the Produdt 2439, is a new Dividend, which being divided by 7 1, the Quotient is 34 f f ; and fo in like manner, if 295 be to be divided by 27, then 27 multi- pi fed by 8, the Denominator of the Fradtion, the Produdt 3 16 is the new Divifor, and 295 the Integers of the Dividend, multiplied by 8, and the Numerator 7,* added to the Produdt, the Sum 2367 is a new Dividend. Now 2367, divided by 216, the Quotient is 10 equal to IT. When there are Fraclions in both Divifor and Dividend. Rule. Firff, reduce the two Fradtions into one Denomination; fecondly, multi- ply the Divifor and Dividend by the Denominator common to both Fradtions, and to their relpedtive Produdts add their Numerators ; and then then Sums being divided as Integers, the Quotient will be the Anfwer required. So if 275 |- be to be divided by 39 f, the two Fradtions reduced into the fame Denomination,, will be ff, and Now 39, the Integers of the Divifor being multiplied by 56, aJfui 40 the Numerator of in Fraction added to is equal to 2224 which is a new jOlYllOTj Of Vulgar and Decimal Fractions. 55 Divifor, and 275 the Integers of the Dividend, multiplied into $6, with 21 its new Numerator, added to the Product, is equal to 1 5421, which being divided by 2224, the Quotient is 6£°C-, which Rradtion is in itsleafi Terms. \ II. Reduction, or rather the changing of Vulgar Fractions into Decimal -Fra Elions, and Decimal Fractions into V ulgar FraBions. Rule . Annex as many Cyphers to the Numerators of the given Fraction, as you would have Places in the Decimal, which, being divided by the Denominator, the Quotient will be the Decimal required. So to reduce 4 into a Decimal of two Places, I add two Cyphers to 3 the Nu- merator, making it 300, which bejng divided by 4 the Denominator, the Quo- tient 75 is the Decimal required. In like manner, if ’twas required to have had the Decimal of 3 Places, then I fliould have added 3 Cyphers to the Numerator 3, making it 3000, which being divided by 4, as before, the Quotient would be 7>o^ which is equal to ,75. For is equal to T g 5 o%, becaule cutting of!" the laft Cyphers in both Numerator, and Denominator, thus T q!{§ the Remains is then the fame as the other Fraction, Vulgar Fractions may be changed into Decimal Fractions by this Analogy , mix. as the Denominator of the Vulgar Fraftion is to its Numerator, fo is the given Denominator of the Decimal Fraction to its Numerator required. So if T p / 6 be a Vulgar Fra&ion given, to be changed into a Decimal, whole Denominator is 100 ; then as 120 : 96 : : 100 : 80, fo that 80 is the Decimal required, and on the con- trary. Decimal Fractions may be changed into Vulgar Fractions by this Ana- logy, mix, as the Decimal Denominator is to its Numerator, lo is the given Vul- gar Denominator, to its Numerator required. Let jQg. be changed into a V r ulgar Fraction whofe Denominator is 120 ; then as too : 80 : : >20 : 96, fo that /V is the Vulgar Fraftion required... AW, It will happen in many Cafes, of changing Vulgar Fractions into Deci- mals, that there will be Hill a Remainder although you fhould annex ten thoufand Cyphers to the Numerator of the given Frabtion; and therefore it is to be noted, that if you make the Decimal to confift of five or fix Places, it will be near enough in almoft every Cafe of Bufinefs, and the Remainder may be reje&ed as of n® Value, Now there only remains to fhewhow to find the Value of any given Decimal Parts of a Foot, Pounds Sterling, lAc. which is done by this Rule, Multiply the gimen Decimal into the Units that are contained in the Integer^ l as in Decimal Multiplication) and the ProduB will he the Value of the Decimal . Example I. Suppofe ,7852 be a given Decimal, whofe Integer is a Foot. Here the Decimal ,7852, multiplied by 12, the Inches or Units ,7832 that are contained in a Foot, which is the Integer, the Produbt is 12 9,4124, which is 9 Inches, and ,41 24 Parts of an Inch. And if we — luppofe an Inch to be divided into 100 Parts, then multiplying 4124 9,41 24 the Remains by ioo, the Produbt is 41,2400, which is 41 hundred 100 Parts of an Inch, and the Remains 2400, is 2400 Parts of one hun- dredth Part of an Inch divided into ten thoufand Parts. So- that re- 41,2400 jebbng this laft Remains 2400, the Value of the given Decimal is 9 Inches and 41 hundred Parts of an Inch. Example II. 1 Suppofe the aforefaid Decimal fignily a Decimal Part of a Pound Sterling. Id a Thws 20 the Shillings in i /. ,7040 12 the Pence in i s. j6 Of Vulgar and Decimal Fractions. Then ,7852, multiplied into 20, the Units, or ,7852 Shillings in the Integer or Pound, the Produft 1 5,7040 is 1 5 Shillings, and 7070 remains ; which being multiplied by 12, the Units in the next lels Integer, viz. the Pence in a Shilling, the Product 8,4480 is 8 Pence, and 4480 remains ; and which being multiplied by 4, the Farthings in a Penny, 8,4480 the Product is 1,7920, which is one Farthing, and 4 the Farthings in id. 7920 Parts of a Farthing, the Farthing being di? vided into ten thoufand Parts. So tbe Value of the 1,7929 Decimal, 7852 Part of one Pound Sterling, is 15 Shillings, 8 Pence, and 1 Farthing, rejecting the laft Remains 7920. Thus, a due Regard being had to the Number of Units, which are contained in the Denomination of the Integer, to which the Decimal Parts belong, any propofed Number of a Decimal may be reduced or changed into the known Parts of what they reprefent, h E C T. IX. The Extraction of the Square and Cube Roots. T O extract the fquare Root, is nothing more than to find the Side of a Geo- metrical Square, whole Area is equal to a given Number of Units, which are generally called a fquare Number. A fquare Number, is that which is pro- duced by any Number multiplied into itfelf: As for Example, 16 is a Iquaie Number, which is produced by 4 multiplied into 4. So in like Manner, 9 is a fquare Number, produced by 3 multiplied into 3. The Side of a geometrical Square, equal to any given Number, is called its Root. In the Margin is a Table of fquare Numbers, whofe Roots are the Ro. Squ. nine Digits, and which being nothing more than a Part of the MuK 1 tiplication Table, it is luppqfed yop have it already by Heart* 4 1 36 49 64 81 1 672 m 672 3 344 47°4 4032 ndef 4U5 8 4 ( b/ 2 a ho p 3b g 1— Let 6 ] 2 he a Root given to find its fquare Number. Rule. Multiply 672 into itfelf as at / whofe Produft is 451584, the fquare Number required, and whofe Root is thus extra&ed, viz. Firft, place a Point under the firft Figure to the right Hand, as at c, and at every other Figure towards the Left, as at h and a ; and ob? ferye, that as many Points as the fquare Num- ber contains, fo many Places of Figures the Root will confift of. Secondly, make a Crotch- et as at n and p on the right Hand Side of the fquare Number, as is done in Divifion ; and note, that every two Figures fo pointed, are called a Pun&ation. Thirdly, find in the Table the neareft fquare Number that is con- 3:2.7)91.5 fifft Refolyend, 889 ) it, r 1 tained in the firft Pvuuftation to the left Hand, 134 . 0=68 4fecondR ? fslvend. ii) , !! _ ^ * which is j6> who fe Root is 6 . 2 fab4 Place 36 under 45, and its Root 6 in the Quo- tient, as at 1/, and fubtra£ting 36 from 45, the 'Remains is q, which place under 36. This is yom’ 0 rem= Of Vulgar., and Decimal Fractions. your firft Work, and is no more to be repeated. Fourthly, bring down the next Pundtaiion 15, and join it to the Remains 9, making it 915, which is your firft Reiolvend, and on its left Side make a Crotchet, as "is done in Divifion to fe- parate the Divifor from the Dividend. Fifthly, double the Root 6, it makes. 12, which place on the left of the Refolvend, as at_g-, Then rejecting the laft Figure 5 in the Reiolvend (which is always to be done) fee how often the Divi- for 12 is contained in the remaining Figures 91, which being 7 Times, therefore put 7 m the Quotient at r, and alfo on the right Hand of the Divifor at / and multiply 127, the Divifor increafed by 7, whofe Product is 889, which place under 915, and being iubtradied from it, the Remains is 26. This being done bring down the next Pun£lation 84* &nd join it to the Remains 26, making it 2684 which is a fecond Refolvend, and then proceed as before, as follows, viz. Firft* double 67 the Root fo far found, makes 134, which place on the left of the fe* cond Reiolvend, as at h y and fee how often 134 us contained in the Refolvend, the laft Figure excepted, viz. in 268, which is two times. Set two in the Quotient at and on the right Hand of that laft Divifor 134, making it 1342, which being multiplied by 2, the laft Figure in the Quotient, its Produ& is 2684, which being placed under the fecond Refolvend, and fubtradted from it as before o re- mains ; which fliews that 45 1 584 is a fquare Number, whofe fquare Root i’s 672 as required. ' * Note , Firft, when the fquare Number contains 4 or more Punftations, as the Remains are produced, the next Punftation is to be brought down, and joined to the Kemaias for a third, (sic. Refolvend ; with which you are to proceed in every Refpe# as before with the firft and fecond Refolvend. Secondly, that if at any time, when you have multiplied the Number Handing in the Place of the Divifor by the Figure laft found in the Quotient or Root, the Produ# be greater than the Reiolvend, then in fuch a Cafe, you are to put a Figure lefs by one, than the former, in the Quotient, and multiply by it as before : and when the Re- mainder be greater than the Divifor, put a Figure greater by one in your Quo- tient, and multiply it as before. Thirdly, if at any Time the Divifor carniot be had in the Reiolvend, then place a Cypher in the Quotient, and alfo on the right Hand oi the Divifor, and to the Refolvend annex the next Pundation for a new Refolvend, with which proceed as before. When it happens,' that, after Ex- tra&ion is made, there is a Remainder, the Number given to be extradted is called an irrational or furd Number, and its Root cannot be exadtly obtained ’al- though by adding Cyphers you may corneas near the Truth as is required, ’but never can come at the Truth itfelf. As for Example, *tis required to extra# the fquare Root of 160. 58 / m 160(12*64911 22 ) 060 iii ft Refolvend. 44 « p ? 4 " 6 ) -ah 1600 iecond Refolvendo 1476 t r—T—cd 252,4) 1 2400 third Rcfoi vend, 10096 5 — —c f 2-28,9) 230400 fourth Refolvend« 227601 zc w— — ~—~gb 55298,1) 279900 fifth Refolvend 252981 : k 2 529821 162079 rem. Of Vulgar and Decimal Fractions* Firft, The firft Pun#ation being 1, the fquare of 1 is 1, which place under i, and fubtrafting 1 from 1 remain? c, fet 1 in the Quotient, and to o, bring down the next Pun#aticn 6c, making the Remains 0. 060, Secondly, double the Quotient, 1, makes 2, w'hich place 'for your Divifor at L Now as 2 is contained 3 Times in 6, if you was to place 3 in the Quotient, and 3 on the right Hand of the Divifor 2, as before taught, to make the Divifor 23, then 23 multiplied by 3, would be equal to 69, which is greater than 60 the firft Refolvend, and therefore cannot be fub- iradted from it : therefore in this Cafe, as was before noted, place a Figure in the Quotient lefs by 1 than the 3, via. 2, and the fameson the right Hand of the Di- vifor 2, as at to, and then multiplying x y 7 "~ 1 r r> r t 1 the Divifor 22, by 2. in the Quotient, 252982,1) 2691900 ix ! ve 0 ten . ^ p roc i u Subducendr, *8 J 15608. Subtrahend, 8 1 1 3 } 575 5»^83. fecond Refolvend, m 56784 4 j Subducends, N7644 /343 57 5 5' 1 83 Subtrahend. Let 146363x83 bea cubed Number given to find its Root, firft, point the firft Figure towards the right Hand, and then every third Figure towards the left, as “at f e d. Secondly, look in your Table of cubed Numbers, and find the neareft lefs cube Number to £46, the firll Pun&ation, which is 2525, whofe Root is 5. Place 5 in the Quo- tient at a , and 125 under .146, and fub- tracting 125 from 246, the Remains is 21, This is your firft Work, and no more to be done. Thirdly, to 21 the Remains, annex 3 63, the next P imitation, making 21,21363, which is your firft Refolvend. Now to find a Divifor, by which you are to divide this’ Refolvend, its two laft Fi- gures excepted, which are always to be rejedtqd, proceed as follows, viz, Firft, fquare the Quotient 3, makes 25, which Triple makes 75, which is the Divifor re- quired, as ad g. Then fay the 75*3 in 213 (the Figures remaining in the Re- folvend, exciufive of the two laft rejected as. afore(a.d) is 2 times, equal to 1 50, which place under 213, as at h, and fet 2 ln Quotient at l. Secondly, treble 5, the firft Figure of the Root, equal >.0 25, which multiply by 4, the Square of 2, the laft Figure in the Quotient makes 60, which place under 150, one Place forward to the right Hand, as at 2 5 alfo Cube 3, the laft: Figure of the Quotient equal to 8, which place under 60, one Place more to the right, as at Then the 3 Subducends, 150, 60, and 8, being added as they ftand, their Sum make a Subtrahend 15608, which being fubtraffed from the firft Refolvend, there remains 5755 ; to which bring down and annex the next Pundlation 183, making 5755183, for a iecond Refolvend, with which you are to proceed, as before ; but to make' the Performance quite eafy, I will explain this Repetition alfo, as follows. lull. Fine a Divifor as follows, viz. Square 52, the Quotient already found, makes 2-704, which trebled makes 8112 the Divifor required. Then fay, how often 8112 in 57551, (for here as before the two laft Figures 83, of the'RefoI- vcnc, are to be rejedted) anfwer 7 times, equal to 56784, which place under 57551, of the Refolvend, and fet 7 in the Quotient at c. Secondly, treble 52, Jie firft and fecond Figures of the Root, equal to 156, which multiply by 49 the ijquise of 7, the laft I'igure in the Quotient makes 7644, which place under 4 567845 o rem® 60 Of Vulgaf and Decimal Fractions, 56784, one Place more to the right Hand, as at n ; alfo Cube 7, the laft Figure ter the Quotient, equal to 343, which place under 7644, one Place more to the right, as at p. Then the three Subducends 5678431 m, 7644 at n, and 343 at j > , being added as they Hand, their Sum make a Subtrahend, 5755183, which being iub- tradted from 5755183, the iecond Refolvend, nothing remains; which Ihews that the given Number 146365 1 83 is a cube Number, whole Root is 527, 3s required. Note I. As many P initiations as any given Number contains , except the fir ft, f» many Times is the Work to be repeated. II. That in all Extractions, when a Divifor cannot be found fo often as once i * its Dividend , or if it can be found , and yet there Jhall arije a Subtrahend greater than the Refolvend, in both thefc Cafes a Cypher muftbe put in the Quotient and annexed to the laft DiviJ'or alfo, for a new Divifor ; and the next Puntiation being brought down and added to the laft Refolvend , makes a new Refolvend . , with which proceed in every Refpetias before. HI. When Numbers remain after the laft Subtrahend is fubtradted from the laft Refolvend, which very often happen, fuch are called irrational or l'urd Numbers, becaufe their Roots cannot be exadtly difeovered. But if to fuch Remainder you annex three Cyphers, continually, as you did two Cyphers in the fquare Root, you may come very near to the Truth, as was there lliewn. To extrati the Cube Root of a Vulgar Fratiion which is commenfurable to its Root * Rule. Extradt the Cube Root of the Numerator for the Numerator of the Root ; and the Cube Root of the Denominator for the Denominator of the faid Root. To extrati the Cube Root nearly, of a V ulgar Fratiion remaining , incommenfur able to its Root. Rule. The Integral Part of your Root being firft found, as before taught, to the Treble thereof add one, and that Sum added to the Square of the faid Root tripled, is a Denominator ; to which the laft Remainder, after Extraction is finilhed, is the Numerator. A Table of the Roots of all fquare and cubed whole Numbers , from 1 to 50, calculated by Thomas Langley. R. Sq. Cube R. Sq. Cube I 1 1 2 4 8 3 9 27 4 16 64 5 2 S 125 6 3 6 216 7 49 343 8 64 512 9 81 729 10 100 TOOO 1 1 121 I 33 I 12 I44 1728 n 1 6qlz 197 l 4 196 2 744 ! 5 225 3375 16 256 4096 '7 289 49 1 3 18 3 2 4 5 8 3 2 l 9 36 1 6859 20 400 800c 2 r 441 9262 22 484 10648 2 3 529 12167 24 576 CO CO 1 -L* 2 5 625 15625 j 6> 676 R . Sq. Cube 2 7 729 19083 28 00 21952 i 9 841 24389 3 C 900 27000 3 1 961 29791 52 1024 32668 33 1089 35937 34 1 1 56 393°4 35 r 225 42S75 lb 1296 46656 37 1369 5 o6 53 38 1444 54652 39 1521 593 1 9 R. Sq. Cube 40 1600 64000 1 68 1 68921 42 1764 74088 43 O' OO 795°7 44 t 93 6 85184 45 2025 9 II2 5 46 21 16 97336 47 2209 103823 48 2304 1 10592 49 2401 1 17649 5 ° 2 500 1 2 5000 Thus INTRODUCTION. 61 Thus have I given all the ufeful Rules in Vulgar and Decimal Arithmetic both in whole Numbers and in Fractions, which if well confidered will be, not only very foon and eafily underftood, but vaftly advantageous to every Work- man in the Execution of his Employs. And as a perfect Knowledge herein may be foon acquired by employing the leifure Hours of Evenings when the Labour of the Day is over, I humbly conceive that every one who will fo employ him- felf will find, not only a very agreeable Amufement, but very great Helps in the Performance of his feveral Works, exclufive of the Reputation that will attend him alfo. But fuch Perfons who will be fo remifs as to lay by this Work in their Chefts, iff c. without taking either Pains or Pleafure herein, cannot expetft that Advantage which others will enjoy. PART II. Of G E O M E T R Y. INTRODUCTION. T he next Science in order after Arithmetic is Geometry, the moll ex- cellent Knowledge in the W orld, as being the Bafis or Foundation of all Trade, on which all Arts depend. Geometry is fpeculative and practical ; the former demonftrates the Proper- ties of Lines, Angles and Figures ; the latter teaches how to apply them to Prac- tice in Architecture , Trigonometry, Menfuration , Surveying, Mechanicks , PerfpeSive , Dialling, AJlronomy , Navigation, Fortification, &c. This Art was firft invented by Jabal the Son of Lamech and Adah, by whom the firft Houfe with Stones and Trees was built. Jabal was alfo the firft that wrote on this Subject, and which he performed, with his Brethren, Jubal, Tubal Cain, and Naamah, who together Wrote on two Columns the Arts of Geometry, Mufick , working in Brafs and Weaving , which were found (after the Flood of Noah) by Hermarinfs, a Defcendant from Noah, who was afterwards called Hermes, the Father of Wifdom, and who taught thofe Sciences to other Men. So that in a fbort time the Science of Geometry became known to many, and even to thofe of the higheft Rank, for the mighty Nimrod, King of Babylon, underftood Geometry, and was not only a Mafon himfelf, but caufed others to be taught Mafonry, many of whom hefent to build the City of Nineve and other Cities in the Eafi. Abraham was alfo a Geometer, and when he went into Egypt , he taught Euclid, the then mofi: worthy Geometrician in the World, the Science of Geometry, to whom the whole World is now largely indebted for its unparalleled Elements of Geome- try. Hiram, the chief Conductor of the Temple of Solomon, was alfo an ex- cellent Geometer, as was Grecus, a curious Mafon who worked at the Temple, and who afterwards taught the Science of Mafoniy in France. England was entirely unacquainted with this noble Science, until the time of St. Alban, when Mafonry was then eftablifhed, and Geometry was taught tomoft Workmen concerned in Building; but as foon after this Kingdom was frequently invaded, and nothing but Troubles and Confufion reigned all the Land over, this noble Science was difregarded until Athelstan, a worthy King of England , fupprefled thofe Tumults, and brought the Land into Peace ; when Geometry and Mafonry were re-eftablifhed, and great Numbers of Abbeys and other ftately Build- ings were erefted in this Kingdom. Edwin the Son of Athelstan was alfo a great Lover of Geometry, and ufed to read Lectures thereof to Mafons. He I * alf# 6 2 Of GEOMETRY. alfo obtained from bis Father a Charter to hold an AfTembly, where they would, within the Realm, once in every Year, and himfelf held the firft at Tori, where he made Mafons ; fo from hence it is, that Mafons to this Day have a grand Meeting and Feail, once in every Year. Thus much byway of Introduction, to Ihew the Ufe, and how much the Science of Geometry has been efteemed rj fome of the greateil Men in the World, and which with regard to the Pubuc Good of my Country, I have here explained, in the moil plain and eaiy manner that I am able to do, and to which I proceed. LECTURE I. Geometrical Definitions. Plate l. T HE Principles of Geometry are Definitions,. Axioms and Poftulates. Defini- tions are the Explication offuch Words and Terms which concern a Propo- fition towards rendering it intelligible and eafy to the Underftanding, avoiding m Demonftration all Difficulties and Objeftions. Axioms arc fuch evident Truths, as are not to be denied, as pne and one are two, two and two arc four, iff c. Popu- lates are Demands, or Suppof tions of things prafticable, and the manner of doing them fo eafy, plain, and evident, that no Man of Senfe and Judgment can deny, or conteft them, fuch as to draw a Line by the Side of a Ruler, from one given Point to another. , , . , Quantity is confidered in three different Manners, viz. ririt, Length wit - out Breadth, as an Interval or Diftance between two Points. Secondly, Lengt with Breadth only, as a Shadow, iffc. Thirdly, Length with Breadth and Thicknefs, or Depth, as a Brick, iffc. The Bounds or Limits of Quantity aie Points, Lines and Superficies. . ' „ n • n r nr A Point, in the Pradtice of Geometry, is the fmalleit Object ot ^ a Sight that can be made, and which is fuppofed to have no geo- metrical Magnitude, capable of being divided to our Sight, and is made by the Point of a Pin, Pen, Pencil, iff c. as the Point A, Plate I. The Varieties of Points, and their particular Denominations, are many ; as for Example, if a Point be affigned, in any certain Place, as the Jjpf 2 • C )J Cl — . • . • -r • in. 9 11 1 - • 7) - ! — x 1 1 rV» Def. I. Point. Point b , in the Line a cl, ’tis called a given Point, from whence the Line b c proceeds, or to which the Line b c is drawn from the End or Point c. Secondly, when the two Lines cut acrofs each other, as xc,yy, ox eh, i f, the Points ixand^, are called Points of InterfeBion ; and when fuch a Point happens to be in the Middle of a fuperficial Figure, as g, ’tis called its Centre, or central Point. Thirdly, when two Lines meet together, and flop in one Point, as km, and ml, in the Point m, fuch a Point is called 511 angular Point. Fourthly, if two Lines touch one another,, but do not cut acrofs each other, asatB, the Point of touch B, is called given Point. Def. 3* Of a Point of In- terjection. Def. 4. Of an angular Point. „ - not cut acrois cacn if the Point of Con tad. a oint oj There are many other Kinds of Points, in the feveral Paits of Lontad. Mathematicks, which at prefent do not concern us ; as for Exam- ple, in Perfpedtive there are Points of Sight, Points of Diftance, vifual Points, .Go. which will be better underftood hereafter, when I come to explain the Principles and Praftice of that Art. Def. 6. Of When Quantities are confidered as Lengths only, they aie Lines, Super- called Lines; thofe of Lengths with Breadths only, are called Su- Jicics and So- perfidies ; and thofe of Lengths, Breadths, and Depths, are calle lids. Solids, or Bodies. ■ Kinds of The Kinds of Lines are three, viz. a. right Line, a curved Line, Line . and a mixed Line. A RIGHT 1 Of GEOMETRY. 6 3 A right Line, is a Length without Breadth, as the neareft jfef. ~ p\f Difiance between two Points ; but in Practice, ’tis a ftraight Line ^ right* Line, defcribed by the Motion of a Pen, Pencil, &c. drawn by the fide of a ilraight Rule, wherein its vifible Breadth is not considered, as a d A curved Line, is any Line that is not a right Line, and p * g pr therefore all crooked, arched, or bended Lines, are curved Lines. Ci fved Lines There are many Kinds of curved Lines, namely a circular or arched Line, as E, Fig. II. an elliptical or ovallar Line, as h /, or i m l, a parabolical Line, as 9?'+8. Of an irregular compound Figure. Bef. 49- Of a regular eompoutid Figure. Bef. 50. Of Imperfebl Fi- gures , Con- centrick and Excentrick. as are ex- • f hibited in Plate II. Of GEOMETRY. 67 "Tier, as the Center of the Voids, in the Square, is not in the fame Points as the Centers of the fhaded Superficies ; that is alfo an excentrick Figure, as th« Lunula. To thefe imperfect Figures I 'mull add Fig. C, which is a Parallelogram di- vided into four Parallelograms, that meet altogether on the Diagonal Line in the Point n. Now if any three of thofe four Parallelograms, as n d, n b, and ^ ^ n a , be taken together, and confidered as one Figure, ’tis called a & Gnty^n * Gnomon; but if the four Parallelograms are confidered feparately, then the Parallelograms n l , and n c, are called Parallelograms defcribed about the Diagonal b c, and the other two Parallelograms a 11, and n d, are the two Supplements thereof, and which are always equal to one another, as will be here- after demonftrated. As fuperflcial Figures are bounded by one or more Lines, fo So- lids or Bodies are bounded by one or more Superficies ; as for Example, a Brick is a Solid, bounded with fix Surfaces, that are all Parallelograms, viz. the upper and the under, the two Sides, and both Ends. . The Number of entire Solids are principally twenty, viz. a The Number Sphere, a Spheroid, a Cylinder, a Cone, a Conoid, a Spindle, a Tetrahedron, a Pyramid, a Pyramis, a Pyramidoid, a Conedoid, a Cylindroid, a Prifm, a Hexahedron or Cube, a Parallelopipedon, an Octahedron, a Dodecahedron, an Icofahedron, the twelve and the thirty Rhombufes. An entire geometrical Solid is a Body from which no Part has been taken, and therefore the Remains of a Body, when a Part thereof is taken away, is called a Fruftum, as the Frultum of a Sphere, or of a Cone, iffc. A Sphere is a round Body, bounded by one convex Superficies, whofe Parts are all at the fame Diftance from the central Point of the Solid ; and is commonly called a Ball, as R, Plate II. A Sph eroid is a round folid Body, bounded by one convex Su- perficies alfo, but its Curvature is not the fame in every Part over its Center, as the Curvature of the Sphere ; becaufe its Length is greater than its greateft Thicknefs, and therefore it is what may be properly called an ovallar Solid, if we confider the Sphere as a circular Solid; as S, Plate II. A Cylinder is a long and round Body of equal Thicknefs, Def.5q.Of as a Garden rolling Stone, or the lowermoft third Part of the a Cylinder. Shaft of a Column, as X, Plate II. and is bounded by threfr Super- ficies, of which one is convex, and two are plane or flat, and whofe Figures de- pend upon the manner of the Cylinder being cut at each End; that is to fay, (1) If the Ends of the Cylinder be both cut fquare to its Length, as X, then the Su- perficies of the two Ends are both Circles (which are equal to each other, becaufe the Cylinder is of equal Thicknefs) and the convex Superficies is no more than a Parallelogram whofe Length is equal to the Length of the Cylinder, and Breadth to its Circumference, being bended about the fame. (2) If a Cylinder as D (on the right-hand Side of the Plate) have its Ends cut obliquely and parallel to each other, the fuperflcial Figure of each End will be an Ellipfis, and the convex Superficies will be a double Rhomboides. ( 3 ) If a Cylinder, as E, have its Ends cut obliquely, and not paral- lel to each other, they will be both Ellipfes, but unequal, (as not being parallel, which caufes the tianfverfe Diameter to be longer in the one than in the other) and the convex Superficies will be an irregular Hexagon ; a Demonftration of which you will fee in the Meniuration of Solids and their Superficies. Def 52 .Of the Bounds cf Solids or Bodies. and Names of Solids. Dfsi- °f an entire Solid. Def. 54. Of the Frujlum of a Sphere. mr-ss.of a Sphere. Def. 5 6.0f a Spheroid. Def 58. Of the various Kinds of Su- perficies that bound regu- lar and ob- lique Cylin- ders. A Cone 68 Def. 6o. Of the Vertex and Axis of a Cone. Of GEOMETR Y. ^ - A Cone is a round Solid, which rifes either from a Circle Of Def. 59 . Ctf an Elliplis, with a gradual and equal Diminution until it termi- * Cone. nates or ends in a Point, as Fig. T, on the left Side of Plate II. and therefore is bounded by two Superficies, of which that of the Outfide is convex, and that of its End or Bottom is a circular or elliptical Plane. In every Cone there is an imaginary Line fnppofed to be drawn from its Top or vertical Point, unto the centrical Point of its Bafe, which is called the Axis of the Cone, and which is fo called becaufe it paffes directly through the Middle of the Solid, and on which the Body may be made to revolve or turn about, as that every oppofite Part is equidiftant therefrom. The fame is alfo to be underftood of a h, the Axis of the Sphere R, alfo of e c, in the Spheroid S, and of all other regular Solids. Now when a Cone hath its Bottom cut fauare to its Axis, as T, ’tis called. a regular Cone, and its Bottom, which is called its Bafe, will be a Circle. But if its Bot- tom be cut obliquely to its Axis, as G, on the right-hand Side of the Plate, it is then called an oblique Cone, and its Bafe will be an Ellipfis. . A Conoid is a Solid, diminifhing in its upper Parts nearly the Def. 61. UJ p ame as a c one> anc [ takes its Rife fiom a Circle alfo ; but as the a Conoid. side 0 £ a £ one j g flight from its Bafe to its Vertex, this of a Co- noid is either the Semi-curve of a Parabola or of a Hyperbola, or the Segment of a Circle, or an Ellipfis ; and therefore terminates at its Vertex either in a Point, as the Cone doth when the outward Curve is of a Circle or an Ellipfis, as B L, or with a curved Top, like unto a Sugar-Loaf, as A, when a Semi-parabola, or Semi-hyperbola. . , . . . _ A Spindle is a Solid, thus to be conceived ; fuppofe ag in B, to be the Diameter of a Circle, on which a Semi-lpindle is to be raifed, whofe Axis is d ; alfo fuppofe the Curve a d to be the Semi- curve of a Parabola ; now if from every Part of the Circumfe- rence of a Circle, of which a g is the Diameter, a Solid be raifed with a Curvature equal to the Semi-parabola a d, that Solid will be a Semi-fpin- dle, and therefore two fuch, being equal and applied together, as B, will form that Solid which is called a Spindle. And as the outward Curve may be either a Hyperbola, ora Parabola, therefore a Spindle maybe Hyperbolical or Parabolical. Def. 6 3. Of A Tetrahedron is a triangular Solid, which rifes from, an a felrahe - equilateral triangular Bafe, with a gradual and equal. Diminu- Aron. tion, until it terminates in a Point, as a Cone doth, which Point is alfo called its Vertex. This Solid is terminated by four equilateral Triangles, as B F, on the left-hand Side of the Plate. A Pyramid is a Solid, which rifes from a geometrical Square, Def. of Of w * th a g ra d ua l Diminution, (as the Tetrahedron rifes from an equi- ® Pyramid. lateral Triangle) and terminates in a vertical Point alfo. I his Solid hath its Height at Pleasure, and is bounded by four Equilaterals or Ifofceles Triangles on its Sides, and a geometrical Square at its Bafe, as Fig. V. A Pyramis is the fame Solid as a Pyramid, only with this Dif- Def. 65. Of f erence> t h at whereas a Pyramid (lands on a geometrical Square, * Pyramis. an q has but four Sides, which are all equilateral, or Ifofceles Tri- a«gles, a Pyramis has fome regular Polygon, as a Pentagon, Hexagon, ,JAc. for its°Bafe, with five, fix, ii fc. Sides which are all Triangles, as in a Pyramid, and meet in a vertical Point alfo. ... A Pyramidoid is a pyramental Solid, whofe Bottom is a tn- angule geometrical Square, or . fome regular Polygon, and Sides are the Curve of a Circle, Elhpiis, Parabola or Hypeibola, as Fig. IV. A Cylindroid is a Solid, fomething like B I, the Fruftum of a Cone, but with this Difference, that as the Fruftum of a Cone, is terminated at its Ends either with two Circles, if cut fquare to its Axis, or with two Ellipfes, if cut oblique, or with a Circle and an Ellipfis, if one Def. 62. Of a Paraholick and Hyherho • lick Spindle. Def. 66. Of a Pyrami- doid. Def. 67. Of a Cylindroid. Of GEOMETRY. 69 Def. 68. Of the various Kinds of Prifms. one End be cut fquare, and the other oblique, the Ends of the Cylindroid are both cut fquave to its Axis ; but the one is an Ellipfis, and the other a Circle, as Fig, C at the Top on the right Hand. The next Kind of Solids in order, are Prifms. A Prism is a folid Body, of equal Thicknefs as a Cylinder * but as a Cylinder is round, and its Length is thereby bounded by one Superficies only ; fo a Prifm is bounded by three, five, fix, or more Parallelograms, and its Ends are either Triangles, geome- trical Squares, Trapeziums, or fome Kind of Polygon, as a Pen- tagon, Hexagon, £ sV. So B C is a triangular Prifm, bounded by two Triangles at its Ends, and three Parallelograms on its Sides. B A is a Trapezium Prifm, bounded by two Trapeziums at its Ends, and four Parallelograms on its Sides. B E is a pentangular Prifm, bounded by two Pentagons at its Ends, and five Pa- rallelograms on its Sides. And laftly, B D is a hexangular Prifm, bounded by two Hexagons at its Ends, and fix Parallelograms on its Sides. It is alfo to be noted, that if the aforefaid Prifms have their Ends cut oblique f o their Sides, then their Sides will be either Trapezoids or Rhomboids, and their Ends will be changed into different Kinds of Triangles, Parallelograms, and un- equal-fided Polygons. A Cube or Hexahedron, is an exaft fquare regular Solid (as a Dice), and is bounded by fix equal geometrical Squares, as Fig. Y. A Parallelopipedon, is alfo called a long Cube, and by fome a Prifm ; but as its Ends, as well as its Sides, are bounded by Parallelograms, which are nevermore nor lefs than fix in Num- ber, as Fig. Z, it is therefore, with refpeft to its Surfaces being all Parallelograms, properly a Parallelopipedon. An Octahedron, is a regular Solid, bounded by eight equi- lateral Triangles, and is compofed of two equal Pyramids, having their Bottoms applied together, fo as to make but ©ne Solid in the whole, as Fig. P, Plate II. A Dodecahedron, is a regular Solid, bounded by twelve Pentagons, as Fig. O, Plate II. An Icosahedron, is a regular Solid alfo, and is bounded by twenty equilateral Triangles, as Fig.Q, Plate II. — The twelve Rhombs, and the thirty Rhombs, are Solids, bounded by as many Rhombufes ; but though they have a Uniformity in themfelves, yet they are not regular Solids. The regular Bodies are the Tetrahedron, the Hexahedron or Cube, the Oflahedron, the Dodecahedron, and the Icofahedron, which fyeing the only Bodies that can be infcribed within a Sphere, are therefore called regular Bodies. A Body isfaid to be infcribed, when, being inclofed within ano- ther Body, every of its folid Angles terminates at the Superficies thereof ; and that Body which contains the infcribed Body is called the circumfcribing Body. A solid Angle, is the meeting together of three or more right- lined Superficies. A Frustum, as in Df. 54, is the Remains of a Body, when a Part is taken away ; fo if from the Sphere B G, the Part A be ta- ken away, the Part B G remaining, is the Fruftum of a Sphere 5 and if from the Spheroid B N, the Part A be taken away, the Part N B is the Fruftum of a Spheroid : and fo the fame of B I, and B K, which are the Fruftums of a Cone, and of a Pyramid, when the top Parts D and A are taken from them. Fruftums of Bodies are cut obliquely, and that not only at their upper, but alfo at their under Parts, as H I K L M, and are then called oblique Fruftums. When a Part is taken fror» the Bottom of a Pyramid* or of a Cone, as the Parts a and x, in K Fand Def. 6 (j. Of a Hexahedron or Cube . Def. 70. Of a Parallelo- pipedon. Def. 71. Of an OBahe- dron. Def 72. Of a Dodecahe- dron. De f- 73 ’ Of an Icofahe- dron. The 12 and 30 Rhombs. What Solids are Jlr icily regular Bo • dies. FheReafon - . D f; 74 - Of infcribed and circumfcribed Figures and Bodies. B > f . 7 S ’ Of a folid Angle , Frufums of a Sphere, Sphe- roid, Cone, & c. explain- ed. 7 G Of GEOMETRY. Def. 76. Of the Segments of a Cone , Pyramid c. Def 77. Of the Segment of a Frujlum. ‘The Frujlum of a Cube. The Frujlum of a Tetrahe- dron F and G, then the remaining upper Parts being confidered feparately, become entire Bodies with oblique Bafes ; but if they are confidered with the Parts a and x, then they are no more than the greater Seg- ments, and the Parts a and x are the leffer Segments, which to- gether do but complete the two Solids ; and when the upper Parts are confidered as entire oblique Bodies, and the Parts a and x are confidered by themfelves, the Parts a and .* are called Segments of Fruftums, whofe Axis is equal to their perpendicular Height. If all the folid Angles of a Cube be fo taken away, as to make every fquare Face, of the Cube an O&agon, then the Remains will be the Fruftum of a Cube, contained under fourteen Superfi- cies or Faces, of which eight will be equilateral Triangles, and fix will be O&agons. If the folid Angles of a Tetrahedron be fo ta- ken off, as to make each of its equilateral triangular Faces a Hexagon, the Remains will be the Fruftum of a Tetrahedron, bounded by eight Superficies; of which four will be equilateral Triangles, and four will be Hexagons. I mention thefe Fruftums, only to give a Hint, that by this Method of cutting off the folid Angles of Bodies, there may be a very great Variety of uncommon Bodies produced. The Body or Shaft of a Column, is compofed of two Kinds of Solids, that is to fay, the lower one third Part of its whole Height, up to S B, is a Cylinder, and R, the Remainder, is the Fruftum, of a Conoid. A Section of a Solid, is a fuperficial Figure, produced by cutting off a Solid, diredly through, in any Part ; fo if from a Sphere a Segment was to be cut, the flat Surface, or Superficies of that Cut, which is a Circle, is called its Se&ion. And in like Manner, if any Cone be cut quite through its Axis, from the Top to its Bottom, the flat Superficies of that Seftion will be a Triangle. The Bafe of an upright Line, is a Point. The Bafe of a Circle is a Point alfo, as the Point g, of the Cir- cle E, ( Fig. V. Plate I. ) ftanding on the tangent Line h z, which by its Curvature can touch the Line h z, but in the Point g ; for as every Point in the Circle’s Circumference is at the fame Di- ftance from the Center, and as the very next Point to g, in the Line h i, is at a greater Diftance from the Center a than the Point g, therefore the Circle cannot touch the tangent Line in two Points, and confequently the Bafe of the Circle is the Pointy. The fame is to be underftood of the Bafe of an Ellipfis. Right-lined Figures may have a Point for their Bafe alfo, by being fet on angular Points, as the Hex- agon B, Plate I. which refts on its Angle 2, on the tangent Line h i. As Points and Lines are the Bafes of Lines and Superficies ; fo Points, Lines, anft Superficies, are the Bafes of Solids ; as for Example : Firft, I he Bafe of a Sphere is a Point, for the fame Reafon, as it is the Bafe of a Circle ; the fame is alfo to be underftood of the Bafe of a Spheroid. Secondly, If we conceive the curved Superficies of a Cylinder to be an infinite Number of Circles, like Hoops fet dole together, ’tis very eafy to conceive, that the Bafe of a Cylinder lying down, is a right Line, becaufe every Circle can touch the Plane it lies on, but TheShaftof a Column , is a Cylinder , and Frujlum of a Conoid. Def. 78. Of the Sedion of a Solid . Def. 79 - °f the Bafe oj a Line ; Def 80. Of the Bafe of a Circle and Ellipfis. End it Hands on is its Bafe ; as, indeed, i3 every Surface on which any Body Ra ids. Fourthly, The Bafe of a Cone, Conoid, Pyramid, Pyramis, Pyramidoid, is that Superficies which is oppofite to the Vertex, and on which they com- monly ftand *, but in their Fruftums, the Superficies of both Ends are called Ba- fes , a6 the leffer Bafe and the greater Bafe : But though Cuftom has thus diftin- 6 guifhed Of G E O M E T R Y. yr f uiftied the fmall End from the greater, I muft own, I think it a very improper farmer of Diftin&ion, becaufe one Body cannot ftand on two oppofite Ends at the fame Time, and therefore cannot be conlidered as two Bafes, but as two Ends, as they really are, and which may be diftinguifhed by the Names of Greatemkmd Leffer, by only making ufe of the Word End y inftead of the Word Bafe ; for, ftrictly fpeaking, except the Fruftum of a Cone Hands on one of its Ends, neither of the Ends is a Bafe ; for when a Fruftum is laid on its Side, its Bafe is a right Line, contained between the two lowed; Points of the Superficies of its Ends. LECTURE II. On the Formation , Names , Kinds , and Menfuration of Jingles, T HE Angles I am now going to explain, are Angles on Superficies, or rather fuperficial Angles. A Superficial Angle is a Space contained between two Lines, of which one muft be oblique, and which meet each other in the fame Point ; as for Example, Fig. I. Plate II. If the oblique Line d r, be continued forward, fo as to meet the Line g f, in the Point f ; the Space that is contained between them is called an Angle. There are three Kinds of fuperficial Angles, that is to fay ; (i) Right-lined, as onp, Fig. II. Plate II. (2) Curvilineal, as xy z, and 3 2 3, of which x y z is a convex Angle, and 123 is a concare Angle. ( 3 ) Compound, or mixtilineal, as q r s, or t an Angle given. ' First, Make ah equal to the given Line g, and bifeft it in n, whereon ereft the Perpendicular n c equal to h the given Height ; by Problem XIII. Left. Ill, diaw d paiallel to ah, bifeft e J in z, and make c b and c d each equal to ze\ draw the Lines d a and b a, and they will complete the Trapezoid a b d h, as re- quired.' Secondly, Make a g, Fig. O, equal to the given Line d, by Prob. XI. Left. III. make the Angle e ag, equal to the given Angle Q, and make e a equal to the given Line d. On the Point e with an Opening equal to the given Line i, and on the Point g with an Opening equal to the fourth given Side, find the Point °‘ Intei lection Draw the Lines c J , and f g, and they will complete the Tra-* pezia, as required. 1 ^ °lp If the Angle had been required to have been made an internal Angle, tnen pie tw o Sides f e and f g, mult have been drawn to the Point of Interfec- tion h, as in Fig. P, which is a quite different Figure from Fig. O, although the given Angle and Sides 'are the fame. Ir is alfo to be noted, when four right Lin^s are propofed to be the Bounds oi a Trapezium, that thofe two Lines which make the Interfeftion, mui be longer than the Diftance contained between the Extremes of thofe Sides which make the given Angle, othenvifo there cannot be a Trapezium made ; for if the mo re find two Lines, f e and f g, Fig. O, were but equal to the Dif- tance contained between g and e, the Extremes of the Angle g a e, they would n ]- r J ut °ne Line, and confcquently the Figure would be a Triangle, inftead ol a I rape? Rim ; and if thofe two Lines were Rfs than the Diftance from e to F then there could not be any Figure produced. Therefore his plain, that to make a irapezium^ the two Sides which make the interfeftional Point, mull be greater than the Diftance contained between the Extremes of thofe Sides which contain the £.iven Ansrle* Prob. 8 * Of GEOMETRY. Prob. V. Fig. A, B, C, D, and S. PlatcTV. "To defer tie a Circle of any given Diameter , fuppofe ten Feet , and to defril t Ovals of the firfl, fee on d, third, and fourth Kinds, to any Length required. Operation, Firit, make a Scale of equal Parts, as Z, and let each Part repre- fent one Foot. Take 5 Parts in your Compares, and on a deferibe the Circle, whofe Diameter c d, will be equal to ten Feet, as required. Secondly, Divide a f. Fig. B, the given Length of an Oval, into 3 equal Farts at e and h, whereon with the Radius b f deferibe two Circles interfer- ing each other, in c and g, from which two Points, through the Centers e and b, draw the Lines g e d, g b k, c b m, and c e n ; on the Points g and c, with the Radius g d, deferibe the Arches d k, and n m, which will complete an Oval of the firfl Kind. Thirdly, Let df, Fig. C, be a given Length, as before. Divide d f into four equal Parts, at c e h ; on the Points c h, with the Radius c d, deferibe two Circles, touching each other in the Point e ; on f /j make the two equilateral Triangles a c h, and n ch, continuing their Sides out both ways at pleafure, as to 586 and 7,' on the Points a and n, which with the Radius n 5, deferibe the Arches 5 6, and 8 7, which will complete an Oval of the fecond Kind. Fourthly, Let ah be a given Length, as f Fore. Di vide a h into 24 equal Parts, an: ! < aw b d and f i, parallel thereto, each at the Diftance of 10 Parts ; draw e h ^through the Middle of a h, at right Angles to a h, and make c b , c d, alfo^/, and^ i, each equal to 10 Parts, and then will you have completed two geometrical Squares, viz. b c f g and c d g i. Draw their Diagonals, and on their Centers y and z, with the Radius of s d, or 2; i, defqribe the Arches fab and d h i. On the Points c and g , with the Radius g d, deferibe the Arches bed, and f h i, which will complete an Oval of the third Kind. ■ It is here to be noted, That as the Proportion that tire Side of a geometri- cal Square bears to its diagonal Line is yet unknown to alf Mathematicians, the Difference between them cannot be afeertained. But however, the nearelt Pro- portion that the Side has to the Diagonal, is, as Five is to Seven ; that is, if the Side be five, the Diagonal is feven, and a little more. And therefore when the Length of the Oval is divided into 24 equal Parts, or twice 12, then c d, life, be- ing 5 , 2. k will be 7, and a little more; and therefore when the Arches d k i % and b a f, are deferibed on the Centers y z, they will exceed the Points a and k , fome fmall Matter. Fifthly, Let e 4, Fig. S, be a given Length, as before. Divide the Length e f, into four equal Parts, at the Points 1,2,3, an d through them draw the Lines r t, b n, and s v, at right Angles, to the Line e 4 : make 1 r, 1 t, alio 2 b, 2 n, and 3 s, 3 v, each equal to one-fourth of e 4, viz. to e 1, and complete the three geometrical Squares, e r 2 /, b 1 n 3, and s 2 v 4, continu- ing the Sides n 1, and b x, as alfo the Sides b 3, and n 3, out at pleafure. On the Centers 1 and 3, with the Radius e 1, deferibe the Arches m e h, and d 40. On the Centers b and 11 , with the Radius n h or n d, deferibe the Arches h d, ajad m 0, y/hich will complete an Oval of the fourth Kind, as required. Prob. VI. Fig. V, W, X, R, T, and Y. Plate IV. To male an Oval of any Length and Breadth required, by divers Methods » Let the Lines z z, x x, Fig. V, be the given Length and Breadth. Operation. Firfl, make d l equal to z z, and by Prob. VI. Lect. lit. divide d l in two equal Parts, by the Line a r. Make x c and x n , each equal to half x x. Make d e equal to x c; divide e x into 3 equal Parts, and make e h equal to 1 Part. Make x t equal to x h, and by Prob. I. hereof, on the Line h t, complete the two equilateral Triangles, h d t, and r h t, con- tinuing their Sides through the Points h and t, at pleafure. On the joints b and tj with 8a Of GEOMETRY. /, with the Radius t l, defcribe the Arches k l m, and b d q ; alfo on the Points a and r, with the Radius r b, defcribe the Arches b c k, and q n w, /which will complete the Oval, as required. , Secondly, by a D'mifion of two Circles, Fig. W. Let the given Length and Breadth be the Lines x x, z z, as before. Operation. Make the Line I 2, equal to the given Line z z, and divide it into two equal Parts by the Perpendicular 3 6. On a, the Point of Interfedtion, with the Radius a 1, defcribe the Circle, 1326; alfo on a, w r ith a Radius equal to half the Line x x, defcribe the concentrick Circle 7,4, 8, 5. Divide the Circum- ference of each Circle into any and the fame Number of equal Parts, (the more the better) as in the Figure where each Circle is divided into 24 Parts. Draw right Lines from the Divilions in the fmail Circle, parallel to the Line 1 2, to the right and to the left, at pleafure. Alfo draw right Lines from the Divifions as s r t x z, in the outer Circle parallel to the Line 3 6, and through the Points of Interfedlion, that they make with the other Lines before drawn, as c deh i, IFc. trace the Circumference of the Oval, whofe Leifgth 1 2, is equal to z z ? and Breadth equal to x x, as required. Thirdly , by the Ordinates of a Circle , Fig. X. Let the given Length and Breadth be as before. Operation. Make b e and ci d, at right Angles to each other, and equal to the given Length and Breadth. On the Point of Interfe&ion with the Radius c d, defcribe the Circle a 5 d, &c. Divide the Semi-diameter c f, into any Number of equal Parts, fuppofe 4, as at the Points 123; through which draw' right Lines, parallel to ad, as I g, 2 i, 3 k, which are called Semi-ordinates of the Circle. Divide b c and c e, each into the fame Number of equal Parts, as / c , at the Points 4, 5, 6, through which draw Lines parallel to a d. Make 4 7, 4 in, (which are Semi-ordinates of the Ellipfis) each equal to 1 g, the Semi-ordinate of the Circle. Make 5 8, and 5 n, each equal to the Semi-ordinate 2 i ; alfo 6 9, and 6 o, each equal to the Semi-ordinate 3 k ; then from the Point a, through the Points 7, 8, 9 , eon irud, trace one half Part of the Ellipfis. In the fame Manner fet off Ordinates on the other Side, and complete the Ellipfis, as required. Fourthly, by the Help of a Line, or String, Fig. T. Let the given Line h be the Length, and the Line a Decagon 36, of an Undecagon 32 T s r> and of a Duodecagon 30. To prove that the aforefaid Degrees are the Quantity contained in an Arch, whofe Chord Line is the Side of a Triangle, geometrical Square, &c. div’de 360, the Number of Degrees in a Circle by the Number of Sides contained in the Figure propofed, and the Quotient is the Number of Degrees contained in tire Arch of every fuch Chord Line, which is the Side required. Let it be required to defcribe a Pentagon, as Fig. A D. Operation. With 60 Degrees of your Line of Chords, on 2 defcribe the Circle a b d i h, make a b, b d, di, i h, and h a , each equal to 72 Degrees, and draw the Lines a b, b d, d i, i h, and h a, they will complete the Pentagon, as required. Note, If your Line of Chords Jfhonld be of tod large or too fmall a Radius, then proceed as follows, viz, fuppofe ’tis required to defcribe the fmall Pentagon pk In m. ‘ s First, complete the Pentagon, a b d i h, as before taught, and draw the Lines s h, z a,zb,z d, and 2 i. Bifeft any Side of the Pentagon, as b d, in u : make u t and w each equal to half one Side of the given fmall Pentagon, and draw t k and avp, at right Angles, to a b, meeting the Lines a 2, and b 2, in the.Points / and k. Make z l, z n, z m, each equal to 2 k, or 2 p, and drawing the Lines / k, k /, p m, m n, and n l, they will complete the Pentagon, as required. Example II. Again, fuppofe the fmall Pentagon pk l n m is given, and ’tis required to de- ftribe the large Pentagon a b di h, with a fmall Line of Chords. First, go 0/ GEOMETRY, First, Complete the fjnall Pentagon, and from its Center draw right Lines through the angular Points at pleafure. Continue any Side of the lmall Pen- tagon at both Ends, at pleafure, as the Side k p, towards q and r ; bifeft k p in s : make s q, and s r, each equal to half of one Side of the large Pentagon. Draw the Lines q b, and r a, at right Angles to q r, and continue them to meet the Lines z a, and z b, in the Points a and d ; make z d, z i, and z h , each equal to z b, or z a, and draw the Lines a b, b d, d i, i h, and h a, which will complete the large Pentagon, as required. Pros. XXXIV. Fig. R. Plate VI. To defcribe any Polygon, on a given Side, having the Number of Degrees given , that are contained in each Angle of the Polygon. The Number of Degrees in the Angle of a regular Pentagon are 108, in a Hexagon 120, in a Septagon 128 f, in an Odtagon 135, in a Nonagon 140, in a Decagon 144, in an Undecagon 147 A, and in a Duodecagon 150. Let a b be the given Side. Operation. On the Points a and b, with 60 Degrees of Chords, defcribe the Arches^ f and h i ; make h z, and g x, each equal to go Degrees, and z i, and x f, each equal to 18 Degrees , then will the Arches g f, and h i, be each equal to 108 Degrees ; through the Points f and i, draw the Lines a e and b a, each equal to a b, by Prob. XL Lect. III. make the Angles a e m, and b a m, each equal to the Angle aba, and draw the Lines e m and a m, which will meet in m, and complete the Pentagon, as required. And fo the like for any other Polygon. The Number of Degrees, that are contained in the Angle of any Polygon, is found by fubtradling the Number of Degrees contained in the Arch, whofe Chord is a Side of the Polygon, from 108, and the Remains is the Quantity of the Angle required. Prob. XXXV. Fig. Plate VI. To find the Radius of a Circle capable to contain any Polygon , whofe Sides Jhall be each equal to a given Line, as a c. Operation. Bifeft a c in b, whereon eredl the Perpendicular b m ; make a h equal to a c, and on h, with the Radius h a, defcribe the Arch a d c, which divide into 6 equal Parts at the Points 121/34, make h n, n 0, h p,pg, g r, r s,s t, and tm, each equal to the Chord Line of the Arch an, a p, a g, a r,a s, at, am, a 1, and draw the Lines a 0, which are the Semi-diameters of Circles that will contain all the Polygons from a geometrical Square unto a Duodecagon, viz. the Line a 0 is the Radius of a Circle that will contain a geometrical Square, the Line a n, the Radius for a Pentagon ; a h, for a Hexagon ; a p, for a Heptagon ; ag % for an Odlagon ; a r, for a Nonagon ; a s, for a Decagon ; a t , for an Undeca- gon ; a m, for a Duodecagon. In the like Manner any greater Number of equal Parts being fet above m , ail other Polygons of more Sides than 1 2 may be deferibed. LECTURE V, On the inferibing and circumfcribing of Geometrical Figures. P r o b. I. Fig. T. Plate VI. To inf c rile a Circle , as c ah, in any right-lined Triangle, rz-f ilk. QPeration. By Prob. XI. Lect. III. divide any two Angles of the Tri- angle, by Perpendiculars, as i d and h e , interfering each other in/ ; from whence, by Prob. VIII. Lect. III. let fall a Perpendicular, as f a, ©n f: with the Radius f a, defcribe the Circle a b c, which will touch the Sides! / i and k /, in the Points of ContaA b and c, and therefore is inferibed, as required. Prob. II. Fig. S. Plate VI. To infer the a Circle , as n 1 m e, 'within a geometrical Square , as b c a d. 6 Operation . Of GEOMETRY. Q1 , operation. Lraw the diagonal Lines b d, and a c, from the Center h ; let fall' t;se Perpendicular h e ; on the Center h, with the Radius he, defcribe the Circle 1 m ch will touch the Sides in the Points n Im e, and therefore is infcribed as required. ’ Prof. III. Fig. V and W. Plate VI. 1 ° tn J crt ° e a (du de, as h k 1 l g, 'within any regular Polygon , as the Pentagon Operation. Let fall a Perpendicular from the Center d, to any Side, as d g, on f e , with the Radius d g defcnbe a Circle, which will touch the Sides of the quire? 0 "’ “ * ' ^ h . U 1 S ' and therefore is infcribed, as re- .y 7 ? 1S a y. cond Ex ampie of a Hexagon, which hath a Circle infcribed Tftttnm it, in the lame manner. P R o b. IV. Fig. X. Plate VI. To infer toe a geometrical Square , as e f d z, within any right-lined Triangles , as Operation. On the Point c ereft the Perpendicular c x, equal to c b. From the angular 1 dint a, draw a g, parallel to x c , meeting the Bafe b c in g. Draw * p- parallel to 6 tr f di e % %>-ia„gk ^ r/J reqSed! f S ^ibed within the . ' P R o b. V. Fig. Y. Plate VI. o wfer the an equilateral Triangle , «abe, in a geometrical Square , as cad?. P er f t0 .!f E»raw the Diagonal <2 g, which bifeft in n. On n, with the Radius L!V tV he , C ' T rC, e C r d Z' OI ) with the Radius gn , defcribe the Arch i j: p Draw "S ht Llp 1 s fl P m « t0 and toy; which will interfeft the Sides of the and d J> ln , the joints b and e. Draw the Line b e, and the Triangle n o e will be equilateral and mlcribed, as required. " P R o b. VI. Fig. A D. Plate VI. To infer ibe an equilateral Triangle as ^ beg, within a regular Pentagon, as alf?d 7 S”th B A ed i any , S ‘ de .’ aS /j In two ’ and ereft the Perpendicular 2 b ; in y be r 1 a t 1 ’ mt ° tW ° equal Pa ' ts > b ? the Line L, cutting b a Arrb h Cen y e a,° the Pentagon. On with the Radius b 2, deferibf the p . h * * * * divide the Arches .v 2 and 2 c, each into tw and make P 0 ec l ual to one Sued to ? C LinCS * ? and ? blfe c^h equal to the Arch L ° n ] th i C r° intS ^ and , throu g h the Points/ and 0, draw Right Lines both Ways at pleafure; which will meet the Lines a 0, and a f, in the Points candy. Make 0 r, and fv, each equal to a f, or a 0, and join „ r, then will J a 0 r v, be the circumfcribing Pentagon, as required. P R O e. XVII. Fig. Z, and A B. Plate VI. J o circumfcribe a geometrical Square, about any Scalenum, or Ifofceles Triangle. I his may be done two Ways. ' 5 Let enb. Fig. Z, be a Scalenum Triangle given. Operation I. Continue the Side en towards d, and through the angular Point b draw the right Line a c A parallel to ed. On e ereft the Perpendicular * a, co meet the Line a c, in the Point Make a r, and <• d, each equal to a r, and ciaw c d, which will complete the circumfcribing geometrical Square, as required. 1 Operation II. Fig. A B. Draw ca through the angular Point b, and parallel to the oide x n. From the Points « and x let foil Perpendiculars to the ‘Line « «. 1 lace c m, and a h, each equal to c a, as required, which will complete the circum.cnbing geometrical Square, as required. L E C T. VI. Of proportional Lanes. ^ Prob. I. _ Fig. N. Plate VII. ■I o fnd a mean proportional Line, between t wo given Lines. A Mean proportional Line, is that which being multiplied into itfelf, its Pro- dud is .equal to the Product of the two given Lines multiplied into each other ; or it is the Side of a geometrical Square, whofe Area is equal to the Area of a Parallelogram, whofe Length and Breadth is eoual to the two given Lines. 6 Lf.t d and g be the two given Lines. Operation. Draw a right Line, as a c, at pleafure, make a b equal to the Lme rd, and b c equal to the Line e. Bifed a c in x, and defcribe the Semi- ^ circle 54 Of GEOMETRY. circle ah c\ on h ereft the Perpendicular h h, which is the mean proportional Lmc required. P r o b. II. Fig. O. Plate VII. To cut from a given Line , a Part that Jhall be a mean Proportional between what remains , and a Line propojea , as tot Line n. Let n be the given Line, and m the Line propofed. Operation. Draw a right Line, as a g, at pleafure ; make a e equal to the Line », and e g equal to m. Bifeft a g in r, and on r deicribe the Semi-circle ax gi and on e ereft the Perpendicular e x. Bifeft eg in h, make h c tqua o h x then e e, the Part cut off from « e, equal to the given Line », is a mean Pro- portional between e a , the Part remaining, and m, the Line piopofed. T making //, in Fig. Q, equal to a c, and i k equal to m ; and the Semi-circle k h / being deferibed, the Perpendicular i h (which by the Lft Prob. is a mean Proportional to the Lines k i, and i l) will be equal to c e, the Part cut oft. P r o b. III. Fig. P. Plate VII. Two Lines being cmneBcd into one Line, and their mean Proportion fparate, being given, to find the Lengths of the given Lines, which are called Extremes. Let Jr be the given Extremes, connefted together without Diftm&ion, and the Line d, the mean Proportional. „ . . . . Operation. Bifeft « c in b ; on b deferibe the Semi-circle a * r ; on r emff the Perpendicular c i, equal to the Line d\ draw t g parallel to a c, cutting Semi-circle in g. Draw g h parallel to i c, which will divide a c m h ; then are a h, and h r,°the two extreme Lines required ; for by Prob. I. h g is a xnpon Proportional to a h and h c, and is equal to the Line d alio. Prob. IV. Fig. R. Plate * VII. Two right Lines being given, to find a third Proportional. Let k and m be two given Lines. . » , Operation. Make an Angle at pleafure, as cl n e. Make nfi equal to h , and n //and f a each equal to m, and draw the Line / h ; alfo draw the Line a , parallel to h f, then will a i be the third Proportional required. Prob. V. Fig. S. Plate VII. 7 he right Lines being given, to find a fourth Proportional. Let the Lines I, 2, 3, be three given Lines, and ’tis required to find a fourt 1, which will be to 3, the third, exaftly the fame, as 2, the fecond, is to the ^Operation. Make an Angle at pleafure, as n g h, make gf equal to the Line j , and g i equal to the Line 2, and / » equal to the Line 3. Draw , /, and parallel thereto, the Line n m ; then will 1 m be the fourth Propoi tional requn , for i m is to i g, the fame as n f is to / g, and therefore m t is to nf, exactly the " Note , This Problem is nothing more than the Golden Rule, or Rule of Three , geo- metrically performed. Prob. VI. Fig. T. Plate VII. ^ . The Mean of three Proportionals, and the Difference of the Extremes being given, to find the Extremes. Let b c be the mean Proportional, and g e the Difference of the Extremes. Operation. On e ereft the Perpendicular e d, of Length- equal to b c. Bifebt *. e in h ; on h, with the Radius h d, deferibe the Semi-circle k day and then i e, and e a, are the Extremes required. Prob. 95 ' Of GEOMETRY. Prob. VII. Fig. V. Plate VII. To find the Extremes b and f, having t b, and therein affume two Points, as a and b, whofe Diftance muft be equal to the given Diftance q r ; on the Point a deferibe the Semi-circle b i, and on b the Semi-circles a c, and i d ; then on the Point a deferibe the Semi- circles c h and d /, and on the Foint b the Semi-circles k e t and If. ' Proceed in like manner, as in the laft Problem, to make as many other Revolutions as may be required. Pros. XIX. Fig. V. Plate VIII. To deferibe an Artinatural Line. Operation. Firft trace by Hand the feveral Curvatures or Turnings at pleafure, which divide into as many Parts as feem each to be the Segment of a Circle, as e c a, n b g, &r. This done, in each Arch aftiime 3 Points, as e c a, and n h g, and then by Prob. XIX. Lect. III. find the Centers f and m, and deferibe the Curves e c a, and n k h g 0. In the like manner proceed throughout the whole, to deferibe all the various Meanders remaining, which will appear with the utmoft Beauty. Serpentine Rivers, and Walks through Wilderneffes, &c. being laid out in this Manner, are the neareft to Nature, and the moft agreeable of all others. PART III. Of Architecture. LECTURE I. Of the DefcOption and ConfruHion of Moldings. T HE feveral Members or Moldings of which the five Orders are compofed, are of three Kinds, viz, fquare, circular, and compound. Flrjl, Square Members are Plinths, Fillets, Dados, Cinftures, Annulets, Aba- cufes, Fafcias, and Tenias of Architraves, Freezes, Denticules, Dentuls, and Regulas. Secondly, Circular Members are Beads, Torufes, Aftragals, Ovolos, Cavettos, and Apophyges. Thirdly , Compound Members are thofe which are compofed of two or more Arches, as Scotias, Cyma Reftas, Cyma Reverfas, Plancers of Modillions, iAc. As fquare Members are nothing more than Parallelograms, I need not fay 98 Of ARCHITECTURE. fay any Thing of their ConftruCtions, and therefore I dial! proceed to fiagle and compound Moldings, and give the Etymology of fquare Members as they come in their Order. Pros. I. Fig. B. Plate VIII. To defcribe a Torus . Let to x be the given Height. Operation. Draw x r at pleafure, and the Line w parallel thereto, at the Di- ftance of the given Height ; in any Part, as at «, ereCt the Perpendicular n a, make n c equal to half the given Height, and on c, with the Radius n c, defcribe the Torus required. This Member is called a Torus from the Greek Toros , a Cable, which its Swelling refembles, or rather from the Latin Torus, a Bed, or Culhion, becaufe it feems to fwell by the impofed Weight. It is generally placed on a Zocolo or Plinth , D, which is fo called, from Plinihos , a fquare Brick or Table, placed the very lowermoft of all, to preferve the Foot of the Column from rotting ; for ori- ginally Columns were made of the Tapering Bodies of Trees. Prob. II. Fig. C. Plate VIII. To defcribe an Ajlragal with its Fillet. Let d f be the given Height. Operation. Draw f z at pleafure, and in any Part, as at f, erect the Perpen- dicular f d , equal to the given Height f d, which divide in 3 equal Parts at e and a , through the Points da e, draw the Lines d iv, a c, and e x, parallel to f z ; make f h and f g each equal to e f. On a defcribe the Semi-circle de, and on g the Quadrant f k , which will complete the Altragal, as required. This Member is called an Altragal from the Greek Aflragalos, the Bone (or more properly the Curvature) of the Heel, and for which Reafon the French call it Tcilon , either of which I think is very proper, when employed in a Pedellal or Bafe of a Column, but not when placed on the Shaft of a Column, when it does the Office of a Collar, and is therefore by many called Collarino. Prob. III. Fig. O. Plate VIII. To defcribe the Apophyges of a Pilafler or Column. The Apophyges of a Column or Pilafter is that curved Part of the Shaft, which rifes or flies from the CinCture, and ends in the Upright of the Shaft, as the Arch b d ; it is alfo by fome Mailers ufed at the lower Part of the Corinthian Freeze, and of the Dado of a Pedellal. This Member takes its Name from the Greek Word ’A vrotpvyv, becaufe in that Part the Column feems to emerge and fly from its Bafe. In the Tufcan Order, this Member is nothing more than a Quadrant, as h a, Fig. B, whole Height is equal to its Projection, but in all other Orders it is not fo, and is thus defcribed. Operation. Divide the Projection of the CinCture e d. Fig. O, before the Upright of the Column into 5 equal Parts, make its Height e b equal to fix of thofe Parts ; draw a b parallel to e d, alfo draw b d y which bifeCt in g, whereon ereCt the Per- pendicular g a, cutting b a in a ; on a defcribe the Arch b d, the Apophyges required. Note, The fame Rule is to be obferved in deferibing the Hollow under the Fillet of the Collarino, at the Top of a Shaft of a Column in every of the Orders. . Prob. IV. Fig. F and G. Plate VIII, To defcribe an Ovolo of any given Height, Let a c, Fig. F, be the given Height. Operation. Firjl , draw c d at pleafure, on any Point, as c, ereCt the Ferpen. dicular c a equal to the given Height, through the Point a draw b e , parallel to d c t on a, with the Radius a c defcribe the Arch c b ; which is the Ovolo re- quired. Secondly, Let b c , Fig. G, be the given Height. Operation. Divide the given Height into 4 equal Parts, and give 3 of thofe Parts to the FrojeClion. Draw the Lines- 3 c, which bifeCt in d, on which ereCt the Perpendicular cl a, on a defcribe the Arch c 3, which is the Ovolo required. 6 " This 0 / ARCHITECTURE. 99 This Member is called an Ovolo , from the Latin Ovum , an Egg, which tis generally carved into, intermixed with Darts and othei Devices, fymbolizing Love, &c. It is alfo called Echinos, or Echinus, from the Greek, as being fome- thim>' like the thorny Hulk of a Chefnut, which being opened, difcovers a Kind of Oval Kernel, fomething dented a little at the Top, which the Latins call Ue- cacumlnata Ova, and Workmen Quarter Round. P. I remember, that , in the lajl Problem, you was /peaking of the Apophyges taking its Rife from the Cindure ; pray what is a Cincture ? . M. A Cindure is the firft Part of a Shaft of a Column, as a w, in Fig. 15, Plate VIII. which always is placed on the Bafe of every Column, and anciently was nothing more than a broad Iron Ferril or Hoop, to coniine and {Lengthen the lowermoft Part of the Shaft, which the Italians call Lijlello, or Girdle. The Shaft of a Column is that round plain Part, which is contained between the Bafc and the Capital, of which I fliall gi ve you a more full Account, when I come to treat of the Parts of an Order. It is alfo called Fuji from the Latin l 1 uf is, a Club ; Vitruvius calls it Sc a pis, and by fome Mailers ’tis called Vivo, Fige , and Trunk. P r o b. V. Fig . D and E. Plate VIII. To defcribe a Cavetto of any given Height. Let a c. Fig. D, be the given Height. « Operation. Firfl, Draw e /at pleafure, and in any Part thereof, as at c, erect the. Perpendicular c a, equal to the given Height, and through the Point q draw the Line b g, parallel to e f; make c e equal to c a, and on e with the Diltance e c, defcribe the Cavetto b c, as required. . Note, If ’tis required to make a billet on the Cavetto, as b n , then the given Height mull be divided into 4 equal Parts, and the Fillet made equal to one Part. The Projection of its under Part c el is equal to one 8th of the whole Height, which is half of h cl, or of one Part. . This Member is called Cavetto , from the Latin Cavus, a Hollow; and Woik- men call this Member a Hollow alfo, though I believe not with Refped to the Latin , but becaufe it is a real Hollow ; and as an Ovolo is generally made a Qua- drant, they therefore call that Member a Quarter Round. To defcribe a Cavetto a fecond Way. Secondly , Let h y, Fig. E, be the given Height. Operation. Divide. h y into 5 equal Parts, and give the upper I to the billet, make the Projedion 1, 3, equal to 4 Parts, andy n equal to 1 Pait, and draw the Line a n parallel to by, continue y n out at pleafure, and draw the Line 3 xn, which bifed in jc, and thereon ered the perpendicular x p. On p defcribe the Cavetto n 3, as required. P r o b. VI. Fig. H. Plate VIII. To defcribe a Bed- Molding of any Height required. Let a .xbe the given Height. . _ Operation. Divide the given Height into 8 equal Parts, give 3 t° the Cavetto, 1 to the Fillet, and four to the Ovolo, and then by Problems IV. and V. defcribe their Curves, as required. P r o b. VII. Fig. I. Plate V III. To defcribe a Cymatium of any given Height. Let a r be the given Height. , . , Operation. Divide the given Height into 4 equal Parts, as at 4 h, and give the upper I to the Height of the Regula. Draw right Lines from the Points 4, 3, and h, at right Angles to the Line 4 h, of Length at pleafure, and draw a g at any Diftance from 4 h, and parallel thereto make n c equal to n g, and draw the Line r which bifed in e, on«, and * gt make the equilateral Sediqns d and /, whereon defcribe the Arches c e and eg, which completes the Cymatium, ^ This Member with its Regula is called a Cymatium, from the Greek Undula, a rolling Wave, which it resembles, or Kymation, a W ave. Vitruvius calls IOC. Of ARCHITECTURE. it EpitthcateSy and the Italians and French, Gala, Gen/e, or Douche. But when we ipeak of this Molding- fingly, without its Regula or Fillet, we call it a Cyma Re 8 a, and Workmen oftentimes call it a Fore Ogee, to diftinguifh it From Cyma Inverfa, which they call a Back Ogee. P R o B. VIII. Fig. K. Plate VIII. To defcrihe a Cyma inverfa, as b r, of any given Height* Operation . Draw the Line nr, at pleafure, in Part, as at r, ere A the Perpen- dicular r l equal to the given Height, which divide into 4 equal Parts, and give the upper I to the Fillet. Through the Points a andT draw right Lines, as d b, and c a, parallel to n r, and of Length at pleafure. Make a c equal to a r, divide c a in 6 equal Parts, and make n r, and e c, each equal to one of thofe Parts 5 draw the Line eg n, which bifeft in g, on the Points n g, and g e ; make equila- teral Se&ions, and delcribe the Arches eg, and g n, which completes the Cyma Inverfa , as required. Prob. IX. Fig. L. Plate VIII. To defcrihe a fngle Cornice of any given Height. Let ah be the given Height. Operation. Firft, divide the given Height into 5 equal Parts, give the lower I to the Cyma Inverfa f ; one third of the fecond to the Fillet e, and the upper I to the Regula c j and the remaining two Parts and f , to the Cyma Refta d. Secondly, by Prob. VII. and VIII. deferibe the Curves of the two Cymas, and the Cornice will be completed, as required. Note, That the Projection of the Cyma Rc&a, and of the Cyma Inverfa, which is alfo called Cyma Reverfa, is always equal to their own Height. Prob. X. Fig. B A. Plate VIII. To divide and proportion Dentals to any given Height . Let n x be the given Height. Operation. Divide the given Height into 8 equal Parts, give the upper one to n s, the Height of the Fillet, the next fix to j- v, the Pleight of the Dentuls, and the lower one to v x, the Margin of the Denticule * To proportion the Breadths of the Dentuls and Intervals between them, make “o q equal to j v, and dividing v q into 3 equal Parts, give tw o to the Breadth of a Dentul, and one to its Interval, which is called Metoche, which with two Pair of Compafles, the one opened to the Breadth of a Dentul, and the other to the Breadth of an Interval, fet off thofe Dillances reciprocally throughout the whole Length of your Molding. If it is required to make Eye-Dentuls in the Intervals, as A A, divide the Height of the Dentul into 5 equal Parts, and give the upper one to the Height of the Eye-Dentul. Note, This Ornament is generally begun at the proje&ing Angle, over an angular Column, with the Form of' a Pine- Apple ; or rather, the Cone of a . Pine-Tree, as at k g, which is thus deferibed. Make its Breadth zn, equal to the Breadth of a Dentul, which divide in 4 equal Parts; make k g equal to n z, and draw z g ; make n d, z b, each equal to half n z ;.and draw db, which bifeft in e. On e, with the Radius e d, deferibe the Semi-Circle dm b. On the Points df, and b f, with the Ra- dius f d, deferibe the dotted Se&ions next above the Line d b, on which, with the fame Opening, deferibe the Arches b f, and f d, which will complete the whole, as required. Thfse Ornaments are called Dentuls, from Denteh 7 , Teeth, which they repre- fent. The Denticulus is that flat or fquare Member, on which the Dentuls are placed. Prob. XI. Fig. I k, next under Fig. A B, aforefaid Plate VIII. * Vo proportion and defcrihe an Isnick Mqdilliov, of any given Height required. Let a b be the given Height. Operation. Divide the Height into 8 equal Parts, as r q, give the upper 2 to the Height of the Cyma Inverfa, with its Fillet, and the next 5 to the Depth Of ARCHITECTUR E.. lot Depth of the Modillion. Draw d c , for the Side of a Fiont Modillion, make c e equal to c d, and d f equal to d c, then is d f the Breadth of the Modillion in Front. Divide d f in 4 equal Parts ; make f l, the Projedtion of the Modillion in Profile, equal to 6 of thofe Parts. Divide the Projection of the Modillion in Profile into 6 equal Parts, at the Points 1, 2, 3, 4, 5. Through the Points 2 and 5, draw the Lines 0 m , and 5 t , parallel to f p. Make 5 t equal to two Parts and half, and 2 0 equal to one Part : Alfo make 0 m equal to 5 t, and draw the Line m s t. On the Points m and t, with the Radius t 5, defcribe the Arches 0 s , and s 5 ; alfo on 2, with the Radius 2 1, defcribe the Arch I o, which will complete the Modillion, as required. This Member is called Modillion, from the Italian Modiglioni , a plain Support to the Corona of the Corinthian and Compofite Cornice, to which they only be- long, although now falfely introduced into the Ionick. P R o b. XII. Fig. N and M. Plate VIII. 1 0 defcribe Scotias of any given Heights. Firjl. Let a g, Fig. M, be the given Height. Operation. Draw the Line/ g, and on any Part thereof,' as at g, ere ft the Per- pendicular g a equal to the given Height, and through the Point a draw the Line a x, parallel to gf. Divide a g in 3 equal Parts, at the Points d z, and through the Point d draw the Line c d e , parallel to a x. Make d e equal to d a. On the Point d defcribe the Quadrant a c ; and on the Point e the Quadrant c.f, which together form the Curve of the Scotia, as required. This Member is called Scotia , from the Greek Xjw-na, Skotos, Darknefs, which the upper Part caufes by its Projedture. : Tis alfo, by lome, called Trochiius, from the Greek Frochdos , T gosxv, or Tgo*a, a Rundle or Pully, whofe hollow Part within the Rope-works hath fome Refemblance of this Member ; and with re- fpedt to its Darknefs, ’tis by many, though improperly, called a Cavetto. The Italians call it Baflone. This kind of Scotia is adapted to the Attick Bafe. Secondly. Let ad, Fig. N, be the given Height. Operation. Draw the Lines k a and n d, parallel to each other, at the Diftance 'of a d, and draw a d at right Angles thereto. Divide ad in 7 equal Parts, and through £, the third Part down, draw h c, parallel to a k. Make c h , and d n y each equal to a c ; and draw i h n, parallel to a d. Make h i equal to h n , and from i through c, draw the Line i c m. On the Point c defcribe the Arch a m , and on i the Arch m ri, which completes the Scotia, as required. Prob, XIII. The Diameter, or Breadth of a Door , or Window , being given, to find the Breadth of an Architrave , that will be proportionable thereto. A General Rule. Divide the Diameter, or given Breadth, into 6 equal Parts, and take one for the Breadth of the Architrave required ; and that you may alio know how to dh vide the Architrave into its proper Members, I have given you in Plate VIII. and IX. thirty and one kinds of Architraves, of which thofe marked A B C D E F, are Tufcan ; G H IKLM N O, are Dorick ; P Q__R S T V, are Ionick ; W X Y Z, AB, AC, are Corinthian ; and A D, A E, A F, A G, and A H, are Compofite, which in general have the Heights of their feveral Members propor- tioned by equal Parts. As for Example. In Fig. A, the Height or Breadth of that Architrave is divided into 10 equal Parts, of which the upper 2 and \ is the Height of the Tenia a, and the Remainder is the great Fafcia, with its Hollow. In Fig. D, the Height is divided into fix equal Parts, of which the upper 1 is the Height of the Tenia ; the lower 2 the Height of the fmall Fafcia c, and the other 3 is the Height of the great Fafcia b. In the fame manner you are to underfland all the others ; and as the principal Parts into which the Height of every Example is divided, are figniiied by the equal Divilions and Figures again!! them ; and as the Manner of deferibing all the Moldings of which they are eompofed has been already taught, to fay any thing further on the manner of deferibing them is O needlefs j iot Of ARCHITECTURE. rseedlefs ; a'a indeed is what L have already faid, the whole being fo very plain, as' to be underftoodby the meaneft Capacity, at the firft View. L E C T. II. ' Of the making of Scales of equal Parts , for the delineating of Plans and Elevations of Buildings. T HE neceffary Scales for our Purpofes, are thofe reprefenting, firft, Feet ; fc- condly, Feet and Inches; thirdly, Modules and Minutes; and fourthly, Chains and Links. Thofe of Feet, and Feet and Inches, are ufed in the making of Plan6 and Uprights, or geometrical Elevations of Buildings. Thofe of Modules and Minutes are for proportioning of the feveral Members of the five Orders of Columns in Archite&ure ; and thofe of Chains and Links are for making Surveys of Lands, as Farms, Parks, csV. whofe feveral Ufes will be fully illuftrated in their proper Places. P r o b. I. Fig. I. Plate IX. 7 o make a Scale of Feet. Operation. Make a Parallelogram at pleafure, as a d tn e ; open your Compares to any fmall Diftance, and fet off 10 equal Parts, from m to x b ; alfo make * h , and b e> &c. each equal to in x b c ; then will the Line m e be a Scale of equal Parts, which may reprefent Inches, Feet, Yards, &c. and which muft be thus numbered, viz. As x b is equal to the io Parts between m x, therafore at b place the Number io, at e the Number 20, life. being fo many Parts from x. To take off any Number of Feet, lefsthan 10, fet one Foot of your CompafTes on x, and extend the other to the Number of Feet required. To take oflF any Number of Feet more than 10, fet one Foot of your Com- pafTesin b , and extend the other to the Number of odd Feet that is contained in the given Length more than 10. Suppofe 17 was the given Length: extend your CompafTes from b to 7 Parts beyond x towards m, which is 1 7 Feet, as re - quired ; and fo the like of any other Number of Feet, more than 10, 20, c fc. To make a Variety of Scales of equal Parts, which it is neceffary to have, a» fome Works require a leffer or a greater Scale than others ; therefore, if from the 10 equal Parts, in m x, you draw right Lines unto the Points, and afterwards draw right Lines parallel to m e, at any Diftances, as f r, g q , 'h p, i 0, k n, and / w, you will have made other Scales of equal Parts, of various Sizes, which may fit all Purpofes required. P R o B. II. 7 b make a Scale of Feet and Inches. Fig. VI. Plate IX. Operation. Make a Parallelogram, as abed, fet off 12 fmall equal Parts, from e to e, reprefenting the Inches in a Foot; make e 10, 10 20, 20 30, &c. each equal to the 12 Parts, then is your Scale of Feet and Inches completed; for e 10, 10 20, are Feet, and the Parts in c c , are Indies. To take off a Length of Feet and Inches, is the fame here, as before in the Feet : fo the Diftance of 3 1 o, is 15 Inches, of 6 10, is 18 Inches, of 9 10, 21 Inches. Scales of Feet and Inches are alfo made on two-foot llules, as Fig. II. in manner follow- ing, Make a Parallelogram, as cazb, at pleafure, and let the Diftance of z/be made to reprefent one Foot. Make / 3, 3 1, and 1 b, on the Line z b, each equal to z f; that is, each equal to one Foot. Draw / g, parallel to c z. Bifeft eg in e, and draw the Lines e z, and e f Divide ^ / in 6 Parts, at the Points I k i h g, and draw right Lines through them, parallel to z b, and then is the Scale completed ; and the Diftance of z f t which is the given Foot, is divided into 12 Inches, viz. The Diftance of g 1, is one Inch ; h 2, two Inches; i 3, three Indies ; k 4, four Inches ; / 5, five Inches ; g 6 , fix Inches ; / 7, feven Inches; k 8, eight Inches; i 9, nine Inches; h ic, ten Indies; g 11, eleven Inches ; and f z, one Foot, as before. These kind of Scales may be made either bigger or lefs, at pleafure, in the very Of ARCHITECTURE. ' 103 very fame manner, as may be feen at the End a b, where the Foot is made but half the aforefaid. Pros. III. Fig. IV. Plate IX. To make a Scale of Chains and Links , for the plotting of Lands., Sic. Operation. Make a Parallelogram, as a v b Fig. I. Plate XIX. is 8 Diameters, 45 Minutes ; as alfo may the Diameter of the entire Order, whofe Height a h is 11 Diameters, 3 Minutes, and -J-, as exprefled on the Line / w. This being underftood, and a Diameter being thus found and divided, the delineating of this Order is eafily performed, as follows. Prob. II. To delineate the Tufcan Pedejlal , by Modules and Minutes . Let A, Plate XIX. be a Diameter found, or given (which is alfo called a Module), and divided into 60 Minutes. Before we proceed to this Operation, it is to be obferved, that the Heights of the Members are exprefled on the central Line, to be read upwards, and their Projedtures are placed againft them, to be read level with the Eye, either on the right or left Hand Side. Operation. Firft, Draw a bafe Line, as h r, Fig. III. Plate XIX. and in any Part, as at k, eredt the Perpendicular kk. Make i f equal to 37 Minutes and ■5, as exprefled between k and f ; alfo make f e equal to 2 ^ Minutes ; e d to 5 Minutes ; d c to one Diameter, 9 Minutes, 4 » c a to 4 Minutes, { ; a b to 2 Minutes, j, ; k k to 1 7 Minutes, * ; and through the Points kb ac d e f y draw right Lines to the right and left, parallel to the bafe Line k r. Secondly, Make Jr, and f s, each equal to 47 Minutes and \ ; and draw the Line s r. Make f and e v, each equal to 45 Minutes, and draw the Line v t. Make d w equal to 41 Minutes. Make d x, and c y, each equal to 40 Minutes, and draw the Liney x. Make c 41 equal to 41 Minutes. Make a % , and b 45, each equal to 45 Minutes, and draw the Line 45 a. Make b r, and k h, each equal to 47 Minutes and and draw the Line h r. Then by Prob. V. of Lect. I. hereof, deferibe the Cavettos y 2, and z v ; and the very fame being repeated on the left Hand Side of the central Line, will complete the Pedeftal, as re- quired. And as the Members in the Bafe and Capital of the Column, as alfo the Members in the Entablature, are all delineated in the very fame Manner, there needs no more to be faid thereof, and therefore the next Work is, How to diminifh the Shaft of this, or any other Column. But before we can proceed to this Work, it muft be obferved, Firft, that the Pleights of the Bafes of Columns in general are all equal to half a Diameter, or 30 Minutes; as is alfo the Height of the Tufcan and Derick Capitals. Second- ly, That the Cindture b, Fig. I. Plate X. and the djlragal, or Collerino h k, are both Parts of the Shaft. Thirdly, That fince the whole Column in the Tufcan Order, including its Bafe and Capital, is 7 Diameters high ; therefore taking the Bafe and Capital from it, which together are equal to one Diame- ter, the Remains, 6 Diameters, is the Height of the Shaft. Fourthly, That Columns in general are diminifhed but in the two upper third Parts of their Height, the lower third Part being a Cylinder. Fifthly, That the Tufcan Co- 6 lumn of architecture:. IG7 iumnisdiminilhed one fourth of the Diameter of its cylindrical Part; the Dorkk one fifth, the Ionic/: one fixth, the Corinthian and Compofite one feventh, and therefore the Dtameter of the Tufcan Column, at its Top, is but 45 Minutes, the Dorick 48 Minutes, the Ionich 50 Minutes, the Corinthian and Compofite each ei Minutes, j. j i P r o b. III. Fig. I. Plate X. To diminjjh the Shaft of the Tufcan, or any other Column. Operation. Draw/ b for its Height, } of which is its Diameter. Divide lb into three equal Parts, at q and C; through the Points / 6 ’ and h draw right l^ines, at right Angles, to the central Line l b. Make C y, and C 7 , each equal to 30 Minutes, and l h. If and C jD, C F, each equal to 22 Minutes and a ha!f, and draw the Lines h JD, and h E on the Point C; with the Radius 6 y, defcribe the Semicircle y w 7. Divide / C, into any Number of equal harts, hippo te four, at the Points n q v, and through them draw the right Lines m o,p r, and j t, of Length at pfeafure. Divide the Arches y 2, and f .7. each into as many equal Parts, as you divide the Lin e l C, which here is L as at the Points 1 2 x, and 456, and draw the Ordinates 1 4, 2 5, * 6. Make and v t, each equal to the half Ordinate B 6 ; alio q p, and q r, each equal to the half Ordinate A 5 ; and n m, and n 0 , each equal to the half Ordinate o 4, from the Points h k, through tiie Points m p s, and or t, unto the Points y draw the Lines hy, and 7, fo as not to make an Angle at any Point, and thev wiU diminiih tne upper Part of the Shaft, as required. As this Method is general tor dimindhing the Shaits of all the other Orders, no more need be faia on this •Subject. • ? 3te L and IL are e ^ibited the particular Members of every prin- cipal Part of tins Order, with their refpe&ive Meafures of Heights and Proieo- tions. 0 J _ Pros. IV. Fig. II. Plate XIX. To proportion the Heights of the principal Parts of the Tufcan Order , by equal Parts. Operation . Divider/, the given Height, into 5 equal Parts; the lower one C ’■ j S , th - e Hei S ht the 1 edeftal, and the remaining 4 Parts, a g, equal to n r, diviued into 5. equal Parts, the upper one is the Height of the Entablature, and the lower 4, the Height of the Column, which being divided into 7 equal Parts, 1 mined 3 ^ ^ ^ iametei ’ and tilUS are tiie Heights of all the principal Parts deter- P R O B. V. jo divide tie HCtghi of the Tufcan P«kJ)al,ht° it, Safi Die and Cora!,,, and them into their refpellme Members. Operation. Divide*/, Fig. II. Plate XIX. the given Height, into a equal Parts, as * v ,• give the lower t, to the Height of the Plinth, one third Part of the next 1, to * k The Height of the Moldings to the Bafe, and half the upper r to g h , the Height of the Cornice. i F To divide the Moldings of the Bafe and Cornice of the Tufcan Pedeflal n ■ r» /r r.. IV * Plate XX - Option. Firft Divide k 3, the Height of the Moldings on the Bafe, into « equal Parts , give t-e upper two to the Cavetro, and the lower one to the Fillet. Secondly, Divided, tne Height ofthe Cornice, into three equal Parts ; alfo the lipper i, b c, into two Parts, and the lower 1 , eg, into three Parts. Then gning the upper 1 of be, to the Ilegula, and the upper 1 of e g, to the Fillet, tne two Remains will be the Plat-band and Cavetto. „ To determine the ProjeSions of thefe Members, irtRST, Make the Projection of the Dado k k, equal to half the Height of the Dado and Moldings on the Plinth taken together, thereby forming a geo- metrical. Square, as in Fig. II. Plate XIX. wherein is a Circle infcribjd iaroW/y, Make the Projection of the Plinth and Regula, before the upright si the Dado, equal to the Height of the Cavetto and Fillet on the Piimh. Thirdly io8 Of ARCHITECTURE. Thirdly, Divide / h, the aforefaid Proje&ion, into 6 Parts, the firft I ftops the two Cavettos at n and o ; the third, the upper Fillet m, and the 5th the Plat-band and lower Fillet /. P R O B. VI. To divide the Height of the Tufcan Column , into its Safe, Shaft and Capital , and them into their refpettive Members. Operation . Firft, Divide b g into 7 equal Parts, and take I for the Diameter. Make eg, and bf, each equal to half a Diameter, for the Heights of the Late and Capital. This done, fuppofe G q, and a c, in Fig. XX. to be the Heights of the Bafe and Capital, as before found. To proportion the Bafe of the Tufcan Column. Divide df, equal to its Heights e, into 7 equal Parts ; give 4 to the Height of the Plinth, and 3 to the Height of the Torus ; alfo make e a, the Height ot the Cincture, equal to 1 Part. To determine the Projections of the Members of the T ufcan Bafe., Divide c 3, equal to the Semi-diameter, into 3 equal Parts, and make c 4, equal to 4 of thofe Parts. Divide the Part 3 4, into 5 equal Parts, and a Line, as ch L being drawn from the fecond Part, parallel to the central Line of the Order, will cut the central Line of the Toms in 1, its Center, and hop the Cine- ture at n. This being done, and the Shaft of the Column erected on the Bale, as before taught, proceed we now To proportion the Tufcan Capital. Divide its Height G q, equal to A B, into 3 equal Parts. Divide the upper i.mJEF, into 4 Parts, give the upper 1 to the Regula, and die lower 3 to the Abacus. Divide the middle 1 into 6 Parts ; give the upper 5 to the Ovolo, and lower 1 to the Fillet. The lower x is the Height of the Hypotrache- liutn, or Neck of the Capital. Now to find the ProjeSures of tbefe Members, make g i equal to half Gg , and divide X /, equal to^ 1, into 6 Parts ; the firft l hops the Fillet, the 4 Parts and 4 the Ovolo, the fifth Part the Abacus. The Aftragal, to the Top of the Shaft, is thus proportioned. - . Make q r its Depth, equal to half* «, the Height of the Necking, which divide into 3 p/rts; giveato die Aftragal, -and i to the Fillet The Pmjeaure of the Aftragal o, is equal to m n , viz. to hall the Height of the Neck, which is equal t ' 4 of the whole Capital’s Height, and its Fillet to | thereof. P r o b. VII. _ To divide the Height of the Tufcan Entablature, into its Architrave, Freeze, and Cornice, and them into their refpettive Members. . • Operation. Divide a A, equal to its Height X G, Fig. IIL Platt -XX. into ■ 7 Parts: give 2 to the Height of the Architrave, 2 to the Height of the Freeze, and 3 to the Height of the Cornice. To divide the Architrave, divide u .D, its Height, into 6 Parts, and give the upper 1 to the Tenia, which is alio called Diadema , a Bandlet or Fillet to bind the Head, whofe Projection d c, is equal to its own Height. Continue its Face to /and lb, making , each equal to its Projedion, and deferibe the Quadrant a c, above the Tenia, for die nn- mediate carrying the Rains from it, and the other below it, to ftrengthen its Projection. „ . . .7,^-7 To divide the Tufcan Cornice into its Members. Its Height being before divided into 3 Parts, divide the lower 1, it, mte .2 Parts, give the upper . to the Height of the Ovolo and the lower I, if. di- vide into 4 Parts ; give the upper 1 to the Fillet, and the lower 3 to the Caietto. The^e three Members taken together, form that which Woikmen call the Be Molding- of a Cornice. Divide the .upper two Parts of the Cornice into 24 equal Parts, as b x ; give nine Parts and a half to the Height of the Coiona, fend to the Height of the Ovolo, and the Remains between them ig, being divided into sParts, give , to the Aftragal, and . to the Fillet. The Proton 0/ ARCHITECTURE. 10 n !»« 4 Parts, the firft , hops the Fillet £ and the next , S^Aft „ ? e firft ' Part - S' *!• Order completed, by equal Pam, £ ; “ dthu5 R.£r^. Pr ° POrti0n “ y Patt ° f thi6 ° rd “ to a " y gi ™ Hc « ht - ** ™ the I. To proportion the Column and Entablature only, to any given Height, and to > frd the Diameter. \ „ f Vi V ~t', gi , vc ? Height into 5 equal Farts, the upper i is the Height Ld at f If e Va-" d tHe °, wer 4 of the Column, which divide into 7 Pafts and take i for the Diameter of the Column. ‘ -tarts, II. To proportion the Pedejlal and Column only, to any given Heigh, and to find n . . the Diameter. J P ; n , 1Vld 5 th f 8 lv l n Height into 21 equal Parts, give 5 to the He.Vht of Diamet d e e r fta ’ and ^ ^ ^ C ° lumn? WhlCh divide fn 7 Parts/and take 1 for the III. To proportion the Height of the Tufcan Cornice to any given Height Thzs admits of two Varieties, Firft, being confided as the CorHce 0 * an^ enure Order; and laftly, as the Cornice of an Entablature, to I Column SSlatum ‘ nt ° 3 PartS ’ 3! brf ° re direaed the Cornice ° { * oTufian. The Intercolummation of this Order that 4. 1 • t i Kmdf tZl 1 C r' Um ” S t0 ^ Won^to^erriS^lS: Kinds, and thofe according to the Ufes they are applied to. As for Examole ! a Colonnade, as Ftg I. Plate XXII. the Diftance between the centrai ffS is 5 Diameters. In the Frontifpieces, Fig. I. and II Plate Y r i • , Arcades A BC, Plate XXII. wife Columns ‘are on P/a^XYT I ?)' lanCC * An T d - in Arcades of Columns on Pedeftals, as 7 Fig. IV Plate XXI. they are at 7 Diameters Diftance. S V * When Tufcan Columns are placed in Pairs as n h * f TT 1 j r 1 • ** XXIP ., TH E -If ercolumniation of Columns, in Tufcan Porticos, are of two kinds viz “fc S a„?r terS ’ “ P,J ‘ XXI1 - "*** Sidcs° 4 S^ 1 E C T. V. Of the Manner of compofing Frontifpieces , Arcades , Colonnades , and Porticos of the Tufcan Order. J or c!rcuIar headed - «“* “ » wh^li^i? 00 " n L m0re gracrful than tbof e that are Semi- nof admit of a h S . fe d0m bu \ at fuch times when the Height will not admit of a Semi-circle, as being either too high or too low When the given Height that an Arch muft rife above the Impofts from which ft forint is more than half the Breadth of the Openihg, the Arch muft be a Semi-eftmiis made on the conjugate Diameter, as Fig. X . Plate EXIII. But wh«n the gfven P Height TIO yJJ Height islefs than half the Breadth of the Opening, the Arch mull he a Serin- ellipsis, made on the tranfverfe Diameter, as Fig. IX. Plate XXIII. It is always to be obferved in making of Doors with arched Heads, that their Impofts be placed fuSciently above a Man's Height, that they may not obitrutt a :y Part of the Entrance. w T y-vr Pros. x. r ig. x. I late aaI. To male a Tufcan fquare-headed Door , with a circular pitched Pediment . Draw the Bale Line, and at any Point, as b, ered the Perpendicular b e, anu draw * and k i, parallel to the central Line h e, each at 3, Diameters Diitance. Set up the Subplinths g and each 1 Diameter m Height, and on them erect two Columns with their Entablature, by Prob. II. or IV. Lect. I V . and give the Subplinths 42 Minutes Projedionson each Side of their central Lines.- Make the Margins m m 30 Minutes in Breadth, from the cylindrical Parts oi tne Columns, and from the under Part of the Architrave. Divide the whole Extent of the level Cornice into 9 equal Parts, as is done in Fig. D, Hate AV . and let up two of thofe Parts, from a to e, and draw the line e i, for the upper i art or the raking Cornice. VAr To proportion the raFmg Members to the raking Cornice, dig. V J 1 . Plate AV . From the Point y draw dy, parallel to 1 10, alio x z, parallel to cl y. On any Part of x z, qs at a, ered the Perpendicular a t , which continue through the level Moldings. Make a b equal to 0 p; b c equal top q; c d equal to q r; d c equal to r s ; and e f equal to s t ; and through the Points a b e d e /, draw right Lines parallel to * z, which' will be the Members required ; and which w fll have the fame Proportion to the raking. Cornice, as the level Members have to the level Cornice. To male a circular Pediment. '• Lxr * i. Fig. E, Plate XV. veprefent the Extent of the whole Entablature. Make x e equal to 2 Ninths of g i, draw e g, or e i, which bifed mf or h, whereon ered the Perpendicular / h, or h i, which will cut e x, continued m the Center, which In Fig. I. Plate XXL is the Point/, on which deienbe the Members found as aforefaid. ' P r o b. II. Fg. IV. Plate XV. * To find the Curvature or Mold of the raking Ovoid, that fie all mitre with the level Ovolo. Let np be a Part of the level Cornice, and a n the Points from which the raking Cornice take* its' rife ; alfo let / a and g n, represent a Part ol the raning Cornice. On n ered tire Perpendicular n b, and continue l a to b ; divide b n into any Number of equal Parts, at the Points 1 2 3, by. and from them draw the Ordinates 1 2, 3 4, 5 6, be. In any part of the raking Ovolo as at c, draw the Perpendicular c ra, and. make c d equal to b a, the x 1'QjeCtion o t le eve Ovolo. * Divide c m into the fame Number of equal Parts as are m b v, as at the Points 1357, Cfc. from which draw Ordinates equal to the Ordinates mb and through the Points 2 4 6 , &c. trace the Curve required. In the fame Man- ner the Curvature or Mold may be found when the upper Member is a CavetLO, Cyma Recta, or Cyma Reverfa, as is exhibited in Fig. V. VI. and V II. Pro b. III. Fig. IV. Plate XV. To find the Curvature or Mold of .the returned Molding, in an open or broken Pediment. Let the Point/ be the given Point, at which the raking Molding is to return. Continue n p towards h at pleasure, and from the Point f, let fall the Perpen- dicular f h; draw / 4 parallel to h p, and make/e equal to h a, the Projecaon of the level Cornice. Draw e i parallel to / h, and divide .eg into the mme Number of equal Parts, as are contained in b n, as at the Points 135 7, oy. from which draw the Ordinates 21, 43, 65, bV. equal to the Ordinates m bn, through the Points 2468, be. trace the Curve required. In the lame manner the Curvature or Mold may be found when the upper Member is a Cavetto, Cyma Reda, or Cyma Reveria, as is exhibited in Fig. V. \ Land Vii. Plate XV. P R 0 B. 3 I I Of ARCHITECTURE. Pros. IV. Fig. II. Plate XXI. 7 o male a Tufcan circular headed Door ’with a pitched Ped'meenf , or Balujlrade. Set up two Columns with their Entablature as before taught, making the Dif- tance of the central Lines equal to 6 Diameters. Divide n b f the Height of the Columns, into 3 equal Parts, and fet down 1 Part from n to g, for the Center of the Arch, and draw the Line g t. Make the Breadth of the Pilafters p q, each 30 Minutes, from the cylindrical Part of the Columns, and delineate the Impofts and Architrave of the Arch as follows, viz. In Fig. III. Plate XXL a 3 reprefents the Breadth of a Pilaff er ; make a h equal to a 3, and divide ah in 3 equal Parts at i and g, then the upper 1 is the broad Regula or Fillet, and the lower 1 the Neck of the Impoft. Divide the Middle Part in 4, give the upper 3 to the Ovolo, and the lower j to its Fil- let. Make b c equal to half g b, and divide be in 3 Parts, give 2 to the Af- tragal and 1 to the Fillet : and thus are the Heights of all the Members deter- mined. The Projection of the Regula on the Ovolo is equal to its Height, as is the Fillet under the Ovolo. The Projection of the Aftragal is equal to the Height t v, and its I diet to | thereof. To divide the Architrave of the Arch, divide a 3 into 3 Parts, the inward 1 is the Breadth of 2 3, tire firft Fafcia, half the outer one is the Breadth of a ti y the Fillet, and the Remains is the Breadth of n 2, the great Fafcia. The Breadth of the Key -done n nr, on the lower Part of the Architrave, is one eleventh Part of the Semicircle. Now if ’tis required to finifh this Door with a Pediment either ftraight or circular, proceed therewith as before taught in Prob. I. hereof, and if with a Balnftrade as on the left Side, then by Prob. V. Lect. IV. divide d s , the Height, which is equal to the Height of the Pediment, into the fame Parts as the Tufcan Pedeftal, making the Breadth of the Dado of the Pedeftal, equal to the Diameter of the Column at its Aftragal, then the Comice and Bafe being continued, and the Dado Part filled with Bani- lters, the whole will be completed, as required. To divide the D fiances of the Banijlers. Divide the Diftance between the Dado of the Pedeftal and the central Line a h, into 33 equal Parts, give 2 to the half Banifter againft the Pedeftal, 2 to the Intervals or Diftances between the Bani- fters, 4 to the Breadth of each Banifter, and 1 to the half Interval at the central Line a h. The Banifter proper to this Order is exhibited in Fig. A B C, Plate LXVIII. with the Proportions of their Members adjufted by equal Parts. Note, If ’tis required to complete this Frontifpiece lbicftly, according to Andrea Palladio’s Meafures, then, inftead of the preceding Impoft, we mult infert either of the Impofts A or B, in Plate XLII. where are exhibited all the impofts to the five Orders by this great Mafter. Note atfo , If to fuch a Semi-circular-headed Door, ’tis abfolutely neceffary to fet the Columns on Pedeftals, then the Diftance of the central Lines of the Co- lumns mull be increafed unto 7 Diameters, as in Fig. IV. Plate XXL Prob. V. Plate XXII. To make a Fufc an Arcade. Arcades are made in three different Manners, viz. Firft, of fingle Columns as ABC; fecondly, with Columns in Piers as D E ; and laftly, with Rullick Piers inftead of Columns as F G and H I K. To form the two firft Kinds of Arcades is no more than to place Columns at fuch Diftances as is expreffed between their central Lines, and to complete them with their Pilafters, Impofts, and Arches, as taught in the laft Problem. Arcades with Piers have their Piers of the fame Breadths as are equal to the Breadths of the Pilafters and Columns in the two former Kinds, as is evident by the dotted Lines continued down to them ; and the Height of the level Ruf- ticks from which the Arches fpring, is the fame as the Height of the Impofts in the former. The Rufticks in the Arches are divided in different Manners, as Firjl, Fig. D, where the Arch is divided into 1 1 Parts, and their Length made equal to half the Breadth of the Pier. Secondly , Fig. E, where the Key-itone b P 2 is 112 Of ARCHITECTURE. is I eleventh Part of the whole ; the Sides a b, and c d , each equal to half h c , and then the Side o a, divided into 4 Parts, give i to each Ruftick. Thirdly , Fig. C is divided in the fame manner as E, but its Pier G being but half the Breadth of the Pier H, the lower Ruftick on each Side is therefore omitted. Fig. B is divided the fame as Fig. D, with its lower Rufticks emitted for the aferefaid Rea- fon. Fig. A is divided the fame as Fig. E, and hath its lower Rufticks omitted as in Fig. C, but its Side Rufticks are fquared on their Sides by the central Line . of each Pier, and at their Tops, by a Line drawn level from the upper Part of the circular Architrave. The circular Architraves in Fig. A B and C have their Heights equal to half the Thicknefs of their Piers, and their Fillet is equal to I fourth of their Height, as exprefled by the Divifions on the right Side of the Key-ltone in Fig, B. P R o b. VI. Fig. I. Plate XXII. To make a Tufcan Colonnade. To form a Colonnade is no more than to range Columns with their Entablature, at 5 Diameters Diftance, as exprefled between the central Lines of the Columns. The Interc. lumniation of this Colonnade is called Araojlyle , from the Greek Aracos , Rare, and Stylos , a Column, by which Vitruvius fignifled the greateft Diftance that Jhould be made between Columns that have not Arches between them to afiift the bearing of the Architrave. Prob. VII. Fig. II. Plate XXII. To make a Tufcan Portico. Porticos were anciently Porches formed by Columns, fupporting Parts of Roofs, continued out beyond the Uprights of the Ends of Temples, as the Portico of St. Paid’s , Covent Garden. But now they are oftentimes placed againft the Fronts of Buddings fupporting a Pediment, to difeharge the Rains, and alfo in Gardens, to terminate the View of a grand Walk, life. Divide the given Breadth into 35 Parts, and take 2 of thofe Parts for the Diameter of the Column. This done, fet out the central Lines of the Columns as exprefled between them, and complete the feveral Columns with their Enta- blature. But as the four middle Columns are finiftied with a Pediment to .make the Portico, they muft advance 3 Diameters forward before the Range of the Columns a and / and Pilafters muft be placed behind the Columns b and c, in lange with a and/, which indeed ftiould be Pilafters alfo. A Pilastfr is called by the Greeks, Para/late, and by the Italians , Mem - Iretti, and is nothing more than a fquare Column, and is dimiiiiftied the fame as a round Column, when Handing with Columns; but when alone, it muft not be diminiftied, nor indeed even when with Columns, as in this Example When Hand- ing at an Angle, as thofe of a and f ; becaufe the Quoins of all Buildings ftiould be eredt. • • - , ; Examples for Practice in the T ujean Order. I. The Height of the Tufcan Architrave being given , to find the Height of its Freeze , and of its Cornice. Rule, Make the Height of the Freeze equal to the Height of the Architrave, and the Height of the Cornice, to" 3 fourths of the Height of the Architrave and Freeze taken together. * • " . II. The Height of the Tufcan Cornice being given, to f.nd the Height of the Archi- trave and of the Freeze. Rule, Divide the Height of the Cornice in 3 Parts* and make the Height of the Architrave, and of the Freeze, each equal to two Parts thereof. ?■ III. The Height of a Tufcan Cornice being gtoen, to fnd the Diameter of the Column . Rule, By Example II. find the Height of the Architrave and Freeze, and add them to the Cornice ; multiply the FI eight of the Architrave, Freeze, and Cor4 nice by 4, and divide their Produdl by 7, the Quotient is the Diameter required. IV. The Diameter of a Tufcan Column being given,' to fnd the Height of the Cornice. Rule, As 12 is to 9, fo is the givea Diameter to the Height of the Cornice required* « V. The 0/ ARCHITECTURE. 113 V. The Height of a Tufcan Architrave being given, to fnd the Diameter of tlje Column. Rule, Double the Height of the Architrave, and it will be equal to the Diameter required ; and fo on the contrary, if the Diameter was given and the Height of the Architrave required, then half the given Diameter is the Height of the Architrave. VI. The Height of the Tufcan Entablature being given, to fnd the Height of the Capital. Rule, Divide the Height of the Entablature into 7 Parts, and make the Height of the Capital equal to 2 of thofe Parts; and fo on the contrary, if the Height of the Capital was given to find the Height of the Entablature, divide tfie Height of the Capital into 2 Parts, and make the Height of the Entablature equal to 7 of thofe Parts. VII. ’The Height of the Capital and Entablature being given, to fnd the Diameter . Rule, Divide the given Height of both Capital and Entablature into 9 equal Parts, the Diameter will be equal to 4 of thofe Parts. * LECTURE VI. Of the Manner of proportioning the particular Parts of the Dorich Order by Modules and Minutes, according to Andrea Palladio ; and by equal Parts , compofedfrom the Mqfers of all Nations. T HE principal Parts of this Order by Andrea Palladio are exhibited in Fig. I. audits Pedeftalin Fig. III. Plate XXIII. The Bafe, Capital, En- tablature, and Plancere of the Cornice are exhibited by Fig. I. and lll.*Plate XXIV. and as they are all proportioned by Modules and Minutes in the fame Manner as the Tufcan Order, it isneedlefs to fay any more thereof. P R 0 B. 1. To proportion the Heights of the principal Parts of the Dorick Order by equal Parts. Let a b, Fig. II. Plate XXIII. be the given Height, divide e f equal to a b t into 5 equal Parts, give the lower 1 to the Height of the Pedeftal. Divide the 4 remaining Parts into 5 equal Parts, the upper 1 is the Height of the Entabla- ture, and the lower 4 the Height of the Column, which divide into 8 Parts, and take 1 for the Diameter of the Column. Prob. II. To divide the Height of the Dorich Pedeftal into its Bafe, Die, and Cornice , and them into their refpective Members. Let a b. Fig. IV. be the given Height and central Line of the Pedeftal, di- vide c d, equal to a b, into 4 equal Parts, give d 1, the loweft Part to b L, the Height of the Plinth. Divide the next Part into 3, as r s, and give I to / s, the Height of the Moldings on the Plinth. Divide t s into 8 Parts, give 3 to the Cavetto G, 1 to the Fillet I, 4 to the inverfed Cyma Refta K, and the lower 1 to its Fillet L. Make e f equal c to half the upper 4th Part of the Pedeftal’* Height, which divide into 2 Parts ; divide h g equal to 1 Quarter of e f into 3 Parts, give 1 to the Fillet E, and 2 to the Aftragal D. Divide k i, equal to half ef, into 4 Parts, give the upper 1 to the Regula A, and the other 3 to the Fafcia B. The Remains is the Ovolo C. ; ■ To determine the Projections of the Members. In Fig. II. a Circle being inferibed within the Dado of the Pedeftal, fhews that its Height and Projeftion are equal, therefore draw the Line q x, parallel to a b, at the Diftance of half th!T' Height of the Dado F. Make 22, 3 . t, &c. which continue upwards through the Cornice unto i h, &c. and downwards through the Tenia and Fillet of the Architrave ; make a b and n z, each equal to 2 of the 1 2 Parts in a n , and draw the Line a- ss ; make b f, i e, k m, and 2; p, each equal to I of the 12 Parts, and draw the Miter Lines cf , d e, e g, h m , m k t and 0 p , which will Complete the Triglyph as required. To form the Drops under the Tenia of the Architrave. From the Points* 2, 4,6,8, 10, 12, draw Lines towards the Points / /, bV. ftopping them at the Fillets w, and they will form the Drops, as required. To form a Metope as n b r a. Make n b and r a , each equal to n r, and draw the Line b a , then nbr a is the Metope required. If it is required to make a hollow Pannel therein, as d eh divide r a in 6 Parts, and make the Margin about the Pannel equal to I of thole Parts ; alfo divide the Margin into 5 Parts, as at A c, and make the Breadth of the Molding within the Pannel equal to 1 of thofe Parts ; then drawing the Diagonals d i and c e, their InterfeCtion is the Center, about which Place a Rofe, ©r any other Ornament at pleafure. To divide the Cornice into its refpe 8 ive Members , Fig. II. Plate XXIV. The Height of the Cornice being 3 Eighths of the whole Entablature, as afore- faid, divide the lower 1 into 3 Parts, give the lower 1 to the Height of the Cap- ping to the Triglyph ; divide the remaining Height equal to b 0 in 4 Parts, and the lower 1 thereof into 6, then the lower 1 is the Height of the Aftragal under the Ovolo, and the next 4 is the Height of the Ovolo ; the fecond Part of b » being divided into 3, the loweft I is the Height of the Bells or Drops, the next 1 of their Fafcial. and the upper 1 divided into 3, the upper 1 is the Height of the Fillet, and the lower 2 of the Cyma Reverfa ; the third 1 of bo divided into 6, the upper 1 is the Height of the Fillet to the Corona, and the lower 5 is the Height ©f the Corona ; laftly, the upper 1 of bo, divided into 4, the upper I is the Height «f the Regula, and the lower 3 of the Cyma Reverfa. To determine the Projections of the Members in this Cornice. The Upright of the Column and Freeze * O S being before drawn, make x M equal to half the Height of the whole Entablature, and from any Part of the Upright of the Freeze draw a Line, as O P, equal to the Projection x M, which divide into 4 equal Parts at 123; divide the ill Part into 3, the firft 1 is the Projection of the Tenia in Profile againft the Return, and of the Aftragal, under the Ovolo, which divide into 4, the firft 2 is the Projection of the Triglyph in return, the next I of the Capping to the Triglyph over the Freeze, and of the Fillet, and Drops under the Tenia of the Architrave. The remaining 2 Parts of the firft I of O P, divided into 6, the firft 3 terminate* the Ovolo, and the next 1 , the Platform K, againft which the Mutules are placed. The 3d Divifion of O P terminates the Fillet of the Cyma Reverfa, that crowns the Mutules, and this third Part divided into 3, and the laft 1 into 3, the firft r terminates the projecting Mutule L. Laftly, the laft Part of O P equal to Q_R, divided into 9, the firft 4 terminate the Projection of the Corona, and the next L its Fillet. Mutules are a Kind of Modillions, that are always placed perpendicularly over the Triglyphs, to fupport the Corona, as well of Pediments as of ftraight or level Cornices, andwhofe Breadths are always equal to the Triglyphs, as exhibited in Plate XXVI. The Word Mutule comes from Mutuli , the Latin for Modillions. The Figure D E F G is the Plancere or Cieling, which the Italians call Soffto y of a Mutule, whofe Sides are each divided into 6 equal Parts, and parallel Lines drawn from them, divides the whole into 36 geometrical Squares, in whofe Centers the Drops or Bells are placed ; and if from their Centers right Lines be drawn up to the projecting Mutule K L, they will be the central Lines, over which the 6 Drops between K and L are to be placed. Of ARCHITECTURE. 117 The central Lines of the Drops to H I the Mutule in Front, are determined by the Continuation of the twelve Lines from the Triglyph, which alfo makea the Bieadth of the Mutule equal to the Breadth of the Triglyph, vide Fig. IV. Plate XLIV. where e dab is a complete Mutule in Front, and Fig. III. a Mutale in 1 rolile, divided as aforefaid, whofe Drops are drawn to the Points n n , Sff’, at the Interfedlions of their central Lines, with the Line c d drawn through the midft of the Fafcia a 0. In Plate XXV. are exhibited various Manners of making the Returns of the Planceres of the Dorick Cornice, wherein ’tis to be noted, that Fig. I, and V. which are Returns at external Angles, have but 18 Bells or Drops each, accord- ingto Palladio , and Fig. II. which is a Return at an external Angle, has 36, as at F. 2dly, That fometimes Mutules are made fquare, and fhew but 28 Bells, as at B and D, Fig. IV. which is a Return at an internal Angle, as alfo is Fig. III. whole (haded Parts A B G C E F G reprefent Parts of Columns, whereby tis feen, that the Mutules D E in Fig. III. D F in Fig. II. and B D in Fig. IV. Rand direftly oyer their refpeftive Columns. The Coffers or hollow Pannels! E A B C in Fig. II. and A C in Fig. IV. are to be enriched with Rofes, as A t Fig. I. Examples of which are given in Figures A, B, C, D, E, Plate XXXVIII. Prob. VI. To determine the Intercolumniations of the Dorick Order. Operation. As the Breadth of a Triglyph is always equal to 30 Minutes, and the Breadth of a Metope to 45 Minutes, therefore the Sum of the Minutes con- tained in the T riglyphs and Metopes, that are required between the central Lines of two Columns, is always the Intercolumniation, or Diffance at which the Columns are to be placed. Therefore to have 1 Triglyph between, as f/f- ° r f P* ^ ate XXVII. the Diffance muff be two Diameters, 30 Minutes ; if 2 Triglyphs between, as b c, and d e, 3 Diameters, 45 Minutes ; if three Triglyphs, as c d, 5 Diameters ; if 4 Triglyphs, as over each of the Arcades, Fig. ABC, DV. 6 Diameters, 15 Minutes, &c. Hence ’tis plain, that m the making of h rontifpieces, &c. to any given Height, the Breadth cannot be confined ; and therefore when fuch a Cafe happens, the Triglyphs and Mutules muff be omitted ; and the Diffance between the Columns (hould not ex- ceed 4 Diameters. ^ ^ XX//I. Fig. I. and II. are Deiigns of Doors, the fir ft with a fquare riead, with both circular and pitched Pediments over it, the other with a Semi- circulai Head, w lth a Balluftrade and pitched Pediment, which are given for Ex- amples, as alio is Fig. III. which is half of an Arcade on a Pedeftal. dig. IV. is the Dortch Impoft at large, whofe Height, ab , divided into 3, the ower 1 is the Height of the Neck, the upper t divided into 4, the upper I is the "r* 1 ? j • ^ ie or ^ e S' ula > an dth e lower 3 of the Fafcia. The middle t, diviced into 3, the upper 2 is the Height of the'Ovolo ; and the lower 1 divided into 3, the upper 2 is the Aftragal, and the lower 1 its Fillet. The Diffance a i> reprefents the Breadth of the Pilafter, and is its upright. Make i h y the Pro- jection, equal to one third of a u Make 3 g equal to i h, which divide into 4, then^the firft one determines the Proje&icn of the two Fillets, to the two Aitragals ; the third Part the Ovolo ; and half the laft Part, the Fafcia of the Abacus. T"! P e P th of . the Aftra g al hi is equal to half the Height of the Neck, divided into 3, give 2 to the Aftragal, and 1 to the Fillet. In Plate X^VII. Pig. I. is a Colonnade ; Fig. II. a Portico ; Fig. ABODE Arcades, with hngle Columns, and Columns in Pairs, and F G id I K, are rufticated Arcades, which are given as Examples for Practice. Q- II. The !iS Of ARCHITECTURE. II. The Height of the Dorlck Cornice being given, to find the Height of the Hr chi- trave, and of the Freeze. Rule, Divide the Height of the Cornice into 3 equal Parts ; make the Pleight of the Freeze equal to the Height of the Cornice, and the Height of the Architrave to two thirds of the Cornice. III. The Height of the Dorick Cornice being given, to find the Diameter of the Column. Rule, Divide the Height of the Cornice into 3 equal Parts, and make the Diameter equal to 4 of thofe Parts. IV. The Diameter of a Dorick Column being given , to find the Height of the Dorick Cornice. Rule, Divide the Diameter into 4 equal Parts, and make the Height of the Cornice equal to 3 of thofe Parts. V. The Height of the Dorick Architrave beipg given, to find the Diameter of the Column. Rule, Double the Height of the Architrave, and ’twill be equal to the Diameter required. ':' j VI. The Height of the Dorick Entablature being given , to find the Height of the Capital. Rule, Divide the Height of the Entablature into 4 Parts, and make the Height of the Capital equal to x of thofe Parts ; and fo on the contrary, if the Height of the Capital was given, and the Height of the Entablature required, ’tis no more than to make the Entablature equal to 4 times the Height of the Capital. VII. The Height of the Entablature and Capital being given , to find the Diameters Rule, Divide the Height of the Capital and Entablature into zo Parts, and take 4 of thofe Pajts for the Diameter required. LECT. VII. Of the particular Parts of the Io nick Order, proportioned by Modules and Minutes , according to Andrea Palladio, and by equal Parts, compofed from the Maffers of all Nations. T HE principal Parts of this Order are exhibited by Fig. I. Plate XXVIII. and the particular Parts by Fig. I. and II. Plate XXIX. which in general are determined by Minutes, as the preceding Orders. P r o b. I. Fig. II. Plate XXXII. To proportion the Heights of the principal Parts of the Ionick Order , by equal Parts. First, Divide d t , equal to the given Pleight, into 5 equal Parts ; give the lower 1 to / s, the Height of the Pedeftal. Secondly, divide a m, equal to the Remains, into 6 equal Parts ; give the upper 1 to t he Height of the Entablature, and'the lower 5 to the Height of the Column, which being divided into 9 equal Parts, take 1 for the Diameter of the Column. Prob.II. Fig. IV. Plate XXVIII. To divide the Ionick Pedejlal into its principal Paris, arul them into their refpeclive Members\ First, Draw q !y%c- in Fig. H, and tin everj P-t of the elliptical Volute N will affeft the 56 Parallelograms in the very fame manner as tue circiuar Vo- lute H doth the e6 geometrical Squares : and as what is here faid of ttie outward Line is to be alfo underftood of the inward; therefore, when you have found all the preceding Points through which the Curyes are to pafs, apply unto tnem a thin pliable Ruler, or with a free Hand trace their Curves, as required. This Ornament is called a Volute, from the Latin Voluta , a volvendo, as tnat it feems to be rolled upon an Axis or Staff ; and the Eye is by fome, from the Latin, called Oculus. yyty Prob. IX. Fig. III. Plate XXIX. divide the Height of the Ionick Entablature into its Architrave , Freeze and Cor- i cl. g ^ . nto the . r re ^ aive MemberS ' Divide a x equal to the given Height, into 10 equal Parts, give 3to tk Height of the Architrave, 3 to the Height of the Freeze, and 4 to the Height of the Cornice. _ , . r , ... T 9 divide the Architrave. Divide the lower 1 of the Architrave into 4 Parts, give the upper 1 to the Bead and lower 3 to the fmall Fafcia. Divide the upper 1 into 4 i arts, give the upper 1 to the Tenia, the next 2 to the Cyma Reverfa, and the Remains to the great Fafcia ; make D H the Projeftion of the Tenia equal to the Height of the Tenia and Cyma Reverfa, which divide into 3 Parts, and give the firit 1 to the Projeftion of the great Fafcia. > , i p • Divide C D, -the Height of the Freeze, into 4 equal Parts, and on the Point. C and D, with the Radius of 3 Parts, make the Seftion E, on which, with the Radius E D, defcribe the fwelling Freeze. To divide the Cornice. The Height of the Cornice conufting of four Parts, divide h k equal to the two lower Parts into 3 Parts, and the lower and upper Parts thereof each into 6 Parts, as h m and i k ; give the lower 5 of i k to the Height 01 the Cavetto, and the upper 1 to the Margin of the Denticule below the Dentines : Give the upper 5 K of h m to the Height of the Ovolo, and the lower . to tts F.llet. Divide 2 f, equal to 1 Quarter Part of the Height of the Coimce, into 4 Pai s > give the lower 3 FartTto the Height of the Corona, and the upper , to the Height of its Cyma Reverfa. Divide b n, equal to the upper 4th Part of the Cor- nice? into 4 Parts, give the upper 1 to the Height of the Regula, and then i dg, equal to the lower x, being divided into three Parts, give the lower 1 to the Fillet between the two Cymas. And thus are the Heights of all the Members deier- mined. _ , . T 0 determine their Projetiures. The Upright of the Column B C D 19 being before drawn, make B A the Projeftion of the Regula equal to B C the Height of the Cornice, an ^. Part of C D, as from v, draw a right Line, as v or 3 Diameters 12 Minutes : If nma^Ifo M^”ir 56 M r“ e5 ’ cq w- ‘° 8 t,meS , 3 2 > or 4 Diameters and 16 Mi- and ft MiLto, S° DS ’ 3Z0 ” UteS ’ equal t0 10 ,imes 5 Dittos Ip v - -P^fcXXX. are three Examples, wherein Flo. I. contains M- P f S - ModlIllon > and II. and V. 14 each, whofe ModiUmns are at 30 Minutes Diftance, as is feen by the Number of Diameters contained in their re- lpective Intercolumniations. . vvhif, P 7 T XXXL ^Vv exhiblted tbe Intercolumniation for the Colonnade, whofe Columns are at 3 Diameters, 44 Minutes Diftance, not 45 Minutes as in’ the ^ t" fj F atC , by - Mulake of the Engraver, and have 7 ModiUions between the central Lines of every two Columns each, at 32 Minutes Diftance between their central Lines. The Portico, Fig. II. and the Arcades, Fig. III. and IV have their Intercolumniations proportioned, fo as to have the Diftances of the cential Lines of their Modilhons each 30 Minutes. Secondly, To proportion Intercolumniations when Dentules are employed Fin- II T Plate XXIX. b ' As xy is equal to 25 Minutes, and being divided into 10 Parts, as aforefaid two of which is the Breadth of a Dentule, and 1 of an Interval ; his therefore evident, that each Part is equal to two Minutes and a half: And therefore to make the Divifion of Dentules eafy, the Diftance between the central Lines of Columns muft always contain fome Number of Parts, each of five Minutes, as the Occafion may require ; as_ one Diameter and f, wherein there are 18 fuch Par s r ter N WnCre 7 ^ ^ 48 / Ucb PaFtS 5 and 5 Diai ^ters, 60 fuch Parts, as m the feveral Intercolumniations of the Portico, Fig. II. Flat XXXI Now, if each of thefe Parts be divided into 2 Parts, then each Part will be equal o 2 Minutes and a half; and then giving 2 of thofe Parts to the Breadth of each 'i -° In i erVa1 ’ the Wh ° le wiI1 be corn P lete d, as required. thjftAn the L akl ^ DentuleS ’ , ,n a]1 K i nds of Pediments ’ muft ft and exa&ly over thofe in the level Cornice, m the very fame Manner as the Mutules in the Dorick Soted wfth e a ‘ C 13 a i °p t0 be , obferv J d of ModiUions ; and as ModiUions are always capped with a Cyma Reverfa, or fome other Molding, whofe Curvatures, or M .° ds ’ PP tbe u PP T er an(i l ower Sides, are both different from thofe of the Front raking Molding; I muft, before I proceed any further, fliew how to deferibe thofe returned Moldings to the Caps of raking ModiUions, Pro#. 124 0/ ARCHITECTURE. P E o B. I. Fig. I. II. III. Plate XV. To defcribe the returned Moldings of the Caps of raking Modillions in Pediments « I ft. Suppose the Ovolo C, Fig. III. to be the raking Molding in Front, with ■which a raking Modillion is to be capped ; draw the Chord Line a c , and divide it into any Number of equal Parts, iuppofe 8, as at the Points 2, 4, 6, 8, 10, &c. and from them draw the Ordinates 1, 2 ; 3, 4; 5, 6; tSV. 2 dly, Suppofe the Lines h b and i c to be the Bounds of the Front raking Ovolo, and let the Line y i reprefent the upper Side of a raking Modillion, and d f its lower Side. Prom the Point y draw the horizontal Line ny, and from the Point <7, the Line 0 p ; make 0 p , and n y, each equal to a b , the Projection of the Front Ovolo, and through the Points n and p draw the perpendicular Lines h l and epl, cutting the tipper Line h b, in h, and e, draw the two Chord Lines h i and f e, and divide each into the fame Number of equal Parts, as the Chord Line a c , and from thofe Parts draw Ordinates equal to the Ordinates in C. Through the Points I, 3 > 5 » 7 » tefc. in Fig. A and B, trace the Curves h 7 i, and / 7 e, which are the true Curves of the returned Moldings on the upper and lower Side of the Modillion, as required. Note, The fame Method of working wall find the Curvatures of all other Kinds of returned Moldings ; as for Example, when the Front Molding is a Cavetto, as C, Fig. II. then A and B are the upper and lower Mold, or when a Cyma Reverfa, as C, Fig. I. where A is the upper, and B the lower, as in the two other Examples. Pros. XII. To proportion the Ionick Frontifpieces , Colonnades, Porticos and Arcades. As by the PraCtice of the two preceding Orders, it is very reafonable to believe that my Reader is now capable of infpeCting into this and the two fucceea- ing Orders, that is, to readily underftand what is meant by the Meafures affixed to each Part with refpeCt to the Intercolumniations, Number of Modillions, Breadth of Pilafters, Height of Impofts, CsV. I fhall therefore only explain the Imports, Fig. VI. Plate XXX. and then recommend him to the feveral Figures in Plate XXX. and XXXI. for his further PraClice. To proportion the Ionick Impojl by equal Parts. Divide a k, its given Height, into 3 equal Parts, the lower 1 is the Height of the Neck. The lower Half of the middle Part divided into 4, the upper I is the Height of the Fillet, and the lower 3 of the Cavetto ; the upper Half is the Height of the Ovolo, as is the lower Half of the upper x, the Height of the Faf- cia. Divide the upper Half into 3 Parts ; give the upper 1 to the Regula or upper Fillet ; and the lower 2 to the Cyma Reverfa. To determine their Projections. Let a b reprefent t;he Breadth of the Pilafter, and b p the Upright thereof $ divide 0 p, equal to the Breadth of the Pilafter, into 3 Parts at t and v, make/* r equal to p _. 125 !• irst, The Dorick Cornice. Divide the given Height^into coequal Part* tl"e Cor„U°, ‘ h ° ie P'V°n ^ ^ ° f & Comic!, "Bchi CoScrS t ° a " entlre .°, rdcr - But being confidered as a Cornice toan Enta- Parts and a . Col “ mn » w J} h .°« a Pedeftal, then divide the Height into 40 equal rl In thTn^h / ° f the C0rnice - Secondly, the Ionick CoJce. v ^ d . the He %f °J a Cornice to an entire Order. Divide the Height of the ,n r V lT 1 1 \ he "PP' r 4 to the Height of the Cornice re! Stthf /” hf' P“C‘ 0 f the . Omiu of an Entablature on a Column only. He!gt‘cfth^ic°e! R °° m m '° 60 Pa,tS ’ an<1 S ‘ Te ‘ hc “PP« 4 4 r cr’r tt ■ r Examples for PraElice in the Ionick Order. Freeze aiifofttc^ Io '" ck Architrave being given, to find the Height of the Height of t e A h -l' Hei g ht of the Freeze e|ual to the p ft oi the Architrave, divide the Height of the Architrave into * eaual 1 'Jr’ ^, d ^ aKe , the , ¥ ei S ht . ^ the Cornice equal to 4 of thofe Parts. 3 q trave and [nh^F ^ ^ iv'^f hei fg S™en, tofnd the Height of the Archi- trave and of the Freeze. Rule, Divide the Height of the Cornice into a. eaual foslhrfe Pat! ghtS ° f theArchitOTe “ d o f the Freeze, each eqj to CoZt »/'& Ionick Cornice bent? given, to fad the Diameter of the Ster^t 3 " t0 5 °’ 18 thC A,ght 0fthe given C “™-> to the T IV ,' V 1C Dmvu if r °f the Ionick Column being given, to find the Height of the AS5 ° i8t ° * tho given Diameter, clmF'dlit °l t ■°" ick Arc ! h \ ,r T e •?'“”> <°M 1 ° ls the Height of the given Architrave, 4 to cl L ” J S*' tt • l0 . n!ck f n ! al l atu " to jind the Diameter of the Diamet'er reqX’d 9 ‘ S 5 ’ f ° R the H “«>“ * ' given Entablature, to the VII. The Height of the Ionick Entablature being given, to find the Height of the Capital of 20 Minutes in Height, according to AL.ul Palladio. RuJ £ A St °.5 , fois the given Height of an Entablature, to the Height of the Capi- as is the Height of tht Cap;tal of 20 Mi " utes ' 45 AtfJt iv'zzti ,o - PA h LA T of the Capital and Entablature, to the ° " ** H '‘S ht L E C T. VIII. Of proportioning the particular Part, of the Corinthian Order, by Module, and Mi. Pallad '°' w <™A7ri>L Kl ate XXX II- exhibits the Proportions and Meafures of all . the principal Parts of this Order, by Andrea and £ TlL the particular Parts of the Pedeftal. Fig. I. and II Plate YYY1 TT °« un v .1 particular Parts of the Bafe to theCoIumn, with its Capital anSaL^whkh ftXoth g e e F r 3 d ? err T, d b ^ Modules a " d Minutes, nothing p 4 be aad 1 « p— * «T 1 , , Prob * !• IP Plate XXX II. To proportion the principal Parts ofjhe^ Corinthian Order, «„, 0 ^ ^ ^ Divide 126 O/ ARCHITECTURE. Divide dnv, equal to h % the given Height, into 5 equal Parts; the lower s I is the Height of the Pedeftal. Divide a, equal to b r the remaining Part, into 6 equal Parts 5 the upper 1 is the Height of the . Entablature, the lower 5 Parts is the Height of the Column, and which being divided into ic equal Parts, take 1 for the Diameter of the Column, which divide mto 60 Minutes, viz. pirft, into 6 equal Parts, which will each contain 10 Minutes, and then the hr ft. one of them into 1 o Parts. Pros. II. Fig. IV. Plate XXXII. To divide the Height of the Corinthian Pedflal into its Bafe, Die dud Cornice, and them into their refpeclive Meaf fires. To proportion and divide the Bafe, draw mb, the bale Line, .and k r, the central Line. Divide / k, equal to ck the given Height, into 4 equal Parts. Divide d e, equal to the fecond Part, into 3 parts ; and c z, equal to the lower 1 Part, into 4 Parts, and make b a, x y, and x eac h equal to two Parts of a y or 20 Minutes. Through the Points O P, QJR., draw the Lines O a, P g, Qji, and R z; then O a, and R 2, will determine the Proje&ions of the two Sides of the Abacus, and the Lines P g, and Q ji, will be the two upright Lines of the Shaft of the Column. Di- vide O 10, on the let t Hand Side, into 8 equal Parts ; then O w, the firft three Parts, determines the. Projection of the' Fillet in the Abacus at r; Ox , the firft 5 Parts, the Projection of the Fafcia at t, and Ovolo at d : O y, the firft i ai ts, tne i lojecuon of tiie Fillet at j / and O z , the firft 7 Parts, the Pro- jection of the Faicia at v. Make the Projections on the right Hand, equal to thole on the left, and then the Abacus will be completed. Make q, the Projection of the Aftragal, equal top 0; and s r, the Fillet, unto 2 thirds thereof. Divide pt into 3 Parts, and make p v equal to 4 of thoie Parts. Draw v x parallel to p t. Draw t v, which bifed in w, whereon raife the Perpendicular equal Parts, the lower 1 Part is the Height of the Plinth. Divide the middle 1 into 5 equal Parts, the lower 3 Parts is the Height of the lower Torus, the next 1 or the Aftragal, and half the next 1 of its Fillet. Divide the upper 1 of n J nito 5 equal Parts, the upper 2 is the Height of the upper Torus : half the £ ,V 7 if f ei S ht °b the tfcf Torus, J* Remains is Z . lught of the bcotia. ro determine the Projedures of thefe Moldings . Draw i h, parallel to «/, at the Diftance of 30 Minutes, and make** h equaUo 12 Minu^ e A n int0 5 e q ual Parts ; the brft 1 Part and half determines the Projedion tbf Altragai on the lower lorus, the fecond Part its Fillet, the third Part the I diet under the upper Torus, and its Center alfo ; and the third Part and fe , , he Pfff th . e Aftra S al on the upper Torus, and its Fillet alfo. The eight of the Aftragal on the upper Torus is equal to half the Height of the upper I orus, and the Fillet on the Aftragal to half the Height of the Aftragal. P R o b. V, Plate XL. To proportion the Parts of the Compofite Capital by equal Parts. fet UP Ai the l ll r Ight ° f . the Ga P‘ tal > proportion its Aftragal, Leaves, and Abacus, exadly the fame as in the Corinthian Capital ; and the 20 Minutes contamed between d, the lower Part of the Abacus, a'nd //the Top of the upper Range of Leaves divide as follows, viz. Divide g s into 8 equal Parts, give between To * artS ’ t0 the , ? C, £ ht ° f the FiUet E ’ Divide the 5 Minutes between 50 and 55 into 2 equal Parts at/; then gf is the Height of the Af- ragal D, which is alio the Height of the Eye of the Volutes N and N. Di- vide the upper 5 Minutes contained between 55 and 6 q into 4 equal Parts; S give , 34 0/ ARCHITECTURE. crive the upper i to the Height of the Fillet under the Abacus, and the remain- f n a- Part e f to the Height of the Ovolo C. Now as. the Volutes N N are elliptical, and have the Centers of their Eyes in that Point of the Line / X, t e uurieht Line of the Shaft that is cut by the central Line of the Aftragal D, and as they are comprized within a Parallelogram, formed by the upright Innes pro- ceeding from a;, "the Projection of the lower Part of the Abacus and w F, as alio bv d tv, the under Line of the Abacus, and i r the Top of the fecond Range oi Leaves ; therefore by Prob. VL or VIE LeCt. VII. hereof, defcnbe a circular Volute 4 whole Height is equal to the Breadth of your Parallelogram; and then from that Volute fo made, by Prob. VIII. Lect. VII. aforefaid, defcnbe an elliptical Volute in the aforefaid Parallelogram, which will be the Volute to this Capital, and which being in like Manner performed on both Sides, the Capital will he completed, as required. Prob. VI. Fig. III. Plate XLI. and Fig. I. Plate XLII. To divide the Height of the Compofite Entablature into its Architrave, Freeze, and Cornice. As I have given two Examples of Entablatures in this Order, the one for the Infide of Buildings, to be feen at a fmall Diilance, and the other for the Out- fides of Buildings, to be feen at a confiderable Diftance, I Avail therefore fpeak L Of the Compolite Entablature, to be ufed within Buildings. Fig. III. Plate XLI. 'Divide / A, equal to the given Height, into 8 equal Parts; give 2 to the Height of the Architrave, 3 to the Height of the Freeze, and the fame to the Height of the Cornice. To divide the Height of the Architrave. Divide / c, its Height, into 50 equal Parts ; give 8 to the Height of Z the lower Fafcia, 1 and half to its Bead, 10 to Y the middle Fafcia, 4 to the double Bead X, 1 c to the upper Fafcia, of which 5 mull be given to the Drops V , 3 to the Cavetto T, x to its Fillet, 2 to the Altragal S, 4 to the 1 ema k, and I to its Fillet. T 0 divide the Height of the Freeze. Divide n v, equal to its Height, into 12 equal Parts, and give the upper x to P, its Capital. ^ _ . To divide the Height of the Cornice. Divide km, equal to its Height, into 70 equal Parts ; give I to the lower Fillet, 2 to the Aftragal O, 4 and half to the Cavetto N, 1 to its Fillet, 6 to the Denticule, of which the upper 5 is the Pleight of the Denudes ; <-hen give 1 to their Fillet, 2 to the Aftragal L, 4 and half to the Ovolo Iv, and 6 to the Platform of the Modillions, of which the upper 5 is the Height ot t ie Modillions. Give 2 to the Cyma Reverfa H, 7 to the Super-Modilhons G, and I to the Fillet. Give 2 to the Aftragal F, 4 to the Super- Aftragal E, and 1 to its Fillet. Give 8 to the Corona 1 ), 3 to the Cyma Reverfa C, and 1 to its Juliet. Give 2 to the Aftragal B, 8 to the Cyma R«fta A, and 3 to its Regula. To determine the Projections of theft Moldings. Make q E, and C D, each equal to the Semi-diameter of the Column at its Aftragal, and draw the Line e d for the Upright of the Freeze, which continue l up through the Cornice. Make the utmoft Projection before the Upright of the Freeze, equal to k m the Height oi the Cornice. , . , From any Part of the Upright of tlie Freeze, as at E, draw a horizontal Line, as E F, which divide into 4 equal Parts. Divide the hrft I Part into 3 Parts : then tlte firft 1 Part thereof determines the Projection ot the Cavetto and Aftragal at w, and two thirds thereof, the Capital of the breeze, who e Fillet Projects equal to its Height. The fecond Part of the hrft Part E F, de- termines the Projeftion of the billet v S and one fourth of the next thud Part the Denticule t. E b, one fourth Part of E F, determines the Projection of the 0 / ARCHITECTURE. 135 Fillet s , and Center of the Aftragal r ; as alfo the Bottom of the Ovolo Iv. Divide b d, the fecond Part of E F, into 8 Parts ; or be its Half, into 4 Parts ; then the fecond Part determines the Projection of the Outfide of the Modillion at n. BifeCt d f the third Part of e f, in e. Divide d e into 4 Parts, then the firft Part determines the Projection of the Modillion in Profile at m ; the fecond Part, the Super-Modillion at /, and d e the Super- Allragal at i. Divide f F, the fourth Part of E F, into 7 equal Parts ; then f 2, equal to 3 of thofe Parts, determines the Projection of the Corona, and f h, equal to ‘J of f F, the Fillet of the Cyma Reverfa C. Make y z the Tenia of the Architrave, equal to | of of E b. Make q r, and t x in the Freeze, equal to half the Diameter at the Bafe of the Column. Divide t x into 6 Parts, and give 2 Parts to the Breadth of each Drop, as in the Doiick Order. To divide the I) entitles in the Cornice. Divide a b into 24 equal Parts ; give 2 Parts to the Breadth of each Dentule, and 1 to each Interval. The Breadth of an upper Modillion is equal to 10 Minutes, and of an under Modillion unto 5 Minutes. The Diftance in the Clear between the upper Modillion is 30, and between their central Lines 40 Minutes; fo that to adjult the Diftances of Columns in this Order, we muft place them at 3, 4, 5, £sV. times 40 Minutes, and then the Modillions will happen at their true Diftances. This Entablature, without Oftentation, is the riclieil and molt mag^ nificent, that has yet appeared in the World. II. Of 'ihe Compofite Entablature , to be ufed againjl the Outjides of Buildings. Fig. I. Plate XL II. Divide r s, equal to the given Height, into 20 equal Parts ; give the lower 3 to the Height of the Architrave ; the next 3 to the Height of the Freeze, and the upper 4 to the Height of the Cornice. Fo divide the Height of the Architrave. Divide t v , equal to the given Height, into 5 equal Parts,; divide the lower I Part into 4 Parts ; give the lower 3 to C the lower Fafcia, and the upper I to B the Bead. The fecond Part of t v is the Height of A, the middle Fafcia, Divide the third Part of t v into 3 equal Parts, and give the lower I to z the Cyma Reverfa. Divide y x, the 4th Part of t v, into 4 equal Parts ; give the upper t to the Height of the Bead x, and the Remains, with the Remains of the third Part, will be the Height ofy the upper Fafcia. Divide the upper Part of t v into 3 equal Parts ; give the lower 2 to the Height of the Cyma Reverfa, and the upper 1 to the Height of q the Regula. To divide the Height of the Freeze. Divide the upper third Part into 5 equal Parts, and the upper 1 of thofe Parts into 3 Parts ; give the upper 2 Parts to the Height of the Aftragal n, and the lower 1 to the Height of its Fillet 0. This Freeze may be made either up- right or {welling, at, the Pleafure of the Architect. To divide the Height of the Cornice • The Height, confifting of 4 principal Parts, divide i n, the firft Part, into. 8 equal Parts ; give the lower 4 Parts to the Cyma Reverfa m, and the upper 4 Parts to the Platform of the under Modillion, of which the upper 3 Parts, muft be given to the Height of the Modillion. Divide, f i, the fecond Part of the Height, into 4 Parts ; give two thirds of the lower 1 to the Height of the Cyma Reverfa i, and the upper 1 being divided into 3 Parts, give the upper z to the Ovolo, and the lower I to the Fillet. Divide c f, the third Part of the Height, into 4 equal Parts, and the upper 1 thereof into 2 Parts ; give the under I to the Height of the Fillet d, and the 3 remaining Parts will be the Height of the Corona e. Divide the upper fourth Part of the Height of the Cornice into 4 equal Parts, and the lower 1 thereof into 3 equal Parts, add the lower 1 to the Remains cf the third principal Part, which together make the Height of the Aftragal C. The upper 4th Part is the Height of the Regula a s 136 Of ARCHITECTURE. To determine the Projections of thefe Moldings. Draw F O parallel to the central Line Q_R, make F G equal to F M, from any Part of the Upright of the Freeze, as at K ; draw the horizontal Line KL equal to F G, which divide into 4 equal Parts, and each Part into 6 equal Parts, then the ill Part of K 1 determines the Prqjeftion of the Fillet and Center of the Aftragal, the 4th Part the under Modillion 5 the 5th Part the upper Modillion, and K 1 the Ovolo or Capping of the upper Modillion ; the 2d Part of K L being divided into 6 Parts, 4 Parts and ^ determines the Projection of the lower Modil- lion in Profile, 5 Parts and £ the Super-Modillion in Profile, and 5 Parts f- its Fil- let ; the %ft half Part, between 2 and 3, determines the Projection of the Ovolo under th^lorona, whofe Projection is determined by the 3d Part of K L, and its Fillet by the next half Part. The Projection of the Tenia O P is equal to 4 Parts of K 1, and which being divided into 5 equal Parts, give -1 of the firft 1 to the Projection of the middle Fafcia, and the firil 2 to the upper Fafcia. The Breadth of a Super-Modillion is 10 Minutes, and the Interval between every two is 25 Minutes, and which being in every refpeCt equal to the Modillions of the Corinthian Order ; therefore when this Entablature is ufed, the Intercolmnniations muft be the fame as thofe of the Corinthian Order, of which Fig. I. II. and IV. Plate XLIII, are Examples, and as the firil and lail of thefe Examples are arched Doors, I muft therefore proceed tq explain the Impoft and circular Architrave, Fig. V. which is ufed therein. To divide the Compofite Impojl and fir chit rave. Divide a h the Height into 3 Parts, the lower 1 is the Height of the Neck, or Freeze of the Impoft. Divide the middle 1 into 3 equal Parts, and the lower I into 3, give the lower 2 to the Cavetto, and the upper 1 to its Fillet ; divide the upper I into 3, and giving the upper 1 to the Fillet, the two lower Parts, to- gether with the middle Part, is the Height of the Cyma ReCfa. BifeCl a h in i ; divide a i into 3 Parts, give the lower two Parts to the Cyma Reverfa, and the Vpper one to the Regula. The Aftragal and its Fillet is equal to half m h the Neck of the Impoft. The Projections of thefe Members are thus found. Draw l e for the Upright of the Pilafter ; divide r/ IV. Of the Corinthian Order. Plate XXXII. hp 20. Is not the Plinth to his Pedeftal much too low for the Stateliaefs of the Older r 21. Is it good Architecture to make the Shaft of the Corinthian Co- L 20 Shorter than the Shaft of the Ionick Column, Fi?. I. Plate XXVIII.? A V. Of the Compofite Order. Plate XXXIX. If 22. Is not the Plinth to his Pedeftal much too low for the Stature of the Order r 23. As the Corinthian Order, which is more delicate than the Compofite Julcr, has its Shaft made 20 Minutes fhorter than the Shaft of the Ionick , why aotn he make the Shaft of the Compofite Order, whole Capital and Entablature aie more mqffive than the Corinthian, 30 Minutes higher than the Shaft of the Ionick ?' ‘ * •: : ■ e - • & 2 4 r 138 Of ARCHITECTURE. 24. Has the double Aftragal d, in Fig. I. Plate XLI. any Similarity or Proportion to the other Members of the Bafe ? 6). 25. Is it good Architecture to proportion the Architrave and Freeze of thisT~Order the fame (i a Minute only excepted) as the Tufcan ? <9. 26. Can any Perfon believe, that the Fillet on the Freeze, and its Aftragal, Ihould be made equal ? 27. ArC not the Greatnefs of the Members in the whole Entablature more proportionate to a Tufcan Column of feven Diameters in Height, than to a {lender Column of ten Diameters, which he has affigned ? To thefe I could add much more ; but let thefe fuffice to (liew, that this great Mailer is no more free from Millakes than another, although fo very much ap- plauded by many, who, for want of knowing better, have believed him inimit- able. LECTURE XI. Of the Grotefque Order, Fig. I. Plate XLYII. T HIS Order is a degree below the Tufcan: It confifts chiefly of Square Members, and is to be ufed in Grottos, CsV. To proportion the Parts of this Order. Divide a 1 , equal to the given Height, into 3 equal Parts, and the lower 1 Part into 7 Parts, give two Parts and \ to the Subplinth ; divide the upper I Part of a l into 7 equal Parts, and give the upper 1 to the Height of the Ovolo ; divide b k into 5 Parts, the lower 4 Parts is the Height of the Column, and which being divided into 7 Parts, is the Diameter of the Column; divide b e, the upper 1 Part of b k, into 3 Parts, the upper 1 is the Height of the Corona and Fillet, which is of the whole; divide/^ into 7 Parts, give 3 to the Archi- trave and 4 to the Freeze ; make g h the Capital equal to \ the Diameter, as alio the Height of the Bafe ; make the Height of the CinClure on the Bafe, and the Fillet under the Capital, each equal to -t of the Height of the Bafe. To rvflicate the Shaft. Divide its Height into 7 equal Parts, and make each Ruftick and each Inter- val equal to one Part. The Projection of the Bafe is 40 Minutes, and of the Sub- bafe 45 Minutes, from the cential Line of the Column. The Projection of the CinClure, from the Upright of the Column, is equal to its own Height, and the ProjeClion of the Rullicks is equal to that of the CinClure. The Shaft is dimi- nifhed h of its Diameter at the Bafe, and its Capital projeCts, before the Upright of the Shaft, * of its Diameter at the Capital. The Projection of the Ovolo, from the central Line c m, is 1 Diameter 37 Minutes and L LECTURE XII. Of the Attick Ordr, Fig VIII. Plate XLV. T HIS Order is never ufed, but when an Attick Story is placed over the Cor- nice of fome one of the preceding Orders, and is thus proportioned. Divide D G the Height into 9 equal Parts, give the upper x Part to the Height of the Cornice. To divide the Members of the Cornice , Fig. II. Divide the Height into 10 equal Parts, give the firft 3 Parts to k m the Height of the Denticule, the next 2 to the Height of the Cavetto x, the next 3 to the Height of the Corona av, and the upper 2 to v, the Cyma Reverfa, with its Fillet t. ... Note, In the Plate the Cyma Reverfa is, by Miflake, made a Cyma Recta, which, the Reader is defer ed to correct. The Height of the Denticule, divided into 6 Parts, the Depth of the Dentulcs mull be made c of thofe Parts, their Breadths 3 Parts, and the Intervals each x 2 Part O/ ARCHITECTURE. 139 Part and \. The Project ion of the Cornice is equal to its Height. The Height of the Plinth is 12 Parts and as alfo is the Breadth of the Pilafter, of thofe to Parts into which the Height of the Cornice is divided, and the fmall Torus and Fillet on the Plinth is 2 Parts and f-. If ’tis required to place Balls on the Necks over the Pilafters of this Order, the Height of the Neck mull be equal to the Height of the Cornice ; which being divided into 5 Parts, give 2 to the Plinth, \ the next 1 to its Fillet, and | of the upper 1 to the upper Fillet. The Diameter of the Ball is equal to the Diameter of the Pilafter, and the Dillances of the Pilafters are always the fame, as of the Columns over which they Hand. LECTfJRE XIII. Fig. II. Plate X. Of wreathed. Columns. A S at fome Times, the Shafts of the Ionick and Corinthian Columns have been wreathed or twilled, it is therefore neceffary to Ihew, How to defcribe a wreathed or twijled Column. Let ah c r be a given Shaft, ill, Bifedl a h in G, and draw the Line G c y make r p equal to r c, and draw w p parallel to c r. Draw the Diagonal Lines t p, and d r, and make the Triangle d z c equal to the Triangle pgr, on the Points z and g ; with the Radius g r, defcribe the Arches p r, and d c; 2dly, Make p 0 equal to p w, and draw e 0 parallel to c r, alfo draw the Diagonals e p, and o w. Make the Triangle 0 p g equal to the Triangle e h w, on the Points h and g, with the Radius h d, defcribe the Arches d e, and p 0. 3dly, Make 0 n equal to 0 v, draw the Diagonals s 0 , and e n, make the Triangle s f e equal to the Triangle n i 0 , and on the Points f and/, with the Radius / o, defcribe the Arches s e, and no; 4thly, Make n l equal to n t, itfc. and fo proceed to repeat thefe Operations, until the whole be completed, as required. LECTURE XIV. Plate XI. Of the Manner of dividing the Flutes and Fillets, on the Surfaces of real PilaJ- ters, and Columns. P ILASTERS are fluted in two different Manners, viz. either with Fillets only, as Fig. N. or with Fillets and Beads at their Angles, as in Fig. M. The Number of Flutes in the Front of a Pilafter ftiould be' feven precifely, al- though fome make lefs, and others more, but thofe are never done by an Artill or Workman. The Breadth of a Flute is to the Breadth of a Fillet, as 3 is to I. In Fig. N. there are 8 Fillets, and 7 Flutes, which are thus found, viz. divide the given Breadth of your Pilafter into 29 equal Parts, give 1 to each Fillet, and 3 to each Flute. In the other Example, Fig. M divide the given Breadth into 31 equal Parts, give 1 to each Bead, and the other 29 to the 8 Fillets and 7 Flutes, as in Fig. N. Fo readily divide the Flutes and Fillets of a Pilajler. Draw a Line at pleafure, ag a l. Fig. N, and therein fet off 29 any equal Parts from a to b. Make the equilateral Triangle a zb, and from the 29 Divi- fions draw Lines to the Point z : This being done, fet the given Diameter of your Pilafter from % to d, and tor, and draw the Line c d, which will be divided at the Points efg h i, &c. into its Flutes and Fillets, as required. For as c d is parallel to a b, therefore the Triangle c d z is fimilar to the Triangle « a b, and confequently the Line c d is divided in the fame Proportion as the Line a b. In the fame Manner a Pilafter with Beads and Fillets is readily divided by an equilateral Triangle of 3 1 Parts, as dab. Fig. M. F 0 divide at once the jujl Breadths of Flutes and Fillets, on the Surface of a real Column. Let Fig. F. Plate XI. be the Plan of the Bafe, and Fig. E. of the Top of a given Column, to be fluted with Fillets, Operation. 140 Of ARCHITECTURE. Operation. Draw a right Line, as p q, Fig. I. at pleafure, and having two Pa i ^ of Compaffes, open one Pair to any fmall Diftafice, fuppofe q r, and the other Pair to -one third Part thereof; now thefe two Openings of the Compaffes are to one another, as the Breadth of a Fillet is to the Breadth of a Flute, therefore irom p towards q, fet off the two Openings, each 24 Times reciprocally, that is interchangeably, as firfty> r, then r s- then s t . equal to p r, &, c. but you muft obferve that the two Openings aforefaid are fach, that when you have fet each 24, Times from p to q, that the Length from p to q be lefsfhan the Girt or Circumference ©f your Column that is to be fluted, otherwife your Labour will be in vain. From the feveral Livifions fo fet off, on the Line p q, draw right Lines perpendicular to p q, of Length at pleafure, and then you may proceed to the finding of the true Breadths of your Flutes and Fillets as following. lft, Strike a perpendicular Chalk Line from the Altragal to the Cin&ure on the Surface of the Column, and being provided with a narrow ftreight-edged Pieie of Parchment, 5 ’ & c ‘ vv ’hich will cut the central Lines cf the Inftrument, in t le t oints 1 2 3 4. 5 6, &c. from which draw right Lines with Black-lead, at right Angles to eg, and they will divide d a into unequal Parts, which are the true Appearances of the Breadths of the feVeral Flutes required. And the Ldge d a, being applied to the Diameter of the Column in your Drawing, prick oft the feveral Diviiions, which will be the Breadths of your Flutes and Fillets, as required. dig. III. is another Inftrument of the fame Kind, made for fetting off the I lutings Oa Dorick Columns, according to the Manner of the Ancients* \ LECTURE XV. Of the Manner of placing Columns again/} W, alls , and over one another, as tht Doiick on the Tufcan, the Ionick on the Defick, efed C OLUMNS are placed either againft Walls, with a fourth Part of their Dia* meters inferted, as Jug. III. and IV. Plate XXX. when three Quarters of the I ody of tue Shaft project before the Upright of the Walls ; or entirely clear fiom the Wall, as Fig. III. Plate XLIII. in which laft Cafe, a Pilafter is always mieiteci in the Wall, as C and E, before the Columns D E; and the Inter- colummation or Diftance of the Column from the Pilafter, is always the fame as when Columns are placed in Pairs. The Quantity of Infertion of Pilafters muft ™ ch as Wl11 be agreeable to the Parts of their Capitals* In the Tufcan and Dorick Orders the Pilafter may project before the Wall, a half, a third, a fourth, ahtth, afixth, or feventh Part of its Diameter: but in the Ionick, Corinthian, unc Compcfte Orders, they ftiould be half a Diameter precifely, otherwife the Ornaments of their Capitals will be unevenly divided, and have a vert bad Appearance. J When Columns are to be placed over one another, as was the Cuftom of the Ancients, who placed an Order in every Story, we are to obferve, firft, That . e p'. iameter the Column in the fecond Story be at its Bafe,. equal to the Diameter of the lower Column at its Aftragal : and that they ftand ex- at Jy peipendicular over each other, that the upper Solid may ftand on the J? v . er : btxondly, To P !ace the u PP er Columns on a continued Pedeftal, whofe iV 'j a 1 j , "1 agreeable to the Windows, as to make the Cornice of the 1 edeftal do the Office of Stools to the Windows; for when Columns have their Bafts placed belotv the Bottoms of Windows, fo that their Stools being con- tinued ltop againft the Shafts of the Columns, as thofe do at the Royal Banquet* trng-houf zx Whitebaf they have a very ill Effect. The Intercolunmiation of Orders praced over one another muft be governed by the Triglyphs and Mo- di -lnuis, and therefore to place the Dorick over the Tufcan , regard muft be had to the Number of Triglyphs in the upper Order, to which the Tufcan muft be conformable, as indeed muft the Ionick to the Dorick in feme Cafes, when the D: Lances of its Modiilions muft be made a little more or lefs: to bring them into Order ; and when the Corinthian is placed over the Ionick, the Modiilions of the' ionick muft be conformable to thofe of the Corinthian * When an open Gallery is made over an Arcade, the Openings between the Columns may be quite down to the Bottom of the Pedeftal in the upper Order, 1! p jPn i a f‘- , bet at fuch Times ’t'.s belt to place a Baiuftrade betweea the. 1 ede.tals, which will be a Security and an Ornament alXo.' LECTURE I4 2 Of ARCHITECTURE. LECTURE XVI. Of the various Kinds of Ornaments for the Enrichment of the feveral Members of < which the Five Orders of Columns are compofed. rp HE Ornaments that are, and may be invented for the Enrichments of Mold- X ings, are endlefs ; but thofe that are now in the greatift Efteem, I have in- troduced in the feveral Members of the laft four Orders ; not that every Order mult be fo fully enriched as I have exprelfed, but fuch Parts or them only, as fhall be judged fufficient ; and that the Learner ftiould not be at a lofs to know what Ornaments are proper for fuch Members, as he' may be inclined to enrich, I there- fore have been fo profufe, as to give every Member an agreeable Enrichment. And as oftentimes ’tis required to enrich Pannels, Pitture-frames, and other Parts of Buildings, I have therefore, in Plates XVI. XVII. and XVII 1 . given a great Variety of Ornaments at large, together with the Sections of divers curious Mold- ings for fuch Purpofes, of which take the following Account : I. The Figures E F I are Ornaments called Vi truvian Scrolls, I fuppofe from Vitruvius , who might be the Inventor of them. The Diftances of the Spirals is at pleafure ; but their Height being divided into two Parts, their Diftance is generally equal to 3 of thole Parts, and their Spirals are defcribed by the hIethod3 before taught. . II. The Figures GHKL M are Juterlachigs, or Giulochis of. various Kinds, of which GHK and L are compofed of the Arches of Circles, as is evident by In-- fpe&ion, and thatof FigM, of parallel right Lines, which form geometrical Squares of any Magnitude connected together, by Quadrants on the Outlides. The fiet Ornament of the Ancients is by fome called Guilochi , of which in Plate XVIII. I have given Examples of 15 Kinds, for the Pradice of the young Student, and whofe Number of Parts into which the Breadth of each is to be divided are fig- nified by Divifions, and Numerical Figures againft each. Hr. The Eggs and Darts, commonly called Eggs and Anchors, as Fig. I. Plate XVI. are thus defcribed. Divide the Height 7 P into 9 equal Parts, at the Points 123456789* F irft, Draw a C and k B, parallel to 7 P» each at the Diftance of 7 Parts ; and divide a 7 and 7 k, each into 7 Parts. Through the Point / draw e m, parallel to C B ; make/ e and f m, each equal to 4 Parts, and draw the Lines e 3 and m h. Through the Point 3, mi the central Line 7 P, draw the Lines e 3 y, and m 3 v. On the Point /, with the Radius / 12, defcribe the Semi-circle o 12 p. On the Points e and in, with the Radius m 0 , defcribe the Arches 0 v and p y ,* and on the Point 3 . with the Radius 3 y, defcribe the Arch v 1 y, which will complete the Outline of the Egg. Secondly , Draw the Line d l, through the Point 7 ■> on ^he Line. 9 ^ » and divide the Diftance between the Points 3 and 4 > on the. Line Q P, into 2 equal Parts, and draw the Lines d z and / nv. On the 1 oints r/.and /, with the Radius d h, defcribe the Arches h z and h w; and on the middle I omt, between 3 and 4, on the Line 9 P> defcribe the Arch av z. Thirdly, Diaw c & through the Point 8 ; make c 8 and 8 g each equal to 3 Parts. From the Points c and g draw the Lines g x and c A, through, the middle I oi.nt be- tween the Points 4 and 5, on the Line 9 P* On the Points c and g, with tue Radius c i, defcribe the Arches * A and 1 x; alfo on the middle Point be- tween 4 and 5 aforefaid, defcribe the Arch x P A. Fourthly, Through, the Point 2 on the Line 9 P, draw the Line 2 r s ; as alfo draw the Lines 1 B, flopping at r ; alfo 1 2 B, and a r ; then one half Part of a Dart will be com, pleted ; and in the fame Manner complete the other Half, and all others. Now from hence ’tis plain, that to fet out the Diftances of Eggs and Darts, you muft fir ft divide the Height of the Ovolo into 9 equal Parts. Secondly , Take 7 of thofe Parts, and fet that Diftance along your Molding, and then Lines be in " drawn from thofe Points, fquare to the I op and Bottom of the Ovolo, 0 / ARCHITECTURE. 143 Qvolo, every other Line will be the central Line of an Egg, and the others of the Darts, which divide as aforefaid. Eggs in Ovolos are oftentimes enriched with Leaves, Hulks, &c. inftead of Darts, as between NOP, Plate XVI. IV. The fever al Moldings for Pannels and Picture Frames , Plate XVII. are thus divided. I. Of Moldings for Pannels, Fig. I. divide the Height into 3 Parts ; give two Thirds of the upper 1 to A the Regula ; the remaining 3d Part, and the middle great Part to B the Cyma Reverfa ; half the lower Part to C the Aftragal ; and the remaining half Part, divided into 3 Parts, give 2 to E the Cavetto, and I to D its Fillet. The Diftances of the central Lines a k, c d, e f, £sV. of the Leaves, itfe. is equal to the Height of the Cyma B. Secondly , Fig. II. Divide the Height into 4 Parts, give the upper 1 to A the Regula ; the next 2 to B the Cyma Redla ; and the lower 1 divided into 3 Parts, give 'the upper 1 to C the Fillet, and the lower 2 to D the Cavetto. Divide b d into 5 Parts, and fet off the central Lines of the Leaves, as a c , Itjc. each at the Diftance of 7 Parts. Thirdly ^ Fig. IV. Divide the Height into 5 Parts; give the upper 1 to A the Regula, two Thirds of the next 1 to B the Cavetto ; the next 2 Parts, with the Remains of the 4th Part, to C the Cyma Reverfa, and the lower Part divided into 3 Parts, give 1 to the Fillet E, and 2 to the Aftragal D. The Diftance of the central Lines of the- Leaves, &c. b d, a e, c f iffc. is equal to the Height of the Cyma Reverfa. Fourthly , Fig. V. Divide the Height into 5 Parts, g'ive the upper 1 to the Regula, the next 1 to the Ovolo, I Third of the next to its Fillet, the rerpain- ing 2 Thirds, and the next 1 to the Cavetto ; and laftly, the lower 1 divided in- to 3, give the upper 2 to the Aftragal, and the lower 1 to the Fillet. Divide a d into 9 Parts, and make the Diftance of a b, b c, &c. equal to 7 of thofe Parts, as aforefaid. Fifthly, Fig. VI. Divide the Height into 3 equal Parts, and the upper 1 into 3 ; give the upper 2 Parts to the Regula A, and the Re- 1 mainder, with the middle great Part, to the Ovolo B. The lower great Part divided into 2 Parts, give the upper x Part to the Aftragal C, and the lower Part being divided into 4 Parts, give the lower 3 Parts to the Cavetto D, and the other 1 Part to its Fillet. The Diftances of the central Lines of the Eggs, tfc. are to be found as aforefaid. II. Of Moldings for Picture Frames. First, Fig. III. Divide the Height into 4 Parts ; the upper 1 divide into 3, f ive 1 to the Regula A, and 2 to the Cyma Reverfa B. Divide the upper lalfof the next Part into 2 equal Parts ; give the lower Part to the Cavetto E, and the upper Part being divided into 3 Parts, give the upper 2 to the Aftragal C, and the lower 1 to the Fillet D. Divide the lower 4th Part into 3 equal Parts, and the lower 1 Part into 3 Parts; give the lower 2 Parts to the Cavetto K, and the upper 1 to the fillet I. Divide the upper 3d Part into 2 Parts ; give the upper 1 to the Fillet G, and the Remains to the Aftragal H. Divide b d into 5 Parts, and make the Diftance of the central Lines of the Leaves, as a c , &c» equal to 6 of thofe Parts, the central Line of the Rofes to the Fitruvian Scroll in the Freeze F, is direftly in the Midft of the Freeze, and the Diftance of the Centers of each Rofe, as e f, is equal to the Height of the Freeze. Secondly, Fig . VII. Divide the Height into 3 equal Parts, and each Part into 4 equal Parts ; give the upper x Part to the Regula A, the next 2 Parts to the Ovolo B, and the next I to the Fillet C and Cavetto D. Give the middle great Part, and 1 Fourth of the lower great Part, to E the Freeze. Give the next fourth Part of the lower great Part to the Cavetto F, and Fillet G ; and then the Remains x being divided into 4 Parts, on sr deferibe the Quadrant y x, and then making c b equal to y z, deferibe the Curves y a , and a b, which with the Qua- drant y x, forms that Molding which Workmen call the Weljl Ogee. The Man- ner of deferibing the Guilcchi in the Freeze is plain to Infpedlion, as alfo are the Diftances of the Eggs, in B the Ovolo, and Leaves in H the Weljh Ogee. i 2- Thirdly, 144 V. Divide the Height into 4 equal Parts, and the upper 1 Part mto 4 Parts, then the upper 3 Parts is the Height of the Architrave, Freeze, and Cornice, and the Remainder is the Height of the Door, whole Breadth is equal to 1 great Part and a Half, and its Architrave to { of the Breadth. The Breadth ot the open I ilailers h x, againfl which Truffes are fixed as at k, to fupnort the Cornice, is equal to f of the Breadth of the Architrave. Divide the lower 4th Part, of the upper great Part, into 2 equal Parts, and that gives the Depth from the Cornice, at which the Foot, of the Trufs is to be placed. The proper Trufs ior the Support ofthefe Kinds of Cornices is exhibited in Fig. I. Plate XIV and its thus defcribed. T° defcribe a fpiral Trufs, for the Support of Cornices over Doors, Windows , and Niches. Pi™ A B, the given Height, (including the Height of the Architrave, Freeze and Cornice) mto 15 equal Parts, give the upper 4 to the Height of the Cornice, and the lower 1 1 to the Height of the Trufs. Let the Line M a reprefent the U plight of the face of the open Pilafter, againft which the Trufs is fixed. Draw We » parallel to M s, at the Durance of two Parts and * ; alfo draw B * the Ba.e Line at Right Angles to M z. From the Points 8, 4 and 2 in the Line A B, draw the Lines $g, 4 G and 2 E, parallel to B 3, and of length towards the Right Hand at pleafure . thefe Lines hill drawn determine the Heights of the greater and leffer Spirals or Scrolls. Divide a r, the under Part of the Cornice, into 8 equal Parts, and into 7 equal Parts; alfo divide G E into 7 equal Parts, and make G y and E $ equal to 8 of thofe Parts ; this being done, proceed in e ,7 re ^eft to defcribe the two Spirals, as you did thofe in the Corinthian Mo- dilkon, Fig. V. Pxob. IX. Lect. VIII. hereof. • FlS ' Is t ' 1C pront View of this Trufs, whofe Breadth H I is equal to B F viz to -i Part and f of the Parts in A B, and which being divided into 8 equal I arts, is defcribed in every Particular the fame as kn, mi op l, in Fir. III. the Face or Front of the Corinthian Modiliion, rp u . f° d ™ « upper Part, t0 ji 415°° The 3d ff reat Part 18 the Height of the Corona P, and the next and lad Part is the Height of the Fillet O, the Cyma Reel a N, and Regula M. I he Projedtion oithe Cornice W X is equal to its Height VV e. h -- To divide the Denudes. Dmoe x x the Height of the Denticule into 6 Parts, and make the Length oj a Dentule equal to 5 Parts, Make the Breadth of aDentule and an Interval equal i 4 6 0/ ARCHITECTURE. iqual to the Height of a Dentule, which divide into 3 Parts, give 2 to a Dentule and 1 to the Interval. II. F 0 proportion Windows and Niches to any given Height , Fig. I. II. and III. Plate XLVI. Divide the given Height into 5 equal Parts, the lower t Part is the Height of the Pedeftal, whole Parts are to be divided according to the Peddfal of any Order required. The remaining 4 Parts being divided into 5 equal Parts, the upper 1 Part is the Height of the Entablature, and their Breadths, if for Win- dows, into 2 Parts. The Breadths of their Architraves, as m n, Fig. III. is equal to g of the Opening, and of their open Pilafter, to § of the Architrave, as likewife are the Margins op and q r, Fig. II. when made into Niches. I he pro- per Entablatures to be placed over Doors, Windows and Niches, are exhibited by Figures A B C D EF and G, Plate XLVII. But as fometimes the Quoins and Heads of Windows are rufticated, I have therefore in Plate XLV. given five Examples thereof, with the Divifions of their Parts, which explains them to the meanefl Capacity. LECTURE XIX. OJ Pediments. P Ediments are employed either for Ornament and Ufe, or for Ornament only. Pediments for Ornament and Ufe are thofe which are made on the Out- fides of Buildings, and which mult be entire, that thereby the Buildings under- neath may be wholly protected from the Injuries of Rains. Entire Pediments are made in three different Manners, viz. ijl, Straight, as Fig. II. Plate XLI 1 I. which Workmen call a raking Pediment. 2 dly, Circular, as Fig. I. Plate XLIIJ. And, 3 dly, Compounded of three Arches, as Fig. II. Plate XLIX. The Manner of finding the Height of the Fajiigium, or Pitch of a raking and circular Pediment, being already taught in 1 rob. I. Lect. V. hereof, I ftiall therefore proceed to {hew How to deferibe a compound Pediment, as Fig. IT. Plate XLIX. A Com pound Pediment has the fame Pitch as a raking Pediment, therefore to deferibe a Pediment of this Kind, draw the raking Bounds of a pitched Pediment, as B A and A C, bifeft B A in b, and A C in d, alfo bifect A d in c, and thereon ereft the Perpendicular c F, cutting the central Line A F in F. Bifedf !B h in a, and d C in e ; on the Points a and e erect the Perpendiculars a E and e D, which will cut the Perpendiculars C D and B F in the Points F and D. On the Points E D and F, with the Radius E B, deferibe the Arches B h, h A d, and d C, and concentrick thereto, at the refpeftive Heights of the feveral Members of the Pediment, deferibe the whole as required. Pediments for Ornament are thofe which are imperfect, and are vulgarly called Hr alert or Open Pediments , as Fig. I. II. III. Plate XLVIII. and Fig. 1 . and III. Plate XLIX. Thefe Sort of Pediments fhould never be ufed without Buildings, becaufe being open in the Middle, they let in the Rams on the Cornice, in the fame Manner as if no Pediment was there. It is therefore that thefe Kinds of Pediments muff be ufed within Doors for Ornament only, and whofe Opening is generally made for the Reception of a Bufto, Shield, obeli, iyc. Now feeing that to make an open Pediment without Doors is abfurd, to make an entire 1 edi- ment within Doors, where no Rains come, muff be abfurd alfo. In the Tufcan Order, the Length of the raking Cornice, as A G, Plate XLVIII. being divided into 5 equal Parts, as at 1, 2, 3, 4; the Length of the Regula J G is equal to the 4 lower Parts. The fame is alfo to be obferved in a circular open Pediment, as Fig. I. Plate XLIX. But in a Dortch Pediment, the Length of the raking Cornice is to be regulated by the Mutules, for as' the raking Mutules, as H I, in the Pediment, mud be dire&ly over A B, in the level Cor- nice, therefore the Diftance//-, the Projection of the Cornice beyond the Upright of the level Mutule K, being fetfrom 4 to b, and the Line b 20 being drawn, it cuts Cuts the raking required. Of ARCHITECTURE. i 47 Line s a into 9, making the Length of the raking Cornice The Length of the raking Cornice of an Ionic k Pediment is determined by- placing a Modillion in Profile againft a raking Modillion, as H againft G, equal in Projection toy 5, the level Modillion in Pi'ofile, and making 5, 1 ; the Projection of the raking- Cornice beyond the Upright of 1, 13, the Upright of the raking Modillion in Profile, equal to the Projection of the level Cornice beyond the level Modillion in Profile. The laft raking Modillion in the Pediment is always at pleafure, according as the Breadth of the Opening of the Pediment is required ; and therefore it might have been either that over E or F, inltead of at G over A. Note, The fame is alfo to be underftood of Pediments of the Corinthian and Compo/ite Orders. Pediments are fometimes finifhed with Scrolls, as Fig. Til. Plate XLIX. which are thus defcnbed. Let A B C be the Extent and Pitch of a raking Pedi- ment. Bifecf B A and A C, in h and g, find the Centers H D G, in the fame Manner as you found the Centers E F D, in Fig. II. Draw the Lines h D, and g D, and on the Points H and G defcribe the feveral Members on each Side as was done in Fig. II. Divide b A into 8 equal Parts. From the third Part draw the Line C D, and on the Center D defcribe the Arch b c , and Members concentrick thereto ; make c e equal to 3 Parts and d of b A. Divide c e into 8 equal Parts,' and on I t ^ ie SjP Part from s defcribe a Circle, as the Eye of a Volute or Spiral, and therein find the Centers as before taught, on which turn about the two Cymas | and finifli the Eye with a Rofe, &c. at pleafure. Note , Sometimes the Cyma ReCta is left out of the Scroll, and the Cyma Re- | verfa with the Corona only, are turned about to form the Scroll, which has a very good Effeft ; and then in fuch a Cafe the Cyma ReCIa is ftopt, and returned ■ as in an open Pediment. LECTURE XXI. Of trujjcd Partitions. W FIEN Partitions have folid Bearings throughout their whole Extent, they have no need to be truffed ; but when they can be fupported but in fome 1 P art j cu ^ ar Places, then they require to be truffed in fuch a Manner that the whole Weight fhall reft perpendicularly upon the Places appointed for their Support i and no where elfe. As Partitions are made of different Heights, to carry one, two i or more Floors, as the Kinds of Buildings require , therefore in Plate L, I*have given fix Examples, of which Fig. II. V. and VI. are of one Storv in Heip-ht and Fig. IIL IV. and VII. of two Stories. } S ’ The firit .Things to be coniidered in Works of this Kind, is the Weight that I * s t° be fupported; the Goodnefs and Kind of Timber that is to be employed * and proper Scantlings neceffary for that Purpofe. The Strength of limber in general is always in proportion to the Quantity of folid Matter it contains. The Quantity of folid Matter in Timber is always more or lets, as the Timber is mere or lefs heavy; hence it is, that all heavy Woods, as Oak, Box, Mahogany, Lignum Vita, t^c. are ffronger than Enter, Deal, Sycamore, iFc. which are lighter, or (rather) lefs heavy, and indeed, Vor the fame Reafon, Iron is not io ftrong as Steel, which is heavier than Iron ; and Steel is not fo ftrong as Brafs or Copper, which are both heavier than Steel/ To prove this, make two equal Cubes of any two Kinds of Timber, fuppofe the one oi Fir, the other of Oar ; weigh them fingly, and Note their refpeftive Weights • this done, prepare two Pieces of the fame Timbers, of equal Lengths, fupoofe each 5 Feet in Length, and let each be tried up as nearly fquare as can be/but to fuch Scantlings, that the Weight of a Piece of Oak may be to the Weight ot the Piece of lit' , as the Cube of Oak is to the Cube of Fir ; then thole two Pieces \ i 4 8 Of ARCHITECTURE. Pieces being laid horizontally hollow with equal Bearings, and being loaded in their Middles with increafed equal Weights, it will be feen, that they will bend or fag equally, which is a Demonftration, that their Strengths are to each other, as the Quantity of folid Matter contained in them. As the whole Weight on Partitions is fupported by the principal Poll, their Scantlings mull be firft conlidered ; and which fhould be done in two different Manners, w. Firft, when the Quarters, commonly called Studs , are to be filled with Brick Work, and rendered thereon ; and laftly, when to be lathed and plaiftered on both Sides. . When the Quarters are to be filled between with Brick Work, the Thicknefs or the principal Pods Ihould be as much lefs than the Breadth of a Brick, as twice the Thicknefs of a Lath ; fo that when thofe Polls are lathed to hold on the rendering, the Laths on both Sides may be flufh with the Surfaces of the Brick Work ; and to give thefe Polls a fufficient 'Strength, their Breadth muft be in- creafed at Difcretion ; but when the Quarters are to be lathed on both Sides, or when Wainfcotting is to be placed againlt the Partitioning, then the Thicknels of the Polls may be made greater at pleafure. The ufual Scantlings for princi- pal Polls of Fir, of 8 Feet in Height, is 4.01- 5 Inches fquare ; of 10 Feet in Height, 5 or 6 Inches fquare ; of 12 Feet in Height, 6 or 7 Inches fquare ; of 14 Feet in Height, 7 or 8 Inches fquare ; of i6‘Feet in Height, from 9 to 10 Inches fquare. But thefe lafl, in my Opinion, are full large, where no very great Weight is to be fupported. As Oak is much llronger than Fir , the Scant- ling of Oak-Pojls need not be fo large as thofe of Fir j and therefore the Scant- lings affigned by Mr. Francis Price , in his Treatife oi Carpentry, are abfurd ; as being much larger than thofe that he has affigned for Fir Pojls. To find the juft Scantling of oaken Polls, that lhall have the fame Strength of any given Fir Pojls , this is the Rule : As the Weight of a Cube of Fir is to the Weight of a Cube of Oak of the fame Magnitude, fo is the Area of the fquare End of any Fir Pojl , to the Area of the End of an oaken Poll; and whofe fquare Root is equal to the Side of the oaken Poll required. The Diftances of principal Polls is generally about 10 Feet, and of the Quarters about 14 Inches, but when they are to be lathed on both Sides, the Diftances of the Quarters Ihould be fuch as will be agreeable to the Lengths of the Laths, otherwife there will be a very great Walle in the Laths. The ..Thicknefs of ground Plates and Raifings are generally from 2 Inches and Half to 4 Inches, and. are fcarfed together, as expreffed in Fig. I. K L M N O P QJL In the feveral Examples aforefaid the principal Polls have their Inter-ties and Braces framed into them, as expreffed in Figures F B G H C D A k E, whole refpedlive Places the feveral Letters in each refer to. LECTURE XX I L Of naked Flooring . T H E principal Things to be obferved in naked Flooring are, firft, the Difpo- fition of Girders, or Manner of placing them in the moll fecure and ad- vantageous Manner. Secondly, their Scantlings ; and laftly, the Manner of truffing them, when their Lengths require it. There are fome Carpenters, who infill that Girders Ihould be laid on ftrong Lentils over Windows, and who allege that Girders, being laid on Lentils in. Piers, the Piers are endangered at the Decay of thofe Lentils. Others infill, that ’tis bell to lay Gilders in Piers, as being the moll folid Bearings, and that if found oaken Lentils are laid under them, they will endure as longastiie Brick Work will remain found. In Buildings, whofe Piers are narrow at the renewing of Lentils, the Pier9 will be endangered in both .thefe Cafes; for Lentils laid over Windows mull be laid into the Piers, on both. Sides of a Window, and which, when taken out, will, make large Fraftures, that will be very little lefs dangerous than the other,, Of ARCHITECTURE; 149 ether, and therefore I fhall fubmit this Point to the Difcretion of the Work- men. - . Lentils laid in Piers between Windows, For the Support of Girders, fhould have their Lengths equal to the Breadths of the Piers : and thofe laid in Party- Walls, or Gable Ends of Building* fhould be equal in Length to the Diftance that is tontained between every two Girders. The Thicknefs of Lentils fliould al- ways be equal unto the Height of 2 or 3 Courfes of Bricks, and their Breadth unto' a Brick’s Length ; fo that in every of thofe Particulars, they may be con- formable to the Brick Work in which they are placed, and to that which is raifed on them. And for the better difpofing of the Weight impofed on Girders,. Lentils fliould always be firmly bedded on a fufficient Number of fhort Pieces of Oak , laid acrofs the Walls, vulgarly called Templets, which are of ex- cellent Ufe. Let Girders be laid in Piers, or in Lentils over Windows, it will, in both thefe Cafes, be commendable to turn fmall Arches over their Ends, that in cafe their Ends are firfl decayed, they may be renewed at pleafure, without difturbing any Part of the Brick Work ; and for their Prefervation; anoint their Ends with melted Pitch and Greafe, viz. of Pitch 4, of Greafe i ; and indeed, were Len- tils to be covered with Pitch and Greafe alfo, it would contribute very greatly to their Duration. It is always to be obferved, that the fhortefl Girders bend down, of fag, as Workmen term it, the leaft, and therefore ’tis always bell to lay Girders over the narrow Parts of Rooms, and whole Ends fhould always have, each, at leaft 14 Inches bearing in the Walls, excepting in fmall Buildings, where the Front, CSV. W alls are but a Brick and half in Thicknefs, when to prevent the Ends of the Girders from being feeri without-'fide, their Bearings cannot much exceed 1 1 Inches; It. is alfo to be obferved, that Girders he fo difpofed of, that the Boards of every Floor be parallel throughout the whole Floor ; for ’tis as difagreeable to the Eye, to fee the Joints of Boards in the fame Floor, lie different ways, as ’tis to fee Steps out of one Room into another, which fhould always be avoided. In the carrying up the fcveral Walls of Buildings, it fhould be carefully ob- ferved, to lay in Bond Timbers on Templets, as aforefaid, at every 6 or 7 Feet iu Height, cogged down, and braced together with diagonal Pieces at every Angle, which will bind the whole together, in the moft fubflantial Manner, and preveat Fraftures by unequal Settlement; I he DiflancC3 of Girders fhould never exceed 12 Feet, and their Scantling# muftbe proportioned according to their Lengths ; as by Experience ’tis known, that a Scantling of 11 Inches, by 8 Inches, is fufficient for a Fir Girder of 10 Feet in Length, the Area of whofe End is 88 Inches, it is very eafy to find the proper Scantling for a Girder of any greater Length, fuppofe 20 Feet, by this Rule i As 10 Feet, the Length of the firfl Girder, is to 88, the Area of its End, fo is 20 Feet, the Length of the fecond Girder, to 17 6, the Area of its End. Now, to find its Scantlings, that being multiplied into each Other, fliail produce 176 Inches, the Area found, one of them muil be giver, viz; either the Depth, Or the Thicknefs; In this Example, the given Depth' fliail be \i Inches; there- fore divide 176 by 12, arid the Quotient is 14 Iriches and 2 Thirds, which is the Other Scantling or Breadth required. To prevent the fagging of fhort Girders, ’tisufual to cut them Camber, that is, to cut them with an Angle in the Mid(l of their Lengths, fo that their Middle# fhall rife above the Levels of their Ends, as many half Inches as the Girder con- tains Times 10 Feet. And indeed, Girders of the greatefl Length, although tvufied; fhould be cut Camber in the fame Marnier. In Plate LII. I have given three differe'rit Examples for the truifing of Cr refers } and in Plate LIII, pig. I. a fourth, which being in general plain to Irifp'edtion, I therefore fubmit the Gnoice- to the Difcretion of the Workman.- The r S o Of ARCHI TiB C T U R E. • ThE next in Order are Jnifls, of which there are five Kinds, tvz. Common* Top, Binding-Jop, Trimming- Joi/ls, Bridging- Joi/ls, and deling- Jop. Firft, Common Joifts are ufed in ordinary Buildings, whofe Scantlings ,n ® re generally made as follow, sm*. Joifts of 6 Feet in Length, to be 6 and half by % and half ; of 9 Feet, 6 and half by 2 and half ; of 1 2 Feet, 8 by 2 and half. But rn laro-e Buildings, the Scantlings are made larger, where ’tis common to maKe Joifts of 6 Feet, 5 by 3 ; of 9 Feet, 7 and half by 3 ; of 12 Feet, loby 3. As Oak is much heavier than Fir, ’tis euftomary to make the Scantlings of Oak- Jop larger than thofe of Fir ; but I believe it to be entirely wrong, for the Reafon before given, relating to the Strength of Timber. Secondly, Binding- Joifls are generally made half as thick again as Common- Jop of the fame Lengths, which are reprefented in Fig. V. and VI. Plate LI. by n m q p , £SV. and which are framed flufh with the under Surfaces of Girders, to re- ceive the deling- Jop, and about three or four Inches below their upper Sur- faces, for to receive the Bridging- Jop ; fo that the upper Surfaces of the Bridging- Top may be exaftly flufti or level with the Girder to receive the Boarding. In Fig- IV. Plate LI. A reprefents the Se&ion of a Girder ; b b, &c. Parts of two Binding-Top, tenon’d into the Girder, a a, &c. the Ends of Bndging- Toifts} ee boarding on the Bridgings; d d, b J r. Mortifes m the Binding- Jofls to receive the Tenons of deling- Jop ; as alfo are the Mortifes be, b e, be. But thefe laft are thofe which are called Pulley Mortifes, into which the 6 tel- ing-Top are did. To underftand this more plainly, the Figures fjfj are added, which reprefent the Seftions of fo many Binding- Jop ; g g, c5r, the Sec- tions of fmall Joifts between them; * * a Side View of a Bndgwg-Jot/i, and h hh Cieling-Jop , tenon’d in the Binding-Jop, flulh with their Bottoms, as aforefaid, to receive the Lath and Plaifter. The Diftance that Bmdtng-Jofs Ihould be laid at, ftiould not exceed 6 Feet, though fome lay them at greater Lii- tances, which is not fo well, becaufe the Bridging and Luhng-Jopmxxil be made of larger Scantlings, to carry the Weights of the Cieling and Boarding, and confequentlv a greater Quantity of Timber muft be employed. But how- ever, as this Particular is at the Will of the Carpenter, I lhall only add, that the Scantlings for Bridgings of Fir, having 6 Feet Bearing, Ihould be 4 by * Inches ; thofe of 8 Feet bearing, 5 and half by 3 ; and thole of 10 Peet, 7 by*. Their Diftance from each other is generally abouQ 12 or 14 Inches, The Fig. ABCDEFGHI, exhibits different Kinds of 1 enons for Btndmg- T op, which are to be practifed as Oceafions require. The Figures V. anff VI. exhibit the View of a Floor over two Rooms, wherein the Girders P P are laid in the Piers C ABB. In Fig. VI. the Binding-Jop ft mqp,&c. and Trimming-Top are reprefented lingly, without the Bridging- Joi/ls ; and in fig. V. the B ridging- Joijls are laid on the Binding-Jop, as when ready for to receive the Boarding. This Example is given, only as a Specimen of thefe Kinds of llans, ' that from thence the young Student may the better know how to reprelent Plans of Floors, when required. . T , ... The Figures II. and III. are Examples of Floors made of lhort Lengths, which I have given for the Diverfion of the curious. tEC T. XXIII. Of Roofs and their Coverings. B EFORE we can proceed herein, a Plan of the Building to be covered muft be made, by which we may acquire a juft Knowledge_of the Dimen- fions of every Part that will be contained ifi the whole Defign, before any Part of the real Work be begun ; and by which we (hall alfo be taught, how to perform every Operation at once in the leaft Time, and to account for, oreftimate the '^Sup p ost IlJ^L 6 LI I. be the Plan of a reguIar Buildi^ to b^o. vered, which, is 50 feet by 25 Feet iuthe Clear within i firft, make a Parallelogram, Of ARCHITECTURE. 15,1 by a Scale of equal Parts, whofe Length {hall be jo of thofe Parts, and Breadth 25 Parts, which will reprefent the Infide of the Building. Secondly, without the Side and Ends of this Parallelogram, draw right Lines parallel thereto, at the Dillance of the Breadth of the Raifing, fuppofe I Foot, equal to I Part of the « Scale. Thirdly, as the Dillance at which Beams are laid, fliould not exceed ip Feet, on account of the Lengths of deling- jfoifls which are framed in between them ; therefore divide the Length of the Plan, with as many Beams as are ne- celfary-, as at the Points bill , and tvxy; and draw the central Lines of the Beams bt, tv, k x, and ly ; as likewife the central Lines of the Plan I 10, and z w, and the Bafes of the Hips a 2, r 2, and 8 m, 8 j. Fourthly, confider the Height of the Pitch, which let be equal to 6, 5 ; then the Lines 5 k, 2nd 5 .v, are the Lengths of a Pair of principal Rafters, the Angle 5 k 6, is the Angle or Mold for their Feet, and the Angle 6 5 for their Tops, On the Points 3 and 8, eredt the Perpendiculars 2, 3 ; 2, 4 ; and 8, 7; 8, 9. Draw the Lines a 3, r 4, m 7, s 9, which are the Lengths of the four Hip Rafters; the Angle 2 a 3, is the Angle or Mold for all their Feet, and the Angle a 32, for all their Tops, and which, with the Lengths of the principal Rafters being meafured on your Scale of equal Parts, will give you their true Lengths in Feet and Parts of Feet. This being done, make your Raifing equal to the Magni- tude of the Building, and brace its Angles, as n n, &c. which will be a very- great {Lengthening to them. Divide out the Diftances of the Beams, and cog them down on the Railings, as at c cl e f, which is a fecure Method to tie the Building together. Set out the Mortiles for the Cielhig-JoiJls in the Beams, fo that the under Surfaces of the Joifts may be flufli with the under Surfaces of the Beams, and obferve, that the Diltances of the deling - 7 oi J s be agreeable to the ufual Lengths of Laths, that no Waite be made thereby in the Lathing. The like Caution Ihould alfo be taken in the Dillance of Rafters, for very often the Tiler is injured very greatly in the Waite of his Laths. When the Lengths and Angles of the principal and Hip Rafters are thus difeovered by the Plan, we mult then confider the proper Scantlings for them, and for the Beams on which they Hand. When Beams exceed 20 Feet Extent, *tis always belt to trufs them up in one or more Places, as their Lengths may- re- quire. Beams Ihould never exceed 15 Feet in their Bearings, nor Rafters more than 10 Feet, and efpecially in Roofs of very low Pitch, whofe Covering has a much greater PrelTure on their Rafters, than thole of higher Pitches, and which may therefore in fome Cafes exceed 10 Feet. The Height or Pitch of a Roof fhouldbe agreeable to the Building it covers, and to the Kind of Materials it is to be covered with. T he Kinds of Covering in England are four, viz. Lead, Pantiles , Plain Piles, and Slates. Firfl, Coverings of Lead are, of all others, the molt beautiful, but the Expence being the greatelt, it is therefore never ufed, but for to cover mag- nificent Buildings. The Height -of Proofs, covered with Lead is at pleafure, but now ’tis generally ufed for Roofs that are very low, and which is commonly 2 Ninths of the Building’s Breadths, which is called Pediment Pitch. Secondly, Coverings of Pantiles may be alio ufed to low Roofs, but the general Pitch is 3 Eighths of the Building’s Breadth. Thirdly, Coverings of plain Tiles and Slates have generally the highell Pitch, on account, that when they are laid on low Roofs, the driving Rains will enter between them. The Pitch allowed forthefe Kinds of Coverings is that, whofe Rafter’s Length is equal to 3 Fourths of the Building’s Breadth, and which is called true Pitch. To form the Truffes for principal Rafters, we mud divide the Length of the Rafter into fome Number of equal Parts, each to contain about 10 Feet ; and at thofe Parts place fuch Collar.Beqms, Prick-Polls, and Struts, as are fufficient to fupportthem. In Plate LIII. are 15 Deiigns for the trufiing of principal Rafters, whofe Beams extend 15, 30, 45, 60, and 75 Feet, and whofe feveral Pitches are made agreeable to the aforefaid Coverings. Fig. Q_and R are Extents 15 Feet each, the fir it for Lead, the lafl for Pantiles, winch U 2 require »S 2 Of ARCHITECTURE. require no Help from Collar- Beams, iff c. but Fig. T, of the fame Extent, being higher, and confequently has longer Rafters, mull be helped by a Collar-Beam placed between them ; and for the fame Reafon, Fig. K, whofe Beam extends 30 Feet, mull have two Collar-Beams, whilft Fig. C and D, of the fame Extent, whofe Pitches are lower, and Rafters are Ihorter, will each do with one Collar - Beam. When the Extent of Beams is Ifuch, that the Length of Collar-Beams will be too great, which Ihould never exceed 15 Feet at the moft, the Weight of the Rafters and their Coverings mull be fupported by Prick-Polls and Struts, framed into King-Polls, by means of which the Beams will be truffed up fecure, and the whole Weight ftrongly fultained. For this Purpofe all the remaining Examples in this Plate, and thofe in Plate LIV. are given, which being in general confpicuous, requires very little more Explanation. In Plate LIV* the Figure E exhibits the Manner of framing the Fo^t of a principal Rafter into the End of a Beam, where a is a Part of the Rafter, f f, a Part of the Beam, and c d, the Tenon of the Rafter’s Foot in its Mortife. The Fig. C exhibits the upper Part of a King-Pofi, with its Joggle d d , into which e c 9 the upper Parts of two principal Rafters, are framed, whofe Shoulders b b muft be made truly fquare to the Joggle. The Fig. B exhibits the Manner of fram- ing the lower Parts of Struts, as h e , into the Joggle of a King-Poll, as at a b d , whofe Shoulders Ihould alfo be fquare to the Joggle, or as nearly fquare as polfi- ble ; n n is an Iron Strap, to bind the Beam g g unto the King-Poll B, which is bolted through the King-Poll at n n. As the common Method of framing the Trufles of principal Rafters of large Roofs, is to lay the whole Weight of the Beam and Covering upon their Feet, they therefore Ihould be fecured at the Beam with Iron Straps, to prevent their flying out, in cafe that their Tenons Ihould fail. According to this Method all the Trulles in Plate LIU. are made; but as I apprehend this Method was capable of Improvement, I therefore confidered, that if under the lower Parts of principal prn Rafters, there be difeharging Struts framed into the Beams and Prick-Polts, nsab t *h Fig. A, P late LIV. they will difeharge the principal Rafters from the greatefl Part of the whole Weight. The Trufs, Fig. F, hath its Struts turned the contrary Way to all the pre- ceding, and the whole Weight is taken off the Rafters, by the difeharging Struts e c and h g, for the whole Weight that hangs on the King-Poll is ful- tained by the Struts a d and b /, which are fuftained by the Prick-Polls c d and h f, which are fullained by the difeharging Struts c e and h g. In the fame Man- ner the Weights of the Trulles, Fig. G, M, R, P, S, and T, are dilcharged by their difeharging Struts, which are fhaded to diftinguifh them from the others. The Trulles HLN are for Buildings that have arched Cielings, which are tied in by their Hammer-Beams l i, in Fig. H,e k, and f i, in Fig. L, and di, and dg, in Fig. N, which mull be made very fecure by Straps and Bolts, as at k and r, in Fig. H. The Trulles G and I admit of Garrets. But the Top of Fig. I, which is called a Trunk Roof, muft be covered with Lead. The Trulles O Q^R and S are Trufles for M Roofs; thofe of O R and S are wholly fupported by their King-Polls and Struts, but that of Q^mult have its Gutter at a, fup- ported either with a Party-Wall, or truffed Partition, as Fig. K, whofe princi- pal Polls are a a, iffc. the Gutter Plate d d , iffc. and Struts c c. The Trufs, Fig. I), as alfo Fig. B, Plate LV. are for the Roofs of Churches, which are fuppofed to be fupported within-lide by Columns at l and c. The next and laft Kind of Roofing whofe Timbers are llraight, is that of Spires on the Towers of Country Churches, as Fig. G, Plate LVI. The Height or Pitch of Spires is from 4 to 5 of the Towers Diameter on which they Hand. And as the feveral Hips have an equal Inclination, they do therefore trufs up each other. The Bafe of a Spire is generally an Oftagon, whofe Manner of framing is exhibited by Fig. A, which if made of good Oak , and fecurely bolted down on the Heads of eight principal Polls, fixed in the Sides of the Tower, O/ ARCHITECTURE. t 5i Tower, will (land unto the End of Time, could the Materials endure fo long. The fecond Example, Fig. C, has its Spire placed on an Ogee Roof at e /, framed together as Fig. B, which is reprefented at large, and whofe Bafe g h is framed together as Fig. D. The third Example Fig. H, whofe Spire is placed on a Lanthorn, is fomething more difficult than the preceding, and therefore Fig. F is given to fhew the Manner of framing the Lanthorn, and Figure E the Kirb to the Lanthorn’s Head. As I have thus given a brief Explanation of thefe feveral Sorts of Trufies for ftraight Rafters, it will be neceffary to fay fomething of the Scantlings of Beams and Rafters before I proceed any further. If the Length of a Beam of Fir be I. Of the Scantlings for Feet. ' 3 ° " ' 60 Its Scantlings {hould be * 75 V 9 ° J l Indies, Feet. r 24 If the Rafter 36 be of Fir, J 41 and its Length 1 60 l 72 II. Of principal Rafters. Inches . Its fcantlings at Top fnould J 9 be III. Offmall Rafters . Feet. f 5 by 6 1 | | 6 I _ 1 [ and at its j 0 9 7 | [ Bottom j IO 7* l 10 9 J 1 j Inches. 7 by 6 Inches. If the Length of the Rafter be Its Scantlings fhould be Circular Roofs are the next that come under our Confideration, which arc Fir fly Cylindrical, as Fig . A, Plate LV. Secondly, Spherical, as Fig. G and N. Thirdly, Spheroidical, as Fig. D, which two fait are vulgarly called Domes. Fourthly , Trumpet-mouth’d, as Fig. C A. Fifthly , Bell Roofs, as Fig. I K* Sixthly , Bottle or Ogee Roofs, as Fig. M. And Lajlly , Compound Roofs, as Fig. C and L. And as by lnfpe$ion ’tis plain, that thefe Roofs in general have their Trufies formed by the fame Principles as the preceding, I need only add, that Fig. F is the Plan of a Spheroidical Dome whole feveral Trufies are con- nected together at their Tops, by the horizontal Braces, ab c d % on which the Lanthorn D is ere&ed. Fig. H is a half Plan of the fpherical Roof or Dome, Fig. G, whofe Pur- loins c f d, and e h g i k, are reprefented by the concentrick Semi-circles 5348, and 6127, and the Bafe of each Trufs by the central Lines q tv , r z, s x, t a, and y v. The feveral Ribs, or principal trufled Rafters, mull diminilh as their Bafes a t , xj-, itfe. and may either be framed into a horizontal Kirb at Top, as d e, and a f, f c> are each equal to s L*he Length ofa Hip? Rafter. Continue the central Lines of the Beams zj’ apd p p to /and x, and to k and iv, making k z, l m } y £ / and / i, alfo c no, iv x and x d : This being done, draw in fuch other principal Raf- ters as are requifite, and between them the Purloins, as 8, 9, 6, 5, 7, tsfe. at Difcretion, obferving not to place any two Purloins directly oppoiite, whofe two Mortifes would weaken the principal very much. Laftly, between the principal Rafters draw in the fmall Rafters, and then the Lengths and Angles of every par- ticular Part of the whole Roof will be determined, and from which a juft Efti- mate of the Quantity of Timber that will be employed therein (Regard being had to the Dimenfions or Scantlings of the feveral Parts as aforefaid) may be made. In Fig. VI. the Angle OPR being equal to the Angle op r i n Fig. IV. therefore the Angle at P is the Bevel of the Feet of the principal Rafters, as the Angle at O, for the fame Reafon, is the Bevel of their Tops ; and the Angle S B R, Fig. V. being equal to the Angle s h r in Fig. IV. therefore the Angle at B is the Be- vel of the Feet of the Hip-Rafters, and S is the Bevel of their Tops. The Fi<*. A B ,on the Left-hand exhibits a Joint made by a Purloin and a Hip, as by a and the Purloin 12, 14, the Meafure of whofe Angle is the Arch 13, 15. Fig. VII. reprefents a Pair of principal Rafters trufted up, on whofe Prick-Pofts is. placed a Cupola, as efig h. The next in Order is, to find the Angles of the Jack-Rafters againft the Hips, and to back the Hip-Rafters. As Jack- Rafters are parallel to one another, therefore all their Angles againft the Hips are the fame. Tq make the F.nd of a fach- Rafter ft to the upright Side of a Hip- Rafter. • There are two Angles to be formed, that is, the one upon the upper Surface of the Jack-Rafter, the other on its Sides from the Ends of the former. The Angle on its upper Surface is the Angle made by the upper Edges of the Jack and Hip ; and which is that, that every Jack- Rafter makes with the Hip-Rafter in the Ledge- ment, as every of the Angles between e and d. Therefore from your drawing in Ledgement, let your Bevel to one of thofe Angles, and the feveral Jack-Rafters being cut to their relpeftive Lengths, at their upper Ends on their upper Surfaces, apply that Bevel, and deferibe the upper Angles. This done, take the Mold S, made for the Tops of the principal Rafters, and apply it againft the Sides of each Jach,- Rafter, at the Ends of the Angle on their upper Surfaces, and by its upper Edge draw Lines ; then from the Line of the upper Angle, through the Lines on the Sides, faw through the Rafter, and that Cut will be the Angle required. To fnd the Single of the Back of a Hip- Rafter. From the Point c let fall a Perpendicular, as chy on the Hip fa\ make e gy equal to c h ; alfo make ai equal to a c ; draw the Lines eg and g i t and the Angle eg i will be the Angle or Back of the Hip required. Example II. Fig. V. Plate LIX. This fecond Example is of a regular double Roof, which is hip’d as the pre- ceding, with Valleys within-fide. The Outlines of this Plan are a f g k, wherein h B, B E,Ei and i h, are the Ridges, i k and h g are the Hips, h A C, B A C, D A E and DAi are the Valleys, A C, D A the Gutter, f q the Height of the Pitch, p q and q r a Pair of principal Rafters, ■y/'and t k Hip-Rafters. By the laft Example, lay out the Endsy c i h , and a B hg y alfo the Skirts a b e f and ^AH; continue^ H t0 f) & I to d, a b to c } and fe to d, and becaufe the Lengths of the Valleys are equal Of ARCHITECTURE. 155 ttjual to the Lengths of the Hips, therefore make H c, I d, be, and d e , each equal to one of the Hips, as I k, and draw the Lines c d and c d: this being done, draw in all the principal and imall Rafters at Difcretion, and then the whole will be completed, as required* Example III. Plate LX. This third Example is of an irregular double Roof, whofe Ends are hip’d, and whofe Plan is tpzzy z, wherein r s, s o, o I, l m, m v and v r are its Ridges, t s, p 0 , 21, z in, vy and r x are its Hips, r q, s q and 9 » & c ' the Line kf, equal to the Lines 1, 7; 2, 8 ; 3,0, hoc. on the Line k a ; and hom j, through the 'Points 13, 12, 11, Iffc. to h, trace the Curve//). In the fame Manner trace the Curve f g. Then the Piece Jg h, being bended up, and laid on the two Flips that Hand over the Line g a and a 58 Of ARCHITECTURE. and h a, will be the Covering for that Side of the R.oof or Niche, as re- quired. Note , The Coverings to the two Oge 30 Feet ; that of Fig. I. to 45 Feet ; and that of Fig. V. to 60 Feet. The Fig. II. is a Section of the feveral Profiles, vvhofe Breadth is equal to 50 Feet. The Piles that fupport the Trulfes of thefe feveral Defigns are fuppofed to rife a fufficient Height above the Flowing of the Water, fo that the Joints in the feveral Trulfes eredted thereon may not be affefled thereby; and when the Depth of a River is fo great, that the Length of Piles above the Bed of the River mull exceed, wdien driven, 25 or 30 Feet, then Super-Piles mull be eredled upon horizontal Beams, mortifed down upon the Heads of the lower Piles, as in every of thefe Examples. The Scantlings proper for Piles to fuch Bridges Ihould not be Iefs than one Foot in Diameter, at the Middle of their Lengths. The Fig. III. reprefents Part of the Plan, with the Bale of two Trulfes, a and b, whofe Diftances in the Clear Ihould not exceed 10 Feet ; be- caufe on them the Joifts which carry the Floor of the Bridge are laid. The under Piles mull be fnod with Iron, that they may the better penetrate through the feveral Stratums of Earth, into which they are to be driven. Before Piles are driven, the whole Weight of the Framing that is to come on them, and the Weight of the Planking on the Joifts, Clay, Gravel, Pavement, &V. Ihould be eflimated nearly to the Truth ; otherwife the Piles cannot be driven with any Certainty, and which is thus to be performed, viz. Divide the total Weight to be fuftaiired, by the necelfary Number of Piles, and the Quotient will be the Weight that each Pile is to fupport. Then each Pile being driven until it re- fill a Force much greater than the Weight it is to fupport; it may be depended on, that afterwards there cannot be any Settlement by the Weight it is to fuftain. The Scantlings for the Beams of Trulfes Ihould be about 12 Inches by 9 Inches, as alfo Ihould be the feveral King-Polls. But the Struts and Joifts need not exceed 9 by 6 Inches, and the Plank on the Joifts being made 3 Inches in Thicknefs, will be fufficient. Before the Timbers are worked (which is fup- pofed to be of the bell Oak), ’tis belt to cut them out to their Scantlings, and lay them in a running Water for a Month at the leaft, to foak out the Sap, which is very deftrutlive, and then dry them thoroughly over a Saw-dull Heat, &s. before they are worked. If this be carefully done, and the Work kept dry vvhilft working, and being truly framed, there will be no fagging in the Work, as ufually happens by the fhrinking of the Timbers, when they are not thus fhrinked before working ; nay, 1 have experienced, that Timbers fo prepared have always fwelled afterwards, and made 1 the Joints much clofer than when firft put together. It is alfo advifable, for the better preferving of the Te- nons, that every Mortile and Tenon be well covered over with a good Body of White Lead, and boiled Linfeed Oil, which will endure along Time, and will not permit any Rains to enter the Mortifes; to the Prejudice of the Tenons. The Ends of the Joiils Ihould alfo be covered with brown Paper, dipped in Pitch, and Sheet Lead laid over the Paper. And for the more efteftual preferving of the Plank and Joifts, the Plank ought to be covered with a ftrong Clay, firmly rammed down unto about 9 Inches in Depth, on which the Road of Gravel and Pavement, or Gravel only, of a fufficient Thicknefs Is to be laid, with a Rifingj jn the Middle; to difeharge hafty Rains to the Sides, as exhibited by B, m Fig. I; Plate LXII. In Plate LXII. are two other Defigns, each of 100 Feet Opening; which I. made for the New Bridge at Wejlminjler ; but believing that Inferell was pre- dominant to real Merit, 1 therefore declined to trouble the Honourable Comm if- X 2 feonerS 160 Of ARCHITECTURE. fioners therewith, as I have now the Public, in hopes that they may be of form? Help to Invention, if not worthy of being put into Practice, over Rivers, where large Openings are required. The Delign, Fig. II. is of prodigious Strength, as being a double Trufs, and whofe Timbers are fo iixed together, that not any Part of the whole can fag the hundredth Part of an Inch, they being prepared, before worked, as afore- faid. Fig. I. is a Sedfion of the Breadth of the Bridge, wherein A A, &c. repre- fents the feveral T ruffes, for the Support of the Joifts and Roads. A and C reprefent the Foot-ways, each io Feet in Breadth ; and B, the Horfe-way, 30 Feet in Breadth. As the Offices ol the Struts a d e t b l c k i, &c. are obvious to every difeerning Eye, I need not fay any Thing thereof. The Fig. V. contains a double D.efign, the Struts on the Side G being dif- ferent from thofe on the Side II. Both thefe Defigns are of immenfe Strength ; and as the whole is laid on Stone or Brick Piers, which rife above the Flowing of the higheft Tide, a Bridge of this Kind will be of very great Duration. As there isfome Difficulty to lay Foundations for Stone Piers in Rivers that are affedted by Tides, and as in wooden Bridges the moft early Decay is in that Part of the Piles that are affetted by the riling and falling Waters of the Tides, there- fore to avoid both thefe Inconveniences, fucli Piers may be thus erected, viz. Con- fider the Weight of a Pier, and the Weight that the Pier is to carry. Af- fign the Place in the River where the Pier is to Hand : bore the Ground for 15 or 20 Feet in Depth, that a Judgment may be formed, how long the Piles nnfftbe. This done, drive a Range of Piles, dove-tailed together, at about 15 Inches, without the Upright that the Stone Pier is to be erefted, all round the Limits of the Pier, and the like exactly under the Upright of the Pier. Thefe two Ranges of Piles form within the Ground a llrong Enclofure, about the en- compaffed Earth on which the Pier is to ftand. Within the Limits enclofed drive as many Piles as fhall be thought fufficient to cany the Weight, and which fliould be driven nearly all equally ; that is, Firft, to drive them all to fuch a Depth, as to keep them upright in their Places. Secondly, to drive them all about 2 Feet lower, and then all two Feet lower again ; and fo on, until each Pile be firmly driven, as aforefaid. By this regular driving down all the Piles together, they will caufe the enclofed Earth into which they are driven to be equally compreffed, and of much greater Compaftnefs than it was before, as being confined by the double Ranges of Piles firft driven. When all the Piles are thus driven, their Heads mult be fa wed level, at about 18 Inches be- low the Surface of the low Water ; and to render them imperifliable, the whole mull be filled up with ftrong Clay, let down in large fquare Pieces, worked very ftiff, and well rammed, which is a Work eafy to be performed, although the Depth of Water fliould be 20 Feet. When this is done, prepare a double Floor of Oak Timbers, free from Sap, each Floor about 10 Incites in Thicknefs, pinned down one on the other, fo that the upper Timbers lie at right Angles acrofs the lower. Fix this Floor on the Piles, and thereon eredf the Stone-work, to any Height required. The next Work is to fill up the Space between the outer Range of Dove-tailed Piles, and the next inner Piles, to pre- ferve the inner Range from being injured by the Flux and Reflux of the Tide; and which being firmly performed, the whole Foundation will be rendered as imperifhable, as were all the Piles driven iuto the very Bed of the River, as be- ing fccured from the Aftions of both Air and Water. The outward Range of Dove -tailed Piles are all that are liable to decay; and as their Office is no more than to fupport the outward Cafe of Clay, which is there placed to preferve the next inner Range of Piles, they are eafily and foon repaired, as their Decays occur. Note, The outer Range of Piles muft be made of fuch a Length, as to rife fomething above the Level of High-Water; and horizontal Beams being mortifed dewa on their Heads, with horizontal Ties laid through the Thicknefs of the Pier Of ARCHITECTURE. i6t ^icr In frriall Arches turned for that Purpofe, being cogged down on the Beams, they will be a lading Prefervative and Defence to the Piers, againft all the Infults of tempeftuous Weather and Navigation that can happen. Note, If the Depth of Low-Water be any Thing confiderable, it will be a very fecure Way to drive a Range of oblique Piles, juft within the Limits of the up- light Piles, as Braces, to fteady the next within, from inclining either Wav bv the Weight of the Pier, If inftead of Timber Trufles, ’tis required to make Arches of Stone, a fuf- ficient Number of Piles mull be added within every Pier, that, with the others, will be capable to carry the additional Weight of the Arches. Note alfo , That Piers built with well burnt Bricks, laid in Terrace, on a Bafe- ment of large Blocks of Stone, about 3 Feet in Height, will be much cheaper than being made entirely of Stone, and of longer Duration : For well burnt Bricks do not decay fo faft as Portland Stone, which is very evident by St. Paul’s Cathedral, where the Stone, in many Paris of the South Side, is already decayed more than the icth Part of an Inch. LECTURE XXVII. Of Brick and Stone Arches to Windows , Doors , See. I. Of Jl ralght, circular , elliptical, Gothlck and rampant Arches in flraipht Walls - ' Plate LXIII. I N this Plate are exhibited 13 Kinds of Arches, of which Fig. I. II. HI. JV- V. VII. VIII. and IX. are Arches of Brick Work, and the others of rufticated Stones. In Fig. I. and III. the Diftance of the Center, to which all the Joints have their Sommering, is equal to the Breadth of the Window ; but thofc of dig. II. and IV. i$ the Center of a geometrical Square, whofe Side is equal to their Breadth. Fig. V. is a femi-circular Arch, whofe Joints fominerto its Center. Fig. VII. and IX. are femi-elliptical Arches, the firft on the conjugate Diameter, and thelaftonthe tranfverfe Diameter. The Courfes in Fi%. Vllfare divided on the inner Curve h f m , and outer Curve a e n, into the fame Number of equal Parts, as alfo is the right-hand Side of Fig. IX. whofe left-hand Side has its Courfes fommering to c and / the Centers of the Ellipfis. Fig. VIII. is a Gothlck Arch, whofe Courfes have the fame Sommerings asthofeof Ff. IX. In all thefe Cafes the only Thing to be obferved is, that the NumberW Courfes into which each is divided be an odd Number, that thereby the Middle Courfe maybe perpendicular, and that the Breadth of each Courfe on the upper Part of the Arch be fomething lefs than the Thicknefs of a Brick, to allow for rubbimr. The rufticated Arches, Fig. VI. X. XL and XII. have the fame Sommerinrf as thofe of Fig, V. VII. VIII. and IX. b Fo divide their Key Stones and Rn flicks. Divide each half Arch into 9 equal Parts, as in Fig. V. give 1 to half the Key Stone, the next 1 g to its Counter Key, and 2 to each Ruftick and Interval, as the Figures exprefs. The like is alfo to be oblerved in all the other Arches. Fhe Arch, Fig. XIII. is a rampant Semi-circle, whofe Curvature maybe de- fciibed by Prob. XIX. Lect. IV. Part II. or as following. Let f h be the Breadth, rniAfg the Height of the Ramp; draw g h, and in the Middle of fh ered the Perpendicular q a, of Length at pleafure ; alfo draw the Line q r parallel t° /"• From the Point of Interfe&ion made by the Lines g h and/ h, fet up half the Breadth of the Opening to a, arid draw the Lines a g and a h. Biftd^ a in m, and a h in 0, and ered the Perpendiculars m n and op; then 'the Point n is the Center of the Arch g d, and p is the Center of the Arch dh, which divide in- to Rufticks, as in Pig. VI. "1 hen the Length of the Rufticks mult be equal to £ ot the Opening, and of the Intervals to $ of the Ruftick, as exhibited by k l \ Pig. VI. 7 JF °f fl raight, circular and elliptical Arches in circular Walls', Plate LXIV. Ihe firft Work to be done is the making of the Centers to turn thefe Kinds of Arches upon, which may be thus performed. Let GHIKbe the Plan of a pifculaf 162 Of ARCHITECTURE. circular Building, and at Fig. VI. ’tis required to make a Centef for a fef&i-fcnrctl* lar Arch to the Window, whofe Diameter without is a d, and within n m. Bife& a d in /, and defcribe the Semi-circle a p d. Divide a d into any Number of equal Parts at the Points 642, fee. and draw the Ordinates 6, 6 ; 4, 4 ; 2, 2 ; &c. Divide n ;n into the fame Number of equal Parts, and make the Ordinates 6, 5 ; 4, 3 ; 2, 1, feV. equal to the Ordinates 6, 6; 4, 4; 2, 2, fee; hnd through the Points 5 3 1 h, fee. trace the Curve n k m, then ap d and n l m will be the two Ribs for the Center : This being done, place the Ribs perpendicular over the Lines a d and n m, and cover them, as Centers ufually are, and then applying the Edge of a Plumb-rule to the divers Parts of the Infide and Outfide of the Window’s Bottom, the Top of the Rule will give the feveral Points at which the Infide and Outiide of the Covering is to be cut off, fo as to ftand exactly over the Infide and Outfide of the Building, and then the Center will be completed as required. T 0 divide the Courfes in the Arch of this Window. On a flat Pannel, fee. draw a Line, as b e, Fig. VII. make a f 0 equal to the Curve ac d, alfo make a l and 0 e each equal to the intended Height of the Brick Arch. Make fp in Fig. VII. equal to e p in Fig. VI. alfo make a b and d e in Fig. VI. each equal to b a in Fig. VII. then the Points b and e will be the Ex- tremes of the Arch. Make p r in Fig. VII. equal to b a the given Height of the Arch, and through the Points b re and ap 0 defcribe two Semi-ellipfes, which divide into Courfes as before taught, and which will be the Face of the Arch required. To find the Angles or Bevels of the Under-part of each Courfe. Continue the Splay-Backs of the Window m d and n a until they meet in F. On F, with the Radius F n and F a, defcribe the Arches n y v and a f s y mak-ng ny v equal to the Girt of the Arch n k m. Make n 6, n 4, n 2, n y, fee. on the Arch n y t>, equal to n 6, n 4, n 2, n y, fe c, on the Curve n k m, and draw the Lines 6 F, 4 F, 2 F, y F, fee. make the Ordinates 6, 5 ; 4, 3 ; 2, I ; y *, feV. on the Lines 6 F, 4 F, fee. equal to the Ordinates y, 6 ; 3 > 4 > L 2 } ^ r, &c. on the Line n m, and through the Points 5, 3, 1, x, fee. trace the Curve v x n. . In the fame Manner transfer the Ordinates 5, 6 ; 3, 4 ; 1, 2 ; c,f, fee. on the Line a d to the Arch sfa y as from 5 to 6, from 4 to 3, fee. and trace the Curve sea-, and then will the Figure n y v sea be the Soffito of the Window laid out, and which being divided into the fame Number of equal Parts, as the under Part of the Arch ap 0, Fig. VII. and Lines drawn to the Center F, as is done in Fig. II. to the Center A, by* the Lines 2, 2, 2, fe c. thofe Lines will give the Bevel of every Courfe in Soffito, as required. Fig.^J. is another Example of a femi-elliptical Arch, whofe Fiont is Fig. IV. Alfo Fig. II. is a third Example of a Scheme Arch, whofe Front is Figi I. And Fig. VIII. is a fourth Example of a llraight Arch, which in gent'* ral are performed by the aforefaid Rule. To find the Curvature of every Courfe in Front. Suppose the rufticated femi- circular-headed Window, Fig. IX.be Handing in the Side of a Cylinder, whofe Sides are the Lines QJT and P V, continue out the Sides of each Ruftick until they cut the Sides of the Cylinder in the Points QRST and NOP, fee. then the Lines QN, R O, Q_N, feV. will be tranf- verfe Diameters of fo many Ellipfes, whofe conjugate Diameters are each equal to the Diameter of the Cylinder, which defcribe as in Fig. X. and draw their con- jugate Diameters k /, i m and n 0 ; make the Diftances 0 ni 3, / 1, on each Ellip- fist equal to a ^ the Semi-diameter of the Window, Fig. IX. alfo make the Diftances. 5 6, 3 4, 1 2, on each Ellipfis, equal to g 10 the Height of the ruftick Arch ; then the Segments of the feveral Ellipfes, 5,6 ; 3, 4; 1,2; at Z X A, w$l be the Curves of the feveral Courfes, . as required. ' Fur. III. reprefents the Manner of covering the Outfide of a Cone, the Arch c a being made equal to the Circumference of the Circle e, which is equal to the Bafe of tire Cone : This Figure is exhibited here to ftievv, that the Soffito of a femi-ciFCular-headed Window, whofe Splay is continued all round, is no more than Of ARCHITECTURE. 163 than the lower Superficies of a Semi-cone ; for if the Splay was continued in every Part, it would meet in a Point, as the Lines k d h and i e h, Fig. VIII. and form a Semi-cone as aforefaid. This is i Hull rated by F'ig.'V. Plate LXVII. where s v w reprefents the Sec- tion of a Wall, in which is placed a circular Window, as Fig. A, whofe Splay is expreffed by a c and f h : Now, if c a and h f be continued, they will meet in i t on which, with the Radius i c, defcribe the Arch c /, alfo the Arch b m. Make the Length of the Curve cl equal to the Circumference of dp, the outer Circle of the Splay, and draw the Line l m : then the (haded Figure k being bent about and fixed within the Splay, it will exactly fit every Part thereof : But as the bend- ing of Stuff of any confiderable Thicknefs is impracticable, tnerefore divide the whole into Parts, as at i, 2, 3,4, 5>6, cF c. which glew, or otherwife fix together* equal to the Curvature of the Window, at pleafure. _ Fig. XI. exhibits the ancient Manner of making ftraight Arches of Stone, in Places where no Abutments can be had, whofe Vouffoirs are joggled together, and their fpreading prevented by Iron Bars toothed into the Head of each, run in with Lead, as at e c e, and c. LECTURE XXVIII. Of Centering to Arches and Groins , Plate L X I \ . T O defcribe the Curvatures of Groins is the chief Thing to be done in Works of this Kind, which is moll eafily performed, as follows : Example I. Fig. A. Let a c e f be a fquare Plan, whofe Vault is to be interfered by two concave Semi-cylinders. Defcribe the Semi-circle a b c, which divide into Ordinates, as 1, 2, 3, &c. Draw the Diagonal a 5 > which continue until they meet de the Side of the Bafe of the fmall Arch, and from thofe Points draw Lines perpendi- cular thereto, of Length at pleaiure. On d i, the given Breadth of the fmall Arch, defcribe the interfering Curve of the fmall Vault of any Kind, as re- quired, as a h i ; divide the Bafe of one Groin, as e i, into die fame Numbei of equal Parts as d x, the 4 Breadth, and erer Ordinates thereon, equal to the Ordinates on d x, and through their Extremes trace the Curve if, which is the Curve of that Groin required. By the fame Rule all other Kinds of interfering Arches may be found, although they cut the llraight Vault on any oblique Angle inilead of a right Angle, the Bafe of the fiiorter and of the longer Groin being divided into the fame Number of equal Parts, and the Oidinates in each being made refpey?, the Height of the * °° r ’ £ Wh , lch 7 * afcend - the Rife, and Number of Steps that are neceffary for the Height. . Thirdly , To divide the Number of Steps by fuch ia f Spaces (or breathing Races) that are necelTary for repofing on the Way, ,Z h} '\ l 5 at / ^ S i paCe , ab0VC tho H ead, commonly called Head Way, be fpa- p> -I 8 ;* 11 j ^ ol . at tke Breadth of the Afcent be proportionable to the whole Building and fufficient for the Purpofe intended ; fo as to avoid Encounters by ■ ei /? n8 i a ; cendin S' a T nd defcending at the fame Time. The Height of Steps fhould S WI* 5 l In f ie8 ’ n ° r niore than 7 Inches, except in fuch Cafes where T u 1 ^ ° [ g es a higher Rife; The Breadth of Steps fhould not be lefsthan 10 Inches, nor more than 15 or 16, although fome allow 18 Inches, which I think is 00 much. The Light to a Stair-cafe fhould always be liberal, to avoid Slips, Falls, rf, and which may proceed from the Sides, from a Cupola or Sky-Light at the i op ; as the Situation will belt admit, Before this Kind of Work is begun, ’tis belt to make a Plan, and to lay oilt the whole in Ledgement, as follows, E/t t,o, 9, II, Fig. D G, Plate LXVL be a given Plan. , ;Y7* E dst £ ual to ' k Breadth of the Afcent, which may be made from 2 Feet TV • V to . l ? V eet ‘ P ravv db,b a, and a m, parallel to the Outlines of the Plan, Hivicie d b, b «, and a m , each into fuch a Number of Steps, whofe fevera! Hugh mare equal to the whole Height to be afeended; within the Parallelogram aim d> draw the Thicknefs of the Hand Rail. Add into one Sum the Heights the feveral Steps, between b and d, and at that Diftanee, draw q r, parallel to 6 S ;braW Hypothenufal Line and continue out the Plan of each Step to meet the Line rs at s; fet up the Pleight of the firft Step, and draw it pa- raUe 1 to 0 s, until it meet the Bafe Line of the 2d Step ; then fet up the Height of the 2d Step, and draw it parallel to * .r ; proceed in like Manner to fet up the .Heights of all the remaining Steps unto r • make op equal to 0 q, and draw 2 p paiallel to to; at the Point 2, begin to fet -up the Steps unto the Point I, and draw v I parallel to t o : make / w equal to t v, and draw c. I he Plan of the Stairs being divided, continue out the Diameter d a, towards the Left- hand, as to/, of Length at pleafure. Make a f equal to the Gut of the Semi- circle b d, which divide into the fame Number of equal Parts as there are Stairs in the Plan of the Semi-circle a b d, as at the Points 1 2 3 4, Or. from which erect Perpendiculars, as 1 a , 2 a, 3 a, Cc. of Length at pF a -iue. Confider the Rife of a Stair, and make the Perpendicular / £, equal to the Kile of all the 12 Stairs that go round the Semi-circle a bd; and divide the ler- pendicular / g into 12 equal Parts, as at the Points X 2 3 4, GV. from which draw Lines parallel to / d, continued out towards the Right-hand, atpleafure, which will interfedt the Perpendiculars 011 the Line / a d, in the Points a c, a c, a c, &c. and which are the Breadths and Heights of the i reads and ki.es of the 12 Stairs, at the Side of the Semi-cylinder a b d-, for was the whole Figure g f a applied about the Semi-cylinder, then the 1 arts a c, a c, U. would be in the refpeaive Place of each Stair. Let a e reprefent the Breadth of the Hand-rail, and the Semi-circle e 10 c it? Bafe, over which its Inhde is to Hand. Divide its Diameter e c into any Number of equai Parts as aL 1234, &c. and draw the Ordinates 1,6; 2, 7 ; 3, 8 » 4 » 9 » whm con- tinue 3 upwards, fo as to meet the horizontal Lines drawn from the Perpendicu ar gf, in the Points 28, 27, 26, 25, fcf*. through which trace the Ogee Curve 28, 14, it, which is the Sedional Line of the Cylinder, over which it Hand- Make the Diftances 15, 21 ; 19, 14? 18, 13 ; 17, ^ ; and 16, n, equal to the Ordinates 10, 5 ; 9, 4 ; 8, 3 ; 7, 2 ; and 6, 1 ; and through the Points 20, 10. 18, 17, 16, to a, on the Line / d, trace the Curve, 20,16, a, which Mtfae infide Curve of the Mold, and whofe Out-curve 21 a, being made con- ' Y 3 centric i68 0/ ARCHITECTURE, centric thereto, will be the Mold required, whofe End 21, 20, when fet up in it$ Place, will (land perpendicular over its Bafe b io. Note, This Mold, though made but for one 4th Part of the Cylinder, will fervc for the whole, by repeating the fame, or adding three or more others of the fame Kind, tp the Ends of each other as often as there are Revolutions in the Cylinder. Fig. VIII. is the Plan of an Elliptical Stair-cafe, whofe Mold \ k is defcribed in the fame Manner, and therefore needs no other Defcription. LECTURE XXX. Of Compartments for Ivlonumental Infcriptions and Shields ; alfo divers Ornaments for Buildings and Gardens. A S in the preceding LeCtures I have explained the principal Parts of Buildings, I ftiall now conclude this Part with fome particular Ornaments, which are in common Ufe, and which are as neceilary for the Enrichment of Drawings, as of Buildings themfelves. In Plates LXIX. and LXX. are contained fourteen Defigns of Compartments, for Monumental Infcriptions, Coats of Arms, to be placed in open Pediments, (He. In Plate LXXI. are contained, firft, eleven Kinds of Vafes, as A B C D E F GHI KL, for the Enrichment of Piers to Gates, Parapet Walls, &c. as alfp are the Balls P and Pine-apple R. The Figures M O S are Defigns for Flower-pots, which are to be employed as Ornaments, in fuch Places where Vafes will be too large. As the principal Parts of thefe Ornaments are proportioned by equal Parts, as exprefled in divers Places between them, the young Student wilf fee how eafy it is to make them to any given Height. The Fig. W Y, A B, A C, have their principal Parts determined by equal Parts alfo. Figures W and Y are Defigns for Chriftening Fonts ; and A B, A C, for Pedeftals to horizontal Dials; and indeed, when horizontal Dials are very large, the Figures W and Y may be employed to their Pedeftals, Fig. X is a Kind of Pedeftal, called a Terme, from Terminus, the God of Bounds or Land-marks, who being anciently made handing ip a Sheath, thefe Kinds of Pedeftals were taken for the Support of Buftos, and are thus pro- portioned to any given Height. Divide the given Height into 10 equal Parts; give the upper 1 to the Height of the upper Aftragal, Fillet, and Cavetto ; and the lower 1 to the Height of the Plinth, Fillet and inverted Cima, The Projec- tion of the great Aftragal is two Parts on each Side the central Line, and of the imall Aftragal in the Bafe, one Part on each Side, from which the other Moldings take their Projections, as common in Columns. • To Jluie thefe Tcdejlals. Divide the Breadth into twenty r one' equal Parts, give one to each Fillet, and three to each Flute. The Fig. N reprefents a Harpy , a fictitious Monfter, faid to have the Plead of a Maiden, and Body oi a Bird ; and if fuch are made in Stone or Metal, having the Bodies of T urtle-doves, Owls, and Magpies, they vyill be pretty Emblems of the Innocency, Wifdom, and babbling Nonfenfe of Women. The Figures Z, A D, A E, and A F, reprefent the Monfter called Sphinx , whofe Head and Breaft are like thofe of a Woman, its Voice like a Man’s, its Body like a Lion’s, and Wings as aBird ; but fometimes their Wings are omitted, as Fig. A D and A E. The Figures T and Y are two Kinds of Obelifks, for Lamp-polls, (He. the one fquare, the other oCtangular ; and Fig. A G is the Deiign of a Shell, for to enrich the Head of a Niche, (He. ¥ PART i6§ PAR T IV. Of the Mensuration of Supei-jicies and Solids . A S the Toot is the Standard Meafure of moft Nations, I fhall therefore pre- fix to the following Rules a Table of Foreign Feet, carefully compared with the Englifh Foot, wherein ’tis fuppofed, that the Englfh Foot is di- vided into 10 po equal Parts, as alfo into 12 Inches, and each Inch into 10 equal Parts, Engl. Feet. Decim. F.Inc.ioths, Englifh Foot, Paris, the Royal Foot. Paris Foot, by Dr, Bernard. Amfterdam Foot. Antwerp Foot. Leyden Foot. Strafburg Foot. Frankfort ad Mien am Foot. Spanifh Foot. Venice Foot, Dantzick Foot. Copenhagen Foot, Prague Foot. Roman Foot. Old Roman Foot , Greek Foot. China Cubit. Cairo Cubit. { Babylonian Cubit. Greek Cubit. Roman Cubit . Turkilh Pike. Perfian Arafh. 1.000 1,068 1,066 ,942 ,946 *>°33 ,920 ,948 1.001 1,162 >944 >965 1,026 >967 >970 1,007 1,016 1,824 2,200 3>*97 12 00 11 11 00 1 1 11 00 01 1 1 11 00 11 00 00 09 °6 06 o 5 3 3 3 o 6 o 9 3 6 3 6 x 2 9 14. To’ff t'A °5 tVA 02 4. 02 3 LECTURE I. A Of Rules for meafuring the Superfcies of geometrical Figures , Plate LXXII. Rule I. To meafure any plain Triangle , Fig. ABC I). I : half the Bale c d or h i. Fig. A or B : : b d or g i, the Perpendicular, the whole Bafe m s, or z y , Fig. C or D the : the Area ; or as 1 Perpendicular : Area. To fnd the Area of any plain Triangle , having the Sides only given. Add the three Sides together ; from the half Sum fubtraft each Side feverally, and note their Differences. Multiply any two of the Differences together, and their Product by the other Difference. Multiply the lafl Product by the half Sum of 3 Sides, and the fquare Root of their Product is the Area required. Rule II. To meafure a geometrical Square, or Parallelogram, as the Figures E F. As 1 : c d the Length : : a c the Breadth : Area. Rule HI. To meafure a Rhombus or Rhomboides, as the Figures G and H. As 1 : a d, equal tp c e the Length : : b c the Perpendicular Height : Area. Rule IV. To meafure a Trapezoid, as Fig. I. As 1 : {-the Bafe_/'e, : : the perpendicular Height b f : Area. Rule V. To meafure a Trapezia , as Fig. K. As 1 : a Diagonal, as b g : : half the Sum of the 2 Perpendiculars a d and c e ; Area. Ruj *7° Of Mensuration of Superficies and Solids. Rule VI. To meafure any Polygon , as the Hexagon L. As i : | the Circumference the Diameter e g, equal to h ax Area. Rule VII. To meafure any irregular right-lined Figure , as Fig. M. _ Divide the Figure into Trapeziums, as d e, e f, c d, b e, and the Triangle b a e, whofe Areas find by Rules I. and V. and their Areas added together is the Area required. Rule VIII. To find the Length of an Arch of any Circle , as a c d, Fig. S. Divide the Chord Line into 4 equal Parts, make the Chord Line of a b equal to 1 Part, then h d is nearly equal to half the Arch Line required : Or thus Arithmetically : Multiply a c, the Chord of half the Arch, by 8 ; from the Pro- duft fubtraft a d. Divide the Remains by 3, and the Quotient will be equal to the Length of the Arch Line a c d required. Or thus : from the Chords a c and c d , fubtradl the Chord a d. Divide the Remains by 3, and then the Quo- tient added to the Chord Lines a c and c d , the Sum will be nearly equal to the Arch Line a c. d, required. Rule IX. To meafure a Quadrant, as b c e, Fig. O, As 1 : f the Arch c e, : : a Side, as be: Area, Rule X. To meafure a Semi-circle, as a d c, Fig. O, As 1 : \ the Arch ad c, : : the Diameter a c : Area, Rule XI. The Diameter of a Circle being given, to find its Circumference. As 7 : 22, : : the given Diameter : Circumference required. Or, as H3 ; 255, : : the given Diameter : Circumference required. Or, as I : 3,141593, : : the given Diameter to the Circumference required. Or, as 1 ,00000, 00000, 00000 , oooqq, 00000, 00000,00000 : is to 3* ■ 4*59>^5 35 >^ 9793 > 23846,26433,83279,50288, fo is the Diameter given, to the Circumference re- quired. _ Rule XII. The Circumference of a Circle being given, to find its Diameter. As 22 : 7, : : the Circumference given : Diameter required. Or, as 355 ; xi 3, : : the Circumference : Diameter. Or, as 3 > ! 4 l 593 : 1 • • t ^ ie Circum- ference to the Diameter. _ Rule XIII. The Diameter of a Circle being given , as a c, Fig. N, to find its Area. I. By Van Culen’j Analogy. As 1 : ,7854, : : the Square of the Diameter : Area. II. By MetiuS’j Analogy. As 452 : 355, : : the Square of the Diameter : Area. III. By Archimedes’j Analogy. As 14 : 11, :: the Square of the Diameter : Area. Rule XIV. The Circumference of a Circle being given, to find its Area. As I : ,07958 : : the Square of the Circumference : Area. Rule XV. The Area of a Circle being given , to find its Diameter. As 1 : 1,2732, : : the Area : Diameter required. Rule XVI. The Area of a Circle being given, to find its Circumference, As 1 : 12,56637, : : the Area : Circumference required. Rule XVII. The Diameter of a Circle being given, to fnd the Side of a Square nearly equal to the given Circle. As I : ,8862, : : the Diameter : Side required. Rule XVIII. The Circumference of a Circle being given, to fnd the Side of a Square nearly equal to the given Circle. As I : ,2821, : : the Circumference :,Side required. • Rule XIX. The Diameter of a Circle being given, to fnd the Side of a Square infcribed. s As I : ,7071, : : the Diameter : Side required. Rule XX. The Circumference of a Circle being given, to fnd the Side of a Square infcribed. As 1 : ,2251, : : the Circumference : Side required. Rule Of the Mensuration of Superficies and Solids. 171 Rule XXI. The Area of a Circle Icing given, to find the Side of a Square in- fcribed. As I : ,6366, : : Area : Side required. Rule XXII. The Side of a Square being given, to find the Diameter of its circum - fcribing Circle. As 1 : 1,4142 ; : the Side of the Square : Diameter required. Rule XXIII. The Side of a Square being given, to find the Circumference of its circumfcribing Circle. As I : 4,443, : : the Side of the Square : Circumference required. Rule XXIV. The Side of a Square being given, to find the Diameter of a Circle , nearly equal to the Square. As i : 1,128, : : the Side of the Square : Diameter required. Rule XXV. The Side of a Square being given, to find the Circumference of a Circle , nearly equal to the Square. As : 3,545, : : the Side of the Square : Circumference required. Rule XXVI. To find the Diameter of a Circle, as c e, Fig. T, having the Chord Line a b, and Height c d, of the Segment a c b, given. Square a d, and divide the Product by c d, the Quotient will be equal to d e , then c d, mere d e, is the Diameter required. Ru le XXVII. To meafure the Settlor of a Circle, as c b a, or d a e f, Fig. R. As 1 the Arch Line, : : the Radius da, or c a : Area. Ru le XXVIII. To meafure the Segment of a Circle, as a b c, Fig. P. Imagine Lines to be drawn from a and c, to the Center P ; and a b c P, will be a Sedor ; -which being meafured by Rule XXVII. and the fuppofed Triangle a c P, being deduced from it, the Remains will be the Content of the Segment required. To meafure the great Segment of a Circle, as d e f. Imagine Lines drawn from d and e, to the Center P, as da and a e, in Fig. R. Then to the Area of the Sedor d a e f, found by Rule XXVII. add the Area of the Triangle da e, by Rule I. and their Sum is the Area of the greater Segment required. Hence ’tis plain, that the Center of a given Segment of a Circle mull be known, before its Area can be found. Rule XXIX. To meafure the Zone of a Circle , as adef b c, Fig. To the Parallelogram d f ab, add the Segments d e f and a b c, and their Sum is the Area of the Zone required. Rule XXX. To meafure the Superficies of any irregular curvilineal Figure , as the Figure V. Divide the curved Bounds into Segments, as n p a, ab c, c d e, e f g, g h i, i hi, l m n. To the Area of the right-lined Figure nacegiln, add the Area of the Segments n p a, c d e, g h i , i k l, and from the Sum fubtrad the Areas of the Segments ab c, efg , and n m l, and the Remains will be the Area of the irregular Figure required. Rule XXXI. To meafure an Ox Eye, as Fig. W. Draw the Line a d, then add the Area of the Segment a c d, to the Segment a b d. Rule XXXII. To meafure any fpherical Triangle, as X Y Z, and A, Fig. II. First, Figure X, to the plain Triangle ace, add the Segments a b c, c d e, and a em, their Sum is the Area required. Secondly, Fig. Y, to the Area of the plain Triangle ab f, add the Segments a c b, and b d J, and from the Sum fubtrad the Segment e a f, and the Remains is the Area required. Thirdly, Fig. Z, from the plain Triangle a e c, fubtrad the Segments e ad, and b a c, and to the Remains add the Segment e c n, the Sum is the Area required. Fourthly, Fig. A, from the plain Triangle, a d f, fubtrad the Segments c b e, the Remains is the Area required. Rule XXXIII. To meafure any mixtilineal Triangle, fliBCD E, Fig. II. First, the Triangle C, from the plain Triangle, cad, fubtrad the Seg- ments acb, and e c d, the Remains is the Area required. Secondly, the Triangle 72 Of the Mensuration of Superficies and Solids, D, to the Triangle c a e, add the Segments b c a, and c e d, the Sum is thfi Area required. Thirdly, to the plain Triangle E, add the Segment b a c, the Sum is the Area required. Rule XXXIV. To meafure compound regular Figures , as F G H, Fig. II. First, the Fig. F, to the geometrical Square abed , add the Semi-circles e and f ? the Sum is the Area required. Secondly, the Fig. G, from the geometrical Square 1234, fubtrad the Quadrants 1 a b+ 2 c d, h 3 g f ef^ t the Remains is the Area required. Thirdly, the Fig. H, from the Parallelogram 1234, fub- tradf the T ri angles T b c, d 2 e, a 3 h, and fg 4, the Remains is the Area required. Rule XXXV. To meafure Egg and Heart Ovals , as Fig. O P Q. First, the Egg Oval, Fig. O. To the Trapezoid a df add the Semi-circle a c d, and the Segments afifbg , and d h t the Sum is the Area required. SeJ condly. Fig. P. To the plain Triangle a c d<> add the Semi-circle a b c, and the two Segments a d, aftd c d, the Sum is the Area required. Thirdly, the Heart Oval Q. To the plain Triangle a b g, add the two Semi-circles ad c, c e b, and two Segments a f g> and bgfi the Sum is the Area required. Rule XXXVI* Tv meafure an Ellipfis, as the Fig. I K< As I : ,7854- ; : the Square of two Diameters : Area* The Area of every Ellipfis is a mean proportional between the Areas of its circumfcribing and in- feribing Circles, as in Fig. N* For as the Area of the circumfcribing Circle abfm : the Area of the Ellipfis a g p x : 1 the Area of the Ellipfis a gpz : Area of the inferibed Circle bg ox. Rule XXXVII* To meafure the Segment of an Ellipfis , as e f i, Figi M, or d g n* Fig. N. First, The Segment of an Ellipfis whofe Bafe is parallel to the conjugate Dia* meter, as e f i. Fig. M, is in proportion to the Segment d f n, of the fame Height of the circumfcribing Circle ; as b m, the Diameter of the circumfcribing Circle : c k the conjugated Diameter of the Ellipfis i : the Area of df n, the Circle’s Seg- ment : efi, the Area of the Segment of the Ellipfis* Secondly, the Segment of an Ellipfis whofe Bafe is parallel to the tranfverfe Diameter as d n, Fig. N, is in proportion, as the Area of the inferibed Circle b g 0 x 1 the Area of the Elliplis a g p x : : the Area of the Segment of the inferibed Circle : d g n the Area of the Elliplis req nr; d. Or as g x the Diameter of the inferibed Circle : a p the tranfverfe Diameter : : the Area of the Segment of the inferibed Circle : Area of the Segment of the Ellipfis. The Fig. K and L, are each a Semi-Ellipfig-, the firft on the tranfverfe, and the laft on the conjugate Diameter, whofe Areas are to be found by confid --ring each of them as a whole Ellipfis, and take f the Area fo found, for their Areas required. The Fig. I K ffhews hew to deferibe any Ellipfis by the Help of three Jlraight Laths, &c* as following : Draw the 2 Diameters a f and b n at right Angles, to their given Lengths* Make n d, and n e, each equal to half the tranfverfe Diameter, then a and e are the two focus Points, whereon fix two Laths, as on Centers, as d g and e h, each equal to the tranfverfe Diameter. To their Ends h and g fix a third Lath, equal to the Diftance of d e, fo that the Ends at h and g may be moveable as the Joint of a two-foot Rule. Then the three Laths being moved about the two focus Points, their feveral Points of Interfe&ion will trace out the Ellipfis required. Rule XXXVIII. To meafure the Area of a Parabola , as Figures R or S. Every Parabola is equal to two thirds of its inferibing Parallelogram. There* fore as 1 : df, Fig. R, or af. Fig. S : : a d, Fig. R, or b a , Fig. S : a 4th Number, two thirds of which is the Area required. LECTURE I. Of Rules for meafuring the Solidity of all Kinds of Bodies , and their Superficies * Rule I. To meafure the Solidity of the Cube R, or the Parallelopipedon W*>- A S 1 : the Area of any End or Side : : the Depth or Length from that End or Side : the Solidity required. The Of the Mensuration of Superficies and Solids. 173 The Superficies of the Cube R, is the geometrical Squares I 23456, Fig. S, and ©f the Parallelopipedon W, the Parallelograms 1X^45 and geometrical Squares 2 3, Fig. X. Rule II. To meafure the Solidity of any Prifrn as the Figures V, A B, and A. D. As 1 : the Area of one End : : the Length : Solidity required ; the Super- ficies of the triangular Prifrn V, is the Parallelogram 125, and Triangles 3, 4, Fig. Z.. Of the hexangular Prifrn A B, the Parallelograms 123456, and Hexagons 7 8. And of the Trapezia Prifrn A B, the Parallelograms 2, 3, 4, 5* and Trapezias 1,6. k Rule III. To meafure the Solidity of a Cylinder, whofe Safe is a Circle , as Fig. A, Plate LXXIV. or an Ellipfis h as Fig. I, Plate LXXIII. As 1 : the Area of one End : s the Length : tue Solidity required. The Su- perficies of the elliptical Cylinder I, is the Parallelogram l nm 0, (whofe Length is equal to the Circumference of the Cylinder) and the 2 Ellipfes c k d a, and e i gf. And the Superficies of the circular Cylinder A is the Parallelogram a , and two Circles D C. Rule IV. To meafure the Solidity of a Tetrahedron , as Fig, T, Plate LXXII. the Pyramis A G and A F, and Cone, Fig. R. Plate LXXIV. In- every of thefe Bodies, as I : the Area of its Bafe : : of its Altitude : the Solidity required. The Reafon hereof is, that every Cone is equal to ~ of its cir* crumfcribing Cylinder ; that is, to a Cylinder of the fame Bafe and Altitude. So likewife every Tetrahedron and Pyramis is equal to f of its circumfcribing Prifm, whofe Bafe and Altitude is the fame as tliofe of the Tetrahedron and Pyramis, and therefore it follows, that as 1 : the Area of the Bafe of a Cylinder or Prifm : : the Length of its Axis : a 4th Number, one 3d of which is equal to the Solidity of the Cone or Pyramis inferibed therein. The Superficies of the Tetrahedron is the equilateral Triangles I, 2, 3, 4, The Superficies of the fquare Pyramis A F is the geometrical Square A E, and the Ifofceles Triangle a e b, b g d, d h c, and a c f. The Superficies of the octangular Pyramis A G is the Octagon A F, and Ifofceles Triangles a be d e f g h ; and the Superficies of the Cone is the Sedtor h h if, and Circle k l. Note, The Length of the Arch k if is equal to the Circumference of the Bafe of the Cone. And the Radius l h, to b f, the Side of the Cone. Rule V. To meafure the Solidity of a Sphere, as Fig. T, Plate LXXIV. As 21 : 1 1 : : the Cube of the Sphere’s, Axis : Solidity required, or as 1 : ,52 36 : : the Cube of the Sphere’s Axis : Solidity required ; for if the Axis of a Sphere be II, its Solidity is ,5236. Every Sphere is equal to a Cone, whofe Axis is equal to .the Radius of the Sphere, and its Bafe to the Area of the Sphere. Or every Sphere is equal to two Thirds of its circumfcribing Cylinder. Therefore, as 1 : the Area of a great Circle of the Sphere : : the Diameter : 4th Number, two Thirds of which is the Solidity of the Sphere. As a Cone is equal to { of a Cylinder of equal Bafe and Altitude, and as a Sphere is equal to | of a Cylinder of equal Diameter and Altitude, ’tis therefore evident that a Cone whofe Bafe is equal to a great Circle of a Sphere, and its Axis equal to the Axis of the Sphere, its Solidity is equal to | the Solidity of the Sphere. And a Cone, whofe Axis is equal to the Semi-axis of the Sphere, and the Di- ameter of its Bafe to twice the Diameter of the Sphere, will be equal to the Sphere ; as alfo is a Cone whofe Axis is equal to twice the Diameter of the Sphere, and the Diameter of its Bafe equal to the Diameter of the Sphere. Rule VI. To meafure the Superficies of a Sphere. The Area of every Sphere is equal to four great Circles thereof, fo the Area of the Sphere, Fig. T, Plate LXXIV. is equal to the Circles VWXY. Or as I ; the Diameter : : the Circumference to the Area required. Note, The Area of a circumfcribing Cylinder is to the Area of the inferibed Sphere, as 3 is to 2 ; and, which is the fame Proportion that the Solidity of the Cylinder has to the Solidity of the Sphere. Z Note, 174 Of the Mensuration ^Superficies and Solids. Note, If the Covering or Area of a Semi-fphere be laid out, as taught in the Covering of the Heads of Semi-circular Niches, in LECT. XXV. hereof, as is exhibited in Fig. M, by t v no x, &c. and the Area of a Part, as of Z, be multiplied by twice the Number of Parts laid out, the Product will be the Su- perficies of the Sphere required. Note alfo, The feveral occult Arches in this &g • arc n ° more than a Repetition of thofe in Fig. K, Plate LVI. which I have infeited here again, for the eafier underftanding the Manner of defcnbing the feveral Parts, t v w x, &c. which are the Superficies of the Semi-fphere laid open. Rule VII. To meafure the Solidity of any Segment of a Sphere, as f I, 7 ; Fig". M, Plate LXXIII. I. The Diameter and Altitude of the Frujlum being given. To 3 times the Square of/ 3, the Semi-diameter of its Bafe, add the Square of 3,1, its Altitude. Multiply the Sum by the Height, and that Produft again by ,5236, the Produft, cutting off 4 Decimals to the Right-hand, is the Solidity required. II. The Axis of the Sphere k g, and I 3 , the Height of the Segment given. From 3 Times the Axis, fubtraft twice the Height of the Segment. Multiply the Remainder by the Square of the Segment’s Height, and that Produft by 5236, the Produft, cutting off 4 Decimals to the Right-hand, is the Solidity required. Rule VIII. To meafure the Solidity of any Frujlum of a Sphere, as h k b, Fie. A L, Plate LXXIV. From the Solidity of the whole Sphere, dcduft the Segment h m h, and the Remains is the Solidity of the Fruffum required. Rule IX. To meafure the Tone of a Sphere, as h k d e, Fig. A, L, Plate LXXIV. _ From the Solidity of the whole Spliere, dedud the two Segments h m k and deb, the Remains is the Solidity of the Zone required. Rule X. To meafure the Zone of a Spheroid, as Fig. L, Plate LXXIII. Multiply the Square of h k, the conjugate Diameter, bv a n, the tranfverfe Diameter, and that Produft by ,5236, the Produft, cutting off the 4 Decimals, is the Solidity required. Note, Every Spheroid, as a c e g. Fig. (/, Plate LXXIII. is equal to two Thirds of a Cylinder, as a dnf, whofe Diameter is equal to the conjugate Dia- meter, and Height to the tranfverfe Diameter. Rule XL To meafure ■ the Solidity of the Segment, or Frujlum of any Spheroid. Inscribe the Spheroid in a Sphere ; then as the Solidity of the Sphere is 'to the Solidity of the Spherdid, fo is any Part of the Sphere to the like Part of the Spheroid. Rule XII. To meafure the Solidity of a parabolich Conoid, as Ficr. N, Plate LXXIII. This Solid is generated by the Revolution of a Semi-parabola, on its Axis, and is thus meafured, viz. Multiply the Square of its Diameter, by ,785.1, and its Produft by Half the perpendicular Altitude, the Produft (cutting off the 4 Decimals) is the Solidity required. Rule XIII. To meafure the Solidity of the Fnidum of a Parabolich Conoid, as f a c o- Fig. N, Plate LXXIII. b Multiply the Sum of the Squares of a c and f g, the leffer and greater" Diameters, by ,3927, and that Produft by the perpendicular Height of the Fruffum, the laft Produft is the Solidity required. Rule XIV. To meafure the Solidity of a parabolich Spindle, as Fig". W, Plate LXXIII. 6 Multiply the Square of g l its greateft Diameter, by ,41888, (being p T of ,7854) and that Produft by h q its Length, the lait Produft, cutting off the Decimals, is the Solidity required. Rule Of the Mensuration ^Superficies and Solids. 175 Rcce XV. To meafure the Solidity of a Fmjhm of a Paraiolici Spindle, os d ft; 1, or oj a Zone , as d f m o. the greateft -Dimeter, by 1,5708; alfo multiply ■?/ TY effer D ' amcter > b y >7854 ; alfo multiply the Square of the Difference of the Diameters, by ,3 141 6 ; then from the Sum of the two former 1Ubt ? a , tH r ait J 1 r ° d f a ’ aild multiplying the Remainder by one Third the peipendicular Length, that Pro dud is equal to the Solidity of the Zone dfm 0 , whofe half Part is equal to the Fruftum elf * l RfhSfLxln“ LXXI1 ' rhe Hakdnn ’ folLwt^Sf ° f each B ° dy be Gonfldered as 1 or ^ity, their Solidities are as Solidities. Tetrahedron Oftahedron Hexahedron Icofahedron Dodecahedron Superficies. 0,1178511 — 1,732051 0,47x4045 — 3,464x02 1,0000000 — 6,000000 8,660254 20,645729 2,181695 7,663119 Act m ^0 find the Solidities of either of thefe Bodies. Bot to be e ± d flT n s ‘If ™ C ’ • ! ‘'“Tube ° f ‘I« Side of the like the Bafe of a Pyramis, 'whofe' Verlex'Ph/ the Cento of “Se B a 0 "'^ fuel. Pyratnis being meafured f.ngly, and R S SolSity multiphed by L NuVber r^mVed. “ ‘ n ** * he Pmia& -H be the lohdit/oT the My Act r 1 ° the Sl ‘P^ies of either of thefe Bodies. tot Lfe S!d f of th v 1 !k t B ? d i gons 1 2 3 4 r 6 7 8 0 10 II 12’ Pm' " d iffrf Py, -°r A , • p,a,e lxxiil and edf, in 4 B. ?’«£ anV/sdV U’ D 7 ‘Ali + V 4 ^g-neuS octangular 4 l/ttetr gular Fruftum E, whfre the Trapezoid! ‘ *x 2’ 3 sT ? °£ °^ Oftagon F its Bafe, an.d the Ocftagon 4 its Top. 4 5 7 8 9 Sides ’ rhe 2 rp A HE- 176 Of the Mensuration of Superficies and Solids. The Superficies of the Fruftum of a Cone, Fig . S, Plate LXXIV, is the im- perfeft Superficies d e qf c r, and the Circles n n, and h k I. Rule XVIII. To meafure the Solidity of a Prifmoid , or Frujlum of an irregular Pyramid, whofe Ends are difproportionahle, Fig. G, Plate LXXIII. To / h add half b d, which multiply by h g, ■ the greater Breadth, and referve the Product. To b d, add half/ A, which multiply by c d the leffer Breadth, to which add the former Product referved, and the Sum being multiplied by one Third of the perpendicular Height, the Product is the Solidity required. The Superficies of this Fruftum is the 4 Trapezoids 1245, and the 2 Paral- lelograms 3 and 6, Fig. H. Rule XIX. To meafure the Solidity of an oblique Fragment of a Cylinder , as a c, Fig. P, Plate LXXIV. As 1 ? the Area of its Baft fl, : : half its Length : the Solidity required. The Superficies of this Fragment is the Ifofceles Triangle/^ e, the Ellipfis'^, and the Circle d. Note, Fig. Z is a double Fragment, whofe Superficies is the two Ellipfes e and /, and geometrical Square 1 gmn ; and Fig. O is the Outiide of d e a b, which is a Fragment of a Fragment of the Cylinder h d gb. Rule XX. T 0 meafure the Solidity of a Cylinder, whofe Ends are oblique to its Axis , as Fig. L, Plate LXXIV. By Rule XIX. meafure the Fragments a and b feparatcly, and add their Soli- dities to the Solidity of the Cylinder p q, the Sum is the Solidity required. The Superficies of this Cylinder is the double Trapezoid e f h g, h g k i, Li/the two Ellipfes c and d. The Figures E I K are other Examples of this Kind, whofe Superficies produce different Figures, according to the various Sections of their Ends, which I have added for further Examples of this Kind. ' Rule XXI. To meafure the Fragment of a Cone, ■ as b d c, Fig. A B, Plate LXXIV. _ ' As I : the Area of its Bafe, : : of its Altitude : its Solidity required. The Superficies of this Fragment is the curved Figure c 8 e i, the Circle qor, and the Ellipfis a b d e, Fig. A X. Rule XXII, To meafure the Fruftum of a Cone, whofe Ends are oblique to the Axis , ai Fig. A C, and A D, Plate LXXIV. First, meafure the Fruftum, as a Fruftum whofe Bafe is right-angled to the Axis, and from that Solidity dedueft the Fragments that are deficient at the Ends, and the Remains will be the Solidity required. The Superficies of thefe Fruftums are laid out as following, Fig. A Bu On a deferibe the Arch c m 1 , &c. e, equal to the Circumference of the Baft of the Cone, which divide into 8 equal Parts, at the Points m l k i, cSV. and draw the Lines a m, a l, a k, &V. Draw b 1 parallel to d c, and divide 1 c into four equal Parts. Make a 5, a 1 1, each equal to a 4 ; make a 6 , a to, each equal to a 3 ; make a 7, a 9, each equal to a 2 ; make a 8 equal to a 1. Through the Points 11, 10, 9, 8, and 7, 6, 5, trace the Curves e 8, and 8 c; then the Figure c 8 e i c is the Superficies of the Side. The Superficies A C, and A D are defqribed in the lame Manner. PART *77 PART Y. Of Plain Triponometry, Geometrically performed. LECTURE I, Of the Solution of plain Triangles. I. Definitions. F IRST, plain Triangles are right-angled or oblique-angled. Secondly, a right-angled Triangle is fuch a Triangle as hath one right Angle and two acute Angles, as the Triangle A, Plate LXXV. whofe Angle b c a is a right Angle, and the Angles c b a, and b a c, are both acute Angles. Third- ly, an oblique Triangle is fuch a Triangle as hath one obtufe Angle, and two acute Angles, as the Triangle B, whofe Angle b c a is obtufe, and the Angles c b a, and cab , are both acute Angles. Fourthly, in every right- angled plain Triangle, that Side which fubtendeth (or is oppofite to) the right Angle, as b a, in Figure A, is called the Hypothenufe ; and of the other two Sides, the one, as c a , is called the Bafe ; and the other, as c b, is called the Per- pendicular. Fifthly, in every oblique plain Triangle, as Fig. C, the longeft Side is generally called the Bafe , as c a j but fometimes one of the other two Sides is made the Bafe. Sixthly, in every right-lined Triangle, the Sum of the Degrees contained in the three Angles, are equal to 1 80 Degrees ; therefore if you have any two Angles given, you have alfo the third given, it being the Complement to 180 Degrees. Seventhly, and as in a right-angled plain Triangle, the right Angle contains 90 Degrees, therefore if any one of the two acute Angles be given, the other acute is alfo given, becaufe it is the Complement of the other acute Angle to 90 Degrees '; or of the other acute Angle and right Angle to 180 De- grees. Eighthly, in all plain Triangles whatfoever the Sides are proportional to the Sines of their oppofite Angles. The Solution of plain Triangles has always confilled of 12 Cafes, but herein I have reduced them unto 8 Cafes, of which 4 are of Triangles right-angled, and 4 of' Triangles oblique ; and which anfwer every particular exactly the fame a* thofe of other Authors divided into 12 Cafes. I. Of right-angled plain Triangles. In the Solution of right-angled plain Triangles, there are always two Parts given, as two Sides ; or an Angle and one Side ; to find a Side or an Angle required. Case I. Fig. A, Plate LXXV. The Bafe, c a 80 Feet, and Perpendicular c b 60 Feet, being given, to find the acute Angles c b a and b a c, and the Hypothenufe. Make c a (by a Scale of Feet) equal to 80 Feet, and c b equal to 60 Feet, and draw b a, which is the Hypothenufe required. With 60 Degrees of Chords, on the angular Points b and a, deferibe the Arches e d and g f, which being meafured on the Scale of Chords, e d will contain 52 Deg. 30 Min. and gf 37 Deg. 30 Min. which are the Angles required. Case II. Fig. A, Plate LXXV. The Hypothenufe b a 100 Feet, and the Bafe c a 80 Feet, being given , to find the acute Angles , and Perpendicular b c. •' > Make 178 Of Plain Trigonometry, Geometrically performed. Mark c a equal to 80 Feet ; eredl the Perpendicular c b of Length at pleafure j on a , with the Length of loo Feet, interfeft the Perpendicular at b, and draw the Line b a ; then meafure the Degrees in each Angle, as in Case I. and b c will be the Perpendicular required. Case III. Fig, A. Plate LXXV. The Bafe c a 80 Feet, and the Angle c a b, oppofite to the Perpendicular 37 Decrees 30 Min. being given, to find the Perpendicular c b, and Hypothenufe ba. * Make c a equal to 80 Foot j ered the Perpendicular e b of Length at pleafure ; make the Angle b a c equal to 37 Deg. 30 Min. and draw the Line .2 b, which will cut the Perpendicular in b , then be is the Perpendicular, and b a is, the Hy. pothenufe required, 1 Case IV. Fig. A, Plate LXXV. The Hypothenufe b a 100 Feet, and the Angle cb a 52 Deg. 30 Min. oppofitetothe Bafe, being given , to find the Length of the Safe c a, and of the Perpendicular c b. Draw b a equal to 10c Feet ; make the Angle bac equal to 52 Deg. 30 Min* and draw b c of Length at pleafure ; make the Angle bac equal to the Comple- ment of the Angle c b a, and draw the Line a c , which will cut b c in c ; then c a Is the Bafe, and be the Perpendicular required. II. Of oblique-angled plain Triangles. . I N t ^ ie Solution of oblique-angled plain Triangles, there are always three Parts given, as two bides and an Angle, or two Angles and a Side, to find a Side or an Angle required. Case I. Fig. B. Plate LXXV. Two Sides and an Angle oppofite to one of the Sides , being given , to find the third Side. This admits of three Varieties, as. Full, The Bafieb a 100 Feet , and Side b c 50 Feet, with the Angle b a c 28 Dev. oppofite to the Side b c, being given , to find the Side ca 60 Feet. Make ba equal to ipo Feet ; on b, with the Length of 50 Feet, deferibe the Arch d c at pleafuie ; m any Part of b a, as at h, make an Angle, as b h e, equal to the given Angle 28 Degrees ; from draw the Line a c parallel to h e, which will cut the Arch d c m c, then the Line c a is the Length of the Side required. Secondly, The Bafe c a Fig. C, too Feet, and Side b a 50 Feet , with the Angle c b a 1 10 Degrees , oppofite to the Bafe c a, being given, 1 0 find the Side c b 60 Feet. Make b a equal to 50 Feet ; make the Angle c b a equal to 1 10 Degrees, and draw b c of Length at pleafure ; on c % with the Length of the Bafe 100 Feet in- terfed the Line be in c , tlienr b is the Length of the Sjde required. I hi idly, The two Sides c b 60 Feet , and b a yo Feet, with the Angle b c a 28, Degrees, oppofite to the Side b a, being given, to find the Length of the Bafe ioo Feet. Draw c a at pleafure; on c make the Angle a c b, equal to the given Angle 28 Degrees, and make c b equal to 60 Feet ; on b, with the Length of co Feet interfea the Line c a in a, then c a is the Le'ngth of the Bafe required. * Case II. Fig. C, Plate LXXV. The Bafe c a ioo Feet , and the Side c b 60 Feet , with the Angle b c a 28 Dev contained between them, to find the third Side b a, and the Angles c b a and b a c. * * Make c a equal to ioo Feet ; make the Angle b ca equal to 28 Deg. and the Side cb equal to 60 Feet ; draw the Line b a, which is the third Side required • then meafuie the Angles c b a and b 0 c, as in Case I. of right-angled plain TrL angles. r Case III. Fig. C, Plate LXXV. The three Sides c a IOO Feet, c b 60 Feet, and b a 50 Feet, being given, to find all the Angles. By Prob. I. Lect. IV. Part II. complete the Triangle b c a, and by Case I. ot right-angled plain Triangles, find the Quantity 'of each Angle. Case Of Flam Trigonometry, Geometrically performed, iyo . . , Case IV. Fig. C, Plate LXXV. 3 W Angles, as b c Deg. b a c 42 i%. *,cb6o Feet, being given, to find the other two Sides b a 50 Feet, and c a 100 Feet ^ An M W * 1 CqUa tG 60 ^ 5 ma ^ C the An & le b c a e ^ ual t0 28 Deg. and the Angle £ a c equal to 42 De^. continue out the Lines b a and c a, and thev will Note The D A ;n th f ^° 1Rt n? ? th f n camdba the two Sides required. e;f!V h I ? pa T e c of pIam T r . ian S Ies ’ performed by the Tables of Logarithms .mes, Tangents and Secants, being more difficult to be underftood by Learners* than the preceding, and as to have added thofe Tables would have fwelled the who e beyond its intended Bulk and Price, I therefore omitted the Analogies and %ame Voffime be favourably accepted, I wiUpubliffi hereafter in a LECTURE II. Of Menfuration of Heights and Di/lances. T a”., thCfc ***** 3 Q - Uadra ”’ “ F * D ' a “ d P R o b. I. Fig. F, Plate LXXV To tale the Alii,. udeof an ObjeS, at the Olelift b n, l, the Help of a Suadranl as ~~ Msss z Ssaiitf Deg 34 Min. the Diftance is double the Altitude ^b-lme cuts 26 If any Obitrudtion is between you and the Ohiee> fr, - to its Safe, then go nearer, or Tlw" as at /, and there make a Mark on the Ground : move backward in a S ' h? t^ 1 ’* With your hrft Station, and the Objeft, until the Plumb-line cut 18 Deg.*^ Min' Aid tuS required " ^ tW ° S “ ions /a " d «• * equal to the' r c J J,. P , R0 “- IL F 'S- G > Plate LXXV. To find the Altitude of an Object, by knowing the Length of its Shadow the Shadow of the Stick! " to the Heiu-ht oOhe Stl V ' r •" u of Shadow of the Objeft the Heigto f f the ObjA ‘ “ *’ ^ cr . , , ... , Prob - ID- %. H, Plate LXXV. TotahetheAlut.de of an OljeS that it accejtlle, by the Help of a ten Feet Rod and a Stick only. Erect a d “% f M ‘ °^ S > mU f c AU ’' tudc " "I'^d. th. R • L f 1 S Rodm an y piace » as at m, and a Stick, as* f eaual to gsf . , a P£, r U P ^be ten Feet Rod ; fo that, looking from the Top of the Stick f t y °“ ^ % « which Vial": Feet If Setr f /i TtoTd Tfe 77^7 tance of the Suck from tire Objeft, to r a the Hmght of the OhiedUbove the tod^requhid. *° Whlch add the Hd « ht of «ko Stick nf, and the Sum is the Alii- P Ei! O 1 80 Of PlalhTk I g o no M et R V, Geometrically performed . p R o B. IV. Fig. G, Plate LXXV. To tale the Altitude of an Objett that is inacceffible, by its Shadow. Suppose the Shadow of the Objedt reach from b to e , and, at the fame I une* the Shadow of a Staff reach from e to,?; at about two or three Hours after, when the Sun is rifen confiderably higher, place down a Mane at the End o* the Obiedt’s Shadow, which fuppofe to be at c ; alfo, at the fame Time, make a Mark at the End of the Shadow of the Stick, fuppofe at f; now, as the I nangle dfg is fimilar to the Triangle a c c, and as the Triangle d ef is fimilar to the fn angle a c b, therefore, as fg is to the Height of the Staffs e, fo is ce to the Height of the Object required. P r o b. V. Fig. I, K, Plate LXXV. To meafure the Altitude of a Hill or Mountain , by the Help of a Spirit-Level and Station - Staffs. (i.) Erect your Level truly horizontal on the Top, as at 5, and directly again & the Inftrument, let a fecond Perfon hold up a Aiding Station-ftaff,- with * Vane fixed thereon, which he is to move up, until, looking through the Sights of vour Level, you fee its upper Edge, as at n : This done, let the fecond Perfon write down the Number of Inches and Parts of Inches that his Vane is above the Ground at m, let a third Perfon write down the Number of Inches and 1 ai ts ot Inches that your Inftrument is above ihe Surface of the Ground at 5. ( 4 .) Re- move your Level down the Hill, as to 4, and your 2d Affiftant to h, and let your ad Affiftant ered his Station-ftaff at m, the Place where your 2d Affiftant Lft flood* This done, fix your Inftrument truly horizontal, and looking to your qd Affiftant at m, let him' Aide up his Vane until you fee its upper Edge, at which Time he is to fet down,- under the Height of the Inftrument obferved at 5, the Inches and Parts of Inches that his Vane is then above the Ground ; alfo lock to the Station-ftaff of vour 2d Affiftant, and caufe him to Aide up his Vane, until you fee its upper Edge, as at /, and let him place down the Inches and Parts that his Vane is above the Ground, under his firft Height obfei ved at m. Proceed in like Manner at every other Obfervation, as may be required to defcend unto the Bottom at b. (3.) Let each Affiftant add mtoom Sum the Heights of his feveral Obfervations, and then that of your 3d Affiftant s being fubtrafted from that of your 2d Affiftant’s, the Difference is the Altitude of the Hill required. P r o b. VI. Fig. P, O, M, Plate LXXV. To meafure an inacceffible Di/lance. Inaccessible Diftances may be meafured by many Methods, as, Firft To find the Di fiance of the two Trees 7 and 8, Fig. P, which are rendered inacceffible by the River b b. Assign any Point on the Ground, from which you can meafure direaiy un- to the two Objedts 7 and 8, as the Point 9 ; continue 7, 9 unto 11, and 8, 9 unto 10, making the Diftance of 9, 11 equal to 7, 9, and the Diftance of 9, 10 equal to 8, 9, then the Diftance from 10 to 11 is equal to the Diftance ot SecondlT , 1 To fnd the Diftance of the Tree at r, in Fig. M, from the Point v, which is rendered inacceffible by the River b b. Imagine a Line to be drawn from v to r, and thereon ereft the Perpendicular v w ‘ of any Length, and let rv be continued at pleafure towards y, winch may be done by ftraining a Pack-thread Line from v towards y, in a right Line with ‘v r In any Part of the Perpendicular v w, affign a Point as w, and at any Diftance from you, place a Stake in a right Line between and r, as zts; alfo another on the Perpendicular, at any Diftance from w, as at t. Make the Triangle w t x equal to the Triangle w s t ; and continue w x, until it meet the Line vy in v / then the Diftance t y is equal to the Diftance v r, required. . J I inrdly. Of Plain Trigonometry, Geometrically performed, 1 8 1 Thirdly, To find the D fiance of the two Trees, 12 and 13, Fig. O, which is rendered inacceffible by the River d. Afilgn a Point as 1 6, from which you can meafure to both the Objects. Place two Stakes at any Diftance in right Lines, from the Point 16, to the two Objedts, as at the Points 14 and 15, and meafure the Sides of the Triangle 14, 15, 16, alfo the Diftances from the Point 16, to the Objedls 1 2 and 13. On Paper', with a Scale of Feet, make a Triangle, whofe refpedtive Sides are equal to the Meafures of the Sides of the Triangle 14, 15, 16, and continue out the Sides, refpedting the Sides 16, 14, and j 6, 15, each equal to the Meafures of 16, 12, and 16, 13. Then the Diftance between the Extremes of thofe Lines, being meafured on your Scale of Feet, will be the Diftance required. Pros. VII. Fig . N K L. Plate LXXV. To meafure an inacceffible Dijiance, by Help of a geometrical Square y right-angled or equilateral Triangle. Firft, To meafure the D fiance 5 b, Fig. N, which is rendered inacceffible by the River c, by Help of a geometrical Square. Imagine a right Line to be drawn from 5, to the Objedt b, which continues towards 4. On the Point 5 eredt the Perpendicular 5 2 of Length at pleafure, and therein affign a Point, as z, where with a Piece of Board make a geometrical Square, apply its Angle over the Point z, and direct its Side k i, to the Objedt j alfo at the fame Time caufe an Affiftant to move along the Line 5, 4, until by the Side of the geometrical Square z 3, you fee his Station-ftaff eredt,' at 4. .Phis done, meafure the Sides of the Triangle 524; and then as the Side 5, 4, is to the Perpendicular 25, fo is the Perpendicular z 5, to 5 b, the Diftance re- quired. Secondly, To meafure the Dflance 1 k, Fig. L, which is rendered inaccejfihle by *0 the River b. Being furniftied with a Piece of Board that is an equilateral Triangle, as / m n y apply one of its Angles over the Point /, and diredt a Side, as l m to h, and at the fame Time diredt an Affiftant to fix up a Station-ftaff in a diredt Line with the other Side / n , at any Diftance from you, as at p, and then fet up a Mark in the Poiiit /. This being done, move along the Line / p, until by the ^ides of the equilateral Triangle you can fee both the Mark fet up at/, and the Objedt at h, which you will do at the Point p ; then the Diftance of Ip is equal to the Diftance Ik required. Note , In the fame Manner, an inacceffible Diftance, as f a, Fig. K, may be found by a right-angled plain Triangle, as efg, whofe Sides e f and fg, are equal, as is evident to Infpedtion. Pros. VIII. Fig. Plate LXXV. To meafure the D fiances of divers Objects, that are inacceffible at two Stations, by the Help of a common [mall Table, or Joint fool \ and a Jlraight Rule , with perpendicular Sights fxed at each End thereof. Let the feveral Objedts be a b c d, and the two Stations i k, at 100 Feet, Yards, c 1 c. Diftance. Being furnifhed with a ftraight Rule, about two Feet, or two Feet and a half, in Length, with perpendicular Sights fo fixed at each End, that the Slits of the Sights ftand perpendicular over the thin Edge of the Rule (which is generally called an Index), and a fmall Table or Stool, that hath a fmooth and even Sur- face, proceed as follows, viz. With a Scale of Feet, &c. draw a Line in the Middle of the Table, as i k, equal to 100 Feet, the Diftance between the two Stations ; and then being at one of the Stations, as at i, lay the Edge of tne Index to the Line i k y and. move the Table, until through the Sights of the Index you fee the other Station k, and there fix your Table fait. On the Point i on your Table fix a Pin, and applying the Edge of your Index to the Pin, look through the Sights, to the firft Objedt at a ; and draw a Line from the Pin, by the Edge of the Index at pleafure, as i a. Move your Index in like Manner to every of the remaining Objedts, drawing Lines from the Pin, towards each Objedt, as at firft. A a This 1 82 Of Surveying LANDS, &e. This done, remove your Table unto k, your fecond Station, and placing the Point i on your Table, towards the firft Station, lay your Index to the Line k i on your Table, and move the Table, until through the Sights you fee your fir ft Station, and there fix your Table fall. Fix a Pin in the Point k on your Table, and then applying the Side of the Index to the Pin, direift the Sights unto every of the Objcfts, and draw Lines, as before, at the fir If Station, which will intarfeft the former in the Points abed, and whofe Diftances (or the Difiances from the two Stations / and k) being meafured on the fame Scale by which the Line i k was drawn on the Table, will be the true Difiances of each Objedl required. Note, By the fame Method of working, the Plan of any open Field may be taken, if the Angles are confidered as fo many different Objects, and can be all feen at each Station. PART VI. Of Surveying LANDS, & 1 c . T HE ufual Inftruments for this Furpofe are generally the Plain Table, Theo- dolite, Circumferentor, and Chain : but as the three firft are Inftruments of great Expence, beyond the Reach of common Workmen, for whofe fake I have publifhed this Work, I fhall therefore give fome few Examples, to Ihew how, by the Help of a ten Feet Rod, or Chain, and a Joint-ftool or Table, they may make the Plan of any Piece of Land, that is not of very great Dimenfions, with the uttnoft Exaftnefs. N. B. The Chain is that which is called Gunter ’s Chain, whofe Length is equal to 4 Statute Poles, or 66 Feet, divided into ioo Links, each. 7 Inches in Length. P r o b. I. Fig. S. Plate LXXV. Fo make the Plan of an irregular Side of a Field, /wihgfedcab. Make an Eye-draught on Paper, expreffmg the feveral Angles, and therein draw the occult Line b a ; as alfo the feveral perpendicular Off-fets 1 2 h, 42 g, 56 f, &c. This done, in the Field, meafure in a right Line from i, towards a ; and when you come againft the Angle h, as at the Point 12, -write down on your Eye-draught the Diftance meafured from/', as alfo the Length of the Off-fet 12 which place on the Off-fet. Proceed in like Manner to meafure the remaining Diftartces to every Off-fet, and the Length of each Off-fet. This done, draw a Line on Paper, and with a Scale of Feet fet off from i all the feveral Diftances, as i 12,142, i 56, life, and from thofe Points erect Perpendiculars, making each equal to their refpec- tive Meafures in the Eye-draught, and then right Lines, as i h, h g,gf,&c. being drawn from i to h, from h tog, from to f, &c. they will be the Plan of the irre- gular Side of the Field, as required. Note, If the Side of the Field be curved, as Fig. R, then take Off-fets at every remarkable Bending, as at h g e i k, &c. which meafure and plan as before, and through their Extremes trace the Curve, as required. Pro b. JI. Fig. V. Plate LXXV. Fo make the Plan of a Field, by the Help of a Chain only, as Fig. a c d g f e. ■ Make an Eye-draught of the Field, and divide it into Triangles. Meafure the Sides of the Field, and of every imaginary Triangle, which place on each refpec- tive Side, with a diagonal Scale of Chains and Links, as expreffed by Fig . IV. Platt Of Surveying LANDS, &c. 183 Plate IX. By Prob. I. Lect. IV. Part II. delineate all the feveral Triangles, as reprefented in your Eye-draught, and they will complete the Plan of the Field, as required. Prob. III. Fig. Y. Plate LXXV. To male the Plan of an irregular curved Field , by Help of the Chain only , as bcdefghik. First fix up Marks, fuch as Pieces of Paper fixed into the flit Ends of Sticks, at proper Places, as at bed e f g h i k, and imagine Lines to be drawn from one to the other, as b c, c d, d e, e f, 1c. Affign a Station towards the Middle of the Field, as at a, and imagine right Lines to be drawn from thence, unto the feveral Marks at b c d e f, iff c. which will divide the whole into ima- ginary Triangles. Make an Eye-draught as before directed, expreffmg every Triangle, &c. By Prob. II. hereof meafure and delineate the feveral Triangles ; and by Prob. I. meafure and delineate the Off-fets on the Out-lines of the feveral Triangles, neceffary for deferibing the curved Boundaries, which will complete the whole, as required. Note, Chains and Links are thus written, viz. 3 Chains, 75 Links, as from h to a, thus, 3 : 75, and two Chains, and 10 Links, as from c to a , thus, 2 : 10, $3V. Prob. IV. Fig. AC. Plate LXXV. To male the Plan of a Field, vjhofe Angles cannot be allfeen under three Stations , as at ad c, by Help of a Table and Cham. Assign 3 Stations in the Field, as a d c, at any Diftances, fuppofe ad, tat 3 Chains Diftance, and d c, at 3 Chains, and 35 Links. Draw a Line on your Table, by your Scale of Chains and Links, to reprefent 3 Chains, the Diftance between the Stations a and d. Place your Table in the Field, over the ftationary Point a, and laying your Index on the Line a d, move the Table about, until you fee the Station d, and there make your Table faft. Fix a Pin in your Table, at the Point a, and laying your Index thereto, diredt the Sights to the feveral Angles m n 0 v w x 3, and draw right Lines from the Pin, towards each Angle. Mea- fure the Diftances from your Station a, unto every of the Angles, and from your Scale of Chains and Links fet from the Pin, on each Line, as a m, an,a 0 , a v, 1Ac. their refpe&ive Lengths, as 2 : 75 ; 3 : 75 ; 3 : 65 ; &c. and draw the Lines mn, no, 0 v, v w, w x, and x 3. Move your Table to the fecond Station d, and laying your Index on the Line a d, move the Table about, until through the Sights you fee your firft Station at a, and there make it faft. Fix a Pin in your Table at the Point d, and laying your Index to the Pin, turn it about, until through the Sights you fee your third Station at c ; and by the Side of the Index draw the Line dc, which make equal to 3 Chains, 25 Links, the Diftance of the third Station c from d. Alfo, from the Pin on the Table, direct the Index to the Angle y, and draw the Line d y, equal to its meafured Length, and join the Side 3 y. Remove your Table to c, the third Station ; lay the Index on the Line dc, and move the Table about, until through the Sights you fee the Station d, and there make it faft. Fix a Pin in your Table, at the Point c, and laying your Index thereto, direct the Sights to,, the Angles z h i l, and draw Lines towards each Angle, equal to their refpe&ive Meafures, from the Station c. Then the right Lines y z, z h, hi, il , and l m, being drawn, they will complete the Plan, as required. Prob. V. Fig. A C. Plate LXXV. To make the Plan of a Field, by going about it without fide , by Help of a Table and Chain. First, go about the Field, and at proper Diftances make choice of Stations, as at a, p, q , r, s, g, whereat fix up Sticks with Paper as aforefaid. Then beginning at any one Station, as at a, meafure the Diftance from a to g, and from a to p. Draw a Line on one Side of your Table, on which fet from your Scale of Chains Aa2 and 184 Of Surveying LANDS, &c. and Links, the Length from a to g,, place your Table over the Point a. Lay y«ur Index on the Line reprefenting the Line a g, and move the Table about until through the Sights you fee the Mark at g, and there make it fall. Fix a Pin in your Table at the Point a, and laying your Index to the Pin, direftthe Sights to the Mark at p, and by its Side draw the Line a p, equal to its Length before mea- fured. By Pros. I. hereof, on the Line a g , meafure and delineate the Off-fets b 0, c n, d m, alfo the Off-fet k l, from the Oft-fet d m ; then e i, and f h, alfo the OlF-fets t v and i and Of M E C H A N I G K S. igy and in the firfl: Second it falls 1 5 Feet as from a to b, at the End of the fecond Second of its falling, it will have fell 4 times a b equal to 60 Feet, as to 4 which is equal to 2 multiplied in 2, the Square of the Seconds or Times in falling. So in like Manner at the End of the third Second it will have fell 9 times 1 5 Feet, equal to 135 Feet, which is equal to 3 multiplied into 3, the Square of the Seconds or Times in falling ; and in the fourth Second, 16 times 15 Feet, equal to 204. Feet, as to 16. Hence *tis plain that the Increafe of Motion in every Minute, &c. is according to the Series of the uneven Numbers, viz. 1, 3 i r, 7, 9, n, fsn. which are the Differences of the Squares, 1, 4, 9, 16, 25, Iffc. . 1 2 * As the Motions of Bodies are accelerated in falling, their Forces are thereby increafed in the fame Proportion. And therefore if the Body a, in falling from® tob, has a Force at b equal to 1 Pound Weight, it will have a Force at 4, equal to 4 Pounds Weight ; for as its Velocity from a to 4 is three Times as great as from a to b , it will therefore have a Force three Times greater at 4 than when at b, and lo in like Manner in its falling to 16 its Force will be equal to 1 6 Pounds, and at 2t to 25 Pounds, &c. J 13. And it is alfo to be obferved, that equal Bodies falling on inclined Planes whofe lowed Parts are in the fame Level, have the fame Force and Velocity at the End of their Falls, as when let fall perpendicular, but employ a longer Time in their Defcents. . So if the Body b , Plate LXXVI. defcend in the perpendicular Dine b g r or in either of the oblique Lines b f or b h, it will have the fame Force at f or h, as at^, but it will be longer in falling from b to /, than from b to^, and longer from b to h, than from b to f &c. > 14. If a Body deicend on an inclined Plane, as db , Fig. C, it will by its acquired Velocity afcend another Plane of equal Inclination, as b c, unto the fame Height, allowing for the Refiltance of the Air, and Fridtion of the Plane. 15. If Bodies fall in the Lines cf, d f e /, bf, a /, feV. defcribed in the Circle, Fig. B, they will from the Points in the Circumference a b c d e t come to the Bafey at the lame Time. For as the Lengths of their Lines of Defcent are to one another, fo are their Velocities to each other. j6. If. a Body, as £ Fig. E, be thrown perpendicularly upward with any Force, the Velocity wherewith the Body afcends, will continually diminifh, till at length, it be wholly taken away ; and from that Inftant of Time, the Body will defcend in the fame Line, with fuch an increafing Velocity, as to fall from a to c , with the fame force and in the fame Time as it was thrown up from c to a. The like is alio in Bodies thrown up on inclined Planes ; for if in Fig. C. the Body a be thrown from b to d, with a certain Force, and in a certain Time, it will by its own Weight return again to b, with the fame Force and in the fame Time as it afcended. 17. If . a Body defcend in the Arch of a Circle, as c Fig. D, in the Arch d e, the.Velocity will always be anlwerable to the perpendicular Height b e , from w Inch the Body fell ; but the I ime of the Body’s Delcent will be greater from c to e, than from b to e. 18. Now from hence it follows that the Body a Fig. F. to defcend the Arch Line a c, or the Chord Line a r, will require more Time than were it to fall rn the Perpendicular b c, but will in all the Defcents have an equal Force C% LECTURE I. Of the Laws of Nature. I T is to be obferved, that all the Varieties of Motion of Bodies in general are conformable to the following three Laws. Law I. All Bodies continue in their State of Rejl , or Motion , uniformly in a right Line , excepting they are obliged to change that State , by Forces mpreffed ; and therefore it follows, First, ISS 0/ M E C H A N I C K S. First, If a Body be abfolutely at Reft, and unfurnifhed with any Principle, whereby it could put itfelf into Motion, it will for ever continue in the fame Place, till afted upon by an external Body. Secondly, When a Body is put into Motion, it has no Power within itfelf, to make any Change in the Direftion of that Motion, and therefore muft move for- ward in a right Line, as I have before obferved, without declining any Way whatever. Thirdly, All Bodies endeavour to remain in their State of Reft or Motion, and therefore fome aftual Force is required to put Bodies out of a State of Reft, into Motion, or to change the Motion which they before received. This Quality in Bodies, whereby they fo preferve their prefent State of Motion or Reft, till fome active Force difturb them, is called the Vis Inertia of Matter. It is by this Property, that Matter unaftive of itfelf retains all the Power impreffed upon it, and will not ceafe to aft, until oppofed by as great a Power as that which iirft moved it. Law II. All Change of Motion is proportional to the Power of the moving Force impreffed \ and is always made according to the right Line in which that Force, is impreffed. That is to fay, firft, If in one Minute of Time, two Bodies, as a c, Fig. G, move from a and B, towards / and d, with equal Velocities, fo that when the Body a is arrived at b, the Body c, which moved from B, may aft its full Force againft the Body at b ; then will the Line of Direftion of the Body < 2 , which was in the Line a d, be changed into the diagonal Line b e, of the geometrical Square fbed; and by the A6tion of the Body c, on the Body b , the Velocity of the Body b will be fo accelerated, as to pafs, in the fecond Minute, through the Dia- gonal b e , the Side of whofe Square is equal to a 6, the Space which the Body b travelled through F the firft Minute. Again, if at the End of the fecond Mi- nute, when the Body b is arrived at e, another Body ftrike againft it at g, with the fame Velocity as b then has, then will the Line of Direftion of the Body b> in the fecond Minute, which is b h, the Diagonal continued, be changed into the Diagonal e n, of the Square nik e ; and by the Force of this fecond Body, the Velocity of the Body at e will be fo accelerated, as to pafs, in the third Minute, through the Diagonal n e , the Sides of whofe Square is equal to the Space which the Body b travelled through in the fecond Minute. If at the End of the third Mifiute, when the Body b is arrived at the Point n, it be again afted upon by a third Body at m , with the fame Velocity as the Body at n then has, then will the Line of Direftion of the Body at «, in the third Minute, which is the Diagonal e n , continued to />, be changed into the diagonal Line n r, of the Square r o p n ; and by the Force received from this third Body, the Velocity of the Body at n will be fo accelerated, as to pafs, in the fourth Minute, through the Diagonal n r t the Sides of whofe Square is equal to the Space which the Body travelled through in the third Minute. And if at the End of the fourth Minute, when the Body is arrived at r, it be again afted upon by a fourth. Body, as s, whofe Velocity is equal to that which the Body b then hath, the Lirie of Direftion of the Body at r, which then is the Diagonal n r continued to "x 9 will be changed into the diagonal Liner v, which is direftly retrograde, Or contrary to its firft Line of Direftion from a to b ; and by this laft additional Force, the Velocity of the Body at r will be fo accelerated, as to pafs through the Diagonal r v, of the Square scvrt, in the fifth Minute. , In this Manner, by the continual Aftions of Bodies, whofe Velocities are alike increafed, at the End, of every Minute, the Velocity of a Body may be fo increafed, as to travel ten thoufand Millions of Millions of Millions of Miles in a Minute. * ' Secondly, That the Change of Direftion is qlways proportional to the Force impreffed, is evident by all the preceding Lines of Direftion of the Body b, for the diagonal Line b e is the fame to the Line b d y , as it is to the Lin tf b. That is. the Angles / b e, and e b d, are equal, and confequently the Diagonal b e, 4. which O/MECHANICKS. 189 which is the focond Line of Dire&ion of the Body b, is perpendicular to the Angle f b d, and therefore is proportional to the Force impreffed at b. Thk like is to be underftood of the L)jagonal n e, which is perpendicular to the Angle i e k all'o of the Diagonal r n, which is perpendicular to the Angle *n p ; and pf the Diagonal r v, which is a Perpendicular to the Angle t rx, 'i hat the Increafe or Diminution of Motion* or the Velocity with which any Body is moved by the Action of a Power upon it, is proportional to that Power,. k evident ; for if I apply a certain Power to a Body, that will make it move with fuch Velocity, as to pvfs in one Minute 500 Yards ; to make two fuch Bodies pafs 5 Yaids in one Minute, will require a Power double to the former, becaufe there is double the Quantity of Matter to be removed in the fame Time. And, «n the other Hand, if this double Force be applied to either one of the aforefaid Bodies, which are fuppofed to be equal, its Velocity will be doubled, and confe- quently it will travel a thoufand Yards in one Minute. Hence ’tis plain, that the Degree of Motion, into which any Body is put out of a Stateof &eft by any Force or Power, will be proportional to that Power ; that is, a double Power will give twice the Velocity, a treble Power three times the Velocity, a quadruple Power four times the Velocity, DV. L A w III. Repulfe, or Re-aSion, is always egua! y and in contrary DireSion to Impulfe or Action ; The Actions oj two IS 0 die r upon each other are always egual y and m contrary Directions . When any E other than a Beam divided into two equal Parts, as b f, at c, Fig. O (and by the enfuing Le&urc will appear to be a Leaver of the firll Kind), which inltead of reiling on its Fulcrum at c, the Center ©f its Motion, is there 1 'ufpended. The tWo half Parts b r, and c f \ are called Brachiasi To have the Balance horizontal, the Center of Motion mull be forriething above the Center of its Gravity ; for were they to be both in one Point, which they would be, was the Beam to be a right Line, as a e, then thofe Weights which equiponderated when the Beam hung horizontally, would alfo equiponderate in any other Pofition ; whereas, when the Center of Motion is placed a little above that of Gravity, as aforefaid, if the Beam be inclined either way, the Weight moll elevated Will furmount the other, and defcend, caufing the Beam to fwing, tintil by Degrees it recovers its horizontal Pofition. The Reafon is very plain. Suppofe a i, Fig. P, be the Beam of a Balance put into an oblique Pofition, and the Perpendiculars a c , and i g , be drawn from its Extremes a aftd /, to the horizontal Line c /;, ’tis evident that e e, the Dillance of the Perpendicular a c, is greater than e g , the Dillance of the Perpendicular gi ; &nd as the Weight m is ''qual to the Weight o, the Weight m will therefore raife tip the Weight o. But was the Balance a right Line, as b h , having its Center 6f Motion and of Gravity both in the Point e , theh the Dillances d e, and e h, of the Perpendiculars b d and h k, would be equal, and the equal Weights / and n would equiponderate in that oblique Pofitioh } which the Beam a e 't cannot do, becaufe the Center ofits Motion is above the Center of its Gravity, which caufes the upper Point a to be the Dillance of c d, without the Perpendioular b d ; and the lower Point i to be the Dillance of g h, within the Perpendicular h h , and therefore c e 5 s longer than e g, by twice c d. The Proportion that the Power has to the Weight in the common Balance, is as i, the Length of one Brachia, is to I, the Length of the other Brachia ; fo is the Power applied, to the Weight required fao equipoife it. II. The Statera Romana, or Roman Balance-, commonly called the Steel-yard, Fig. R and Q, Plate LXXVI. This Sort of Balance is called the Roman Balance , from its being ufed in common at ; and it being originally made about 3 Feet in Length, and of Steel, ’twas therefore called a Steel-yard, and is thus Made : Prepare a imall fquare Bar of Iron or Steel, as 12 < 3 , Fig. R, of any Length, and ol equal Thieknefs, and let the Point a be the Center of Motion. Make the flat End be of fuefy Solidity, as to balance the Part 12 a. At any Dillance from a fix a Point, as c, on whiqh the fisveral Things to be weighed are to be fufpeuded. Note, Of MECHANICK S. 93 Note, The Point c is here fixed below the / Jlraight Line 12 b, for the fame Reafon at in the common Balance . Draw c b perpendicular to the Line 12 b ; make the Divifions, a I ; 1, 2 ; 2, 3 ; 3, 4 ; (sfc. each equal to«« b. Then 1 Pound Weight, applied at I, will equipoife 1 Pound at c ; alio 1 Pound Weight at 2, will equipoife 2 Pounds at c ; alfo 1 Pound Weight at 3, will equipoife 3 Pounds at c ; and I Pound at 12, will equipoife 12 Pounds at c, &c. For as a b , equal to one Part, is to a 12, 12 Parts; fo is 1 Pound Weight at 12, to 12 Pounds (as the Body_/'), at c; and therefore the Point a is the common Center of Gravity of the two Weights, be- caufe 13, the Sum of the two Weights, is to x, the lead Weight, as the Length of the Balance is to one Part, the Diftance of the great Weight from the Center of Gravity. To find the common Center of Gravity of two Bodies applied to a Beam of a known Weight and Length , which is not balanced , as Fig. R was fuppofed to be , by the more folid Part b c. Let d b, Fig. be divided into 13 Parts ; let the Body x be 1 Pound, and the Body k 12 Pounds ; and let the Point a be their common Center of Gravity, and the Weight of the Beam equal to 3 Pounds. On a , the common Center of Gravity, hang the Weight /, equal to the Weights x and k ; and at h, the Center of Gravity of the Beam, hang the Weight g , equal to 3 Pounds, the Weight of the Beam. Then as the Sum of the Weights g and /, 16 Pounds, is to 3, the leffer Weight g ; fo is the Diftance h a, of thofe two new Weights, 5 to I the Diftance of a from the true Center of Gravity required. III. A falfe Balance, as Fig. S, has its Beam unequally divided, as c e, and e d> which are to one another as 9 is to 10, £s V. and its Scales being alfo in the fame Proportion, they will therefore equiponderate as the juft Balance ; and whatever is weighed in the Scale hanging on c, will be lefs Weight than it really ought to be ; but this Cheat is immediately difeovered by changing the Scales. LECTURE V. Of the Lever, commonly called the Leaver. T HERE are three Sorts of Leavers, which are diftinguifhed by the different Manners of applying the Power and Weight. A Leaver of the firft Kind is that, whofe Fulcrum is between the Power ap- plied, and the Weight that is to be raifed, as Fig. A Q^, Plate LXXVI. where the Power is applied at d, the Weight at c , and the Fulcrum at a. Hence ’tis plain, that the common Balance Fig. O, the falfe Balance Fig. S, and the Roman Balance Fig. R, are all Leavers of the firft Kind, becaufe their Centers of Motion, as Fulcrums, are between their Powers and Weights. To know what Weight ca?i be raifed by a Leaver of the jlrjl Kind , this is the Analogy : As the leffer Brachia a c is to the greater Brachia da ,fo is the Power applied at d to the Weight it will equipoife at c. Therefore a little more being added to the Power at b , will raife the Weight required. The Length of a Brachia is the Diftance of a Power, or of a Weight, from a Fulcrum, and is always equal to a Perpendicular let fall from the Fulcrum, upon the Line of Direction of the Power or Weight. So b /, Fig. A N, is the Diftance of the Power at d , becaufe ’tis perpendicular to the Line of Direction d b, of the Power at d ; in like Manner the Line i e, which is perpendicular to e b, the Line ©fDiredlion of the Powers, is the Diftance of the Power at e ; as alfo is a i the Di ftance of the Power at c. Hence ’tis plain, that the greateft Power is that at d, whofe Line of Direction is right-angled with the Leaver £ k ; and which is yet more evidently fo by the Power applied at£, whofe Diftance from the Fulcrum is no more than h i, equal to the Perpendicular i f. The like is alfo to be under- flood of bended Leavers, as Fig . A F, A E, AG, and A L. 194 O/MECHANICKS. It matters not whether the Brachias of a Leaver be ftraight or curved, as Fig . A M, and A I ; for in both thefe Cafes the Diilances of the Powers and of the Weights from their Fulcrums are the Chord Lines of the Arches, and not the Arches themfelves. The nearer the Weight is to, and the farther the Power is from, tire Fulcrum, the lefs will be the Power, and the lefs wxll be the Height that the Weight can be raifed ; .for if the Body k, in Fig . W, be removed nearer to the Fulcrum from o p unto n m, it will not require fo great a Power at j to raife it, as when at o p, nor can it be railed fo high as when at op; foi if two equal Bodies be placed at n m and op, and j, the End of the Leavers p, be forced down to t, the Body o p will be raifed to a q, and the Body n m but to c b. When a Body is on the End of a Leaver, as the Body n o l c, Fig. A K, fo as to have its Center of Gravity above the Leaver, and is equipoifed by a Power at but the Weight of a Column of the Atmofphere, on a fquare Foot of the Earth’s Sur- face when the Air is the heavied, is found to be equal to 2259 Pounds Avoir - dupoife (at which Time the Mercury will rife to 31 Inches), which is 15 Pounds and 11 Ounces on every fquare Inch. But when the Air is lighted, fo that the Mercury is raifed but to 28 Inches, then the Weight of the Atmofphere on every fquare Foot is but 2025 Pounds, and on every fquare Inch 14 Pounds and I Ounce. . The greated Extent of that Part of the Air which is called Atmofpherey from the Surface of the Earth and Seas, is about 45 Miles in Height. The Weight of the Air is greater, the nearer it is to the Earth’s Surface, which is cauled by the great Weight of the Air next above it. CC2 200 O/HYDROSTATICKS. Fo find the Weight of a Pillar of the Atmofphere . Take a glafs Tube, of about 3 Feet in Length, and about fs or jV ° 1 * an Inch in Diameter, hermetically fealcd atone End: fill it full of Quickfilver ; immerfe the open End in a fmall Bafon of Quickfilver ; and then, holding the Tube perpendicular, the Quickfilver within the Tube will fubfide or run out into the Bafon, until it be lufpended at fome Height above 28 Inches perpendicular Height. The Reafon why the Quickfilver will be fo fufpended, is, that the Top of the Tube being fealed, the Preifure of the Pillar of the Atmofphere, perpendicularly over the Top of the Tube, is made on the Top of the Tube only, and not on any Part of the Quickfilver within it ; and if it be confidered, that every Part of the Quickfilver’ s Surface, in the Bafon about the Tube, equal to the Bafe of the Tube, is preffed by the fame Weight of Air as that on the Top of the Tube, ’tis evident that the Preflure of any one of thofe Parts is equal to the Weight of the Quickfilver prefiing on its own Bafe ; therefore the Quickfilver cannot defcend lower ; and therefore the Weight of the Quickfilver in the Tube is equal to the Weight of a Pillar of the Atmofphere of its own Diameter. On this Principle depends the railing of Water out of Wells, by the Help of a common Pump. In Page 24 may be feen, that a Cube Foot of Quickfilver weighs 874 Pounds and a Cube Foot of River Water 62 Pounds ; therefore Quickfilver is fomething more than 14 times heavier than River Water; and therefore, in are- curved Tube placed with the Ends upwards and open, l Inch of Quickfilver will keep in Equilibrio 14 Indies of Water. Now to find how high Well Water can be railed by a Pump in any Place, obferve how many Inches the Quickfilver will rife in the Tube as aforefaid ; and fo many times 14 Inches Water may be railed by a Pump, becaufe every 14 Inches Height of Water is but the Equipoife of an Inch of Quickfilver.. There- fore when a Pillar of the Atmofphere is equipoifed by a Pillar of Quickfilver, whofe Height is 30 Inches, to equipoife a like Pillar of the Atmofphere with a Pillar of Water of the fame Bafe, its Altitude mult be 35 Feet, which is 30 times 14 Inches, and which is generally the greateft Height that Water can be made to rife by the Help of a Pump. The Antlia , or common Pump, Fig. Q ^Plate LXXVII. is a Machine. of a very long Date, which is faid to be the Invention of Ctefebes , a Mathematician of Alexandria, about 120 Years before Chrijl. This Machine made of Lead confifts of a fucking Pipe, as 0 p, foldered to the Bottom of a larger Pipe or Barrel, as at n m> but, being made of Wood, is no more than a common Pipe, open at both Ends; but, be it made either of Lead or Wood, at a proper Diftance below itsTop, as at l m, is placed a Valve as/, which opens upwards ; within the upper Part of the Barrel is fitted a Pijlon or Bucket, as^, juft as big as the Bore of the Barrel, in which alfo is a Valve, that opens upwards. To this Pifton or Bucket is fixed an Iron Rod, as c h , which by a Pin is fixed, to the End of the Handle e f ; but as thereby the Rod is drawn out of a Perpendicular, tho’ there may be a Joint in the Rod near the Pifton, the Power rauft be greater than was the Rod to rife up and down perpendicularly, which may be eafily effe&ed by the Arch b d , fixed to the upper Part of the Handle, and by two Chains fixed from a tor/, and from c to b, which will rife up and force down the Pifton truly perpendicular, and with the leaft Friction. Now the Manner of the Pump’s Performance is eafily underftood ; for when the Pifton is forced down towards n, and a Quantity of Water poured in at the Top, the 2 Valves being then fhut, and the external Air being feparated from that within the fucking Pipe op , whofe End p is before immerfed in Water, therefore as foon as the Pifton with the Water poured on it is raifed, the Air within the fucking Pipe by the Force of the Atmofphere on the Surface of the Water in the Well is puihed up through the Valve at /, and fills that Part of Of H YDRO STATIC K S. 201 the Barrel, in which the Pifton afcended, at which Inftant the Valve at / is fhut* Now as muffh Air as is contained between the Valve at n m, and the Bottom of the Pifton, fo much Water at the fame Inftant afcended at the lower Part of the fucking Pipe. The Pifton being again forced down the Barrel towards ntn, the confined Air under it is compelled to force open the Valve at g, as the Pifton defcends ; and it being lighter than the Water, is by the Water pulhed up inta the external Air, and the Valve of the Pifton is inftantly fhut. Then the Pifton being railed, the Air fucceeds, and the Water below afcends after the Air, by the Preflure of the Atmofphere aforefaid ; and fo by a few Repetitions the whole Air is pumped out, and the fucking Pipe and Barrel filled with Water. Now to raife the Water as the Pifton is forced down the Barrel, the Valve at n m being then fhut, the Water under the Pifton, as before was faid of the Air, in that Part is compelled to open the Valve of the Pifton, and admit the Pifton to defcend into it, which Valve is fhut the very Inftant that the Pifton is down ; and then the Pifton being raifed as its Valve is then fhut, that Water cannot return back, and is therefore lifted up by the Pifton, in the upper Part of the Barrel, fo as to be received at the Spout i, and at the fame Time the Valve at n m is forced open by the afcending Water in the Pipe^o p ; and the lower Part of the Barrel being again filled, the Valve at n m fhuts, and retains it for the next Defcent of the Pifton ; and thus the A&ion of the Pump may be continued in raifing Water at pleafure. The Syphon or Crane^'a. b, Fig. R, Plate LXXVII. is nothing more than a recurved or bended Pipe, having one Side longer than the other. And as the afcending Liquid is forced up into the fhorter Side (the Air being firft exhaufted), by the Preflure of the Atmofphere as before in the Pump, therefore Mercury will run from one Veflcl to another by the Means of this Inftrument, provided that the Bend of the Syphon is not more than 30 or 31 Inches above the Surface of the Mercury, and Water, or Wine, if the Height of the Bend doth not exceed 35 Feet ; but in boththefe Cafes the Mouth of the defcending Tube mult be fomething lower than the Surface of the Mercury, or Water, into which the iliort Tube is immerfed ; for if the defcending Tube be equal to the afcending Tube, the Fluid will remain in the Syphon, unlefs fome external Caufe more than the Air force it out ; becaufe the Weight of the Fluid on both Sides is equal. By this Method, Water may be carried over Hills, as exprefled in Fig. V, Plate LXXVII. if their perpendicular Height above the Surface of the Water, as q r, be lefs than 35 Feet. By the Preflure of the Atmofphere it is, that Mercury will afcend to the fame Altitude in all Kinds of Veffels, and in any Situation, as is fhewn in Fig. S, Plate LXXVII. provided that their upper Parts be perfectly clofe, fo as not to admit any Air to enter in ; and by the Preflure of the Atmofphere it is, that Water in Refervoirs is forced to enter the Conduit-Pipes for conveying of Water to any Fountain, £ffr. that is below the Horizon or Level of the Refervoir, be the Diftaiwe .ever fo great. 2 I N I S. BOOKS on ArchiteBure ,