W^'^^^y/iA:t^/iJ c/6 dy^y r tcU'^y^t^ 
 
 :y3. 
 
 >trK* 
 
 9 . /f^ /s>j 
 
 tA 
 
 / ^(/tj. 
 
THE 
 
 COMPENDIOUS MEASURER. 
 
C ani R. Baldavtny Pnnters^ 
 
 New Brtdge-Jlrtety Linden, 
 
THE 
 
 COMPENDIOUS MEASURER; 
 
 BEIKG A BRirr, VET COM^R EHEN8IVK, 
 
 TREATISE ON MENSURATION, 
 
 AND 
 
 PRACTICAL GEOMETRY, 
 
 WITH 
 
 A>J INTRODUCTION TO DECIMAL AND DUWJECIMAT. 
 
 ARITHMETIC. 
 
 ADAPTED TO >RACTICB, AND THE USE OF SCHOOLS. 
 By CHARLES HUTTON, LL.D. and F.R.S, &c. 
 
 THE SIXTH EDITION, CORRICTED AND ENLARGEBj 
 
 Illustrated with the plan of a new rittu-Mo&it 
 
 ENGRAVEN ON COfPER PLATfi. 
 
 LONDON: 
 
 PHINTEt) FOR J. JOHNSON; R. BALJ)W1N; G. WILKIEJ AK» 
 
 J. ROBINSON; J. WALKER ; G. R0B1NS9N i SCATCHIR* 
 
 ANT LETTERMANJ AND LONGMAN, MVrST, 
 
 REES, AND ORME. 
 
 UNIVERSITT 
 
PEEFACE.- 
 
 COME years fince I publifhed a complete Tread fs 
 on Menfuration, both in 'I lieory and Pradtice ; in 
 which the Elements of that Science are demonflratedr 
 and the Rules applied to- the various pradlical purpofes 
 of life. That work has been well received by the 
 Public^ and honoured with the high approbation of 
 the more ifearned Mathematicians. 
 
 It has however been often reprefcnted to me, by 
 Tutors and others, that the great fize and price of that 
 work, as well as the very fcientific manner in which 
 it is formed, prevent it from being fo generally ufefiil in 
 fchools, and to prai^ical meafurers, as a more compen- 
 dious and familiar little book might be, which they 
 could put into the hands of their pupils, as a work 
 containing all the pradical rules of that art, in a form 
 proper for them to copy from, and unmixed with fucli 
 geometrical and algebraical demonflrations as occur in 
 the large work. 
 
 In compliance therefore with fuch reprefentations, I 
 have drawn up this CompeHdium of Menfuration, 
 Pra<^ical Geometry, and Arithmetic, expref^ly with 
 
 A 3 the 
 
 8675!. 
 
Vi P»-EFACE. 
 
 the view of accommodating it to practical matters, anti^ 
 the ufe of fchools, I have, for that end, here brought 
 together all the moft uilful rulles and precepts j have 
 arranged them in an orderly manner, proper for the 
 pupil to copy ; and delivered ihem in plain and familiar 
 Janguage. An example, worked out at full length., is 
 • et down to each rule, together witli drawings or re- 
 prefentations of the geometrical figures proper to 
 illuftrate each problem ; and then are* fubjoincd foraa 
 'Hore queftions to each rule, as examples propofed for 
 (he pni(ftice of the Icarner-s with the anfvver ftt down, 
 by which he may know when his work is right. 
 
 The IntroduvSlion to Decimal and Duodecimal 
 Arithmetic will be found ufef'ul, by going over thofe 
 branches before entering on the Menfuration, that 
 the learner may be very ready and expert in numeral 
 calculations. 
 
 The Pradical Geometry contains a great number 
 of geometrical conftrudlions and operations; by the 
 prafbite of which, the learner will acquire the free and 
 eafy ufe of his inftruments, and fo become prepared 
 for making the drawings that are ufcful for illuftrating 
 the various branches of Menfuration folfowing. ^^ 
 
 The Menfuration itfelf next fuccecds, and is divided 
 into various parts; firfl:, Menfuration in general, and 
 then as applied to the fcveral. pradical ufes in lifci 
 
PREFACE. Vil 
 
 The whole being arranged in fuch order as the learner 
 may properly take in fucceflion; or diftinguifhed inta 
 feveral branches, of which he may felc£l and ftudy 
 any peculiar ones that may be more to his purpoftf 
 than the reft:, when he has not either leifure or induce- 
 ment to go over the whole in a regular gradation. And 
 notwithftanding the compendious fize of the book, 
 and the great number of praftical branches here 
 treated, it will be found that each one is much more 
 full and complete than the firft appearance of fo fmall a 
 form may promifc to admit of. However, if further 
 fatisfadion be delired by any one, either concerning 
 the fcience in general, or the demonftrations of tlie 
 rules, or the more curious and copious difplay of pro- 
 perties, he may apply to my large treatife before men- 
 tioned, where he will find every part delivered in the 
 inoft ample form. 
 
 To this edition is added the new method of furvey- 
 ing, now pradifed by the bed furveyors, illuftrated 
 with a map or plan, and an engraved form of the 
 Field Book. 
 
 A 4 COI^- 
 
CONTENTS. 
 
 ARITHMETIC. 
 
 ?age 
 
 "TVECTM AL Fraaions 1 
 
 '^ Addition of Decimals 4 
 
 Subtraftion of Decimals , » ^. 5 
 
 Multiplication of Decimals 5 
 
 Divifion of Decimals S 
 
 Reduction of Decimals 1 i 
 
 Circulating Decimals 1^ 
 
 Rednftion of Repetends 17 
 
 Addition of Repetends 18 
 
 Subtraftion of ^Repetends , 19 
 
 Multiplication of Repetends 20 
 
 Divifion of Repetends 21 
 
 Involution, or Raifing of Powers . » », 23 
 
 Evolution, or Extradlion of Roots 23 
 
 To extraft the Square Root 30 
 
 Ditto of Vulgar Fradions or Mixt Numbers 34 
 
 To find a Mean Proportional 36> 
 
 To extraa the Cube Root 58 
 
 To find two Mean Proportionals 40 
 
 To extrad any Root whatever ,,..... 43 
 
 Ditto of Vulgar Fradlions or Mixt Numbers 43 
 
 Duodecimals, or Crofe Multiplication , . . . 4^ 
 
 Table of Square Roots , 50 
 
 MENSWRATiON* 55 
 
 fRAC- 
 
« CONTENTS. 
 
 PRACTICAL GEOMETRY. figC 
 
 Definitions 56 
 
 Problems 63 
 
 To divide a Liive into two equal Parts 63 
 
 To divide an Atrglc into two equal Farts 63- 
 
 To divide a Right Angle into three equal Parts 63- 
 
 To draw one Line parallel to another 64 
 
 To raife a Perpendicular on a given Line 64" 
 
 To let fall a Perpendicular on a given Line 66 
 
 To divide a Line into a-ny Number of equal Parts . . . ,67 
 
 To divide a Line in a given Pro{->ortion 6j 
 
 To make one Angle equal to another 6^6 
 
 To make an Angle of any Number of Degrees .68 
 
 To meafure a gi^'en Argle 69 
 
 To find the Centre of a "Circle 69 
 
 To draw a Circle through three Points ,^ 69 
 
 To draw a Tangent to a Circle 70 
 
 To find a third Pro^-rortional or two Lines 70 
 
 To find a fourth Proportional to three Lines 7i 
 
 To find a Mean Proportionalbetwcen two Lines 71' 
 
 To make an Equilateral Triangle ........... 71' 
 
 To make a Triangle with three given Line* . .... . . ... 72 
 
 To make a Square ► . 72 
 
 To make a Reftangle 72 
 
 To make a Pentagon 73 
 
 To make a Hexagon 74 
 
 To make an Od^agon 75 
 
 To make any regular Polygon 75 
 
 To infcribe a- Polygon in a Circle 76 
 
 To defcribe a Polygon about a Circle 77 
 
 To find the centre of a Polygon 78 
 
 To infcriba a Circle in a Triangle .79 
 
 To defcril>e a Circle about a Triangle 79 
 
 To defcribe a Ciicle within or about a Square 7g 
 
 To defcribe a Square or Odagon in or, about a Circle 79- 
 ^o infcribe a Trigon, or a Hexagon^, in a Circle .... 80. 
 loinfcrilse a Di^decagon in a CiKle ...,...*..,... 80> 
 
 To^ 
 
Page 
 Fo defcribe a Pentagon or a Decagon in.a^Circle .... 81 
 
 1 o div.iile a Circle mto 1:2 equal Parts ,.... 81 
 
 To draw a Line equal to the Circurni". ot a Circle 8'i 
 
 To dravv a Right Line equal to any Arc o{ a Circle 82 
 
 To divide a Circle into a^iy Number of equal Parts . . 83 
 
 To make one Triangle iimilar to another 83 
 
 To make a Figure fi-milar to a given one 8^ 
 
 To reduce a figure from one Scale ro another ... S4f 
 
 To make a Triangle eijiial to a Trapezium 85 
 
 To make a Triangle equal to a Circle 86 
 
 To make a Parallelogram equal to a Triangle 86 
 
 To make: a Square equal to a Rcdangle .8/ 
 
 To n.Hke a Square equal to the Snm of two Squares . . 87^ 
 To make a Square equal to the Diff. of two Squares ..88 
 To make a Square equal to any Number of Squares . . m 
 
 To tivake plane Diagonal.Scales S9 
 
 Geometrical Remarks ,.9^ 
 
 MENSURATION OF SUPER.P1 CI ES. 92 
 
 To find the Are.i of a Parallelogram 93 
 
 To find the Area of a Triangle .95 
 
 To find any Side of a Right-angled Triangle 97 
 
 To find the Area of a Trapezoid 99 
 
 To find the Area of a Trapezium loa 
 
 'Vo find the Area of ^rvy irregular Figure 103 
 
 To find the Area of a Regular Polygon 104- 
 
 To find the Radius, or Chord, &c. of an Arc 106' 
 
 To find the Diameter^nd Circumference of a Circle 1Q6 
 
 To find the Length of any Circular Arc lOSf 
 
 To find the Area of a Circle 109 
 
 To find the Area of a Circular Sedor 11 
 
 To find the Area of a Circular Segment 112 
 
 To find the Area of a Circular Zonr. 115 
 
 To find the Area of a Circular Ring 115 
 
 Xo meafur.c Long Irregular Figures 1 15 
 
 MEN- 
 
XU CONTENTS. 
 
 Page 
 
 UBNSVRATXOW OF 80fcI0S» 1 1<S 
 
 Definition* , 1 1 6 
 
 To find the SoYuVny of a Cube K'l 
 
 To find the Solidity of a Parallclopipcdort V2l 
 
 To find the Solidity of any Prifm 12^ 
 
 To find. the Convex Surface of a Cylinder 1^3 
 
 To find the Convex vSnrface ©f a Cone l'-24 
 
 To find the Surface of a Conical Fruftutn l•2^ 
 
 To find the Solidity of any Pyramtc! or Cone IQG 
 
 To find the Solidity of the Fruft. of a Pyr. or Cone . . 1-27 
 
 To find the Solidity of a Wedge .130 
 
 To find the Solidity of a Prifmoid r30 
 
 To find the Surface of a Sphere or Globe 132 
 
 To find the Solidity of a Sphere or Globe 135- 
 
 'To find the Solidity of a Spherical Segment IS-t 
 
 To find the Solidity of a Spherical Zone 134 
 
 To find the Surface of a Circular Spindle ^ ... 135 
 
 To find the Solidity of a Circular Spindle 1 37 
 
 To find the Solidity of the Zone of a Circ. Spindle .. 138. 
 To find the Surface or Solidity of any regular Body lil 
 To find the Surface of a Cylindrical Ring ^^vi**)*-. 143 
 To find the Solidity of a Cylindrical Ring .,,,,,,. 149 
 
 OF THE carpenters' RULE. 144 
 
 To Multiply Numbers by the Rule 145 
 
 To Divide by the Rule U6 
 
 To Square any Number 1 4^ 
 
 To extraft the Square Root 146 
 
 To find a Mean Proportional 14/ 
 
 To find a Third Proportional 147 
 
 Tc find a Fourth Propmtional 147 
 
 TINfBER MEASURE. 
 
 To meafure Board or Ptank 148 
 
 To noeafore Squared Timber 149 
 
 To meafure Round Timber 1 6 i 
 
 ARTI. 
 
CONT«rTS» XIU 
 
 P.gC 
 ARTIFICERS* WO«K. 154. 
 
 Bricklayers' Work 155 
 
 Mafons' Work 157 
 
 Carpenters' Werk 159 
 
 Slarers and Tilers' Work l62 
 
 Plafterers' Work , . . . l63 
 
 Painters' Work ,.<,«.. l64 
 
 Glaziers' Work ,.., ... . , ?65 
 
 Pavers' Work 16/ 
 
 Plumber:.' Work 1 68 
 
 VAULTED AND ARCHE0 ROOFS. 1^9 
 
 To find the ^vrface af a Vatilted Roof .... l6.9 
 
 To find the Solid Content of a Vaulted Roof 170 
 
 To find the Surface of a Domie 171 
 
 To find thf Solid Content, of a' Dome 172 
 
 To find the Superficies of a Saloon 172 
 
 To find the Solid Content of a Saloon 173 
 
 Tc find the Concave Surface of a Groin 17* 
 
 To find the SoKd Content of a Groin 175 
 
 tAND SVRVSYINC. Vj^S 
 
 Of the Surveyino: Chain 176 
 
 Of the Plain Table 178 
 
 Of the i he odolite 18J 
 
 Of the Crofs 182 
 
 Of the Field-book 18* 
 
 THS rAACTieB GJ fiVRVSYIKO. 186 
 
 To meafure a Line , . . , 1 g6 
 
 To meafure Angles and Bearings 188 
 
 To meafure Uffvets ) gp 
 
 To meafure a Triangthr Field 15Q 
 
 To meafure a Fonr.fided Field l^i 
 
 To furvey any Field by the Chain only , \gQ 
 
 To furvey a Field with the Plain Table J94 
 
 To 
 
^iv C^KTEXTS> 
 
 •To furvey a Field with ibc ThcodcUle 1^)6 
 
 To furvey a Field having crooked Fences 193 
 
 "To furvey a Field by two Stations ID^ 
 
 To furvey a Large l*:fta e 'JOl 
 
 The New Method cf Purveying '2i)A- 
 
 To fufAey'a Couhiy, ^r a Large Traft 2J 3 
 
 To furvey a City olr Town 2 lO 
 
 or tiAViiiKG.cotJi^vrivt}, and ditidikg. 217 
 
 To lay down the Plan of any Survey 217 
 
 To compute the Contents of Fields »215^ 
 
 To transfer a Plan to another Paper 220 
 
 CONIC SECTIONS AND THEIR SQLIDS. 2^^ 
 
 refinitions . , . . ! .' . . VV.'. C29 
 
 I'o dcfcribe an Ellipfc .V., . . . ., -31 
 
 To find the Tranfverfe, Cohj^ .5cc. in an Ellipfc .^ . . 232 
 
 To find the Circumference of an Ellipfe , 236" 
 
 To find the Area of an Ellipfe 337 
 
 To find the Area of an Elliptic Segment 237 
 
 TTo dcfcribe a Parabola 23S 
 
 Tt find a Parabolic Abfcrfs or Ordinate 239 
 
 To find the length of a Parabolic Curve 239 
 
 To find the Area of a Parabola .,.,^,f, ,^... . 240 
 
 To find the Area of a Parabolic Fruftum . . *r>» .->^>. 241 
 
 To conftruft or defcribc an Hyperbola 242 
 
 To find the Tranfverfe, Conj. &c. in an Hyperbola . . 242 
 
 To find ihe length of an Hyperbolic Curve 246" 
 
 To find the Area of an Hyperbola 248 
 
 To find the Solidity of a Spheroid , i . , . 250 
 
 To find the Solidity of a Spheroidal Segment 264 
 
 To find the Solidity of a Spheroidal Fruftum 253 
 
 To find the Solidity of an Elliptic Spindle 256' 
 
 To find the Fruftum or Seg* of an Elliptic Spindle . . 2.59 
 
 To find the Solidity of a Paraboloid 26'l 
 
 To find the Solidity of a paraboloidal Jt'^uftarn . , . . 26"42 
 
 To 
 
P?.gC 
 
 To find tht Solidity of a Paracolic Spindle 2^2 
 
 To £«d the Frnftum of a Parabolic Spindle , , 263 
 
 or GAUGING, 265 
 
 Of the Gauging Rule 265 
 
 The Ufe of the Gauging Rule 267 
 
 Of the Diagonal Rod 26.9 
 
 Of Cafks as Divided into Varieties 270 
 
 To find the Content of a Cafk of the Firft Form . . . . 27'1 
 To fold the Content of a Caflc of the Second Form .. 273 
 To find the Content of a Cafk of the Third Form . . 274. 
 To find the Content of a Caflc of the Fourth Form , . 275 
 To find the Content of a Cafk by Four Dimenfions . . 277 
 Tx> find the Conteat of aCaik by Three DimeHfioas 278 
 
 OF ULLAGING CASKS. 280 
 
 To find the Ullage by the Sliding Rule 280 
 
 To Ullage a Standing Calk by the Pen 281 
 
 To Ullage a Lying Cafk by the Pen , S82 
 
 OT SPECIFIC gravity; 283 
 
 A Table of Specific Gravities 283 
 
 To find the Magnitude of a Body from its Weight . . 284. 
 To find the Weight of a body from its Magnitude . . 285 
 To find the Specific Gravity of a Body 286' 
 
 5pei 
 Qu 
 
 To find the Quantity of Ingredients in a Compound 287 
 
 WEIGHT AND DIMENSIONS OF BALLS. 289 
 
 To find the Weight of an Iron Ball 289 
 
 To find the Weight of a Leaden Ball 290 
 
 To find the Diameter of an Iron Ball 29 1 
 
 To find the Diameter of a Leaden Ball 293 
 
 To find the Weight of an Iron Shell 294. 
 
 To find how much Powder will fill a Shell 295 
 
 To find the Powder to fill a Reaangular Box 296 
 
 To find the Powder to fill a Cylinder 29^ 
 
 To 
 
XVI CaNTENTS, 
 
 Page 
 To find the Shell from the Weight of the Powder . . 297 
 To find a Cube to any Weight of Powder ........ 298 
 
 To find a Cylinder to any Weight of Powder 299 
 
 OF PILING BALLS AND SHELLS. 300 
 
 To find the Kamber of Balls in a Triangular Pile .... 301 
 
 To find the Number of Balis in a Square Pile 501 
 
 To find the Number in a Redangular Pile 302 
 
 To find the Number in an Incomplete Pile 303 
 
 OF tilSTkSCEB BY THE YKLOCITT OF SOUND 305 
 
 m.scellankous questions 306 
 
 A TA6LS OF CI&CULAR SECMSMTt 3 i6 
 
 THC U4« •F TUB TABLE OF SEGMINTS. 521 
 
INTRODUCTION. 
 
 A 
 
 DECIMAL FRACTIONS. 
 
 Decimal Js a fradlon whofe denominator is a« 
 unit, or 1, with forae number of ciphers annexed ; 
 
 Decimals are written ^own without their denomina- 
 tors, the numerators being fo diftinguifhed as to Ihow 
 what the denominators are ; which is done, by feparat- 
 ing, by a point, fo many of the right-hand figures from 
 fhe reft, as there are ciphers in the denominator; the 
 figures on the left fide of the point being integers, and 
 :hofe on the right decimals. 
 
 Thus, 0*5 is underftood to be -^^ or | 
 And 0-25 is - - ^Voor i 
 And 0-75 is - - - _y_ or 3 
 
 And ] -3 is - - 41 or 1 j^Q 
 
 Arid 24*6 is - - 24-/*o or 24 1 
 
 But when there is not a fufiicient number of figurs: 
 n the numerator, ciphers are prefixed to fupply the 
 Icfeft, ^^/ 
 
 B So 
 
INTRODUCTION, 
 
 So, '62 is -,1-js or/^ 
 And •001.3 is xV.*o^ or ^^, 
 
 ;^a7 
 
 So that ihc denominator of a decimal is always I, with 
 as many ciphers as there arc figures in the decimal, 
 
 A finite decimal, is that which ends at a certain num- 
 ber cf place?. But an infinite decimal, that which no 
 where ends, bnt is undcrftood to be indefinitely con- 
 tinued. 
 
 A repeating decimal, has one figure or fisveral figures 
 
 coniinually repeated. As 20*2 4-33 &c, which is a finglc 
 
 or fimplc repctend. And 20'2i24' &c, or '20'2i624-6 &c. 
 which are compound reperends ; and are otherwife called 
 circulates, or circulating decimals. A point is firt over ^ 
 lingle repctend, and a point over the firll and laft figuie« 
 of a circulating decimal. 
 
 The firft place, next after the decimal marT<, is lOih 
 part , . the fecond is 1 00th parts, the third is lOOOth 
 parts, and fo on, decreafing towards the right hand by 
 lOths, or increafing towards the left by lOihs, the fame 
 as whole or integer numbers do. As in the following 
 i'cale of Notation, 
 
 3 J2 
 
 Zi-O 
 
 SSS8888-8. 88888 
 
 Ciphers 
 
DECIMALSr ^ 
 
 Ciphers on the right hand of decimals do not alter 
 their value. 
 
 For '5 or ^^5 is -J 
 
 And -50 or -^5^% is i 
 And -500 or -r'^%% is l 
 
 Sec, are all of equal value. 
 
 But ciphers before decimal figures, and after the fepn- 
 rating poinf, diminifl; their value in a ten-fold proportion 
 for every cipher. 
 
 And '005 is j^^^^ or ^i^ 
 
 A 'id f > on. 
 
 So that, in any mixed or fraVionai nambcr, if the 
 fepa rating point be moved 
 
 one , two , three , Sec, phicfs to the right-hand, every 
 figure will be 
 
 10, 100, 1000, 5cc, timea greater than before. 
 
 But if the point be movod towards the left hatid, then 
 every figure will be diiBiiiifhed in the fame manner, or 
 the whole quantity will be divide] by 
 
 10, 100, 1000, &c« 
 
 b2 ADDi» 
 
IKTRODUCTIOK. 
 
 ADDITION OF DECIMALS. 
 
 Set the numbers under each other according to the 
 value of their places, as in whole numbers, or fo that the 
 decimal points (land direftly below each other. Then add 
 as in whole numbers, placing th« decimal point in thefum 
 'ilraight below the other points. 
 
 
 EXAMPLES. 
 
 
 ( 1 ) 
 
 ( 2 ) 
 
 (3 ) 
 
 276 
 
 7530 
 
 312-09 
 
 3.g-213 
 
 16 -201 
 
 3-5711 
 
 72014-9 
 
 3*0142 
 
 4195-6' 
 
 4J7' 
 
 957-13 
 
 71*498 
 
 5032- 
 
 6-72819 
 
 9739215 
 
 22i4*298 
 
 •03014 
 
 179- 
 
 79993-411 
 
 8513-10353 
 
 14500-9741 
 
 Ix. 4. What is the fura of -014, '9816, '32, -15914, 
 72913, and -0047? 
 
 Ex. 5. What is the fum of 27-148, 918-73, 14016, 
 294304, -7138, and 221 •7.? 
 
 Ex. 6. Required the fum of 312-984, 21*3918, 
 2700-42, 3*153, 27*2, und 681-06. 
 
 SUB- 
 
SISCI^IALS. 
 
 SUBTRACTION of DECIMALS. 
 
 Set the less number under the greater in the fame 
 manner as in addition. Then fubtraft as in whole num- 
 bers, and place the decimal point in the remainder ftraight 
 below the other points^ 
 
 EXAMPI^ES, 
 
 ( 1 ) ( 2 ) ( 3 ) 
 
 From '9173 '2'73 214'81 
 
 take -2138 1-9183 ^'901^2 
 
 rcro. 7036 - 0'8115 2i)9'y0S58 
 
 Ex. 4. What is the difference between 91 '713 and 
 
 i07 ? • 
 
 Ex. 5. What is the difference between 27 U* and 
 •916? 
 
 Ex. 6, What is the difference between l6*37 and 
 «0O-135. 
 
 MULTIPLICATION of DECIMALS. 
 
 Set down the fadors under each other, and multiply 
 them as in whole numbers, then from the produft,, to- 
 wards the right hand, point off as many figures for 
 decimals, u there are decimal places in both factors to. 
 gether. 
 
 B 3 But 
 
1NTR©DUCT10N< 
 
 But if there be not as many figures in the proilinfl 39 
 there ought to be decimals, prefix the proper r.uinbcr of 
 ciphers to fupply the defeat. 
 
 (1 ) 
 
 620 '3 
 •417 
 
 EXAMPLR8. 
 (2) 
 
 91-78 
 •381 
 
 (3) 
 
 •217 
 
 •0431 
 
 36421 
 5203 
 20812 
 
 9178 
 73424 
 27534 
 
 217 
 651 
 S68 
 
 2l6i)051 
 
 34'ij6'8l8 
 
 •C093o27 
 
 Ex. 4. What is the produft of 51-6 and 21 ? 
 
 Ex. 5. What is the produd of 314 and •02.9 ? 
 
 Ex. 6. What is the piodoa of '051 and '{)6()l ? 
 
 Notr, When decimals are to be multiplied by 10, or 
 100, or 1000, &c, that is by 1 with any number of 
 ciphers, it is done by only moving the decimal point at 
 many places farther to the right hand, as there arc cipher* 
 in the faid ncukiplier; fabjoining cipher* if there be not 
 fo many figures. 
 
 EXAMPLES. 
 
 1. The producl of 51*3 and ]0 is 513 
 
 2. The produa of 2*714 and 100 is 
 
 3. The produa of \<j)163 and 1000 is 
 
 4. The produa of. 21'«l «nd 10000 is 
 
 CONTRACTIOK, 
 
 When the produa would contain feveral more decimals 
 than are neceflary for the purpofe in hand, the work may 
 be much contradcd thus, retaining only the proper num- 
 ber of decimals. 
 
 Set 
 
PF.CIMALS. 
 
 Set the units figure of the multiplier flrafgbt under 
 fuch decimal place of the multiplicand as you intend the 
 laft of your produfl fliall be, writing the ether figures 
 of the multiplier in an inverted order: then in multi- 
 plying, reject all the figures in the multiplicand which 
 are on the right of the figure you are multiplying by ; 
 fetting the produfts down fo, that their right-hand 
 figures fall iiraight l)eIow each other; and carrying to 
 fuch right-hand figures from the proJuft of the two 
 preceding figures in the multiplicand thus, viz. 1 from 5 
 to 14-, 2 from 15 to 2-1, 3 from 25 to 34, &c, inclu- 
 lively; and the fum of the lines will be the produd to 
 the number of decimals required, and will commonly be 
 to the neartft unit in the laft figure. 
 
 EXAMPLES. 
 
 3. Multiply QJ'lArQhG by 92'-l-10S5, fo as to retain 
 only four phiccs of decimals in the product. 
 
 Contracted. Common "ay, 
 
 27*Uf}i36' 27*uci85 
 
 53014'29 92-41035 
 
 I 
 
 24434-874. 
 
 13 
 
 574930 
 
 542997 
 
 81 
 
 44958 
 
 108599 
 
 2714 
 
 985' 
 
 2715 
 
 108599 
 
 44 
 
 81 
 
 542997 
 
 2 
 
 14 
 
 24434874 
 
 
 25089280 
 
 $508'9280 
 
 650510 
 
 2. Multiply 480* 14936 by 2'7241(), retaining four 
 Uccimals in the produdi, 
 
 3. Multiply 2490-3048 by .573286, retaining five 
 decimals in the produ<ft, 
 
 4. Multiply 325-701428 by '7218393, retaining three 
 cccimals in the produd, 
 
 B 4 DIVI- 
 
INTRODUCTION. 
 
 DIVISION pF DECIMALS. 
 
 DiYiDE as In uhole numbcis. And to know how 
 many decimals to point off in the quotient, obfcrve the 
 following rules : 
 
 1. There mud be as many decimals in ihc dividend, as 
 in both the divifor and quotient, together; therefore, 
 point off for decimals in the quotient, as many figures, 
 as the decimal places in the dividend exceed thole in the 
 divifor, 
 
 v;. If the figures in the quotient be not fo many 
 a« the rule requires, fupply the defed by prefixing 
 ciphers. 
 
 3. If the decimal places in the divifor be more than 
 thofe in the dividend, add ciphers as decimals to the 
 dividend, till the number of decimals in the dividend be 
 equal to thofe in the divifor, and the quotient will be 
 integers till all thefe decimals are ufed. And in cafe of 
 a remainder after all the figures of the dividend are 
 ufed, and more figures are wanted in the quotient, annex 
 ciphers to the remainder, to continue the divifion as far as 
 ncceffary. 
 
 4. The firft figure of the quotient will poffefs the 
 fame place, of integers or decimals, a« that figure of 
 the dividend which ftands over the units place of the firft 
 produi^. 
 
 EXAMPLES. 
 
 I.Divide3424.'6056by43-6. 2.Divide3877875by675. 
 43'0' ) 3424-6056' { 78-546 '675 ) 3877875 000 ( 5745000 
 3726 5028 
 
 2380 3037 
 
 2005 3375 
 
 26l6 ....000 
 
 3. Divide 
 
DECIMALS. 
 
 3. Divide '0081892 by '347. 5. Divide 3-15 by 375. 
 4;. Divide 7M3 by M8. ' 6. Divide 109 by -215. 
 
 CONTRACTIONS. 
 
 1. If the devifor be an integer with any number of 
 ciphers at the end ; cut them off, and remove the decimal 
 point in the dividend fo many places farther to the left as 
 there were ciphers cut off, prefixing ciphers if need be ; 
 then proceed as before, 
 
 EXAMPLES. 
 
 1. Divide 953 by 21000. 2. Divide 41020 by 32000. 
 21;000 ) '953 . 32-000 ) 41*020 
 
 * 7 ) 31766' 8 ) 10'255 
 
 •04538 &c. 1-281875 
 
 Here, firft divide by 3, and Here, firft divide by 4, 
 then by 7, becaufc 3 times and then by 8, becaufe 4 
 7 is 21. times 8 is 32. 
 
 3. Divide 45-5 by 2170. 4. Divide 6l by 79000. 
 
 2. Whence, if the divifor be 1 with ciphers, the 
 quotient will be the fame figures with the dividefti^^ 
 having the decimal point fo many places farther to the 
 ]eft as there are ciphers in the divifor. 
 
 EXAMPLES. 
 
 2173 by 100 = 2-173 419 by 10 rz 
 5* 1 6 by 1000 =: '21. by 1000 =: 
 
 3. When the number of figures in the divifor is great, 
 the divif on at large will be very troublefome, but may 
 be coHiraded thus : 
 
 Having by I he fourth general rule, fou»d what place 
 
 of decimals or integers the firft figure of the quotient 
 
 B 5 will 
 
10 INTRODUCTION, 
 
 wfll poflcfs ; coofidcr how many figures of the quotient 
 will fervo the prefcnt pnrpofe ; then take the fame num. 
 ber of the left-hand figures of the divifor, and as many 
 of the dividend figures as will contain them (lefs than 
 10 times) J by thefe find the firft figure of the quotient; 
 and for each following figure divide the laft remainder 
 by the divifor, wanting one figure to the right more 
 than before, but obfcrving what muft be carried to the 
 firft produ<J^ for fuch omitted figures, as in the contraftion 
 of Multiplication; and continue the operation till the 
 divifor is exhaufted. 
 
 When there are not fo many figures in the divifor as 
 arc required to be in the quotient, begin the divifion 
 with all the figures as ufual, and continue it till the ^num- 
 ber of figure* in the divifor, and thofe remaining to be 
 found in the quoticat, be equal, ^ftcr which ufe the coii- 
 tradion. 
 
 EXAMFLES. 
 
 1. Divide 2hOS'92S06 by t)2-41035, fo as to have four 
 decimals in the quoiienr,— In this cafe the quotient will 
 contain fix figures. Hence 
 
 92 4103,5 ) 2308-c)28,06 ( 27-U98 
 66';)721" 
 13849 • • 
 4608 • • ' 
 
 912 
 
 80 .... . 
 
 6 •• 
 
 2. Divjde 410S»C35l by 230*409 fo that the quotient 
 aav contain four decimals. 
 
 4. Divide 37*10438 by 5713*96 that the quotient may 
 contain five decimals. 
 
 4. Divide 91308 by 2137*2 that tht quotient may 
 
 contain three dccinwla,. 
 
 4 R^UC. 
 
DECIMALS. 
 
 iX 
 
 REDUCTION OF DECIMALS. 
 
 1. To reduce a vulgar fraf^ion to a ©rcimal. Divide 
 the numerator, with as many decimal ciphers annexed, as 
 may be neceffary, by the denominator; and the quotient 
 will be the decimal fought. 
 
 EXAMPLES. 
 
 ]. Reduce ^'-^ to a decimal. 
 9") 1- 000000 
 11 ) 0. nil 11; 
 
 0*0 10 101 &c = ^ 
 
 H«r« divide by 9 and 11, 
 becaufe 9 times 11 is gg. 
 And the decimal value of 
 ■5^ is the circulate 'Ol'. 
 
 2. Reduce y^ to a decimal. 
 5 ;i-o 
 
 5 ) 0-20 
 3 ) O'O* 
 
 0'01333 &c =:y«^ 
 
 Here divide by 5, 5, and 3, 
 becaufe 5 x 5 x 3 = 75, 
 And the decimal value Of 
 
 ■y'-y is the repctcnd '01 3. 
 
 I = -3 
 
 J =-25 
 
 s — 
 
 y — 
 
 OTHER IXAMPLES, 
 
 ,Z H •i'^2857 
 
 7sir — 
 
 9 
 I 
 
 To 
 
 = •125. 
 = -1. 
 
 So that whenever we meet with the repctend '2, 
 in any operation, we may fubftltute ^ for it: in like 
 manner we may take f for *6, and ^ for • 16, and i for 
 i, and f or 1 for '§, &c, 
 
 B 6 JV,/tf 
 
12 INTEODUCTION. 
 
 Note, When a great many figures are required in the 
 decimal, and the denominator of the given fraftion is a 
 prime number greater than II, the operation will be bcft 
 performed jts follows. 
 
 Suppofe, for inftancc, we would find the reciprocal of 
 the prime number 2.9, or the value of the fradion ^'^ ia 
 decimal numbers. Pirft divide J'OOO by 29. in the com- 
 mon way, fo far as to find two or three of the firft figures, 
 or till the remainder becomes a fingle figure, and then 
 affume the fupplement to complete the quotient. Thus wc 
 fhall have ^*,y -^ 0*034'4 8/^ for the complete quotient; 
 which equation multiply by the numerator 8, and it will 
 give /^=0-27584.|| or rather ^^=^'2758^^^. Subfti- 
 tute this inftead of the fradion in the firlt equation, 
 and we fhall have ^'^ r= 0-0344827586V9. Again, 
 multiply this equation by 6, and it will give -^-^ =z 
 0"i06S9655]7^-^y and then by fubditution ^V=^*0344S2 
 7586206S965517j^. Again, multiply this equation by 
 7, and it l^ecomes -/^ z= 0-24 13793 1034.482758620||, 
 and thfn by fuhftitutibn ^'-^—O-OS^^S^? 5S6206S9655l7 
 2413793U)344827586'20;|; where every operation will 
 at leaft deuble the number of figures found by the pre- 
 ceding operation. Arid this will be an eafy expedient for 
 converting divifion into rnultiplication in all cafe?. For 
 this reciprocal of the divifor being thus found, it may be 
 multiplied by the dividend to produce the quotient, 
 
 . JI. "To reduce a Decimal to » Vulgar Fradion, 
 
 Under the figures of the given Decimal write its proper 
 denominator ; which fradion, abbreviated as much as it 
 can be, will be the vulgar fradion fought. 
 
 EXAM- 
 
PICIMALS* 
 
 
 EXAMPLES. 
 
 
 So -5 = ^ = 
 
 And -^5 = tVo = 
 And 75 zz /,V = 
 And .6 rr ^*j = 
 And .6-25 ^ -^eVo = 
 And •56'25 = -fVoVo= 
 
 I 
 
 13 
 
 III, To find the Value of a decimal, in the Lower 
 Denominations. 
 
 Multiply the given Decimal by the number of parts in 
 the next lower denomination; from the produft cut off as 
 many decimals as are in the given number. 
 
 Multiply ihefe by the parts in the next lower denomi- 
 nation again, cutting off the fame number of decimals as 
 before. 
 
 And proceed in the fame manner to the loweft denomi- 
 nation ; then the feveral integer parts cut off on the left 
 hand will give the value of the decimal propofcd. 
 
 EXAMPLES, . 
 
 J. For the value of •39 Ul. 2. For the value of 
 
 •2 1 39lb. avoir, 
 •3914 -213^ 
 
 20 16 
 
 s 7-8280 
 12 
 
 d 93960 
 
 4 
 
 q 37440 
 
 Anf. 7s. 9id 
 
 12834 
 2139 
 
 02, 3*4224 
 16 
 
 dr. 67584 
 
 Anf. 3 oz 6 dr. 
 
 OTHEK 
 
14 
 
 INTRODVCTION. 
 
 OTHE& EXAMPLES. 
 
 1. 
 2. 
 3. 
 4. 
 5. 
 6. 
 7. 
 8. 
 
 p. 
 
 30. 
 11. 
 12. 
 
 IV. 
 
 ^efttous* 
 
 ' '775 1 
 ' •6"25 s 
 •8635 I 
 
 • '0125 lb troy 
 
 ■ -469-1 lb troy 
 . '625 cat, 
 
 • •009943 mile 
 
 • -6875 yd cloth 
 •3375 iicr 
 
 • •2083 hhd wine 
 
 ■ -40625 qr corn 
 
 • -42857 month 
 
 Aff/kven, 
 16s 6d 
 7i 
 17 3 
 3 dwts 
 
 5 or. 12 dwts J 5 gt 
 2 qr. 14 lb 
 17 yd 1 f 6in almoft 
 
 2 qu 3 nl 
 1 rd 14 pi 
 13 gl 
 
 3 bu 1 pk 
 
 1 wk 5 day nearly 
 
 To bring Quantities to Decimals of Higher 
 Denominations. 
 
 CASE 1. 
 
 If a fingle integer or decimal be propofed, reduce it to 
 the higher denomination, by dividing as in redudlion of 
 Vfhoh numbers. 
 
 IXAMPLES. 
 
 1. Reduce 9d to the de- 
 cimal of a pound. 
 
 Anf. 
 
 12 
 20 
 
 2. Reduce 1 dwt to the 
 decimal of a lb. 
 20 1 dwt 
 12 0-05 oz, 
 Anf, 0-00416 lb 
 
DECIMALS. 
 
 15 
 
 ^uejihftst An/werr, 
 
 3. Reduce 26 d to 1 fieri . - 'OOIOSS I. 
 
 4. Reduce 7 drams to lb avoird - •027'34375 lb 
 
 5. Redacc 2*13 lb to a cwt . - 'Oipif^G cwt 
 (j. Reduce 24 yds to a mile - - •013636 mile 
 
 7. Reduce -056 pole to an acre • '00035 acre 
 
 8. Reduce 1-2 pint to hd wine - '00238 hd 
 f). Reduce 14 raiB to a day - - '009722 day 
 
 10. Reduce '21 pint to a peck - •013125 peck 
 
 CASE II. 
 
 A compound number may be reduced to a fuperior name 
 by reducing each of its parts, and taking the fum of the 
 decimals: the beft way ^o do which is thus: 
 
 Write the given numbers under each other, proceeding 
 orderly from the leaft to the greateft name, for dividends; 
 draw a perpendicular line on the left of thefe, and on the 
 left of it write oppofitc to each dividend fuch a number, 
 for a divifor, as will reduce ir to the next fuperior name; 
 then begin with the upper divifion, and affix the quotient 
 of each to the next dividend, as a decimal part of it, 
 before it is divided, and the latt fum will be the anfwer. 
 
 EXAMPLES. 
 
 1. Reduce 3l 12s 6|d to 2. Reduce 5 oz 12 dwt l6 gr 
 
 the denomination of !• 
 
 4 
 
 3 
 
 12 
 
 6-75 
 
 20 
 
 1 2^5625 
 
 .nf. 
 
 •628125 
 
 to the denom. of lb. 
 
 24 
 
 20 
 
 12 
 
 Anf, 
 
 \6 
 
 12'66 
 5-633 
 0'4^4 
 
16 
 
 INTftOPUCTIOK. 
 
 ^ejlioau AnfyDfiTt, 
 
 3. Reduce 19 I l/ s 4| d to 1 ll)*86354.l6' 1 
 
 4. Reduce 15 s 6 d to 1 775 I 
 
 5. Reduce l^Alo^ ftiil '625 s 
 
 6. Reduce 3 cwt 2 qr 14 lb to cwt 3*6*25 cwt 
 
 7. Reduce 17 yd 1 ft 6' in to a mile '00994318 mil 
 
 8. R."duce2 qr 3 nls to a yard ^ '(iSZSyd 
 
 9. Reduce 13 ac 1 r 14 pel to acres 13*3375 acr 
 lOu Reduce 13 gal 1 pint ro hd wine -2083 hd 
 
 11. Reduce 3 bu(h 1 pec to a qr *406'25 qr 
 
 12. Reduce 3 mo 1 we 5 da to mon 3^42857 mon 
 
 CIRCULATING DECIMALS. 
 
 It has already been obfcrved, that when an infinite de- 
 cimal repeats always one figure, it is a fmgle rcpetcnd; 
 and when more than one, a compound repetend, or a cir- 
 culate: alfo that a point is fet over a fingle repetend, and 
 a point ovtr the firft and laft figure* of a circulate. 
 
 It may further be cbferved, that when other decinjal 
 figures precede a repeiend, in any numbers, it is called a 
 
 mixed number or quantity, as '23, or '104123: otherwife 
 it is a pure repetend, as •3 and •123. 
 
 Similar repetend s begin at the fame place, and confift 
 
 of the fame number of figures: as '3 and '6, or 1*341 
 and2-*156T 
 
 Diffimilar repetends begin at different places, ?.nd con- 
 fift of an unequal number of figures. 
 
 Similar and conterminous repetends l^egin and end at ' 
 
 the fame place, as 2*9104 and •OO'ia,- 
 
 REDUC. 
 
REPETENDS. 17 
 
 REDUCTION OF REPEATING DECIMALS. 
 
 CASE I, 
 
 To reduce a fingle Rcpejend to a Vulgar Fraftion. 
 
 Make the given decimal the numerator; and for a de- 
 nominator take as many nines as there are recurring places 
 in the given repetend. 
 
 If one or more of the left-hand places, in the given de- 
 cimsl, be ciphers, annex as many ciphers to the right-hand 
 of the nines in the denominator. 
 
 1XAMFLB3. 
 
 1 So •3=1= ^. 4And S'Ss = $4|=2xV. 
 
 2 And -03 =^ =_>^. 5And'Q59U05=^%^^%9^z=^^j%, 
 
 3 And-isa --^^z=^\\.6And 'JQO^So = 5|.ff3| -|J, 
 
 CASE IT. 
 
 Td reduce a mixed Repetend to a Vulgar Fraflion. 
 
 To as many nines as there are figures in the repetend, 
 annex as many ciphers as there are finite places, for the 
 denominator of the vulgar fraction. 
 
 Multiply the nines in the denominator by the finite part 
 of the decimal, and to the produ<5l add the repeating part, 
 for the numerator. 
 
 Or find the vulgar fraftion as before, anfwering to the 
 repetend, then join it to the finite part, and reduce them 
 to a comraoii denominator, 
 
 EXAM* 
 
1ft 1NTK0PCCT'!0V# 
 
 1. So -03 = 
 
 5. And 'jsj !:r 
 
 3. And '\tii ^ 
 
 4. And '59 2d = 
 
 EXAMPLES, 
 
 5) X :> 4- 3 _ ii _ ^ 
 
 yu — f;o — 15 
 
 58 X .9 -f J _ ^ I 
 
 13 X fj + 8 I2j J) 
 
 A X 9 <)') - f <>'■? ^ '')9'20 16* 
 
 ADDITION OF REPETENDS. 
 
 Make every line fo begin snd end at the Tame place, 
 by extending ilic repetend.', and filling up the vacanciea 
 with the proper figures and ciphers. Then add as in 
 common nuitibers; only increafe the fum of the right- 
 hand row, or laft row of the repetcnds, by as many units 
 as the firft row of repetends contains nines. And the fum 
 will circulate at the fame places as the other lines, 
 
 IXAMPLEI* 
 
 ( 1 ) . { 2 ) 
 
 59*5548 = 39*<>54SO 91*3.57 = 91 '3^70 
 
 8l'04-5 = 8I-0466b" 72-3» = 72'3«88 
 
 42'36 = 42-35555 7'2'i = 7*211 1' 
 
 9'837 = 9*83777 4'2965 = 4-2965' 
 
 Sum 172'894SO Sura 175-2535 
 
REi-r/ii.vrn, 1^ 
 
 , < "^ > . . ^ -^ > 
 
 p-yiir: 9«S14M4-Sl 2-41 -.^ 2*41 
 
 1*0 :::; 1 ViOOOOOOO 13»216 -r. 13'l.M 5 1 5*15 1 
 
 0\s;5 r: 0'3333333*3 •iZ-O.gO* =r 27 •0.9696^)69 
 1 C6-0<) = 1 :25 'QfmoQOO 0'9 1 3 r= O'.0 1 39 1 391 
 
 Sum CCJ-JO.572390 Sum 49'43603j12 
 
 SUBTRACTION of REPETENDS. 
 
 Make the rqictemls to bfgin ami end togcthrr, as in 
 addition, Then fubtradl as ufualj onl}', it' the rcpctend 
 of the number to be fubtraL^led exceed tlie rcpctend of 
 the other number, tnak© the laft figure ot' the remainder 1 
 lefs than it otherwife wculd be. 
 
 EXAMPLVS* 
 
 76*32 ^ = 76*3222 89-576 w 69*6760 
 
 54-7617 =: 54*7617 ^2-584^ « 12*5846 
 
 Diff. 21-5604 DifF. 76-9913 
 
 . ( ^ ^ r T . < * ) 
 
 1S?9'21 c= 29;2i2i2i 87-ii6i = 87'4r6ii 
 
 S'k>l = 3*561561 3-532 = 0-5323*2 
 
 DifF. 25*650559 Diff; 8f;-8«38*l 
 
 MUL- 
 
^0 Ifil^JVOpDUCTIOV. 
 
 MULTIPLICATION of REPETENDS. 
 
 T. When a rcpetcnd is to be multiplied by a finite 
 number: iMuItiply as in common numbers; only obfcrvc 
 what muft he carried tVom the beginning of the repelend 
 to the cjnd of it. And make all the lines begin atid end 
 together when they are to be added. 
 
 2. In multiplying: a finite decimal by a fmgle rcpetcnd; 
 multiply by the repetenH, and divide by -9 or /j. 
 
 In more complex cafes, reduce the repctends to vulgar 
 fractions; then divide thefe, and reduce the quotient to a 
 decimal, if ncceflary. 
 
 716-2935 
 •27 
 
 EXAMPLES. 
 
 ( 2 ) ( 3 ) 
 
 2-loi 3 023 
 1*2 17 
 
 50140548 
 143258711 
 
 4208 21202' 
 21044 30288 
 
 393.3-99266 
 
 2-5253 51-491 
 
 (4 ) 
 
 27'1241 
 3-6 
 
 Or 3-6 = 3| = 3f = ^ 
 Then 27-1241 
 11 
 
 9 ) 162744(J 
 
 1808273 
 813723 
 
 3 ) 298-3651 
 9945503 
 
 99«45503 
 
 
 ( 5 ) 
 
HEPETENDS.* 21 
 
 Mult. 1-266 by 5-5 
 
 ■JTO 
 
 P206 
 
 = 
 
 ].2^ = 1 2/t = 1t^ = 
 
 and 3*5 
 
 = 
 
 3| = V; 
 
 then ff 1 
 
 X 
 
 3^2 = '^V^« = 4-^882 15 
 
 DIVISION OR REPETENDS. 
 
 1. If the dividend only be a repetend, divide as in 
 common numbers, bringing down always the recurring 
 figures, till the quotient become as exaft as requifite. 
 
 2. And if the divifor only be a repetend, it will be beft 
 to change it into its equivalent vulgar fraftion, then mul- 
 tiply bv its denominator, and divide by its numerator, 
 
 3. But if both divifor and dividend be repetends, change 
 them both to vulgar fraftions. 
 
 EXAMPLES, 
 
 (1) (2) ^ 
 
 1-2) 2-5263 8)27-912 
 
 2*10i 3'489027' 
 
 (3) (4) 
 
 17 ) 51-491 ( 3*028 27 ) 193-399''^ 
 
 49 9) 64'46'6'42 
 
 151 7-162935 
 
 151 
 
 5. Divide 
 
INfRODUCTIOK 
 
 5. Divide p9-450:3 by 3'0 cr 3j = 3} 
 
 vJ 
 
 11 ) i?9S-36jio 
 27 -i '241 
 
 6, Divide 4'2b82154 by 3-5 or 3-J- = V* 
 9 
 
 32 38'i939363 
 
 i"^o6' 
 7. Divide 4-'?82 154 by 1-200*. 
 
 Here l-2oG = 1-2^ = 1-2/y = ||« 
 
 And 4-2882 1 5i = 4-2|4i^*-$ = tf||ili^. ^ 
 
 Alien -jiv^sjjs^o + 3*¥ — "Jo3'«TX:3T? — *^ -> 
 
 Or rather thus: 
 
 Having found 1«206' = ff | = ^^2 J-2tt. then 
 4'28S2154 
 10 
 
 42-882154 
 3 
 
 12S-6*i646'4 
 11 
 
 398) 1415-11 (3-5 
 
 2211 
 
 221 
 
 IN- 
 
tHVULt I ION. 25 
 
 INVOLUTION; 
 
 OR 
 
 RAISING OF POWERS. 
 
 A Power, is a number produced by multiplying any 
 given number continually by itfelf a certain number of 
 times. 
 
 Any number is called the firft power of itfelf; if it be 
 multiplied by itfelf, the produ<5l is called the fecond 
 power, and fometimes the fquare; if this be miiltipliod 
 by the f.rit power again, the produft is called the third 
 power, and fometimcs the cube; and if this be multiplied 
 by the firft power again, the prodiift is called the fourth 
 power, and fo on: that is, the power is denominated 
 from the number vvhich exceeds the multiplications by 1, 
 
 Thus: 3 is the firft power of 3, 
 
 3 X .0 r= .9 is the fecond power of 3. 
 
 3 X 3 X 3 =: 27 is the third power of 3. 
 
 3x3x3x3 = 81 is the fourth power of 3, 
 
 Sec, &€, 
 
 And in this manner may be calculated the following 
 table. 
 
 TABLE 
 
TABLE of the Firfi Twelve Powers of Numbers* 
 
 o> " 
 
 a. 
 
 „ 
 
 05 
 
 
 
 ^ 
 
 Sd 
 
 1 ** s 
 
 ,1,' 
 
 
 CD 
 
 o» 
 
 « 
 
 
 1 
 
 
 04 
 
 
 ao 
 
 
 
 t-» 
 
 >c 
 
 
 
 1— 
 
 
 § i 
 
 •^ 
 
 
 
 
 «o 
 
 o> 
 
 
 Q>| 
 
 O 
 
 o 
 
 CO 
 
 
 
 
 
 
 C»5 
 
 
 •^ 
 
 
 1 QO m 
 
 
 
 
 
 
 
 »o 
 
 ^ 
 
 ^ 
 't 
 
 
 34867 
 313810 
 
 
 
 
 
 
 
 
 
 
 ' ' 
 
 «N 
 
 oo~ 
 
 
 2i 
 
 i 
 
 
 •^ 
 
 -!»• 
 
 2 
 
 CO 
 
 
 •* 
 2 
 
 1 
 
 
 
 
 
 ^ 
 
 o» 
 
 o» 
 
 r— 
 
 t- 
 
 r- 
 
 
 ^ 
 
 CO 
 
 
 
 
 
 M 
 
 «5 
 
 
 
 s 
 
 
 o? 
 
 s 
 
 
 
 
 
 J 
 
 
 
 «o 
 
 "1 
 
 
 S 
 
 i 
 
 OS 
 
 i 
 
 t- Oi 
 
 o*? 
 
 _ 
 
 f^ 
 
 a> 
 
 Ci 
 
 ^ii^ 
 
 t— 
 
 1 
 
 H2 
 
 v< 
 
 
 •^ 
 
 « 
 
 § 
 
 Xl 
 
 s 
 
 S 
 
 s 
 
 s 
 
 
 § 
 
 
 
 
 0>< 
 
 
 r- 
 
 
 3 
 
 
 »n 
 
 CO 
 
 1— 
 
 
 
 
 
 
 
 ^ 
 
 
 »— 
 
 
 ao 
 
 
 
 
 
 
 ,_4 
 
 t- 
 
 (J>) 
 
 •* 
 
 fl 
 
 O' 
 
 
 
 
 
 
 
 1 
 
 1 
 
 o 
 
 ? 
 
 1 
 
 s 
 
 QO 
 
 «C ts 
 
 cs 
 
 o 
 
 (D 
 
 «3 
 
 CO 
 
 2 
 
 o 
 
 CO CD 
 
 "~2 
 
 
 W 
 
 
 o 
 
 V 
 
 
 en 
 
 
 05 
 
 t- irj 
 
 Vi 
 
 
 
 0>J 
 
 w 
 
 r— 
 
 
 C3 
 
 o 
 
 «c 
 
 
 o 
 
 
 
 
 
 
 t- 
 
 <o 
 
 a. 
 
 a> 
 
 r- 
 
 CO 
 
 t- 
 
 OD 
 
 
 
 
 
 
 
 t- 
 
 h- 
 
 
 CO 
 
 o 
 
 
 
 
 
 
 
 CJ 
 
 co 
 
 
 
 
 r- 
 
 
 
 
 
 
 
 
 
 2 
 
 1 
 
 CO 
 
 
 
 
 
 
 
 
 
 
 
 ^ ! 
 
 
 ^ !^ 
 
 2 
 
 i 
 
 cS 
 
 i 
 
 2 
 
 i 
 
 i-1 
 
 i £!l 
 
 i 
 
 
 
 
 
 ;5 
 
 
 QD 
 
 i 
 
 
 
 §5 
 
 
 rj. ,o" 
 
 s 
 
 s 
 
 ^ 
 
 o 
 
 "* 
 
 1^ 
 
 1 •«♦« 
 
 1 -* 
 
 152 .S 
 
 c^ 
 
 
 
 
 O) 
 
 
 ^ 
 
 
 »f5 
 
 
 
 !M 
 
 
 
 
 
 ,^ 
 
 O 
 
 «f5 
 
 c^ 
 
 00 Tf 
 
 t- 
 
 
 
 1 
 
 
 
 
 " 
 
 to 
 
 CO 
 
 01 
 
 
 5 
 
 ' 
 
 1 
 
 
 
 
 . 
 
 
 
 1 
 
 "" 
 
 11 ' d 
 
 1 r- 
 
 ^, 
 
 cr; 
 
 O 
 
 t- 
 
 
 CO 
 
 1 C5 , r- 
 
 ^. 
 
 
 
 ! S5 
 
 1 
 
 QO 
 
 ^ 
 
 
 2 
 
 
 i 
 
 5904 
 17714 
 
 1 
 
 55 — 5?- 
 
 00 
 
 1 « 
 
 • 0* 
 
 -f 
 
 1 * 
 
 CO 
 
 o* 
 
 :? . ££ 1 
 
 
 
 
 r 
 
 <« 
 
 CO 
 
 '2 
 
 5 
 
 lO 
 
 S IS 1 
 
 O 
 
 
 1 
 
 
 
 
 
 
 — oi 
 
 "* 
 
 ^,^,„,„,^|^,_|^|-,,^.-.„j 
 
 
 
 
 
 
 
 
 
 I — k- 1 
 
 
 i 
 
 S 
 
 $ 
 
 ^ 
 
 i* 
 
 ^ 
 
 E 
 
 ^ 
 
 l( 
 
 ^ 
 
 %- 
 
 ^ 
 
 a. 
 
 o, 
 
 a 
 
 1 a 
 
 a. 
 
 g. 
 
 s. 
 
 Q. 
 
 &. 
 
 E. 
 
 §. 
 
 -s. 
 
 - 
 
 -t3 
 
 s? 
 
 1 -c 
 
 »5 
 
 jC 
 
 
 do 
 
 ( 
 
 o 
 
 ^ 
 
 J3 
 
 ~~" 
 
 
 
 
 
 
 
 
 
 
 
 
1 he number which exceeds the multiplications by 1 , is 
 ■called the index or exponent of/the power : fo the index 
 of the firft power is 1, that of the fecond power is 2, that 
 of the third is 3, and fo on. ^ ^ . ^ . 
 
 Powers are commonly denoted by. writing their indices 
 above the fiift power: fo the fecond power of 3 is de- 
 noted thus, 3^; the third power thus. 3^; the fourth 
 power thus, 3^* j and. fo on: alio the 6th. power of 503, 
 thus, 503^. 
 
 Involution is the finding of powers j to do which, from 
 their definition there evidently conies this rule. » 
 
 Rule. 
 
 Multiply the given number, or firft power, continually 
 by itfelf, till the number of naultiplication be 1 lefs than 
 the index of the power to be found, and the laft prodadl 
 will be the power required. 
 
 Noie 1. Becaufe fradlions are multiplied by taking 
 the produds of their numerators and of their denorainn- 
 . tors, they will be involved by raifing each of their terms 
 to the power required. And if a mixed number be pro- 
 pofed, either reduce it to an improper fraftion, or reduce 
 the vulgar fra<5lion to a decimal, and proceed by the 
 rule. 
 
 2. The raifing of powers may be fometimes. fhortened 
 by working according to this obfervation, viz, wiia.tevcr 
 two or more powers are multiplied together, their pro;^ 
 dud^ is the power whofe index is the fum of the indices oT 
 the fadors; or if a power be multiplied by itfelf, the 
 produft will be the power whofe index is double of that 
 which is multiplied ; fo if wc would find^the fixth power, 
 we might multiply the given number twice by itfelf for 
 the third power, then the third pqywer into itfelf would 
 give the fixth pqwer ; or if we ^ould find, the fevenih 
 power; wc might firft find the third and fourth, and rbeir 
 
 Krodu^ would be the feventhj or laftly, if we would 
 C find 
 
26 INTRODT^CTIOIC. 
 
 find the eighth power, wc might firft find the fecond, 
 then the fecond into itfclf wouldbc the fourth, and this 
 into itfclf would be the eighth. 
 
 tXAMrXE 1. 
 
 EXAMrLE 2. 
 
 For the fquarc of 45. 
 
 For the fquare of '027 
 
 45 1ft power. 
 
 45 
 
 225 
 180 
 
 •027 
 •027 
 
 189 
 54 
 
 2025 = 45» 
 
 *ooo729 = -e^?*' 
 
 EXAMPLE 3. 
 
 EXAMPLE 4* 
 
 :F.ox the cube of 5*5 
 
 Tor the fottith power of 51 
 
 3-5 
 3-5 
 
 175 . 
 
 105 
 
 5-1 
 •6*1 
 
 51 
 
 255 
 
 12*25 
 3-5 
 
 26-01 ^ 5*t* 
 26-01 ditto 
 
 6125 
 36f6 
 
 2601 
 15606 
 5202 
 
 42'875 = 3-5» 
 
 676*5201 = 5'1* 
 
 IJI- 
 
BXAMFLE 5. tXAMTLt 6, 
 
 For the fifth power of *29* For the iixth power of 2*S, 
 
 2-6 
 2-« 
 
 •29 
 
 
 
 
 261 
 
 
 ^8 
 
 
 •0841 = '29* 
 •0841 ditto 
 
 
 
 841 
 
 3364 
 6728 
 
 
 '00707281 = 
 '29 = 
 
 *29* 
 lit 
 
 6365529 
 1414562 
 
 
 •0020511149 = 
 
 •29* 
 
 156 
 
 52 
 
 6-76 =: 2 6» 
 2-6 
 
 4056 
 1352 
 
 17-576 = 2-63 
 17'576 dit. 
 
 105456 
 123032 
 87 8. SO 
 
 123032 
 
 17576 
 
 308-915776 - C»6* 
 
 £x, 7. The fquare of f is f x y = | 
 
 Ex. 8. The cube of | is I x 5 X t = tI? 
 
 £x»^. The fijuarc of 3| or y is ^ x V-V? 
 = ll||=ll-56. 
 
 >c $ 
 
 EVCV 
 
^5 INTIODUCTIO-N-. 
 
 EVOLUTION; 
 
 OR 
 
 EXTRACTION OF ROOTS. 
 
 The root of any given number, or power, is fuch a. 
 fiumber, as being multiplied by itfclf a certain number of 
 times, will produce the power; and it is denominated 
 the firft, fecond, third, fourth, &c. root, refpc^ively, as 
 ihe number of multiplications made of it to produce the 
 given power, is 0, 1, 2, 3, &c ; that is, the name of 
 the root is taken from the number which exceeds the 
 multiplications by 1, like the name of the power in invo- 
 lution. 
 
 The index of the root, like that of the power in invo- 
 lution, is 1 more than the number of the multiplicaiions 
 rieceflary to produce the power or^iven number. So 2 is 
 the index of the fecond or fquare root ; and 3 the index of 
 4he 3d or cubic root j and 4 the index of the 4th root; 
 and fo on. 
 
 Roots are fometimes denoted by writing ^ before the 
 power, with the index of the root againft it : fo the third 
 root of 50 is \/ 50, and the fecond root of it is ^ 50, 
 the index 2 being omitted ; which index is always under- 
 Hood when a root is named or written without one. But 
 if the power be exprefled by feveral numbers with the 
 lign -f or — , &c# between them, then a line is drawn 
 from the top of the fign of the root, or radical fign, over 
 all the parrs of it ; fo the third root of 47 — 15, is 
 8/4.7 — 15. And fometimes roots are defigned like 
 powers^ with the reciprocal of the index of the niot 
 2 above 
 
I 
 
 EVOLUTION. "5 
 
 above the given number. So the root of 3 is 3% the 
 
 root of 50 is 50*, and the third root of it is 50^ ; alfo 
 
 the third root of 47 — 15 is 4? — 1 5^ or (47 — ]5) ^. 
 And this method of notation has juftiy prevailed in the 
 modern a'gehra ; becaufe fuch roots, being confidcred as 
 fractional powers, need no other diredions for any ojera- 
 tions to be made with them, but thofe for integral powers, 
 
 A numbtr is called a complete power of any kind, whetr 
 its root of the fame kind cm He accurately cxtratled ; but 
 if not, the number is called an imperfcd power, and its 
 lood a furd or irrational quantity. So 4 is a comple'c 
 y,ower of the fecond kind, its root being 2; but an im- 
 perfeft power of the third kind, its third root being a furd 
 quantity, which cannot be accurately extradled. 
 
 Evolution is the finding of the roots of numbers, 
 either accurately, or in decimals to any propofcd decree 
 lif accuracy. 
 
 The power is firft to be prepared for extradlion, or 
 evolution, by dividing it, by m.eans of points or commas, 
 from the place of units, to the left hnnd in integers, 
 and to the right in decimal fradions, into period?, con- 
 faining each as many places of figures as are denoted 
 by the index of the root, if the power contain a complete 
 number of fuch periods; that is, each period to have 
 two figures for the fquare root, three for the cube root, 
 four for the fourth root, and fo on. And when the laft- 
 period in dccitrals is not complete, ciphers are added to 
 complete ir. 
 
 Noie. The root will contain juft as marry places g^ 
 figures, as there are periods or points in the given power; 
 and they w 11 be integers or decimals, refpc^cively, as the 
 periods are (o, from which they are found, or to which 
 they correfpond ; that is, there will be as many integer 
 or decimrd figures in the root, as there are periods of in- 
 tegers or decimals in the given numbe», 
 
 C 3 T« 
 
5d. lUTRODUCTlOir. 
 
 TO IXTRACT THE S(^ARS ROOT, 
 
 1. Having divided the gircn number into periods of 
 two figures eaeh, find, from the table of powers in page 
 24, or otherwife, a fquare niHTiber either equal to, or the 
 next 1 fs than the firft period, which fubtradl from it, and 
 place the root of the fquarc on the right of the given 
 number, after the manner of a quotient in divifion, for 
 the firit figure of the root required. 
 
 2. To the remainder annex the fecond period for a di- 
 vidend; and on the left thereof fet the double of the root 
 already found, after the manner of a divifor. 
 
 3. Find hovv often the divifor is contained in the divi~ 
 dend, wanting its laft figure on the right hand; place thaC, 
 number for the »ext figure in the quotient, and on the right 
 of the divifor, as alfo below the Uwc, 
 
 4.. Multip/y the whole increafed dirifor by it, placing 
 the produd ix'low the dividend, and fubtraft it from it, 
 and to the remainder bring down the next period, for a 
 new dividend; to which, as before, find a divifor by 
 doubling the figures already found in the root; and fr.»m 
 thefe find the next figure of the root, as in the lafl article; 
 eot.inuing the operation ftill in the fame manner till all 
 tlie periods be ufed, or as far as you pleafe, 
 
 jSofc, Inftead of doubling the root, to find the new 
 diviiors, you may add the lalt divifor to the figure belovr 
 it. 
 
 T« prove the work,, multiply the root by itfelf, and to 
 t^e produd add tbe>refflainder« and the funa will be the 
 given nurabgr.. 
 
 Kx, 
 
StQtfARX. ROOT,- 
 
 E«r 1 . To extraa the root of 17'3055. 
 
 ;u 
 
 17*30,56 ( 4*l6- 
 16 
 
 81 
 
 J 
 
 130 
 81 
 
 821) 
 6 
 
 4956 
 
 Having divided the given 
 number into three periods, 
 namely 17) and 30, and 56, we 
 find that l6 is the next fqoare to 
 17, the firft period, which fct 
 below, and fubtrading, 1 re- 
 mains, to which bring down SO, 
 the next period, and it makes 1 30 
 for a dividend. Then 4, the root 
 of l6, is fet on the right of the. 
 given number for the firfl: figure of the root, and m 
 double, or 8, o-n the left of the dividend for the firlt 
 figure of the divifor; which being once contained in ]'3, 
 the dividend wanting its laft figiire,^ gives 1 for the next 
 figure of the root, which 1 is accordingly fct in the root, 
 roaking 4*1, and in the divifor making S), as alfo helnT<r 
 tin fame. Thffe multiplied, make alfo 81, which f-t 
 below tlie divider;d, ;ind fuhtradling, we hr.>'e 49 rcn-nin- 
 jng, to which the lad period 56 being brought down, it 
 makes 4956 for the new dividend. Then, for a new di- 
 vifor, either double the root 4*1, or clfe, which is eaiicfr, 
 to the laft divisor add the figure 1 ftanding below it, and 
 either way gives 82 for the iirft part of the new divifor. 
 Thjs 82 is 6 times contained in 495, and therefore 6 is 
 the next figure, to fet in the root, and in the divifor, as 
 alfo below the fame; which being then multiplied by it, 
 gives 4956,. the fame as the dividend; therefore nothing, 
 j«mains, ami 4*l6 i« the root of 17'$056, ascequired. 
 
 e-4^ 
 
 X- 
 
32" JVTRODUCTTOK- 
 
 EXAMPLL 0. 
 
 For the Root of. 2025, 
 
 20,25 ( 45 rooV 
 16" 
 
 EXAMPLE 3. 
 
 For the root of *000729. 
 
 •rO,()7,2f) ( -027 root 
 4. 
 
 S5 425 
 
 5 425 
 
 47 
 7 
 
 329 
 329 
 
 Note, When all the periods of the given number are 
 !)rought down and ufed, and more figures are required to 
 be found, the operation may be continued by adding as 
 many periods of ciphers as we plcafe, namely, annexing 
 always two ciphers at once to each dividend. And when 
 the root is to be extracted to a greater number of places, 
 the work may be much abbreviated thus: having pro- 
 ceeded in the extraftion after the common method tillyoa 
 have found one more than half the required number of 
 figures in the root, the reft may be found by dividing the 
 laft remainder by its correfpondin^ divifor, annexing a 
 cipher to every dividaal, as in divilion of decimals ; or 
 rather, without annexing ciphers, by omitting continually 
 the right hand figure of the divifor, after the manner of 
 the third contraftion in divifion of decimals in page 10. 
 
 So the operation for the root of 2, to 12 or 13 places, 
 may be thus. 
 
 EX- 
 
SQUARK ROOT. 
 
 EXAMPLE 4. 
 
 2 ( 1-414213562373 root, 
 
 1 
 
 3a 
 
 24 
 4 
 
 100 
 
 96 
 
 281 
 1 - 
 
 4ca ' 
 
 281 
 
 2824- 
 4 
 
 1 lyoo 
 
 28282 
 2 
 
 60400 
 56564 
 
 2a2841 
 1 
 
 ^ 383600 ^ 
 282841 
 
 282842 
 
 3 10075900 
 3 8485269 
 
 2828426 ) ' 1590631 ( 562373 
 ...... 170418 
 
 6712 
 1055 
 
 S 
 
 Here having found the firft (even figures 1*414213 by 
 the common extraction, bv adding always periods of ci- 
 phers, th' laft fix figures 562373 a-e found by the method 
 of contrai^ed divifion in decimal^, without adding ciphers 
 to the remainder, but only pointing off a figure at eaeli 
 time from the laft divifor. 
 
 And the fame hi the two following examples, 
 
 C 5 EX- 
 
 u 
 
5* IN'TROD-JCTTOir. 
 
 EXAMPLE 5. EXAMPLE 6, 
 
 For the rocPt of 3. For the root of 5. 
 
 3 ( 1-732051 root 5 ( 2*236o6S r«or 
 
 1 4 
 
 57 
 7 
 
 200 
 189 
 
 343 
 3 
 
 1100 
 1029 
 
 3-^62 
 2 
 
 7100 
 692* 
 
 42 I 100 
 2 I 94.' 
 
 443 I 1600 
 3 I 1329 
 
 4466 1 27100 
 6 I 26796 
 
 3464 ) 176 ( 051 4472 ) 304 ( 06S- 
 
 ... 3 *« • 3^ 
 
 In like manner may be found the foffowing Rocfs, 
 
 The root of 6 is 2*449490 
 The root of 7 is 2*64o751 
 The root of 10 is 3-162278 
 The root of 11 is 3*316625 
 
 Rules for the Square Roots of Vulgar Fwftions anil 
 Mixed Numbers. 
 
 Firft prepare all vulgar fraflions,. by reducing them 
 to their leall terms, both for this and all other r#ots» 
 Then, 
 
 1. Take the root ef the numerator and of the deno- 
 minator, for the refpeftive terms of the root required. 
 And this is the bed way if the denoniinator be a complete 
 power. Bat if it be nor, 
 
 2. Multiply the numerator and denominator toMther ; 
 take the root of the pro^u^; this root being made the 
 
SQUARE ROOT, S5 
 
 numerator to the dcnoniinator of the given fraction, or 
 made the denominator to the Humcrator of it, will form 
 the fractional root required. 
 
 ^ If 6 ^ah 
 
 And this rule will ferve whether the root be finite or 
 infinite. Or, 
 
 3. Reduce the vulgar frad^ion ta a decimal, ^ndextraft; 
 its root. 
 
 4. Mixed numbers may be either reduced to improper 
 fradions, and extrafted by the firii or fecond rule : or 
 the vulgar fradion may be reduced to a decimal, then 
 joined to ihe integer, and the root Qi[ the whoip extraded. 
 
 Ex. 1. vl|is| 
 
 Ex. 2. ViVf or V^ '^ I 
 
 Ex. 3. For the root of j% 
 
 Here 
 
 ^orfis 
 
 75 ( -866025 
 64 
 
 root 
 
 
 166 1 
 61 
 
 1100 
 S)96 
 
 
 
 1726 J 
 61 
 
 104UO: 
 
 10356 
 
 
 
 1732 ) 
 ... 
 
 44 ( 02 5 
 9 
 
 
 tx. 
 
 4fr For the 
 
 root of r/^ 
 c 6= 
 
 
 flere 
 
3j 
 
 iNTRObtrcTioir. 
 
 Hore T»^ IS = •4160' ( -64.549/1001 
 '36 
 
 124 
 
 4i 
 
 566 
 496 
 
 1285 
 5 
 
 7066 
 6425 
 
 1290) 641 (497 
 *•• 125 
 9 
 
 TO FIND A MEAN PROPORTIONAL. 
 
 There are varioHs ufes of the fquare root; one of which 
 isrto find a mean proportional between any two numbers, 
 wliich is perfbrnoed thus : Multiply the two given numbers 
 together, then extrad the fquare root out of their product, 
 and it will be the mean proportional fought. 
 
 Ex. 1. To findra Mean Proportional between 3 and 12. 
 
 Here 3 x 1 2 = 36. 
 
 And ^36 is 6, the mean proportional fought. 
 
 For 3 : 6 ; : 6 : 12. 
 
 Ex, 2. Tafinda Mean between 2 and 5.. 
 
 Here 2 x 5 1= 10 ( 3-162.278 the mean required. 
 9 
 
 61 I 100 
 1 I 61 
 
 626 I 39CO 
 6 3756 
 
 0322 
 2 
 
 6524 
 
 14400 
 12644 
 
 1756 ( 
 49* 
 49 
 
 278 
 
 Noie* 
 
SQUARE Root. 
 
 S7 
 
 Noie» By means of the fquarc root alfo we readily find 
 the 4th root, or the 8th root, or the l6'th roof, &c. that 
 is, the root ©f any power whofe index is fome power of 
 the number 2: namely, by extrading fo often the fquare 
 rOdt as is denored by the index of that power of 2; that 
 is, two extra^ions tor the 4fih root, three for the 8th. 
 root, and fo on. 
 
 Thus for the 4th root of 97*41. 
 
 ^7'-^y,00,00 ( 9'86,C)6,50,oO ( 3«14159999 anf, 
 81 9 
 
 18S 
 8- 
 
 1^41 
 1504- 
 
 61 I 86 
 1 61 
 
 
 13/00 
 11796 
 
 624 
 4 
 
 2596 
 2496 
 
 19729 
 
 • 9 
 
 190400 62s 1 
 177561 1 
 
 lOOoO 
 6281 
 
 19738 ) 12839 62825 
 9 
 
 376950 
 314125 
 
 62830 ) 62825 ( 9999 
 6278 
 623 
 
 58 
 
 So that the 4th root of 97*41 is 3*]4159999, which 
 expreffes the circaraference of a circle whofe diameter is 1 
 iieaily, 
 
 OTHER EXAMPLES, 
 
 The 4th root of 21035*8 is 12-0431407. 
 Jfee 4th root of . 2 is M89207, 
 
 TO 
 
9f XXT](,ODUqTION. 
 
 TO 5XTH4CT. THB CUBE ROOT. 
 
 RULB 1. 
 
 1. Pomt the given number into periods of tbree placca 
 each, beginning at units ; and there wiJI l>c a& many inte- 
 gral places in the root, as there are points qvcr the int^era 
 in the given number. 
 
 Q, Seek the greateft cnl^ in the left-hand period ; write 
 the root in the quotient, and the cube under the period ; 
 from which fubtraft it, and to the remainder luring down 
 the next period : Call this the refolvend, under which 
 draw a line. 
 
 3. Under the refolvend, write the triple fqyare of the 
 root, fo that units in the latter may ftand under the place 
 of hundreds in the former ; under the triple feiuarcof the 
 root, write the triple root, /eoioved one place to thp right; 
 nn<i the fum of thefe two lines caJla diviforj under which 
 draw a line. 
 
 4. Seek how often this divifor may be had in, the re- 
 folvend, its right-hand place excepted, iind write the refult 
 in the quotient. 
 
 5. Under the divifor, write the product o.f the triple 
 fquare of the root by the laft qnotient figure, fetting the 
 units place of this line, under that of tens in the divifor; 
 under this line, write the prpdu(5t of the triple root by 
 the fquare of the laft quotient figure, let this line be re- 
 moved one place beyond the right of the former: and 
 tinder this line, removed one place forward to tht; right, 
 fct the cube of the laft quotient figure ; the fum of thefe 
 three lines caH the fubtrahend, under which draw, a line. 
 
 6. »Sttbtraft the fubtrahend from the refolvend; to the 
 remainder bring down the next period for a new refolvend; 
 the divifor to this, rauft be the triple fquare of all the 
 quotient added to the triple thereof, and fo on as in th« 
 third articJc Sc^, 
 
CUBE »00T# SJI 
 
 1>XAMPLE 1. 
 
 What is the cube root of 48228544.? 
 
 48228*541 ( 3^4 
 27 
 
 21228 Refolvend 
 
 .A A / 27 Triple fquare of 3 > .y^ ._^ 
 
 *^^l 09 Triple of 3 jt,iCfoor. 
 
 279 Divifor 
 
 r 162 Triple fquare of 3 multiplied by 6 
 
 add< 324 Triple of 3 mullipli«d by fquare of 6 
 
 \ 216 Cube©f6 
 
 Subtrahend 
 
 Refolvend 
 
 Triple fquare of 36 7 . 
 Triple of 36 - j the root 
 
 Divifor 
 
 Triple fquare of 36 mulr, hy 4 
 Triple of 3llf mulf, by fquare of 4 
 Cube of 4 
 
 1572544 Subtrahend 
 
 If the work of this example be well confidered, and 
 eomp^ed with the foregoing luie^ it will be cafy to con- 
 eeive how any other example of the fame kind may be 
 wrought. And here obferve, that when the cube root is 
 extrafted to more than two places, there is a neceffity of 
 doteg feme work upon a fpare piece of paper, in order 
 
 to 
 
 19656 
 
 1572544 
 
 add. 
 
 388» 
 10$ 
 
 
 389S5 
 
 add/ 
 
 16552 
 172s 
 64 
 
4^ XNTRODUCTIOW. 
 
 to come at the root's triple fquare, and the produtfl of the 
 triple root by the fquare of the quotient figure, &c. 
 
 Jn this example, the given number is a cubic number; 
 and therefore at the end of the operation there remained 
 nothing ; for 3()4 multiplied by 36*4-, and the produd mul- 
 tiplied by 36'4 again, gives 48228344, the given number. 
 
 But if the number given be not a cubic number ;^ then, 
 to the laft remainder always bring down three ciphers,., 
 and vroik anew for a decimal fraftion if needful, 
 
 MORE EXAMPLES. 
 
 What is the cube root of 
 
 SH90]7\ 
 
 l()5)-2727/ 
 
 27054O36'008 > Anfwei 
 
 21936532779^ \ 
 
 122615327232J 
 
 Thefe examples are all performed in the fame manner as 
 tlae foregoing one. 
 
 TO FIND TWO MEAN PROPORTIONALS. 
 
 There are many ufes of the cube root: one is to find 
 two mean proportionals between two given numbers; 
 which is performed thus : 
 
 Divide the greater extreme by the lef<>, and the cube 
 root of the quotient multiplied by the lefs extreme, gives* 
 the lefs mean. Multiply the faid cube root by the lefs 
 mean, and the produdl is the greater mean proportional. 
 
 No/e. This is only und<*rllood of thofe numlngrs 
 that are in continued g;eometric propoilion. 
 
 EXAMPLE 1. 
 
 What are the two mean proportionals between 4 and 
 
 108? 
 
 10& 
 
€UEE ROOT. <!*• 
 
 108 Divided by 4 gives 27, whofe cube root is 3 : and 
 the lefs extreme 4, multiplied by 3, .gives 12 for tr.e lefs 
 mean ; and l^ multiplied by the faid root 3, gives 36 for 
 the greater mean. 
 
 For i is to 12 as 12 to 36' and as 36 to 108. 
 I 
 
 EXAIvIFl-E II, 
 
 To find tvi'o geometrical means between 8 and 1728 ? 
 
 Here 8 ) 1728 ( 2l6, whofe cube root is 0'. Then 6 
 times 8 is 48, the kfs mean, and 6 times 48 is 2Sy, the 
 greater mean. ^ 
 
 For 8 is to 48 as 48 to 288 and as 288 to 1728. 
 
 If the rule already given for the cube root be thought 
 too tedious, the following one will be found much more 
 eafy and ready for ufe. 
 
 RULE ir. 
 FOR THE CL'BE ROOT. 
 
 1. By trials take tbe nearefl rational cube to the giveft 
 •eube or number, and call it the afTumed cube. 
 
 2. Then fay, as the fum of the given number and double 
 the afTumed cube, is to the fum of the afTumed cube- 
 and double the given number, fo is the root of the afTumed 
 cube, to the root requireti, nearly. Or as the firfl 
 fum is to the difference of the given and afTumed cube, 
 fo is the afTuraed ropt, to the diflerence of the roots 
 nearly. 
 
 3. Again, by ufing, in like manner, the cube of ike 
 root lait found as a new afTumed eube, another root will 
 be obtained ftill nearer. And fo on as far as we p'eafe; 
 ufing always the cube of the lafl-found root, for the af- 
 fumed cube, 
 
 EXAM* 
 
To find the cube root o{2W3S'9, 
 
 Here we foon fiml. that the root lies between QO and 
 ;iO, and then betuccn 2? and 28. Taking therefore w7» 
 m cube is 1 9(>\^3 the afTumcd cube. Then 
 15683 i! 1113 5 '8 
 
 2 2 
 
 3i)365 4 207V(i 
 
 i: 1635-5 106S3 
 
 As 60401*8 : 6l754-() : : 27^: 27-6f tjr 
 22- 
 
 4 .'^2? 8 2 2 
 123509'^ - 
 
 eo^OJ'S ) H;0, 374-2 C 2?'50i7 the r©otn?8ilf. 
 4'^(;33S 
 36'525 
 
 42 
 ^ Agahi, forafccond operation^ the culic of this root 
 IS 210KV318645155823, and the piocc{s by the Utter 
 method will be thus: 
 
 21035»318645&c. 
 2 
 
 4207O-6372i;0 2103.5»a 
 
 21035-8 2 1035-31 8645 &c, 
 
 -' " ' '" ■ — >— « ' -I ' m m, 
 
 As63\06'-^a7'^9O : dif. •48135.5 : : 27-6047 : 
 
 the dif, •(K)02 1083^4. 
 
 confcq. the root req. is 27'6'045J0834, 
 
•EWERAL ROOT. 45 
 
 TO EXTRACT ANY ROOT WHATEVER. 
 
 Let G be the given power or number, n the Index of 
 the power, a the aflumed power, r its root, r the required 
 root of G, Then 
 
 As the fum of » + 1 times a and n — 1 times o, is. 
 to the fum of « -j- 1 times c and n — 1 times a, fo is 
 the aflumed root r, to the required root r. 
 
 Or, as half the faid fum of w -|- 1 times a and n — 1 
 times G, is to the difference between the giren and affumed 
 powers, fo is the affumed root r, to the difference between 
 the true and adumed roots: which dii&rence added or 
 fubtraded, gives the true root nearly. 
 
 That is, 7z -f- 1, A •{- /z — 1. G : « 4- 1. c -f w — J, 
 A : : r : R. 
 
 Or, »+ ].^A + »— rl.|:G:Acr) •:ir:Rc/3r, 
 
 And the operation mav be repeated as often as we pleafe, 
 Vy uling always the laft found rn(it ior t!ie aiiiimcd roor» 
 and its «ih power for the affumed power a. 
 
 EXAMPLE. 
 
 To extraa the fifth root of 21035'». 
 
 Here it appears that the 5th root is between 7'Z and 
 7*4. Taking 7-3, its 5th power is 2073071 593* Mcacc 
 then we have 
 
41. 
 
 l\TRODUCTIO?r. 
 
 G = 2103.5VS; r =: 7*3;« = 5; {. » 4- 1 = 3; 
 
 and ;. « — . 1 = 2 
 
 21035*8 
 
 2 
 
 4?U7 1 -6* 
 
 A» 10420*;57 : 305-084 : : J'3 : •051060* 
 7*3 
 
 
 G- 
 
 = 20730 7 1() 
 
 
 - A = 305 084 
 
 
 A 
 
 A 
 
 C 
 
 = 20730-7 1(> 
 3 
 
 
 - 4207 1 -6; 
 
 913232 
 2135588 
 
 
 104^637 ) 2227*1132 ( 
 
 •0213(505 
 
 
 14184 7-3 =r add 
 
 
 3758 7 
 
 •321;5(;0 rr R thf 
 
 
 630 root true to the 
 
 
 5 laH figure.. 
 
 
 OTHER EXAMPLES, 
 
 
 K 
 
 What IS the 3d. root of 2? 
 
 Anf. 1-259921. 
 
 2. 
 
 What is the 4th root of 2? 
 
 Anf. 1-189207.. 
 
 3. 
 
 What is the 4th root of f)7-41? 
 
 Anf. 3-14159.9. 
 
 4. 
 
 What is the 5th root of 2? 
 
 Anf. 1- 148699. 
 
 5. 
 
 What is the 6"th root «'f 21035*8? 
 
 Anf. 5-254037. 
 
 6. 
 
 What is the 6"th root of 2? 
 
 Anf. IM224()2, 
 
 7. 
 
 What is the 7fh root of 21035*8? 
 
 Anf. 4-145392. 
 
 8. 
 
 What is the 7th root of 2? 
 
 Anf. 1-104089. 
 
 9. 
 
 What is the 8th root of 21035*8? 
 
 Anf. 3-470323. 
 
 10. 
 
 What is the 8rh root of 2? 
 
 Anf. 1-090508. 
 
 11. 
 
 What is the 9th root of 21035*S? 
 
 Anf. 3 •022239. 
 
 12. 
 
 What is the 9th root of 2? 
 
 Anf. I-O8OO59. 
 Gei<bral 
 
EvoLUTiey, 
 
 45 
 
 General Rules for extrading any Root out 
 Vulgar Fraftion or Mixed Number. 
 
 of 
 
 If the given fradion have a finite root of the kind re- 
 quired, it is beft to extrad the root out of the numerator 
 and denominator, for ihe terras cf the root required. 
 
 2. But if the fraftion be not a complete power ; it may 
 'be thrown into a decimal, and then extrnfted. Or, 
 
 3. Take either of the terms of the given fra<5^ion for 
 the correfpnnding term of the root; and for the other 
 term of the root, cxtracf^ the required root of the produd, 
 arifing from the multiplication of fuch a power of the 
 firft afligned terra of the root whofe index is lefs by 1 
 than that of the given power, by the other terra of the 
 given number. 
 
 This rule will do when the root is either finite or 
 infinite. 
 
 That 
 
 V 1 - 
 
 J^fih^^ _ 
 
 ^ba^-^- 
 
 4. Mixed numbers may be reduced either to improper 
 fra£lions or decimals, and then ex traded. 
 
 EXAMPLES. 
 
 What is the cube root of -^j} Anf. - 
 
 What is the 4th root of ^^L? Anf. - 
 
 What is the cube root of {} Anf. 
 
 What is the cube root of 2 iy? Anf. - 
 
 What is tlie third root of 7 -• ? Anf. 
 
 •753700?! 
 
 I "r 1 f. 
 
 1-9^0979^ 
 
 PUO- 
 
•4^ INTR<»I>DCTX«K* 
 
 DUODECIMALS; 
 
 OR 
 
 CROSS MULTIPLICATION. 
 
 Tr\v ©DECIMALS arc the calculations by feet, inches, and 
 •*-^ parts, and are fo called, bccaufe they decreafe by 
 twelves, firom the pbcc of fcer, towanis the right-hand. 
 Inches are fometimes called primes, and are marked thus'^ 
 the next divifion after inches are called parts, or feconds, 
 and are marked thus"; the next are thirds, and marked 
 thus"'; and fo on , 
 
 This rule is othcrwife called Crofs Multiplication, bc- 
 caufe the faftors are fometimes multiplied crofs ways. And 
 it is commonly ufed by workmen and artificers in com- 
 puting the contents of their work; the dimenfions being 
 taken in feef^ inches, and parts; though a much better 
 way would be by a decimal fcald of diviiions. 
 
 RVLE U 
 
 1. Under thcraultipHcand write the fame names or de- 
 nominations of the multiplier ; that is, feet uader fcef, 
 inches under inches, parts under parts, &c, 
 
 2. Multiply each term in the multiplicand, beginning at 
 the loweft by the feet in the multiplier, and fet each refutt 
 under its refpeflivt term, obferving to carry an uuit for 
 «very J 2, fram each lower denomination to its next fu* 
 fwrior, 
 
 3. In the fame manner moltiply every term in the mul- 
 tiplicand by the inches in the multiplier, and fet the refult 
 ot each term one placo removed to the right of tbofe ia 
 tke iii«ltipUca&d« 
 
 I 4, Proi. 
 
-» U O » EC I M A LS. ^:Vf 
 
 4.. Proceed in Tike manrner with the fcconds, and all 
 the reft of the denominations, if there be any more, 
 fetting the produft of each Hne always one place more to- 
 wards the right-ha«d than the line next before, and the 
 fum of all the lines will be the whole produ(f^ required. 
 
 Or the dcnooiinations of the particular produds will 
 be as follow : 
 
 Feet by feet, give feet. 
 ¥€et by primes, give primes, 
 feet by fecoads, give feconds, 
 &c. 
 
 Primes by primes, give feconds. 
 Primes by feconds, give thirds. 
 Primes by thirds, give fourths, 
 &c. 
 
 Seconds by feconds, give fourths, 
 tieconds by -thirds, give Efths. 
 Seconds by fourths, give lix'ths, 
 
 Thirds by thirds, give fixth?. 
 Thirds by fourths, give fevenths. 
 Thirds by £fths, givis eighths, 
 &c. 
 
 Jn general thus: 
 
 When feet are concerned, the produft is of tht fame 
 denominiition with the term multiplying the feet. 
 
 When feet are not concerned, the name of the produ^ 
 is cxpreffed by the fura of the i»idiccs of the two faftori; 
 
 Ex, 1. 
 
4» INTRODUCTION. 
 
 f / /» f ' ' 
 
 Kx, 1. Multiply lOt' 4- 5 bv 7 8 6 
 . 7 ti 6 ' 
 
 72 1) 1 1 
 
 '6 10 11 4 
 
 5 'i 2 G 
 
 ^ 7<) 11 6 6* Anfwer. 
 
 RULE 11. 
 
 When the feet in the multiplicand are expreffed by a 
 large number. 
 
 Multiply firft by the feet of the multiplier, as before. 
 
 Then, inftead of multiplying by the inches and part?. 
 Sec proceed as in the Rule of Pradice, by taking fueh 
 aliquot parts of the multiplicand as correfpond with the 
 inches and ficonds, Sec. of the multiplier. Then the 
 fum of them all will be the prod u ft required. 
 
 £ J II £ / // 
 
 Ex, 2, Multiply 240 10 8 by 9 4 6 
 9 4 6 
 
 6 = 
 
 Y - 
 
 2168 
 
 
 
 
 
 
 - 80 
 
 3 
 
 6 
 
 8 
 
 - 10 
 
 
 
 5 
 
 4 
 
 2258 
 
 4 
 
 
 
 
 
 RULE III, 
 
 If the feet in both the multiplicand and multiplier be 
 large numbers, v /• 
 
 Multiply the feet only into each other : then, for the 
 inches and fcconds in the multiplier, take parts of the 
 
 multi- 
 
DUODEClMALii. 
 
 ^9 
 
 multiplicand ; and for the inches a«d feconds of the mul- 
 tiplicands take aliquot parts of the feet only in the mul- 
 tiplier. Then the fum of all will be the whole produft, 
 f ' " f . ., 
 
 Ex. 3. Multiply 3(38 7 5 by 137 8 4 
 137 8 4 
 
 
 257(3 
 
 
 - 
 
 
 
 1104. 
 
 
 
 
 
 36'8. . 
 
 
 
 
 6' = 4 
 
 . - 184 
 
 3 
 
 8 
 
 6 
 
 2' = 4 
 
 - - 61 
 
 5 
 
 2 
 
 10 
 
 4"= ^ 
 
 - - 10 
 
 2 
 
 10 
 
 5 
 
 6^ = 4 
 
 - - 68 
 
 6 
 
 , 
 
 , 
 
 1' = i 
 
 - - 11 
 
 5 
 
 , 
 
 , 
 
 <:=^ 
 
 - - 3 
 
 9 
 
 8 
 
 , 
 
 l"=z' 
 
 1 
 
 1 
 
 5 
 
 , 
 
 50756 7 10 9 8 Anfwcr. 
 
 ^ I II HI 
 
 4. Mult. 4 7 . 
 by 6 4. 
 
 5. Mult. 14 9 9 
 by 4 6,. 
 
 6. Mult. 4 7 8 
 by 9 6. 
 
 7. Mult. 7 8 6 
 by 10 4 .5 
 
 8. Mult. 39 10 7 
 by 18 8 4 . 
 
 9. Mult. 44 2 9 2 
 by 2 10 3 . 
 
 10. Mult. 24 10 8 7 
 
 by 9 4 6 
 
 OTHER EXAMPLES. 
 
 Anfwers. 
 
 J If m W r 
 
 f 
 
 29 4. . , 
 66 4 6. . . 
 44. 10 , . 
 
 79 11 6 6 
 745 6 iO 2 4 
 126 2 10 8 10 11 . 
 
 233 4 5 9 6 4 . 
 
 A TABLE 
 
TABLE 
 
 OF 
 
 SQUARES AND CUBES, ALSO SQUARE ROOTS AKD 
 
 CUBE ROOTS. 
 
 Num. 
 ber. 
 
 1 
 52 
 3 
 A 
 6 
 6 
 7 
 8 
 
 9 
 10 
 
 11 
 
 12 
 13 
 14 
 15 
 16 
 17 
 18 
 
 J9 
 20 
 21 
 22 
 23 
 24 
 25 
 
 Square. 
 
 Cube. 
 
 1 
 
 1 
 
 4 
 
 8 
 
 . 9 
 
 27 
 
 16" 
 
 64 
 
 Q5 
 
 125 
 
 56 
 
 216 
 
 4-9 
 
 343 
 
 64 
 
 512 
 
 81 
 
 729 
 
 100 
 
 J 000 
 
 121 
 
 1331 
 
 144 
 
 1728 
 
 169 
 
 2197 
 
 «96 
 
 2744 
 
 -225 
 
 3375 
 
 256 
 
 4096 
 
 289 
 
 4913 
 
 324 
 
 58.'i2 
 
 361 
 
 6859 
 
 400 
 
 8000 
 
 441 
 
 9^'6l 
 
 484 
 
 10648 
 
 529 
 
 12167 
 
 576" 
 
 13824 
 
 625 
 
 16625 
 
 Sqtiarc 
 
 Cube 
 
 Root. 
 
 Root. 
 
 1 OOOOOCX) 
 
 1 -000000 
 
 1 •4-142 136 
 
 i -259921 
 
 1-7 520308 
 
 1-442250 
 
 2-0000000 
 
 1-587401 
 
 2-2360680 
 
 1709976 
 
 2'4494S97 
 
 1-817121 
 
 2-6457513 
 
 I 912933 
 
 2-8284271 
 
 2 0(^0000 
 
 3-00(:000() 
 
 2-080084 
 
 3-i6>2777 
 
 2- 15^435 
 
 3-3166248 
 
 2 223980 
 
 3-464 10 1 6 
 
 2'2 89428 
 
 3 6055513 
 
 2-351335 
 
 3-7416574 
 
 2-410142 
 
 3-8729833 
 
 2-466212 
 
 4-00(.0o00 
 
 2-519842 
 
 4-12310..6 
 
 2-571282 
 
 4'2426407 
 
 2 620741 
 
 4-35685)89 
 
 2-668402 
 
 4-472 U.6O 
 
 2-714418 
 
 4-5825757 
 
 2-758923 
 
 4-6904158 
 
 2-802039 
 
 i'795S3\5 
 
 2-843867 
 
 4-8:.89795 
 
 2 881499 
 
 5 '0000000 . 
 
 2.924018 
 

 A TABLE or SQITARES, &r:. 
 
 51 
 
 Num. 
 ber. 
 
 Square. 
 
 Cube. 
 
 Square 
 Root. 
 
 Cube 
 Root, 
 
 2(5 
 
 676 
 
 17576 
 
 5-0990195 
 
 2-962496 
 
 27 
 
 729 
 
 19683 
 
 5-1961524 
 
 3-000000 
 
 28 
 
 784 
 
 21952 
 
 5-2915026 
 
 3-0365S9 
 
 29 
 
 841 
 
 24389 
 
 5-3S51648 
 
 3-072317 
 
 30 
 
 900 
 
 27000 
 
 5-4772256 
 
 3-107232 
 
 31 
 
 961 
 
 29791 
 
 5-5677644 
 
 3-141381 
 
 32 
 
 1024 
 
 32768 
 
 5-656S542 
 
 3-174802 
 
 33 
 
 1089 
 
 35937 
 
 57445626 
 
 3-207534 
 
 34 
 
 II06 
 
 3930i< 
 
 5*830ij519 . 
 
 5-239612 
 
 35 
 
 1225 
 
 42875 
 
 5-9160798 
 
 3-271066 
 
 36' 
 
 1296 
 
 46656 
 
 6-0000000 
 
 3-301927 
 
 37 
 
 1369 
 
 50653 
 
 6 08 27 625 
 
 3-332222 
 
 38 
 
 1444 
 
 54872 
 
 6-1644140 
 
 3-361975 
 
 59 
 
 1521 
 
 59319 
 
 6*2449980 
 
 3-391211 
 
 40 
 
 1600 
 
 64000 
 
 6'32^5553 
 
 3*419952 
 
 41 
 
 1681 
 
 68.921 
 
 6-4031242 
 
 3*448217 
 
 -42 
 
 1764 
 
 74088 
 
 6*4807407 
 
 3-476027 
 
 43 
 
 1849 
 
 79307 
 
 6-5574385 
 
 3-50 398 
 
 44 
 
 1936 
 
 85184 
 
 6*6332496 
 
 3-530348 
 
 4.5 
 
 20J5 
 
 91125 
 
 67082039 
 
 3 '5 56893 
 
 46' 
 
 2116 
 
 97236 
 
 6782:5300 
 
 3'58304-8 
 
 47 
 
 2209 
 
 103823 
 
 6-8556546 
 
 3*608826 
 
 48 
 
 2304 
 
 110592 
 
 6-9282032 
 
 3*634241 
 
 49 
 
 2401 
 
 117649 
 
 7*0000000 
 
 3*659306 
 
 50 
 
 2500 
 
 125000 
 
 7*0710678 
 
 3*684031 
 
 51 
 
 2601 
 
 132651 
 
 7*1414284 
 
 3*708430 
 
 52 
 
 2704 
 
 I40608 
 
 r21 110-6 
 
 3*7325 1 1 
 
 53 
 
 2809 
 
 148877 
 
 7*2^01099 
 
 3756286 
 
 54 
 
 2916 
 
 157464 
 
 7*3484692 
 
 3'779763 
 
 55 
 
 3025 
 
 166375 
 
 7•4l6l9^5 
 
 3 8O2953 
 
 56 
 
 3136 
 
 175616 
 
 6-4893148 
 
 3-825862 
 
 57 
 
 3249 
 
 185193 
 
 7-5498344 
 
 3-84 h50 1 
 
 58 
 
 3364 
 
 195112 
 
 7-6 157 731 
 
 3-870877 
 
 59 
 
 3481 
 
 205379 
 
 7-6811457 
 
 3*8929<,'6 
 
 60 
 
 3600 
 
 216000 
 
 7-7^59667 
 
 C'JUbGr 
 
52: 
 
 A TABLE or 
 
 Num- 
 ber. 
 
 Square. 
 
 Cube. 
 
 Scjiiare 
 Roor. 
 
 Cube 
 Root. 
 
 6*1 
 
 3721 
 
 22698 1 
 
 7-81024.97 
 
 3-936497 
 
 62 
 
 3844 
 
 2:J8328 
 
 7-8740079 
 
 3-957892 
 
 63 
 
 3969 
 
 250047 
 
 7-937^539 
 
 3-97.9057 
 
 64 
 
 AO96 
 
 262144 
 
 8-0000000 
 
 4-000000 
 
 '65 
 
 4225 
 
 274625 
 
 8-0622577 
 
 4-020726 
 
 66 
 
 4356 
 
 287496 
 
 8-1240384 
 
 4-041240 
 
 67 
 
 4489 
 
 300763 
 
 8-1853528 
 
 4-061548 
 
 68 
 
 4624 
 
 314432 
 
 8*24621 13 
 
 4-081656 
 
 69 
 
 4761 
 
 32S509 
 
 "8 3066239 
 
 4«10 1566 
 
 70 
 
 4900 
 
 343000 
 
 8-3666003 
 
 4- 1 2 1 285 
 
 71 
 
 5041 
 
 357911 
 
 8-4261498 
 
 4-140818 
 
 72 
 
 5184 
 
 373248 
 
 8*485281* 
 
 4- 160 168 
 
 73 
 
 5329 
 
 389017 
 
 8«54t0037 
 
 ^-179339 
 
 74 
 
 5476 
 
 405224 
 
 8'6()23253 
 
 4-1.98336 
 
 75 
 
 5625 
 
 421875 
 
 8«6602540 
 
 4-217163 
 
 76 
 
 5176 
 
 438976 
 
 8*7 177.979 
 
 4-235824. 
 
 77 
 
 5929 
 
 456533 
 
 8-7749644 
 
 4-254321 
 
 78 
 
 6084 
 
 474552 
 
 8-8317609 
 
 4-272659 
 
 7i) 
 
 6241 
 
 493039 
 
 8-8881944 
 
 4-290841 
 
 80 
 
 6400 
 
 5 1 2000 
 
 ,8-9-^42719 
 
 4-308870 
 
 81 
 
 6061 
 
 53144.1 
 
 9-0000000 
 
 4-326749 
 
 82 
 
 6724 
 
 551368 
 
 9-0553851 
 
 4-344481 
 
 83 
 
 6S89 
 
 571787 
 
 9*1104336 
 
 4-36-07 1 
 
 84 
 
 7056 
 
 592704 
 
 9-J651514 
 
 ■i-'3795l9 
 
 85 
 
 7225 
 
 614125 
 
 9-2195445 
 
 4-3.96830 
 
 86 
 
 7396 
 
 636056 
 
 9*2736185 
 
 4-414005 
 
 87 
 
 7569 
 
 658503 
 
 9-3273791 
 
 4-431047 
 
 88 
 
 77^* 
 
 68 127 2 
 
 9-3828315 
 
 4-447960 
 
 SP 
 
 7921 
 
 704969 
 
 9*43398 1 1 
 
 4-461745 
 
 90 
 
 8100 
 
 729000 
 
 9 4868330 
 
 4-481405 
 
 91 
 
 8281 
 
 753571 
 
 9*5393920 
 
 4-497942 
 
 f)2 
 
 8464 
 
 778688 
 
 9-59^6630 
 
 4-514357 
 
 f)3 
 
 S649 
 
 804357 
 
 9-6436508 
 
 4-530655 
 
 <M. 
 
 8836 
 
 830584 
 
 9-6953597 
 
 4-546836 
 
 i-' • ' 
 
 9025 
 
 857375 
 
 974679^3 
 
 4-562903 
 
SQUARES^ CUBES, A"S P ROOTS. 
 
 53 
 
 Square, 
 
 Cube. 
 
 Square 
 Root. 
 
 Cube 
 Roof. 
 
 92 1 6 
 
 88473^ 
 
 9-7979590 
 
 4-578857 
 
 9^09 
 
 912673 
 
 9*8488578 
 
 4-594701 
 
 0604 
 
 941192 
 
 9-«9.9^.949 
 
 4 610436 
 
 ()80l 
 
 970299 
 
 9-9493744 
 
 4-626065 
 
 ]0000 
 
 1000000 
 
 10*0000000 
 
 4-641589 
 
 10201 
 
 1030301 
 
 10-0^98756 
 
 4-657010 
 
 10404 
 
 106 1208 
 
 " 10-0995049 
 
 4-672330 
 
 106"09 
 
 1092727 
 
 10-1488916 
 
 4-687548 
 
 10816 
 
 1124864 
 
 IO-I9SO39O 
 
 4-702669 
 
 11025 
 
 1157625 
 
 10-2469508 
 
 4-717694 
 
 11236 
 
 1191016 
 
 10-2956301 
 
 5732624 
 
 11449 
 
 1225043 
 
 10-3440804 
 
 4-747459 
 
 11664 
 
 1259712 
 
 10-3923018 
 
 4-762203 
 
 118S1 
 
 1295029 
 
 10-4403065 
 
 4-776856 
 
 12100 
 
 1331000 
 
 10-4880885 
 
 4-791-^20 
 
 12321 
 
 136/631 
 
 10-5356538 
 
 4-305896 
 
 125i4 
 
 1404928 
 
 .10-5830052 
 
 4-S202S4 
 
 12769 
 
 1442897 
 
 10-6301458 
 
 4-S3458S 
 
 12996 
 
 1481544 
 
 106770783 
 
 4-848808 
 
 13295 
 
 1520875 
 
 10-7238053 
 
 4-862944 
 
 13456 
 
 1560896 
 
 10-7703296' 
 
 ^'&76999 
 
 13689 
 
 1601613 
 
 10-8166538 
 
 4-890973 
 
 13924 
 
 l6i3032 
 
 10-8627805 
 
 4-904868 
 
 14161 
 
 I6S6I59 
 
 10-9087121 
 
 4-9 18685 
 
 14400 ' 
 
 17280CO 
 
 10-9544512 
 
 4-932424 
 
 14641 
 
 1771561 
 
 11 -0000000 
 
 4-946O88 
 
 14884 
 
 1815848 
 
 11-0453610 
 
 4^-959673 
 
 15129 
 
 IS6OS67 
 
 11-0905365 
 
 4^-973190 
 
 15376 
 
 1906624 
 
 11-1355287 
 
 4-986631 
 
 15625 
 
 1953125 
 
 1 1 • 1 803399 
 
 5-000000 
 
 15876 
 
 2000376' 
 
 11-2249722 
 
 5-013298 
 
 16129 
 
 2048383 
 
 11-2694277 
 
 5-026526' 
 
 16384 
 
 2097152 
 
 11-3137085 
 
 5-039684 
 
 16641 
 
 2146689 
 
 11-3578167 
 
 5052774 
 
 16900 
 
 219700b 
 
 11'4017543 
 
 5-065797 
 
J4 
 
 A TABLE OF 
 
 Niim 
 bcr. 
 
 Square. 
 
 Cube. 
 
 Square 
 Roof. 
 
 Cube 
 Root. 
 
 131 
 
 17161 
 
 224}509I 
 
 11-4155231 
 
 5-078753 
 
 152 
 
 17424 
 
 229.990 8 
 
 11*4891253 
 
 5-091643 
 
 133 
 
 I768t) 
 
 2352637 
 
 11-5325626 
 
 5-1044(;9 
 
 13-t 
 
 119^>6 
 
 240()104 
 
 1 1*5758369 
 
 5-1J7230 
 
 136 
 
 18225 
 
 2460375 
 
 11*6189500 
 
 5*12.9928 
 
 1:^6 
 
 18^96 
 
 2515456 
 
 11-6619038 
 
 5-142563 
 
 137 
 
 1876<; 
 
 2571353 
 
 11-7046999 
 
 5»155137 
 
 J 38 
 
 19044 
 
 262 8 07 2 
 
 11-7473444 
 
 5' 1 67 6-^9 
 
 139 
 
 iyj2i 
 
 2685619 
 
 11*7 898261 
 
 5 180I0I 
 
 140 
 
 ic)6(0 
 
 2744000 
 
 11-8321596 
 
 5-1.92494 
 
 141 
 
 i<;88i 
 
 2803221 
 
 11-8743421 
 
 5-204828 
 
 142 
 
 20 J 64 
 
 2803288 
 
 11-9163753 
 
 5-217103" 
 
 143 
 
 20449 
 
 2}' 24207 
 
 11-9582607 
 
 5-22.9321 
 
 144 
 
 20736 
 
 2935984 
 
 12-OOOCOOO 
 
 5*241482 
 
 145 . 
 
 21025 
 
 5048625 
 
 12-0415946 
 
 5*253588 
 
 146 . 
 
 21316 
 
 3112136 
 
 12-0830460 
 
 5-265637 
 
 147 
 
 2 J 609 
 
 3176523' 
 
 12-1243557 
 
 5*277632 
 
 148 
 
 2 1 904 
 
 3241792 
 
 12-1665251 
 
 5 289572 
 
 14,<^ 
 
 22201 
 
 3307949 
 
 12-2065556 
 
 5-301459 
 
 150 
 
 22500 
 
 3375000 
 
 12-2474487 
 
 5-313293 
 
 151 
 
 22801 
 
 «44295 1 
 
 12-2882057 
 
 5*325074 
 
 152 
 
 23104 
 
 3511808 
 
 12-32Si^280 
 
 5*336803 
 
 143 . 
 
 23-109 
 
 S58a577 
 
 12*3693169 
 
 5-348481 
 
 154 
 
 237 16 
 
 3652264 
 
 124096736 
 
 5*360108 
 
 J 55 
 
 24025 
 
 3723875 
 
 12-4498.996 
 
 5-37 1 6S^ 
 
 156 
 
 24336 
 
 37<)64I6 
 
 12-4.S.99960 
 
 5-383213 
 
 157 
 
 24649 
 
 3S69>^93 
 
 12-529.9641 
 
 5-5.946.90 
 
 158 
 
 24964 
 
 3944312 
 
 1 2-56.9805 1 
 
 5-406120 
 
 J 5.9 
 
 25281 
 
 4019619 
 
 12 60.95202 
 
 5-417501 
 
 l6o . 
 
 ,25600 
 
 4096000 
 
 12-6491 106 
 
 5 -428835 
 
 161 
 
 "25921 
 
 .4173281 
 
 V2'6SS5775 
 
 5-440122 
 
 162 
 
 26241. 
 
 4251528 
 
 12*727.9221 
 
 5*451362 
 
 163 . 
 
 26569 
 
 4330747 
 
 12-767 J 453 
 
 5 -4625^6 
 
 164 
 
 2/)*S96 
 
 4110944 
 
 12-8062485 
 
 5-473703 
 
 10 5 
 
 27225 
 
 4492125 
 
 12-8452326 
 
 5*4.^4806 
 
SQUARES, CUBES,. AND ROOTS. 
 
 55 
 
 Num- 
 ber. 
 
 Square. 
 
 Cube. 
 
 Square 
 Root. 
 
 16-6 
 
 27550 
 
 4574:96 
 
 12-8S40987 
 
 lOZ 
 
 2788y 
 
 4657463 
 
 12-9228480 
 
 168 
 
 28224 
 
 4741632 
 
 12-9614814 
 
 36'9 
 
 28561 
 
 4(S26"809 
 
 13 '000 000 
 
 170 
 
 28900 
 
 4 13000 
 
 13-0384048 
 
 371 
 
 2924 1 
 
 500021 • 
 
 13 07669 -8 
 
 172 
 
 29o84 
 
 5088418 
 
 13 1148770 
 
 173 
 
 299'29 
 
 51777U 
 
 13-1529464 
 
 174> 
 
 30.76 
 
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MENSURATION. 
 
 Tl/|ENSURATION is tbe meafuring and eftimating the 
 •^' *- magnitude and dimenfions of bodies and figures: 
 and it is either angular, lineal, fuperficial, or folid, ac- 
 coiding to the ohjeds it is concernccf with. It is 
 ncccrdingly treated in feveral parts: as 1(1, Praftical 
 Geometry, which treats of the definitions and con- 
 Ihuftion of geometrical figures ; 2d, Trigonomelr)', 
 which teaches the calculation and conftrudion of tri- 
 angle}:, or three.fided. figures, and, by application, of 
 other figures depending on them : 3d, Superficial Menfu- 
 ration, or the meafuring the furfaces of bodies; 4th, 
 Solid Menfuratiop, or meafuring the capacities or folid 
 contents of bodies. Bcfide thefe general heads, there arc 
 feveral other fubordinate divifions, as alfo the applicatioa 
 of them to the pradical concerns of life. Of each of 
 which in their order; excepting Trigonometry, which is 
 fully treated of in my large book ot Menfu ration, as alfo 
 iu my New Courfe of Mathematics. 
 
 PRACTICAL GEOMETI^Y. 
 
 DEFINITIONS, 
 
 1. AP^INT has pofition, but 
 -^^ no pirts nor dimenfions, 
 neither length, breadth, nor thick- 
 nefs. 
 
 2. A line is length, without 
 breadth er thicknefe. 
 
 3. A 
 
PRACTICAL GEOMETRY. 
 
 57 
 
 \ ^\ 
 
 
 V 
 
 3. A furface or fuperficies, is an 
 cxtenfion, or a figure of two dimen- 
 fions, length and breadth ; but with- 
 out thicknefs. 
 
 4. A body or folid, is a figure of 
 three dimenfions, namely, length, 
 breadth, and thicknefs 
 
 Hence furfaces are the extremities of folids ; lines the 
 extremities of furfaces; and points the extremities of lines, 
 
 5. Lines are cither right, or curv- v*— »v 
 ed, or mixed of thefe two. — — — a^N 
 
 6. A right line, or ftraight .line, 
 
 lies all in the fame diredion, between . 
 
 its extremities; and is the fhorteft 
 diftance between two points. 
 
 7. A curve continually changes its 
 diredion between its extreme points, 
 
 S. Lines are either paTallel, ob- 
 lique, perpendicular, or tangential. 
 
 9. Parallel lines are always at the 
 fame diftance ; and never meet though 
 ever fo far produced* 
 
 10. Oblique right lines change 
 their diftance, and would meet, if 
 produced, on the fide of the leaft 
 diftance. 
 
 11. One line is perpendicular to 
 another, when it inclines not more 
 on the one fide than on the other. 
 
 12. One line is tangential, or a 
 tangent to another, when it touches 
 it without cutting, when both are 
 produced, 
 
 5 13, An 
 
-6S 
 
 rHACTICAL OEOMLTRY. 
 
 13. An angle is the inclination, 
 or opening oF two lines, having 
 different dire<flions, and meeting in a ^^-^ 
 
 point. '' - - 
 
 J 4. Angles are right or oblique, acute or obtufe 
 
 15. A right angle, is that which 
 is made by one line perpendicular to 
 another. Or when the angles on each 
 fide are equal to one another, the/ ■ 
 
 arc right angles. 
 
 Iv?. An oblique angle, is that 
 which is made of two oblique lines ; 
 and is either lefs or greater than a 
 right angle. 
 
 17' An acute angle is lefs than a 
 right angle. 
 
 18. An obtufe angle is greater 
 than a right angle. 
 
 19. Superficies are either plane or curved. 
 
 20. A plane, or plane fufierficics, is that with which a 
 right line may every way, coincide. Rut if not, it is curved, 
 
 21. Plane figures are bounded either by right lines or 
 curves. 
 
 2^2. Plane fi^u-es that are bounded by right lines, have 
 names according to the number of their fedes, or of their 
 angles ; for they have as many fides as angles ; the leaft 
 number being three. 
 
 23. A figure of t^iree fides and angles, is called a tri- 
 angle. And it receives particular denominations from 
 the relations of its fides and angles. 
 
 24, An equilateral triangle, 
 whofeuiHit -fides are all equal. 
 
 is that 
 
 25 An 
 
DEFINITIONS* 
 
 59, 
 
 12 J. An irofceles triangle, is that 
 which has two fides equal. 
 
 (26. A fcalene triangle, is that 
 whofe fides are all unequal, 
 
 27. A right-angled triangle, is 
 that vyhich has one right angle. 
 
 28. Other triangles are oblique- 
 angled, and are either oblufe or acute* 
 
 29. An obtufe-angled triangle has 
 on^ obtufe angle. 
 
 30. An acufe-angled triangle has 
 all its three angles acute. 
 
 31. A figure of four fides and angles, is called a 
 quadrangle, or a quadrilateral. 
 
 32. A parallelooram is a quadrilateral which has both 
 its pairs of oppofite fides parallel. And it takes the tol- 
 lowing particular names. 
 
 33. A reilangle is a parallelo- 
 gram, having all its angles right 
 ones 
 
 3i„ A fquare is an equilateral 
 red^angle; having all its fides equal, 
 and'all its angles right ones. 
 
 w6 
 
 35, A 
 
60 PRACTICAL GEOMETRY. 
 
 35. A rhomboid is an oblique / / 
 
 angled parallelogram, Z / 
 
 $6, A rhombus is an equilateral 
 rh mboid; having all its fides equals 
 but its angles oblique. 
 
 37* A trapezium is a quadrilate- 
 ral which hath not both its pairs of 
 oppofue fides parallel. 
 
 38. A trapezoid hath only cnc 
 pair of oppofue fides parallel. 
 
 39. A diagonal is a right line 
 joining any two oppofite angles of a 
 quadiiiateral. 
 
 40. Plane figures that have more than four fides are, in 
 general, called polygons ; and they receive other particu- 
 lar names aciording to the number of their fides or angles. 
 
 41. A pentagon is a polygon of five fides; a hexagon 
 hath fix fides ; a heptagon, feven ; an oftagon, eight ; a 
 nonagon, nine ; a decagon, ten ; an undecagonj eleven ; 
 and a dodecagon hath twelve fides. 
 
 42. A regular polygon hath all its fides and all its angles 
 equal. If they are not both ecjual, the polygon is irregular. 
 
 43. An equilateral triangle is alfo a regular figure of 
 three fides, and the fquare is one of four : the former 
 being alfo called a trigon, and the latter a tetragon 
 
 44. A circ!« is a plane figure 
 bounded by a curve line, called the 
 circumference, which is every where 
 ecpi-diftant from a certain point 
 within, called its centre. 
 
 Nou, The circumference itfclf is often called a circle, 
 
 45. The 
 
DEFINITIONS. 
 
 6l 
 
 45. The radius of a circle is 
 a right line drawn from the centre to 
 the circumference. 
 
 46, The diameter of a circle is a 
 right line drawn through the centre, 
 and terminating in the circumference 
 on both fides. 
 
 47. An arc of a circle, is atny part 
 of the circumference. 
 
 48. A chord, is a right line join- 
 ing the extremities of an arc. 
 
 49. A fegment, is any part of a 
 circle, bounded by an arc and its 
 chord. 
 
 50. A femicircle, is half the 
 circle or a fegment cut off by 
 a diameter. 
 
 5 1 . A feflor, is any fart of a 
 circle, bounded by an arc, and two 
 radii dravin to its extremities. 
 
Gt 
 
 PRACTICAL GEOMETRY 
 
 
 
 V y 
 
 52. A quadrant, or quarter.of a 
 circle, is a fedlor having a quarter 
 of the circumference for its arc, 
 and its two radii are perpendicular 
 to each other. 
 
 53. The height or altitude of a 
 figure, is a perpendicular let fall 
 from an angle, or it:- vertex, to the 
 oppofite fuie, called the l)afe. 
 
 54. In a right-angled triangle, the fide oppofite the 
 light angle, is called the hypothenufe ; nnd the other two 
 fides, the legs, or fomciimes the bafe and perpendicular. 
 
 i-'SS. When an angle is denoted by ^) 
 
 three lettcr5, of which one Hands at \ 
 
 the angular point, and the other two 
 
 on the two "fides, that which (lands __ 
 
 at the angular point is read in the ^ AC 
 
 middle. 
 
 56. The circumference of every circle is fuppofed to 
 te divided into 3(0 equal parts, called degrees; and 
 each degree into 60 minutes, each minute into 60 
 feconds, and fo on. Hence a femicircle contains ISO 
 degrees, and a quadrant go degrees, 
 
 57. The meafure of a light-lincd 
 angle, is an arc of any circle con- 
 tained between the two lines which 
 form that angle, the angular point 
 being the centre; and it is eftimated • 
 by the nun>ber of degrees contained 
 in that arc. Hence a right angle is 
 an angle of 90 degrees. 
 
 The definition of folids, or bodies, will be given 
 afterwards^ when wc come to treat ol the nicnfuration of 
 folids. 
 
 TROBLEM^; 
 
PRACTICAL GEOMETRY. 
 
 PROBLEMS. 
 
 PROBLEM I, 
 
 To divide a Ginjen Line AB into Tnvo E^ual Par/s» 
 
 111 
 Frctm the centres A and B, with 
 any dirtance greater than half AB, 
 
 defcribe arcs cutting each other in m a 
 
 and n. Draw the line mCn, and it 
 will cut the given line into two 
 equal parts in the middle point C, ^ ... 
 
 6$ 
 
 PROBLEM ir. 
 
 To divide a Given Angle ABC into T^wo Equal Par ts» 
 
 From the centre B, with any dif- 
 tance» defcribe the arc AC. From 
 A and C, with one and the fame 
 radius, defcribe arcs interfering in 
 m. Draw the line Bm, and it will 
 bifeft the angle as required. 
 
 PROBLEM III, 
 
 To divide a Right Angle ABC into Three Equal Parts* 
 
 From the centre 
 tance, defcribe the 
 the centre A, with 
 crofs the arc AC in 
 centre C, and the 
 the arc AC in m. 
 the points m and 
 Bn, and they will 
 angle as required, 
 
 ^R0« 
 
 B, with any difl 
 arc AC. From 
 the fame radius, 
 n. And with the 
 fam& radius, cut 
 Then through 
 n draw Bm and 
 trifeft the ri^ht 
 
 » C 
 
^4 PRACTICAL GEOMETRT, 
 
 PROBLEM IV^, 
 
 To drarw a Lint Parol It I to a Given Line AB. 
 
 Case i. When the Parallel Line is to ie at a Given 
 Dijlance C, 
 
 From any two points m and n, • 
 
 in the line AB, with a diftancc C ; - ^ - -. ...' ~- ... D 
 
 equal to C, defcribe the arcs r and 
 
 o; — Draw CD to touch thefe a ^ 
 
 arc?, without cutting them, and i^^ u 
 
 it will be -the parallel required. C 
 
 Case 2, When the Parallel Line is' to fafs through a 
 Gvven Point C, 
 
 From any point m, in the line . c x 
 
 AB, with the diftancc mC, de- ^ / TC 
 
 fcribethearcCn. — Fromthecen- 
 
 ,tre C with the fame radius de- A-L- j_^ 
 
 fcribethcarcmr. Take the arc Cn ^ ^ 
 
 in the coropalTcs, and apply it 
 from m to r. — Through C and r 
 draw DE, the parallel required. 
 
 Note. This problem is more cafily effe<Sed with a 
 parallel ruler. 
 
 PROBLEM V. 
 
 To ere^ a Perpendicular from a Given Point A in a Gi'ven 
 Line BC. 
 
 Case 1. When the Point is near the Middle of the Line^ 
 
 On each fide of the point A take r 
 
 any two equal diftances Am, An. •••.'•♦' 
 
 From the centres m, n, with any -'y 
 radius greater* than Am or An; 
 
 defcribe two arcs cutting in r. — JK I C 
 
 Through A and r draw the line ^^^ A ^^ 
 Ar, and it will be the perg^ndiciu 
 lar as required. 
 
 Case 
 
PROBLEMS. ■ 6*5 
 
 Case 2. When the Point is mar the end cf the^tjmjlkf 
 
 
 With the centre A, and any diftanc! 
 defcribe the arc m n s. — From the"^ 
 point*m> with the fame radius, turn ^f 
 
 the compafles twice over on the arc, as 
 at n and s. — Again, with the centres n 
 and s, defcribe arcs interfering in r. — > 
 Then "draw Ar, and it will be the per- 
 pendicular as required. 
 
 Another Method, 
 
 From any point m as a centre, 
 ^ith the radius or dlftance m./^, de- 
 fcribe an arc cutting the given line 
 in n and A.— -Through n and m draw 
 
 a right line cutting the arc in r,— ^*^r- Jf 
 
 Lalily, draw A r, and it will be the ,.•'*-♦* 
 
 perpendicular as required. 
 
 Another Method* 
 
 From any plane fcale of equal 
 parts, fet off Am equal to 4 parts. J.':.^. 
 
 —With centre A, and diftance of 3 
 parts, defcribe an arc — And with 
 
 centre m, and radius of 5 parts, . , 
 
 cross it at n. — D.»aw An for the "fe ^^ JZ 
 
 perpendicular required. 
 
 Or any other numbers in the fame proportion, as 3, 4, 
 5, will do the fame j fiich as 6", 8, 10, &c. 
 
 PRO- 
 
66 
 
 PRACTICAL CEOMETRf. 
 
 PROBLEM VI, 
 
 Frotfi a G:ve» Pohtt A, ouf of a Given Line BC, ta let fait 
 a Perpendicular, 
 
 Case 1. When the Point is nearly 6J>pofitt the Middle of 
 the Lit:e, 
 
 With the centre A, and any dif- 
 tance, dtfcribe an arc cutting BC in 
 m and n.— -With the centres m and 
 n, and ihe fame, or any other ra- 
 dius, dcfcrihe arcs interfering in 
 r. — Draw ADr, for the perpendi- 
 cular requited. 
 
 A 
 
 y>-'v::r: 
 
 birc 
 
 Case 2. When the Point is nearly c^pofite the Etfd of the 
 Line, 
 
 From A draw any line Am to 
 meet BC, in any point m. — Bifcfi 
 Am at p, and with the centre n, and 
 dillar.ce An or mn, defcribe an aro, 
 cutting BC in D. — Draw AD the 
 perpendicular as required, 
 
 Jnof/jer Method, 
 
 From B, or any point in BC, as 
 a-cer.tre, defoiihe an arc through 
 the point A. — From any other 
 centre m in B'J, deftribe another 
 arc through A, and cuuing the 
 former ac again in n. — Through 
 A and n dra-w the line ADn; and 
 AD will be tlie perpendicular as 
 required. 
 
 «.<■' 
 
 B.. 
 
 li 
 
 -C 
 
 '<..\ 
 
 15-^ i 
 
 Notf 
 
PROBLEMS. 
 
 6Y 
 
 Nou, Perpendiculars may be more readily raifed and 
 let fall, in praftice, by mt-ans of a fquare, or by the 
 common parallelogram protradlor. 
 
 PROBLEM VII. 
 
 To diiude a Giveti Line AB into any propoftd Numbgr of 
 Equal Parts» 
 
 From A draw any line AC at 
 random, and from B draw BD 
 parallel to it. — 0\\ each of, thefe 
 line?, beginning at A and B, fet off 
 as many equal parts of any length, 
 as AB is to be divided into. Join 
 the oppofite points of divifion by 
 the lines A 5, 1 4, 2 3, &c, and 
 they will divide the given line 
 AB as required. 
 
 3-JL4.C 
 
 PROBLEM VIII, 
 
 To divide a Given Line AB in the fame "Proportion as 
 another Line CD is Divided % 
 
 From A draw any line AE 
 equal to CD, and upon it tranf- 
 fer the divifions of the line CD. 
 — Join BE, and parallel to it 
 draw the lines 1 1, 'I '2, 3 3, Ic^. 
 and. the*y will divide the line AB 
 as r^uired. 
 
 ?. 4- 
 
 PRO- 
 
€i 
 
 PRACTICAL GEOMETRY. 
 
 PROBLEM IX, 
 
 Jt a Given PohitA, in a Gi^,„ Li»e AB, to mah an Af,glt 
 Equal to a Given Angle C. 
 
 With the centre C, and any «(!ancc. A 
 defcribe an arc mn.— With the centre 
 A, and the fame radius, defcrihe the 
 ^l^ ^^•~-'^«^e ^he diliance rnn between 
 
 the corapafTts, and apply it from r to \ 
 
 s — Then a line drawn through A and \ 
 
 s, will make the angle A equal to the IV C 
 
 angle C as required. 
 
 PROBLEM X, 
 
 At a Given Pcint k.ina Given Line AB, to make an Angle 
 of any propofed Number of Degrees, 
 
 With the centre A, ahd radius equal 
 to 60 degrees, taken from a fcalc of 
 chords, defcribe an arc, cutting A^ 
 m m.— Then take between the cora- 
 pafTes the propofed number of de- 
 grees from the fame fcale of chords, 
 and apply them from m to n. Through 
 the point n draw An, and it will make 
 the angle A of the number of degree* 
 propofed. 
 
 Note. Angles of more than 90 degrees are ufually 
 taken off at twice. 
 
 ^ Or the angle may be made with the protraftor or other 
 jnftrument, by laying the centre to the point A, and its 
 radius along AB; then make a mark n at the propofed 
 numr.er of degrees, ihroueh wliich draw the line An as 
 before. . 
 
 PRO. 
 
PROBLEMS, 
 
 69 
 
 PROBLEM XI. 
 
 I'd meafure a Given Angle A. 
 
 {See the laj} Figure.) 
 
 Defcrlbe the arc mn with the chord of ^0 degrees, as 
 in the Ia(l problem. — Fake the arc mn between the com- 
 paffes, and that extern, applied to the chords, will ^aQv^r 
 the degrees in the given angle. 
 
 Note. When the diftance mn exceeds <)0 degrees, it 
 muft be taken oiF at twice as before. 
 
 Or the angle may be meafured by applying the radius 
 of a graduated arc, of any inOrnment, to AB, as in the 
 laft problem ; and then noting the degrees cut off by the 
 other kg An of the angle. 
 
 PROBLEM xri. 
 To Jin d the Centre of a Circle, 
 
 Draw any chord AB; and 
 bifed it perpendicularly with 
 CD, which will be a diameter. 
 Eifeft CD in the point O; and 
 that will be the centre. 
 
 PROBLEM XIII, 
 
 91? difcrihe the Circumference of a Circle through Three 
 Gii}en Points, 
 
 From the middle point B 
 draw chords to the two other 
 points. — Bifed thefe chords 
 perpendicalarly by lines meet- 
 ing in O, which will be the 
 <:entre, — Thenfromthecentre t 
 
 O, at the didance O A, or OB, 
 or OC, defcribc the circle. Note, 
 
?o 
 
 TRACTICAL GEOMETllY. 
 
 h^ote. In the fame manner may the centre of an arc 
 of a circle be found. 
 
 PROBLEM XIV. 
 
 through a Given Po'wt A, to draiv a Tangent to a Given 
 Circle, 
 
 Case 1 . JVhen A is in the Circumference of the Circle^ 
 
 From the given point A, i\ ,\ c, 
 
 tlraw AO to the centre of the 
 circle. — Then through A draw 
 BC perpendicular to AO, and 
 it vvill be the tangent as re- 
 quired. 
 
 Case 2. When A ij out of the Circumference, 
 
 From the given point A, 
 draw \0 to the centre, which 
 bifed in the pojnt nr:. — With 
 the centre m, and radius mA 
 or mO, defcrihe an arc cutting 
 thegiv<incirckinn. — Through 
 the points A and n, draw the 
 tangent BC. 
 
 PROBLEM xrv 
 
 To fnd a Third Proportional to T'wo Given Lines, AB, AC. 
 
 Place the two given Lines, 
 AB, AC, making any angle at 
 'A, and j » n BC. — In Aa take 
 AE) equal to AC, and draw 
 DE pjraDel t(^ BC. So fliall 
 AE be the thiid proportional 
 to AB and AC. 
 That is, AB : AC : : AC ; AE. 
 
 A 
 
 A 
 
 B 
 
 DB 
 
 TRO. 
 
PROBl-Ii.Mii. 
 
 n 
 
 B 
 
 TACBL-EM XVI. 
 
 ^0 find a Fourth Prepcrtional to Three Given Lines, AB, 
 AC, AD. 
 
 Place two of them AB, AC. 
 making any angle at A, and 
 join BC. Place AD on AB, 
 and draw DE parallel to BC. 
 So fliall AE be the fourth pro- 
 portional r'^quired. 
 
 That is, AB : AC : : AD : AE. 
 
 A~ 
 
 A:- 
 
 JO 
 
 K 
 
 -C 
 
 PROBLEM XVir, 
 
 To find a Mean Proportional letiveen Tivo Given Littes^ 
 AB, EC. 
 
 Join AB and EC in one 
 ftraight line C, and l)i feci it 
 in the poinr O — With the 
 centre O, and radi(i«! < )A or 
 OC, dcfcribe a femicircle. — 
 Erift the perpendicular BD, 
 and it will be the mean pro- 
 xjortlonal required. 
 
 That is, AB : BD : : BD : Ba 
 
 A— 
 
 _C 
 
 — , — 1... 
 
 15 
 3) 
 
 O B C 
 
 PROBLEM XVIII, 
 
 To maice an Equilateral Triangle on a Gi'ven Line AB. 
 
 From the centres A and B, ( 
 
 with the diftance \B, defcribe 
 arcs, interfacing i' C. — Draw 
 AC and EC, and it is done. 
 
 Note, rui ifofceles triangle 
 may be made in the-farae man- 
 ner, taking for the diftance the 
 given length of oiic of the 
 equal fides. 1 
 
 me. 
 
PRACTICAL GEOMITRT, 
 
 PROBLEM XIX. 
 
 To make a Triangle ixfith Three Given Lines, AB, AC, BC, 
 
 With the centre A and diftancc 
 AC, dcfcribc an arc. — With the 
 centre B, anddiftance BC, dc- 
 fcribe another arc, cutting the 
 former in C, — Draw AC and 
 BC, and ABC is the triangle re- 
 quired. 
 
 PROBLEM XX. 
 
 To make a Square on a Given Line AB. 
 
 -C- 
 
 A / 
 
 \u 
 
 A 
 
 ' 
 
 B 
 
 c « 
 
 D 
 
 Draw BC perpendicular and 
 equal to AB. FromA and C, with 
 the diftance AB, defcribe arcs in- 
 terfeding in D — Draw AD and 
 CD, and it is done. 
 
 Another Waj\ 
 
 On the centres A and B, with 
 the diftance AB, defcribe arcs 
 croffing at o. — Bifeft Ao in n.-~ 
 With centre o, and radius on,' 
 crofs the two arcs in C and D. — 
 Then draw AD, BC, CD. 
 
 PROBLEM XXI. 
 ^0 defcribe a ReSinngle^ or a Parallelogram^ of a GUaen 
 Length nnd Breadth. 
 Place EC perpendiciil'^r to D C 
 
 AB. — With centre A, and dif- 
 tance AC, defcribe an arc. — 
 With centre C, and radius AB, 
 d&fcribe another arc, cutting the 
 former in D, — Draw AD and 
 
 B 
 
 CD, and it is done. 
 
 C 
 
 'Note* 
 
PRACTICAL GEOMETRY. 
 
 73 
 
 Note, In the fame manner is defcribed any oblique 
 parallelogram, only drawing BC, to make the given ob- 
 lique angle with AB, inftead of perpendicular to it, 
 
 PReBLEM XXII. 
 
 To make a Regular Pentagon on a Given Line AB» 
 
 Make Bm perpendicular and 
 equal to half AB. — Draw Am, 
 and produce it till mn be equal 
 to Bm.— With centres A and B, 
 and diftance Bn, defcribe arcs 
 interfering in o, which will be 
 the centre of the circumfcribing 
 circle.— Then with ' the centre 
 o, and the fame radius, defcribe 
 the circle; and about the cir- 
 cumference of it apply AB the 
 proper number of times. 
 
 Another Method* 
 
 Make Bm perpendi- 
 cular and equal to AB. D 
 — Bifcdl AB in n; then 
 with the centre n, and 
 diftance nm, crofs AB 
 produced in o. — With 
 the centres A and B, and 
 diftance Ao, defcribe arcs 
 interfering in D, which 
 will be the oppofite angle 
 of the pentagon. — Laft- 
 ly with centre D, and 
 
 radius AB, crofs thofe arcs again in C and E, the other 
 two angles of the figure.— Then draw the liaes from 
 angle to angle, to complete the figure. 
 
 K , A Third 
 
74 
 
 PRACTICAL OF.O^IETRT. 
 A Third Method fieatlj true. 
 
 On the centres A and 
 P, with the ciiftanceAB, 
 defcribe two circles in- 
 terfering in m and n. 
 \Vith the fame ra- 
 ti ins, and the centre m, 
 defcribe rAoBS, ^ and 
 draw ran cutting it in 
 
 o.. Draw roC and 
 
 SoE, which will give 
 two angles of the pen- 
 tagon. Laftly, with 
 
 radius AB, and centres 
 C and E, defcribe arcs 
 interfering in D, which 
 will be the other angle 
 •f the pentagon nearly. 
 
 PROBLEM XXIII. 
 
 To make a Hexagon eu a Given Line AB. 
 
 Wiih the diftance AB, and 
 the centres A and B, defcribe 
 arcs interfering in o. — With 
 the fame radius, and centre o, 
 defcribe a circle, which will 
 circumfcribe the hexagon.— 
 Then apply the line AB fix 
 times round the circumference, 
 marking out the angular points ; 
 and connea them with right 
 
 PRO- 
 
PRACTICAL GEOMETRY. 
 
 7 J 
 
 PROBLEM XXIV. 
 
 To make a?i OBagon on a Given Line AB. 
 
 Ereft AF and BE per- 
 pendicular to AB. — Pro- 
 doce AB both ways, and 
 bifeft the angles m A F and 
 dBE with the lines AH and 
 
 BC, each equal to AB. 
 
 Draw CD and HG parallel 
 to AF or BE, and . each 
 equal to AB.— -With the dif. 
 tance AB, and centres G and 
 D, crofs AF and BE in F 
 and E.— Then join GF, FE, 
 KD, and it is done. 
 
 PROBLEM XXV. 
 
 To make any Regular Polygon 9H a Given Lhte AB, 
 
 Draw Ao and Bo making 
 the angles A and B each 
 equal to half the angle ol 
 the polygon. — With the cen- 
 tre o and diftance oA, de. 
 fcribe a circle, — Then apply 
 the line AB co-tinually 
 round the circumference the 
 proper number of times, and 
 it is done. 
 
 Note. The angle of any ^x)Iygon, of which the 
 
 angles oAB and oBA are each one half, is fouud 
 
 thus : Divide the whok 360 degrees by the number of 
 
 E 2 fides, 
 
76 
 
 PRACTICAL GEOMETRY. 
 
 fides, and the quotient will be the angle at the centre o ; 
 then fubtradl that from 180 degrees, and the remainder 
 will be the angle of the polygon, and is double of oAB 
 or of oBA. And thus you will find the numbers of the 
 following table, containing the degrees in the angle o, 
 at the centre, and the angle of the polygon, for all the 
 regular figures from 3 to 12 fides. 
 
 No. of 
 fides 
 
 Name of tht 
 Polygon 
 
 Angler 
 at the 
 centre 
 
 Angle 
 of the 
 polyg. 
 
 Angle 
 
 OAB 
 or OB A 
 
 3 
 
 Trigon 
 
 120«' 
 
 eo** 
 
 30*' 
 
 4 
 
 Tetragon 
 
 90 
 
 90 
 
 45 
 
 5 
 
 Pentagon 
 
 72 
 
 108 
 
 54 
 
 6 
 
 Hexagon 
 
 60 
 
 120 
 
 60 
 
 7 
 
 Heptagon 
 
 51^ 
 
 128f 
 
 642 
 
 8 
 
 Oaagon 
 
 45 
 
 135 . 
 
 67 
 
 9 
 
 Nonagon 
 
 40 
 
 140 
 
 70 
 
 10 
 
 Decagon 
 
 36- 
 
 144 
 
 72 
 
 11 
 
 Undecagon 
 
 32A 
 
 Wt\ 
 
 73/r 
 
 12 
 
 Dodecagon 
 
 30 
 
 150 
 
 75 
 
 PROBLEM XXVr. 
 
 In a Given Circle to Ivfcribe any Regular Polygon; or to 
 dhjide the Circumference into any Number of Equal Farts* 
 
 {See the laft figure,) 
 
 At the centre o make an angle equal to the angle at 
 the centre of the polygon, as contained in the third 
 column of the above table of polygons. — Then the dif- 
 tance AB will be one fide of the polygon; which being 
 carried round the circumference the proper number of 
 times, will complete the figure. Or, the arc AB will be 
 one of the equal parts of the circunaference. 
 
 Another, 
 
PRACTICAL GEOMETRY. 
 
 77 
 
 Another Method, nearly true. 
 
 Draw the diameter AB, 
 which divide into as many 
 equal parts as the figure hag 
 fide?.-— With the diftance AB, 
 and centres A and B, defcribe 
 arcs crofling at n : from thence 
 draw nC ihroagh the fecond 
 divifion on the diameter; fo 
 ihall AC be a fide of the poly- 
 gon, nearly. 
 
 Another Method^ fiill nearer, 
 Divide the diameter AB, as 
 before, into as many equal parts 
 as the figure has fides. From 
 the centre o raife the perpendi- 
 cular om, which produce till mn 
 be equal to three fourths of the 
 radius om. — From n draw nC 
 through the fecond divifion of 
 the diameter, and the line AC 
 will be the fide of the polygon 
 flill nearer than before; or the 
 arc AC one of the equal parts 
 into which the circumference is 
 to be divided, 
 
 PROBLEM XXVir, 
 
 About a Gi<ven Circle to Circumfcrihe any Polygon, 
 Find the points m, n, p, &c. a 
 
 as in the laft problem ; to which ^ 
 
 draw radii mo, no, &c. to the 
 
 i~ centre of the circle,- Then 
 through thefe points m, n, &C. 
 jand perpendicular to thefe radii, 
 praw the fides of the polygon, 
 I E 3 
 
 PRO- 
 
78 
 
 PRACTICAL GEOMiTAY. 
 
 rROfiLEM XXVII 
 
 To find the Centre of a Ginjen PcIygCfiy or the Centre of its 
 Infcribtil or Circutnfcribed Circle, 
 
 Bifeft any two fides with the 
 rer[)enfliculars mo, no \ and their 
 interfe^ion will be the cenrre. — 
 Then with the centre o, and the 
 diftance om, defcribe the infcribed 
 circle; or with the diftance to one 
 of the angles, as A, defcribe the 
 circumfcribing circle. 
 
 Ntie, This method will alfo circumfcribc a circle 
 abtut any given oblique triangle. 
 
 PROBLEM XXIX. 
 
 In arty Given Triangle to hfcribe a Citclt; 
 
 Bifeft any two of the angles 
 viih thft lines Ao, Bo; and o 
 will be the centre tf the circle. 
 —'I hen with the centre c, and 
 radius the nes^rell diftance to any 
 one of the fides, defcribe the 
 circle. 
 
 PRO- 
 
P 11 A (• 1 I C A L G I : O M K T II "N' « 
 r!<OBLEM XXX. 
 
 Ahouf any Given Triangle to Circurnfcrihc a Circle^ 
 
 Bifeift any two of rhe flies 
 AB, £C, with the perpendi- 
 culars mo, no. With tli£ 
 
 cemre o, and diftance to any 
 one of the angles^ dcfciibe the 
 circle. 
 
 PROBLEM XXXI. 
 
 //;, or About ^ a Given SquarCy to defcrihe a Circle, 
 
 Draw the two dia^^oi a's of 
 the fquare, and their interfv.<ition o 
 will l>e the centre of both the 
 circles, — ^Then with that centre, 
 and the neareft diftance to one 
 lide, defcribe the inner circle; and 
 with the diftance to one angle, 
 defcribe the outer circle. 
 
 r^ 
 
 -r 
 
 PR.X)BLEM XXXII. 
 
 Iftj er About, a Given Circle, to defcrihe a Square ^ or an 
 Octagon, 
 
 Draw two diameters AB, CD, 
 
 perpendicular to each other. ^ 
 
 Then connect their extremities, 
 and that will give the infcribed 
 
 quare A.CBD. Alfo through 
 
 tkelr extremities draw tangents 
 parallel to them, and they will 
 {arm the outer fquare mnop. 
 E 4 
 
 MA 
 
 Note, 
 
80 
 
 PRACTICAL OEOMET»y. 
 
 Note. If any quadrant, as AC, be bifefted in q, it 
 will give cne.eighth of the circumference, or the fide of 
 the odlagoA. 
 
 JPROBLSM XXXIII( 
 
 In a Grveft Circle, to Infcrile a Trigon, a Hexagon, er a 
 Dodecagon, 
 
 The radius of the circle is the 
 fide of the hexagon. Therefore 
 from any point A in the circum- 
 ference, with the diftance of the 
 radius, defcribc the arc BOF. 
 Then is AB the fide of the hex- 
 agon; and therefore carrying 
 it fix times round will form the 
 hexagon, or will divid« the cir- 
 cumference into fix equal parts, 
 each containing 60 degrees — 
 The fecond of thefe C, will give 
 AC the fide of the trigon, or equilateral triangle ACE, 
 and the arc AC one-third of the circumference, or 120 
 degrees. — Alfo the half of AB, or An, is one-12th of 
 the circumference, or 30 degrees, which gives the fide of 
 the dodecagon. 
 
 i\W. If tangents to the circle be drawn through all 
 the angular points of any infcribcd figure, they will form 
 the fides of a like circumfcribing figure. 
 
 PRO- 
 
I 
 
 PRACTICAL GEOMETRY. SI 
 
 PROBLEM XXXIV. 
 
 In a Given Circle to Injcrihe a PentagGrty or a Becageti, 
 
 Draw tke two diameters 
 AP, mn perpendicular to each A 
 
 other, and \A(t^ the radius 
 on at q. — With the centre 
 q and diftance qA, defcribe 
 the arc Ar; and with the 
 centre A, and radius Ar, de- 
 fcribe the arc rB. Then is 
 AB one-fifth of the circum- 
 ference; and AB carried five 
 times over will form the pen- 
 tagon, Alfo the arc AB bi- 
 fefled in s, will give A s the 
 tenth part of the circumfer- 
 ence, or the fide of the deca- 
 gon, t 
 
 Note, Tangents being drawn through the angular points, 
 will form the circumfcribing pentagon or decagon, 
 
 PROBLEM XXXV. 
 
 To divide the Circumference of a Given Circle into 1 2 Equal 
 PartSy each of 30 Degrees, 
 ' Or to Infcribe a Dodecagon bj another Method* 
 
 Draw two diameters 1 7 
 and 4 10 perpendicular to 
 each other. — Then with the 
 radius of the circle, and the 
 four extremities, 1, 4, 7, 10, as 
 centres, defcribe arcs^ through 
 the centre of the circle; and 
 they will cut the circumference 
 in the points required, divid- 
 ing it into 12 equal parts, at 
 the points marked with the 
 numbers in the figure. 
 
 E 5 PRO- 
 
 L. 
 
82 
 
 PRACTICAL GEOMETRY. 
 PROBLEM XXXVI. 
 
 To dranx) a Right Line equal to the Circumference of a Ghen 
 Circle* 
 
 m 
 
 II 
 
 Take III 1 equal to 3 times the diameter and y part 
 more: and it will be equal to the circumference^ very 
 nearly, 
 
 rROBLEM XXXVII. 
 
 To find a Right Line equal to any Given Arc AB of a Circle* 
 
 m 
 
 Through the point A and 
 the centre draw Am, making 
 mn equal to -^ of the radius 
 n o. — Alfo draw the indtfiaite 
 tangent AP perpendicular to 
 ir. — Then through m and B 
 draw raB: fo (hall AP be 
 equal to the arc AB very, 
 nearly. 
 
 Otherimife. 
 
 Divide the chord AB into 
 4 equal prts. — Set one part 
 AC on the arc from B to D.— 
 Draw CD, and the double of it 
 will be nearly equal to the arc 
 ADB. 
 
 A C 
 
 FRO- 
 
PRACTICAL GrEOMETRY, 
 
 S3 
 
 PROBLEM XXXVIII. 
 
 Ta divide a Given Circle into any propo/ed Number of Vanf 
 by Egual Lines ^ Jo that theje Farts /hall be mutually 
 Equal both in Area and Perimeter, 
 
 Divide the diameter AB 
 into the propofed number of 
 equal parts at the points a, 
 b, c, &c. Then on Aa, 
 Ab, Ac, &-C, as diameters, 
 defcrihe femicircks on one 
 fide of the diajneter AB ; and 
 on Bd, Be, Bb, &c. defer! be 
 femicircles on the other fide 
 ef the diameter. So Ihall the 
 Gorrcfponding joining femi- 
 circles divide the given circle 
 in the manner propofed. 
 
 And in like manner we may proceed when the fpacea 
 are to be in any given proportion. — As to the perimeters, 
 they are always «qual, whatever be the proportion of the 
 fpaces. 
 
 PROBLEM XXXIX. 
 To make a Triangle Similar to a Gi'ven Triangle ABC. 
 
 Let ah be one fide of the 
 requiredTriangle, Make the 
 angle a equal to the angle A,. 
 and the angle b equal to the 
 angle B;. then the triangle 
 abc will be fimilar to ABC 
 as propofed. 
 
 Note, If ab be equal to 
 AB, the triangles will alfo 
 be equals as well as fimilar. 
 
 a b 
 
 PRO- 
 
 - '/ERSI 
 
u 
 
 PRACTICAL OEOMETRV. 
 
 PROBLBM XL. 
 
 Te mtike afigure Similar to any othtr GiviM Figuft ABCDE. 
 
 d 
 From any angle A draw 
 diagonals to the other angles. 
 —Take Ah a fide of the 
 figu re requ i red . Then d raw 
 be parallel to BC, and cd to 
 CD, and de to DE, &c. 
 
 Otherwife 
 
 Make the angles at a, b, 
 c, refpcdively equal to the 
 angles at A, B, E, and the 
 lines will interfeft in the cor- 
 ners of the figure required. 
 
 PROBLEM XLI, 
 Sr<? reduce a Complex Figure from one Scale to another , aljo 
 to copy/uch a Figure of the fame Size, mechanicalfy, ij 
 means of Squares. 
 
 \ 
 
 
 
 
 / 
 
 
 
 
 
 
 
 y 
 
 / 
 
 
 
 r 
 
 
 
 
 ^-^ 
 
 
 
 ^ 
 
 J 
 
 --> 
 
 
 
 , / 
 
 
 y 
 
 
 
 T 
 
 r" 
 
 ^ 
 
 1 
 
 
 
 Divijc 
 
PRACTICAL GE0MI;TRY. 
 
 «5 
 
 r" 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 
 / 
 
 -^ 
 
 ■»^ 
 
 
 ^■■* 
 
 
 
 T 
 
 ^ 
 
 > 
 
 
 Av-H 
 
 "/ 
 
 — ^ 
 
 / 
 
 
 E 
 
 
 ' T? 
 
 
 
 
 \ 
 
 Divide the given figure, by crofs lines, into fquares, as 
 fmall as may be thought neceflary, — Then divide another 
 paper into the fame number of fquares, and either greater, 
 equal or lefs, in the given proportion. — This done, ob- 
 ferve what fquares the fcveral parts of the given figure are 
 in, and draw with a pencil, fimilar parts in the corref- 
 ponding fquares of the new figure. And fo proceed till 
 the whole is copied. 
 
 PROBLEM XLII. 
 
 To make a Triangle Equal to a Given Trapezium A BCD. 
 
 Draw the diagonal DB, D 
 
 alfo CE parallel to it, meet- r^^r--^ 
 
 ing AB produced inE. — Join 
 DE; fo (hall the triangle 
 ADE be equal to the trape- 
 zium ABCD. A BE 
 
 PROBLEM XLIII. 
 
 T9 make a Triangle equal to the Figure ABCDEA. 
 
 Draw the diagonals DA, 
 DB, and the lines EF, CG 
 parallel to them, meeting the 
 bafe AB, both ways produced, 
 in F and G.— JoinDF, DG; 
 and DFG will be the triangle 
 required equal to the given 
 figure ABCDE. 
 
 Q 
 
8(> 
 
 PaACTlCAL GEOMETRY* 
 
 Kou. Nearly in the fame manner may a triangle be 
 made equal to any ri^ht-lined figure whatever* 
 
 PROBLEM XLIV. 
 
 Sr« mah a Triangle Equal to a Given Circle, 
 
 Draw any ra- 
 dius AOy and the 
 tangent AB per- 
 pendicular to ir, 
 — On which take B 
 AB equal to the 
 
 circumference of the circle by Problem xxxvi. — Joia 
 BO; io Ihall ABO be the triangle revquired, equal to the 
 gHren circle, nearly. 
 
 PROBLEM XLV, 
 
 7i make aj^edangle, or a Parallelogram^ Equal t a a Given. 
 Triangle ABC. 
 
 Bifea the bafc AB in m 
 
 Through C draw Cne parallel 
 to AB. — I h rough m and B 
 draw mil and BsJ ptralkl to 
 each other, and enher per; en- 
 dicufar to A B, or making any 
 angle with it. And the redan- 
 gle or parallelogram mnoB will 
 be equal to the triangle, as re- 
 quired. 
 
 PRO, 
 
PRACTICAL GEOMETRY. 
 
 sr 
 
 PROBLEM XLVI. 
 
 To make a Square Equal to a Gi'ven Re£lattgle ABCD, 
 
 F G 
 
 / 
 
 \ 
 
 D 
 
 Produce one fide, AB, till 
 BE be equal to the other fide y 
 
 BC. — BKeft AE in oj on which ^/'q 
 as a centre, with radius Ao, 
 defcribe a feinicircle, and pro- 
 duce EC to meet it at F. — On E B o HA 
 BF make the fquare BFGH, 
 and it will be equal to the reftangle ABCD, as required. 
 
 *^* Thus the circle, and all right-lined figures, have 
 been reduced to equivalent fijuares. 
 
 PROBLEM XLVII. 
 
 i|jj_. To make a Square Equal to Tnvo Given Squares P and Q^ 
 
 ■^ c.^ . _ ri ATI T>/> _r ^ 
 
 Set two fides AB, BC, of 
 the given fiquares, perpendicu- 
 lar to each other. — Join their 
 extremities AC ; fo fhall the 
 fquare K, conftrufted on AC, 
 be equal to the two P and (^ 
 taken together. 
 
 Note, Circles or any other fimilar figores are added in 
 the fame manner. For, if AB and BC be the diameters 
 of t>vo circle?, AC will be the diameter of a third circle 
 equal to both the other two. And if AB and BC be the 
 like fides of any two fimilar figures, then AC will be the 
 like fide of another fimilar -hgure equal to both the two 
 former, and on which the third figure may be conftru(5ted 
 by Problem xl. 
 
 pao- 
 
88 
 
 PRACTICAL GEOMETRT« 
 
 rROBLEM XLVIII. 
 
 To mah a SquareEqual t4 the Difference beinveen Tivo Given 
 Squares P, R. 
 
 {See the laji Figure,) 
 
 OnthcfidcAC of the greater fquarc, as a diameter, 
 defcribe a femicircle; in which apply AB the fide of the 
 lefs fquare. — Join BC, and it will be the fide of a fquarc 
 equal to the difference between the two P and R, as re- 
 quired. 
 
 PROBLEM XLIX. 
 
 To make a Square Equal to the Sum of any Number of Squares 
 taken together. 
 
 Draw two indefinite lines 
 Am, Ah, perpendicular to 
 each other at the point A, 
 On the one of thele fet off 
 AB the fide of ottc of the 
 given fquares, and on the 
 other AC the fide of another 
 of them. Join BC, and it 
 will be the fide of a fquare 
 equal to the two together. 
 
 Then take AD equal to BC, and AE equal to the fide of 
 the third given fquare. So ftiall DE be the fide of a fquare 
 equal to the fum of the three given fquares. — And fo on 
 continually, always fetting more fides of the given fquares 
 on the line An, and the fides of the fucceffive furas on 
 the other line Am. 
 
 Note. And thus any number of any fort of figures 
 may be added together. 
 
 D B 
 
 pno- 
 
PRACTICAL GEOMETRY. 
 
 S9 
 
 PROBLEM L. 
 To make Plane Diagonal Scales, 
 
 
 A C 
 
 
 B 
 
 
 1 \n \\\ I ^ 
 
 
 
 
 1 M W \\ ^ 
 
 
 
 s 
 
 I W \\- 
 
 
 
 6" 
 
 TlimTTT 
 
 — " 
 
 
 
 \ n 
 
 
 
 
 1 n M 
 
 
 
 4 
 
 \y\ n 'Ti . 
 
 
 
 
 _ , \ 1 ' ■ 
 
 
 
 H 
 
 llUilU 
 
 
 
 D8 6 4 2 
 
 Draw any line as AB, of any convenient length. 
 Divide it into 11 equal parts*. Complete thefe into 
 redangles of a convenient height, by drawing parallel and 
 perpendicular lines. Divide the altitude into 10 equal 
 parts, if it be for a decimal fcale for common numbers, or 
 into 12 equal parts,- if it be for feet and inches; and 
 through thefe points of divifion draw as many parallel 
 lines, the whole length of the fcale. — Then divide the 
 length of the firft divifion AC into 10 equal parts, both 
 above and below ; and conneft thefe points of divifion by- 
 diagonal lines, and the fcale is finilhed, after being num- 
 bered as you pleafe. 
 
 Note, Thefe diagonal fcales ferve to take off large di- 
 ■lenfions or numbers of three figures. If the firft large 
 divifions be units; the fecond fet of divifions along AC, 
 will be 10th parts; and the divifions in the altitude, 
 along AD will be 100th parts. If CD be tens, AC will 
 be units, and AD will be the 10th parts. If CB be 
 hundreds, AC will be tens, and AD units. If CB be 
 thoufands, AC will be hundreds, and AD will be tens. 
 And fo on, each fet of divifions being tenth parts of the 
 former one. 
 
 For example, fuppofe it were required to take off 243 
 from the fcale. Fix one foot of the compafles at 2 of 
 tjift greateft divifions, at the bottom of the fcale, and 
 
 * Only 4 parts are here drawn, for want of room. 
 
 extend 
 
DO 
 
 PRACTICAL GEOMI/ntV 
 
 extend the other to 4 of the fecond divifionf, along ihc 
 bottom ; then, for the 3, Aide vp boti) poims of the coin. 
 pafTes by a parallel motion, till they fall upon the third 
 longituiHnal linp; and in that pofirion extend i he fecoud 
 point of the conipailVs to the fourth diagonal line, and 
 you have the extent ot three figures as required. 
 
 Or, if you have any li ^e to uieafure the Itngth of. — 
 Take it between the conipalTes, and applying it to the 
 fcair, fuppofc it fall between 3 and 4- of the large divi- 
 fions; or, more nearly, that it is 3 of the large divilions, 
 or J. hundreds, and between 5 a»d 6 of the iecond divi- 
 fions, or 6 tens or 50, and a little more. Slide up the 
 points of the compafl'es by a parallel motion, keeping one 
 foot always on the vertical divifion of 3 hundred, till the 
 other point fail exadly on one of the diagonal line:», 
 which fuppofe to be 8, being 8 iinitt,, which fnowj thac 
 ihe length of the linc^ propofed to be ineafurcd, is 358t 
 
 fLANI SCALBI FOK TWO FIGURES. 
 
 • .... 1 ...1 1 
 
 
 1 
 
 1 
 
 1. / 
 
 10 8 6 4- 2 
 
 1 
 
 *y 
 
 3 
 
 4 
 
 imirlnr,,! , „ i ■ i i 
 
 IS 29 IS $ 4 £ <) 
 
 
 «? 
 
 
 
 ... /v I 
 
 1 1 
 
 / 
 
 -Hi/- — 
 
 / 
 
 ^ ^J ' 
 
 
 
 
 
 
 
 
 
 V ) 
 
 
 
 M 1 
 
 
 >. 
 
 <> ] 
 
 i 
 
 i a 
 
 The above are three other forms of fcales, the firft of 
 which is a decimal fcale, for taking off common number* 
 confifting of two figures. H he other two are duodecimal 
 fcales, and ferve for feet and iaches, &c. 
 
 Tbefc 
 
rilACTICAL GEOxMETRY. 
 
 9i 
 
 Thefe and other fcalcs, ei^graved on ivory, arc fltleft 
 for pradical ufe. And the moft convenient form of a 
 plane fcale of equal divifions, is on the very edge of the 
 ivory made thin at the edge for laying along any line, and 
 then marking the j apcr oppofite any divifion required: 
 which is better than taking kogths off a fcale with com- 
 pafTes. 
 
 RLMARK8. 
 
 No/£ 1. That in a circle, 
 the half chord DC, is a mean 
 proportional between the feg- 
 ments AD, DB of the dia- 
 meter AB perpendicular to iN 
 That is AD : DC : ; DC : 
 DB. 
 
 2. The chord AC is a mean proportional between AD 
 and the diameter AB. And the chord BC a mean pro- 
 portional between DB and AB. 
 
 That i-, AD : AC : : AC : AB. 
 and BD : BC : : BC : AB. 
 
 3. The angle ACB, in a femicircle, is always a right 
 angle, 
 
 4. The fquare of the hypothenufe of a right-angled 
 triangle, is equal to the fquare of both the fides. 
 
 That is, AC* = AD^ + CDS 
 andBC^ = BD^ + DG% 
 and AB* == AC^ + BC*. 
 
 5. Triangles that have all the three angles of the one 
 refpcdively equal to r.U the three of the other, are called 
 equiangular triangles, or fimilar triangles. 
 
 6» In timilar triangles, the like fides, or fides oppofitc 
 the equal angles, are proportional. 
 
 7. The areas, or fpaces, of fimilar triangles, are to 
 each other, as the fquarcs of their like fides. 
 
 MEN- 
 
MENSURATION 
 
 OF 
 
 SUPERFICIES. 
 
 n^HE area of any figure, is the ineafure of its furface, 
 -^ or the fpace contained within the bounds of the fur- 
 face, without any regard to thick nefs. 
 
 The area is elHmated by the number of fquarcs con- 
 tained in the furface, the fide of thofe fquarcs being either 
 an inch, or a foot, or a yard, &c. And hence the area is 
 faid to be fo many fquare inches, or fquare feet, or fquare 
 yards, &c. 
 
 Our ordinary lineal meafures, or meafurcs of length, 
 are as in the firtt table here below; and the annexed table 
 of fqnare meafures is taken from it, by fquaring the feveral 
 numbers* 
 
 Lineal Meafures, Square Meafures, 
 
 12 inches - 1 foot 144 inches . 1 foot 
 
 3 feet - - 1 yard f) feet - 1 yard 
 
 6 feet - - 1 fathom 3^ fcec - 1 fathom 
 
 l6-^ feet, or") r 1 pole 272^: feet or") f 1 pole 
 
 5^ yards J \ or rod 2>0^ yards S \ or rod 
 
 40 poles - 1 furlong l600 poles - 1 furlong 
 
 8 furlongs 1 mile 64 furlongs 1 mile 
 
 PRO. 
 
MENSURATION OF SUPERFICIES. 93 
 
 PROBLEM I. 
 
 To find the Arta of a Parallelogram ; ^whether it he sc 
 Square, a ReBavgle, a Rhomhusy or a Rhomboid, 
 
 Multiply the length by the breadth, or perpendicular 
 height, and the produd will be the area. 
 
 EXAMPLES. 
 
 1. To find the area of a fquare, whofe fidejs 6 inches, 
 or fix feet, &c. 
 
 6 
 
 36' 
 Anfwer Z^ 
 
 
 2. To find the area of a reftangle, whofe kngth is 9, 
 and breadth 4 inches, or feet, &c. 
 
 9 
 
 4. 
 
 mmm 4.'. 
 
 36 
 
 Anfwer 36 
 
 
 X To 
 
9^ NEN9UUATI0N' 
 
 3. To find the area ef a rhombus, whofe length > 
 ۥ20 chains, and perpendicular height 5 '45 
 
 5-4S 
 6*20 
 
 10.900 0'2Q 
 
 3270 
 
 10)337900 
 3-379 
 4 
 
 1-516 
 40 
 
 20*640 Anf, 3 acres, 1 rood, 20 perches. 
 
 Note, Here the fquare chains are divided by 10 to 
 bring them to acres, becaufe 10 fquare chains raal^e an 
 acre. Alfo the decimals of an acre are multiplied by 4 
 roods, and thefe by 40 perches, becaufe 4 roods make 1 
 acre, and 40 perches 1 rood. 
 
 4. To find the area of the rhomboid, whofe length Is 
 X2 feet 3 inches, and breadth 5 feet 4 inches. 
 f i 
 12 3 
 5 4 
 
 61 3 ^ 7 
 
 * ^ t. / 
 
 65 4 12 f 3 in 
 
 Anfwcr 65 j fquare feet, 
 
 4 5. To 
 
OP SUPERFICIES. 6)3- 
 
 !). To find the area of a fquare, whofe fide is 35 25 
 ehains, Anf. 124 ac 1 ro 1 perch. 
 
 6". To find the area of a parallelogram, whofe length 
 is 12'25 chains, and breadth S'5 chains. 
 
 Anf. id ac 1 ro 2G perch. 
 
 7. To find the area of a reftangular board, whofe 
 length is 12*5 feet, and breadth f) inclies. Anf. 9|- feet. 
 
 8. To find the fquare yards of painting in a rhomboid, 
 whofe length is 57 feet, and breadth 5^ feet. 
 
 Anf. 21/2; fquare yards. 
 
 PROBLEM II. 
 
 To find the Area of a Triangle, 
 
 Rule 1 . Multiply the bafe by the perpendicular height, 
 and take half the produft for the area. 
 
 Rule 2. When the three fides only are given: Add the 
 three fides all together, and take half the fum j from the 
 half fum fubtrad each fide feparately ; multiply the half 
 fum and the three remainders continually together; and 
 take the fquare root of the laft pr«dud for the area of the 
 triangle. 
 
 EXAMPLES. 
 
 1. Required the area of the triangle, whofe bafe is ^'25 
 chains, and perpendicular height 5*20 chains. 
 
9^ 
 
 M»KSURATIOK 
 
 6-25 
 5-20 
 
 12500 
 
 312^ 
 
 20 ) 32-5000 
 1»625 
 4 
 
 2-500 
 40 
 
 20»000 
 
 Anf. 1 ac 2 ro 20 perches, 
 2, To find the numWer of fquare yards in the triangle 
 whofc three fides are 13, 14, 15 feef. 
 13 
 
 15 
 
 2)42 
 
 ^ fum 21 
 13 
 
 remainders 8 
 
 21 
 14 
 
 7 
 
 1( 
 nf.l 
 
 21 21 
 15 6 
 
 6 126 
 
 7 
 
 
 - 882 
 8 
 
 9) 
 
 7056 ( 84 feet 
 
 64 9i fq. yds, 
 
 
 54 
 4 
 
 656 
 65(5: • 
 
 A 
 
 l^f 
 
 q, ya/dfi. 
 
 3. How 
 
OF SUPERFICIES. 97 
 
 3. H©w many fqirare yards arc in a right-angled triaaglc, 
 •whafc bafe is 40, and perpendicular 30 feet ? 
 
 Anf. 66 J Tquarc yards. 
 
 4. To find the area of the triangle, whofe three fide» 
 arc 20, 30, 40 chains. Anf. 29 ac ro 7 per. 
 
 5. How many fquarc yards contains the triangle, whofc 
 bafe is 49 feet, and height 25 i feet ? 
 
 Anf. 6841 or 68*73(3 1. 
 
 6. How many acres, &c. in the triangle, whofe three 
 fides are 380, 420, 765 yards ? Anf. 9 ac ro 38 per. 
 
 7. To find the area of the triangle, whofe bafe is IS 
 feet 4 inches, and height 11 feet 10 inches. 
 
 Anf. lOS feet 5 inches 8". 
 
 8. How many acres. &c, contains the triangle, whofe 
 three fides are 49*00, 5C*2j,25*69 chains? 
 
 An, 6l ac 1 ro 39' 6$ per. 
 
 PROBLEM 1II« 
 
 9"o /V/i one Side -cfa Right-aighd Triangle j hanjing ihs 
 other tivo Sides gi'ven. 
 
 The fquare of the hypotherufc is equal to both the 
 fquares of the two legs. Therefore, 
 
 1. To find the hypoihenufe ; add the fquares of the 
 two legs together, and extrad the fquare root of the 
 fum. 
 
 2. To find one leg; fubtrafl the fquare of t^e other 
 leg from the fquare of the hypothenufe, and cxtraft the 
 
 Square root of the difference. 
 
 EXAMPLES. 
 
 1. Required the hypothenufe of a right angled triangle 
 Fhofe bafe is 40, and perpendicular 30, 
 
 F 40 
 
9^ MENIURATION 
 
 40 30 
 40 30 
 
 l600 $00 
 
 900 
 
 2500 ( 50 the hypothcnufc AC 
 S5 
 
 00 
 
 *?. What is the perpendicular of a right-angled triangle, 
 whofc bafc AB is 66, and the hypothcnufe AC 6*5 ? 
 
 56 
 
 
 65 
 
 56 
 
 
 65 
 
 336 
 
 
 325 
 
 280 
 3136 
 
 
 3yO 
 
 4235 
 3136 
 
 1089 ( 33 the perp. BC. 
 9 
 
 
 63 
 
 189 
 
 
 3 
 
 189 
 
 3. Required the length of a fcaling ladder to reach the 
 top of a wall whofc height is 28 feet, the breadth of the 
 ditch before it being 45 feef. Anf, 53 feef. 
 
 4. To find the length of a (hoar, which, ftrutting 12 
 feet from the upright of a building, may support a jaumb 
 I'D feet from the ground, " Anf. 23*32380 feet, 
 
 5. A line of 320 feet will reach from ihe top of a pre- 
 cipice, (landing clofe by the fide of a brook, to the oppo- 
 fite bank : required the breadth of the b«-ook ; the height 
 «f the precipice being 103 feet. Anf. 302\9703 feet. 
 
 6» A 
 
• F SVPERFICIB^ 
 
 99 
 
 0. A ladder of 50 feet long being placed in a ftreet, 
 reached a window 28 feet from the ground on one fide; 
 and by turning the ladder over, without renudving th« 
 foot out of its place, it touched a moulding 36 feet high 
 on the other fide; required the breadth of the ftreet ? 
 
 Anf. 76«1233335 feet, 
 
 PROBLEM IV. 
 To find the Area of a Trapczofd. 
 Add together the two parallel fides; multiply thtl fum 
 by the perpendicular diftance between them, and take half 
 the produd for the area. 
 
 EXAMPLES. 
 
 1. In a trapezoid the parallel lines are A B 7*5, and I>C 
 12'25, alfo the perpendicular diftance AP or Cn is 15*4 
 chains -, required the area. 
 
 12-2i 
 75 
 
 E n 
 
 D V 
 
 20 ) 30i-J.50 
 
 iD'2075 
 
 4 
 
 '8300 anf. 15 ac ro 33 per 
 40 
 
 33*2000 
 
 2. How many fquare feet contains the plank, wliofe length 
 
 is 12 f«et () inches, the breadth at the greater end 1 f^Mt 3 
 
 inches, an4 at the lefs end 1 1 inches ?. Anf. ISjJ feet. 
 
 f2 :.. Re- 
 
100 
 
 MEK-8URATI0H 
 
 3. Required the area of a trapezoid, the pnmllel fides 
 being '2\ fert 3 inches and IS feet 6' inches, and the dif- 
 tance between them 8 feet 5 inches. 
 
 Anf. 107 feet 3 inches 4" 6^". 
 
 4. In meafuring along one fide AB of a quadrangular 
 fteld, that fide and the two perpendiculars upon it from 
 the oppofite corners, meafurcd as below: required the 
 content, Anf. 4 ac 3 r 17'92 p. 
 
 chains X) 
 
 • AP = IMO 
 AQ^= 7*45 C 
 
 AB =z 11-10 
 VC = 3-52 
 QP = 5-95 
 
 PROBLEM V, 
 
 To find the Area of a Trapezium, 
 
 CASE 1. 
 
 For any Trapezium. 
 
 Divide it into two Triangles by a diagonal ; then find 
 the areas of thefe triangles, and add them together. 
 
 Note. If two perpendiculars be let fall on the di.igonal, 
 from the other two oppofite angles, the fum of iheie per- 
 pendiculars being multiplied by the diagonal, half the 
 produd will be the area of the trapezium. 
 
 CASE 2. 
 
 When the Trapezium can be injcrihed in a Circle, 
 
 Add all the four fides together, and take half the fum ; 
 next fubtrad each fide frparately from the half fam: then 
 iTiuliiply the four remainders continually together, and 
 take the fquare root of the laft produdl for the area of the 
 trapezium. 
 
OF SUPERFICIES. 
 
 101 
 
 EXAMPLES, 
 
 1. To find the area of the trapezium ABCD, the diagonal 
 AC being 42, the perpendicular BF 18, and the perpendicu- 
 lar 1>E 10'. 18 Ui 
 16 
 
 34. Sum 
 42 
 
 68 
 136" 
 
 2 ) 142* 
 
 714 the anfwer. 
 2. In the trapezium ABCD, the fide AB is I5^BC 13, 
 CD 14, AD 12, and the diagonal AC h l6: required 
 the area, 
 
 AC 1<> AC 16 
 
 AB 15 CD 14 
 
 BC IS AD 12 
 
 2 ) 44 2 ) 42 
 
 32 22 22 half fum 21 21 21 halffum 
 
 16 
 
 15 
 
 13 
 
 16 
 
 14 12 
 
 6 
 7 
 
 7 
 
 9 
 
 5 
 7 
 
 7 9 
 
 42 
 9 
 
 
 
 35 
 9 
 
 
 378 
 
 22 
 
 
 
 315 
 21 
 
 
 75G 
 7o6 
 
 
 
 315 
 630 
 
 
 83 1(). 
 
 (.91' 
 
 Jp2l 
 
 6"6l5<.81'3326 
 
 The 
 
 \ 
 
Kyi MENStTRATlOV 
 
 The triangle A BC - - - <)T1<)51 
 The triangle ADC - . - 81 'ia^^ 
 
 The trapezium ABCD 172'6'247 the anfwer. 
 3. If a trapezium can be infcribed in a circle, and have 
 Its four fides 24-, 26, 28, 30 j required its area. 
 24 
 26 
 28 
 30 
 
 5f 54 54 half Aim 
 
 26' ,28 30 
 
 26 l6 24 
 
 24 
 
 840 104 
 
 624 
 S40 
 
 24^60 
 
 524160 ( 723-98vS9488 anTwer, 
 
 4. How many fquarc yards of paving are in the trape. 
 ziom, wbofe diagonal h t>5 feet, «nd the two perpendicu- 
 lars let fall on it 28 and 33-5 feet ? Anf. '3'22-,V yar^^s. 
 .5. What is the area of a trapezium, vvhofe fonth fide 
 is ^7*40 chairis, eaft free 35v3 chains, north fide 37*55 
 chains, w'eft fide 4 1 '05 chains, and the diagonal from 
 fouih-weft to ftorth-eaft 48*35 chains? 
 
 Anf. 123 ac ro WS(i72 per. 
 6". What 
 
€P StJPERFICIES. 103 
 
 6*. What is the area of a trapezium, whoTc diagonal is 
 108| feet, and the peri>endicalars 56^ and 6"0^ feer? 
 
 Anf. 034-71: feet. 
 
 7. What is the area of a trapezium infcribcd in a circle, 
 the four fides being 1'2, 13, U, 15? 
 
 Anf. lSO-9.972372. 
 
 8. In the four-fided field ABCD, on account of ob- 
 ftrudions in the two fides AB, CD, and in the perpendi- 
 culars BF, DE, the following meafures only could be taken: 
 namely, the two fides EC '265 and AD 220 yards, the dia- 
 gonal AC 378 yards, and the two diilances of the perpen- 
 diculars from the ends of the diagonal, namely A£ ]00, 
 and CF 70 yards: required the area in acres, when 48-tO 
 Square yards make an acre. Anf. 17 ac 2 ro 21 per. 
 
 PROBLEM TI. 
 
 To Jind the Area of an Irregular Polygon. 
 
 Draw diagonals dividing the figure into rmpezlums and 
 triangles. Then find the areas of all thefe feparately, and 
 add them together for thecuateni of th« whole figure. 
 
 EXAMPLE. 
 
 To find the content of the irregular figure ABCDEFG A, 
 in which are given the following diagonals and perpendi- 
 culars: namely, 
 
 B 
 
 AC 
 
 5-5 
 
 FD 
 
 5'2 
 
 GC 
 
 4-* 
 
 Gm 
 
 13 
 
 Ba 
 
 1-8 
 
 Go 
 
 1-2 
 
 Ep 
 
 0«8 
 
 Dq 
 
 2-3 
 
^^4f MENSUmATlOK 
 
 1ft, 2d, Jd. 
 
 For trapcz. AECG. For trapcz. GDEF. For triangle GXD. 
 1-3 1-2 4-4. 
 
 1*8 0-8 Q'3 
 
 J'l 2-i) J3'2 
 
 55 5'2 88 
 
 155 10'4 10-12 
 
 1705 double ABCG 
 
 10-40 double GDEF 
 
 10-12 double GCD 
 
 2 ) 37 '57 double the whole 
 
 18-785 theanfwer. 
 
 PROBJ EM VII. 
 
 Tojind the Area of a Regular VtJygon* 
 JIULE J. 
 
 I^nd the perimeter of the figure, or Aim of its iide«, 
 
 and multiply it by the perpendicular falling from its centre 
 oo one of its Ijdcs, and take half the produft for the area, 
 
 RULE 2. 
 
 Square one iide of the polygon; multiply that fquare 
 by the multiplier fet againft its name in the following 
 table, and ilie produd will be the area. 
 
 No, 
 
OF SUPERFICIES. 
 
 105 
 
 [ No of 
 fides. 
 
 Names. 
 
 Muliipliers. 
 
 3 
 
 Trigon or equ. tri. 
 
 0-43^0127 
 
 4 
 
 Tetragon or fquare 
 
 1-0000000 
 
 5 
 
 Pentagon 
 
 1-7504.774. 
 
 6 
 
 Hexagon 
 
 2-5980762 
 
 7 
 
 Heptagon 
 
 3'6'339124 
 
 8 
 
 Odagon 
 
 4-8284271 
 
 9 
 
 No n agon 
 
 (5-1818242 
 
 10 
 
 Decagon 
 
 7 -6942088 
 
 11 
 
 Undecagon 
 
 9'36'56399 
 
 12 
 
 Dodecagon 
 
 n- 1961524 
 
 EXAMPLES. 
 
 1. Required the area of the regular pentagon, whof« 
 fide AB is 25 feet, and perpendicular CP 17 204774, 
 ^j the \Ji Rule. 
 17*204774 perp. 
 125 pcrim. 
 
 86023870 
 34409548. 
 17204774? 
 
 2) 21 50*596750 
 
 107 5*298375 anf.. 
 
 By the 2d Rule, 
 Firft 25 Then 17204774 
 
 25 625 
 
 125 
 50 
 
 625 
 
 . 86023870 
 
 34409548 
 
 103228644 . 
 
 1075-2983759 anfwer* 
 
 tl^^ 
 
 ts 
 
 To 
 
106 MENSURATION 
 
 ^. To find the area of the hexagon, whofe fide is 20. 
 
 Anf. 1039-23048. 
 
 3, To find the area of the trigon* or equilateral trian- 
 gle, whofe fide is '20. Anf. 173-20508. 
 
 4, Required the area of an o^lagon, whofe fide is 20, 
 
 Anf. 1931-37084. 
 
 5, What is the area of a decagon, whofe fide is 20? 
 
 Anf. 3077*68351. 
 
 PIOBLEM VIII, ^ 
 
 To find the Diameter agd Circumference of a Circle, the one 
 from the other, 
 
 RUL? 1; 
 As 7 is to 22, fo is the diameter to the circumference. 
 As 22 is to 7> fo is the circumference to the diameter. 
 
 RULE 2. 
 
 As 113 is to 355, fo is the diameter to the circumf. 
 As 355 is to 113, fo is the circumC to the diameter, 
 
 RULE 3. 
 
 As I is to 3* 1 41 6, fo is the diameter to the circumf. 
 As 3*14l6 is to 1, fo is the circumf. to the diameter. 
 
 EXAMPLES. 
 
 1. To find the circumference of a circle, wbofe diameter 
 AB is 10. 
 By Rule 1. 
 7 : 22 : : 10 : 31| 
 10 
 
 7 ) 220 
 
 or 31-42857 anf. 
 
OF SUPERFICIES, 
 
 i<if 
 
 By Rule 2. 
 113 : 355 : : 10:31//! 
 10 
 
 By Rule 3. 
 
 1 : 3-14.16 : : 10 : 31*416 
 the circumference nearly, 
 the true circumference 
 
 being 
 31-4155263358979 &c. 
 
 So that the 2d rule is neareft 
 the truth. 
 
 113) 3350(31-41593 
 
 l60 the anf. 
 
 470 
 
 180 
 
 670 
 
 1050 
 
 330 
 
 J?, T« find the diameter when the circumference is 50. 
 By Rule 1. 
 
 22:: 7 : : 50 : L2L££ = !!£=: 15|$- = 15-9090 anf. 
 11 11. 
 
 By Rule 2. By Rule 3. 
 
 355 : 113 • : 30 : 15^f: 34416 : 1 : : 30 : 15-9i36 
 50 30 
 
 355 I 3650 3«1416 ) 50-000 { 15-9156 
 71 I 1130 ( 1>9155 ...... 18484 
 
 420 2S76 
 
 650 49 
 
 110 15 
 
 39Q 2 
 
 330 
 
 3. If the diameter of the earth be 7958 miles, as it is 
 very nearly, what is the circumference, fuppofing it to be 
 cxaaiy round? Anf. 2500b-8528 miles. 
 
 4. To find the diametfer of the globe of the earth, fup- 
 pcCng it^eiicumfexcnceto be 25000 miles. 
 
 Anf, 7957i nearly. 
 
 f 6 
 
 I^RO. 
 
103 
 
 MENSUIIATIOW 
 
 PROBLSM IX. 
 
 ^0 find the Length of any Arc of a Circle^ 
 RULE 1, 
 
 As 180 IS to the number of degrees in the arc, 
 So is S*14l6* times the radius, to its length. 
 Or as 3 is to the number of degrees in the arc. 
 So is 'Ob'l^^ times the radius, to its length. 
 
 ^' Ex. 1 . To find the length of an arc ADB of 30 degrees^ 
 
 the radius being [) feet. 
 
 3-U16' I> 
 
 As 1 80 : 30- 
 
 1 A 
 
 Or 6': 1 : : 28-2744 : 471 04 / 
 Or 3 : 30 : : »06'2J^ X [) : 47 124 / 
 .90 
 
 47 124 the anfwer. 
 
 RULE 2. 
 
 From 8 times the chord of half the arc fubtraft the 
 chord of the whole arc, and take \ of the remainder for 
 the length of the arc nearly. 
 
 Ex. 2. The chord AB of the wJiolc arc being 4*65874, 
 and the chord AD. of the half arc 2'a4i;47 ; required: the 
 length of the arc. 
 
 2-34947 
 
 187i)37() 
 4-6j874 
 
 3 ) 14.13702 
 
 471234 anfwer. 
 
 Ex. 3. Required the length of an arc of 12 degrees 10* 
 minutes, or 122 degree*, the radius being 10 feet. 
 
 Anf. 2-l234|. 
 Ex. 4. 
 
OF SUPERFICIES. 
 
 109 
 
 Ex. 4. To find the length of an arc whofe chord is 6, 
 and the chord of its half is 3|-. Anf. 7^^ 
 
 Ex. 5. Required the length of the ar«, whofe chord is 
 8, and the height PD 3. Anf. lOf. 
 
 Ex. 6, Required the length of the arc, whofe chord is 
 6, the radius being 9. Anf. 6*1 17G(5. 
 
 PROBLEM X. 
 
 ^0 find the Area of a Circle^ 
 
 The area of a circle may be found from the diameter 
 and circumference together, or from either of them alone, 
 hy thefe rules following. 
 
 Rule 1. Multiply-, half the circumference by half the 
 diameter. Or multiply the whole circumfe- 
 rence by the whole diameter, and take i of 
 thc^jcoduik. 
 Rule 2- Maltiply: tl)^ fquare of the diameter by 
 
 •7834. 
 Rule 3. Multiply the fquare of the circumference hy 
 
 . '07968. 
 Rule 4. As 14 to i 1, fo is the fquare of tl^ diameter 
 to the yrea. 
 ,!j5«/« 5, As, 8.8 to 7, fo is the fquare of the circumfe- 
 rence to the area. 
 
 EXAMPLES. 
 
 1. To find the area of a circle whofe diameter is 10, 
 and circumference 31*41 5926"5. 
 
 By Rule \. 
 31.'41592()5 
 
 . '. .: ; V 10 
 
 4 )314-15926'5. 
 area 7 8-53981 6" 
 
 By Rule 2. 
 
 :, 100 
 
 78-54 
 
 By Rule 4. 
 100;' 
 11 
 
 14 
 7 
 
 1100 
 550 
 78-57 
 
fy Rult 3. 
 
 fq. circ. 986-96044 
 
 invcrf. 83970 
 
 ATK 
 
 :7 
 
 8 
 11 
 
 ON 
 
 By Rule 5. 
 31-4159265 circum. 
 562951413 invert. 
 
 65(>fi723 
 
 888264 
 
 49348 
 
 7896 
 
 9424777?^ 
 
 3141593 
 
 1256637 
 
 31416 
 
 157O8 
 
 2827 
 
 63 
 
 J^ 
 
 2 
 
 78-54231 area 
 
 »8 
 
 : ; 986-96044 
 7 
 
 
 1 69O8-723O8 
 
 1 863-59038 
 
 78-50821 
 
 Ex. 2. Required the area of tfee circle, whofe diameter 
 is 7, and circumference 22. Ans. 38p 
 
 Ex. 3. What is the area of a circle, whofc diametcB is 
 1, and circumference 3'14l6? Anf. -7854. 
 
 Ex. 4. What is the area of a circle, whofe diameter 
 is 7 ? Anf. 38*4846.. 
 
 Ex. 5. How many fquare yards are io a circle whofe 
 diameter is 3^ feet ? Anf.. 1 •O69. 
 
 Ex. 6. How many fquare feet does a circle contain, the 
 circumference being 10*9956 yards ^ Anf. 86*19266. 
 
 PROBLEM XI» 
 
 7oJind the Area af the SeBor •f a Circle. 
 
 RULE 1. 
 
 Multiply the radius, cr half the diameter, by half the 
 arc of the feftor, for the area. Or, multiply the diame- 
 ter by the arc of the feftor, and take ^ of the produft. 
 
 Note* The arc may be found by problem ix, hulk 
 
OF SUPIRTICIES, 
 
 111 
 
 RUEB 2. 
 
 As 3^0 is to the degrees in the arc of the fe£lor, fo is 
 the whole area of the circle, to the area of the fedor. 
 
 liote. For a femicircle take one half, for a quadrant one 
 quarter^ &c. of the whole circle. 
 
 EXAMPLES. 
 
 1 . What is the area of the feftor CADB, the radius 
 being 10, and the chord AB 16".^ 
 
 100 1= AC* D 
 
 64 = AEa 
 
 K^ fE ::^B 
 
 36 ( 6 = CE 
 10 = CD 
 
 4 = DE 
 
 "16 = DE* 
 64 ~ AE« 
 
 80 ( 8»9442719 = AD 
 8 
 
 71-5541752 
 16 
 
 3)35-5341752 
 2) 18-3180384 arc ADB 
 9-2390297 = half arc 
 10 =: radius 
 
 52'590297 anfwer.' 
 
 Kx.2. 
 
11-2 WE!I8trillTn>N 
 
 Ex. 2. Required the area of the fe(f^or, whofc arc con- 
 tains 18 degrees; the diameter being 3 feet, 
 
 9 
 
 Then, as ']60 : 18 : : 7*o6"86 the area of the whole circle. 
 
 Or as 20 : 1 : : 7-0686 : '353V3 the anrvver. 
 
 Ex 3. What is the area of ttie fedor, whofe radius U 
 10, and arc 20 ? Anf. 100. 
 
 Ex, 4. What is the area of the fed^or, v^hofe radius is. 
 9, and the chord of iis arc 6? Anf. 27*.52678. 
 
 Ex. 5. Required ihc area of fhe fcdior, whofe radius is 
 25, its arc containing 147 degrees 29 minutes. 
 
 Anf. .S01.'4017. 
 
 Ex, 6. To find the area of a quadr^inf and a femicircle,. 
 to the radius, 13» Anf. 132-732e" and 265-'i052.. 
 
 PROBLEM XII. 
 
 To find the Area of a Segment of a Circle. 
 RULE 1. 
 
 Find the area of the feftor having the fame arc with 
 the fegraenr, by the laft problem. 
 
 Find alfo the area of the triangle, formed by the chord 
 of the fegment and the two radii of the fefior. 
 
 Then add thef<j two together for. the anfwer when the 
 fcgmeni is greater than a fcmicircle ; but fubtrad them-- 
 for the anfwer when it is lefs than a femicircle. 
 
 . EXAMPLE K 
 
 Required the area of the fegreient ACBDA, its chord 
 AB being 12, and the radius AE'or CE 10. 
 
 100 
 
• F SUPZRTIClfiS* 
 
 11$ 
 
 J 00 AE^ 
 36' AD* 
 
 its root 8 DE 
 from 10 CE 
 
 leaves 2 CD 
 
 4CD2 
 36 AD* 
 
 40 chord AC* 
 
 \ 
 
 X 
 
 l)~~7\ 
 
 9 AD 
 
 — 8 DE 
 
 Its root 6-324555 chord AC — t 
 
 8 48flreaof A £l^B 
 
 50*5S6440 
 12* 
 
 3 ) 38*5^644 
 2) 12'86548 arc ACB 
 6''43274.tialf arc 
 10 radius 
 
 64*3274 area of fc^, or EACB 
 .48 '0000 area of triangle EAB 
 
 anf. l6*3274area«f fegm» ACBA. 
 
 . ", rul-e' 2. . . 
 To the chord of the whole arc add -f of the chord of 
 half the arc, or add the latter chord and f of it mare. 
 Multiply the fum by the verfed fine or height of the feg- 
 ment, and take ■j'V of the produ^ for the area of the 
 fegment, ' 
 
 Ex, 
 
lli MINSVRATIO:* 
 
 Kx. Take the fame example, in which the radius is 10, 
 and the chord AB 12. 
 
 Then, as before, are found CD 2» and the chord of 
 the half arc AC 6-3^4555 
 Hence 4 is 2-108185 
 AB 12- 
 
 20'4327-10 
 CD - . 2 
 
 40-86548 
 
 •4 
 
 Anf. l6'»346l92 area nearlf. 
 
 RULE 3. 
 Divide the height of the fegment by the diameter, and 
 find the quotient in the column of hcightior versed fines, 
 at the end of the book. 
 
 Take out the correfponding area in the next column on 
 the right hand, and muliiply it by tht fquarc of the diamc* 
 ter, for the anfwer. 
 
 Ex. The example being the fame as before, we have 
 CD equal to 2, and the diameter 20. 
 Then 20 ) 2 ^ •! 
 And to •! anfwers '040 S7 5 
 Sq. of diam. . . 4C0 
 
 Anfwer 16-3500 
 
 OTHER EXAMPLES. 
 
 Ex. 2. What is the area of the fegment, whofc height 
 is 2, and the ch»rd 20? Anf. 26'878787. 
 
 Ex. 3. What is the area of the fegment, whofe height 
 is 18, and diameter ©f the circle 50? Anf. 6ti6'375, 
 
 Ex. 4. Required the area of the fegment whofe chord 
 is l6, the diameter being 29. Anf. 44.7292. 
 
 rao- 
 
• F STTPE11FTCIE8. 
 
 115 
 
 PROBLEM XIII. 
 
 To find the Area of a Circular Rifig, or Space included be* 
 tnueen t'wo Concentric Circles, 
 
 Take the difference between th« two circles, for tlie 
 ring ; or multiply the fum of the diameters by their 
 difference, and multiply the produd by '/Sd^, for the 
 anfwcr. 
 
 EXAMPLES, 
 
 1 , The diameters of the two concentric circles being 
 AB 10 and DG 6', required the area of the rinjg contained 
 between their circumferences AEBA, and DFGD, 
 
 JO -785-1 E 
 
 6 64 
 
 fum l6 
 
 31416 
 
 47124 
 
 64 50* 2^*56' anf. 
 
 Kx. 2. The diameters of two conccQtric circles being 
 20 and 10; requited the area of the ring between their 
 circumferences. Anf. 235*62. 
 
 Ex. 3. What is the area of a rins, the diameters ^ 
 
 whofe bounding circles arc 6 and 4 ? 
 
 ring. 
 
 Anf. 15-708. 
 
 PROBLEM XI7. 
 
 To meafure long Irregular figure » 
 
 Take the breadth in fever-^l places, at equal diftances. 
 Add the firft and laft two breadths together, and divide the 
 fum by 2. for the half fura, or arithmetical mean between 
 thofe two. Then add together this mean and all the other 
 breadths, omitting tha firft and laft, and divide their fum 
 
 by 
 
116 
 
 MENSURilTION OF SUPEHFICIIS. 
 
 by the number of parts Co added, which will give a me- 
 dium breadth ameng the whole; then muhiply it by th» 
 length, to give the true area. 
 
 If the breadths be not taken at equal diflances ; then 
 compute all the little trapezoids feparately, and add thctn 
 all togethcr.~Or, add all the breadths together, and di- 
 vide the fum by the whole number of them for the mean 
 breadth, to multiply by the length for the whole area, 
 which will not be far from the truth. 
 
 EXAMPLES. 
 
 1. The breadths of an irregular figure, at five cqui- 
 diftant places being AD 8-2, mp 7*4, nq j)-2, or 10*2 
 BC S'6; and the length AB 39 i required the area* 
 
 8-2 
 
 B-6 
 
 5 ) 16''8 
 
 8 '4 mean of firft and hd 
 
 4 ) 35'2 
 
 8'S mean of all 
 39 length 
 
 m 
 
 B 
 
 343 '2 anfwer. 
 
 Ex. 2. The'length of an irregular figure being 84, and 
 th« breadths at 6 equi-diftant places 17*4, 20*6', 14*2, l6'v5, 
 CO ] , 24*4 ; what is the area? Anf. 1550*64» 
 
 MEK^ 
 
MENSURATION OF SOLIDS. 
 
 DEFINITIONS, 
 
 SOLIDS, or bodies, are figures having length, breadth, 
 and thicknefs, 
 
 2. A prifm is a folid, or body, whofe ends are any 
 plane figures, which are parallel, equal, and fimilar; and 
 
 ' its fides are parallelograms. 
 
 A prifm is called a triangular 
 one, when its ends are triangles; 
 a fquare prifm, when its ends arc 
 fquares ; a pentagonal prifm, when 
 its ends are pentagons; and fo 
 on. 
 
 3. A cube is a fquare prifm, 
 baving fix fides, which are all 
 fquares. It is like a die, having 
 its fides perpendicular to one an- 
 other. 
 
 4. A parallelepiped on is a fo- 
 lid having fix rei^angular fides, 
 every oppofite pair of which arc 
 equal and parallel. 
 
 5. A cylinder is a round 
 prifm, having circles for its 
 ends. 
 
 6. A pyramid is a folid having 
 any plane figure for a bafe, and 
 its fides are triangles whofe ver- 
 tices meet in a point at ihe top, 
 called the vertex of the pyramids 
 
 The 
 
Hi MJIN8URATI0K 
 
 The pyramid takes names according to the figure of its 
 bafe, like the prifm; being triaBgular, or i^uare, or hex- 
 agonal, Sec 
 
 7. A cone is a round pyramid, /\ 
 
 having a circular bafe, Mi \ 
 
 S. A fphere is a folid bounded mB^ ""^^ 
 
 by one continued convex furface, WOuh j-'—M 
 
 every point of \vhich is equally dif- ^^liL— -"^ 
 
 tant from a point within, called the >i#3S^^ 
 
 centre. — The fphere may be con- »^^^^ 
 
 ceived to be formed by the revolu- ffi ,^B 
 
 tion of a femicircle aVjout its diame- ^^^^^y 
 
 ter, which remains fixed, ^^^^^ 
 
 9. The axis of a folid, is a line drawn from the middle 
 of one end, to the middle of the oppofile end j as between 
 the oppofite ends of a prifm. Hence the axis of a pyramid, 
 is the line from the vertex to tlie middle of the bafe, or 
 the end on which it is foppo'ed to ftand. And the axi« 
 
 , of a fphere, is the fame as a diameter, or a line pafling 
 * throHgh the centre, and terminated by the furface on both 
 fides, 
 
 10. When the axis is perpendicular to the bafe, it is a 
 right prifm cr pyramid ; otherwife, it is (/b iqup. 
 
 ! 1 J. The height or altitude of a folid, is f» line drawn 
 from its vertex or top, perpendicular to its bafe. — This is 
 equal to the axis in a right prifm or pyramid ; hut in an 
 oblique one, the I eight is the perpendicular fide of a 
 right-angled triangle, whofe hypothenufe is the axis. 
 
 12. Alfo a prifm or pyranud is regular or irregular, as 
 its bafe is a regular or an irregular plane figure. 
 
 13. The fegmeni of a pyramid, fphere, or any other 
 folid, is a part cut off the top by a plane parallel to th« 
 bafe of that figure. 
 
 4 U. A 
 
OI »OLII>S. 13JI 
 
 14. A fraftrum or trunk, is the part that remains at the 
 bottom, after the fegrrfent is cut oft*. 
 
 15. A zone of a fp^ere, is a part intercepted betweeix 
 two parallel plaaes p ^d is the difference, betjs^een two 
 fegments. When the ^nds, or plar es, are equally diftant 
 from the centre, oa both fides, tfee figure is called the 
 middle zone. ' ' i ' ' * 
 
 16. The fedor of a fphere, is concpofed of a fegment 
 lefs than a hemifpiiere or half fphere, and of a cone having 
 the fame bafe with ihe feginent, ajfui its vertex in the 
 centre of the fphere, ^ ^' 
 
 17. A circular fpindle, is a foHd ge- 
 nerated by the revolution of a fegment 
 of a circle about its chord, which re- ^^' 
 mains fixed, 
 
 18. A regular bod)-, is a folid contained under a cer- 
 tain number of equal and regular plane figures «f the fame 
 fort. 
 
 1<). The faces of the folid arc the plane figures under 
 which it is contained. And the linear fides, or edges of 
 the folid, are the fides of the plane faces. 
 
 20. There are only tive regular bodies: namely, ift, 
 the tetraedon, which is a re.'ular pyramid, having four 
 trianguliir faces : 2d, the hex.^edron, or cube, which has 
 6 equal fquare faces; 3d, the oQaedron, vhich has 8 
 triangular faces j 4th, the dodecaedron, wnich has 12 
 pentagonal faces ; 5th, the icofaedron, which has 20 tri- 
 angular face'^. 
 
 Note, if the following figures be exaftly drawn on 
 pafteboard, and the lines cut half through, fia that the 
 parts be turned up and the'r edges glued together, they 
 will reprefcnt the five reiJ^uiar bodie?: na^neiy, figure 1 
 the tetraedron, figure 2 the Jiexaedron, figure 3 the oc- 
 taedron, figure \ the dodecaedron, and figure 5 the ico- 
 faedron. ^^^...H^— «- - 
 
 
12» 
 
 >I£NSURATXaN 
 
 ■ 
 
 to 
 
 Koie alfo, that, in cubic meafure, 
 
 17 '2 8 inches make 1 foot 
 
 27 feet - - 1 yard 
 
 166" J yards - 1 pole 
 
 1)4000 poles - - 1 furlong 
 
 5 1 2 furlongs 1 mile. 
 
 PRO. 
 
OF somds. 121 
 
 PAOSLEM I. 
 
 To find the 'Sdidity of a Cube. 
 
 ^wJ^K"^?^i''^^^'^^^'*^«^«'^t«nt; that is, multiply 
 the fide by itfelf, and that produd by the fide agaili ^ ^ 
 
 EXAMPLES. 
 
 ^ 1. If the fide AB, or AC, or BIX of a cube be 24 
 mch^fi, what is its folidity or content ? 
 
 2-t 
 
 48 
 
 24 
 
 2304 
 1152 
 
 13824 anfwcr* 
 
 iIllit:o. ""^ ''^^^ '^^^ '^^ - ^^^-b^ where 
 
 Ex.3 Required how many folid frr^ ,r. ^"^^^0648. 
 whofe fide is 18 inches? ^ ^''^ *'" ^" '^Z-^be 
 
 Anf. 3|, 
 
 PROBLEM II. 
 To find the Solidity ,f a Paral/elo^ipedon 
 
X2t MRNSVIlATIdN 
 
 BXAKPLEff. 
 
 1. Rcquirc<! the content of the parallelopipedon, whofc 
 length AB is ^fect, its breadth AC 2^ feet, and altitude 
 BD 1 J feet? 
 
 1«75 BD 
 6AB 
 
 10-50 
 2-5 AC 
 
 5250 
 2100 
 
 *• is 
 
 26-250 anfwer. 
 
 Ex. 2. Required the eontent of a parallffton'pe^on, 
 5vhofc length is 10-5, breadth 4*2, and height b • 
 
 Anf. J 
 
 Ex. 3. How many cubic feet are in a block of mai \ , 
 whofe length is 3 feet 2 inches, breadth 2 feet 8 inches 
 and depth 2 feet 6 inches? Anf. 21f. 
 
 raoBLEM III. 
 
 To find thi Solidity of any Prifm^ 
 
 Find the area of the bafe, or end ; which multiply, by 
 the height, or* length, and it will give the content. 
 
 Which nilfe will do, whether the prifm be triangular, 
 cr fquare, or pentagonal, Ac, or round, as a cylinder. 
 
 EXAMPLES. 
 
 1. What is t1)e content of a triangular prirm, whofc 
 length AC is 12 feet, and each fide AB of its equilatetal 
 bafe infect? 
 
 Hcrc±xl=H£ = (?i.' 
 
 2 2 4 Then 
 
Oi? SOLIDS, 
 
 Then '4330 13 tabular n» 
 H 
 
 2-598078 
 108253 
 
 2706331 area of end 
 12 length 
 
 32*475972 anfwcr. 
 
 Ex. 2. Required the folidity of a triangular prifin, 
 whofc length is 10 feet, and the three fides of its trian- 
 gular end or bafc, are 5, 4, 3 feet ? Anf, 60. 
 
 Ex. 3. What is the content of a hexagonal prifni, the 
 length being 8 feet, and each fide of its end 1 foot 6 in- 
 ches? Anf. 46''76"5368. 
 
 Ex. 4, Required the content of a cylinder, whofe 
 length is 20 feef, and circumference 5| feet ? 
 
 Anf. 48»1459. 
 
 Ex, 5. What is the content of a round pillar, whof« 
 height is iS feet, and diameter 2 feet 3 inches ? 
 
 Anf. 63-6174. 
 
 rEOBLBM IV. 
 
 To find the Convex Sur/ate of a CjrUnder, 
 
 Multiply the circumference by the height kr length o£ 
 the cylinder. 
 
 Note. The upright furface pf any prifm is found in the 
 feme manner, viz. by multiplying the perimeter of the 
 end by the length. And the folidity of a cylinder is 
 found as the prifm in the laft problem, 
 
 EXAMPLES. 
 
 1, What is the convex furface of a cylinder, whofe 
 length is i6 feet, and its diameter 2 feet 3 inches, 
 
 G 2 3-1416 
 
in 
 
 MEKSFRATION 
 
 3-U16' .,, 
 2^ diameter 
 
 6-2S32 
 •7854 
 
 
 ( 
 
 
 
 4241 16 
 
 706S6 
 
 
 113'()976anfwer. 
 
 
 Kx. 2. Required t^e convex furface of the cylinder; 
 whofe length is CO feet, and its diameter 2 feet ? 
 
 Anf. 1 25-664. 
 Ex. 3. What is the convex furface of a cylinder, whofc 
 length is 18 feet 6 inches, and circumference 5 feet 4 in- 
 ches? PROBLEM V. Anf. 98|. 
 ^ofind the Conrvex Surface of a Right Cone,. 
 Multiply the circuaiferenc* of the bafe by the flant 
 height, f r length ot the fide, and take half the produd 
 for the furface. 
 
 Ex. 1. If the diameter of the bafe be AB 5 feet, and the 
 fide of the cone AC 18, required the convex furface ? 
 3i4l6 
 
 5 diameter C 
 
 15*7080 circumf, 
 18 
 
 125664 
 15708 
 
 2)282-744 
 
 14 1 '372 anfwer. 
 
 Ex. 2. 
 
OF SOLIPS. 125 
 
 Ex. 2. Wkat is the convex furfacc of a cone, whofe 
 fide is 2Q4 and the circumference of its bafe 9? 
 
 Anf. 50. 
 
 Ex, 3. Required the convex furface of a cone, whofc 
 fl-ant height is 50 feet, and the diameter of its bafe 8 feet 
 6 inches f Anf. 66? -51). 
 
 PROBLEM VI* 
 
 to find the Cott'uex Surface of the Frufium of a Right Cone, , 
 
 Add together the perimeters of the (wo ends; then 
 multiply that fum by the fl«nt height, or fide of the 
 fruftuoii and Cake half the produfl for the furfice. 
 
 tXAMPLES. 
 
 1. If the circumfcfcnccs of the two ends be 12*5 and 
 10'3 and the flant height AD 14, re<juir§d the COBvex 
 inrfacc of the fruRum ABCD ? 
 
 J2-5 
 
 10-3 A 
 
 ill ^^% 
 
 2 ) 319*2 ' 
 159'6anfwer, 
 
 Ex. 2. What is the convex fur face of the fruftum of a 
 cone, the flant height of the fruftum being 12*5, and the 
 circumferences of the two ends 6 and 8*4 f Anf. 90. 
 
 Ex. 3. Required the convex fnrface of the frul^una of 
 
 a cone* the fide f f the fruftum being 10 feet 6 inchep, 
 
 and the circumferences of the two ends 2 feet 3 inches, 
 
 and 5 feet 4 inches? Anf. 39 r^. 
 
 G 3 r&o. 
 
fJlOBLlM VII. 
 
 Ta find the Solidity of a Cene, or any Tjramtd* 
 
 Ceinputr the area of the bafe, then multiply that area 
 by the height^ and take \ of the product for the content. 
 
 EXAMPLBS, 
 
 1. What is the folidity of a cone, whofe height CD is 
 12 J feet, and the diameter AB of the bafe 2^? 
 
 ncfe$ix2i = | ^lzz^zz6i, 
 
 Then 7854 
 64 
 
 4-7124 
 
 4*S0875 area of bafe, 
 12^ height. 
 
 58905CO 
 2454375 
 
 3 )6l '359375, 
 
 20'453125 anfwer. 
 
 Ex.2. What is the foiid content of a pentagonal pyramid, 
 
 its height being 12 feet, and each fide of its bafe 2 feet? 
 
 1720477 tab. area 
 
 4 fq. fidc 
 
 6-*J^ 1908 are* b^fe 
 
 4 i of height 
 
 27*527632 
 
 Ex. S< 
 
©F /SOLID*. 137 
 
 Ex. 3. What is the content of a cone, its height being 
 lOl feet, and the xiircumferenca of its bafe 9 feet? 
 
 ' Anf. 22- 56093. 
 Ex. 4. Required the content of a triangular pyramid, 
 its height being 14 feet 6 inches, and the three fides of iia 
 t)arc5, 6. 7? Anf. 71 -0302. 
 
 - Ex. 5. What is the content of a hexagonal pyramid, 
 whofe height is 6*4, and each fide of its bafe 6 inches ? 
 
 Anf. I'JSofil, 
 
 PHO^LEM VIII. 
 
 Tiijind the Solldiij of the FruJtuM (f a Cone »r mny 
 pyramid, 
 
 KVLES. 
 
 1. Add into oi>e fum, the areas of the two ends, and 
 the mean proportional between them, or the fquare root 
 of their produft; and take i of that fam for a mean area ; 
 which multiplied by the height of the fruftum, will give 
 the content. 
 
 2. When the ends are regular plane figures 3 the mean 
 area will be found by multiplying \ of the correfponding 
 tabular number belonging to the polygon, either by the fum 
 arifing by adding together the fquare of a fi4e of each end 
 and the produd^ of the two fides, or by tTie quotient of 
 the difference of their cubes divided by rheir difference, or 
 by the fum arliing from the fquare of their half difference 
 added to 3 times the fquare of their half fum. 
 
 3. And in the fruftum of a cone, the mean area is 
 found by multiplying •iJb'l 8, or | of 7854, either by 
 the fum arifing by adding together the fquares of the two 
 diameters and the produ<^ of the two, or by the difference 
 of their cubes divided by their difference, or by the 
 fquare of half their difference added to 3 times the fquare 
 of their half fum. 
 
 64 Or^ 
 
12ft IIENSUHATieK 
 
 Or, if the circumfrrenccs be uffd in like mann«jr, in* 
 ftcad of their diameters, the multiplier will be 'OSflAlp, 
 inBeadof '26 1 8. 
 
 KXAMFLES* 
 
 1. What is the content of a Iruftum of a cone, whofe 
 height is 70 inches, and the diameters of its two ends i^ 
 and 20 inches? 
 
 58 20 
 
 20 20 
 
 560 4«0 
 
 784 
 
 784 
 
 400 
 
 174* 
 
 •2618 
 
 J3962 /# Pr*^. vi. 
 
 1744 
 J0464 
 3488 
 
 456-6792 
 20 
 
 131 '5840 anfwcr. 
 
 ' Ex. 2. Required the content of a pentagonal fruftum, 
 whofe height is 3 feet, each fide of the bafc I foot 6 ia- 
 ches, and each fide of the lefs end 6 inches? 
 
 ]S 
 
18 
 18 
 
 144 
 18 
 
 324 
 
 144 
 
 12 
 
 fl2 
 112 
 
 OF SOLIDS. 
 
 18 
 
 6 
 
 G 
 6 
 
 108 
 
 324 
 
 36 
 
 36 
 
 3 ) 468 
 
 
 1 56 1 of Aim 
 1-720477 tab. area 
 
 129 
 
 10322862 
 8602385 
 1720477 
 
 268-394412 mean area 
 5 height. 
 
 1341 •972060 
 lll'83100o 
 9*319250 anfwer in cubic feet. 
 
 Ex. 3. What is the foHdity of the fruftum of a coac, 
 the altitude being 25, the circumference at the greater 
 end 20, and at the lefs end 10 ? Anf. 464'i30,5, 
 
 Ex. 4. How many folid feet are in a piece of timber, 
 whofe bafes are fquares, each fide of the greater end being 
 15 inches, and each fide of the lefs end 6 inches ; alfo the 
 length, or perpendicofar alritude is 24 feet? Anf. 19^-. 
 
 Ex. 5. To find the content of the fruftum of a cone, 
 the altitude being 18, the greatcft diameter 8, and the 
 leaft 4. Anf. 527*7888, 
 
 Ex, 6. What is the folidlty of a hexagonal fruftum, 
 the height being 6 (ect, the fide cf the greater end 18 in- 
 chei, and of ihc left 12 inches? Anf. 2-i*4)81722. 
 
 G $ PRO- 
 
130 
 
 MENkURATIOV 
 
 PROBLEM IX. 
 
 Te find the SaliJity 9J a Wedge. 
 To the length of the edge acid twice the length of the 
 back or bafe, and refervc the fum ; multiply the height 
 of the wedge by the breadth of the bafe; then tnultiply 
 this produA by the rcferved fum, and take \ of the laft 
 produft fof the content. 
 
 EXAMFLES. 
 
 1. What is the content in feet of a wedge, whofe altitude 
 AP IS 14 inches, its edge AB 21 inches, and the length 
 of its bafe DE 32 inches, and>its breadth CD 4j inches? 
 
 21 U 
 
 32 41: 
 
 32 — • 
 
 S5 
 
 ^55 
 ,-«o # '* 892*5 anf. in cubic inches 
 J/28^,. 74375 
 
 6-197516 
 •5 1 6493 anf. in feet, or little more 
 — " ■ than half a cubic foot. 
 
 F.x, 2. Required the content of a wedge, the length 
 and breadth of the bafe being 70 and 3D inches, the 
 length of the edge 1 10 inche?> and the height 34-2901 6' ? 
 
 Anf. 24-8048> 
 
 PROBLEM X. 
 
 Tofiud tht Solidity of a Pri/moid, 
 
 Definition, 
 
 A prifmoid differs only from the fruftum of a pyramid,, 
 
 in not baving its oppodtc ends fitniiar planes. n tf Ls« 
 
I 
 
 OF S01IZIS< 
 
 RULE*. 
 
 131 
 
 Add into «nc fura, tke area* of the two ends and 4 
 times the middle fedion parallel to them, and ^ of that 
 futn will be a mean area; which being multipjied by the 
 height, win give the content. 
 
 riote. For the length of the middje fcdion, take half 
 the fara of <he lengths of the two ends; and for ita 
 breadth, take half the fumof the breadths of the two ends. 
 
 E3CAMPLES. 
 
 1. How many cubic feet are there in a ftone, whofc 
 ends are reftangies, the length and breadth of the one 
 being 14 and 12 inches; and the correfponding fides of 
 the other 6 and 4 inches : the perpendicular height beinj?^ 
 SOifcct?^ ^^ 
 
 14 
 12 
 
 les 
 
 10 
 
 8 
 
 80' 
 4 
 
 320 
 
 16*8 
 
 24 
 
 6)7l2 
 
 6« 
 
 4 
 
 24> 
 
 85 J mean area in inches, 
 30i height 
 
 256^0 
 42V 
 
 144 
 
 2602 -6 
 
 2l()-8 
 
 18-074 anfwefi 
 « 6 
 
 Bx. 2. 
 
132 
 
 MENSlfRATIOV 
 
 Ex. 2. Required the content of a reftangular prifmoid, 
 whofc greater end meafure5 12 inches by 8, the lefler ^nd 
 8 inches by 6, and the perpendicular height 5 feet ? 
 
 Anf. 2*453 feef. 
 
 Ex. 3. What is the content of a cart or waggon, whofe 
 
 infide dimenfions are as follow: at the top the length and 
 
 breadth 81^ and 55 inches, at the bottom the length and 
 
 breadth 41 and Sjyi inches, and the height 47 -^ inches? 
 
 Anf. 126340-5(j375 cubic inches. 
 
 PROBLEM XI. 
 
 To fndthe Convex Surface »f a Sphere or Chbe, 
 
 Multiply its circumference by its diameter. 
 
 Note, in like manner the convex furface of any zone 
 or fegment is found, by multiplying its height by the 
 whole circumference of the fphcre. 
 
 EXAMPLIS. 
 
 1. Required the convex fuperficies of a globe, whofe 
 iliameter or axis is 24 inches. 
 3M416 
 
 24 diam* 
 
 125664 
 62832 
 
 75*3<)84 circumf. 
 2+ 
 
 30 1 5936 
 1507.968 
 
 I80p-56l6anfwei. 
 
 Ex. 2. What is the convey furface of a fpherc, whofc 
 diameter is 7, and circumference 22 ? Anf. 154, 
 
 Ex. 3* 
 
OF SOLIDS, 133; 
 
 Ex. 3. Required the area of the furface of the earth, 
 its diameter, or axis, being 79^7^ n»iles, and its eircura- 
 fcrence 25000 miles? Anf. 1989437.50 fq. miles. 
 
 Ex. 4. The axis of a fphere being 42 inches, what is 
 the convex fuperficies of the fegmenr, whofe height is 
 pinches? ^ Anf. 1187-5248 inches. 
 
 Ex. 5» Required the convex furface of a fpherical 2^mc, 
 whofe breadth or height is 2 feet, and cut from a fphere 
 •f 124 feet diameter ? Anf. 78-34 feet. ' 
 
 PROBLEM XII. 
 
 To find the Solidity of a Sphere or Globe, 
 Find the cube of the axis, and multiply it by •523G. 
 
 EXAMPLES. 
 
 1. What is the folidity of the fphere, whofe axi»-is 
 12? /. ) 
 
 Or thus 
 12 •523(5 
 
 12 12 
 
 144 6-2832 
 
 12 12 
 
 1728 75-3984 
 
 •5236 12 
 
 10368 904-7808 anf. 
 5184 
 
 3456. 
 «640 
 
 9O478O8 anf. 
 
 Ex. 2. To find the content of the fphere, whofe axis ^ 
 
 is 2 feet 8 inches. Anf. 9*9288 ifeet. ', 
 
 4 Ex. 3. i 
 
1S« 
 
 MEVrtVRATlOH 
 
 Ex. 3. Required the folid content of the earthy fop- 
 poGng its circiimfcrcnccto be 25000 miles? 
 
 Anf. 263858149120 miles. 
 
 rROBLBM Xfll. 
 
 To find the SotUity of a Spherical SigmenU 
 
 To three times the fquare of the radius of its bafe, add 
 the fquare of its height ; then muliiply tbc fum bj the 
 height^ and the produ^ again by *5236'. 
 
 EXAMPLES. 
 
 1. Recjulred the content of a fpherical fcgmcnt, its 
 height being 4 inches* and the radius of its bafc 6 ? 
 8 4 •5236' 
 
 8 4 832 
 
 64 
 3 
 
 192 
 
 16 
 192 
 
 208 
 4 
 
 832 
 
 10472 
 13708 
 4188& 
 
 435*6352 anf. 
 
 Ex. 2. What is the foKdity of the fegraent of a fphcr«, 
 whofe height is 9, and the diameter of its bafe 20? 
 
 Anf. 1795-4244, 
 
 Ex. 3. Required the content of the fpherical fegment, 
 
 whofe height is 2^, and the diameter of its bafe 8 61084? 
 
 An£ 71:5695. 
 
 T.ROBLBM XIV. 
 
 T»find the Soliditj of a Spherical 2^ne or Entjium^ 
 
 Add together the fquare of the radius of each end, and 
 ^ of the fquare of their dillance, or of the height; then 
 multiply the fiiin by the faid height, and the produft again 
 by 1 -5708. li- 
 
IXAMPLlf.' 
 
 19^ 
 
 1. What is the folid content of a zone, wliofe greateff 
 diameter i» 12 inches, the lefs S, and the height 14» laohes? 
 
 6 
 
 B6 
 
 4 
 4 
 
 16 
 36 
 33f 
 
 1«5708 
 
 125664 
 
 5236 
 
 > ■ ■ ■ 
 
 134-p41j^ 
 
 10 
 
 1340-416 anf. 
 
 10 
 10 
 
 3) 100 
 
 83^ 
 
 Ex. 2. Required the content of a zone, whofc greater 
 diameter it 12« lefs diameter 10, and height 2 ? 
 
 Anf. 195*8264. 
 Ex, 3. What is the content of a middle zone, whofe 
 keight is 8 ktt, and the diameter of each end 6 ? 
 
 Anf. 494-2784 feet. 
 
 PKOBLIM XV. 
 
 To find the Surfact of a Circular SpinJie* 
 
 Multiply the length AB of the fpindle by the radius 
 OC of the revolving arc. Multiply alfo the faid arc 
 ACB by the central diftance OE, or diftancc between 
 
 tiit 
 
13^ I MEKSVftlTION 
 
 the centre of the fpindle and centre of the revolving arc, 
 SubtraA the latter prodiiO from the f )rmer, and multiply 
 double the remainder by 3*14i6, or the fmgle remainder 
 by 6"2&32, for the furfacc. 
 
 Kite, The fame rule will ferve for any fegment or 
 zone cut off perpendicular to the chord of the revolving 
 arc, only ufing the particular length of the part, and the 
 part of the arc which defcnbes it, inftead of the whole 
 length and whole arc, 
 
 IXAMFLES, 
 
 1. Required the' furface of a circular fpindle, whbfe 
 length AB is 40, and its rhicknefs CD 30 inches? 
 
 Here, by the noi es at pa . yi. 
 
 The chord AC = V^EM- CE^ =r ^20* + 15* = 25, 
 
 oca 
 
 and 2CE : AC : : AC : CO = ^^ = 20J, 
 
 ow 
 
 hence OE = OC — CE = 20| — 15 = 5». 
 
 Alfo, by ptoblcmfx, rule 2, ^ne^Mtu-c. o^^/ t/vyV i 
 
 C 
 
 25 AC 
 8 
 
 200 
 40 AB 
 
 o; l60 
 
 534 arc ACB 
 
 Then, 
 
•r 9thl99i f^i 
 
 The«, by the rule, 
 
 40 5 J 
 
 800 266'|- 
 
 33| 4iJ 
 
 3nf 
 
 31 1| 
 
 522|.or522-2o» ♦y* 
 6*2832 
 
 »— — « Or thus^ 
 
 10444. . 6-2832 
 
 156666 , 4700 
 
 4177777 439824 
 
 10444444. 251323 
 
 313333333 . ■ 
 
 . -. • 9)29531»04^ 
 
 328 1'22666* 328 1 '226 anf. nearly. 
 
 Ex. 2. What is the furftce of a circular fpindle, whofe 
 lengih is 24, and ihickuefs in the middle 18? 
 
 Anf. 1177-4485, 
 
 fkfiBLEM XTI. 
 
 Toj^ftdiht Solidity of a Circular Spindle • 
 
 Find the arc;i of the revolving fegment ACBEA, whick 
 multiply by half the central diftance OE. Subtraft the 
 produdl from \ of the cube of AE, half the length of the 
 fpindle. Then nndtiply the remainder by 12*5664, or 4 
 times 3*14l6, for the whole content, 
 
 EXAMPL-ES. 
 
 1. Required the content of the circular fpindle, wbo^ 
 length AB is 40, and middle diameter CD 30 ? . 
 
 [See the lafi Figure] By 
 
iW MlNSUR4Tn>V 
 
 By the work of the laft problem, 
 wehavcOE= 6| 20 half lenijth 
 
 and arc AC = 26} qq * 
 
 and rad. OC = 201 , "^ 
 
 22 
 
 400 
 H 20 
 
 '* _ 
 
 3 ) 8000 
 
 ScAorOACB 555* 
 AExOE=:OABii6| 
 
 2 ) 438| .-HZ 
 
 Ifcg.ACEsiPI 
 
 10571 
 
 183 ncail/ J 3 8644 
 
 fi7739 
 
 12801 ^532 
 
 -— 832 
 
 83 
 
 5 
 
 46'65-2l mtilt, imrcr. 
 
 17423-5 Anf. 
 
 Kx. 2. What is the foKdity of a circular irrindlcwhofe 
 length IS 24, and middle diameter 18 ? 
 
 Aflf. 3739-93. 
 
 PROBLEM XVII, 
 
 7> A4' /*r ^V/^'//^' cf the Middle FfvfiMm or Zone {/ a 
 Circular Spi»4U, 
 
 From the fquare of half the length of the whole fBin- 
 
 die take | of the fquare of half the length of the middle 
 
 |r«ftora, and multiply the remainder by the faid half 
 
 *cn^th of the ini^ura.-^Moltiply t<hc ocntral d^fteocc 
 
 i •• by 
 
• F S«LUI9« 
 
 l^pJ 
 
 hy the revolving area, which generates the middle fruf- 
 tum»—Suhtraft this latter produft from the former; then 
 the remainder multiplied by 6*2832, or 2 times 3'Ul6, 
 will give the content. 
 
 EXAMPLES. 
 
 1. Required thefoUdity of the fruftum, whofe length 
 mn is 40 inches, alfo its greateft diameter EF is 35, ani 
 kaft diameter AD or BC 24 ? 
 
 Draw BG parallel to mn, thpn we 
 have DG = 4^ran = 20, 
 and EG z= ^EF — |AD = 4, 
 chordDE* = DG*4- GE* = 4l6, 
 
 and DE» ^ EG = ii£ 
 
 4 
 
 104 the diameter of 
 the generating circlf. 
 
 or the radius OE = 52, 
 
 hence OI =: 52 — l6 =36 the central diftancc, 
 
 and HI* rzOH*— OI« =52*-36^ = 1408, 
 
 i DG« = i of 400 = ... 1331- 
 
 PG 
 
 1274f 
 
 20 
 
 254934- 1ft prod. 
 GE 
 
146 
 
 MCNSVAATIO!! 
 
 104 26 
 Its tab. fcgmcnt . - 'OOPfM 
 but 104* is . - loiVc 
 
 aret of {eg. DECGD • 
 mD X niA s 12 X 40 
 
 goier. area mDECa 
 OI 
 
 4326'4 
 P7344 
 P7344 
 
 107'HliOi 
 
 408* 
 
 687*5U04 
 
 352306(524 
 
 JII50*3.9744 
 25493' 33333 
 
 4342*93589 i 
 2382'6 n. 
 
 27287*5 anfweu 
 
 ^ Ex. ?• What is the content of the middle fruflum of a 
 circular fpindle, whofe Itmgth is 20, grcateft diameter 1 8, 
 and leaft di^njeter 8? Anf, iQ57'l6i 3. 
 
 tKQ' 
 
t? SOLIBA. 141 
 
 r&oBLBM xriii. 
 Ttjittd tbt Su^r/icUs or Solidity of any Regular Body* 
 
 1, Multiply the proper tabular area {taken from the 
 following table) by the fquare of the linear edge of the 
 folid, for the fuperficies. 
 
 2. Multiply the tabular folidity by the cube of the 
 linear edge, for the folid content. 
 
 Surfaces and Solidifies ot Regular Bodies-. j 
 
 No. of 
 faces 
 
 Names. 
 
 Surfaces 
 
 Solidities 
 
 4 
 
 6 
 
 8 
 
 12 
 
 20 
 
 Tetraedron 
 
 Hexaedron 
 
 Odaedron 
 
 Dodecaedron 
 
 Icofaedron 
 
 173205 
 6-00000 
 3-46410 
 20-64573 
 8-6V.02 
 
 0«11785 
 1 '00000 
 0-47140 
 7*66312 
 2* 18 169 
 
 
 EXA\ 
 
 IPLES. 
 
 
 1 . If the linear edge or fide of a tetracdron hy 3, re- 
 quired its furface and folidity ? ' 
 
 The fquare of 3 is 9, and the cube 27» Then, 
 ta , furf, 173205 0*11785 tab. fol. 
 
 9 27 
 
 fuperf. 15-58846 
 
 «2495 
 23570 
 
 folidity 3U8195 
 
 Ex. 
 
U2 
 
 Vl^rtVieATtoN 
 
 Ex. 2. What Is the fiipefficiesl aitcf folidlty of the hex. 
 aedron, whofe lineal fide is 2? 
 
 ^ r S Superficies "Z* 
 ^"'- I folidity 8 
 
 Ex. 3. Required the fupferftcies rfnd folidity of the oc* 
 Ucdron> whofe linear (ide is 2.^ 
 
 . f Cfuperficies 13*85640 
 '*"*• I folidity 377120 
 
 Ex. 44 Whaf it the fuperficies and folidity 6f the dodc- 
 tledron, wliofd linear fide is 2^ 
 
 .1 * f fuperficiei 8*»58292 
 ^"** Ifohdity 61-30496 
 
 Ex. 5. Required the fuperficies and folidity of the ice- 
 faedron, whole linear fide is 2f 
 
 M r ^ fuperficies .34*6'4100 
 ^'' 1 folidity 17-45352 
 
 rRo, 
 
tKO^LEM XtX, 
 Tnfitli ilie 'Stiff ace of a Cylifrdrkal Kivg, 
 
 This figure being only a cylinder bent round Into a 
 ring, its forfitee and folidity may be found as in the cy- 
 linder, namely, by multiplying the axi?, or length of the 
 cylinder, by the circumference of the ring, or of the 
 feftinn, for the furface ; and by the area of a fedion, for 
 the folidiry. 
 Or ufe the foHowirig rules: 
 
 For the furface,' — To the thicknefs of the ring add the 
 inner diameter; multiply this fum ly the thicknefs, and 
 the produft again by p* 8696, or the fquare of 3*14i(f. 
 
 EXAMPLES, 
 
 1. Required the fuperficies of a ring;, whofe thicknefs 
 AB is 2 inches, and inner diameter BC is 12 inches? 
 
 12 9*^696 
 
 2 28 
 
 14 7^568' 
 
 2 197392 
 
 2S 27^*3488 Anf. 
 
 Ex.2; What is the furface of the ring whofe inner 
 ^amcter b l6i and thicknefs 4? Ahf. /S^'jes; 
 
 PROBLEM X5C.- ' 
 
 To find the Solidity of a CylindrUai Ring* 
 
 To the thicknefs of the ring, add the iftncf. diafrret^r 5 
 then multiply that fum by the fquare of the thicknefs'; 
 and the produd again by 2*4()7'4, or ^ of tKc fqaare of 
 3 ♦U46', for the folidity, 
 
 2 ix« 
 
144 MSNfiURATlOK 
 
 EXAMPLES. 
 
 1. Required the foiidity of the ring, whofe thickaefs if 
 finches, and its inner diameter l^i 
 
 12 2*4674 
 
 2 5i} 
 
 14 148044 
 
 4 123370 
 
 56 138 1744 anf. 
 
 Ex. 2. What b the foiidity of a cylindrical ring, whofe 
 thicknefs is 4« and inner diameter l6.^ 
 
 Anf.789-56i. 
 
 OF THE 
 
 CARPENTERS' RULE. 
 
 THIS inftrnment is otherwife c:il]ed the Gliding rule; 
 and it h much ufed in meafuring of timber and ar. 
 tificers' works, borh for taking the dimeniions, and com- 
 puting the contents. 
 
 The inftrument confifts of two equal piece?, each a 
 foot in length, which are connedled together by a folding 
 
 One fide or face of the role, is divided into inches, 
 and half quarters or eighths. On the fame face alfo are 
 feveral plane fcafes, divided into 12th parts by dia. 
 gonal lines; which are ufed in planning dimenfions that 
 are taken in feet and inches. I he edge of the rule is 
 commonly divided decimally, or into tenths ; namely, each 
 foot into lO equal parts, and each of thofe into 10 part3 
 
 again; 
 
SWDING RUL&. 145 
 
 «^ain: fo that, by means of this laft fcale, dimenfions are 
 taken in feet and tenths and hundredths, and then multi- 
 plied as common decimal numbers, which is the beft way. 
 
 On the one part of the other face ar^^- four lines, marked 
 A, B, C, D; the two middle ©ncs, B and C, being on a 
 Aider, whieh runs in a groove made in the ftock. The 
 fame numbers ferve for both thefe two middle lines, the 
 ©ne being above the numbers, and the other below. 
 
 Thefe four lines are logarithmic one«, and the three 
 At B, C,. which are all e<]ual to one another, are double 
 lines, as they proceed twice over from 1 to 10. The 
 other or loweft line D, is a fingle one, proceeding from 
 4 to 40« It is alfo called the girt line, from its ufe im' 
 computing the contents of trees and timber. Upon it are 
 alfo marked WG at 17*15, and AG at 18;95, the wine 
 and ale gage points, to make this inftrument ferve the pur - 
 pofe of a gagifig rule. 
 
 On the other part of this face there is a table of the 
 value of a load, or 50 cubic feet of timber, at all prices, 
 from G pence to 2 lliillings a-foot, , 
 
 When 1 at the beginning of any line is accounted 1, 
 or unit, then the 1 in the middle will he 10, and the 1 at 
 the end 100; and when 1 at the beginning Is accounted 
 10, then the 1 in the middle is 100, and the 1 av the end 
 1000; and fo on. All the finallcr divifions being altered 
 proportionally. 
 
 PROBLEM I. 
 
 To multiply Numbers tsgeihei\ 
 
 Suppofe the two numbers 13 and 24. — Set- 1 on B,t« 
 13 on A; then agalnft 24 on B {lands 312 on A,' which 
 is the required produdl of the two given numbers 13 and 
 
 Note, In any operations, whep a number runs beyond 
 
 the end of the line, fcek it on ih^i other iadius, or other 
 
 part of the lifie; that is, take the 10th part of it, or the 
 
 H lOOtk 
 
lit) SLIDIHO RVL-E. 
 
 lOOtli part of it, &c, and incrcafc the rcfult proportionally 
 10 Told, or 100 fold, dec. 
 
 In like mamier ihc produft of 35 and 19 is 665. 
 and ihc produdt of 270 and 54 is U580, 
 
 7ROBLBM II. 
 
 To di'viJe by the Sliding Rule* 
 
 As fuppofc to divide 312 by 24.— Set the divifor 24 on 
 B to the dividend 312 on A; ihen againll 1 on B ftands 
 13, the qaotienr, on A. 
 
 Alfo 396 divided by 27 gives 14*(J, 
 
 And 741 divided by 42 ^.ives W6. 
 
 PROBLEM 111. 
 
 To/quare any l^umher. 
 
 Suppofe to fquare 23.--Set 1 on B to 23 on A; then 
 againft 23 on B, ftauds 529 on A, which is the fquare of 23, 
 Or, by the other two li^e^, fet 1 or 100 on C to the 
 30 on D, then .'gainft every number on D, Hands its 
 iquare in the line C. bo againft 23 iUtds 329 
 a^ain!^ 20 ftands 400 
 against 30 liands i^OO 
 ana io on. 
 If the given number be hundreds. Sec. reckon the I on 
 D for 100, or 1000, &c« then the conefponding 1 on G 
 is 10000, or 1000000, &c. So the fquare of 23U is foun4 
 to be 52^00* 
 
 PROBLIM IT. 
 
 To txtraR the Square Roof, 
 
 Set 1 or 100, &c. on C to 1 or 20, &c. on D; then 
 igainft every number found on C, ftands it5 fquare root 
 
SLIBING RULE. 147 
 
 So, againft 529 ftands its root 23 
 againft 400 ftands its root 20 
 againlt 900 ftands its root 30 
 agaiull 300 ftands its root 17*3 
 and (o on. 
 
 PROBLEM V. 
 
 ^ojlnd a Mean Vroforiional between t«wo Numbers, 
 
 As Aippofe between 29 and 430, — Set the one number 
 29 om C to the fame on D; then againft the other number 
 430 on C, ftands their mean preportional 111 on D, 
 Aifo the mean berween 29 and 320 is 9o-3» 
 And the mean between 71 and 274 is 139. 
 
 PROBLEM VI. 
 
 To find a Third Proportienal to i^o Numbers* 
 
 Siippofe to 21 and 32.— Set the firft 21 on B to the 
 fecond 32 on A ; then againft the fecond 32 on B, ftands 
 48*8 on A ; which is the third proportional fought, 
 
 Alfo the 3d proportional to 17 and 29 is 49*4. 
 
 And the 3d proportional to 73 and 14 is 2*/;. 
 
 PROBLEM VII. 
 
 To find a Fourth Proportional to thru Numbers, 
 Or, to perform the Rule»of.Three, 
 
 Soppofe to find a fourth proportional to 12, 28, and 
 114. — Set the finl term 12 on B to the 2d term 28 on A; 
 then againft the third term 114 on B, ftands 26'6 on A, 
 which is the ftmrth proportional fought. 
 
 Alfo the 4th proportional to 6, 14, 29, i« (yi'(d. 
 
 And the 4th proportional to 27, 20, 73, is 54*0. 
 
 H 2 TIM. 
 
 ft. 
 
TIMBER MEASURING. 
 
 PROBLEM I. 
 
 To find the Area, or Superficial Content, of a Board or 
 Plank. 
 
 JVluLTiPLY the length by the mean breadth. 
 f_ Note, When the board is tapering, add the breadths 
 at the two ends together, and take half the fum for the 
 mean breadth. 
 
 By the Sliding Rule, 
 
 Set 12 on B to the breadth in inches on A ; then againfl: 
 the length in feet on B, is the content on A, in feet and 
 fradional parts. 
 
 EXAMPLES. 
 
 1. What is the value of a plank, at l^d. per foot, whofe 
 length is 12 feet 6 inches, and mean breadth 11 inehes ? 
 
 By Decimals, By Duodecimals, 
 
 12-5 12 6 
 
 11 11 
 
 ly.isi 
 
 12 1 137*5 
 1145 
 Is. 5d anf. 
 
 l^d.isl 
 
 5 is 
 
 11 
 
 4id 
 
 Is 5d anf. 
 
 By the Sliding Rule, 
 AslSB: 11 A : : 124:B: Hi A. 
 That is, as 12 on B is to 11 on A, fo is 12^ on B to 
 1 lion A. 
 
TIMBER MEASURIN«» lif 
 
 Ex. 2. Required the content of a board, whofe length 
 is 11 feet 2 inches, and breadth 1 foot 10 inches, 
 
 Anf. 20' 5' S'' 
 
 Ex. 3. What is the value of a plank, which is 12 feet 9 
 inches long, and 1 foot 3 inches broad, at 2-3-d. a foot? 
 
 Anf. 3s. 3|d. 
 
 Ex, 4. Required the value of 5 oaken planks at 3d. per 
 foot, each of them being 17t ^^^^ long J and their feveral 
 breadths are as follow, namely, two of 13^ inches in the 
 middle, one of 14i inches in the middle, and the two re- 
 maining ones, each 18 inches at the broader end, and 11^ 
 at the narrower. Anf. ^1 3 8i. 
 
 PROBLEM II. 
 To find the Solid Content of Squared »r Four-fided Timber, 
 
 Multiply the mean breadth by the mean thicknef?, and 
 the product again by the length, and the laft produdt wiU 
 give the content. 
 
 By the Sliding Rule, 
 
 G D DC 
 
 As length : 12 or 10 : : quarter girt : content. 
 
 That is, as the length in feet on C, is to 12 on D when 
 the quarter girt is in inches, or to 10 on D when it is in 
 tenths of feet; fo is the quarter girt on D, to the conteat 
 on C. 
 
 Note 1. If the tree taper regularly from the one end to 
 the other, either take the mean hrtadth and thicknefs ia 
 the middle, or take the dimenlions at the two ends, and 
 half their fum for the mean dinaenfions. 
 
 2. If the piece do not taper regularly, but is unequally 
 thick in fome parts and fraall in others; take feveral dif- 
 ferent dimenfion.s, add them all together, and divide their 
 fum by the number of them, for the ra«an dimenfions, 
 
 3. The qaarter girt is a geopietrical mean prop( rtional 
 between the mean breadth and thicknefs, that is the 
 fq^uare root of their produft. Sometimes unlkiliul 
 
 H. 3 mea.. 
 
150 
 
 aiMBER MSASUKIN9. 
 
 meafurcrs uie the arithm£rk»l mean inflead •f it, that is 
 half their fum ; but this is always attended with error, 
 and the more {o, as the breadth and depth differ the more 
 from each other. 
 
 EXAMPLES. 
 
 1. The kngth of a piece of timber is 18 feet 6 inches, 
 the breadths at the greater and Iffs end 1 foot 6* inches and 
 1 fcv^t 3 inches, and the thicknefs at the greater and Icfsend 
 1 foot 3 inches and 1 foot: required the folid contcntt 
 Dicimals, Duidecimalsm 
 
 1-5 1 6 
 
 1-25 IS 
 
 2)2-75 
 1-375 
 
 mean breadth 
 
 mean depth 
 mean breadth 
 
 length 
 
 > 2) 2 
 I 
 
 9 
 
 4 
 
 6 
 
 
 1-25 
 1- 
 
 1 
 1 
 
 3 
 
 
 
 
 « ) 2-25 
 
 2 2 
 
 3 
 
 
 1-125 
 1'375 
 
 1 
 1 
 
 1 
 4 
 
 6 
 6 
 
 
 '5625 
 7875 
 3375 
 1125 
 
 1 
 
 1 
 4 
 
 6 
 6 
 
 6 
 
 6 
 
 
 18-5 
 
 1 
 
 18 
 
 6 
 
 9 
 
 7734-375 
 2375000 
 11546375 
 
 27 
 
 ' 10 
 
 1 
 
 3 
 
 4 
 
 2»'6i7l875 contont 
 
 C8 
 
 7 4 10 
 
 By 
 
TIMBER MEASURING. 151 
 
 By the Sliding Rule. 
 
 B A B A 
 
 As 1 : l.'ii : : lOi : 223, the mean fquare,. 
 CD CD 
 
 As 1 : 1 : : 223 : 14-9, quarter girt. 
 CD DC 
 
 As I8i : 12 :: 14-9: 2.S-6, the content. 
 
 Ex. 2. What is the content of the piece of timber, 
 whofe length is 24^ feet, and the mean breadth and thick- 
 nefs each 1*04 feet? Anf. 2G-^ teet. 
 
 Ex. 3. Required the content of a piece of timber, 
 whofe length is 20*58 feet, and its ends unequal fquares, 
 the fide of the greater being 191, and the fide of the lefs 
 9^ inches ? Anf. 29*7 So feet. 
 
 Ex. 4, Required the content of the piece of timber, 
 whofe length is 27*36" leet; at the greater end the breadth 
 is 1»78, and the thicknefs 1»23; and at the lefs end the 
 breadth is 1*04, and thicknefs 0*91? Anf. 41*278 feet, 
 
 PROBLEM III, 
 
 Tijind the Solidity of Round or tin/quared Timber, 
 
 Rule 1 , or Commm Rule, 
 
 Multiply the fquare of the quarter girt, or of -^ of the 
 mean circamferenee, by the length, for the content. 
 
 By the Sliding Rule. 
 
 As the length upon C : 12 or 10 upon D : : 
 quarter girt, in 12ths or lOths, on D : content on C. 
 
 Note 1. When the tree is tapering, take the mean di- 
 menfions as in the former problems, either by girting it 
 in the middle, for the mean girt, or at the two ends, and 
 ta^e half the fum of the two. But when the tree is very- 
 irregular, divide it into fevcral lengths, and find the con- 
 tent of each part feparately : or elfe, add all the girts t®- 
 getherj and divide the fum by tlie number of them, for 
 the mean girt. 
 
 H 4 2. Tlis 
 
15^ TIMBER MEASURTNi. 
 
 2. This rule, which is commonly ufcd, gives the anfwer 
 about I lefs than the true quantity in the tree, op nearly 
 what the quantity would be after the tree is hewed fquare 
 in the ufual way; Co that it feems intended to make an 
 allowance for the fquaring of the tree. When the true 
 quantity is dcfircd, ufe the 2d rule, given below. 
 
 EXAMPLES. 
 
 1. A piece of round timber being 9 feet 6 inches long, 
 and its mean quarter girt 42 inches ; what is the content ? 
 
 Duodecimals* 
 3 6 
 3 G 
 
 Decimals^ 
 
 
 3-0 
 
 quarter girt 
 
 3-5 
 
 
 175 
 
 
 105 
 
 
 12-25 
 
 
 9-s 
 
 length 
 
 6l25 
 
 
 11025 
 
 
 ll6'-375 
 
 content 
 
 10 
 
 1 
 
 6 
 9 
 
 
 12 
 9 
 
 3 
 6' 
 
 
 110 
 6' 
 
 3 
 
 6 
 
 110" 
 
 4 
 
 6 
 
 By the Sliding Rule* 
 
 C D 
 
 D 
 
 C 
 
 As 9-5 : 10 : 
 
 r35 : 
 
 :116- 
 
 Or 9-5 : 12 : 
 
 : 42: 
 
 116> 
 
 Ex. e. The length of a tree w 24 feet, its girt at the 
 thicker end 14 feet, and at the fmaller end 2 feet; re- 
 quired the content ? Anf. 9^ feet. 
 
 fix. 3. 
 
TIMBER MBASURINO*- 160 
 
 Ex, 3. What is the content of a tree, whofe mean girt 
 is 3*15 feet, and length l-i feet 6 inches? 
 
 Anf. 8*9922 feet- 
 Ex, 4. Required the content of a tree, whofe length is 
 17^ feet, which girts in five different places as fullows, 
 namely, in the firft place 0-4.3 feet, in the fecond 7*92, in 
 the third 6-15, in the fourth 4'74-, and in the rifth 3'l6? 
 
 Anf. 42'519o» 
 
 RULE II, 
 
 Multiply the fquare of -i- of the mean girt by double 
 the length, and the produft will be the content, very near 
 the truth. 
 
 Bj the Sliding Rule, 
 
 As the double length on C : 12 or 10 on D : : 
 ~ of the girr, in liihs or lOths, on D : content on C. 
 
 EXAMPLES, 
 
 1. What is the content of a tree, its length being 9 
 feet 6 inches, and its mean girt 14 feet? 
 
 D-ecimals, 
 
 2-8 I of girt 
 
 2-S 
 
 ' ■ 224 
 56' 
 
 7-84 
 
 7o:)6 
 
 784 
 
 Duodecim 
 2 9 
 2 9 
 
 7 
 7 
 
 5 7 
 
 2 1 
 
 1 
 
 2 
 3 
 
 8 
 
 7 10 
 19 
 
 1 
 
i^^ artificers' work. 
 
 Bj the sliding Rule, 
 
 C D D C 
 
 As 19 : 10 : : 28 : M9 
 Or 19: 12 : : 33^^: 1*9 
 
 Ex. 2. Required the content of a tree, which is 24 feet 
 long, and mean girt 8 feet? Anf. 122*88 feet. 
 
 Ex. 3. The length of a tree is l^l- feet, and mean girt 
 3*15 feet; what is the content? Anf, 11 '5 1 feet. 
 
 Ex. 4. The length of a tree is 17^ htt, and its mean 
 gin 6*28; what is the content? Anf. 54'4065 feet. 
 
 . Other curious problems relating to the cutting of tim- 
 ber, fo as to produce uncommon efFefts, may be found ia 
 my large Treatife on Menfuration, 
 
 ARTIFICERS' WORK. 
 
 ARTIFICERS compute the contents of their works by 
 fcveral different meafures. 
 
 As glazing and mafonry by the foot. 
 
 Painting, plaftering, paving, &c. by the yard, of 9 fq«are 
 
 Flooring, partitioning, roofing, tiling, &c, by the fquare, 
 of 100 fquare fee*. 
 
 And brick-work, either by the yard of 9 fquare iezt. 
 or by the perch, or fquare rod or pole, containing 272^ 
 fquare feet, or 30^ fquare yards, being the fquare of the 
 rod or pole, of ife-J feet or 5-^ yards long. 
 
 As 
 
BRICKLAYERS* WORK. 1.55 
 
 As this number 272A is a troublefome number to divide 
 by, the ^ is often omitted in practice, and the content in 
 feet divided only by the 272. But when the exatft divifor 
 27 2|: is to be ufcd, it will be eaficr to multiply the feet by 
 4, and then divide fuccefiively by 9» 11> and II. Alfo 
 to divide fquare yards by 30.^, firft multiply them by 4, 
 and then divide twice bv 11. 
 
 All work?, whether fuperficial or folid, are computed 
 by the rules proper to the figure of them, whether it be 
 a triangle, or redangle, a parallelopiped, or any other 
 figure. 
 
 BRICKLAYERS' WORK, 
 
 "Drick-wobk is eflimated at the rate of a ])riek and a 
 -*-^ ha?f thick; fo that ifa wall be more or lefs than this 
 ftandard tliicknefs, it muft be reduced to it, as follows: 
 Multiply the fuperficial content of the wall by the number 
 of half bricks in ihc thicknefs, and divide the product by 3. 
 And to find the fuperficial content of a wall, multiply 
 the length by the height, for the content. 
 
 Chimneys are by feme mcafured as if they were folid, 
 deduding only the vacuity from the hearth to the mantle, 
 on account of the trouble of them* 
 
 And by others they are girt or meaftired round for 
 their breadth, and the height of the ftory is their height, 
 taking th. depth f the }.»mbs for their thicknefs. And 
 in this cafe no iiedadion is made f )r the vacuity from the 
 floor to the mantle-tree, becnufe < f the g-itherisg of the 
 breaft and wings, to m^ke roon> lor the hearth io the next 
 ftory. 
 
 H 6 An 
 
15^ 
 
 BRICKLATSRS TV^OKK, 
 
 All windows, doors, &c. arc to l)e deduced out of thc^- 
 contents of the walls in which they are placed. 
 
 EXAMPLES^ 
 
 1. How many yards and rods- of ftartdard brick- work 
 arc in a wall whofe length or compafs is 67 feet 3 inches* 
 and height 24 feet () inches; tiic walls being 2^ bricks^ 
 or 5 half bricks thick? 
 
 Decimals, 
 57-15 
 24-5 
 
 thid< 
 
 3 
 9 
 
 11 
 11 
 
 Duodecimals, 
 t>7 3 
 24 6 
 
 
 28625 
 22900 
 11450 
 
 234 
 
 114 
 
 2S 
 
 
 7 
 
 O' 
 
 1402*625 
 
 5 half bricks 
 
 1402 
 
 7 
 
 6 
 
 5 
 
 3 70^3-125 
 9 2337*7084 fq. feet 
 j259*74ol^yds. 
 4 
 
 ,7013 
 2337 
 
 4 
 
 1 
 8 
 8 
 
 6 
 6 
 6 
 
 11 1038 981^7 
 
 11 94'-i5^-^8 
 
 ods 8-J866 Anf. 
 
 1030" 
 
 , 94 
 
 8 r 
 
 2 
 I7yds6fs'6 
 
 By the SliJi/7g Rule* 
 B A B A 
 
 As 1 : 24^: : b7];l 1403. 
 Kx. 2. Required the content of a wall 62 feet 6 inches 
 lon-s and 14 leet 8 inches high, and 2i bricks thick? 
 ** Anf. 169 753 yards. 
 
 llx, 3. 
 
MASONS* WORK. 157 
 
 Ex. 3. A tnangular gable is raifed 17^ feet high, on 
 An end wall whofe length is 24- feet 9 inches, the thicknefs 
 being 2 bricks; required the reduced content? 
 
 Anf. 32-084- yards, 
 
 Ex. 4. The end wall of a houfe is 28 feet 10 inches 
 long, and 55^ feet S inches high to the eaves, 20 feet high 
 is Si bricks thick, other 20 feet high is 2 bricks thick, 
 and the remaining 15 feet 8 inches is 1 a brick thick, 
 above which is a triangular gable of 1 brick thick, which 
 rifes 42 courfes of bricks, of which every 4 courfes make 
 a foot. What is the whole content in ftandard mcafure ? 
 
 Anf,. 253*62 yards*. 
 
 MASONS* WORK. 
 
 •^PO mafonry belongs all forts of ftone-work ; and the 
 -*- meafure made ufe of is a foot, either fuperficial or 
 folid. 
 
 Walls, columns, blocks of ftone or marble, &c. are 
 meafured by the cubic foot; and pavements, flabs, chim- 
 ney-pieces, &c. by the fuperficial or fquare foot. 
 
 Cubick or folid meafure is ifed for the materials, and 
 fquare meafure for the workmanlhip. 
 
 In the folid meafure,. ike true length, breadth, and 
 thicknefs, are taken, and multiplied continually together. 
 In the fuperficial, there muft be taken the length and 
 breadth of every part of the projedion, which is feen: 
 without the general upright f^ce of the building,. 
 
 &!Sm 
 
J 55 XASOKS* WORK. 
 
 EXAMPLES. 
 
 1. Required the folid content of a \vall, 53 feet 6 
 inches long, 12 feet 3 inches high, and 2 feet thick. 
 
 Duodecimals, 
 53 6 
 12 3 
 
 (;42'0 
 13'375 
 
 Anf. 
 
 ()42 
 J3 
 
 
 
 4 
 
 6 
 
 656-375 
 
 655 
 
 4 
 
 6 
 
 2 
 
 1310-750 
 
 1310 
 
 £) 
 
 
 
 ^ the Sliding Rule, 
 
 B A 
 
 B A 
 
 1 : 534 : : 
 
 12^ : ^55 
 
 1 : 655 : : 
 
 2 : 1310 
 
 Ex. 2. What is the foJid content of a wall, the length 
 being 24 feet 3 inches, height 10 feet 9 inches, and 2 
 feet thick? Anf. 521 '375 htt. 
 
 Ex. 3. Required the value of a marble flab, at ?s. per 
 foot ; the length being 5 ftet 7 inches, and breadth 1 foot 
 10 inches, Anf. £^ 1 10^. 
 
 Ex. 4. In a chimney piece, fuppofc the 
 length of the mantle and flab, each, 4f 6in 
 breadth of both together - - 3 2 
 length of each jamb - - 4 4 
 
 breadth of both together - - 19 
 Required the fuperficial content ? Anf. 2 if lOin. 
 
 . 1 CAR. 
 
CARPENTERS 
 
 AND 
 
 JOINERS* WORK. 
 
 nrO this branch belongs all the wood- work of a houfe, 
 -*■ fuch as flooring, partitioning, roofing, &c. 
 
 Note. Large and plain articles are ufually raeafured by 
 the fquare foot or yard, &-c. but enriched mouldings, and 
 fome other articles, are often efti mated by running or 
 lineal meafure, and fome things are rated by the piece. 
 
 h\ meafuring of joifts, multiply the depth, breadth, 
 and length all together, for the content of one jolft, mul- 
 tiply that by the number of the joifts, note that the lengtlt 
 of the joifts will exceed the breadth of the room by the 
 thicknefs of the wall, and |- of the fame, bccaufe each 
 end is let into the wall about | of its thicknefs. 
 
 Partitions are meafured frosn wall to wall for one di- 
 menfion, and from floor to floer, as far as they extend, 
 for the other; then multiply the length by the height. 
 
 In meafuring of joiners' work, the ftring is made to ply 
 clofe to every part of the work over which it paiTes. 
 
 The meafure of centering for cellars is found by making a 
 ftring pafs over the furface of the arch for the one dimcn- 
 fion, and taking the length of the cellar for the other ; 
 but in groin centering, it is ufual to allow double meafure, 
 on account of their extraordinary trouble. 
 
 In roofing, the length of the rafters is equal to the 
 length of a ftring ftrctched from the ridge down the 
 
 rafter. 
 
l60 CAHPENTBRS AN» 
 
 rafter, and al®ng the caves-board, till it meets with the 
 top of the wall. This length multiplied by the common 
 depth and breadth of the rafters, gives the content, and 
 that multiplied by the numbers of them, gives the content 
 of all the rafters. 
 
 for Jiair.cafest take the breadth of all the fteps, by 
 making a line ply clofe over them, from the top to the 
 bottom, and multiply the lengtli of this line by the length 
 of a ftep for the whole area, — By the length of a ftep is 
 meant the length of the front and the returns at the two 
 ends; and by the breadth, is to be underftood the girt of 
 its two outer furfaces, or the tread and rife. 
 
 For the balujiradey take the whole length of the upper 
 pisirt of the hand-rail, and girt over its end till it meet the 
 top of the newel poft,. for one dinienfion; and twice the 
 length of the balufter upon the landing, with the girt of 
 the liand-rail, for the other dimenfion. 
 
 For luainfcottingy take the compafs of the room for one 
 dimenfion; and the height from the floor to the ceiiiBg,. 
 making the ttring ply clofe into all the mouldings, tor the 
 other dimenfion. — Out of this muft be made dedudions, 
 for windows, doors, and chimneys, &c. 
 
 For doors y it is ufual to allow for their thicKnefs, by 
 adding it into both the dimenfions of length and breadth, 
 and then multiply them together for the area. — If the door 
 be pannelled on both fides, take double its meafure for the 
 workmanfhip : but if one fide only be pannelled, take tiie 
 area and its half for the workmanlhip. — For thefurrounding 
 architra'vey gird it about the outermok part for one di- 
 menfion, and meafure over it as far as it can be feca when 
 the door is open, for the other. 
 
 Windoiu-Jhuttersy bafes, &c. are meafured in the fame 
 manner. ■ 
 
 In the meafuring of roofing, the holes for chimney 
 fhafts and (ky-lights are generally deduced. 
 
 EX^ 
 
JGINEHS* WORK. l6i 
 
 EXAMPLES, 
 
 1. Required the content of a floor 48 feet 6 inches 
 long, and 24 feet 3 inches broad. 
 
 Decimals, 
 48-5 
 24^ 
 
 Duodecimals, 
 48 6 
 24 3 
 
 1040 
 
 970 
 12125 
 
 204 . 
 96 
 12 1 6 
 
 1176-125 
 11-76125 
 
 feet 11/6 1 6 
 fquares Anf. 11 '76 1 ^ 
 
 Ex. 2. A floor being 36 feet 3 Inches long, and l6 feet 
 £ inches broad, how many I'quares are in it? 
 
 Anf. 5 fq. 98| feet. 
 
 Ex.3 How many fquarps arc there in 173 feet 10 
 inches in length, and 10 feet 7 inches heipht, of parti- 
 tioning? Anf. 18*3972 fquares* 
 
 Ex. 4. What coft the roofing of a houfe at 10s. 6d. a 
 »fquarei the length, vvithin the walls, I eing 52 feet 8 
 inches, and the breadth 30 feet 6 inches; reckoning the 
 roof I of the flat? Anf. ^12 12 11 1. 
 
 Ex. 5. To how much, at 6s. per fquare yard, amounts 
 the wainfcorring of a room ; the height, tak ng in the 
 cornice and moulding'', being 12 feet 6 inches^ cind the 
 whole compafs 83 feet 8 inches; alfo the three window 
 fhufters are each 7 feet 3 inches by 3 feet 6 inche,, ;md 
 the door 7 feet by 3 feet 6 inches; the door and fii. ters, 
 being worked on both fidef:, are reckoned work and halt 
 work? Anf.j(;36. 12 2^. 
 
 SLA^ 
 
SLATERS 
 
 AND 
 
 TILERS* WORK. 
 
 TN thcfe articles, the content of a roof h found by mul- 
 ■^ tiplying the length of the ridgc by the girt over from 
 caves to eaves; making allowance in this girt for the 
 double row of flatea at the bottom, or for liow much one 
 row of flates or tiles is laid over another. 
 
 When the roof is of a true pitch, that is, farming a* 
 right angle at top; then the breadth of the building with 
 its half added, is the girt over both fides. 
 
 In angles formed in a roof, running from the ridge to the 
 eaves, when the angle bends inward , it is called a valley; 
 but when outwards, it is called a hip. 
 
 Ded unions are made for chiraney fhafts or wiadoMf 
 holes* 
 
 EXAMPLES* 
 
 1. Required the content of a flated roof, the length 
 being 43 feet 9 inches, aud the whole girt 34 feet 3 iaches? 
 
 Decimalt^ Duodecimals ^ 
 
 ^5-75 45 9 
 
 34i 34 3 
 
 18300 
 13725 
 114375 
 
 9) 1566-9375 feet 
 yds J74M04 
 
 
 205 
 
 6 
 
 
 
 135 
 
 
 
 
 11 
 
 5 
 
 S 
 
 9) 
 
 1566 
 
 11 
 
 3 
 
 
 174^ 
 
 IV 
 
 r 
 
 Ex. 
 
PLASTLRERS* WORIC. iGS 
 
 Ex, 2. To how much amounts the tiliag of a houfe, at 
 i5s. 6\i, per fquare; the lengrh being 43 feet 10 inches, 
 and the breadth on the flat 27 feet 5 inches, alfo the eaves 
 projefting l6 inches on each iide, and the roof of a true 
 pitch? Anf. ^'2-t 9 5|. 
 
 PLASTERERS' WORK. 
 
 pLASTERERS' work is of two kinds, n^rmcly, ceiling, 
 ■*- which is plaftering upon laths; and rendering whicli 
 is plaftering upon walls: which are racafured feparately, ' 
 
 The contents are ettimated either by the foot or yard, or 
 fquare of 100 f^-tt» Enriched mouldings, &c. are rated 
 by running or lineal meafure. 
 
 Dedudions are to be made for chimneys, doors, win- 
 dows, &c. 
 
 EXAMPLES, 
 
 1, How many yards contains the ceiling, whick Is 
 43 feet 3 inches long, and 25 feet 6 inches broad ? 
 
 Duodecima!t» 
 43 3 
 
 21()'25 221 3 
 
 86*30 85 
 
 21623 21 7 5 
 
 9) 1102-873 9 ) 1102' 10 6^ 
 
 yds 122*341 Anfwer 122^ 4''l0' 6" 
 
 Ex» 
 
1§'4 PAINTERS* WORK. 
 
 Ex. 2. To how much amounts the ceiling of a room, ;it 
 lOd. per vard; the length lacing 21 feet « inches, and the 
 breadth U fert 10 inches ? Anf. ^1 g 8|. 
 
 Ex. 3. The length of a room is 18 feet 6 inches, the 
 breadth 12 feet 3 inches, and height 10 fe-t 6 inches; to 
 how much amounts the ceiling and rendering, the farmer 
 at «d. and the latter at 3d, per yard ; allowing for the door 
 of 7 feet by 3 feet 8, and a fire place of 5 feet fquare? 
 
 Anf ^1 13 3. 
 Ex. 4. Required the quantity of plaftering in a room, 
 the length being 14 feet > inches, breadth 13 feet 2 inches, 
 and height 9 feet 3 inches to the under fiJe of the cornice, 
 which girts 8^ inches, and projeds 5 inch, s from the wall 
 on the upper part next the ceiling; deducing only for a 
 door 7 feet by 4. 
 
 Anf. 53'" 5^ 3' of rendering 
 1% 5 6 of ceiling 
 29 Of*- of cornice. 
 
 PAINTERS' WORK- 
 
 P' 
 
 PAINTERS* work is computed in fquare yards. Every 
 parf is meafured vvhrc rh* colour lies; and th^! msa- 
 furing line is f »rced into all ihe moulding"^, and corners. 
 
 Windows are done at To -nuch a pi.ce. And it is ufual 
 to allow double meafure for carved .mouldings, &c. 
 
 EXAMPLES. 
 
 1. How many yards of painting contains the room 
 which is 6'5 feet 6 inches in compafs, and 12 feet 4 inches 
 high.? 
 
«r.AZIERS* WORK. l65 
 
 Duodi'cimals. 
 65 6 
 12 4, 
 
 786 
 21 10 
 
 9 ) 807-83 , 9 ) 807 10 
 
 89-7888 Anfwer 89 6 10 
 
 , Ex. 2. The length of a room being 20 feet, its breadth 
 14 feet 6 inches, and height 10 feet 4 inches; how many- 
 yards of painting are in it, deducing a fire-place of 4 feet 
 by 4 feet 4 inches, and two windows each 6 feet by 3 
 feet 2 inches ? Anf. 73rj\ yards. 
 
 Ex. 3. What«ofi: the painting of a roono, at 6d. per 
 yard; its length being 24 feet 6 inches, its breadth l6 
 feet 3 inches, and height 12 feet 9 inches; alfo the door 
 is 7 feet by 3 feet 6, and the window fhutters to two 
 windows each 7 feet 9 by three feet 6, but the breaks of 
 the windows thtmfelves arc 8 feet 6 inches high, and 1 
 foot 3 inches deep: deducing the fire-place of 5 feet by 
 5 feet 6? Anf. ^3 3 10^, 
 
 GLAZIERS' WORK. 
 
 /Glaziers take their dimenfions either in feet, inches 
 ^~* and parts, or feet, tenths and hundreths. And they 
 compute their work in fquare feet. 
 
 In taking the length and breadth of a window, the 
 crofs bars between the fquares are included. Alfo win- 
 dows of round or oval forms are meafured as fquare, 
 meafuring them to their greateft length and breadth, on 
 account of the wafte in cutting the glafs. 
 
liSO glaziers' VORK, 
 
 BXAMPLBS. 
 
 1 . Hotjr many fquare feet contains the window which if 
 4i'25 leet long, and 275 feet broad? 
 
 Decimals, 
 2-75 
 
 Ai 
 
 ifwcr 
 
 Duodecimals, 
 2 9 
 4 3 
 
 ll'OO 
 
 •6875 
 
 11 
 8 3 
 
 11-6875 
 
 118 3 
 
 2. "What will the glazing a triangular flcy-Iight come f 
 at lOci per frot ; tl>e bafe beiny; 12 feet 6' inches, and the 
 perpendicular height G fiet y inches? 
 
 Anf.^I 15 1|. 
 
 3. Thwt: is a hoofe wiih three tier of window.«, three 
 windows in each tier, their con>mon breadth 3 leet 11 
 inches; 
 
 now the height of the firft tier is 7' lO'" 
 of the fecond 6 8 
 
 of ihe third 5 4 
 
 Required the cxpenec of glazing at 14d. per foot? 
 
 Anf. ^13 11 10^. 
 
 4. Required the eKpenre of glazing the windows ©f a 
 houfe at i3d a foot ; there being three ftorief^, and thre« 
 windows in each ftory : 
 
 the height of the lower tier is 7' 9* 
 of the middle 6 6 
 
 of the upper 5 3| 
 
 iftd of an oval window over the door 1 101 
 
 The common breadth of all the windows being 3 (ctt ^ 
 if>€hc«. 
 
 Anf. /12 5 6, 
 PA- 
 
PAVERS* WORK. 
 
 PAVERS* work is done by the fquare yard. And the 
 content is found by multiplyii.g the length by the 
 breadth. 
 
 EXAMPLES, 
 
 1. What coft t^e paving a fnot-path at 3s. 4d. a-ya^d ; 
 the length being 35 teet 4 inches, and breadth 8 teei 3 
 inches? 
 
 Decimals, Duodecimals, 
 
 8i 
 
 9 
 !ontent 
 
 3yd is 
 
 6f is 
 
 Anfwer 
 
 35 
 8 
 
 4 
 3 
 
 282'66' 
 8*S3 
 
 2S2 
 8 
 
 8 
 10 
 
 32-38 C 
 
 )291 
 32 
 
 G 
 3,6 
 
 2s is Vs 3-2388- 
 
 Is is k 1-6194 
 4d is 1 5398 
 
 1 5-3981 
 20 
 
 L^ 
 
 3 4 
 4 
 
 
 
 IZ 4 
 8 
 
 s 7-9620 
 12 
 
 5 6 8 
 
 d 11 '54+0 
 
 Ex, 2. What coft the paving a court, at 3s. 2d. per 
 yard ; the length being 27 feet 10 inches, and the breadth 
 14 f€et pinches? Aaf. 7 4 5i. 
 
 Ex, 
 
l6S PLUMBERS* WORK. 
 
 Ex. 3. What will be the expence of paving a reAangu- 
 lar coirt yard, whofe length is 63 feet, and breadth 45 
 feet; in which there is laid a foot path of 5 feet 3 inches 
 broad, running the whole length with broad ftones, at 3s, 
 a-yard; the reft being paved with pebbles at 2s. fd. a 
 yard? ^ Anf, 40 5 10^. 
 
 PLUMBERS* WORK. 
 
 "pLUMBERS* work IS rated at fo much a pound, or elfe 
 ■*• by the hundred weight, of 112 pounds. 
 
 Sheet lead ufed in roofing, guttering. See, is from 7 to 
 12lb to the fquare foot. And a pipe of an inch bore is 
 commonly 13 or I4lb to the yard in length, 
 
 EXAMPLES. 
 
 1. How much weighs the lead which is 39 feet 6 inches 
 long, and 3 feet 3 inches broad, at 84:1b to the fquare 
 foot? 
 
 Decimalt, Duodecimals, 
 
 39'5 39 6 
 
 3i 
 
 118-5 
 
 ^•875 
 
 3 3 
 
 118 6 
 9 10 
 
 6 
 
 128-375 
 Si 
 
 128 4 
 
 ^ 
 
 1024 
 6'4 
 
 ^ 
 
 1027000 
 64-1875 
 
 
 1091-1875 Anfwer lOpl'lb 
 
 Ex. 
 
ARCHED ROOFS. l6^ 
 
 Ex. 2. What cod the covering and guttering a roof 
 with lead, at 18s. the cwt.; the fength of the roof being 
 43 feet, and breadth or girt over it 32 feet ; the guttering 
 57 feet long, and 2 feet wide ; the former 9*83 1 lb, and 
 the latter 7»373lb to the fquare foot ? 
 
 Anf.^ilS 9 Ih 
 
 VAULTED 
 
 AND 
 
 ARCHED ROOFS, 
 
 "ARCHED roofs are either vaults, domes, faloons, or 
 groins. 
 
 Vaulted roofs are formed by arches fpringing from the 
 oppofite walls, and meeting in a line at ihe top. 
 
 Domes are made by arches fpringing from a circular or 
 polygonal bafe, and meeting in a point at the top. 
 
 Saloons are tornied by arches connefting the fide walls 
 to a flit roof, or ceiling, in the miJ.dle. 
 
 Groins are formed by the interfcdHon of vaults with each 
 other. 
 
 Vaulted roofs are commonly of the three following 
 forts : 
 
 1. Circular roofs, or thofe whofe arch is fome part of 
 the circumference of a circle. 
 
 2. Elliptical or o^oal roofs, or thofe whofe arch is an 
 oval, or fome part of the circumference of an ellipfis. 
 
 3. Gothic roofs, or thofe which are formed by two cir- 
 cular arcs, ftruck from different centres, and meeting in a 
 point over the middle of the breadth, or fpan of the arch. 
 
 PROBLEM I. 
 To find the Surface of a Vaulted Roof. 
 
 Multiply the length of the arch by the length of the 
 vault, and the produd will be thefuperficies. 
 
 I Note. 
 
170 VAULTED AND 
 
 Nofe, To iind the length of the arch, make a line or 
 ftring ply clofe to it, quite acroii. from fide to fide, 
 
 EXAMPLES. 
 
 1. Required the furface of a vaulted roof, the length 
 cf the arch being 31-12 feet, and the length of ihe vault 
 li^Ofcet? 
 
 31-2 
 \10 
 
 Anf. 374-i-*0 fquare fttt, 
 
 Ex. 2. How many fquare yards are in the vaulted roof, 
 whofe arch is 42*4 ittt, and the length of the vault 106' 
 tect ? Anf. 495*37 yds. 
 
 PROBLEM IT. 
 
 ^ojind the Content of the Concwvity of a Vaulted Roof, 
 
 Multiply the lergth of the vault by the area of one end, 
 that is, by the area of a vertical tranfverfe fediion, for 
 the content. 
 
 Note, When the arch is an oval, mnltiply the fpan by 
 the height, and the produd by -7854, for the area. 
 
 EXAMPLE*!. 
 
 ]. Required the conteni of the concavity of a femi- 
 circular vaulted oof, the fpan or diameter being 30 feet, 
 and the length of the vault ]3« kti ? 
 •7«54 
 900 the fquare (/ 30. 
 
 2' ) 7O()-80* 
 
 353*43 area cf the end 
 J50 the length 
 
 I767I0O 
 33343 
 
 5301450 the contcat. 
 
 Ex. 
 
ARCHED ROOFS. 171 
 
 F.x. C. What is the content of the vacuity of an oval 
 vault, whofe fpan is 30 feet, and height 12 feet; the 
 length of the vault being 6'0 feet ? Ar,f. j6"94-64» 
 
 Kx. 3. Required th<; content of the vacuity of a Go- 
 thic vault, whofe fpan is 50 feet, the chord of each arch 
 ,^.0 feet, and the ditUnce of each arch from the middle of 
 lliefe chords 10 feet; alfo the length of the vault 20. 
 
 Anf. 35401 -r. 
 
 PROBLEM III, 
 
 To find the Superficies of a Dome, 
 
 Find the area of the bafe, and double it ; then fay, ai 
 the radius of the bafe, is to the height of the dome, fo is 
 the double area of the bafe, to the fuperficics. 
 
 Note, For the fuperiicies of a hemifpherical dome, take 
 the double area of the bafe only. 
 
 EXAMPLES, 
 
 1. To how much comes the painting of an odiagonal 
 fpherical dome, at Sd. per yard; each fide of the bafe 
 being 20 feet? 
 
 4-828427 tabular area 
 400 fquare of 20 
 
 ]931-370Sareaof the bafe 
 2 
 
 9 ) 386'2-74l6' fuperficies infect 
 429-1S34 yards 
 8 
 
 12 
 2,0 
 
 3433-5472 
 
 28,6 U 
 ^■14 6 l| anfwer. 
 Ex. 2. Required the fuperficies of a hexagonal fphe- 
 rical dome, each fide of the bafe being JO feet. 
 
 Anf. 5JC)(5l52. 
 1 2 Ex. 
 
172 VAULTED AND 
 
 Ex. 3. What is the fuperficies of a dome with a circu- 
 lar hafe, whofe ciicumtcrcnce is 100 feet, and height 20 
 feet ? Anf. ^000 feet, 
 
 FROBLEM IV. 
 To find the Solid Content of a Dome, 
 Multiply the area of the bafe by the height, and take 
 ^ of the produil. 
 
 EXAMPLES. 
 
 1. Required the folid content of an 0(f\agonal dome, 
 each fide of the bafe being 20 feet, and the height 21 fcctf 
 
 4*8284.27 
 400 
 
 ]i)31*3708 area of the bafe 
 1-i I of height 
 
 7 7 '-.'54832 
 19313708 
 
 27039*1912 anfwer. 
 Ev. 2. What is the folid content of a fpherical dome, 
 the diameter of whofe circular bafe is 30 feet ? 
 
 Anf. 706s-6 feef. 
 
 PROBLEM V. 
 To find, the Superficies of a Saloon^ 
 Find its Vreadth by applying a ftring clofe to it acrofs 
 the furface. Find alfo iis length by mcafuring along the 
 middle of it, quite round the room. 
 
 Then mulii^ ly thefe two together for the furface. 
 
 EXAMPLE. 
 
 The girt acrofs the face of a faloon being 5 feet, and its 
 mean coinpafs 100 feet, required the area or fuperficies? 
 100 
 5 
 
 500 anfwer, 
 
 PRO- 
 
ARCHED HOOFS. 175 
 
 PROBLEM Vr. 
 
 To find the Solid Content of a Saloon. 
 
 Multiply the area of a tranfverfe fe<Sion by the compafg 
 taken round the middle part. Subtraft this produd from 
 the whole vacuity of the room, fuppofmg ihe vvalls to go 
 upright all the height to the flat ceiling. And the differ- 
 ence will be the anfvver, 
 
 EXAMPLE. 
 
 If the height AB of the faloon be 3*2 feet, the chord 
 ADC of its front 4*5, and the diftance Dk of its middle 
 part from the arch be 9 inches; required the folidity, 
 fuppofing the mean compafs to be 30 feet? 
 
 2)^'D 
 
 
 0-75 DE 
 0*75 
 
 2"25 
 
 AD 
 
 — Q. 
 
 2-25 
 
 375 ^ 
 
 _- 
 
 
 525 
 
 1125 
 
 
 
 450 
 
 
 '5625 
 
 450 
 
 
 
 5-0625 
 
 AD* 
 
 
 •56"25 D^"" 
 
 
 5'6250 
 
 ( 2'37 AE 
 
 4. 
 
 4 
 
 
 43 162 , 
 
 3 ) 948 
 
 
 3 12i> 
 
 
 
 
 3-16 
 
 = f AE 
 
 46 1 33 
 
 4-50 
 7-66 
 
 AC 
 
 
 •3 : 
 
 = T% T)E 
 
 2-298 area (eg, ADCEA. 
 
 X 3 Again, 
 
i/^ VAULTtlD A.NV 
 
 Again, 0*2 4\'3 
 
 ■3-2 ' ^'0 
 
 J)6 ISO 
 
 lO-;il CO -25 AC* 
 10»2-V AB^ 
 
 JOOi ( 3'l6 = BC 
 i; 1-6" rz ?; AR 
 
 61 I l-oi < 3896" 
 1 61 SlG 
 
 40 5- 056' area of triangle ABC 
 2'29S area feg. 
 
 2738 area of feaion AECBA 
 50 compafs 
 
 137 '900 content of the folid parti 
 
 Then this taken from the whole upright fpace, will 
 leave the content of the vacuity contained within the 
 room. 
 
 PROBLEM Vll. 
 
 To find the Concave Superficies of a Groin, 
 
 To the area of the bafe add 4. part of itfelf, for the fu- 
 peificial content. 
 
 EXAMPLES, 
 
 1. "What is the fuperficial content of the groin arch, 
 raifed on a fquare bale of 1^ feet on each fide/ 
 
 15 
 
ARCilES IlO'Oi&. liT^ 
 
 15 
 15 
 
 le 
 
 75 
 
 15 
 
 7 ) 225 area of the bafc 
 32-7 its 7th part 
 
 '257^ anfwer. 
 
 Ex. 2. Required the fuperficies of a groin arch, raifed 
 ©n a reiiangular bafe, whofc dimenfions are 20 feet by 16', 
 
 Anf. 36'5f'. 
 
 PROBLEM Vlir. 
 
 ^ofind the Solid Contettt of a Groin Arch* 
 Multiply the area of the bafe by the height: from th, 
 produft fubtraft to of itfeif j and the remainder will be 
 the content of the vacuity , 
 
 EXAMPLES. 
 
 1, Required the content of the vacuity within a groin 
 arch, fpringing from the fides of a fquare bafe, each fide 
 •f which is 1 6' feet. 
 
 16' 
 1^ 
 
 16 
 
 256 area of bafe 
 8 height or radius 
 
 204-8 
 204| ,V ^ubtraft 
 
 1843^ anfwer. 
 
 I 4 5. What 
 
i7^ VAULTED AUD AliCHKD aOOFS. 
 
 2. What is tfce content of a vacuity below an oval 
 groin, the fide of its fquara bafc being 24- fret, and its 
 height 8 feet f Anf. 4147^. 
 
 NOTES. 
 
 1. To find the folid content of the brick or ftone-work, 
 which forms any arch or vault : Multiply the area of the 
 bafc by the height, including the work over the top of the 
 arch ; and from the produd fubtrad the content of the 
 vacuity, found by the foregoing problems; then the re- 
 mainder will be the content of the folid materiiili.. 
 
 2. In groin arches, however, it is ufual to take the 
 whole as folid, without deducting the vacuity, on iu count 
 oi the trouble and wade of marerials, attending the cut- 
 ting and fitting them to the arch. 
 
 LAND SURVEYING. 
 
 CHAPTER I. 
 
 Dr/cri/fian and Ufe of tb( Injirumaits^ 
 
 I. OF THE CHAIM. 
 
 T AND is meafured with a chain, called Gunter's chain, 
 -■-^ of 4 poles or 22 yards in length, which confilts of 
 JOO equal links, the length of each link being ^\ of a 
 jrard, or -,-*6^ of a foot, or 7*S)2 inches, that is nearly 8 
 inches or *- of a foot. 
 
 An acre of land is equal to lOfquare chains, that is, 
 10 chains in length and I chain in breadth. Or it is 220 
 X 22 or 4840 fquare yards. Or it is 40 x 4 or l()0 
 fqaare pole^. Or it is 1000 x 100 or 100000 fquare 
 links. 1 hcfe being all the fame quantity, 
 
 Alfo, 
 
SURVEYING. 177 
 
 Alfo, an acre is divided into 4 parts called roods, and 
 a rood into 40 parts called perches, which are fquare 
 poles, or the fquare of a pole of 5\: yards long, or the 
 fquaie of ^ of a chain, or of 25 links, which is 625 
 fquare links. So that the divifions of land meafure will 
 be thus : 
 
 625 fq. links — 1 pole or perch 
 40 perches zz 1 rood 
 4 roods rr 1 acre. 
 
 The length of lines, meafured with a chain, are heft 
 fet down in links as integers, e ery chain, in length being 
 100 links ; and not in chains and d.cifnals. Ther-^fore 
 after the content is found, it will be in fquare links; then 
 cut off 5 of the figures on the right-hnnd for decimals, 
 and the reft will be acres. Thofe decimal^ are then mul- 
 tiplied by 4 for roods, and the decimals of thcfe again by 
 40 for perches. 
 
 EXAMPLE, 
 
 Suppofe the length of a reflangular piece of ground b« 
 792 link?, and its breadth 385 ; to find the area in acres, 
 roods, and perches. 
 
 792 
 
 '385 
 
 3900 
 6336 
 2376 
 
 — — — ac ro p 
 
 3*o+y:o Anf. 3 7 
 
 •li/OoO 
 40 
 
 7-87200 
 
 1 5 2. a» 
 
17* SURVF.YIXC. 
 
 C. OF THE PLAIN TABLE. 
 
 Tihis indrumfnt confifts of a plane re(5tangular board of 
 any convenient fizc, the centre of which, when ufed, is 
 flxcd by means of fcrews to a three-legged Hand, hnving 
 a ball and focket, or joint, at the top, by means of which, 
 when the legs are fixed on the ground, the table is inclined 
 in any direction. 
 
 To the table belong feveral parts ; viz. 
 
 1. A frame of wood, made to fit round Its t^ges, and 
 to be taken off, for the convenience of putting a flieet of 
 paper on the table. The one fide of this frame is ufually 
 divided into equal parts, for drawing lines acrofs the 
 table, parallel or perpendicular to the fides; and the other 
 fide of the frame is divided into 360 degrees, from a cen- 
 tre, which is in the middle of the tabic; by means of 
 which the table is to be ufcd as a theodolite, ^'C. 
 
 C. A needle and compafs fcrewed into the fide of the 
 table, or elfe in rhe middle of the fupporr, to point out 
 the diredions ; and to be a check upon the fights. 
 
 .'3. An index, which is a brafs two-foot fcalc, with 
 cither a fmall telefcope, or open fights erected perpendi- 
 cularly on the ends. Thefe fights and one edge of the 
 index are in the fame plane, and that edge is called the 
 fiducial edge of the index. 
 
 Before iifing this inftrument, take a fiieet of paper which 
 will cover it, 'and wet it to make it expand ; then fprcad 
 it flat on the table, prefilng down the frame on the edges, 
 to ftretch it and keep it fixed there; and when the paper is 
 become dry, it will, by contradling again, ftretch itfelf 
 fmooth and flat from any cramps and unevennefs. On this 
 paper is to he dran'n the plan or Ibrra of the thing mea- 
 fured. 
 
 In ufing this inftrument, begin at any part of the ground 
 you think the moft proper, and make a point on a con- 
 venient part of the paper er table, to reprefent that point 
 t)f the ground; then fix in tliat point one leg of the com- 
 palTc=, or a fine fleel pin, and aj^ply to it the fiducial 
 
 6 ed^e 
 
SURVEYING, 179 
 
 edge of the index, moving it round, till through the 
 fights you perceive fome remarkable objeft, as the corner 
 of a fie]{3, &c. and from the Nation point draw a line with 
 the point of the compaffes along the fiducial edge of the 
 index; then fet another objedl or corner, and draw its 
 line; do the fame by another, and fo on, till as many 
 objeds are fet as may be thought neceffary. Then raeafure 
 from the ftation towards as many of the objeds as may be 
 neceffary, and no more, faking the requifite offsets to 
 corners or crooks in the hedges, 8cc^ and Jay the meafures 
 down on their refpedive lines on the table. ^^T^jhen, at 
 any convenient place, meafured to, fix the t^MeJJ) the 
 fame polition, and {qi the objedls which app^|j:jj,from 
 thence, &c. as before; and thus continue till th^ vvork. i^ 
 finifhed, meafurrng fuch lines av are neceffary, and deter- 
 mining as many as you can by in rerfeciing lines of di- 
 rection drawn from different ftations. 
 
 And in thefe operations, obferve the following parti- 
 Gular cautions and direftions: I. Let the lines on which 
 you make ftaiions be dindted towards objeils as far diftant 
 as poffible; and when ycu hav- fet any fuch objed^ go 
 rouhd the table and look through the fights from the other 
 end of the index, to fee if any other remarkable objed:' 
 lie diredly oppofite: if there be not fuch an one^ endea- 
 vour to find another forward objedl, fuch as fhall have a 
 remarkable backward oppcfite one, and make ufe of it, 
 rather than the other; becaufe the back objed will be o£ 
 ufe in fixing the table in the original pofition, eitlycr when 
 you have meafured too near to the forward objed, or 
 when it may be hid from your fight at any neceffary 
 ftation by intervening hed^s. Sec, 
 
 2. Let the faid bnes, on which the flations are taken, 
 be purfued as far as you conveniently can ; f jr that will be 
 the means of preferving more accuracy in the work. 
 
 3. At each ftation, it will be neceffary to prove the 
 truth of it; that is, whether the table be ftraight in 
 the line towards the objc<^, and a'.fo whether thediftance 
 
 1 6 be 
 
ISO suRTEtma. 
 
 be rightly meafored and laid down on the paper. — To 
 know if the table be fet down ftiaight in the line; lay the 
 index on the table in any manner, and move the tabl« 
 about, till through the fights you perceivp either the fore 
 or back objed; then, without moving the table, go 
 round it, and look through the fights by" the other end of 
 the index, to fee if the other objcfl can be perceived ; 
 if it be, the table is in the line; if not, it muft be fliifted 
 to one fide, according to your judgment, till through the 
 fights both objr^s can be feen. — The aforefaid operation 
 Only irfprms yi u if the ftiuion be ftraight in the line : but 
 to know if it be in the right part of the line, that is, if 
 the difttin'ce h?s been righly Ijiid down ; fix the table in 
 the origin'al pofition, by laying the index along the ftiition 
 line, and turning the table about till the fore and back 
 objefls appear through the fights, and then alio will the 
 needle point at the fame degree as at firft; then lay the 
 index over the ftation point and any other point on the 
 paper reprefenting an objed which can be feen from the 
 ftation J and if the faid obj 6i appear ftraight through the 
 fights, the ftation may be depended on as r'ght; if nor, 
 the diftance (hould be examined and corre<5ted till the ob- 
 jeft can be fo feen. And for this very ufef il purpofe, it 
 is advifable to have fome high objeft or two, which can 
 be feen from the greateft pan of the ground, accurately 
 laid drwn on the paper from the beginning of the furvey, 
 to ferve continually as }>roof ohjt-fts 
 
 When f om any ftation, the fore and back cbjeds can- 
 not both he feen, the agreement rf the needle with one of 
 them may be depended on for placing the table ftraight on 
 the line, and for fixing it in the original pofition. 
 
 Of JhlfttHg the Paper on the Plain Table, 
 
 When cne paper is full, and there Is cccafion forrnore; 
 draw a line in any manner through the fartheft point of 
 the laft ftation liae, to which the work can be convenient- 
 ly 
 
SURVETIN®. 181 
 
 ly hid down ; then take the fheet ofF the table, and fix 
 another on, drawing a line on it, in a part the moft con- 
 venient f )r the rert of the work ; then fold or cut the old 
 (heet by the line drawn on it, apply the edge to the line on 
 the new fheer, and, as they lie in that pofuion, c mtinue 
 the laft ftation line on the new paper, placing on ic the reft 
 of the meafure, beginning at where the old Iheet left oiF, 
 And fo on from ftieet to fheet. 
 
 When the work is done, and you would faftcn all the 
 fheets together into one piece, or rough plan, the afore- 
 faid lines are to be accurately joined together, as when the 
 lines were transferred from the old fheers to the new one». 
 
 But it is to be noted, that if the faid joining lines, on 
 the old antl new fheet, have not the fame inclination to the 
 fide of the t:ible, the needle will not point to the original 
 degree when the table i-. redified; and if the needle be 
 required to refped flill the fame degree of the compafs, the 
 eafieti way ( f drawing the lines in the fame pofition, is to 
 draw them both parallel to the fams fides of the table, by 
 means of the equal divifions marked on the other two fides, 
 
 3. OF THE THEODOLITE, 
 
 The theodolite is a brazen circular ring, divided into 
 S60 degree?, and having an index with fights, or a te- 
 lefcope, placed on the centre, about which the index is 
 moveable ; alfo a compafs fixed to the centre, to point out 
 courfes and check the fights ; the whole being fixed by the 
 centre on a (land of a convenient height for ufe. 
 
 In ufing this inftrument, an exad account, or field-book, 
 of all meafures and things neceffary to be remarked in the 
 plap, muft be kept, from which to make out the plan on 
 returning home from the ground. 
 
 Begin at fuch part of the ground, and meafure in fuch 
 diredions, as you judge moft convenient j taking angles 
 or dircdions to objeds, and meafuring fuch diflances 
 as appear neceflary, under the fame reftridions as in 
 ^ ^ _ ' tht 
 
182 SWRVEYINC. 
 
 the ufe of the plain table. And it is fafeft to fix the theo- 
 dolite in the orij'inal pofition at every ftation by means of 
 f )rc and back objefts, and the compafs, exaftly as in ufin^ 
 the plain table; rcgiflering the number of degrees cut off 
 by the index when direded to each objciJ^ ; and at any 
 ftatjon, placing the index at the fame degree as when the 
 dire*flion towards that flation was taken from the lafl pre- 
 ceding one, to fix the theodolite there in the originnl 
 pofition, after the f '.me manner as the plain table h tixed 
 in the original pofition, by layirig its index along the line 
 of the lalt direftion. 
 
 The bell method of laying down the aforefaid lines of 
 direftion, is to defcribe a pretty large circle, quarter it, 
 and lay on thiC circumference, the fcveral numbers of de- 
 grees cut off by the index in each direftion, marking the 
 points they reach to ; then dra-v lines from the centre to 
 all thefe points in rhe circumference; laAIy, parallel to 
 the faid lines, draw ether lines from ftat ion to llation, 
 
 4* OF THE CRQSS^ 
 
 The crofs confifts of two pair of fights fet at right 
 angles to each other, on a ftafF having a Iharp point at the 
 bottom to tlick in the ground. 
 
 The crofs is very ufciul to meafure fmall and crooked 
 pieces of ground. The method is to meaf;ire a bafe or 
 rhicf line, uAially in the i'ongeft diredlon of the piece 
 from corner to corner ; and while meafuring it, fijiding 
 the places where perpendiculars would fall on this line,, 
 from the feveral corners a. id bends in the boundary of the 
 piece, with the crofs, by fixing it, by trials, on fuch parts 
 of the line as that through one pair of the fights both ends 
 ©f the line may appear, and through the ( tlier pair you 
 can perceive the correfponding bends or corners j and then 
 ireafuring the lengths of the faid perpendiculars, 
 
 ,, REM.ARKS. 
 
 Qf all the inftraments for meafiiring, the plain table is 
 
SURVEYIK*. i&3 
 
 on many occafions the beft; not only becaufe it may be 
 ufed as a theodolite or femi.circle, by turning uppermolt 
 that fide of the frame which has the S60 degrees on it; 
 but becaufe it is, in its own proper ufe» by much the 
 eafieft, fafeft, and raoft accurate for the purpofe; for, by 
 planning every part immediately on the fpot, as foon as 
 meafured, there is not only faved a great deal of writing 
 in the field-book, but every thing can alfo be planned 
 more eafily and accurately while it is in view, than it can 
 afterwards from a field. book, in which many JIttle things 
 may be either negleifted or miftaken ; and befides, the op- 
 portunities which the plain table affords of correfling the 
 work, or proving if it be right, at every ftation, are fuch ad- 
 vantages as can never be balanced by any other inftrument. 
 But though the plain table be the moft generally ufeful in- 
 llrument, it is not alxvays fo ; there being many cafes in 
 which fometimesone inftrument is the propereft, and fome- 
 times another; nor is that forveyor mafter of his bufinefs, 
 uho cannot in any cafe diTtinguifh which is the fitted: in- 
 flrument or method, ajid ufe it accordingly: nay often no 
 inftrument at all, but barely the chain itfelf is the beft 
 method, particularly in regulnr open fi-^Ids lying together; 
 and even when you are ufing the plain tabic, it is often of 
 advantage to meafure fuch large open parts with the chain 
 only, and from thofe meafures Jay them down on the 
 table.. 
 
 The perambulator is ufcd for meafuring roads, and 
 other great diftances on level ground., and by the fides of 
 rivers. It has a wheel of 8^ feet, or half a pole in cir^ 
 cumference, on which the machine turns; and the didance 
 meafured is pointed out by an index, which is moved 
 round by clock work. 
 
 Levels, with telefcopic or other fights, are nfed to find 
 tl^e level between place and place, or how much one place 
 is higher or lower than another. And in meafuring any 
 floping or oblique line, either afccnding or defcending, a 
 fmall pocket level ii ufeful for fliowing how many links 
 
 for 
 
184 89IIVSTIN6* 
 
 for each chain are to be dcduAcd, to reduce the line te 
 the true horizontal length. 
 
 An ofF<;et ftafF is a very ufeful and neceflary inftrument, 
 for meafuring the ofEets and other fhort diftanccs. It is 
 ]0 links in length, being divided and marked at each of 
 the 10 links. 
 
 Ten fmall arrows, or rods of iron or wood, are ufed 
 to mark the end of every chain length, in meafuring lines. 
 And fometimes pickets, or ftaves with flags, arc fet up as 
 marks or objefts of direftion. 
 
 Various icales are alfo uffd, in protra^ing^ and mea- 
 furing on the plan or paper; fuch as plane fcales, line of 
 chords, protraftor, compaffcs, reducing fcale, parallel and 
 perpendicular rules, &c. Of plane fcales, there fliould 
 be fever?l fizes, as a chain in 1 inch, a chain in | of an 
 inch, a chain in ^ an inch, &c. And of ihefe, the bett 
 for ufe arc thofe that are laid on the very edges of the 
 ivory fcale, to prick off diftanccs by, without compaiTcs* 
 
 THE FIELD-BOOK. 
 
 In furveying with the plain table, a field-book is not 
 ufed, as evt- ry thing is drawn on the table immediately 
 when it is meafured. But in furveying with the theodo- 
 lite, or any other inftrument, fome fort of a field-book 
 muft be ufed, to write down in it a regifter or account of 
 all that is dcmc and occurs relative to the furvey in hand. 
 
 This book every one contrives and rules as he thinks 
 fitted for himfelf. The following is a fpecimen of a form 
 very generally ufed. It is ruled into 3 columns: the 
 middle, or principal column, is for the ftations, angles, 
 bearings, diftanccs meafured, &c.; and thofe on the right 
 and left are for the offsets on the right and left, which are 
 fet againft their correfponding diftanccs in the middle 
 column ; as alfo for fuch remarks as may occur, and be 
 proper to note in drawing the p. an, &c. 
 
 Here O 1 is the firft ftation, where the angle or bear- 
 ing is 106® 25'. On the left, at 73 Jinks in the diftance 
 
 CM 
 
StJRTETlN©. 
 
 185 
 
 or principal line, is an offset of 92; and at (J 10 an offset 
 of 24 to a cri)fs hedge. On the right, at o, or the be- 
 gi' ning, an offset Q5 to the corner of the field; at 248 
 Brown's boundary hedge commences; at 6lO an offset 35; 
 and at C}S\i the end of the firit line, the o denotes its ter- 
 minal ing in the hedge. And fo on for the other ftations, 
 A line is drawn undt^r the work, at the end of every 
 flaiion line, to prevent coniufion. 
 
 Form of the Field.BooL 
 
 
 Stations, 
 
 
 Offsets and Remarks 
 
 Bearings, 
 
 Jffse^s and Remarks 
 
 on the left. 
 
 and 
 Diftances. 
 
 on the right. 
 
 
 1 
 
 
 
 105^25' 
 
 
 
 (0 
 
 25 corner 
 
 92 
 
 73 
 
 
 
 248 
 
 Brown's hedge 
 
 crofs a hedge 24 
 
 6lO 
 
 35 
 
 
 954 
 
 00 
 
 
 2 
 
 
 
 53^10' 
 
 
 
 00 
 
 00 
 
 houfe corner 51 
 
 23 
 
 21 
 
 
 120 
 
 29 a tree 
 
 34 
 
 734 
 
 40 a f^yle 
 
 
 3 
 
 
 
 57 "iO' 
 
 
 
 61 
 
 35 
 
 a brook 30 
 
 248 
 
 
 
 ^i.y^9 
 
 l6 a fpring 
 
 foot-rath US 
 
 8iO 
 
 
 crofs h< dge 18 
 
 . 97^ 
 
 20 a pond 
 
 The learner will here draw a plan to this field-book. But 
 
18^ 9URVEYIMe# 
 
 But fome fkilfnl furvcvors now make ufe of a difttffent 
 method for the ficld-bock, namely, beginning at the bot- 
 tom of the pagr, and writing upward; by which they 
 flcetch a neat boundary on either hand, as they pafs along : 
 an example of which will be given further on, in the me- 
 thod of furve) ing a large cllate. 
 
 In fmaller lurvcys and raeafurements, a good way of 
 fctting down the work, is, to draw, by the eye, on a 
 pitce of paper, a figure refembling thac which is to be 
 meafured; and fo writing the dimcnfions, as they are 
 found, againft the corrcfponding parts of the figure. And 
 this merhod may be pradifcd to a confiderable extent, 
 evCD in the larger furveys. 
 
 CHAPTER 11. 
 
 THE PRACTICE OF SURVEYrNG. 
 
 'T'HIS part contains the feveral work? proper to be done 
 ■*■ in the field, or the ways of mcafufing by all the in- 
 flruments, and in all fituations. 
 
 rROBLEM I. 
 
 To meajure a Line or DiJIance* 
 
 To meafure a line on the ground with a chain : Having 
 provided a chain, with U) fraall arrows, or rods, to ftick 
 one into the ground, as a mark, at the end of every chain; 
 two perfons take hold of the chain, one at each ei)d of it, 
 and all the 10 arrows r.r^ taken by one of them, wh© is 
 to go foremofl, and is called the leader; the other being 
 called the follower, for diftin(^ion fake. 
 
 A picket or Itation (lafF, being fet up in the diredlion 
 of the line to be meafured, if there do not appear fome 
 
 marka 
 
SURVEYING, 1-S7 
 
 marks naturally in that cUrcdlion; the follower flands at 
 the beginning of the line, holding the ring at the end of 
 the chain in his hand, while the leader drags forward the 
 chain by the other end of it, till it is ftretched ftraighr, 
 and laid or held level, and the leader direfted, by the fol- 
 lower waving his hand, to the right or left, till the fol- 
 lower fee him exad^Iy in a line with the mark or diredlioa 
 to be meafured to; there both of them ftretching the ch:iia 
 ftraighf, and Hooping and holding it level, the leader liav- 
 ing the head of one of his arrows in the fame hand by which 
 he holds the end of the chain, he there fticks o«e of them 
 down with it while he holds the chain ftretched. This 
 done, he leaves the arrow in the grpund, as a mark for the 
 follower to come to, and advances another chain forward, 
 being direded in his pofition by the follower, ftanding at 
 the arrow, as before; as alfo by himfelf now, and at every 
 fucceeding chain's length, by moving himfelf from fide 
 to fide, till he brings the follower and the back mark into 
 a line. Having then ftretched the chain, and ftuck down 
 an arrow, as before, the follower takes up his arrow, and 
 they advance again in the fame manner another chaia 
 length. And thus they proceed, till all the 10 arrows are 
 employed, and are in the hands of the follower; and the 
 leader, without jjn arrow, is arrived at the end of the 
 1 1 th chain-length. The follower then fends or brings the 
 10 arrows to the leader, who puts one of them down at 
 the end of his chain, and advances with the chain as be- 
 fore; and thus the arrows are changed from the<^ne to the 
 other at every 10 chains' length, till the whole line is 
 finifhed; then the number of changes of the arrows ftiows 
 the number of tens, to which the follower adds the arrows 
 he holds in his hand, and the number of links of another 
 chain over to the mark or end of the line; vSo if there 
 have been 3 changes of the arrows, and the follower hold 
 6 arrows, and the end of the line cut off 45 links more, the 
 whole length of the line is fet down in links thus, o6 \5, 
 When the ground is Hoping, afcending or defcending ; 
 ' ' at 
 
188 SURVEYING, 
 
 at every chain length, lay the offset ftaff, or link OafFc^own 
 in the flopc of the chain, on which lay the f nail pocket 
 Icve', to Ihow how many links or parts the flope line is 
 Jongff than the true level one; then draw the chain for- 
 ward fo many links or parts, which re^luces the line hori- 
 zonta'. Or, holding the chain level every time, will 
 perhaps be the belter way to have the true length of the line, 
 
 PROBLEM II. 
 
 To take Angles and Bearings, 
 
 Let B and C be two objcfts, 
 or two pickets fet up perpendi- 
 cular, and let it be required to 
 take their bearings, or the angle 
 
 formed between them at any 
 
 ftalion A. ^ ^ JS 
 
 1. With the Flain Table. 
 
 The table being covered with a paper, and fixed on its 
 ftand; plant it at the ftation A, and fix a fine pin, or a 
 point of the compafles in a proper point of the paper, to 
 reprefent the point Ar Clofe by the fide of this pin lay the 
 fiducial edge of the index, and turn it about, ftill touching 
 the pin till one objeft B can be feen through the fights : 
 then by the fiducial edge of the index draw a line. In the 
 very fame manner draw another line in the dire^ion of 
 the other objeft C, And it is done. 
 
 2. With the Theodolite, ^c. 
 
 Direft the fixed fights along one of the line*?, as AB, 
 by turning the inftrument about till you fee the mark B 
 through thefe fights; and there fcrew the inftruinent faft. 
 Then turn the moveable index about till, through its 
 fights, you fee the other maik C. Then the degrees cut 
 by the index, on tne graduated limb or ring of the in* 
 ftruraent, Ihcvr the quantity of the angle. 
 
 3. With the Mflgneiic Needle and C-.mpafs, 
 
 Turn the inltrument, or compafs, fo that the north 
 6 end 
 
SURYEYINft, 
 
 ia9 
 
 end of the needle point to the flower-de-luce. Then 
 dired the fights to one mark as B, and note the degrees 
 cut by the needle. Next direft the fights to the other 
 mark C, and note again the degrees cut by the needle. 
 Then their fum or difference, as the cafe is, will give the 
 quantity "bf the angle BAC. 
 
 4. By Meafurement nxjith the Chain, ^c, 
 
 Meafure one chain length, or any other length, along 
 both diredions, as to b and c. Then meafure the dikance 
 b c, and it is done, — This is eafily transferred to paper, 
 by making a triangle Abe with thefe three lengths, and 
 then meafuring the angle A as in Pradical Geometry, 
 prob, XI. 
 
 PROBLEM III. 
 
 To meafure the Offsets* 
 
 A h i k 1 m n being a crooked hedge, or river, &c. 
 From A meafure in a ftraight direftion along the fide of it 
 to B, And in meafuring along this line AB, obferve when 
 you are diredly oppofite any bends or corners of the fence, 
 as at c, d, e, &c. and thence meafure the perpendicular off- 
 fets c h, d i, &c. with the offset-flafF, if they are not veiy 
 large, oiherwife with the chain itfelf. And the work is 
 done. The regifter or field-book of which may be as 
 follows : 
 
 Offs, leti. 
 
 Bale line A B, 
 
 
 
 O A 
 
 ch 62 
 
 45 Ac 
 
 di 8-t 
 
 220 Ad 
 
 ek 70 
 
 340 Ae 
 
 fl 88 
 
 310 A f 
 
 gm 37 
 
 Bn pi 
 
 634 Ag 
 I 785 AB 
 
 Islote. When the ( ffsers are not very large, their place* 
 c, d, c, &c. on the bafe line, can be very well determined 
 
 by 
 
l.Od S0RVEYIN»i 
 
 by the eye, efpecially when afHAed by Inyi'ng down tlic 
 oScr-flafF in a crofs or perpendicular diredion. But 
 when thefe perpendiculars are very large, find their pofi- 
 lions by the crofs, or by the inftrument which you happen 
 to be ufing, in this manner : In meafuring along AB, 
 when you con)e nearly oppofitc C, where you judge a per- 
 pendicular will fland, plant the inftrument in the line, and 
 turn the index till ihe marks A and B can be feen through 
 both the fights, lo(;kingboth backward and forward; then 
 look along the crofs fights, or the crofs line on the index; 
 and if it point diredly to the corner or bend h, the place 
 of c is right. Otherwife move the inftrument backward 
 or forward on the lihe A B, till the trofs line points 
 firaight to h. This being found, fet down the diftance 
 meaiured from A to c : then meafure the offset c h, and 
 {ct it down oppofite the tormer, and on the left hand fide, 
 • Then proceed forward in the line A B, till you arrive 
 oppofitc another corner, and determine the place of the 
 perpendicular as before. And fo on throughout the whole 
 length, 
 
 PROBLEM IV. 
 
 T^/ufjej a Triangular Field ABC. 
 
 J , By the Chain. 
 
 c 
 
 AP 794- y. 
 
 AB 132J y\ 
 
 FC 826 X \\ 
 
 Having fet up marks at the corners, which is to be done 
 in all cafes where there are not marks naturally; meafure 
 wit^ the chain from A to P, where a perpendicular would 
 fall from the angle t, and fet up a mark at P, noting 
 down the diftance AP. Then complete the diftance AB 
 
 by 
 
SITIIVEYIXG. 
 
 151 
 
 hy mcafurfug from P to B. Having fet down this raeafare, 
 return to P, and meafure the perpendicular PC. And 
 thus, having the bafe and perpendicular, the area from 
 them is eafiiy found. Or having the place P of the per- 
 pendicular, the triangle is eafiiy conftrufted. 
 
 Or, meafure all the three fides with, the chain, and 
 note them down. From which the content is ealily found^ 
 or the figure conftrufted. 
 
 2. Bj taking cue or mote of the Angles * 
 
 Meafure two fides AB, AC, and the angle A betweea 
 them. Or meafure one fide Al>, and the two adjacent 
 angles A and B. From either of thefe ways the figure is 
 eafiiy planned; then by meafuring the perpendicular CP 
 on the plan, and multiplying it by half AB, you have 
 the content, 
 
 PROBLEM V. 
 
 To meafure a Faur-fded Field, 
 ]. Bj the Chain* 
 
 AE 
 
 214 
 
 210 DE 
 
 AF 
 
 3b2 
 
 1 60 BF 
 
 AC 
 
 5i)2 
 
 
 Meafure along either of the diagonals, as AC; ami 
 cither of the two perpendiculars DE, BF, as in the laft 
 problem; or elfe the fides AB, BC, CD, DA. From 
 cither of thefe ways may the figure be planned and com- 
 puted, as before direfted. 
 
 Others 
 
fS9 
 
 8T511VET1N9. 
 
 OthernMife by the Chain* 
 
 AP 110 1352 
 AB 1110 
 
 PC 
 Ql) 
 
 A r 
 
 Thus 
 
 
 AC 
 
 59 
 
 
 Cab 
 
 37^ 
 
 >20' 
 
 CAD 
 
 41 
 
 15 
 
 ACB 
 
 72 
 
 35 
 
 ACD 
 
 54 
 
 40 
 
 Mrafure on the longed fide, the diftanccs AP, AQ^ 
 AB i and the perpendiculars FC, QD. 
 
 2. By taking one or more of the Angles, 
 
 Mcafure the diagonal AC (fee the laft fig. but one,) and 
 the angles CAB, CAD, ACB, ACD.— Or meafure the 
 four fides, and any one of the angles at BAD. 
 
 Or thus 
 AB 486 
 BC 3^4 
 CD 410 
 DA 462 
 BAD 78S36' 
 
 PROBLEM VI. 
 To/urvfy any Field by the Chain only. 
 
 Having fet up marks at the corners, where neceffary, 
 of the propofed field ABuDtFG. Walk over the 
 ground, and confi .er how it can bcft he divided into trian- 
 gles and trap<zuims; and meafure them feparately as in 
 the laft two problems. And in this way it will be proper 
 to divide it into tiianyles and trapeziums, by drawing 
 diagonals fr m corner to eorner ; and fo as that all the 
 perpendiculars may fall withm the figures. Thus, the 
 foilouing figure is divided into the two trapeziums 
 ABCG,"Gi>1!:F, and the triangle GCD. Then at the firft 
 trapezium, beginning at A, meafure the diagonal AC, 
 
 and 
 
SVRVEYIN*. 
 
 15a 
 
 and the two perpendiculars G ni, B n. Then the bafe 
 G C, and the perpendicular D q. LalHy, the Diagonal 
 D F, and the two perpendiculars p E, o G, All which 
 meafures write againft the co r re fpo riding parts of a rough 
 figure, drawn to refemble the figure to be furveyed, or 
 fet them down in any other form you choofe. 
 
 Thus 
 
 A m 
 An 
 A C 
 
 133 
 410 
 550 
 
 C q 152 
 CG UO 
 
 130 m G 
 180 n B 
 
 230 qD 
 
 Fo 206 I 120 o G 
 F p 288 80 p E 
 F D 520 1 
 
 Or rhus, 
 Meafure all the fides AB, EC, CD, DE, EF, FG, 
 GAi and the diagonals AC, CG, GD, DF. 
 
 Othenivi/e. 
 
 Many pieces of land maybe very well furveyed, by 
 meafuring any bafe line, either within or without them, 
 together with the perpendiculars let fall on it from every 
 corner of them. For they are by thofe means divided 
 into feveral triangles, and trapezoids, all whofe parallel 
 fides are perpendicular to the bafe line; and the fum of 
 thefe triangles and trapeziums will be equal to the figure 
 propofed if the bafe line fall within it; if not, the fum 
 of the parts which are without being taken from the fum 
 of the whole which are both within and without, will 
 leave the area of the figure propofed. 
 
 In pieces that are not very large, it will be fufficiently 
 exad to find the points, in the bafe line, where the fe- 
 veral perpendiculars will fall, by means of the Cro/s, and 
 K thence 
 
19^ sunvBTaNo* 
 
 thence roeafunng to tlie corners for the lengths of tlve 
 perpendiculars. — And it will be mod convenient to draw 
 the line fo, a:> lh<»t all the perpendiculars may fall within 
 the fijjure. 
 
 Thus, in the fallowing figure, beginning at A, and 
 meafuring along the line AG, the diftances and perpendi- 
 culare, on the ught and left, are as below* 
 
 Ab 
 Ac 
 Ad 
 Ac 
 Af 
 
 315 
 
 440 
 5Sj 
 010 
 990 
 
 AG 1020 
 
 350 bB 
 70 cC 
 
 3*20 dO 
 50 eE 
 
 470 fF 
 
 
 PROBLEM VII. 
 
 Tofufvey any Field nuHh the Plain Taile, 
 1. From one Station* 
 
 Plant the table at any 
 angle, as C, from which all 
 the other angles, or marks 
 fet up, can be feen. 1 hen 
 turn the table about till the 
 needle point to the flower- 
 de-luce ; and there fcrew it 
 faft. M. ke a point for C on 
 the paper on the tablf, and 
 
 lay the edge of the index ^ _ 
 
 to C, turning it about that A. -b 
 
 point till through li.e fights you fee the mark D; and by 
 the edge of the index draw a dry or obfcure line; then 
 meafure the diftance CD, and lay that diftance down on 
 the line CD. Then turn the indejt about the fame point 
 C, till the nwik E be feen thx-cogh the fights, by which 
 
 draw 
 
StrRVETriN€t. 
 
 195 
 
 <Sraw a line, and meafure the diftance to E, laying it on 
 the line from C to E. In like manner determine the po. 
 fitions of CA and CB, by turning the fights fucceffively 
 to A and B ; and lay the lengths of thofe lines down* 
 Then conned the points with the boundaries of the field, 
 by drawing the black lines CD, Dli, EA, AB, EC. 
 
 2. From a Station Within or Withont the Field* 
 
 When all the other parts can- 
 not be feen from one aragle, 
 choofc forae place O within ; or 
 €ven without, if more conve- 
 nient: frona which the other 
 parts can be feen. Plant the 
 table at O, then fix it with the 
 needle north, and mark the point 
 O OH it. Apply the index fuc- 
 ceffively to O, turning it round 
 with the fights to each angle 
 
 A, B, Cj D, E, drawing dry lines to them along the 
 edge of the index ; then meafuring the diftances OA, OB, 
 &c. and laying them down on thofe lines, LaSly, draw 
 the boundaries AB, EC, CD, DE, EA, 
 
 3. By gQi^tg Round the Figure, 
 
 When the figure is a wood, or water, or when from fome 
 other obfiruftion you cannot meafure lines acrofs it; be- 
 gin at any point A, and meafure round it, either within 
 or without the figure, and draw the dJre<flions of all the 
 fides thus: Plant ihe table at A, turn it with the needle 
 to the north of flower-de-luce, fix it, and mark the point 
 A. Apply the index to A, turning it till you can fee the 
 point E, there draw a line; and then the point B, and 
 there draw a line: then meafure thefe lines, and lay them 
 down from A to E and B, Next move the table to B, 
 iay the index along the line AB, and turn the table about 
 ¥. 2 tin 
 
^$S -SURVEYr-NS. 
 
 4ill you can fee the mark A, and fcrew faft the table; In. 
 which pofition aKo the needle will again print to the 
 Hower-de-lucp, as it will do indeed at c\ery ftation when 
 the table is in the rigl t pofition. Here turn the index 
 about B till through the fights you fee the mark C; there 
 draw a line, roeafure BC, and lay the diftance on that 
 Jine after you have fet down the table at C. 1 urn it then 
 again into its proper prfition, and in like manner find the 
 next line CD. And fo on, quite round by £, to A again. 
 Then the proof of the woik will be the joining at A : for 
 if the work be all righf, the laft diredlion EA on the 
 ground, will pafs exa<^ly thri^ugh the point A on the 
 paper; and the meafitfed diftance will alfo reach exactly 
 to A. If thefe do not coincide, or ncarl> fo, fome error 
 has been committed, and the work muft be examined over 
 again, 
 
 PROBLEM VIII. 
 
 J^o/u) vey a Field iviih the Theodolite ^ i^c, 
 
 J. From One Point or Station, 
 
 When aH the angles can be feen from one point, as the 
 angle C, (firii fig. to laft prob.); place the inflrument at 
 C, and turn it aboiit til), through the fixed fights, yoa 
 fee the mark B, and theie fix ir. Then turn the move- 
 able index about, till the mark A is feen through the 
 fights, and note the degrees on the inftrument. Next 
 turn the index fuccefilvely to E and D, noting the de- 
 grees cut eft at eathi which gives all the angles BCA, 
 BCE, BCD. Lallly, mc;^fure tbe Imcs CB, CA, CE, 
 CD; and enter the meafures in a field-book, or rather 
 aoainft the correl ponding parts of a rough figure^ drawn 
 by guefi, to rcfcmhie the 6eld. 
 
 2. From a Point Within or Without, 
 
 Plant the inftrument at O, (!aft fi_.) and «urn it aboiit 
 
 till tie fixed figh.s point to any obj; ft as A; and there 
 
 icrcw it faft. Then turn the moveable index round, till 
 
 2 the 
 
SUllVEYINC. 197 
 
 the fights point fuccefllvely to the other points E, D, C^ 
 B, noting rhe degrees cut ojEF at each of them; which 
 gives all the angles round the point O. Lallly, meafure 
 the diftances OA, OB, OC, OD, OE, noting them dowa 
 as before, and the work is done. 
 
 3. By gomg Round the Field, 
 
 By meafuring round, 
 either within or without 
 i^Q. field, proceed thus. 
 Having fet up marks at 
 B, C, &c. near the corners 
 as ufual, plant the indru- 
 ipent at any point A, and 
 turn it till the fixed index 
 be in the diredion AB,. 
 and there fcrew it fall : 
 
 then turn the moveable index to the dire(^ion AF; and 
 the degrees cut off will be the angle A. Meafure the line 
 AB, and plant the inftruraent at B, and there in the fame 
 manner obferve the angle A. Then meafure BC, and ob. 
 ferve the angle C, Then meafure the diflance CD, and 
 take the angle D. Then meafure DE, and take the angle 
 E. Then meafure EF, and take the angle F. And laftly 
 meafure the diftance FA. 
 
 To prove the wt)rk ; add ail the inward angles A, Bi C, 
 &c.. together, and when the work is righr, their fum will 
 be equal to twice as many rig/it angles, as the figure has 
 fides, wanting 4 right angles. But when thfre is an 
 angle, as F, that bends invvards, and you meafure the ex- 
 ternal angLe, which is lefs than 2 right anghs, fubtract it 
 from 4 right angles, or 36o degrees, to give the internal 
 angle greater than a femicircle or ISO degrees, 
 
 Otherix-'ifr^ 
 Tnfie.id of ohfervjpg the internal angles, you may take 
 the external angles, formed without the figuie by pro- 
 ducing the fides farther our. And in this cafe, when tlic 
 K 3 work 
 
m 
 
 fliriivBYii?«, 
 
 work is right, their fum altogether will be equal to 36# 
 degrees. But when one ©f them, as F, runs inwards, 
 fubtrad it from the fum of the reft, to leave 360 degrees, 
 
 PROBLEM IX, 
 
 To fur'vey a Field nvith Crooked Htdget, 
 
 With any of the inftruments raeafure the lengths and 
 poCticns of imaginary lines running as near the fides of 
 the field as you can ; and, in going along them, meafure 
 the olFsetS) in the manner before taught; and you will have 
 the plan on the pnper in ufing the plain table, drawing 
 the crooked hedges through the ends of the offsets; but 
 in furveying wiih the theodolite, or other inftrument, fet 
 down the meafures properly in a field-book, ©r memo- 
 randum book, and plan them after returning from the 
 field, by laying down all the lines and angles. 
 
 a 
 
 B 
 
 So, in furveying the piece AECDE, ^ei up marks a, b, 
 C| d, dividing it into as few fides as may be, commonly 4. 
 Then begin at any ftation a, and meafure the lines ab, be, 
 cd, da, and take their pofitions, or the angles a, b, c, d; 
 and, in going along the lines, meafure all tlic offsets, as 
 at ix\, n, o, p, &c. along every ftation line. 
 
 And 
 
ttrilTEYlNff. 
 
 199 
 
 And this Is done either within the jSeld, or without, as 
 may be moft convenient. When there are obflru(^ions with- 
 in, as wood, water, hills, &c. then meafuie without, as in 
 the figure here below- 
 
 PROBLEM X, 
 
 Tofurniey a Held or any other ding, by T'wo Stations, 
 
 This is performed by choofing two ftation s, from 
 which all the marks and objects can he {ten ; then mea- 
 furing the diftance between the Rations, and at each Ra- 
 tion taking the angies formei by every objeifl, from th© 
 ftation line or diftance. 
 
 The two Rations may be taken either wirhin the bounds, 
 or in one of the fides, or in the di region ( f two of the 
 obje^is, or quite at a diftance and without the bounds of 
 the objects, or part to be fnrveyed. 
 
 In this manner, not only grounds may be furveyed, 
 
 without even entering them, but a map may he taken of 
 
 the principal parts of a county, or the chief places of a 
 
 town, or any part of a river or coaft fuive ed or any 
 
 other inacceflible objeils ; bj ta k ing two Rations, on two 
 
 towers, or two hills, or fuch like. 
 
 K 4 When 
 
300 
 
 SUEVETIN*. 
 
 jay,<p i\ "TT^^*'.. 
 
 When the plain table is u fed ; plant It at one ftarlon m, 
 i!raw a line mn on it, along which lay the edge of the in- 
 dex, and rum the table about till the fights point directly 
 to the other ftation; and there fcrew it fall. Then turn 
 the fights round m fncceffivcly to all the objcfts A, B, C, 
 See, drawings dry line by the ci\g& of the index at each, 
 as m A, m B, m C, &c. Then meafure the diftance to 
 the other fiation, there plant the table, and ].\y that dif- 
 tance down on the fiation line from m to n. Next lay the 
 index by the line nm, and turn the table about till the fights 
 point to the other fiation m, and there fcrew it fivft. Then 
 dired the fights fuccefiively to all the ohjeds A, B, C, &c. 
 as before, drawing liijes each time, as n A, n B, n C, &c. 
 and their interfedtion with the former lines, will give the 
 places of all the objeft'^, or coirers. A, B, C, Sec, 
 
 When the theodolitf, or any other infirument for taking 
 angles, is ufed; proceed in the fame way, meafuring the 
 ftation diftance m n, planting the innrumcnt firll at one 
 fiation and then at the other; then placing the fixed fights 
 in the direction m n, and direding tiie moveable fights to 
 every ol)jfd>, noting the degrees cut off at each time. 
 Then, thefe obfcrvattons being planned, the interfedioiis 
 of the lines will give the objedls as before. 
 
 When 
 
SURVEYING. 201 
 
 When all the objeds to be furveyed cannot be feen from 
 two ftations; then three ftations may be ufed, or four, or 
 as many as neceilary ; meafuring always the diftance from 
 one ftation to another; placing the inflrument in the fame 
 pofition at every ftation, by means defcribed before; arxl 
 from each ftation obferving or fetting every obje>5l that 
 can be feen from if, by taking its direflion or anguli»r po- 
 fition, till every objeft be determined by the interfedion 
 of two or more lines of direction, the more the better. 
 And thus may very extenfive furveys be taken, as of large 
 commons, rivers, coaft?, countries, hilly grounds, and 
 fuch like. 
 
 PROBLEM XI, 
 
 Tofuwey a Large Efiate, 
 
 If the efiate be very large, and contain a great number 
 of fields, it cannot well be done bv furveying all the fields 
 lingiy, and then putting them together; nor can it be 
 done by taking all the angles and boundaries that inclofe 
 it. For in thefe cafes, any fraall errors will be fo multi- 
 plied, as to render it very much diftorred. 
 
 1. Walk over theeftatc two or three times, in order to 
 get a perfect idea of it, and till you can carry the map of 
 
 ttolerakly in your head. A.nd to help your memory, 
 draw an eye draught of it on paper, or at leaft of the 
 principal parts of it, to guide you ; fetting the names 
 within the fields in that draught. 
 
 2. Choofe two or more eminent places in the eftat^, for 
 your ftations, from which you can fee all the priaoipal 
 parts of it : and let thefe ftarions be as ftr dittant from 
 one another as poftible, as the fewer ftations you have to 
 command the whole, the-more exaft y(^ur uork will bej 
 and they will be fi'ter f'r your purpose, if thefe itati )a 
 lines be in or near the boundaries of the ground, efpecially 
 if two lines or more proceed from one ftation. 
 
 3. Take what angles, between the ftations, yon think 
 neceflary, and iiieafure the diftances from ftatioa to ftation, 
 
 K a always 
 
200 SV^lVITlNCf. 
 
 always in a rigl-t line: thefc things muft be done, till you 
 get as many lines and angles as are fufficient for determin- 
 ing all the ftation points. And in meafuring any of thcfe 
 ftation diftances, mark accurarel}' where thefc lines meet 
 with any hedge*^^, ditcher, roads, lanes, paths, rivulets, &c. 
 and vvbcre any remarkable objed is placed, by meafuring 
 its dillance from the ftation line; and where a perpendicu- 
 lar from it cuts that line; and always mind, in any of thefe 
 oofcrvations, that you be in a right line, which you will 
 know by taking a backfight and forcfight, along the fta- 
 tion line. And thus in going along any main ftation line, 
 take offsets to the ends of all hedges, and to any pond, 
 houfe, mill, bridgp. Sec, omitting nothing that is remark- 
 able. And all thefe things muft be noted down : for thcfe 
 are the data, by which the places of fuch objed^s are to 
 !)€ determined on the plan. And be fure to fet up marks 
 at the interfe<5tions of all hedges with the ftation line, that 
 )'ou may know where to meafure from, when you come 
 to furvey thefe particular fields, which muft immediately 
 be done, as fopn as you have raeafured that ftation line, 
 while they are frefn in memory. In this way all the fta* 
 tion lines are to be meafured, and the fitoation of all 
 places adjoining to them determined, which is the firll 
 grand point to be obtained. It will be proper to lay down 
 the work on paper every night, when you go home, that 
 you may fee how you go on, 
 
 4. As to the inner parts of the eftate, they muft be de- 
 termined in like manner, by new ftation lines: for, after 
 the main ftations are determined, and every thing adjoin- 
 ing to them, then the eftate muft be fubdivided into two 
 or three parts by ne\v ftation lines; taking inner ftation« 
 at proper pkces where you can have the beft view. 
 Meafure thefe iiation lines as you did the firft, and all 
 their interfedions with hedges, and all offsets to fuch ob- 
 je(\s as appear. Then proceed to furvey the adjoining 
 fields, by taking the angles that the fides make with the 
 ftation line, at the ini€rfe(^tions, and meafuring the 
 
 dif. 
 
SURVEYING. 203 
 
 diflances to each corner, from the inter fed^ions. For the 
 ftation lines will be the bafes to all the future operations; 
 the fituations of all parts being enrirely dependent on them ; 
 and therefore they (hocld be taken of as great length as 
 poffible ; and it is befl: for them to run along fome of the 
 hedges or boundaries of one or more fields, or to pafs 
 through fome of their angles. Ail things being determined 
 for thefe ftations, you mud take more inner ftations, arid 
 continue to divide and fubdivide till at laft you come to 
 fingle fields; repeating the fame work for the inner ftations, 
 as for the outer ones, till all be done; and clofe the work 
 as often as you can, aiid in as few lines as pofiible. And 
 that you may choofe ftations the moft conveniently, fo a-j 
 to caufe the leaft labour, let the ftation lines run as far as 
 may be alonjj fome hedges, and through as many corners 
 cf the fields', and other remarkable points, as you can. 
 And take notice how one field lies by another; that you 
 may not mifplace them in the draught. 
 
 5, An eftate may be fo fituated, that the whole cannot 
 be furveyed together; becaufe one part of the eftate may 
 not be (ecn from another. In this cafe you may. divide 
 it info three or four parts, and furvcy thefe parts feparate- 
 ly, as if they were lands belonging to different perfons ; 
 and at laft join them together, 
 
 6\ As it is neceffary to protraft or lay down the work as 
 you proceed in it, you muft have a fcale of a due length 
 to do it by. To get fuch a fcale, meafure the whole 
 length of the eftate in chains; then confider how many in- 
 ches long the map is to be; and from thefe you will know- 
 how many chains you muft have in an inch;, then make 
 your fcale accordingly, or choofe one already made. 
 
 7. The trees in every hedge row may be placed in their 
 proper fituation, which is fc'On done by the plain table; 
 but may be done by the eye without an inftrument; and 
 being thus taken by guefs in a rough draught, they will 
 be exad enough, being only to look at; except it be fuch 
 23 are at any remarkable places, as at the ends of hedges, 
 
 K (S Sit 
 
I!04 &URVEYINO. 
 
 at ft lies, gates, &c. and thefe muft he mcafured, or taken 
 with the plain table. But all this need not he done tfll the 
 draught is finilVied. And obferve in all the hedges, what 
 fide the gutter or ditch is on, and to whom the fences 
 belong. 
 
 iS. When you have long ftations, you ought to have a 
 good inftrument to take angles wirh, and tlie plain table 
 xnay very properly he made ufe of, to rake the ft-veral 
 fmall internal parr^, and fuch as cannot he taken from the 
 main flaiions : as it is avery quick and ready inftrument. 
 
 ^heNeiv Method of Sur'v eying, 
 
 Inftcad of the foregoing method, an ingenious friend 
 (Mr. Abr..ham Crocker), nfter mentioning' the new and 
 improved method of keeping the fici(^-hook, by writing 
 from bottom to top of the pages, obferves that •* In the 
 former method of meafuring a hirge eftate, tl.e accuracy 
 of it depends on the corrtd \tU of the inttrumenfs uft-d in 
 taking the angles. To avoid the errors incident to fuch 
 a mul'itudc ot anglef, other methods hnve of late years 
 been ufed by fome fewfliiiful furveyors: the moft practical, 
 expeditious, and corred, feems to be the following: 
 
 '* A-. was adviffd in the foregoing method, fo in this, 
 choofe two or more eminences, as grand itanons, and 
 ineafure a principal bafe line from one fia'ion to the other, 
 noting every hedge, brook or other remaikable objr'd as 
 you pafs by it ; meafuring alfo fuch fhort perpendicular 
 lines to fu<:h bends of hedges as rray be near at hand. 
 From the extremities of this bafe line, or from any con- 
 venient parts of the fame, go oiT with other hues to fome 
 ren^arka'ole obj^^ft fituated towards the lides of the ellafe, 
 without re^'^aroing the angles they ma'<e w iih the bafe line 
 or with one another; liill rememiiering to note every 
 hedge, brook, or other objeft that you pafs by. 'I hefe 
 lines, when laid down by intcrfedions, w ill with the ' afe 
 line form a grand iriangle en the cflalej feveral of \vhich> 
 
 ii 
 
SURVEYING. 205 
 
 if need be, being; th'is laid down, you may proceed to 
 form other fmaller rriaigles aid trapezoids, on the fides 
 of the form r: and fo on, until you finifti with the en- 
 clofures indiv'dually. 
 
 *' This ^rand triangle being completed, and laid down 
 on the rough plan paper, the parts, exterior a^ well as 
 interior, are to be completed by fmaller triangles and 
 trapez 'id . 
 
 *' When the whole plan is laid down on paper, the 
 contents of each fi'^ld might be calculated by the method* 
 laici down below, at prob. 2. chap. 3. 
 
 " In countries wncre the lands are enclofed with high 
 hedges, and where many Janes pafs through an eftate, a 
 theodolite may be ufed to advantage, in meafiring the 
 angles of fuch lands; by which means, a kind of flceliitort' 
 of the eOate nay be obtained, and the lane lines ferve as 
 the hafes of fuch triangles and trapezoids as are neceflary^ 
 to fiM up the interior parts." 
 
 The method (f meafuring the other crofs line^, offsets^ 
 and inerior parts and enclofure*^, appears in the plan fig, 
 J. pi. 28. Didionary. 
 
 Another ingenious correfponde^it (\'Ir. John Rodham, 
 of Richmond, Yorkfhire), has alfo communicated th« 
 following e-.a np'e of the ne v merhod of fiirveying, ac- 
 companiei by the field-hook, and its correfponding plan. 
 Hjs a count of the mechod is a'^ follows. 
 
 '* The l:e;d-book is ruled into three columns. In the 
 middle one are fet dov n the diftances on the chain line at 
 which anv marl^, offer, or other ohferVation is made; 
 ajid in li.e right and letr hand columns are entered the 
 offetsand obferva ions made on the right and left hand 
 le pedively of the cl^a'.n line. 
 
 ♦* it is of great a ivan age, both for hre/vy and rerfpi- 
 culty, to bt*gin a( the botiom of the leaf and wriie up- 
 wards, denoting; the eroding offences, bylines drawa 
 acrofs the middle column, or only part of fuch a line 
 ©a the right and left oppofue the figures, to avoid con. 
 
 fufioa; 
 
COG SITRTETINC' 
 
 Miotti and th« corners of fierds, and other rtenrtatk- 
 ahle turns in the fences whf^rc offsets are taken to, by lines 
 joining in the manner the tVnccs do, as will be belt (ten 
 by corrparing the book with the plan annexed to the field- 
 book, in p. 'ioS. 
 
 ** The marks called, a, 6, r, &r. are bed made in the 
 fields, by making a fmall hole with a fpade, and a chip or 
 fmall bit of u'ood, with the particular letter upon if, may 
 be put rn, to prevent one mark being taken for another, 
 on any return to it. But in general, the name of a mark 
 is very eafily had by referring in the book to the line it 
 was n^ade ir. After the fmall alphabet is gone through, 
 the capitals may be next, the print letters afterwards, and 
 fo on, uhich anfwer the purpofe of fo many different 
 letters; or the marks may be numbered. 
 
 ** The letter in the left hand corner at beginning of 
 every line, is the mark or place meafu red yr(jw; and, that 
 at the right hand corner at the end, is the mark meafured 
 fo: But when it is not convenient to go exactly from a 
 mark, the place meafured from, is defcribedyl/r^ a dijiance 
 from one 7nark towards another; and where a mark is not 
 meafured to, the exaft place is afcenained by faying, turn 
 to the right or' left hand, fuch a d'Jiance to fuch a mark; 
 it being always underftood that thofe diftances are taken in 
 the chain line. 
 
 " The charaders ufed, are Tfor turn to the right hand, 
 Iffor turn to the left handy and /\ placed over an offset, to 
 fhow that is not taken at right angles with the chain line, 
 but in the lire with feme ftraight fence; being chiefly ufed 
 when croffing their dirtclions, and it is a better way of 
 obtaining their true places than by offsets at right angles* 
 
 '* When a line is meafured whofc pofition is determined, 
 either by former work (as in the cafe of producing a given 
 line, or mejfuring from one known place or mark to an- 
 other) or by itfelf, (,hs in the third fide of a triangle) it is 
 called ^fajl Ibie^ and a double line acrofs the book is 
 drawn at the conclufion of it: but if its pofition is not de. 
 
 termined. 
 
SURVEYING. 507 
 
 terroined, as in the fecond fide of a triangle, it is called a 
 locfe line-, and a Tingle line i? drawn acrofs the book. When 
 a line become.^ determined in pofirion, and is afterwards 
 continued, a double line half through the book i<5 drawn. 
 
 ** When a loofe line is meafured, it beGomes abfolutely 
 neccflary to meafure fome line that w ill determine its po- 
 fition. Thus, the firft line ch, being the bafe of a tri- 
 angle, is always determined ; but the pofition of the fecond 
 lide hjy does not become determined, till the third (idey^ 
 is meafured; then the triangle may be conftrufted, and 
 the pofition of both is determined. 
 
 *• At the beginning of a line, to fix a loofe line \o the 
 mark or place meafured from, the fign of turning to the 
 right or left hand muft be added (as aty in the third line;) 
 otherwife a (Iranger, when laying down the work, may as 
 eafily conftruft the triangle hjb on the wrong fide of the 
 linctf/^, as on the right one; but this error cannot be fallen 
 into, if the fign above named be carefully obferved. 
 
 ** In choofing a line to fix a loofe one, care nauft be 
 taken that it does not make a very acute or obtufe angle ; as 
 in the triangle /Br, by the angle at B being very obtufe, 
 a fmall deviation from truth, even the breadth of a point, 
 at/ or r, would make the error at B, when conllrudted, 
 xtry confiderable; but by conflru(fting the triangle /B^, 
 fuch a deviation is of no confequence. 
 
 " Where the words leave off are written in the field- 
 book, it is to fignify that the ta'<ing of offsets is from 
 thence difcontinucd ; and of courfe fomething is wanting 
 between that and the next offset," 
 
 The field book for this method, and the plan drawn from 
 it, are contained in the four following* pages, engraven en 
 copper-plates. After the manner of which, the pupil muft 
 lay down a plan, to a larger fcale, from the field-book en- 
 tirely ; and then computing the contents of every feparare 
 fi«ld, and adding all the contents together, the fum will 
 amount to between 103 and 104- acres, when the work is 
 all right. 
 

 Pu'ld Mook. 
 
 
 
 
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 i.-^i 
 
 ~l 56 to e 
 
 
 
 
 83 « 
 
 ^ \- 
 
 
 n 
 
 
 
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 .o-T 
 
 
 
 
 1-480 
 
 r cyo to fl 
 
 
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 060 
 700 
 
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 n 
 48 
 
 
 ni 
 
 
 40 
 
 30 
 
 
 
 
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 40 
 
 
 
 
 
 1004 
 f>8o 
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 m 
 ^4 
 
 
 
 h 
 
 
 280 
 
 32 
 
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 1464 
 
 X05 
 
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 3.50 
 
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 60 
 
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 142 
 
 
 
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 ^"J 
 
 
 
 
 '-• 434 
 
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 2000 
 
 
 
 
 i8 8u 
 
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 1840 
 
 
 
 
 _Y^r-r<cr 
 
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 rJ34 + .'>« 
 
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 76 
 
 13 -'8 
 
 
 
 
 96 
 
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 34 
 
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 2080 
 
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 ^ 
 
 
 
 
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 rfT 
 
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 600 
 
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 Br- 
 
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 480 
 
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 r^ 
 
 88.5 
 
 A 
 
 
 
 /44 
 
 6 66 
 
 
 
 
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 310 
 
 z 
 
 
 d 
 
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 236 
 
 
 
 
 
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 183^ 
 
 
 
 
 Y^vs 
 
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 4420 
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 Frf 
 
 
 
 
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 1900 
 i8 4 
 1770 
 1320 
 808 
 
 3«o 
 
 rf / leave eir 
 
 r 
 
 
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 170 
 
 
 
 
 
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 13 
 
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FieldBifok. 
 

 /'/</// //v'/// ////• I'onifoiito h'l.t litu'L- 
 
 
 
 1 
 
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sunrETiNO, 215 
 
 PROBLEM XII. 
 
 To furvey a County ^ or Large TraS! of Land, 
 
 1. Choofe two, thrce^ or fonr eminent placfes for ^a-^ 
 tJons; ftich as the tops oF high htlls or. mountains, to'.vers, 
 or church fteeples, which m*y be feen from one another; 
 and from which mofi of the towns, and other places o^ 
 note, reav alfo be feen. And let them be as far diHant 
 from another as pofTible, On thefe places raise hencons, 
 or long poles, with flags of difK^rcnt colours flying at 
 them: fo as to be vifible from all the other ftations. 
 
 2. At all the places, which ynu would fet down in the 
 map, plant long poles with flags at them of feveral co» 
 lours, to diftinguifh the places from one another ; fixinj^ 
 them on the t'ps of church fl:eep!es, or the tops of 
 houfes, or in the centres of Iefl>r towns. , ' ' 
 
 But you need rxot have (hefe marks at many places at 
 ©nee, as fupnofc half a fcore at a tirne. For wlie'n the 
 angles have been taken, at the two ftations, to all thefe 
 place?, the mHrks m^.y be removed to pew ones; and fo 
 fucceflively to ad the places wanted. • Thefe marks then 
 being fet up at a convenient number of places, and fuch 
 a« may be feen fro-m both ftatio^ns; go to one of thefe fta- 
 tions, and, with an inRrumsnt to take angles, ft;indins^ at 
 that ftation, take all the angles between the other ftarion, 
 and each ot thefe marks, obferving which is blue, which 
 is red, &c. and which hand they lie on ; and fct all down 
 with their colours. Then go to the other ftation, and 
 take all the angles between the firft fta'ion, and t'^:\\ of 
 the former marks, and fet them down with the others, 
 each againft its fellow with the fame colo ir. Y' n may, 
 if you can, alfo ra'<e the angles at fome th rd fta<ion, 
 which niay ferve to prove the work, if the three lines in* 
 
 terfe^ 
 
tI4 BURVtYtlCei! 
 
 tcrfcA In that point where any mark ftands. The marks 
 rauft (land till the obfervations «re finiftied at both fta- 
 tions; and then they muft be t ken down, and fc np at 
 frefh places. The fame operations nnuft be performed, at 
 both ibtions, for thele f n fh places; and rhe like for 
 others. The inftrument for taking angles muft W an ex- 
 ceeding good ore, made on piirpofe wiih tel fcopic fights ; 
 and of a good length of radiu>. 
 
 3. And though ir be n^t abfolutely necefTary to menfure 
 any diftance, becauf: a ftarionary line being once laid 
 down from any fcale, all the other lines will be propor- 
 tional to it ; yet it is better to meafure f >ine of the 1 nes, 
 to afcertain the diftanecs of pi ices in miltes; an i to know 
 how many geometric.il miles there are in any l.ni;th; and 
 from that to make a fcale to meafure any diftance in miks. 
 In raeafnting any diflance, it will not be exa^fl enough (a 
 go along the high roads; by nafon of their turnings and 
 wiadings, and hardly ever lying in a right line between 
 the fta i' ns, which muft canfe infinite reduflitns and 
 create endlefs troul)lc to make it a right line; for which 
 reafon it can never be exatt. But a Inuer way is to mea- 
 fure in a right line with a cl ain, between ftation and 
 fiation, over hill, and dales or Kvcl fif-lds, an i all ohli^acleF, 
 Only in cafe of water, woods, louns, rotks l^anks, &c. 
 where one cannot pafs; fuch parts of the line muft be 
 meafured by the methods of inacceffible diftances; and 
 befidey, allowing for afcents and defcenrs, when we meet 
 with them. And a good compafs, that fhows the bearing 
 ef the two ftations, will always dire<ft you to go ftraight, 
 when you do not fee the two ftations; and in the progrefs, 
 if you can go flraight, you may take off-ets to any re- 
 markable places, l.kevvife note the interfedion of your 
 ftationary line with all roa Is, river^, &c. 
 
 4. And from all the ftation^, and in the whole pro- 
 grefs, be very particular inobferving fea-coafts, river, 
 roouths, towns, caftles, houfes, churches, windmills, 
 wateroiills, trees, rocks, faudf, reads, bridges, fords, 
 
 ferries, 
 
lURVEYIN*. 215 
 
 ferrief, woods, hills, mountains, rills, brooks, parks, bea- 
 cons, fluices, floodgates, locks, &c. and in general all 
 things that are remarkable, 
 
 5. After you have done with the firft and main ftation 
 lines, which command the whole county; you muft then 
 take inner ftations, at fome places already determined ; 
 which will divide the whole into feveral partitions; and 
 from thefe ilations you muft determine the places of aa 
 many of the remaining towns as you can. And if any 
 remain in that parr, you mull take more ftations, at fome 
 places already determined; from which you may deter- 
 mine the reft. And thus we muft go through all the parts 
 of the country, taking ftation after ftation, till we have 
 determined all we want. And in general the ftation dif- 
 tances muft always pafs through fuch remarkable points as 
 have been determined before, by the former ftations. 
 
 6. Laftly, the pofition of the ftation line meafured, or 
 the point of the compafs it lies on, muft be determined by 
 aftronomical obfervation. Hang up a thread and plummet 
 in the fun, over £ome part of the ftation line, and obfervc 
 when the (hadow runs along that line, and at that moment 
 take the fun's altitude; then having his declinati,*n, and 
 the latitude, the azimuth will be found by fpberical trigono- 
 metry. The azimuth being the angle the ftation line 
 makes with the meridian^ therefore a meridian may eafily be 
 drawn through the map. Or a meridian may be drawn 
 through it, by hanging up two threads in a line with the 
 pole ftar, when it is juft north, which may be known from 
 aftrcnomical tables. Or thus; obferve the ftar Alioth, or 
 that in the rump of the great bear, being that next the 
 fquare, by a line and plummet when that ftar and the pole 
 ftar come into a perpendicular; for at that time they are 
 due north. Therefore two perpendicular lines being fixed 
 at that moment, towards thefe two ftars^ will give the po- 
 fition of the meridian. 
 
 FROSLEM 
 
i6 
 
 SURVf.YING. 
 
 PROBLEM XIII. 
 
 To/urvej a Tcmin or City, 
 
 This is bcft performed with the plain table, where ever^ 
 minute part is drawn while in fi^ht. It is belt alfo to 
 have a chain of 50 feet Ion;;, divided into 60 links, each 
 1 foot in length, and anofFset-ftaff of 10 feet long. 
 
 Begin at the meeting of two or more of the principal 
 flreets, through which you can have the longed profpeds, 
 to get the longetl ftation lines. There having fixed the 
 inftrumcnt, draw lines of dired^ioii along thofe ftreets> 
 ufing two men as marks, or poles fet in wooden pedeftals, 
 or perhaps fome remarkable places in the houfes at the 
 fartlier ends, as windows, doors, corners, &c. Meafure 
 thefe lines with the chain,, taking offsets with the Itaff, at 
 til corners of flreets, bendings, or windings, and to all 
 remarkable things, as churches, markets, hails, college?, 
 eminent houfes, &c. Thea remove the inftruraent to 
 another ftation, along one of thefe lines, and there repeat 
 the fame procefs as before. And fo on till the whole ii 
 finiihed. 
 
 Thus, fix the inftruraent at A, and draw lines in the 
 direi^ion of all the flreets meeting there ; then meafure' 
 
 AB^ 
 
SURVEY1N». Sir 
 
 AB, noting the ftreet on the left at m. At the fecond 
 ftation B, draw thedireft ons of the ftrects meeting there j 
 meafure from B to C, noting the places of the ftreets at 
 n and o as you pafs by ihem. At the 3ti ftation C, take 
 the diredtioa of all the ftreets meeting there, and meafure 
 CD. At D do the fame, and meafure DE, noting the 
 place of the crofs ftreets at p. And in this manner go 
 through all the principal ilreets. "ihis done, proceed to 
 the frnaller and intermeiiiate ftreets; and lalUy to the lanes, 
 alleys, courts, yards, and every part which it may be 
 thought proper to reprefent in the plan. 
 
 CHAPTER III. 
 
 OF PLANNING, COMPUTING, AND 
 DIVIDING. 
 
 PROBLEM I, 
 
 To lay do'-wn the Plan of any Suwey, 
 
 T F the furvey was taken with a plain table, you have a 
 -*- rough plan of it alreaily on the paper which covered 
 the table But if the furvey was with any other inftrument, 
 a plan of it is to be drawn from the meafures that were 
 taken in the furvey, and firft of all a rough plan on paper. 
 To do this, you mu!t have a fet of proper inftruments, 
 for laying dov\n both Ines and angles, &c. as fcales of 
 various fizes, rhe more of them, and the more accurate, 
 the better; fcales of chords, protraftors, perpendicular 
 and parallel rulers, &c. biagonal fcales are beft for the 
 linps, becaufe they extend to three figures, or chains and 
 links, which are hundred h parts of chains. But in 
 ufing the diagonal fcale, a pair of compafTes mrft be em- 
 ployed to take off the lengths of the principal lines very 
 
 accurately. 
 
9lt S«]IVET1K». 
 
 accurately. But a fcalc with a thin edge divided, is raucK 
 readier for laying down the perpendicular offsets to crooked 
 hedges, and for marking the* places of thofe offsets on the 
 ftatiou line; which is done at only one application of the 
 edge of the fcale to ihat line, and then |.)rickiag off all at 
 once the diftanccs along it. Angles arc to be Inid down, 
 either with a good fcale of chords, which is perhaps the 
 rooft accurate way; or with a large protraOor, which is 
 much readier when many angles are to he laid down at one 
 point, as they are pricked off all at once round the edge of 
 the protraftor. 
 
 Very particular dIreOions for laying down all forts of 
 figures cannot be neceffary in this place, to any perfon who 
 has learned praftical geometry, and the conlhudiion of 
 figures, with the ufe of his indruments. It may therefore 
 be fuflScient to obferve, that all lines and angles muft be 
 laid down on the plan in the fame order in which they were 
 meafured in the field, and in which they are written in the 
 field-book ; laying down firft the angles for the pofition of 
 lints, then the lengths of the lines, with the places of the 
 offsets, and then the lengths of the offsets themfelves, all 
 with dry or obfcure lines ; then a bl.ick line drawn through 
 the extremities of all the offsets, will be the hedge or 
 bounding line of the field, &c. After the principal bounds 
 and lines are laid down, and made to fit or clofc properly, 
 proceed next to the fmaller objects, till you have enttred 
 every thing that ought to appear in the plan, as houfes, 
 brooks, trees, hills, gates, ftiles, roads, lanes, mills, 
 bridges, woodlands, &c, &c. 
 
 The north fide of a map or plan is commonly placed 
 uppermoft, and a meridian fomewhere drawn, w;th the 
 compafs or flower-de-luce pointing north. Alfo, in a va- 
 cant place, a fcale of equal parts or chains muft be drawn, 
 with the title of the map in confpicuous charafters, and 
 embellifhed with a compartment. All hills muft be 
 fliadowed, to diftinguifti them in the map. Colour the 
 hedges with different colours ; rcprefent hilly grounds by 
 
 broken 
 
SURVEYING. 3lf 
 
 l)r(>kcn hills and valleys; draw fmgle dotted lines for foot- 
 paths, and double ones for horfe or carriage roads. Write 
 the name of each field and remarkable place within it, and, 
 if you choofe, its content in acres, roods, and perches. 
 
 In a very large eilate, or a county, draw vertical and 
 horizontal lines through the map, denoting the fpaces be- 
 tween them by letters placed at the top, and bottom, and 
 fides, for readily finding any fidJ cr other objedl, men- 
 tioned in a table. 
 
 In mapping counties, and cflafes that have uneven 
 grounds of hiils and valleys, reduce dl oblique line?, 
 meafured up hill and down hill, to horizontal ftraight 
 lines, if that was not done during the {uivcy, before they 
 were entered in the field-book, by making a proper allow- 
 ance to Ihorten them. For which purpofe there is com- 
 raonly a fmall table engraven on fome of ihe inftruments 
 for furveying. Or it may be done by holding the chairx, 
 in meafuring, quite level, and then dropping the arrow 
 from the hand. 
 
 PROBLEM II. 
 
 To Commute the Contents of Fields, 
 
 1. Compute the contents of all the figures, whether 
 triangles, or trapeziums, &c. by the proper rule*, for the 
 feveral figures laid down in meafuring; muiiifjly the 
 lengths by the breadths, both in links, and divide by 2 ; 
 the (quotient is acres, after yon have cut oft hve figures on 
 the right for decimals. Then bring thefe decimals to 
 roods and perches, by multiplying firll by 4, and then by 
 40. An example of which is before given, in the de- 
 fcription of the chain. 
 
 2. In fmall and feparate pieces, it is afual to compute 
 their contents from the meafures of the lines taken in fur« 
 veyiflg them, without making a correft plan of them, 
 
 I ^ Thus, 
 
J20 
 
 SURVEY1K», 
 
 Thus, in the triangle in prob. iv. pa^e 190, wliere wt 
 hail AP = 7!)4, and AB = 1321 
 hC =1 826* 
 
 7i)26 
 1?642 
 10.56'8 
 
 2 ) 1()-91146 
 
 3*45573 ac r p 
 4 Anf. 5 1 33 
 
 I -822.02 
 40 
 
 32-£)l680 
 
 Or the firft example to prob. v. page I9I, thus 
 
 AE 214 
 AF 362 
 AC 592 
 
 210 KD 
 306 FB 
 
 516 fnm of perps, 
 5y2 AC 
 
 1032 
 4644 
 2680 
 
 2 ) 3-05472 
 
 1 '32736' ac r p 
 
 4 Anf. 12 4 
 
 2M0Q44 
 40 
 
 4-3:760 
 
 Or 
 
SURVtYING. 
 
 221 
 
 Or tlie 2d example to the Tame prob. v. thus : 
 
 A? no 
 
 AQ^ 745 
 AB 1110 
 PC 332 
 AP 110 
 
 2APC 38720 
 
 352 
 
 595 
 
 PC 
 
 OP 
 PCL 
 
 PC 
 
 OP 
 
 352 
 595 
 
 9^7 
 
 635 
 
 4735 
 2841 
 5682 
 
 QP595 
 QB 3G5 
 
 2.075 
 3570 
 1785 
 
 217175 = 2QPB 
 601345rr2PCDQ^ 
 38720 =: 2 AFC 
 
 2 ) 8*57240 r=:dou. the whole 
 
 4-2S6"2 
 
 2PCDQ^601345 4 
 
 ac r p 
 
 J-1448 Anf. 4 1 5 
 f40 
 
 5-7920 
 
 AP 
 
 3. In pieces bounded by very crooked and winding 
 hedges, meafured by offsets, all the parts between the off- 
 sets are moft accurately meafured feparately as fmall trape- 
 7.oids. Thus, for the example to prob, in. p. 189, where 
 
 Ac 
 
 45 
 
 62 
 
 ch 
 
 Ad 
 
 220 
 
 84 
 
 di 
 
 At 
 
 340 
 
 70 
 
 ek 
 
 Af 
 
 510 
 
 88 
 
 fl 
 
 Ag 
 
 634 
 
 57 
 
 gm 
 
 AB 
 
 785 
 
 91 
 
 Bn 
 
 The^ 
 
SLMIVEYINC; 
 
 Th(n 
 
 Ac 45 
 
 ch (Vj 
 
 ■ 9( 
 270 
 
 ch 6ii 
 
 o.i 84 
 
 14( 
 cd 17;. 
 
 73( 
 1022 
 14-6 
 
 1i 84 
 k 7i> 
 
 154 
 
 ic 12( 
 
 ck 70 
 fl i)8 
 
 168 
 •f 170 
 
 fl 88 
 gm 5? 
 
 145 
 % 124 
 
 580 
 
 2i;o 
 
 105 
 
 I7.9S( 
 
 57f}( 
 
 1 8480 
 
 11760 
 16"8 
 
 2S.50"0 
 
 
 
 
 2655i 
 
 
 
 
 
 2790 
 25550 
 18480 
 28560 
 17980 
 22J48 
 
 gm 57 
 
 148 
 iB 151 
 
 US 
 740 
 14S 
 
 22348 
 
 a ) 1.15708 
 
 ac r p 
 
 •0/854 Content 2 12 
 4 
 
 2-31416 
 40 
 
 12*560"4O 
 
 4. S»metiraes fuch pieces as that above, are corriputcd 
 by finding a mean breadth, by ciividing the Uim of the 
 ofB,ets by the number of them, accounting that for on;; ot 
 riiem where the boundary meets tbe iiation line, as at A; 
 tl tn multinly the rength AB by that mean bieadth. 
 
 Thut: 
 
SURVEYING. 223 
 
 785 A B 
 6*6 mean breadth 
 
 4/10 ac r p 
 
 4710 Content 2 2 by this n^ethod, 
 — — which is 10 perches too little, 
 
 •51810 
 4 
 
 2*0/240 For this method is always erroreous, 
 40 except when the » flsets ftand ateijual 
 _.- — diftances from one another, 
 2*89600 
 
 5. But in larger pieces, and whole ellates, confifting 
 of many fields, it is the common pradioe to make a roii^h 
 plan of the whole, and from it compute the contents quite 
 independent of the meafures of the lines and angles that 
 were taken in furveying. For then new lines are dravvri 
 in the fields in the plan, Co as to divide, them into tr^ipe- 
 ziums and triangles, the bales and perpendiculars of which 
 are meafured on the plan by means > f thefcale from which 
 it was drawn, and fo multiplied together for the contents. 
 In this way, the work is very expeditioufly done, and fuf- 
 ^clently corre(ft ; for fuch dimenfions are t iken, as afford 
 the, mod eafy method cf calculation; and, among a num- 
 ber of parts, thus taken, and applied to a fcale, ir is 
 likely that fome cf the parts will be taken a fnjall matter 
 too little, and others too great; fo that fhey will, upon 
 the whole, in all probability, very nearly b:-'Iince one an- 
 other. After all the fields, and particuhir parts, are thus 
 computed fepara'ely, and added all together into one funi; 
 calculate the whole eilate independent c f the fields, by 
 dividing it into huge and arbitrary triangles and trape- 
 ziums, and add thefe all tc>gether. Then if this Turn he 
 equal to the former, or nearly fo, the work is n^hr; but 
 I- 2 if 
 
SH SUHV EYING. 
 
 the films have any confulcrable difference, it is wrong, and 
 hey mull be exdmined and recomputed till they nearly agree. 
 
 A fpccimen vf dividing into ore triangle, or one trape- 
 zium, which uill do for moll fingle fields, may be Cccn in 
 the examples to the laft problem ; and a fpecimen of divi- 
 ding a large trad into fcveral fuch trapeziums and triangles, 
 in prob. vi of chapter 1 1 of Surveying, page 193, where 
 a piece is fo divided, and its dimensions taken and fee 
 down; and again at prt)b. vi of Menfuration of Surfaces, 
 where the contents of the fame piece are computed. 
 
 6. But the chief fecret in computing, confifts in finding 
 the contents of pieces bounded by ctrved or very irregular 
 ines, or in reducing fuch crooked fides of fields or boun- 
 daries to flraight lines, that fliail inclofe the fame or equal 
 area with thofe crooked fides, and fo obtain the area of 
 the curved figure by means of the right lined one, which 
 will commonly be a trapezium. Now this reducing the 
 crooked fides to Itraight ones, is very eafily and accurately 
 performed, thus: Apply the ftiaight edge of a thin clear 
 piece of lanthorn horn to the crouked line, which is to be 
 reduced, in futh a manner, that the fmall parts cut off 
 from the crooked figure by it, may be equal to thofe which 
 are taken in; which equality cf the parts included and ex- 
 cluded you will prefenily be able to judge of very nicely 
 by a little pradice: then with a pencil, or point of a 
 tr^.ce^, draw a line by the flraight edge of the horn. Do 
 the fame by the other lides of the field or figure. 80 fhali 
 you have a Itraight fided figure equal to the curved one; 
 the content of which, being computed as before direded, 
 vi^ill be the content of the curved figure propofed. 
 
 Or, inllead of the flraight edge of the horn, a horfe- 
 liair line thread may be applied acrofs the crooked fides in 
 the fame manner; and the eafielt way of ufing the hair is 
 to firing a fmall flender bow with ir, either oi' wire, or 
 cane, cr whide-bone, or fuchlike flender elallic matter; 
 for, the bow keeping it always ilretched, it can be eafily 
 and neatly applied with one hand, while the other is at 
 
 liberty 
 
SURVEYING. 
 
 225 
 
 liberty to make two marks by the fije of it, to draw the 
 flraight line by. 
 
 i;X AMPLE. 
 
 Thu-;, let it be required to find the contents of the 
 fame figure as in prob. ix of the laft chapter, page l^S, 
 to a fcale of 4 chains to an inch. 
 
 D 
 
 C 
 
 Draw the tour dotted ftraight lines AB, BC, CD, DA, 
 cutting off equal quantities on both fides of them, which, 
 they do as near as i he eye can judge: fo is th^ crooked 
 figure reduced to an equivalent right lined one of four 
 fides A BCD. Then draw the diagonal BD, which, by 
 applying a proper fcale to it, meafures \Q56 Alfo" the 
 perpendicular or neareft dittance, from A to this diagboal, 
 reeafures 436; and the diftance of C from it, is 4)2«. 
 
 Then 
 
126 
 
 STRVEVINr, 
 
 Tien 456' 
 
 8S4 
 
 6024 
 10048 
 1004 8 
 
 2) n-10304 
 
 6'5ol53 
 
 4 
 
 2* 20608 
 40 
 — — ac r 
 
 8^24320 Content j 2 8 
 
 And thus the content of ihe trapezium, and confequent- 
 ly of the irregular figure, to which it is equal, is eafily 
 found to be 5 acrcF, 2 roods, 8 perches. 
 
 PROBLEM III, 
 
 To transfer a Flan to another "Paper y i^c* 
 
 After the rough plan is completed, and a fair one is 
 wanted; this may be done, either on paper or vellum, by 
 any of the following methods. 
 
 riRST METHOD. 
 
 Lay the rough plan over the clean paper, and keep them 
 always prefled flat and clofe together, by weights laid on 
 them. Then, with the point of a fine pin or pricker, 
 prick through all the corners of the plan to be copied. 
 Take them afunder, and conneifl the prick-ed points on the 
 clean paper, with lines; and it is done. This 
 
SURVEYING, 
 
 2^17 
 
 This method is only to be praiflifed in plans of fuch 
 figures as are fmall and tolerably regular, or bounded by 
 light lines. 
 
 SECOND METHOD. 
 
 Rub the back of the rough plan over with black lead 
 powder ; and lay tKe faid black part on the clean paper on 
 which the plan is to be copied, aud in the proper pofition* 
 Then with the blunt point of fome hard (ubftance, as 
 brafs or fuch like, trace over the lines of the whole plan; 
 prefTing the tracer fo much as that the black lead under 
 the lines raay be transferred »o the clean paper; after which 
 take off the rough plan, and trace over the leaden marks 
 with common ink, or with Indian ink. Sec. — Or, indead 
 of blacking the rough plan, you may keep conltaiiily % 
 blacked paper to lay between the plans. 
 
 THIRD METHOD. 
 
 Another method of copying plans, is by means cf 
 fquares. This is performed by dividing both ends and 
 fides of the plan which is ta be copied, into any conve- 
 nient number of equal parts, and conne6ling the corrcf- 
 ponding points of divilion with lines; which will divide 
 the plan into a number of fmall fquares. Then divide the 
 paper, on v.hich the plan is to be copied, into the fame 
 number cf fquares, each equal to the former when the pi. m 
 is to be copied of the fa ne fize, but greater or lefs than 
 the others, in the proportion in which the plan is to be 
 incrcafcd or diminilhed, when of a difR^rent fize. Laftly, 
 copy into the clean fquares the parts contained in the cor- 
 refponding fquares of the old plan; and )ou will have the 
 copy, either of the fame lize, or greater or lefs in any 
 proportion. 
 
 FOURTH METHOD. 
 
 A fourth method is by the inikument called a penta- 
 graph, which alfo copies the plan in any fize required. 
 
 L 4 FIFTH 
 
228 sukvevij:© 
 
 FIFTH METHOD, 
 
 But the neateft method of any is this. Procure a co\)y- 
 ing frame or glafs, made in this manner ; namely, a large 
 fquare of the bell window glafs, fet in a broad frame of 
 wood, M,hich can be raifcd up to any angle, when the 
 lower fide of it refls on a table. Set this frame up to any 
 angle before you, facing a ftrong light; fix the old plan 
 siui clean paper together with feveral pins quite around, 
 10 keep them together, the clean paper being laid upper- 
 moft, and over the face of the plan to be copied. Lay 
 them with the back of the old plan, over the glaf^, 
 namely, that part which you intend to begin at to copy 
 iirft; and, by means of the light fiiining through the 
 paper?, you uill very diftin^ly perceive every line of the 
 plan through the clean paper. In this (late then trace all 
 the lines on the paper with a pencil. Having drawn that 
 ) art which fucft covers the glafs. Aide another part over the 
 glnfs, and copy it in the fame manner: then another part. 
 And {o on, till. the whole be copied. 
 
 Then, take them afunder, and trace all the pencil-lines 
 over with a fine p?n and Indian ink, or with common ink. 
 
 And thus you may copy the iineft plan, without injuring 
 it in the h';<ft. 
 
 When the lines, &c. are copied on the clean paper or 
 vellum, the next buhnefs is to write in fuch names, re- 
 marks, or explanations as may be judged neceffary: lay- 
 ing down the fcale fi-r faking the lengths of any parts, a 
 flower-de-luce to point <>ut the diredlion, and the proper 
 title, ornamented with a compartment: and dluftrating or 
 colouring every part, in fuch manner as fhall feem moft 
 natural, fui h as flinding rivers or brooks with crooked 
 line?, drawing the reprefentation rf irees, bufhes, hills, 
 Avoods, hedges, houfc-s, gates, roads, &c, in their proper 
 places; running a fingle doited line for a foot path, and a 
 double one for a carriage road ; and either reprefenting 
 the bafes or the elevation of buildirgs, Sec, 
 
 CONIC 
 
CONIC SECTIONS 
 
 AND THEIR 
 
 SOLIDS, 
 
 DEFINITIONS. 
 
 1. /^ONIC Sedions are the plane figurei? formed by 
 
 ^ cutting a cone, "-. 
 
 According to the different pofitions of the cutting plane 
 there will arife five diiferent figures or fedlions. 
 
 2. If the cutting plane pafs 
 through the vertex, and any part 
 of the bafe, the fedion will be 
 a triangle. 
 
 3. If the cone be cut parallel 
 to the bafe, the feftion will be a 
 circle. 
 
 4. The fedion is called an ellipfis, 
 when the cone is cut obliquely through 
 both fides. 
 
 L 5 
 
 5. Thft 
 
!30 
 
 CONIC SECTIONSi 
 
 5. The fcftlon is a parabola, when 
 the cone is cut parallel to one of its 
 iidcs. 
 
 mu 
 
 6. The fedion is an hyperbola, 
 when the cutting plane meets the op- 
 pofite cone continued above the ver- 
 tex, where it will make another fed^ion 
 •r hyperbola, 
 
 7. Tiie vertices of any feftion, 
 are tht points where the cutting plane 
 meets the c^pofite fides of the cone, 
 
 8. The tranfverfe axis, is the line 
 between the two vertices. And the 
 middle point of the tranfverfe is the 
 centre of the conic fedion. 
 
 p. The conjugate axis, is a line drawn through the 
 centre, and perpendicular to the tranfverfe, 
 
 ICU An 
 
A>iD TllEIR S«^LIDS, 
 
 231 
 
 10. An ordinate, is a line perpendicular to the axis, 
 
 11. An abfcifs, is a part of the axis between the ordi- 
 nate and the vertex. 
 
 12. A fpheroid, orelHpfoid, is a folid 
 generated by the revolution of an ellipfe 
 about one of its axis. It is a prnlale one, 
 when the revolution is made about the 
 tranfvcrfe axisj and oblate, when about 
 the conjugate. 
 
 13. A conoid is a f;]id formed by the 
 revolution of a parabol.i, or hyperbola, 
 about the axis. And is accordingly called 
 parabolic, or hyperbolic. — The parabolic 
 conoid is alfo called a paraboloid; and the 
 hyperbolic conoid, an hypt-rholoid. 
 
 14. A fpindle is formed by any of the three feftions 
 revolving about a double ordinate, like the circular 
 fpindle. 
 
 15. A fegment of any of thefe figures, is a part cut off 
 at th« top, by a plane parallel to ihe bafe. 
 
 16. And a frultum is the part left next the bafe, after 
 the fegment is cut off. 
 
 PROBLEM I, 
 
 To de/crile an Elhpfe, 
 
 LetTRbe the tranf- 
 vcrfe, CO the conjugate, 
 and c the centre. With 
 the radius Tc and centre 
 C, defcribe an arc cut. 
 tin'g'TR in the points F, 
 /; which are called ihe 
 tvvo foci of the ellipfe. 
 
 lG 
 
 Affume 
 
232 COKIC SECTIONS 
 
 Affume any peint P in th«; tranfverfe ; then with the 
 radii PT, PR, and centres F, /, defcribe two arcs intcr- 
 fc(f\ing in Ij which will be a point in the curve of the 
 cUipfe. 
 
 And thus, by affuming a number of points P in the 
 tranfverfe, there will be found as many points in the curve 
 as you pleafe. Then, with a Heady hand, draw the curve 
 through all thcfe points. 
 
 Othefwife nuith a Thread, 
 
 Take a thread of the length of the tranfverfe TR, and 
 faften its ends with two pins in the foci K, f» Then 
 Ihetch the thread, and it will reach to I in the curve : and 
 by m -vjjig a pencil round, within the thread, keeping it 
 always Hi etched, it will trace out the ellipfc. 
 
 PROBLEM II. 
 
 In an Ellipfe, to find the Tranfiverfct or Conjugate^ or Or-^ 
 dinatet or Abfcifs : halving the other three given, 
 
 CASE 1, 
 
 To find the Ordinate » 
 
 As the tranfverfe 
 
 Is to the conjugate ^ : 
 
 So is the mean proportional between the two abfciffes 
 To the ordinate^ 
 
 EXAMPLES. 
 
 ]. Tn the ellifpe ADBC, the tranfverfe AB is 70, tlic 
 conjugate CD is 50, and the abfciffes AP U, and BP 56; 
 what is the ordinate PQJ 
 
 Firft 
 
AND THEIR SOLIDS. 
 
 2P.S 
 
 Firft 
 
 ft 
 
 56=1 
 
 PB 
 
 
 
 14 = 
 
 A-P 
 
 
 
 
 
 
 
 224 
 
 
 
 
 56 
 
 
 
 
 
 
 
 
 784 ( 28 mean 
 
 
 4 bel 
 
 ^ween 
 
 56 
 
 
 
 and 
 
 14 
 
 48 
 
 384 
 
 384 
 
 
 
 Then 70 : 50 : : 28 : 20 =: PQjhe ordinate. 
 
 Ex. 2. If tht tnnfverfe be 80, the conjugate 60, and 
 an abfcifs l6 ; required the ordinate? Anf, 24» 
 
 CASE II, 
 
 To^nd the Abfcifs, 
 
 From the fqnare of half the conjugate, take the fquare 
 of the ordinate; and extrad the fquare root of the re- 
 mainder. Then {^jy 
 
 As the conjugate 
 
 Is to the tranfverfe 
 
 So is that fquare root 
 
 To half the difference of the ahfciffes. 
 
 Then add this half difR^rence to half the tranfverfe, for 
 the greater abfcifs; and fubtradl it, for' the lefs abfeifs. 
 
 EXAMPLES, 
 
 1. The tranfverfe AB is 70, the conjugate CD is 5(^ 
 and the ordinate PO is 20 ; required the ahfciffes AP and 
 PB? . 
 
 Firft 
 
SJi- CONIC SECTIONS 
 
 Firft 25 Then 
 
 25 > 35 half AB 
 
 — As 50 : 70 : : 15 : 21 half dif. 
 
 125 — 
 
 50 56 = PB 
 
 < 14 = AP 
 
 625 =r CO* 
 400 = PQ^ 
 
 225 ( 15 
 1 
 
 25 1 125 
 
 125 
 
 Ex 2. V>^hat are the two abfcifles to the ordinate 24^ 
 the axes being SO and 60 ? Anf, l6" and 6-i. 
 
 CASE iir. 
 
 ^tfind the Conjugate, 
 
 As the mean proportional between the abfcifles 
 Is to the ordinate 
 So is the tianiverfe 
 To the eorjjgate. 
 
 "Nof. In the fame manner, the tranfverfe may be found 
 from the conjugate; nfmg here th^' abf« iUes of the conju- 
 gate, and their ordinate perpendiculdjr to the conj jgate, 
 
 EXAMPLES. 
 
 1, The tranfveife being 180, the ordinate \6\ and the 
 greater abfcifs 144; required the conjugate } 
 
 ISOtranf- 
 
AND THEIU SOLIDS, 23a 
 
 180 tranfvrrfe 
 144' greafor abf, 
 36' lefs abf. 
 
 432 
 
 5184< ( 72 : 16 : ; ISO : 40 the conjugate. 
 49 16' 
 
 142 I 284 1080 
 
 I 284 IS 
 
 72 ) 2830(40 
 
 288 
 
 Ex. 2. The tranfverfe being 70, the ordinate 20, and 
 abfcifs 14] what is the conjugate? Anf. 5O9 
 
 CASE IV. 
 
 To find the Tranfverfe, 
 
 From the fquare of half the conjugate, fubtradl the 
 fquare of the ordinate ; and extradl the root of the re- 
 mainder. Next add ihis root to the half conjugate, if 
 the lefs abfcifs be given, but fubtraft it when the greater 
 abfcifs is given, referving the fum or difference. 
 
 Then fay. 
 
 As the fquare of the ordinate 
 
 Is to the redlangle of the abfcifs and conjugate 
 
 So is the referved fum or diiTcrence 
 
 To the tranfverfe. 
 
 EXAMPLES. 
 
 1. If the conjugate be 50, the ordinate 20, and the 
 lefs abfcifs 14 ; what is the tranfverfe ? 
 
 Firft 
 
iki6 
 
 
 CONIC 
 
 SECTIOyS 
 
 
 Firft 
 25 
 25 
 
 Then 
 20 14 
 20 50 
 
 
 125 
 50 
 
 400 
 
 : 700 : : 40 : 
 
 70 the tranf. 
 
 625 
 
 400 
 
 
 
 
 225 ( 
 
 1 
 
 15 
 
 Q5 
 
 
 
 25 1 125 
 
 1 125 
 
 40 
 
 
 
 Ex. 2. The conjugate being 40, the ordinate l6, and 
 the lefs abfcifs 36; required the tranfverfe? Anf, ISa. 
 
 PROBLEM III. 
 
 To find the Circumference of an Elliffe, 
 
 Add the two axes together, and muhiply the fum by 
 1*5708, for the circumference nearly. 
 
 EXAMPLES. 
 
 1. Required the circumference of the ellipfe whofe two 
 axes are 70 and 50 ? 
 70 
 50 
 
 120 fum. 
 1-5708 
 
 188 '4960 clrcumf, nearly. 
 
 wmmmmmmmmm 
 
 Ex.2. 
 
AND THEIR SOLIDS. 237 
 
 Ex. 2. What is the periphery of an ellipfe whofe two 
 axes are 2i and 20? Anf. 6"9»il52. 
 
 PROBLEM IV. 
 
 ^0 find the Area of an Ellipfe. 
 
 Multiply the tranfverfe by the conjugate; then that 
 produft multiplied by vSS^, will be the area. 
 
 Or multiply -/SS^ firit by the one axe, and the produft 
 again by the other. 
 
 EXAMPLES. 
 
 1 . To find the area of the ellipfe whofe two axes are 
 70 and 50. 
 
 •7854 
 50 
 
 39'2700 
 70 
 
 2748-90G0 anf. 
 
 Ex. 2. What is the area of the ellipfe whofe two axes 
 are 24 and 18? Anf. 33(?-2928. 
 
 PROBLEM V. 
 
 To find the Area of an Elliptic Segment, 
 
 Divide the height of the fegment by that axis of the 
 ellipfe of which it is a part; and find, in the table of 
 circular fegments at the ead of the boo'<, a cir "ular feg- 
 ment having the fame verfed fine as this quotient. The« 
 multiply continually together, this ferment, and the two 
 axes, for the area required, 
 
 EX- 
 
538 
 
 COMC SECTIONS 
 
 EXAMPLES. 
 
 1. What is the area of an elliptic Tei^ment RAQ, whofe 
 height AP is 20; the tranfverfe AB being 70, and ihc 
 coDJugaie CD 50? 
 
 70)20( -2831 the tab. verf. 
 The correfpond. fcg, 
 is 'ISolOO" 
 
 70 ' 
 
 12 ' 90 \ 6-20 
 50 
 
 6AS-081000 
 
 Ex. 2. What is the area of an elliptic fegment, cut ofF 
 parallel to the fhorter axis, the height being 10, and the 
 axes 25 and 35 ? Anf. 162-0210. 
 
 Ex. 3. What is the area of the elliptic fegment, cut off 
 pnralicl to the longer axis, the height being 5, and the 
 axes 25 and 35 ? Anf. 97-8458. 
 
 PROBLEM VI. 
 
 To De/cribe or Covjlru^l a Parabola^ 
 
 VP being an abfcif«, and PQ^ 
 its given oidinate; bikft PQjn 
 Ay join AV^ .and draw AP pcr- 
 pendicuhir to it; then transfer 
 FB to VF and VC in the axis 
 produced. So Ibali F be wh 
 IS called the focus. 
 
 Draw feveral double ordinafcs 
 SRS, &'C. perpendieiiiar to VP. 
 .Then with the radii CR, &c. 
 and the centre F, dcfcribe arcs 
 cutting the correfponding ordi- 
 nates in the points 8, &e. Then 
 draw the curve through all the 
 points S, <ic. jao- 
 
AND THEIR SOLIDS. 
 
 239 
 
 PROBLEM VII. 
 
 To fi^id any Varabolk Ahfcifs or Ordinate, 
 
 The abrdfles are to each other as the fquares of their 
 ordinatcs; ihat is. 
 
 As any abfcifs is to the fquare of its ordinate, 
 So is any other abfcifs, to the fquare of its ordinate. 
 Or as the fquare root of any abfcifs, is to its ordinate. 
 So is tlie fquare root of another abfcifs, to its ordinate. 
 
 EXAMPLES. 
 
 ^ . The abfcifs VB is p, and its 
 ordinate AB is 6"; required the or- 
 dinate DE whofe abfcifs VE is 1^. 
 
 Here ^9 is 3, and y^\6 is 4. 
 
 Then 3 : 6 : : 4 ; S = D£ 
 required. 
 
 Or if the ordinate DE were giv- 
 en rz 8, to find its abfcifs VE. 
 Tlien 6"^ = 3(5, and 8* = 64. 
 Hence 36' : 64 : : 9 : 1^ = VE 
 required, 
 
 Ex. C. If an abfcifs be 8, and its ordinate 10; re- 
 quired the ordinate whofe abfcifs is 18? Anf. 15, 
 
 Ex. 3. If an abfcifs be IS, and its ordinate 18; what 
 is the abfcifs whofe ordinate is 10 ? Anf. 8. 
 
 PROBLEM viir. 
 
 To find the Length of a Parabolic Curnje* 
 
 To the fquare of the ordinate, add f of the fquare of 
 the abfcifs ; extraft the fquare root of the fum, and 
 double it for the length of the curve, cut off by the 
 double ordinate, nearly, 
 
 EX' 
 
340 
 
 CONIC SECTIONS. 
 
 EXAMPLES. 
 
 ] . The ahfclfs VB being 2, and the ordinate AB 6 ; 
 required the length oi the curve AVC ? 
 
 2 = VB 
 2 
 
 4 = VB* 
 4 
 
 3)10' 
 
 5'33l]3 
 36' = AB* 
 
 41*3333 (64*291 root 
 36 2 
 
 124 
 1 4- 
 
 533 
 49? 
 
 1282 
 
 
 3733 
 2564 
 
 12-8582 = arc AVC nearl;'. 
 
 1284) 1169(91 
 1156 
 
 13 
 
 Ex, 2. What is the length of the parabolic curve, 
 whofe abfcifs is 3, and ordinate 8 ? Anf. 17'4356. 
 
 PROBLEM IX. 
 
 To find the Area of a Parabola^ 
 
 Multiply the bafe by the height, and take f of the pro- 
 du^ for the area. 
 
 IX. 
 
AND THEIR SOLIDS. 241 
 
 EXAMPLES. 
 
 1. Required the area of the pa;aboIa AVCA, the ab- 
 fcifs VB being 2, and the ordinate AB 6? 
 12 
 
 2 
 
 2-i 
 2 
 
 3) 48 
 lO" anf. 
 
 Ex. 2. What is the area of a parabola whofe abfcifs is 
 10, and ordinate 8 ? Anf. loGf . 
 
 PROBLEM X. 
 
 'To find the Area of a Parabolic Frufium, 
 
 Cube each end of the fruftiim, a: d fabtrnft the one 
 cube from the other; tnen mulriply that difFtreuce by 
 double the altitude, and divide the produd by tripie the 
 diiFerence of their fquares, for the area. 
 
 EXAMPLES. 
 
 1. Required the area of the parabolic fruftum AQFD, 
 AC being 6", DF 10, and the altitude BE 4. 
 Ends Sqrs. Cubes 
 
 10 100 1000 
 
 6' 35 2l6 
 
 64 dif. 784 
 3 8 
 
 1^2 ) ()272 ( 32||| = 32| anf. 
 576 
 
 312 
 384 
 
 128 
 Ex. 2. 
 
C42 
 
 tUMC SECTIONS 
 
 Ex. Q, What is the area of the parabolic fruftum, whofe 
 tw» ends are 6 and 10, and its altitude 3 ? Anf. 2A-^, 
 
 PROBLEM XI. 
 
 To Conjlruil or Dejcrihe an Hjperhcla, 
 
 Let D be the cen- 
 tre of the hyperbola, 
 or the middle of ihe 
 tranfv'crfe AB; and 
 BC perpendicular to 
 AB, and equal to half 
 the conjugate. 
 
 With centre D, and 
 radius DC, dcicribe 
 an arc, meeting AB 
 produced in F and f, 
 uhich are the two 
 focus points of the 
 hyperbola. 
 
 Then affuming feveral points E in the tranfverfe pro- 
 duced, with the radii AE, BE, and centres f, F, defcribe 
 arcs inttrftding in the feveral points G ; through all 
 which points draw the hyperbolic curve. 
 
 PROBLEM XII. 
 
 Jn an Hjperhcia to find the Tran/'verjcy or Cbh jugate y a 
 Ordifiatf, or Abfcijs, 
 
 CASE I 
 
 To find the Ordinate, 
 
 As the tranfverfe 
 
 Is to the conjugate 
 
 So is the mean proper, between the two abfcifles 
 
 To the ordinate. 
 
 Nctf. 
 
AN» THEia SOI.IDS. 
 
 21-3 
 
 Note. In the hyperbpla, the lefs abfcifs added to the 
 axis, gives the greater abfcifs. 
 
 EXAMPLES. 
 
 1. If the tranfverfe be 24-, the conjugate 21, and the 
 lefs abfcifs VB 8 ; what is the ordinate AB ? 
 
 2.4 tranf. 
 8 lefs abf. 
 
 32 greater abf. 
 8 
 
 256 ( \6 mean proper. 
 1 
 
 26 I 1 56 
 156 
 
 Then 24 : 21 : : l6 : 14 rr AR required, 
 
 Ex. 2. The tranfverfe being 6o, the conjugate 36, and 
 the lefs abfcifs 20, required the ordinate? Anf. 24. 
 
 CASE II. 
 
 To pnd the Ah/cifs, 
 
 To the fquare of half the conjugate, add the fquare of 
 the ordinate ; and extrad the fquare root of the futn. 
 Then fay, 
 
 As the conjugate 
 
 Is to the tranfverfe 
 
 So is that fquare root 
 
 To half the fum of the abfciffcF. 
 
 Then, to this fum, add half the tranfverfe, for the 
 greater abfcifs i and fubtrad it, for the lefs abfcifs. 
 
 EX- 
 
2U 
 
 COMC SECTIONS 
 EXAMPLES. 
 
 1. The tranfverfe being 24<, and ihe conjugate 21 j 
 required the two ablcifles to the ordinate AB 14 f 
 
 Firft 10'5 = 4:Conj. 14 ord. 
 10-5 14 
 
 ■ 
 
 525 56 
 1050 14 
 
 
 110-25 
 
 196- 196 
 
 
 306'25 ( 17*5 the fquare root. 
 
 1 
 
 Then 
 
 20 half fura 
 12 half tranf. 
 
 27 
 7 
 
 205 8 
 
 
 32 greater abf. 
 8 lefs abfcifs. 
 
 34-5 
 5 
 
 1725 20 
 
 
 
 Ex. 2. The tranfverfe being 60, the conjugate 06 ; re- 
 quired the two abciffes to the ordinate 24 ? 
 
 Anf. 80 and '2'\ 
 
 CASE III. 
 
 ^0 find the Conjugate, 
 
 As the mean proportion between the abfcifles 
 Is to the ordinate 
 So is the tranfverfe 
 To the conjugate. 
 
 EX- 
 
AifB THEIR SOLIDS. 
 
 245 
 
 EXAMPLES. 
 
 1. The tranfverfe being'24, the IcA abfcifs VB S, and 
 Us ordinate AB 14, what is the conjugdt- ? 
 
 Firft 
 
 24. 
 8 
 
 R R A /t y 
 
 ah run 
 
 32 Then 
 
 8 As 16 : 1-i : : 24 t 21 
 
 236 ( l5 the mean, — . 
 
 1 8) l68 
 
 21 
 
 26 I 156 
 
 6 io6 
 
 
 Ex, 2. What is the conjugnte to the hyperbola, whofe 
 tranfverfe is 60, and ordinate 24, and the Ids abfcifs 2u? 
 
 Anf. 36. 
 
 CASE IV, 
 To find the Tranjver/c, 
 
 To the fiuars of half the conjugate acid the fquare of 
 the ordina e, an i cxtraft the fquare root of the fum. 
 
 Next, to (his root aid the half conjtiga'e when in^- lefs 
 ahfcifs is ufed, but fobtract ir when the grearer a jfcifs is 
 ufed ; refcrving the fum or difference. Then fa) , 
 
 As the fquare of the ord nate 
 
 Is to the produft of he ahfcifs and conjugate 
 
 So is the referved fum or difference 
 
 To the ti anfverfe. 
 
 EXAMPLES. 
 
 1, The lefs abfcifs VB being 8, and its ordinate AB 
 \\\ required the tranfverfe tj ihe cuajuLaie 21? 
 
 M Firft 
 
246 
 
 ^ 
 
 rONIC SECTIONS 
 
 Firft lO'/X 
 
 U 
 
 
 10-5 
 
 U 
 
 
 
 _^ 
 
 
 525 
 
 50 
 
 
 
 105 
 
 14. 
 
 19^ 
 
 
 11 ••25 
 196- 
 
 
 
 
 
 
 306'25 
 
 1 
 
 ( 17--5 
 10-5 
 
 28-0 • 
 
 As 196 : 
 Or 7: 
 
 
 'i\ 
 
 206 
 i89 
 
 Then 
 
 345 
 5 
 
 1725 
 1725 
 
 168 : : ^8 : 
 6 : : 28 : 24 Anf. 
 
 Ex. 2. What is the tranfverfe of the hyperbola, whofe 
 conjugate is 36; the lefs abfcifs being 20, and its ordi- 
 Bate24? Anf. C'O. 
 
 PROBLEM XIII. 
 
 ^ojitid the Length of an Hyperbolic ciir've, 
 
 1. To 21 times the fquare of the conjugate, add 9 
 tiroes the fqaare of the tranfverfe; and to the fame 21 
 times the fquare of the conjugate, add \9 times the fquare 
 of the tranfverfe ; and nDultiply each fum by the abfcifs. 
 
 2. To each of thefe two products add 15 times the 
 produft of the tranfverfe and fquare of the conjugate, 
 
 3. Then as the lefs fum is to the greater, fo is the 
 double ordinate, to the length of the curve nearly, 
 
 EXAMP „E8, 
 
 1. Required the length of the curve AVG to the 
 abfcifs VB 20 and ordinate AB 24 j the two axes being 
 60 and 36" 
 
 36 
 
A^D THEIR SOLIDS. 
 
 24/ 
 
 36 
 36 
 
 216 
 108 
 
 27216 
 32400 
 
 59616 
 20 
 
 1192320 
 1160400 
 
 27216 
 68400 
 
 95616 
 20 
 
 1296 
 60 
 
 77760 
 15 
 
 fq. conju. 1296 
 21 
 
 1912320 
 1 166400 
 
 388800 
 77760 
 
 1296 
 2592 
 
 2358720 
 
 3078720 
 
 1166400 
 
 27216 
 
 Then 2358720 : 3078720 : : 48 : 62*6520 the whole curre 
 48 
 
 2462976 
 
 1231488 
 
 ( 62*6520 Anf, 
 
 a/ 
 
 7 
 
 ft . 
 
 
 2358772 ) 14777856 
 1415232 
 
 
 62553 
 47174 ' 
 
 
 iy^79 
 14152 
 
 \ 
 
 
 v 
 
 1227 
 1179 
 
 48 
 47 
 
 ^ 
 
 Ji 
 
 I 
 
 m2 
 
 Ex. $, 
 
949 
 
 CO VIC SECTIONS 
 
 Fx. 2. What is the length of the whole curve to the 
 ordinate 10, the tranfverfe and conjugate axes bein^ 80 
 
 and 00? 
 
 Anf. 20-60U 
 
 PROBLEM Xrv. 
 
 ^0 find the Area of an Hyperbola, 
 
 •.1. To ^ of the abfcirs add the tranfverfe: multiply the 
 fun\ by tl.e abfcifs; and extract ihc fquare root of the 
 product, 
 
 2. Muhijly the tranfverfe by the abfcifs, and cxtrafl 
 the rot. t of that produdl alfo. 
 
 3. To 21 times the firll root, add 4 tintes the fecond 
 root ; mi 1 iply the fum by douKle the produdl of the con- 
 jugate and abfiifsj then divide by 75 times the tranfverfe, 
 for the ait a nearly, 
 
 EXAMPLES, 
 
 ; 
 
 1. Required the area of the hyperbcla AVCA, whofe 
 abfcifs VB is 10, the tranfverf<f and conjugate being ;3G 
 and IS? 
 
 10 
 
AXD THEIR SOLIDS. 
 
 n$ 
 
 10 
 5 
 
 7 ) 50 
 
 7'14285ri 
 
 30* 
 
 37*1428571 
 10 
 
 SO 
 10 
 
 300 ( ir*3205081 
 1 4. 
 
 97 200 69'2820324. 
 189 
 
 — . 343 
 
 571-428571 ( 15*2724822 3 
 
 1100 
 1029 
 
 1^1 — 
 
 29 I 271 
 9 I 261 
 
 3462 
 
 192724822 2 
 38544j;644 
 
 7100 
 ^924 
 
 382 
 2 
 
 3847 
 7 
 
 1042 
 764 
 
 27885 
 26929 
 
 404722 126'2 
 6>2820524 
 
 474*004 1586 
 
 34040 I 17600 ( 5081 
 17320 
 
 280 
 277 
 
 38542 
 2 
 
 95671 
 
 77084 
 
 58544 ) 18587 ( 4822 
 •••• 15418 
 
 3169 
 3083 
 
 86 
 77 
 
 MS 
 
 IS 
 
J^O CONIC SECTIONS 
 
 474-0041 586 
 720 
 
 J 8 75 .94800831720 
 
 40 30 33180291102 
 
 720 2250 3412829941920 
 4 4 
 
 90C0 ( 1365-131 •97()7 680 
 
 151»6S133 area required. 
 
 Ex. 2. What is the area of the hyperbola to the abfcifs 
 is, the two 3%CB being 50 and 30? 
 
 Anf. 805*090868. 
 
 FROBLEM XV, 
 
 ^0 find the Solidity of a Spheroid, 
 
 Square the revolving axis, multiply that fquare by th« 
 fixed axis, and multiply the produft by '5236 for the con- 
 tent. 
 
 EXAMPLES. 
 
 1 . Itequired the fclidity of the prolate fphcroid ACBD, 
 whof^ axes are AB 50 and CD 30? 
 
 30 -5236 
 
 30 • 45(;00 
 
 900 26180000 
 50 20944 
 
 45000 C3562'0000 Anf. 
 
 Ex. 2. 
 
AND IHI-IR SOLIDS. 
 
 -251 
 
 Ex, 2. Wha* is the content of an oblate fpheroic^, whofe 
 axes are 50 and 30 ? Anf. 39270. 
 
 Ex. 3. What is the folidity of a prolate fpheroid, whofe 
 axes are ^ and 7 ? Anf. 230*S076'. 
 
 PROBLEM XVI. 
 
 ^0 find the So tidily of a Segment of a S^h^r^iid* 
 CASE I. 
 
 When the Bafe is Circular, or Parallel to the Revol'vivg 
 Axis, 
 
 From triple the fixed axe, take double the height of the 
 fegment ; raukiply that difftrence by the fquare of the 
 heig'u, and ihe produft again by 'b'iZ^. 
 
 'I'hen as the fquare of the fixed axe is to the fquare of 
 the revolving axe. fo is the latt proJud to the couient of 
 the fegment. 
 
 EXAMPLES. 
 
 1, Required the content of the fegment of a prolate 
 fpheroid, the height AG being 5, the fixed axe AB 50, 
 and the revolving axe CD 30 ? 
 
 150 
 
 •523(S 
 
 10 
 
 3500 
 
 140 
 
 261S000 
 
 25 
 
 15/08 
 
 700 
 
 1832-()000 
 
 28 
 
 
 3.500 
 
 Ci- 
 
 M 4 
 
 Thca 
 
tSi €«VXC fZCTieMf 
 
 Then as 25 : .9 : : 1 832-6 : 
 Or as 100 ; 36 : : 1832-6 : 659736 
 36 
 
 10.9956 
 5497 » 
 
 100 ) 65SI73-6 ( 659736 Anfwer. 
 
 Ex. 2. If the axes of a prolate fpheroid be 10 and 6, 
 re(]uired the content of the fcgment uhofe height is 1, and 
 its bafe parallel to the revolving axe ? Anf. 5*277888. 
 
 Ex. 3. The axes of an oblate fpheroid being 50 and 
 3Vy vvhat is ihe content ot the fegmmt, the height being 
 6, and its bafe parallel to the revolving axe? 
 
 Anf. 4084'0r. 
 
 CASB II. 
 
 fp^hft ibt Ba/e it EUipticah or "Perpendicular to the Reroh 
 ving Axe, 
 
 From tf'p^e the revolving axe, fake double the height 
 of tlie f<gment ; multiply that difference by the fquare of 
 the heighr, and the produft again by •5236, 
 
 Then as the revolving axe, is to the fixed axe. 
 So is the lafl produd, to the content. 
 
 EXAMPLES. 
 
 1. In the prolate fpheroid ACBD, the fixed axe AB is 
 50, the revolving axe CD 30; required the folidiiy of 
 the fegment CEF^ its height CG being 6 ^ 
 
 90 
 
AND THEIR SOLIDS. 253 
 
 90 •5236 
 
 12 2808 
 
 78 4I88S 
 
 S6 41888 
 
 — — 10472 
 
 468 
 
 234 1470-2688 
 
 2808 
 
 "if" 
 
 Then as SO : 50 : : 1470*2688 : 2450»44S 
 
 5 
 
 3 ) 7351*3440 
 
 2450-4180 Anf^ver 
 
 Ex. 2. In an oblate fpheroid, whofe axes are 50 and 
 30, required the content of the fegment whofe height it 
 5, its bafe being perpendicular to the revolving axe ? 
 
 Anf. lO90'56. 
 
 PROBLEM XVII. 
 
 To/»J the Content of the Middle Frujium of a Spheroids 
 
 CASE I. 
 
 When the Ends are Circular y or Parallel to the Revolving 
 Axe, 
 
 To double the f.juare of the middle diameter, add the 
 fquare of the diameter of one end ; multiply this fum by 
 the length of the frullum, and the produft again by 
 261 8 for the content, 
 
 M 5 EX- 
 
254 CONIC SECTIONS 
 
 EXAMPLES. 
 
 1. Reqnired the folidity of the middle fniflum EGHF 
 cf a fpheroid, the greateft diameter CD being 30, the 
 diameter of each end EF or GH 18, and the length AB 
 40. "" 
 
 18 30 
 
 18 30 
 
 J4-t 900 
 
 18 2 
 
 324 J 800 
 
 324 
 
 2124 
 40 
 
 84960 
 •2618 
 
 679680 
 84§6 
 50976 
 16992 
 
 22242-5280 Anfwer. 
 
 Ex. 2. What is the folidity of the middle frudum of 
 an oblate fpheroid, having the diameter of each circular 
 cod 40, the middle 50, and the length 18 ? 
 
 Anr31101*84. 
 
 CASE 
 
AND THEIR SOLIDS. 255 
 
 CASE II, 
 
 When the Ends are Elliptical^ or Perpendicular to the Re- 
 'volving Axe. 
 
 To double the produft of the tranfverfe and conjugate 
 diameters of the middle feftion, add the produ6l of the 
 tranfverfe and conjugate of one end; multiply the fuin 
 by the length of the fruftum, and the produft again by 
 •2618 for the content. 
 
 EXAMPLES. 
 
 1. In the middle fruftum EFHG of an oblate fpheroid 
 the diameters of the middle or greateft elliptic feftion AB 
 are 50 and 30, and of ore end P.F or GH 40 and 24-; 
 required the content, the height IK being 9? 
 
 24 50 
 40 30 
 
 960 ] 500 
 
 3000 
 960 
 
 39^0 
 -9 
 
 35640 
 •L'6l8 
 
 2851 JO 
 
 3504 
 21384 
 7128 
 
 9330-5520 Anf. 
 : m6 Ex.2. 
 
236 
 
 C051C SECTIOKS 
 
 Fx. 2. In the middle fruftum of an ohiate fphcroid, 
 the axes of the middle ellipfe are 50 and 30, and thofe of 
 each end arc 30 and 18; required the content, the height 
 tcing40? Anf. 37070*88. 
 
 PKOBLEM XYIII. 
 
 ^0 find the Solijilty of an Elliptic Spindle^ 
 
 RULE I. 
 
 1. Take the difference between 3 times the fquare of 
 the middle or ^reareft diameter, and 4 rimes the fquare of 
 the diameter at ^ of the lenpjth, or equally dilhnt between 
 the middle a-^d one end ; alfo take the difference her ween 
 3 times the grr aeft diameter, and 4 limes the faid middle 
 diameter. Then the former difference divided by «he latter, 
 will be quadruple the central diftance, or diftance between 
 the ceniie cf the fpindle and centre of the generating 
 ellipfe, 
 
 2. Then find the axes of the ellipfe by Problem ir, 
 and the area of the fegment which generated the fpindle 
 by problem v. 
 
 3. Divide 3 times that area by the length of the fpin- 
 dle ; from the quotient fubtradt the greateft diameter; 
 and nniltiply the remainder by 4 times the central dif- 
 tance, before found. 
 
 4. 6uhtra<^ this produdl from the fquare of the greateft 
 diame er ; and multiply the remainder by the length of 
 the fpindle, and again by '6236", fjr the folidity, 
 
 EXAMPLES. 
 
 1. Required the folidity of the elliptic fpindle ACBDA, 
 the length AB being 40, the greateft diameter CO 12, 
 and the diameter EF, at ^ of the length, 9-4^546? 
 
 2 1. For 
 
AND THIIU SOLIDS. 257 
 
 1 . For the Central Dijiance, and 4xes of the Ellip/e^ 
 
 y. c 
 
 4EF 
 3CD 
 
 dif. 
 
 37-98184 
 S6"'00()00 
 
 1-98 184 
 
 3 CD* 432'()O0O 
 4EF^ 360-0546 
 
 1-98184 ) 71-3454 ( Z6 — 40G 
 
 69-4582 9 = OG 
 
 . — 6 = CG 
 
 118872 — 
 
 118916 15 =: OC 
 
 »— SO =: CHtheconj. 
 
 24 = GH 
 
 6~ CG 
 
 144 
 its root 12 = mean between CG&GH. 
 Then as 12 : 20 (or AG) : : 30 (or CH) : 50 ^ IK 
 
 the tranfverfe, 
 
 2. For the Generating Elliptic Segment, 
 
 CH 30 ) 6 CG 
 
 •2 tab. verf, 
 •111823 rah, area correfp, 
 50 IK 
 
 5-591150 
 
 • 30CH 
 
 167-734500 area generating kg. ACBA. 
 
 3. For 
 
i5S 
 
 ?. 
 
 For the Solidify of the tpinile, 
 1677345 
 . 3 
 
 AB 40 ) 503-2035 
 
 12-'5800875 
 CD 12 
 
 OG 
 
 0-5800875 
 36 
 
 
 34805260 
 17402625 
 
 prod, 
 front 
 
 rem, 
 AB 
 
 20-8831500 
 144- 
 
 123* 11 685 
 40 
 
 
 4924-67400 
 •5236 
 
 29548044 
 14774022 
 9849348 
 24623370 
 
 257 8 '5593064 Anfvver. 
 
 folidiry 
 
 Ex. 2. Required tlie folldity of the elliptic fpindle, 
 Dvhofe length is 20, the greateft diameter 6, and the dia- 
 meter at \ of the kngth 4*74773? Anf. 322 32. 
 
 RULE II. 
 
 To the fquare of the greateft dia:neter,, add the fquart 
 pf double the diameter at ^ cf the '.ngth ; multiply the 
 fum by the length, and the produdl agciin by '1309 for 
 ijic folidiiy, very nearly, l^ot€» 
 
ANB THETll ?OLTDS, 259 
 
 Note, This rule will aKo ferve for any other folid for- 
 med by the revolution of any conic fedion. 
 
 EXAMPLE. 
 
 What is the folid content of the elliptic fpindle, whofe 
 length is 20, the greateft diameter 6, and the diameter at 
 iof the length 4 •74-773? 
 47-t773 
 2 
 
 5*49546 double the diam. 
 645949 ditto inverted. 
 
 8545914 
 
 379818 
 
 85459 
 
 4748 
 
 380 
 
 56 
 
 90-16375 fq. of double diam. 
 36-00000 fq. of oiher diam. 
 
 126-16375 fum 
 
 20 length 
 
 2523*27500 
 
 9031 or 1309 inrerted 
 
 2523 
 
 22 
 Anf. 330*2 the fo!idity nearly, 
 
 PROBLEM XIX. 
 
 To find the Sdidity of a Fmfium or Segment of an Elliptic 
 Spindle, 
 Proceed as in the laft rule, for this, or any other folid 
 
 formed 
 
f60 
 
 COVIC mCTION* 
 
 formed by the revolution of a conic fe^ion about an axis 
 namely. 
 
 Avid together the ff|uare8 of the grcated and leaft dia- 
 zneters, and ihc fquaie vi d( nble t!ie diameter in the 
 iniddle berwe<n the two; multiplv thefiirnby the length, 
 and the produd ngnin by •i3()f) f -r the T lidity. 
 
 Nuif. ^ or ;ill fuch folid*, this rule is exaO when the body 
 is formed from the conic feftion, or a part of it, revolved 
 about the axi-^ of rhe fcdion. And will always he very near 
 the truth when the figure rt-vohes about anorher line. 
 
 EXAMPLES. 
 
 1. Required the content of the middle fruflum EGHF 
 of anv fpindle, the length AB being 40. the greateft or 
 iniddle diamttrr CD 32, the leaft or diameter at either 
 end EF or GH 24, and the diameter IK, in the middle 
 between EK and CD, 30* 157568? 
 
 32 30' 1 57568 
 
 32 2 
 
 "^ 
 
 1024 
 
 24. 
 *J4 
 
 96 
 48 
 
 576 
 
 00 315136 double 
 51306 invert 
 
 36i8yo^ 
 
 J 809 
 60 
 30 
 3637-89fq. of 2lK 
 10:4*00 fq. of CD 
 576-00 fq. of EK 
 
 5237-89 fum 
 40 length 
 
 SCHiTII^O 
 c)03 1 inverted 
 
 2093 1 
 
 6285 
 
 188 
 
 a7424 Anfwer, 
 
 Ex. 2. 
 
AKD THEIR SOLIDS. S^l 
 
 Ex. 2. What is the content of the fegment of any 
 fpindle, the length being 10, the grcateft diameter 8, and 
 the middle diameter 6? Anf. 272-272« 
 
 Ex. 3. Rf qui red the folidity of the fruftum of an hy- 
 perbolic conoiil, the height being 12, the greateft dia- 
 meter 10, the lead diameter 6, and the middle diameter 
 8i? ^ ^ Anf. 667-59. 
 
 Ex. 4. What is the content of the middle fruftiim of 
 an hyperboiic fpindle, the length being 20, the middle or 
 greatelt diameter l6, the diameter at each end 12, and 
 the diameter at ^ of the kng[h 14-1 ? Anf, 3248«p38. 
 
 PROBLEM XX. 
 
 Tafind the Solidity of a Paracolic ComiJ, 
 
 Square the diameter of the bafe, multiply that hy the 
 altitude, Snd the produift again by *3927t for the content. 
 
 EXAMPLES. 
 
 1. Required the folidity of the paraboloid whofe height 
 £D is 30, and the diameter of its bafe AC is 40? 
 
 40 
 40 
 
 1600 
 30 
 
 48000 
 •3927 
 
 3U16 
 13708 
 
 1 884*96000 Anfwer. 
 
 Ex. 2. 
 
HGii ' CONIC SECTIONS 
 
 Ex. C. What is the content of the parabolic conoid 
 whofc altitude is 42, and the diameter of its bafc 24 ? 
 
 Anf. 9500-19«4» 
 
 PROBLEM XXI. 
 ^nfind the Solidity of the Frujium of a ParaholoiJ* 
 
 ■ Square the diameter of the two ends, add thofe two 
 fquart's together, multiply that fum by the height, and 
 the produft again by '3927, for the content, 
 
 EXAMPLES, 
 
 1. Rcqaired the cpnient of the paraboloidal fruftum 
 ABCD, the diameter AB being 20, the diameter DC 40, 
 and the height EF 224-? 
 l600 DC^ 
 400 AB.» 
 
 2000 fum 
 224- EF 
 
 4-^000 
 •3927 
 
 19^35000 
 15708 
 
 17671-5000 
 
 ^•':c:_ r : 
 
 Ex. 2. What is the centent of the fruftum of a para- 
 boloid, the greateft diameter being 30, the lead 24, and 
 the altitude 9 ? Anf..52 16-6'26'6\ 
 
 PROBLEM XXII. 
 
 To find the Zoliditj of a Parabolic Spindle, 
 
 Take the fquare of the middle or greateft diameter, 
 multiply it by the length, and the product again by '4 188 8, 
 fof the content. ex- 
 
4ND THEIR SOLIDS. 
 EXAMPLES. 
 
 2(53 
 
 1. Required the content of the parabolic fpindle ACBD, 
 whofe length AB is 40, and the greateft diameter CD l6? 
 
 16 CD 
 16 
 
 16" 
 
 256 CD* 
 40 AB 
 
 10240 
 
 •41888 
 
 1675520 
 83776 
 418S8 
 
 4289*33 1 '20 Anfwer, 
 
 Ex. 2. What is the folidity of a parabolic fpindlf 
 whofe length is 18, and its middle diameter Q feet? 
 
 Aaf. 271'4336 
 
 PROBLEM XXIII, 
 
 ^ofind the S»Iidity of the Middle Frufium of a Parabolic 
 
 Spindle, 
 
 ^ Add altogether, 8 times the fquare of the greateft 
 diameter, 3 times the fquare of the leaft diameter, and 
 4 times the produd of the two diameters; multiply th« 
 fum by the length, and the produd again by '05236 for 
 the folidiry, 
 
 EX- 
 
5^^ CdNlC SECTIONS. 
 
 EXAMPLES. 
 
 1. Required the content of the fruflum of a parabolic 
 fpindle liGHF, the length AB being 20, the greatcft 
 diameter CD 16*, and the Icaft diameter EF 12 ? 
 
 i6 12 16 
 
 )6 12 12 
 
 96 
 16 
 
 144 
 3 
 
 192 
 
 4 
 
 256 
 
 432 
 
 7 08 
 
 8 
 
 .^.« 
 
 »— . 
 
 2048 8 CB^ 
 432 3 EF« 
 768 4 CD 
 
 1 
 
 X EF 
 
 
 3248 fum 
 20 AB 
 
 
 
 64960 
 •05246 
 
 Set Fig. in p. 
 
 r. 
 
 .26a 
 
 3897^0 
 19^88 
 12992 
 32480 
 
 
 3401-30560 Anfwei 
 
 
 I 
 
 Ek. 2. What is the content of the fruftiim of a para, 
 bolic fpindle, whofc length is IS, greatcft diameter 18, 
 and Kaft diameter 10? Anf. 340+-23776. 
 
 t^ofe. Tre folidities of the hyperboloid and hyperbolic 
 fpindle, J»re to be found by rule 2 to prob. xviji. And 
 thofe ( f thfir frulluins by prob, xix ; where ioinfi ex. 
 «mple& oi ihem arc given. of 
 
OP 
 
 GAUGING. 
 
 »npHE bufinefs of cafk gauging is commonly performed 
 ■*■ by two inftrumenrs, n:imely, the gauging or fliding- 
 rule, and the gauging or diagonal rod. 
 
 1. OF THE GAUGING RULE. 
 
 This inftrument ferves to compute the contents of cafksj 
 &c. after tl.e dimenfions have been taken. It is a fquarc 
 rule, having vanous logarithmic lines on its four fides or 
 faces; and three Aiding pieces, running in grooves in 
 three of them. 
 
 On the firft face are three lines, namely, two marked 
 A, B, f)r multiplying and dividing; and the third, MD, 
 for malt depth, becaufe it ferves to gauge malt. The 
 middle one B is on the Aider, and is a kind of double 
 line, being marked at both the edges of the Aider, for ap- 
 plying it to both the lines A and MD. Thefe three lines 
 are all of the fa^Tie radius, or diltance from 1 to 10, each 
 containing twice the length of the radias. A and B 
 are placed and numbered exadly alike, each beginning at 
 1, which may be either I, or 10, or 100, &c. or •!, or 
 •01, or '001. &c. but whatever itiis, the middle divi- 
 fioiij 10, will be ten times as much, and the lail divifion 
 
 100 
 
f66 
 
 (SAUaiNC RULE. 
 
 100 times as much. But 1 on theline MD isoppofite 215, 
 or more exadly 2150-4- on the other lines, which number 
 2l!jO'4t denotes the cubic inches in a mah buihelj and its 
 divifions numbered retrograde to thofe of A and B. On 
 thefe two lines arc alfo feveral other marks and letters: 
 thus, on the line A are MB, for mall bulhel, at the number 
 2150-4; and A for ale, at 282, the cubic inches in an ale 
 gallon; and on the line B, is W, for wine, at 231, the 
 cubic inches in a wine gallon ; alfo j /, for fquare infcribed, 
 at '707, the fide of a fquare infcribed in a circle whofe 
 diameter is 1 ; s e, for fauare equal at '886, the fide of a 
 fquare which is equal to the fame circle ; and c, for circum- 
 ference, at 3* 14 16', the circumference of the fame circle. 
 
 On the fecond face, or that oppofite the firft, are a 
 Aider and four lines, marked D, C, D, E, at one end, 
 and root, fquare, root, cube, at the other ; the lines C 
 and D containing refpedivcly the fquares and cubes of the 
 oppofite numbers on the lines D, D ; the radius of D 
 being double to that of A, B, C, and triple to that of 
 E; fo that whatever the firft 1 on D denotes, the firft on 
 C is the fquare of it, and the firft on E the cube of it ; 
 fo if D begin with 1, C and E will begin with 1 ; but if 
 D begin with 10, C will begin with 100, and E with 
 1000; and.fo on. On the line C are marked o r at '0796, 
 for the area of the circle whofe circumference is 1; and 
 </ at •78.54, for the area of the circle whofe diameter is 1 . 
 Alfo on the line D, are WG, for wine gauge, at 17*15; 
 and AG for ale gauge, at 18-95; and MR, for malt round, 
 at 52*32; thefe three being the gauge points for round 
 and circular meafure, and are found by dividing the fquare 
 roots of 231,282, and 2150*4 by the fquare root of 7854: 
 aflfo MS, for malt fquare, are marked at 46'*37, the malt 
 gawge point for fquare meafure, being the fquare root of 
 2150'4. 
 
 On the third face are three lines, one en a Aider marked 
 N ; and two on the ftock, marked S S and SL, for kg- 
 ment ftanding ai d.fegment lying, which ferve for ulliging 
 liaHding and lying c^jfks. And 
 
aAURIN-G RULE. 26f 
 
 And on the fourth, or oppofite face, are a fcale of in- 
 ches, and three other fcales, marked fpheroid, or 1ft 
 variety, 2d variety, 3d variety; the fcale for the 4th, or 
 conic variety, being on the infide of the flider in the third 
 face. The ufe of thefe lines is to find the mean diameters 
 of caflcs. 
 
 Befides all thofe lines, there are two others on the in- 
 fules cf the two firft Aiders, being continued from the one 
 flider to the other. The one of thefe is a fcale of inches, 
 from ]2{- to 06 i and the other is a fcale of ale gallons, 
 between the qorrefponding numbers 435 and 3*6"l ; which 
 form a tab'e tb (low, in ale gallons, the contents of all 
 cylinders whofe diameters are from 12^ 10 36 inches, their 
 common altitude being 1 inch. 
 
 Tie Ufe of the Gauging Rule, 
 
 PROBLEM I. 
 
 To Multiply fivo Numbers, as 12 and^l. 
 
 Set 1 on B, to either of the given numbers, as 12, on 
 A; then againft 2a on B, ftands 3uo on A; which is the 
 produd. 
 
 PROBLEM II. 
 
 To DiiAdt one Number by another y as 300 by ^S, 
 
 Set I on R, to 25 on A ; then againft 300 on A, Paads 
 12 on B, for the quotient. 
 
 PROBLEM in. 
 
 To find a Fourth ? yoportlonaJ^ as .'0 8, 2r, and g6. 
 
 Set 8 on B, to 2\- on A; then againft 96 on B, is 288 
 on A, the 4:h proportional to 8, 24, 9<?, required, 
 
 PRO. 
 
ft6$ eAlTtlNG RULE. 
 
 PROBLEM IV. 
 
 To Extraa thf Square Root, as of 225. 
 
 The firft 1 on C ftandirg oppofife the one on D, on the 
 ftock; then againll 226 on C, (lands its fquaic root 15 
 on U, 
 
 PROBLEM V. 
 
 7a Exira^ the Cubt Roctt as of 3375. 
 
 The line D on the Aide being fet ftraight with E ; then 
 eppofiie 3375 on t> ftands its cube root 15 on D. 
 
 PROBLEM VI. 
 
 To find a Mean Troportionalt as between 4 and S)* 
 
 Set 4 on C, to the fame 4 on D; then againft 9 on C, 
 (lands the mean propoitional 6 on D. 
 
 PROBLEM VII. 
 
 To find Numbers in DupUcate Proportion, 
 
 Js to find a Number 'which Jhall he to 120, as the Square of 
 3 to the Square of 2. 
 
 Set 2 on D, to 120 on C ; then againft 3 on D, ftandj 
 270 on C, for the anfwer. 
 
 PROBLEM VIII. 
 
 To find Numbers in Sub duplicate Proportion. 
 
 As to find a Number <which JhaU be to 2 as the Reot of 270 
 to the Root of 120. 
 
 Set 2 on D. to 120 on Cj then againft 270 on C, 
 ftands % on D, for the anfwer, 
 
 PRO- 
 
GAUGING RULE. 259 
 
 PROBLEM IX. 
 
 1^0 find 'Numbers in Triplicate Proportiofu 
 
 As, to find a Numher nvhich fi:>all he to 100, as the Cube of 
 2,6 is to the Cube of 40. 
 
 Set 40 on D, to 100 on E; then aoainft 36 on D, 
 Aands 729 on K, for the anfwer. 
 
 TROBLEM X, 
 
 To find Numbers in Subtrificate Proportion, 
 
 As, to find a Number nvhich /hall be to 40, as the Cube 
 Root of72'0 i^ t^ ihe Cube Root of 100. 
 ■ » 
 Set 40 on D, to 100 on E; then agalnft 7'2'9 on E, 
 ftands 36 on D, for the anfwer. 
 
 PROBLEM XI. 
 
 To Compute Malt Bujhels by the Line MD. 
 
 As, to find the Malt Bujhels in the Couch, Utor, or Cfiern, 
 
 nvho/e Length is 230, Breadth 58'2, 
 
 and Depth 5*4 Inches. 
 
 Set 230 on B, to 5*4 on MD; then agalnft 58-2 on A 
 ftands 33*6 bulhels on B, for the anfwer, ^ 
 
 Note, The ufes of the other marks on the rule, will 
 appear in the examples farther on. 
 
 OF THE GAUGING OR DIAGONAL ROD, 
 
 The diagonal rod is a fquare rule, having four faces; 
 
 being commonly 4 feet long, and folding together br 
 
 N joints. 
 
270 #AUc.i:, t» i.ojs. 
 
 joinfj, Th's inflrurent is ufcd both for gauging or 
 ircafuriiig . .iflc , and computinij rheir contents, and that 
 fron one i.i IK I lion on!), namrly ihe diagcHial of the calk, 
 or ihf Icn^ h fr 'm the middle of the bung hi»Ic to ihc 
 meiti< g » t the head of the calk 4viih the ftave oppufite to 
 the bu.io ; beiiig (he longed line that can be drawn within 
 the calk frv,m ihe mid.lle of the bung. And, accordingly, 
 on one face ot the rule is a fcale of inches for meafunng 
 this dirgo al ; to which are ['laced the areas, in ale gallons, 
 cf circles to the correfponding di. meters, in like manner 
 as the lines on the under lidts cf the three Aides in the 
 Aiding rule. 
 
 On the oppofiie face^ are two fcales of ale and wine 
 gallons, exprclfing the contcnrs of caiks having the cor- 
 refpondmg diagonals. And ihefe are the lines which 
 chiefly foim the diffc-ence between this inftrutuent and the 
 Aiain>i rule; far all their other lines are the fa.ne, and ara 
 to be ufed in the fame manner. 
 
 EXAMPLE. 
 
 The rod being applied within the calk at the bung-hole, 
 the diagonal was found to be S-^'i inches j required the 
 content in g.dlons, 
 
 Now to Si'^ inches correfponc^, on the rod, f}()| ale 
 gall ns, or 111 wine gallons, the conient required. 
 
 Nofe. The contents exhibited by the rod, anfv-er to 
 the moft common form of calk^, and fall in between the 
 2d and 3d varieties following. 
 
 t OF CASKS AS DIVIDED INTO VARIETIES, 
 
 It is ufual to divide caflcs into four cafes or varieties, 
 which are judged of from the greater or lefs apparent cur- 
 vature if their fides; namd^', 
 
 1. The midd'e frufturn of a fpheroid, 
 
 2. The m ddle fniftum ( f a parabolic fp^ndle, 
 3.'1 he two equal fruftums of a paraboloid, 
 
 4. Tae two equal frullums of a cone. And 
 
• ASie GAUGiyc. 271 
 
 And if the content of any of thefc be computed in 
 inches, by their proper rules, and this be divided by 282, 
 or 231, or2I50'4', the quotient will be the cr)ntent in 
 ale gallons, or wine gallons, or malt buftieL, refpedivtly. 
 Becaufe 
 
 282 cubic inches make 1 ale gallon 
 
 23 1 1 wine g^illon 
 
 2150'4 1 majt bulhel. 
 
 And the particular rule will be for each as in the follovr- 
 ing problems : 
 
 PROBLEM XII. 
 
 To find the Content of a Cajk of the Firfi Form, 
 
 To the fquare cf the head diarriCter, add double the 
 fqoare cf the bung diameter; «nd multiply the fum by 
 the length of the calk. Then let the produft 
 
 be moliipliod by •0009^, or divided by 1077, for a^e 
 
 gallons; 
 
 and multiplied by •0011|, or divided by 882, for wine 
 
 gallons. 
 
 EXAMPLES. 
 
 1. Required the content of a fpheroidal cafk, whofe 
 length fs 40, and bung and head diameters 32 and 24! 
 inoihtts* 
 
 n2 t4 
 
f7'2 
 
 CASK ftAUGlNII* 
 
 24. 
 i)6 
 
 48 
 676' 
 
 32 
 3^2 
 
 64 
 9^ 
 
 1024 
 
 2048 
 576 
 
 20'24 
 40 
 
 104960 
 •0009^ 
 
 944640 
 26240 
 
 104960 
 •00115- 
 
 1154560 
 34987 
 
 104960 ale 97*OS80 gallons 1 18-9547 wint 
 
 Bj the Ga^igifig Rule, 
 Having fet 40 on C, to the ale gauge 32-82 on D, 
 
 a^gainft 
 
 24 on D, ftands 21*3 on C 
 32 OB D, ftands 38-0 on C 
 the fame 380 
 
 fum 97 -3 ale gallons. 
 
 And having fet 40 o« C, to the wine gauge 297 on D, 
 
 againft ^ _ 
 
 2 A on D, ftands 26-1 on C 
 32 on D, ftands 46*5 on C 
 the famt 46-5 
 
 fum 119*1 wine gallons. 
 Ex ^ Required the content of the fpheroidal cafk, 
 
 ». .?;i;,wdi is 20, and diameters 12 and I6 inches, 
 whofe length is .u, ^^^^^^^ ^^^ ^^^^^^^^ 
 
 Anfwer ^ j i^.^(^^ wine gallons. pRO« 
 
€-ASK CAUGINO^* 
 
 !73 
 
 J?ROBLEM XIII. 
 
 To find the Content of a Cajk of the Second Form, 
 
 To the fquare of the head diameter, add double the 
 fquare of the bung diameter, and from the fum take |. or 
 /(J of the fqware of the difTerencc of the diameters; then 
 multiply the remainder by th« length, and the produj^l- 
 again by •0009^ for ale gallons, or by ♦001 ly for wine 
 gallons, 
 
 EXAMPLES. 
 
 1 , The length being 40, and diametf rs 24 and 52, r«- 
 quired the content. 
 
 S2 
 
 24 
 
 8 
 8 
 
 64 
 
 4 
 
 2624*0 
 25-6 
 
 25S8'4 
 40 
 
 1035)36 
 •0009i 
 
 935424 
 23984 
 
 103935 
 •0011^ 
 
 1143296 
 34645 
 
 25-6 103936 ale 9()-l 408 gall. 11 7'7941 wine 
 
 l^y the Gauging Rule, 
 
 Having fet 40 on C, to 32*82 on D, againfl 8 on D, 
 ftands 2-4 on C; the y% of which is 0-96. This taken 
 lifoin the i'%'d in the laft form, leaves 96'*3 ale gallons. 
 
 N 3 And 
 
274 
 
 CASK GAUOIN«. 
 
 And hnving fct40on C, lo QiJ'J on D, againd 8 on D, 
 ftands C'f) on C : ihe -,-\ of which is T-l6. Ibis taktn 
 from the J iy*l in the laft form, leaves 1 J7'9 wine gallons. 
 
 Ex. C. Required the content when the length is 20, 
 and the diameters 12 and l6". 
 
 I 14/24? wine gallons. 
 
 PROBLEM XIV. 
 
 I'o Jind the Contttit of a Ccjk of the Third Form, 
 
 To the fquare of the bung diameter, add the fquare ©f 
 the head diameter; multiply the Aim by the length, and 
 the produ(5\ again by '00 1 4r for ale gallons, or by •0U17 
 for wine gallons. 
 
 EXAMPLES, 
 
 1. Required the content of a criflc of the third form^ 
 vf hen the length is 40, and the diameters 24 and 32, 
 
 1024 
 576 
 
 l600 
 40 
 
 64000 
 •0024 
 
 25(3 
 ^4 
 
 64000 
 •00J7 
 
 449 
 64 
 
 64000 ale 89*6 gallons 108-8 wme 
 
 By 
 
CA'<K G AUG IN" 6. 
 
 275 
 
 ^v the G, uging Rule» 
 
 Set 40 on C, to i.()-.^ on 0, then againft 
 
 V4 en D, flancls 3'i*() on C 
 32 on D, Hands 37 'J on C 
 
 fuii; bc,*3 ale gallons. 
 
 And having fct 40 on C. fn oj..>5 (.^ Dj then againft 
 24 on D 1 ar. '<, .,«;•! on C 
 32 on D, Itapcis (.y 8 on C 
 
 fum i(/S\9 wine gallons. 
 
 Ex. 5. P.eqnireei the content when the length is CO, 
 and the diaii; tiers 12 and lo. 
 
 I 13 (■ vMue 
 
 gallons. 
 
 PROBLEM XV. 
 
 V^ejtnd the Content of a Lojh f the Founh Fonv, 
 
 Add the fquare of the difference of the dianieters, to 3 
 
 times the fquare o{ their fum ; then multiply the fum by 
 
 the length, snd the produfi: again by '00023-^ for aie gal. 
 
 Ions, "or by •0G02«j for wine rallons. 
 
 EXAMPLES. 
 
 1, Required the content, when the length is 40^ aad 
 tlie dianaeters 24- and 32 inches. 
 
 K 4 
 
 d6 
 
2/5 CASK eAur. iN«>. 
 
 56 8 
 
 56* 8 
 
 — 37SS80 37888© 
 
 330* ()4. •000231 •000'28} 
 
 280 9408 
 
 ^ 1130*(ut0 3031040 
 
 3i36 9*72 7577(;0 757760 
 
 3 40 75776 126293 
 
 9408 378880 ale 87'900l6 gall. 107*34933 wine. 
 
 Fj> the Sliding Ruh. 
 
 Set 40 on C, to G5*o4. on D; then againft 
 ^ 8 on D, (lands 0*6 on C 
 £6 on D, ftanus '29-1 on C 
 29-1 
 29*1 
 
 fum 87 '9 ale gallons. 
 
 And fet 40 on C^ to 59*4 1 on D ; then againft 
 
 / 8 on D, ftands lYJ 
 
 56 on D, (lands 355 
 
 35-6 
 
 • 35-6" 
 
 fum 107 '5 wine gal. 
 
 "Px. 2. What is the content of a conical Criflc, the length 
 being '20, and the burg and head diameters 16 and VZ 
 inches? 
 
 - r f 10*985 ale gallons, 
 An.wcr 1 13.^15' ^ine gallons. 
 
 PRO* 
 
CASK GAUGIXO. 2/7 
 
 PROBLEM XVI. 
 
 To find the Content of a Cajk by Four Dimenjionr, 
 
 Add together, tlie fquares of the hung arid head diame- 
 ters, and the fquare of douiile tbe diameter taken in the 
 middle between the bung and bead ; then muhiply the 
 fum by the length of the calk, and the produft again by 
 •00041- lor ale gaHbns, or bj '00051 fcr wine gallons. 
 
 EXAMPLES, 
 
 1 , Required the content of any cafk- whofe length is 40, 
 the bung diameter being 32, the head diameter 24, and the- 
 middle diameter between the bung and head 2S|- inches. 
 
 57'5 
 67-5 
 
 
 24! 
 24 
 
 32 
 32 
 
 2875 
 4025 
 2875 
 
 
 96 
 48 
 
 64 
 
 96" 
 
 1024 
 
 :'306*25 
 1024 
 
 4S06-25 
 40 
 
 ' 
 
 ]<)^250 
 *0C04^ 
 
 
 1962.50 
 •(r005f 
 
 
 7H5000 - 
 130833 
 
 ;alIons 
 
 981250 
 130833 
 
 
 ale 91-5833 § 
 
 111-2083 
 
 wine. 
 
 
 w 
 
 5 
 
 
 Mj 
 
27t CASK GAUGING. 
 
 By the Sliding Rule, 
 
 Set 40 on C, to 46*4 on D; then againft 
 24 on D, flamU 10'5 
 32 on D, ftands ly 
 57 k on D, {lands 62'Q 
 
 fum 91-5 ale gallons, 
 
 Set 40 on C, to 42*0 on D; then againft 
 24 on D, ftands 130 
 32 on D, ftands 23-2 
 57 i on D, (lands 75*0 
 
 fum 1 1 1*2 wine gallon?. 
 
 Ex. 2. What is the content of a cafk, whofe length is 
 20, tlie bung diameter being U), the head diameter 12, 
 and the diameter in the middle between them 14-^? 
 
 Anfwer^ 1 1 -4479 ale gallons, 
 /\niwer| ^y.^^^^^ ^^.j^^g gallons. 
 
 PROBLEM XVII. 
 To find the Content of any Cojk from Three Dimenjious only. 
 
 Add into one fum, 39 times the fquare of the bung 
 <1ianieter, 25 times the fquare of tne head diameter, and 
 26 times the produft of the two diameters : tken multi- 
 ply the fum by the length, and the produd again by 
 
 •^^^^ for wine gallons, or fty 1^52^ or -00003^ ^ox 
 
 9 
 ale gallons. 
 
 EXAMPLES, 
 
 1. Required the content of a calk, whofe length is 40, 
 and the bung and head dian:M^ters 32 and 24? 
 
 32 
 

 CASK 
 
 GAUOING 
 
 32 
 32 
 
 24 
 24, 
 
 32 
 24 
 
 64 
 
 96 
 
 96 
 
 48 
 
 128 
 64 
 
 102-t 
 39 
 
 576 
 25 
 
 763 
 
 S)2l6 
 3072 
 
 2880 
 J152 
 
 4608 
 1536 
 
 74304 
 40 
 
 ] 1888640 
 
 9) 1010-53440 
 
 39936 14400 19968 
 
 39936 
 
 1996s 
 
 297 2 J 60 
 •00034 2972160 
 
 •00003^ 
 
 8916480 89Ki480 
 
 . 270 iy6 
 
 279 
 
 1 12*28 16 wine gal. 91 •8f)676 ale gallons. 
 
 Ex. 2. What is the content of a ctfk, whofe length is- 
 20, and the bung and head diameters 10 and 12? 
 
 Anfwer^ l l -4833 ale gallons, 
 /^niwei I ^^.(3352 wine gallons. 
 
 Nofi. This is the moil exad rule of any, for three 
 dimenfions only; and agrees nearly with the «iiagonal 
 rod. 
 
 h 6 oJ 
 
2S0 ULLAGE 
 
 OF THE ULLAGE or CASKS, 
 
 The ullage of n cnfk, is whnt it contains when only 
 pnrtiv filled. Aiui it is confiJereii in two pofuions, namely, 
 as f};»nding on its end with the axis perpendicular to the 
 horizon, or as lying on its fide with the axis paralleh to 
 th« horizon. 
 
 PROBLEM XViri. 
 Tofndthe Ullage by the SliJhrg Rule. 
 
 By one of the preceding problems fiftd the whole con- 
 tent of the callv. Then fct the length on N, to 100 on 
 SS, ff)r a fegiTicnt (landing, or fet the hung diameter on, 
 N, to ICO tin SrL, for a fegment lying; then agninft the 
 wer inches on N, is a number on SS or SL, to be referx ed. 
 
 Ncxr, Set 100 on B, to the referved number on A; 
 then ;igain(l the whole content on B, will be found the 
 ullage on A. 
 
 EXAMPLES. 
 
 1. Required the ullage anfwering to 10 wet inches of 
 a n.'u-ding calk, the whole coateni of which is 92 gallonsi 
 and jcngth 40 inches. 
 
 HiTving (tt 40 on N, to 100 on SS; then 
 
 againft 10 on N, is 2J on SS, ^he referved numb. 
 
 'Chen fet 100 on B to 23 on A; and 
 
 againrt y2 on B, is 21 '2 on A, the ullage required. 
 
 Ex. 2. What is the ullage of a (landing cafk whofe 
 
 whole length is 20 iixhcs, and content 1 i *: gallons ; the 
 
 wet inches bel.t e 5 .? AnC 2 65ga lo m. 
 
 7 J£x. 3. 
 
OP CASK3. 
 
 2tft 
 
 Ex. 3. The content of a caflc being 92 gallons, and the 
 l»ui>g diameter 32, required the ullage of (he fegment lying 
 when the wet inches are 8 ? Anf. l6'*^ gallons- 
 
 PROBLEM XIX. 
 
 To Ullage a Standing cajk by the Pev*^ 
 
 Add all together, the fqiiare of the diameter at the fur- 
 £ice of the Jitjuor, the fq-jare of the diameter of ther 
 iieareft end, and the fqnare of double the diameter taken 
 in the mid<dle between the other two; then multiply the 
 fum by the length betvveen the furface and ne;reft end,, 
 and the product again by 'OOO^j for ale gallons,, or by 
 '0005^ for wine gallons, in the lefs pare of the calk, 
 whether empty or filled* 
 
 EXAMPLE. 
 
 The three diameters being 24, 27, and 29 inches!, re* 
 quired the ullage for 10 w€t inches. 
 
 24. 
 
 29 
 
 54 
 
 
 2-t 
 
 29 
 
 54 
 
 2916 
 
 ._ 
 
 
 , 
 
 841 
 
 96 
 
 261 
 
 316 
 
 676 
 
 48 
 
 58 
 
 270 
 
 4333 
 
 '70 
 
 841 
 
 2916 
 
 JO 
 
 
 
 
 
 
 
 43330 
 
 43330 
 
 
 
 •0004f 
 
 •0005* 
 
 
 173320 
 
 2l6'650 
 
 
 
 28885 
 
 23885 
 
 Ale 20*2205 gallons 24*5535 wine 
 
 PRO- 
 
>SJ ULLAGE OF CASKS* 
 
 PROBLEM XX. 
 
 To Ullage a Lying Cajk hy the Pen, 
 
 Divide the wet inches by the bung diameter; find the 
 quotient in the column of verfed fines, in the table of 
 circular fegraents at the end of the book, taking our its 
 correfponding fcgmcnt. Then multiply this fegment by 
 the whole content of the caflc, and the product again by 
 ly for the ullage required, nearly. 
 
 EXAMPLE. 
 
 Suppofing the bung diameter 33, and content 92 a-le 
 gallons ; to find the ullage for 8 wet inches. 
 
 32 ) % ('25, whofe tab. kg. is -153546 
 
 92 
 
 307092 
 1381914. 
 
 14 •126232 
 J is 3-531558 
 
 3 7 -657790 Anfwei, 
 
 OF 
 
OP 
 
 SPECIFIC GRAVITY. 
 
 nnHE fpecific gravities of bodies, are their relative 
 "*■ weights, contained un ier the fame given magnitude, 
 as a cubic foot, or a cubic inch. Sec. 
 
 The fpecific gravities of feveral forts of matter are ex- 
 preffcd by the numbers annexed to iheir names in the fol- 
 lowing table : 
 
 j4 Table cf the Specific Gratuities of Bodies* 
 
 Fine gold 
 
 196-W 
 
 Brick 
 
 2000 
 
 Standard gold 
 
 18888 
 
 Light earth 
 
 1984* 
 
 Quick-filver 
 
 13600 
 
 Solid gun-powder 
 
 1745 
 
 Lead 
 
 11 3'25 
 
 Sand 
 
 1520 
 
 Fine filver 
 
 11091 
 
 Pitch 
 
 11^ 
 
 Standard filver 
 
 10535 
 
 Box-wood 
 
 1030 
 
 Copper 
 
 9000 
 
 Sea-water 
 
 1030 
 
 Gun metal 
 
 8784. 
 
 Common water 
 
 1000 
 
 Caft brafs 
 
 8000 
 
 Oak 
 
 9^^5 
 
 Steel 
 
 7850 
 
 Gun-powder, fhaken 
 
 922 
 
 Iron 
 
 7645 
 
 Am 
 
 755 
 
 Caft iron 
 
 7425 
 
 Maple 
 
 800 
 
 Tin 
 
 7320 
 
 Elm 
 
 600 
 
 Marble 
 
 2700 
 
 Fir 
 
 550 
 
 Common ftone 
 
 2520 
 
 Cork 
 
 240 
 
 Loom 
 
 2160 
 
 Air 
 
 li 
 
 Note. The feveral forts of wood are fuppofed to be 
 dry. Alfo as a cubic foot of water weighs juft U GO 
 ounces avoirdupois, the numbers in this table exprefs, not, 
 only the fpecific graviries of the feveral bodies, but alfo 
 the weight cf a cubic foot of each, in avoinlupois ounces; 
 and hence, by proportion, the weight of any other quan- 
 tity, or the quantity of any other weight, may be known, 
 as io the following problems. pro- 
 
i^* SPECIFIC GRAVITY.. 
 
 PROBLBM I. 
 To find the Magnitude of any Body from Its Weight. 
 
 As thr tabular fpecific gravity of the bocly. 
 Is to its weight in avoirdupois ounces. 
 So i^ one cubic f )Ot, or 1728 cubic inches. 
 To its content in feet, or inchp, lefpcaively. 
 
 SXAMFLES. 
 
 1. Required the conte»it of an irregular block ot oom- 
 jnoD ftone which weighs l cwt. or n2lb, 
 1121b. 
 16 
 
 672 
 112 
 
 2520 : 17.W2 : : 172S : 1228|^^*:> 
 1728 
 
 14.336 
 3684 
 
 1254-i 
 
 37i)2 
 
 cubic inch. 
 
 2520 ) 3096576 ( 1228||^-S Anrwcr. 
 252 
 
 504- 
 
 725 
 504. 
 
 2217 
 yoi6 
 
 1016 
 . Ex. 2. 
 
Ex. 2. How many cubic indies of gun-powder are there 
 in lib. weight ? ^ Auf. 30 cubic inches nearly* 
 
 Ex» 3. How many cubic feet are there in a ton weight 
 of dry oak? Anf» 'JBjIj cubic feet* 
 
 PROBLEM It. 
 
 7*0 fnd iii IVeight of a Baiyfnm Ui Magniiudu 
 
 As one cubic foot, or 1728 cubic inches. 
 Is to the content of the body. 
 So is its tabular fpecific gravity. 
 To the weight of the body. 
 
 EXAMPLES, 
 
 1. Required the weight of a bbck of marble, whole 
 length is 6"3 feet, and breadth and thicknefs each 12 ittt; 
 being the dimensions of owe of the ilones in the wall* of 
 Balbeck. 
 
 63 
 12 
 
 12 
 
 — 02. tons ] 
 
 1 : 5072 : : 2700 : (J83A 
 2700 
 
 6350400 
 18144. 
 
 a6{ 
 
 4 
 4 
 112 
 20 
 
 24494400 oz. 
 6123600 
 1530900 lb. 
 13668 cwt. 
 Anf. 6S3/o ton, almoft equal to the burthen of an 
 
 ^- Eaft India fliip. 
 
 Ex, 2. 
 
28^ fJPEGinC GRAVITY. 
 
 Ex. 2. What is the weight of 1 pint, ale mcafnre, of 
 gtin powder' Anf. H) oz. nearly, 
 
 Fx, 3. What ib the weight of a block of dry oak, which 
 mealurcs 10 icet long, 3 fcec broad, and 'i-|. feci deep f 
 
 All". 43354 » lb, 
 
 PROBLEM III. 
 
 ^0 Jind the Sp(.(.iji< Gr/ivitj of a Body. 
 
 Case i. When the bo»lv is heavier than water, weigh 
 it l>()th in water and out of water, and take the difference, 
 which will be the weight loil in water. Then fay. 
 
 As tfee weight loft in water. 
 
 Is to the whole weight. 
 
 So is the fpecific gravity of water. 
 
 To ihi f^jecific gravity of the body. 
 
 EXAMPLI. 
 
 A piece of ftone weighed lOlb, but in water only ^J lb. 
 required its fpecific gravity ? 
 10 
 
 !! 
 
 3^ : 10 : : 1000 j 
 oris : 40 : : 1000 : 3077 anfvrcr. 
 40 
 
 13 ) 400C0 ( 3077 
 100 
 
 90 
 
 Case 2. When the lody is lighter than water, fo that 
 it will not quite finkj aflSii to it a piece of another body 
 
 heavier 
 
srEciric ciiAViTY. 287 
 
 heavier than water, fo that the mafs compounded of the 
 two may (ink together. Wei^h ihe heavier body, and the 
 compound mafs, ft par;^fely, b( th in water and out of it; 
 then fiiid ht)w much e -ch lofes in uater by fubtrafting its 
 weight in water fom its wtight in air; and fubtr3i!:t the 
 lefs of thefe rem.' iaderb from the greater. Then fay, 
 
 As this laft re^nairyler. 
 Is to the weik,hr of the li?>ht body in air. 
 So is the fp cihc gi; vi;\ t f vvater, 
 To the fpecific gravi;) of ihe body. 
 
 EXAMPLE, 
 
 Suppofe a piece of elm weighs l^lbin air, and that a 
 piece of copper, which w< ig'hs ISib in air, and l6!b in 
 water, is afiixel to it, ^nd that the compound weighs Sib 
 in water i required the fpccifi' gravity of the elm? 
 
 Copper Corv^-ound 
 18 in air 33 
 l6 in water 6 
 
 2 lofs 27 
 — . 2 
 
 ' Then As 25 : 15 : : 1000 : 600 anf. 
 
 PROBLEM IV. 
 
 To /ad the ^amities of l^^o I igredients in a Given 
 Compoundm 
 
 Take the three differences of every pair of the three 
 fpecific gravities, namely, the f <ecific gravities of the 
 compound and each ingredient; and. multiply the difference 
 of every two fpecific gravities by the third Then, as 
 
 the 
 
288 8P11CIVIC ftKAVIlY. 
 
 the greateft prodnd is to the whole weight of the com- 
 pound, (o is each of the oiher proUufts, to the two 
 wei^kts ('f the ingredients. 
 
 BXAMPLr, 
 
 A compofition of 1 12lb being made of tin and copper, 
 whofe fpecific gravity is found ta be 8784; required th« 
 quantity of each ingredient, the fpecific gravity of iitk 
 being 7320, and of copper 9000 ? 
 
 90v)0 9000 8784 
 
 7320 8784 7320 
 
 1680 
 8784 
 
 216 
 7320 
 
 4320 
 648 
 1512 
 
 1464 diiF. 
 9000 
 
 
 13176000 
 
 702720 
 52704 
 
 8784 
 
 
 1581120 
 
 
 14757120 : 
 
 112 : ; 13176000 : 100 copper 
 112 
 
 __________ 1 (-» fin 
 
 
 26352000 
 13176 
 13176 
 
 14757120 ) 1475712000 ( 100 
 Anfwer, tfeere i» lOOlb of copper, S • i. * • 
 
 and confequeHtly 12lb of tw | '" ^^"^ compofitio.. 
 
 OF 
 
• r THi 
 
 WEIGHT AKD DIMENSIONS 
 
 BALLS AND SHELLS. 
 
 n^HE weight and dimenfions of balls and flieHs might 
 be found from the problems lad given, concerning 
 fpecific gravity. But they may be f )und ftill eaficr by 
 means of the experimented weight of a ball of a given 
 lize, frooa thd known proportion of fimilar figurai, 
 namely, as the cubes of their diametera. 
 
 PROBLEM I. 
 
 To find the Weight of an Iron Ball, fram its Diameter, 
 
 An iron ball of 4 inches diameter weighs 9lb, and the 
 weights being as the cubes of the diameters, it will be, as 
 6'4 (which is the cube of 4) is to 9, fo is the cube of the 
 diameter of any other ball, to its weight. Or take -j^^ of 
 the cube of the diametpf, for the weight. Or take ^ of 
 the cube of the diamerer, and \ of that again, and add 
 the two together, for the weight. 
 
290 Of lALLi 
 
 EXAMPLES. 
 
 1 . The diameter of an iron (hot being 6'7» requireil 
 its weight? 
 
 6-7 
 6-7 
 
 469 
 402 
 
 4^«88 
 6-7 
 
 31423 
 26934 
 
 8 ) 300'763 
 
 8) ^7*^95 
 4"oV9 
 
 Anf. v2-294> lbs. 
 
 Ex. 2. What is the weight of an iroji ball whofe dfa- 
 meter is 5*54 inches? Anf, 24lb. 
 
 PROBLEM II. 
 
 To find the Weight of o Leaden Ball, 
 
 A leaden ball of 4i inches diameter weighs 17lb; there* 
 fore as 'he cube of 4^ is to 17, or nearly as .0 to 2, fo is 
 the rule of rhe diameter of a leaden ball, td its \*'eighi. 
 
 Or lake 1^ of the cube of the diameter, for the weight, 
 nearly. 
 
 5 EX- 
 
AND SHELLS, 291 
 
 EXAMPLES. 
 
 1. Required the weigh l of a leaden ball of 6*6 inches 
 diameter ? 
 
 6-6 
 6-6 
 
 396 
 
 43'5(5 
 6 6 
 
 26136' 
 
 26136 
 
 287 -i^D 
 •2 
 
 9 ) 57^-992 
 Anl. 63*888 lb. 
 
 Ex. 2. What is the weight of a leaden ball of 5'24 
 inche!> diameter ? Auf. 32lb nearly. 
 
 PROBLEM iir. 
 To find the Diameter of an Iron Ball font its Weight* 
 
 Multiply thf» weight by 7-1 then t<ike the cube root of 
 
 the produd for the diameter. 
 
 E AMPLES. 
 
 1 . Required the diameter of a 42lb iron ball ? 
 
 42 
 
2P2 Of BALLS 
 
 254. 
 ^'666 
 
 The cube root of this is alirofl 7. Suppofe 7, whofe 
 cube is 343. Then, by the 2d rale for the cube root at 
 fage 41, proceed thus: 
 
 343 5p8f 
 
 2 
 
 66s 55)75- 
 
 As 58 if : 940^ : : 7 
 3 3 
 
 Or as Q954, : 2821 :: 7 : 6'6S5 Anf. 
 7 
 
 2i)54) 19747 ( 6-685 inchw 
 
 177 24 
 
 iJ023 
 1772 
 
 251 
 
 236. 
 
 15 
 15 
 
 Ex, 2, What is the diameter of a 24lh iron ball ? 
 
 Anf. 5'54 inches. 
 
 PRO- 
 
AN» SHELLS. '^^J 
 
 PROBLEM IV. 
 
 To find the Diameter of a Leaden Ball from its Weight, 
 
 Multiply the Weight by 9, nnd divide the produ6l by 2; 
 then take the cube root of the quotient for the diameter. 
 
 EXAMPLES. 
 
 ] , Required the diameter of a 64lb leaden ball ? 
 
 9 
 
 2 ) 576 
 
 288 
 
 the cube root of which is almoft 7, whofe cube is 341i 
 Then 34-3 288 
 
 
 
 6'&6 576 
 
 288 343 
 
 As 974 : 919 •• : 7 : <^'S05 Anf. 
 
 974 ) 6433 ( 6'605 inches 
 5844 " 
 
 589 
 584 
 
 Kx. 2. What is the diameter of an 81b leaJen ball ? 
 
 Aaf. 3-303 inche^-c 
 
 O TK9' 
 
294 €¥ BALLS 
 
 PROBLBM V. 
 To find the Wtigbt of an Iron Shtii, 
 
 Take -^^ of the difference of the cubes of the cxferral 
 and imerni'l diameter, for the weight of the (hell. 
 
 That is, from the cube of the external diameter bke 
 the cube of the inrernal diameter, multiply the remainder 
 by 9, and divide the produft by 6'4<, 
 
 EXAMPLES, 
 
 1, The ouifide diameter of an iron fhell being 12*8, 
 and the infide diameter 9*1 inches ; required its weight ? 
 0-1 12*8 
 
 9*1 12-8 
 
 91 
 819 
 
 1024 
 1536 
 
 £2 81 
 9-1 
 
 8281 
 74529 
 
 l63'84 
 12-8 
 
 131072 
 190608 
 
 753-571 
 
 2097-152 
 753-571 
 
 
 
 1343*581 
 9 
 
 Anf. 
 
 12092-229 
 1511-528 
 18«-941 
 
 lb. 
 
 Kx. 2. What is the weight ef an iron Ihell, whofe ex. 
 terual and internal diameters are 9*8 and 7 inches ? 
 
 Anf. 84^ lb. 
 
 PRO- 
 
AN» SMBLLS. 
 
 295 
 
 PROBLEM VI, 
 
 To find ho^ much Po-wder njoillfilla Shell, 
 
 Take the cube of the internal diameter, ivi inches anl 
 divide It by 57-3, for the lbs. of powder. 
 
 EXAMPLES. 
 
 1. How much powder will fill the (heir whofe internai 
 fliameter is 9-1 inches? internal 
 
 lb 
 57-3) 753-571 (I3J-V nearly 
 
 1S0.5 
 1719 
 
 86 
 
 Kx. 2. How much powder will fill the fii^Jl ,„;, r • 
 tcpnil diameter is 7 inches ? ' ^'^^ i'f V^ 
 
 a 12 
 
 paa. 
 
i?£)C OV BALLS 
 
 PROBLEM VII, 
 
 1^9 find h»v3 much IBonjuder nuill Jill a Reilangular Box» 
 
 Find the content of the box in inches, by multiplying 
 tlVe length, breadth, and depth all together. Then divide 
 by 30, for the pounds of powder. 
 
 EXAMPLES. 
 
 1. Required the quantity of powder that will fill a box; 
 the length being 15 inches, and the breadth 12, and the 
 depth 10 inches ? 
 
 15 
 
 12 
 
 180 
 10 
 
 SO) 1800 
 
 Anf. CO lb. 
 
 Ex. 2. How much powder will fill a cubical box, whofe 
 fide is 12 inches? Anf. oZ-^lb. 
 
 PROBLEM VIIL 
 
 Tofi?id hc-vj much Voider 'will fill a Cylinder* 
 
 Multiply the fquare of the diameter by the length; 
 then divide by 38*2, for the pounds of powder. 
 
 EXAMPLES. 
 
 1. How much powder will the cylinder hold, whofe 
 diamet«r is 10 inches, and length ':o inches? 1* 
 
/l:3f« SHELl-S. 
 
 10 
 
 10 
 
 100 
 
 20 
 
 lb. 
 
 .38-2 ) 2000 ( 52} nearly 
 1910 
 
 900 
 136 
 
 Ex* 2. How much powder can be contained in the cy- 
 linder, whofc diameter is 4 inches, and length 12 inches? 
 
 Anf. 5,|x 1'^- 
 
 PROBLEM IX. 
 
 To find the Size cf a Shell to contain a Gi'ven Weight of 
 Pofwder, ^ 
 
 Muhiply the pounds of powder by 57'3; then take the 
 cube root of the produd, for the diameter in inches. 
 
 EXAMPLES. 
 
 1. What is the diameter of a Ihell that will hold IS^lb. 
 
 i^f powder I 
 
 57-3 
 134 
 
 1719 
 
 573 
 955 
 
 754'45 
 o 3 
 
 The 
 
S^A OV £ALL9 
 
 The cube root of this is nearly J), whofc cube is 72f), 
 
 Then 
 
 729 754'4.5 
 
 2 « 
 
 1458 1508*50 
 
 754*45 729* 
 
 As 2212-45 : 2237 '90 : : 9 : 9*1 Anfwer. 
 9 
 
 221,245 ) 2014,10 ( 9*1 inchci 
 1991 
 
 23 
 
 12 vl 
 
 Exi 2. Wrat is the diameter of a flielli to contain 6\b, 
 of powder ? Anf, 7 inchos# 
 
 PROBLEM X, 
 
 ^ofini the Size of a Cuhical B«x, t9 cantain a Oh'm Weigh 
 cf Po<v:der» 
 
 Multiply the weight in pounds by 30, then the cube 
 root of the produft will be the fide of the box in inches. 
 
 EXAMPLES. 
 
 1. Required the fize of a cubical box, to hold 50lb of 
 gun-powder ? 
 
 50 
 
AND SHELLS. 
 
 50 
 30 
 
 5§9 
 
 The cube root of 1500 is 11 nearlf, whofc cube is 1331, 
 Then 1331 1500 
 
 2 2 
 
 2662 
 1 500 
 
 As 4162 
 
 3000 
 1331 
 
 4331 : : 11 : 11*4 1 Anf. 
 11 
 
 4162 ) 47641 ( 11*44 inches. 
 • • 45782 
 
 1859 
 Idtjo 
 
 194 
 
 Ex.2. Reijuired the lizc of a cubical box, to hold 
 400ib. of gun-powder ? Anl", 22-89 inches. 
 
 PROBLEM XI, 
 
 ^0 find what Length of a Cylinder luill be filled hj a Given 
 Weight of Gun-ponvder, 
 
 Multiply the weight in pounds by 3S'2 ; then divida 
 the i^redud by the fquare of the diameter in inches, for 
 the length. 
 
 EXAMPLES, 
 
 1. What length of a 3() pounder gan, of 6| inches 
 diameter, will be filled with 12lb. of powder? 
 
 o 4 
 
 6| 
 
309 riLIN* OF 
 
 6} = V 3S»2 
 
 its fq. = ♦|* 12 
 
 458-4 
 9 
 
 400 ) 4125-0' 
 
 Anf. 10*3 14 iiichei. 
 
 Ex. 5.' Whatlength of a cylinder of 8 inches diameter 
 may be filled with 201b of powder ? Anf. 1 If*. 
 
 • r TUK 
 
 PILING 
 
 Of 
 
 BALLS AND SHELLS. 
 
 ■J RON balls and {hells are commonly piled, by horizontal 
 -*• courfes, cither ia a pyramidical or in a wedgt-likc 
 form ; the bafe being either an equilateral triangle, or a 
 fquare, or a re^angle. In the triangle and fquare, the 
 pile finiflies in a fing^le ball ; but in the reftangle it finilhcs 
 in a lingle row of balls, like an edge. 
 
 In triangular and fquare piles, the number of horizontal 
 rows, or courfes, is always equal to the number of balls 
 in one fide of the bottom row. And in rectangular piles, 
 
 the 
 
BALLff AND SHELLS. 301 
 
 the number of rows is equal to the number of balls in the 
 breadth of the bottom row. Alfo the number in the top- 
 row, or edge, is one or more than the difference between 
 the length and breadth of the bottom row. 
 
 PROBLEM I. 
 
 ^ojind the Number of Balls in a TriaKguIar Pile. 
 
 Multiply continually together, the number in one fide 
 of the bottom row, that number increafed by I, and the 
 fame number increafed by '2 ; then, take ~ of the laft pro- 
 iuft for the anfwer.. 
 
 EXAMPLES^. 
 
 1, Required the number of balls in a triangular pile, 
 each fid^j of the bafe containing 30 bails? 
 32 
 31 
 
 32 
 
 992- 
 30 
 
 6)'297<S0 , 
 Anf. 4t)6o 
 
 Ex. 2. How many balls are in the triangular pile, each 
 ilde of the bafe containing 20? Atif. l^-tO. 
 
 PROBLEM If, 
 To find the Number of Balls in a Square Pih, 
 
 Multiply continually together, the number in one^ fide 
 •f the bottom courfe, that number increafed by ij and 
 o. 5. doubia 
 
302 nuNTfi Of 
 
 double the fame"* number incrcafed by 1 ; then take | of 
 the laft produft for the anfwer. 
 
 EjXAMPLES. 
 
 1 . How many balls are in a fquare pile of 30 rows ? 
 
 61 
 31 
 
 6l 
 183 
 
 IS91 
 30 
 
 6 ) .16730 
 Anf. 9455 
 
 Ex. 2. How many balls are there in a fqiiare pile of 20 
 lews ? Aof. 2870. 
 
 PROBLEM III. 
 
 To -find the Number of Bulls in a Re B angular Tile. 
 
 From 3 times the namber in the length of the bafe row, 
 fubtraft one lefs than tie breadth of the fame, nuiltipiy 
 the remainder by the faivl breadth, and the prodii6l by 1 
 more than the famej and divide by 6" foe the anfwer. 
 
 EXAMPLES. 
 
 1. Required the number of balls in a reftangular 
 jflile, tVe length and breadth of the bafe row being 46 
 and 15 ; 
 
 IJ 46 
 
BALLS AND SHELLS. S03 
 
 111()0 
 
 I860 
 
 6 ) 29760. 
 Anf. 4960 
 
 Kx. 2. How many fhot are in a redlangular complete 
 pile, the length of the bottom courfe being 59» and its 
 breadth 20? Anf. 11060, 
 
 PROBLEM IV. 
 
 To find the Number of Balls in an h complete Vile, 
 
 From the number in the whole pile, confidered as com* 
 plete, fubtraft the number in the upper pile which \% 
 wanting at the top, both computed by the rule for their 
 proper form ; and the remainder will be ti:.e number in 
 the fruilura, or incomplete pile. 
 
 EXAMPLES. 
 
 1 «, To find the number of fhot in the incomplete trian- 
 o 6 gular 
 
S©** PILIN* OF BALLS AND SHELLS. 
 
 gular pile, one fide of the bottom courfc being 40, and 
 the top courfe i20. 
 
 19 
 20 
 
 40 
 41 
 
 3 SO 
 21 
 
 l6iO 
 
 42 
 
 380 
 
 760 
 
 3280 
 6560 
 
 7980 
 
 68880 
 7980 
 
 
 6 ) 609000 
 
 10150 Anfwer. 
 
 Kk. 2. How many (hot are in the incomplete triangular 
 pile, the fide of the bafe being 24, and of the top 8 ? 
 
 Anf. 2516. 
 Ex. 3. How many balls are in the incomplete fquare 
 pile, the fide of the bafe being 24, and of the top 8 ? 
 
 Anf. 4760. 
 
 Ex. 4. How many ftiot are in the incomplete reftangu- 
 
 lar pile of 12 courfes, the length and breadth of the bafe 
 
 feeing 40 and 20 ? Anf. 6l4fc\ 
 
 #P 
 
DISTANCES 
 
 BY THE 
 
 VELOCITY OP SOUND. 
 
 BY various experiments it has been found that founc5 
 flies through the air, uniformly at the rate of about 
 114^? feet in one fecond of time, or a mile in 4j feconds. 
 And therefore by proportion any diftance may be found 
 correfponding to any given time; namely, multiply the 
 given time in feconds, by 1142, for the correfponding 
 diftance in feet ; or take ^^ of the given time, for the 
 diftancc in miles. 
 
 Note, The time for the paffege of found, in the inter- 
 val between feeing the flafh of a gun, or lightning, and 
 hearing the report, may be obferved by a watch or a fmall 
 pendulum. Or> it may be obferved by the beats of the 
 pulfc in the wrift, counting on an average, about 70 to a 
 minute in perfons in moderate health, or 5^ pdfations to 
 a mile, aad more or lefs according lo circumttances.. 
 
 EXAMPLES. 
 
 K After obferving a fiafh of lightning, it was 12 fe. 
 conds before I heard the thunder j required the diftance of 
 the cloud from whence it came ? 
 
 1%. 
 
S06 OF DISTANCES BY SOUND. 
 
 12 
 3 
 
 14 ) 36 ( C^ miles, the aofwer. 
 
 Ex. 2. How long, after firing the Tower guns, may 
 the report be heard at Shooter's Hill, fuppofing the dif. 
 Cance to be 8 miles in a ilraight line ? 
 
 14 
 8 
 
 3) 112 
 Anf. 37 J feconds. 
 
 Ex. 3. After obferving the firing of a large cannon at 
 a diftance, it was 7 feconds before I heard the report; 
 what was irs diftance? Anf. 1^ mile. 
 
 Ex. 4. Perceiving a man at a diftance hewing down a 
 tree with an axe, I remarked that 6 of my pulfations 
 paffed between feeing him ftrike and hearing the report of 
 the blow ; what was the diltance between us, allowing JO 
 pulfes to a minute? ' Anf. 1 mile and 198 yards* 
 
 Ex. 5. How far off was the cloud, from which thunder 
 iffued, whofe report was 5 pulfations after the flafh of 
 lightning J counting Jo to a minute? Anf. 1523 yards* 
 
 MISCELLANEOUS QUESTIONS. 
 
 Qu. l.'O/HAT difference is there between a floor 
 ^^ 28 feet long by 20 broad, and two others 
 each of half the dimenfions ; and what do all three come 
 to at 45s. per lOU fquare feet? 
 
 Anf. dif. 280 fq. feet. Amount 18 guineas* 
 
 2. An 
 
MISICELLANEOUS QU'I&STIONS. 307 
 
 2. An clna plank is 14 feet 3 inches long, and I would 
 have juft a fquarc yard flit off it; at what diftance from 
 the edge mnft the line. be ftruck ? Anf. 7tVt inches. 
 
 3. A ceiling contains 114 yards 6 feet of plaftering, and 
 the room 28 feet broad; what is the length of it? 
 
 Anf. 35| feet 
 
 4. A common joift is 7 inches deep, and 2-^ thick ; but 
 I, want a fcantling juft as big again, that fliall be 3 inches 
 thick ; what will the other diraenfion be ? 
 
 Anf. llf inches. 
 
 5. A wooden trough coft me 3s. 2d. painting within, at 
 6d. per yard; the length of it was J 02 inches, and the 
 depth 21 inches ; what was the width ? 
 
 Anf. 27^ inches. 
 
 6. If ray court-yard be 47 feet g inches fquare, and I 
 have laid S foot-path with Purbeck ftone, of 4 feet wide, 
 along one fide of it ; what vnll paving the reft wi;h flints 
 come to, at 6d. per fquare yard ? Anf. £5 l6' 0|. 
 
 7. A ladder, 40 feet long, may be fo planted, that it 
 flial] reach a window 33 feet from the ground on one fide 
 of the ftreet ; and, by only turning it over, without mov- 
 ing the foot out of its place, it will do the fame by a win- 
 dow 21 ffct high on the other fide; what is the breadth 
 of the flreet ? Anf. 66 feet 7^ iuches. 
 
 8. The paving of a triangular court, at 18d. per foot, 
 came to 1 OOl. ; the longeft of the three fides was 88 feet 5 
 required the fura cf the other two equal fides. 
 
 •Anf. 106-85 feet. 
 
 9. There are two columns in the ruins of Perfepolis left 
 fl;anding upright : the one is 6'4 feet above the plain, and 
 the other 50 : in a ftraight line between thefe, ftands aa 
 ancicHt fmall ftatue, the head of which is 07 feet frora 
 the fummit of the higher, and 80' feet from the top of 
 the lower column, the bafe of which raeafures juft 76 feet 
 to ihe centre of the figure's bafe. Required the diftance 
 between the tops of the two columns? 
 
 Anf. 157 feet nearly. 
 
 S 10. The 
 
 V 
 
 Ojc 
 
ZO^ MISCELLAKIOUS 
 
 10. The perambulator, or furveying wheel, is (b con- 
 trived, as to turn juft twice in the length of a pole, or 
 l6| feet; required the diameter? Anf, 2*626 feet. 
 
 11. In turoing a one-horfe chaifc within a ring of a 
 certain diameter, it was obferved that the outer-wheel 
 made two turns while the inner made but one: the wheels 
 were both 4 feet high ; and, fuppofing them fixed at the 
 ftatutable diftance of 6 ^et afunder on the axle-tree, what, 
 was (he circujnference of the track defcribed by the outer 
 v»heel? Anf. 63 feet nearly. 
 
 12., What is the fide of that equilateral triangle whofe 
 area coft as much paving at 8d. a-foot, as the pallifading 
 the three fides did, at a guinea a yard ? Anf. 72*7 4-() feet. 
 
 13. In the trapezium ABCD are given, AB ~ 13, 
 BC =: 31i^, CD =: 24, and DA rz 18, alfo B a right- 
 angle; required the area? Anf. 4 10* 122. 
 
 14. A re of, which i» 24 feet S inches by 14 feet 6, 
 inches, is to be covered with lead at 8lb, to the fquarc 
 fool: what will it corae to at 18s. per cwt ? 
 
 Anf. ^22 19 10^. 
 
 15. Having a rectangular marble flab, 58 inches by 27> 
 I would have a fcjuare foot cut off parallel to the (horter 
 edge; I would then have the like quantity divided from the 
 remainder, parallel to the longer fide ; and this alternately 
 repeated, till there (hall not be the quantity of a foot Icftr 
 what will be the dimenfions of the remaining piece ? 
 
 Anf. 207 inches by 6-086. 
 
 16. Given two fides of an obtufe-angled triangle, v»hicb 
 are 20 and 40 poles; required the third fide, that the tri- 
 angle may contain juft an acre of land ? 
 
 Anf 58-876 or 23099. 
 
 17. The end wall of a houfe is 24 feet 6 inches in 
 breadth, and 40 feet to the eaves; y of which is two. 
 bricks thick, i more is l:^ brick thick, and the reft 1 
 ferick thick, Now the triangular gable rifea 38 courfes 
 
 oi 
 
tWESTIONS. $0p 
 
 of bricks, 4 of which ufuatly make a foot in depth, and 
 this is but 4~ inches, or half a brick thick : what will 
 this piece of work come to at 51. 10s. per ftatute rod? 
 
 Anf./'20 11 7h 
 
 18. If from a right-angle triangle, whofe bafe is 12, 
 and perpendicular l6' feet, a line be drawn parallel to the 
 perpendicular, cutting off a triangle whofe area is 24» 
 fquare feet ; required the fides of this triangle ? 
 
 Anf. 6, 8, and 10. 
 
 19. The ellipfe in Grofvenor-fquare meafures 840 links 
 acrofs the longeft way, and 612 the ftiorteft, within the 
 fails : now the walls being 14 inches thick-, what ground 
 d« ihey inclofc, and what do they ftand upon? 
 
 . ^ 5 Jnclofe 4 ac r 6 p 
 '^"*- I ftand on 1760^ fq. feet. 
 
 20. If a round pillar, 7 inches over, has 4 feet of ftone 
 in it ; of what diameter is the column, of equal length, 
 that contains 10 times as much? Anf. 22*136 inches. 
 
 21 . A circular fifh-pond is to be made in a garden, thai 
 Ihall take up juft half an acre; what muft be the length of 
 the cord that ftrikes the circle ? Anfwer 27| yards. 
 
 22. When a roof is of a true pitch, the rafters are 1 of 
 the breadth of the building: now fuppofing the eaves- 
 boards to projeft 10 inches on one fide, what will the new 
 ripping a houfe coft, that meafures 32 feet 9 inches long, 
 by 22 feet 9 inches broad on the flat, at 15s. per fqjarc? 
 
 Anf. ^8 15 91. 
 
 23. A cable which is 3 feet long, and 9 inches in com- 
 pafs, we'ghs 22lb : what will a fathom of that cable weigh, 
 which meafures a foot about ? Anf. /S^lb, 
 
 24. My plumber has put 2Slb per fquare foot inio a 
 ciftern r4 inches, and twice the thicknefs of the lead 
 long, 26 inches broad, and 46 deep; he has alfo put three 
 ftays acrofs it within, l6 inches deep, of the fame 
 ftrength, and reckons 22s. per cwt, for work and ma- 
 terials. I, being a mafon, have paved him a worklhop, 
 22 feet 10 inches broad, with Purbeck ftone, at 7d. per 
 
 foot J 
 
SlO MIS€£LI,AKBQUS 
 
 fool ; and upon the balance I find there is 3s. 6d. due t» 
 him. What was ihc length of the workfhop ? 
 
 An f. 3-2 f. OA inches* 
 
 C5. The diftancc of the centres of two circles, vvhofc 
 diameters are each 50, being given equal to 30; what 
 is the area of the fpace inclofed by their circumferences? 
 
 Anf. 555-119. 
 
 q6. If CO feet of iron raib'ng weigh half a ton, when 
 the bars are an inch and a quarter fquare, what will f>0 
 feet con»e to, at S^d, per lb. the bars being put -^- of an 
 inch fquare? Anf. /20 2. 
 
 27. The area of an equilateral triangle, whofc baf« 
 falls on the diameter, and its vertex in the middle of the 
 arc of a femicircle, is equal tOrlUO : what is the diameter 
 of the femicircle? Anf. 26"'32U8. 
 
 28. It is required to find the thickncfs of the lead in a 
 pipe, of an inch and quarter bore, which weighs 14 lb per 
 yard ia length; the cubic foot of lead weighing IJ'325 
 ounces. Anf. •20737 inches. 
 
 29. Suppofing the expence of paving a femicircular 
 plot, at 2s. 4d. per foot, come to lOl. what is the diamc- 
 terofit? Anf. 14*7737. 
 
 30. What is the length cf a cord, which cuts off ^ of 
 the area, from a circle whofe diameter is 289 ? 
 
 Anf. 278*671^. 
 
 31. My plumber has fet me upaciftern, and his (hop* 
 book being burnt, he has no means of bringing in the 
 charge, and I do not choofc to take it down, to have it 
 weighed; but my meafure he fi-ds it contains 6'4 ^'o fl^are 
 feet, and that it is precifely ^ of an inch in thicknefs. 
 Lead was then wrought at 211. per fother of 191 cwt. It 
 is required from thefe items to make out the bill, allowing 
 6|- oz. for the weight of a cubic inch of lead ? 
 
 Anf. ^4 11 2. 
 
 3^. What will the diameter of a globe be, when the 
 
 folidiry and fuperficial content arc cxprcffed by the fame 
 
 number? Anf. 6". 
 
 • 33. A 
 
QUESTIONS, 31 i 
 
 33. A fack that would hold 3 bufhels of corn, is 22^ 
 inches broad when empty; what will that fack contain 
 which, being of the fame length, has twice its breadth or 
 circumference ? Anf, 12 bufhels. 
 
 31'. A carpenter is to put an oaken curb to a round well 
 at 8d. per foot fquarej the hreaith of the curb is to be7|: 
 inches, and the diameter within 3^ feet : whax will be the 
 cxpencc? Anf. 5s. 2|'d, 
 
 35. A gentleman has a garden 100 feet long, and bO 
 feet broad; now a gravel walk is to be made of an equal 
 width all round it : what rauft the breadth of the walk be, 
 to take up juft half the ground ? Anf. 25"i)6"8 feet, 
 
 36. A may-poh whole top, being broken off by a blaft 
 of wind, flruck the ground at 15 feet dillance from tht 
 foot of the pole; what was the height of the whole may- 
 pole, fuppofing thq length of the broken piece to be 39 
 fstt ? ' Anf. 7-> ^Qct^ 
 
 3T» Seven men bought a grinditig ftone, of CO inches 
 diameter, each paying y part of the expence ; what pari 
 of the diameter muft each grind down for his (liare ? 
 
 Anf. the 1ft 4-4508, 2d 4-84.00, 3d 0-3535, 
 
 4th 6'0765, 5th 7*2079, Gth 9*3935, 7 th 22-()778, 
 
 38. A malttter has a kiln, which is i6 feet 6 inches 
 
 fquare : but he wants to pull it dov/n, and build a nevr 
 
 one, that may dry three times as much at once as the old 
 
 one ; what muft be the length of its fide ? 
 
 Anf. 28 feet 7 inches. 
 
 39- How many 3 inch cubes may be cut out of a 12 
 
 inch cube? Anf. 64. 
 
 40. How long muft be the tether of a horfe, that will 
 allow him to graze, quite around, juft an acre of ground? 
 
 Anf. 39i yards, 
 
 41. What will the painting of a conical Ipire come to 
 at 8d. per yard ; fuppofing the height to be 118 feet, and 
 the circumference of the bafe 6"4 feet ? 
 
 Anf. ^li 8|, 
 
 42. The diameter of a, ftandard com buflul is I8i 
 
 inchefij 
 
912 MUCELfANEOVS 
 
 inchcj, and its depth 8 inchrs ; what muft be the diameter 
 of that bulhel be, whofe depth is 7^ inches? 
 
 Anf. 19-l06r. 
 
 43. Suppofe the bnll on the top of St. Paul's church is 
 6 feet in diameter; what did the gilding of it coft, at 3*d, 
 per fquare inch ? Av\i, £Td7 10 1. 
 
 44. What will a fruflum of a marble cone come to, at 
 12s. per folid foot : the diameter of the greater end being 
 4 feet, that of the lefs end Ji, and the length of the flant 
 fide 8 feet? Anf. ^30 1 101, 
 
 45. To divide a cone into three equal parts by fe<5tion8 
 parallel to the bafe, and to find the altitudes of the three 
 parts, the height of the whole cone being 20 inches ? 
 
 Anf. the upper part 13*867, 
 
 the middle part 3*604, 
 
 the lower part 2'528. 
 
 46. A gentleman has a bowling-green 300 feet long, 
 and 200 feet broad, which he would raifc 1 foot higher, 
 by means of the earth to be dug out of a ditch that goes 
 round it ; to what depth muft the ditch be dug, fuppofing 
 its breadth to be every where 8 feet ? Anf. 7^-J. feet. 
 
 47. How high above the earth muft a perfon be raifed^ 
 that he may fee j of its furface ? 
 
 Anf. to the height of the earth's diatneter. 
 
 48. A cuMc foot of brafs is to be drawn into a wire of 
 ^ of an inch in diameter; what will the length of the 
 wire be, allowing no lofs in the metal ? 
 
 Anf. 97784797 yards, or 53 m les 984*797 yards. 
 
 49. Of what diameter muft the bore of a cannon be, 
 which is caft for a ball of 24lb weight, fo that the dia- 
 meter of the b ire may be -^^ of an inch more than that 
 of the b?ll, and fuppofing a ylb bdll lameafure 4 inches 
 in diameter? Anf. 5.757 inches. 
 
 50. Suppofing the ditmeter of an iron 9lb ball to be 4 
 in< hcs, as it is very nearly ; it is required to find the- 
 diameters of the feveral b«lls weighin- 1, 2, 3, 4, 6", 9» 
 12, 18, 24, 36^ and 42 lb, and the caliber of their guns; 
 
 allowing 
 
%lJI»TIONS. 
 
 3i: 
 
 aiiowing -yV 
 windage. 
 
 of the caliber, or ^^ of tlie ball's diameter fot 
 Anfwer, 
 
 Wt 
 
 Diameter 
 
 Caliber 
 
 Uall 
 
 1 
 
 ball 
 
 gun 
 
 1-9230 
 
 l-96"22 
 
 2 
 
 2-4.228 
 
 2-4723 
 
 3 
 
 2-7734 
 
 2-8301 
 
 4 
 
 3-0526' 
 
 3-1149 
 
 6 
 
 3-4943 
 
 3'5656 
 
 9 
 
 4'0000 
 
 4-0S16 
 
 12 
 
 4-4025' 
 
 4-4924 
 
 18 
 
 5-0397 
 
 5-1425 
 
 24- 
 
 5-546'9 
 
 5-^601 
 
 36 
 
 6'-3496' 
 
 6'-4792 
 
 42 
 
 6''6844 
 
 6'-8208 
 
 51 Soppofing the windage of all mortars be allowed t« 
 ^c ^ of the caliber, and the diameter of the hcllow part of 
 the (hell to be /^ of the caliber of the mortar; it is required 
 to determine the diameter and weight of the (hell, and the 
 quantity or weight of powder requifite to fill it, for each oi 
 the feveral forts of mortars, namely, the 13, 10, 8, 5*8, and 
 4-6 inch mortar? 
 
 Anfwer. 
 
 Cahb. 
 
 Diameier 
 
 Wt. Ihell 1 
 
 Wt. of 
 
 Wr. IheH 
 
 mort. 
 
 flitll 
 
 empty 
 
 powder 
 
 filled 
 
 4-6 
 
 4-523 
 
 8-320 
 
 0*583 
 
 8-903 
 
 5-8 
 
 5-703 
 
 16-677 
 
 1-168 
 
 17-S45 
 
 8 
 
 7'!ti()7 
 
 43764 
 
 3-065 
 
 46 829 
 
 10 
 
 9-833 
 
 85-476 
 
 5-986 
 
 91-462 
 
 13 
 
 1 9*7 83 
 
 187791 
 
 13-151 
 
 200-942 
 
 52. How many fhot are in a complete fquare pile, each 
 fide of the bafe containing 29 ? Anf. 8555. 
 
 53. How 
 
514 MUCELLANEOtS 
 
 53. How many (hot are in a complete oblong pile, the 
 length of the bafe containing 49, and the breadth 19? 
 
 Anf. 817O. 
 
 54. How many fhot arc in a triangular pile, each fide 
 ef the bafe being 50? Anf. 22100. 
 
 55. How many Ihct are in an unfinifhed triangular pile, 
 the fide of the bottom being 50, and top 20 ? 
 
 Anf. 20770, 
 
 56. How many (hot arc in an unfiniflied oblong pile, 
 having the corner tow 12, and the fides of the top 40 and 
 10? Anf. 86o6\ 
 
 57. If a heavy fphere, whofe diameter is 4 inches, be 
 let fall into a conical glafs, full of water, whofe diametet 
 is 5, and altitude 6' inches; it is required to determine how 
 much water witl run over? 
 
 Anf. 26**272 cubic inches, or near -J^ parts of a pint. 
 
 58. The dimcnfions of the fphere and cone being the 
 fame as in the Ui\ queftion, and the cone only 3- full of 
 water; required what part of the axis of the fphere is im- 
 merfed in the water ! Anf. '546 parts of an inch. 
 
 59. The cone being ftill the fame, and -y full of water ; 
 required the diameter of a fphere that may be juft all co- 
 vered by the water? Anf. 2-445996'. 
 
 60. -if 1 fee the flafti of a cannon, fired by a (hip in 
 diftrefs at fea, and hear the report 33 feconds after, how 
 far is (he off? Anf. 7ylj- miles, 
 
 61. Being one day ordered to obferve how far a battery 
 of cannon was from me, I counted by my watch 17 fe- 
 conds between the time of feeing the fla(h, and hearing 
 the report J how far was the battery from me ? 
 
 Anf. 3i miles. 
 
 62. An irregular piece of lead ore weighs in air 12 
 ounces, but in water only 7 ; and another fragment weighs 
 in air 1 4 ^ ounces, but in water only 9; required their 
 comparative denfities? Anf. as 145 to 132. 
 
 63. Suppofing the cubic inch of common glafs weigh 
 1'36 ounces troy, the fame of fait water .'5427^ and of 
 
 brandy 
 
QUESTIONS. 315 
 
 brandy •48p25; then a feaman having a gallon of that 
 liquor in a glafs bottle, which weighs 3^1b, troy out of 
 water, and to conceal it from the officers of the cuftoms, 
 throws it overboard. It is required to determine, if it 
 will fink, how much force will juft buoy it up ? 
 
 Anf. 12-896'8 ounces. 
 
 64. Supppofe by meafurement it be found that a fhip 
 
 of war, with its ordnance, rigging and appointments, 
 
 draws fo much water as to difplace oOOOO cubic feet of 
 
 water; required the weight of the veffel? 
 
 Anf. 1395 1^ tons* 
 
 TA^LE 
 
TABLE 
 
 OF THE 
 
 Areas of the Segments of a Circle, 
 
 \Vhofe diameter is Unity, and fuppofed to be divided into 
 1000 equal parts. 
 
 Height 
 
 Area Sfg 
 
 Height 
 
 •027 
 
 Area Seg. 
 
 Height 
 
 Area Seg. 
 
 . '001 
 
 •0. 0042 
 
 •005867 
 
 •053 
 
 •016007 
 
 •002 
 
 •000119 
 
 •028 
 
 •006194 
 
 •054 
 
 •016457 
 
 •003 
 
 •000219 
 
 •029 
 
 •006527 
 
 •055 
 
 •016911 
 
 •004 
 
 •000337 
 
 •o30 
 
 •006865 
 
 •056 
 
 •017369 
 
 •005 
 
 •000470 
 
 •031 
 
 •007209 
 
 •057 
 
 •017831 
 
 •006" 
 
 •00061 8 
 
 •032 
 
 •007558 
 
 •058 
 
 •018296 
 
 •007 
 
 0007 7P 
 
 •033 
 
 •007913 
 
 •059 
 
 •018766 
 
 •008 
 
 000951 
 
 •034 
 
 •008273 
 
 •060 
 
 •019239 
 
 •009 
 
 •001135 
 
 •035 
 
 •008698 
 
 •061 
 
 •019716 
 
 •010 
 
 •001329 
 
 ^036 
 
 •OO9OO8 
 
 •062 
 
 •020196 
 
 •Oil 
 
 •001533 
 
 •037 ■ 
 
 •009383 
 
 •003 
 
 •020680 
 
 •012 
 
 •CO 1746 
 
 •038 
 
 •009763 
 
 •064 
 
 •021168 
 
 •013 
 
 •001968 
 
 •039 
 
 •010148 
 
 •06'5 
 
 •021659 
 
 •014 
 
 •002199 
 
 •040 
 
 •010537 
 
 •066 
 
 •022154 
 
 •015 
 
 •002438 
 
 •041 
 
 •010931 
 
 •067 
 
 •022652 
 
 '016 
 
 •002685 
 
 •Oi2 
 
 •011330 
 
 •068 
 
 •023154 
 
 •017 
 
 •002940 
 
 •043 
 
 •011734 
 
 •069 
 
 •023659 
 
 •018 
 
 •003202 
 
 •044 
 
 •012142 
 
 •070 
 
 •024168 
 
 •019 
 
 •00347 1 
 
 •045 
 
 012554 
 
 •071 
 
 •024680 
 
 •020 
 
 •003748 
 
 •046 
 
 •0 1 297 1 
 
 •072 
 
 •025195 
 
 •021 
 
 •004031 
 
 •047 
 
 •013392 
 
 •073 
 
 025714 
 
 •022 
 
 •004322 
 
 •048 
 
 •013818 
 
 •074 
 
 •026236 
 
 •023 
 
 •00461 8 
 
 •049 
 
 •0142*7 
 
 *075 
 
 •020761 
 
 •024 
 
 •00492 J 
 
 •050 
 
 •014681 
 
 •076 
 
 •027289 
 
 •025 
 
 •005230 
 
 •05 i 
 
 •015119 
 
 •077 
 
 •027821 
 
 •026 
 
 •005546 
 
 052 
 
 •015561 
 
 •078 
 
 •028356 
 
AREAS or THE SEGMENTS OF A CIRCLE- 317 
 
 "flcigb 
 
 Area Sfg. 
 
 Height 
 
 Area Seg. 
 
 Height 
 
 Area Seg. 
 
 •079 
 
 •02S^9-t 
 
 •114 
 
 •049528 
 
 •149 
 
 •073161 
 
 •080 
 
 •02.9435 
 
 •115 
 
 •050165 
 
 •150 
 
 •073874 
 
 •081 
 
 •029979 
 
 •116 
 
 •050804 
 
 •151 
 
 •074589 
 
 •082 
 
 •030526 
 
 •117 
 
 '051446 
 
 •152 
 
 •075306 
 
 •083 
 
 •031076 
 
 •IKS 
 
 •052090 
 
 •153 
 
 •076026, 
 
 •084. 
 
 •031629 
 
 •119 
 
 •052736 
 
 •154 
 
 •076747 
 
 •085 
 
 •032186 
 
 •120 
 
 •053385 
 
 •155 
 
 •077469 
 
 •086 
 
 •032745 
 
 •121 
 
 •054036 
 
 •156 
 
 •07Sl9t 
 
 '087 
 
 •033307 
 
 •122 
 
 ■054689 
 
 •157 
 
 •078921 
 
 •088 
 
 •033872 
 
 •123 
 
 •055345 
 
 •158 
 
 •079649 
 
 •089 
 
 •034441 
 
 •124 
 
 •056003 
 
 '159 
 
 •080380 
 
 •090 
 
 •035011 
 
 •125 
 
 •056'6()3 
 
 •160 
 
 •081112 
 
 •091 
 
 •035585 
 
 •126 
 
 'OoJo'ZG 
 
 •161 
 
 .•081846 
 
 •092 
 
 •036162 
 
 •127 
 
 ■057991 
 
 •162 
 
 •0825 S2 
 
 •093 
 
 •036741 
 
 •128 
 
 •05S658 
 
 •163 
 
 •083320 - 
 
 •09-1 
 
 '037323 
 
 •129 
 
 •059327 
 
 •164 
 
 •084059 
 
 •095 
 
 •037909 
 
 •130 
 
 '^^59999 
 
 •165 
 
 •084801 
 
 •096 
 
 '03845)6 
 
 •131 
 
 •060672 
 
 ■166 
 
 •085514 
 
 '097 
 
 '039087 
 
 •132 
 
 ■01)1348 
 
 •167 
 
 •086289 
 
 •098 
 
 •03968O 
 
 •133 
 
 •062026 
 
 •163 
 
 •087036 
 
 •099 
 
 •040276 
 
 •134 
 
 •062707 
 
 •i69 
 
 •087785 
 
 . -IGO 
 
 •040875 
 
 •135 
 
 •063389 
 
 •170 
 
 •08S535 
 
 •lOl 
 
 •041476 
 
 •136 
 
 •064074 
 
 •171 
 
 •089287 
 
 •102 
 
 •0420SO 
 
 •137 
 
 •064760 
 
 •172 
 
 •090041 
 
 M03 
 
 042687 
 
 •138 
 
 'O65449 
 
 •173 
 
 •090797 
 
 •104 
 
 •043296 
 
 •139 
 
 •O()"6l40 
 
 •174 
 
 •09 J 554 
 
 •105 
 
 •04390s 
 
 •140 
 
 '0G6833 
 
 •175 
 
 •092313 
 
 •106 
 
 •044522 
 
 •141 
 
 •067528 
 
 .176 
 
 •093074 
 
 •107 
 
 •045139 
 
 •142 
 
 •068225 
 
 •177 
 
 •093836 
 
 •108 
 
 •045759 
 
 •143 
 
 •068924 
 
 •178 
 
 •094601 
 
 •109 
 
 •04638 1 
 
 •144 
 
 '•069025 
 
 •179 
 
 •095326 
 
 •no 
 
 •047005 
 
 •14v5 
 
 •070328 
 
 •180 
 
 •0.96134 
 
 •ni 
 
 •047632 
 
 •146 
 
 '07 1033 
 
 •181 
 
 •096003 
 
 •112 
 
 •048262 
 
 •147 
 
 •071741 
 
 •182 
 
 •097^7-4 
 
 '\VJ 
 
 •OV8S94 
 
 •148 
 
 •072450 
 
 •183 
 
 •008 1-47 
 
313 
 
 THE AULAS OF THfi 
 
 Hc.giii 
 
 Aica Srg. 1 
 
 Heigr.r j 
 
 Area se^. 
 
 Hdgit 
 
 Area S^g. 
 
 •184 
 
 •09^:21 
 
 •219 
 
 •127285 
 
 •25 i 
 
 •157019 
 
 •1S5 
 
 •09099r' 
 
 «.>o 
 
 *22l 
 
 •128113 
 
 •255 
 
 •157890 
 
 •186' 
 
 •100774 
 
 •1289+2 
 
 •256 
 
 •158762 
 
 •187 
 
 •101553 
 
 •222 
 
 •129773 
 
 •257 
 
 • 1 59636 
 
 •188 
 
 •102334 
 
 •223 
 
 •130605 
 
 •25» 
 
 •1^0510 
 
 •189 
 
 •1031 16 
 
 •224 
 
 •131438 
 
 •-259 
 
 •16 1 386 
 
 • 1 <)() 
 
 • 1 03<X)0 
 
 •225 
 
 •132272 
 
 •260 
 
 •162263 
 
 •191 
 
 •101085 
 
 •226 
 
 •133108 
 
 •261 
 
 •163140 
 
 •192 
 
 •105472 
 
 •227 
 
 •133945 
 
 •262 
 
 •164019 
 
 •193 
 
 •106261 
 
 •228 
 
 •13478+ 
 
 •263 
 
 •164899 
 
 •194 
 
 •107051 
 
 •22Q 
 
 •135624 
 
 •264 
 
 •1657^0 
 
 ■195 
 
 •107842 
 
 •230 
 
 •136465 
 
 •265 
 
 •166663 
 
 •19(^ 
 
 •10863() 
 
 •231 
 
 •137307 
 
 •266 
 
 •167546 
 
 '197 
 
 •109+36 
 
 •232 
 
 •138150 
 
 •267 
 
 •168430 
 
 ^•198 
 
 •110226 
 
 •233 
 
 •138.995 
 
 •268 
 
 M69315 
 
 •199 
 
 •111024 
 
 •s^3+ 
 1^35 
 
 •139841 
 
 •269 
 
 •170203 
 
 •200 
 
 •111823 
 
 ' 1 40688 
 
 •270 
 
 •17 1089 
 
 •201 
 
 •112624 
 
 •236 
 
 •141537 
 
 •271 
 
 •171978 
 
 •202 
 
 •113426 
 
 •237 
 
 •142387 
 
 •272 
 
 •172867 
 
 •203 
 
 •114230 
 
 •238 
 
 •143238 
 
 •273 
 
 •173758 
 
 •204 
 
 •115035 
 
 •239 
 
 •144091 
 
 •274 
 
 •174649 
 
 •305 
 
 •1I5S+2 
 
 •240 
 
 •1449+4 
 
 •275 
 
 •175542 
 
 •206 
 
 •116650 
 
 •241 
 
 •145799 
 
 •276 
 
 •176435 
 
 •207 
 
 •117460 
 
 •243 
 
 •146655 
 
 'V7 
 
 •177330 
 
 •208 
 
 •118271 
 
 •243 ' 
 
 •147512 
 
 •27s 
 
 •178225 
 
 •5^09 
 
 •110083 
 
 •244 
 
 •148371 
 
 •279 
 
 •179)23 
 
 •210 
 
 '119^97 
 
 •245 
 
 • 1 49230 
 
 •280 
 
 •1800 19 
 
 •211 
 
 •120712 
 
 •246 
 
 •150091 
 
 •281 
 
 •I8O918 
 
 •212 
 
 M 2 1529 
 
 •24)» 
 
 •150953 
 
 •2S2 
 
 •181817 
 
 •213 
 
 •122347 
 
 •248 
 
 •15J816 
 
 •2S3 
 
 •182718 
 
 '•21 + 
 
 •123167 
 
 •249 
 
 •152680 
 
 •284 
 
 •1 83619 
 
 /215 
 
 •123988 
 
 •250 
 
 '153516 
 
 •285 
 
 •184521 
 
 •216 
 
 •124S10 
 
 •251 
 
 •154412 
 
 •2S6 
 
 •185425 
 
 •217 
 
 •135634 
 
 •252 
 
 .•155280 
 
 •287 
 
 •1863291 
 
 1-?18 
 
 • 1 26459 
 
 •253 
 
 •156l4q 
 
 •288 
 
 • 187234 1 
 

 »E«HtfEIfT6 
 
 OJ* A cmcLz. 
 
 ID* 
 
 Height 
 
 Area Seg. 
 
 Htight 
 
 Area Seg. 
 
 Height 
 
 Area Seg. 
 
 •289 
 
 •188140 
 
 •324 
 
 •220404 
 
 •359 
 
 •253590 
 
 •290 
 
 •189047 
 
 •325 
 
 •221340 
 
 •360 
 
 •234550 
 
 '291 
 
 •189955 
 
 •326 
 
 •222277 
 
 •361 
 
 •255510 
 
 •2^2 
 
 •1 90864 
 
 •327 
 
 -223215 
 
 '362 
 
 •256471 
 
 •293 
 
 •191775 
 
 •328 
 
 •224154 
 
 '363 
 
 •257433 
 
 •294 
 
 •192()84 
 
 •329 
 
 •225093 
 
 •364 
 
 •258393 
 
 •29'y 
 
 •19359^^ 
 
 •330 
 
 •226033 
 
 •365 
 
 •25^357 
 
 '29^ 
 
 •I9+0O9 
 
 •331 
 
 •226974 
 
 •366 
 
 •260320 
 
 '297 
 
 •195422 
 
 •332 
 
 •227915 
 
 •367 
 
 •261284 
 
 •298 
 
 •196337 
 
 •333 
 
 •228858 
 
 •368 
 
 •262248 
 
 '299 
 
 •1<>^7252 
 
 •334 
 
 •229801 
 
 •369 
 
 •263213 
 
 •300 
 
 •198 168 
 
 •335 
 
 •230745 
 
 •370 
 
 •264178 
 
 •301 
 
 •199085 
 
 •336 
 
 •231689 
 
 •371 
 
 •2651-44 
 
 •302 
 
 •200003 
 
 •337 
 
 •232634 
 
 •372 
 
 •266111 
 
 •303 
 
 •200922 
 
 •338 
 
 •233580 
 
 •373 
 
 •267078 
 
 •304 
 
 •201841 
 
 •339 
 
 •2345'26 
 
 •374 
 
 •268045 
 
 •305 
 
 •202761 
 
 •340 
 
 •235473 
 
 •375 
 
 •269013 
 
 •306 
 
 •203(^83 
 
 •541 
 
 •236*21 
 
 ■376 
 
 •269982 
 
 •307 
 
 •204605 
 
 •342 
 
 •237369 
 
 •317 
 
 •270951 
 
 •308 
 
 •205527 
 
 •343 
 
 •238318 
 
 •378 
 
 • ,7 1920 
 
 •309 
 
 •206451 
 
 •344 
 
 •239268 
 
 •379 
 
 •272890 
 
 •310 
 
 •207376 
 
 •345 
 
 •210218 
 
 •380 
 
 •3738(>1 
 
 •311 
 
 •208301 
 
 •346 
 
 •241169 
 
 •381 
 
 •274832 
 
 •312 
 
 •209227 
 
 •347 
 
 •242121 
 
 •382 
 
 •275803 
 
 •313 
 
 •210154 
 
 •348 
 
 •243074 
 
 •3S3 
 
 •276775 
 
 *314 
 
 •211082 
 
 •349 
 
 •244026 
 
 •384 
 
 •277748 
 
 •315 
 
 •212011 
 
 •350 
 
 •2449S0 
 
 •385 
 
 •278721 
 
 •316 
 
 •212940 
 
 •351 
 
 •245934 
 
 •386 
 
 •279694 
 
 -.317 
 
 •213871 
 
 •352 
 
 •246889 
 
 •387 
 
 •28066s 
 
 •318 
 
 •214802 
 
 •353 
 
 •247845 
 
 •338 
 
 •281642 
 
 •319 
 
 •215733 
 
 •354 
 
 •24SS01 
 
 •389 
 
 •282617 
 
 •320 
 
 •216666 
 
 '355 
 
 •249757 
 
 •390 
 
 •283592 
 
 •321 
 
 •217599 
 
 •536 
 
 •250715 
 
 •391 
 
 •284368 
 
 •322 
 
 •218533 
 
 '357 
 
 •251673 
 
 . •392 
 
 •285544 
 
 •323 
 
 •219468 
 
 [ -358 
 
 •252631 
 
 1 "393 
 
 •286521 
 
320 AREAS OF THI SSGMSMTS OF A CIRCLE* 
 
 •39+ 
 
 AicaSrg. 
 
 Height 
 
 Area S^g. 
 
 He-ight 
 •466 
 
 Area Scg. 
 
 
 •287498 
 
 •430 
 
 •322928 
 
 •358725 
 
 
 '395 
 
 •288476 
 
 •431 
 
 •323918 
 
 •467 
 
 •35972s 
 
 
 •396' 
 
 •289452 
 
 •432 
 
 ;324909 
 
 •468 
 
 •360721 
 
 
 "397 
 
 •290431 
 
 •433 
 
 •326900 
 
 •469 
 
 •361719 
 
 
 •3.98 
 
 •291411 
 
 •434 
 
 •326892 
 
 •470 
 
 •362717 
 
 
 '399 
 
 •292309 
 
 •435 
 
 •327882 
 
 •471 
 
 •363715 
 
 
 •400 
 
 •293369 
 
 •436 
 
 •328874 
 
 •472 
 
 •364713 
 
 
 •101 
 
 •294349 
 
 •437 
 
 •329866 
 
 •473 
 
 •365712 
 
 
 •402 
 
 •295330 
 
 •438 
 
 •330858 
 
 •474 
 
 •366710 
 
 
 •403 
 
 •296311 
 
 •439 
 
 •331850 
 
 •475 
 
 •367709 
 
 
 •404 
 
 •297292 
 
 •440 
 
 •332843 
 
 •476 
 
 •368708 
 
 
 •405 
 
 •298273 
 
 •441 
 
 •333836 
 
 •477 
 
 '369707 
 
 
 •406* 
 
 •299255 
 
 •442 
 
 •334819 
 
 •478 
 
 •370706 
 
 
 •407 
 
 •300235 
 
 •443 
 
 •335822 
 
 •479 
 
 •371705 
 
 
 •408 
 
 •301220 
 
 •444 
 
 •336116 
 
 •480 
 
 •372704 
 
 
 •409 
 
 •302203 
 
 •445 
 
 •337810 
 
 •481 
 
 •373703 
 
 
 •410 
 
 •30^187 
 
 •446 
 
 •338804 
 
 •482 
 
 •374702 
 
 
 •411 
 
 •304171 
 
 •447 
 
 •339798 
 
 •483 
 
 •375702 
 
 
 •412 
 
 •305155 
 
 •448 
 
 •340793 
 
 •484 
 
 •376702 
 
 
 •413 
 
 •306140 
 
 •449 
 
 •34178? 
 
 •485 
 
 •377701 
 
 
 •414 
 
 •307 1 25 
 
 •450 
 
 ^•342782 
 
 •486 
 
 •378701 
 
 
 •415 
 
 •308110 
 
 •451 
 
 '3^3777 
 
 •487 
 
 •37.9700 
 
 
 •416- 
 
 •309095 
 
 •452 
 
 •344772 
 
 •488 
 
 •380700 
 
 
 •417 
 
 •310081 
 
 •453 
 
 •345768 
 
 •489 
 
 •38 1 699 
 
 
 •418 
 
 •31.1068 
 
 •454 
 
 •346764 
 
 •490 
 
 •382699 
 
 
 •419 
 
 •312054 
 
 •455 
 
 •347759 
 
 •49IJP 
 
 '3^3699 
 
 
 •4'^0 
 
 •313041 
 
 •456 
 
 •3^8755 
 
 •492 
 
 •384699 
 
 
 •4'2l 
 
 •31] 029 
 
 •457 
 
 •349752 
 
 ••^93, 
 
 '3S5699 
 
 
 •402 
 
 •315016 
 
 •458 ^ 
 
 •350748 
 
 •494 
 
 '3^6699 
 
 
 •423 
 
 •316004 
 
 •459 
 
 •351745 
 
 •495 
 
 '3S7699 
 
 
 •424 
 
 •316992 
 
 •460 
 
 •352742 
 
 •496 
 
 •388699 
 
 
 •425 
 
 •317981 
 
 •461 
 
 '353739 
 
 •497 
 
 •38c;699 
 
 
 •426' 
 
 •318970 
 
 •462 
 
 •35^736 
 
 •498 
 
 •3906.99 
 
 
 •427 
 
 "S 19959 
 
 '463 
 
 •355732 
 
 •499 
 
 *39i699 
 
 
 •428 
 
 •320948 
 
 •464 
 
 •356730 
 
 5-00 
 
 '39'2699 
 
 
 •429 
 
 •321938 
 
 •465 ~ 
 
 •357727 
 
 
 -• • ■« ■ '" 
 
 «' 
 
THE USE OF THE TABLE. 
 
 TN tlic foregoing table, each nuihbcr In the column of 
 ■- area /eg, is the area of the circular fegment -whofe 
 height, or the verfe fine of its half arc, is the number 
 immediafely on the left of it^ in the colamn of heights; 
 the diameter of the circle being I, and its whole area 
 •785398. 
 
 The ufe of this table is to find, by it, the area of the 
 fegment of any other circle, whatever be the diameter. 
 And this is done by firft dividing the height of any pro- 
 pofed fegment by its own diameter, and the quotient is a 
 decimal to be fought in the column of heights, and againfl. 
 it is the tabular area to be taken out, which is fimilar to 
 the prcpofed fegment. Then this tabular area, being mul- 
 tiplied by the fquare of the given diameter, will be the 
 area of the fegment required ; becaufe fimilar areas arc to 
 tach other as ihe fquares of their diameters. 
 
 EXAMPLE. 
 
 S© if it be required to find the area of a fegment of a 
 circle, whofc height is Z\, the diameter being 50, 
 
 Here 50 ) 3*25 ( *065 qtw. or tabular height, 
 and the tab. {tg, is •O2l6o.9 
 which multiply by 5500 the fquare of the diam, 
 
 ivcs 54-147500 the arw wquired. 
 
 1 Eat 
 
5!J2 the USB OF THE TAaiK, 
 
 But in dividing the given height by the diameter, if 
 the quotient do not terminate in three places of decimals 
 without a fraftional remainder, then the area for that 
 fraftional part ought lo be pieportioned for, thus ; Having 
 found the tabular area anfwering to the firft three decimals 
 of the quotient, take the difference between it and the 
 next following tabular area, which difference multiply by 
 the fradlional remaining part of the quotient, and the pro- 
 duft will be the corrcfponding proportional part, to be 
 added to the firft tabular area. 
 
 So if the height of a propofed fegment were 3j, to 
 the diameter 50. 
 
 Here 50 ) 3} ( •0()6f 
 to '066 anfwers -022154 
 the next area is •022652 
 
 their difterence is A^^ 
 
 ^ of which is 232 
 
 which added to •022154' 
 
 gives the whole tab, area 
 and this multiplied by 
 
 gives the area. 
 
 •022386' 
 2500 
 
 Then 
 
 55 '965000 fought. 
 
 BIN! S. 
 
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 vSITT 
 
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