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THE 
 
 PRACTICE 
 
 OF 
 
 PERSPECTIVE, 
 
 On the P R I N C I P L E S of 
 
 Dr. BROOK TAYLOR: 
 
 A feiies of examples, from the moft hmple, and eafy, to the molt 
 complicated, and difficult cafes* 
 
 In the courfe of which, his method is compared with thofe of fome, 
 of the moft celebrated writers, before him, on the fubje^t. 
 
 Written many years fince, but now firft publifhed. 
 
 By JOSEPH HIGHMORE. 
 
 LONDON, 
 
 Printed for A. Millar, and J. N o u R s e, in the Strand. 
 
 MDCCLXIII. 
 
% 
 
THE 
 
 PREFACE. 
 
 T here are^ already^ fo fnany treatifes on perfpeEiive^^ 
 that perhaps it may feem needlefs to add to the number 
 and it might juftly he thought impertinent to offer aty 
 thing to the public^ on this fubjeSi-i after Hr, Brook Taylor;, 
 unlefs the end propofed were different from his^^and confequently 
 differe 7 ^t means neceffary. 
 
 He has inventedy and^ in a very fhort compafs^ exhibited > an^ 
 • , i univerfal theory y, the truths and excellence^ of which is. acknow^ 
 ledged by all who have ready and confidered ity at the fafne time 
 that they complain of its obfcurity, ^he attentmiy and application 
 which the readings, and underflanding this little book requirCy eff 
 pecially with fuch as are but little converfant in geometryy has diff 
 couraged< the generality of thofe^ffor whofe fervice it was. chiefly 
 deflgnedy from the attempt ; ,fo that very flew have profited by the- 
 befl treatife that has been publifhed on the JubjeSi^ 
 
 It was firfl printed in 1715, and agam in 1719, with fo 7 ne 
 differencey^ m order to render it lefs diffmlty objeElions having been.' 
 made to the forjner editioUy:. on account of its^ intricacy i. neither- 
 of the wtprejflons is entirely foldy if we are rightly infor 7 ned *. But 
 though this author has been fludied by fewy..yet with thefe he is m 
 the highefl eflee^Uy as the. invetJtor of the true tmiverfal fyflem. 
 Now if thaty (the mofl excellent of all books on the fubjeliy) > 
 has hee?i liable to fuch objeSliojiSy as to 7 nake the labours of later 
 
 * Since the above was written, there has been another edition of Dr. Brook ‘Taylor, 
 publilhed in 1749^ faid-to bejeyjied, and correfted by Mr. Colfon, of Cambridge. 
 
 writers^. 
 
VI 
 
 PREFACE. 
 
 writersy on the fame principles y acceptable to the publicy the author 
 hopes that this traSl (the firfl written after Brook Taylor’r, as 
 he has reafon to believCy though laft publifued) will be received with 
 cafidour: And efpecially becaufe, though his defign^ m ge?teraly 
 be the fame with their Sy his man7ier of treating the fubjeSl has 
 been very differenty as he had conceived it fnight be ?nore naturally 
 adapted to the comprehenfon of learnersy for whofe ufe it was 
 principally intended. 
 
 His purpofcy and endeavour y has beeji to give the fur eft y and 
 ftjorteft rules for reprefenting all forts of objeBsy and this in a 
 populary familiar ma7inery without cojijlant ftriSi mathe7natical 
 demonftrations'y although illuftratio77Sy and eve7t de7nQnftrations-y 
 are 720t 077nttedy where they have been thought ?ieceffary. 
 
 He hady originallyy intended to fupply only what was wantmg 
 in the old perfpeBivCy which 77iight have been acceptable to thofe 
 already experienced m the ar'ty but would have been wholly uje~ 
 lefs to others. And ft77ce 7na77y have bee77 difcouragedffrom the 
 fiudyy by hearmg of the deficiency of the old 772ethQdy ai^d the difficulty 
 of co 77 iprehendmg the neWy he judged, it better to 7nakc his work as 
 CG777plete in its kindy as he could % fo as to e77ahle any oncy with a 
 co77t77ion applicatio77y to reprefent objeBsy m all poffible ftt7iatio77Sy 
 with the feweft Imes that the ttature of the thing will ad/mity and 
 without the affiftance of atiy other booh 
 
 With this view he hathy in the firft party give7t a few exatnples 
 hi a 7na7i7'ier cotntnoti to the oldy attd 72ew j'yftetny o.nd has etideavour- 
 ed to explam even this as clearlyy and comprehenfively as poffibky 
 both to render it eajy. to the learttery and alfo to prepare hwt more 
 efeBually for the other titethod. 
 
 In theJeco7id party objeBs are reprefented in both tnethodsy fe~ 
 paratelyy to fhew the advantage of the neWy 7 Wt only where the 
 old is falfcy but aljo where it is mcuntbered with unneceffary lineSy 
 and points y for waitt of the truey univeijal ptinciples : herCy ex~ 
 attiples are taken from Pozzo, and the Jefuit, the two 77ioft cele¬ 
 brated y and moft ftudied authors'y as alfo from . K.^ddky an 
 
 old 
 
Vll 
 
 PREFACE. 
 
 old French writer^ by fome^ much ejleemedy from whom the Je- 
 fuit has borrowed, with proper acknowledgment to the merit of 
 Monf Defargues, on whofe principles 'Bolii^profejfes to have writ¬ 
 ten. And in the courfe of this part^ feveral mfiakes of thefe au¬ 
 thors are remarked. 
 
 This fecond part may he confideredas a comparative perfpe&ive, 
 and will be acceptable to thofe who are already acquainted with 
 mojl of the methods of projeSHon^ though they may not have taken 
 the pains to make fuch comparifon ; but it is principally defigned 
 to Jhew the great advantages of the new methady and to excite the^ 
 ftudentSy in this fcience, to render themfelves mafiers of it ; whichy 
 although it may require more application at firfl, will ettable them, 
 afterwards, to execute what^er they undertake with more cer¬ 
 tainty y and expedition, than any other^ 
 
 Fhofe whofe curiofity may not detain them to examine the fe^ 
 veralfchemes of this fecond part, may p ofs direEily from the firft, to 
 the third partwherein the five regular folids are proje&ed , which 
 examples are chofeii, as furnifing occafion for almofl every cafe 
 that has any difficulty, in perfpeSlive ; injomuch, that whoever 
 fully comprehends the diagrams, -and can projeSi thefe objeSis, 
 will (it is apprehended) find the projeSlmi of all others eafy. 
 
 In this, and the following parts, are many things which the 
 author prefumes are entirely new, at leaf, he has never met with 
 them elfewhere. 
 
 , Dhe learner, however, is advifed, not to content hunfelf with a 
 mere hfpeSiion of the diagrams, nor even with perfomning the 
 prohle?ns as here exhibited, only, but to projeSl the fame objetls in 
 various fituations, till he finds himfelf perfeB both m the p?inci- 
 ples and praSiice ; he is afo advifed, to begin thefe operations 
 with a fmall difance, that fo all, or 7 nofl, of the vanifbmg points 
 may be found wit bin the limits of his paper , hut when he fall 
 have acquired a facility in the execution, he may take what clij- 
 tance he pleafes, and if any difficulties arije, on that, or any 
 other circumjlance, he will find in the next, 
 
 3 And 
 
viii 
 
 PREFACE. 
 
 And fourth fart^ expedients for them^ this being the place 
 frequently referred to^ in the courfe of the treatife^ for ohviatmg 
 federal inconveniencies that may happen from want of fpace^ as 
 well as for many otherfchemes^ of great utility in pra&ice\ thefe 
 were refervedfor this part on purpofe^ that the learner^ by hav- 
 wg gradually advanced thus far ^ anight he 7nore fetfble of them 
 ufefuhefsy and fo apply hhnfelf %mth the more eagetmefs^ amd 
 pleafure^ to cotnprehend them. 
 
 "The ffth^ and lafl part.^ treats of the 7nan7'ier of fi7idmg the 
 fjadows of objeSls on divers pla7tes.^ attd the hnages of objeCls vi 
 rejleSiing planes., but briefly., as being of lefs ufe than the for- 
 7 ner parts, which are abfolutely neceffary. Both floadows, and re- 
 fleSiions, are wholly otnitted by Pozzo, though fo great a 7}iajler 
 in the praSlice of perfpeSiive. If he Jefuit has exatnples of floa- 
 dows cafl on planes, but is flrangely miflaken m fo7ne of them, as 
 well as in the precepts with which they are acco77tpanied-i as is floewn, 
 where they are particularly 7nentioned, 
 
 ADVERTISEMENT. 
 
 As the Author was near fixty miles from London, while this Work was printing, 
 it is hoped the following errors of the prefs will be the more eafily excufed: 
 And the Reader is particularly defired to corredl them, with his pen, before he 
 begins the book, becaufe the fmalleft errors, in thefe Subjects, perplex the fenl'e, 
 and in fome cafes entirely pervert it; efpecially where letters of reference are 
 miftakcn. 
 
 page Line E R R A T A. 
 
 8 5 from the bottom, for f, read f, 
 
 15 6 —-—-- for diftance o, read dijlance', o, 
 
 10 ' -- hr from nxihich, read and 
 
 11 -- - - for having been, read method vjas, 
 
 20 5 from the top, for D, read D, 
 
 22 8 from the bottom, read, by dravedng from S, through 4, to the ground, 
 
 64 4 from the top, dele iTc, and begin the fentence with PrW 
 
 5 - after the words5 of this nianijhing line, infert, Then 
 
 ig -after 3, 2, 10, make a full flop, and then read, Js 
 
 65 1 I from the bottom, for Fig. T.' jSelow is, read Fig. 1. Is • 
 
 87 6 - for e, read e, 
 
 93 14 from the top, the lafl; letter in the line, for S, read B, 
 
 94 I for V., read U, 
 
 98 24 for V, read U, 
 
 99 8 from the bottom, for (nvhich nuill be perffeUively perpendicular to the 'vanijhing line 
 
 a, C, b, andy of \ ,V, and all its parallels, dravo U, P, cutting the line a, C, b, in V, 
 Read, 9^^ i, P, and of all its parallels ; draw U, P, which will be perfpedlively 
 perpendicular to the aianijhing line a, C, b, and will cut it in V, 
 
 THE 
 
] 
 
 THE 
 
 INTRODUCTION. 
 
 S INCE the completion of this treatifb in the order propofed, it 
 has been thought proper to prefix a few of the firft principles 
 of geometry, to facilitate the progrefs of fuch readers who may not 
 have been converfant in thefe ftudies. 
 
 Definitions from Euc/ufs Elements. 
 
 Fig, I. A point is confidered as having no parts, as A. 
 
 2. A line is confidered as having no breadth, as A, B. 
 
 3. The extremities of a line are points. 
 
 4. A right (or ftraight) line, is that which lies equally between its 
 points, or is the fliorteft that can be drawn from point to point, as 
 A, B, fig. 2. 
 
 5. A fuperficies is that which hath only length and breadth, without 
 depth, or thicknefs, as A, B, C, D, fig. 3. 
 
 6. The extremes or ends of a fuperficies are lines. 
 
 7. A plain fuperficies is that which lies equally between its lines. 
 
 8. A plain angle, B, A, C, is an inclination of two lines in a plane 
 to each other, as A, B, and A, C, the one touching the other, as in 
 the point A, fig. 4. 
 
 N. B. The fecond, or middle letter, is always the angular point. 
 
 9. When the lines which contain the angle are right (or flraight) 
 lines, it is called a right-lined angle. 
 
 If both be curved, it is a curve-lined angle; if one be curved, 
 and the other right, it is a mixed angle. 
 
 a 
 
 Fig; 
 
X 
 
 T:he INTRODUCTION. 
 
 Fig. 10. When a right line, as A, B, ftanding upon aright line, as C, D, 
 makes the angles on each fide equal, then both of them are right 
 angles, and the right line B, is called a perpendicular to C, 
 
 %• 5 * 
 
 11. An obtufe angle is that which is greater than a right angle, as 
 E, B, C, fig. 5. 
 
 12. An acute angle is that which is lefs than a right angle, as E, B, D, 
 
 5 * 
 
 13. A circle is a plain figure comprehended by one line, which is called 
 a circumference, to which all right lines drawn from the point in the 
 middle of the figure (called its center) are equal, as C, A,—C, B,— 
 
 C, D, fig. 6. 
 
 14. The diameter of a circle is a right line, as A, B, drawn through 
 the center C, and being terminated by the circumference, on either fide, 
 divides the circle into two equal parts. 
 
 15. A femicircle is contained by the diameter, and half the circumfe¬ 
 rence, as A, D, B, fig. 6. 
 
 16. Of trilateral, or three fided figures, that which hath three equal 
 Tides, is called an equilateral triangle, as A, B, C, fig. 7. 
 
 17. That which hath only two fides equal is called an ifofceles tri¬ 
 angle, as A, B, C, fig. 8. 
 
 18. And that which hath all the three fides unequal, is called a fcale- 
 num, as A, B, C, fig. 9. 
 
 19. But that which hath one angle right is called a right-angled tri¬ 
 angle, as A, B, G, fig. 10. 
 
 And the fide oppofite to the right angle is called the hypothc- 
 nufe, as A, B, 
 
 20. Of quadrilateral figures, the fquare is that which hath the four 
 fides equal, and the four angles right, as A, B, C, D, fig. ii. 
 
 21. An oblong, or long fquare, is re^fanglcd, but not equilateral, as 
 A, B, C, D, fig. 12. 
 
 .22. A rhombus, is a figure equilateral^ but not right-angled, as 
 A, B, C, D, fig, 13, 
 
 Fig* 
 

 

 ' ■ h' 
 
 / 
 
 
 1 
 
 i 
 
 
 •M ' • '. 
 
 
Xi 
 
 He INTRODUCTION. 
 
 Fig. 23. A rhomboides, hath the oppofite fides and angles equal, but is 
 neither equilateral, nor right angled, as A, B, C, D, fig. 14. 
 
 24. All other quadrilateral figures (being irregular) are called trape¬ 
 ziums, as A, B, C, D, fig. 15. 
 
 25. Parallels are right lines in the fame plane, which being infinitely 
 prolonged on both parts, would never meet, as A, B and C, D, fig. 16. 
 
 26. A parallelogram is a quadrilateral figure, whole oppofite fides arc 
 parallel, as A, B, C, D, fig. 12, and 14. 
 
 27. When in a parallelogram, as A, B, C, D, fig. 17. there is drawn 
 a diameter (or diagonal) A, C, and two right lines G, H, and F, E, 
 parallel to the fides, cutting the diameter in the fame point I, fo that 
 the parallelogram be divided into four parallelograms, thofe two, 
 I, E, D, H, and I, F, B, G, through which the diameter doth not 
 pafs, are called complemeiits, but the two others, I, E, A, G, and 
 I, F, C, H, through which it doth pafs, are faid to be about the dia^ 
 meter. 
 
 Some Propofitions from the firft, fecond, third, and 
 fixth, books of Euclid\ Elements. 
 
 PROP. I. PROBLEM. 
 
 Upon a given right line A, B, (fig. 18.) to make an equilateral 
 triangle A, B, C. 
 
 On the center A, at the diftance A, B, deferibe the circle B, C, D, 
 and on the center B, at the fame diftance B, A, deferibe the circle 
 A, C, E, and from the point C, where the circles interfedl one an¬ 
 other, draw the two right lines C, A, and C, B. Then A, B, C, will 
 be an equilateral triangle. 
 
 For A, C, and C, B, are each equal to A, B, by conlirrudtion. 
 
 P R O P, IX. PROBLEM. 
 
 To divide a given right-lined angle B, A, C, (fig, 19.) into two equal 
 parts. 
 
 'a 
 
Xll 
 
 rhe INTRODUCTION. 
 
 Let there be taken, in the line A, B, a point at pleamre, D, and on 
 A, C, cut off A, E, equal to A, D, (by fetting one foot of the com- 
 paffes on A, and with the other defcribing the arc D, E ;) draw the 
 right line D, E, and on it make an equilateral triangle D, F, E, and 
 draw A, F, which v/ill divide B, A, C, into two equal angles. Or the 
 points D, and E, being found, the right line D, E, may be omitted; 
 and inftead of v/hole circles (as at the hrft Prop. hg. i8.) only mark 
 the interfedlion at F, as in this figure. 
 
 P R O P. X. PROBLEM. 
 
 To divide a given right line A, B, (fig. 20.) into two equal parts. 
 Euclid diredls here alfo to make an equilateral triangle A, C, B, on the 
 given line, and then to divide the angle C, as in the laft propofition ; that 
 is, by means of another equilateral triangle below the line ; but if the an^ 
 gular points above and below are found by interfedlion, it is fufficient. 
 
 P R O P. XI. PROBLEM. 
 
 On a given right line A, B, (fig. 21.) and from a given point therein 
 C, to raife a perpendicular C, F. 
 
 In the part C, A, take any point D, and let C, E, be taken equal 
 to C, D ; then, on D, E, defcribe the equilateral triangle D, F, E, and 
 drav^^ C, F, which will be perpendicular to A, B. 
 
 To raife a perpendicular at the end of a line A, D, fig. ir. With 
 any opening of the compaffes A, D, defcribe the arc D, c,/', and with 
 the fame opening the arc D, c, and again with the fame opening 
 laffly, with the fame c, g, and g, now draw g, A, which will be 
 perpendicular to A, D. 
 
 PROP. XII. PROBLEM. 
 
 On a given right line A, B, (fig. 22.) and from a given point Cj 
 which is not in it, to draw a perpendicular line C, F. 
 
 Let any point H, be taken on the other fideof A, B, and from the 
 point C, as a center, at the diftance C, H, defcribe the circle D, H, E, 
 
 cutting 
 
p. XII. 
 
 E 
 
 D 
 
f 
 
 
 \ 
 
 
 i 
 
 ‘i.. 
 
 /■ 
 
 
 «lV- 
 
 'i'\. 
 
Xlll 
 
 rhe INTRODUCTION. 
 
 cutting A, B, in the points D, and E, and divide D, E, into two 
 equal parts in F, and draw C, F, which will be perpendicular to 
 A, B. 
 
 PROP. XIII. THEOREM. 
 
 When a right line E, B, (fig. 5.) falls on another right line C, D, ei¬ 
 ther it makes two right angles, or tvv^o angles equal to two right angles. 
 
 Dem. For if the angle E, B, D, be equal to E, B, C, they fhall 
 be both right angles; but if it be unequal, let A, B, be drawn at 
 right angles to C, D, then A, B, D, and A, B, C, fhall be right 
 angles. Now fince E,* B, D, and E, B, A, (taken together) are 
 equal to the right angle A, B, D, if the common angle A, B, C, be 
 added, then the three angles E, B, D,—E, B, A, and A, B, C, fhall 
 be equal to the two right angles A, B, D, and A, B, C. And fince 
 the angle E, B, C, is equal to the two angles E, B, A, and A, B, C, 
 if you add the common E, B, D, the two angles E, B, D, and E, B, C, 
 fliall be equal to the three angles E, B, D,~E, B, A, and A, B, C. 
 But thefe three have been Oiewn to be equal to two right angles; there¬ 
 fore E, B, D, and E, B, C, fhall be alfo equal to two right angles. 
 \¥hich was to be demonflrated. 
 
 PROP. XXII. PROBLEM. 
 
 To conftitute a triangle F, G, K, (fig. 23.) of three right lines equal 
 to three given right lines A, B, and C. 
 
 Draw an indefinite right line D, E, and on it make D, F, equal to 
 A,—F, G, equal to B, and G, E, equal to C j and from F, as a center, 
 with the length F, D, deferibe a circle D, K, L: again, from the 
 center G, with the length G, E, deferibe the circle E, K, L, and draw 
 F, K and G, K; then the triangle F, G, K, is made of three lines 
 equal to A, B, and C. 
 
 PROP. XXIII. PROBLEM. 
 
 On a given right line A, B, (fig. 24.) and at a point given, A, to 
 3iiake an angle F, A, G, equal to a given angle D, C, E. 
 
 a Set 
 
XIV 
 
 ne INTRODUCTION. 
 
 Set one foot of the compaffes on C, and with the other foot, at 
 any diftance, defcribe the arc E, D; then, with the fame opening, fet 
 one foot on A, and defcribe the arc G, F. Take, with the com¬ 
 pares, the difrance D, E, and fet it off from G, to F, and draw 
 A , F j then the angle F, A, G will be equal to E, C, D. 
 
 P Pv O P. XXXI. P R O B L’E M. 
 
 Through a given point A, (fig, 25.) to draw a parallel to a given 
 right line B, C. 
 
 From A, draw an oblique line A, D, to the line B, C, and from D, 
 with the diftance D, A, defcribe the arc A, B j then from A, v/ith 
 the fame diftance, defcribe the arc D, E, make D, E, equal to A, B, 
 and draw A, E, which will be parallel to B, C. 
 
 PROP. XXXH. THEOREM. 
 
 Of every triangle, as A, B, C, (fig. 26.) (one fide B, being pro¬ 
 longed) the exterior angle A, C, D, is equal to the two interior, and op- 
 pofite angles A, and B. 
 
 And the three angles of any triangle, as A, B, C, are equal to two 
 right angles. 
 
 For having drawn C, E, parallel to A, B, it is evident that E, C, D, 
 muft be equal to A, B, C, and alfo that the angle A, C, E, mull be 
 equal to C, A, B. ^his is not here JlriBly demonfirated^ nor is that 
 necejfary in this introdudlioHy but the reader is referred to the pre¬ 
 ceding propofitionsy hi Euclid, for farther fatisfaSlion. Therefore 
 the exterior angle A, C, D, (compofed of them both) mufl be equal 
 to A, and B; which is the firft affertion. 
 
 Again. Since the angle A, C, D, and the angle A, C, B, taken to¬ 
 gether, are equal to two right angles (by Prop. XIII.) and fmce the 
 angle A, C, D, is equal to the angles A, and B, (as above) it follows 
 that the angles A, B, and A, C, B, which is common, (the three an¬ 
 gles of any triangle,) are equal to two right •, which was the fecond 
 affertion. 
 
 PR O P3 
 
INTRODUCTION. 
 
 XV 
 
 PROP. XXXV. THEOREM. 
 
 The parallelograms A, C, D, B, and F, C, D, E, (fig. 27.) confli- 
 tuted on the fame bafc C, D, and between the fame parallels A, B, and 
 C, D, are equal to one another. 
 
 For the demonjiration of this^ and the three following propoftions^ the 
 reader is referred to Euclid. 
 
 PROP. XXXVI. THEOREM. 
 
 Parallelograms on equal bafes, and between the fame parallels, are 
 equal. 
 
 PROP. XXXVII. THEOREM. 
 
 Triangles (being the halves of parallelograms) conftituted on the 
 fame bafe, and between the fame parallels, are equal. 
 
 PROP. XXXVIII. THEOREM. 
 
 Triangles on equal bafes, and between the fame parallels, are equaL 
 
 PROP. XLVI. PROBLEM. 
 
 On a given right line A, D, (fig. ii.) to deferibe a fquare. Draw 
 the right line A, G, perpendicular to A, D, make A, B, equal to A, D, 
 through B, draw a parallel to A, D, and through D, draw a parallel 
 to A, B. 
 
 PROP. XLVII. THEOREM. 
 
 In any right-angled triangle, (fig. 28.) the fquare of the hypothenufe, 
 (/. e.) the fide oppofite to the right angle, is equal to both the fquares 
 of the other fides taken together. 
 
 For demonftration, the reader is referred to Euclid j but to affifi: 
 the imagination, a regular figure is here exhibited, which win 
 make the propofition evident, on infpedtion only. 
 
 PROP. XXXI. of the third book oi Euclid, THEOREM. 
 
 The angle in a femicircle (fig. 6.) is a right angle. For demon= 
 ftration, fee Euclid, D E F I- 
 
XVI 
 
 INTRODUCTION. 
 
 DEFINITION in. BOOK VI. 
 
 A right line is faid to be cut in mean, and extreme proportion, 
 when the whole is to the greater fegment, as the greater fegment is to 
 the lefs. 
 
 LEMMA. Fig. 29. 
 
 To divide a given line A, B, in extreme and mean proportion. 
 
 Through the extremity A, draw F, D, perpendicular to it, bifecl 
 A, B, in X, take A, D, equal to A, X, and from D, as a center, 
 with the radius D, B, defcribe the arc B, F; then from A, as a center, 
 with the radius A, F, defcribe the arc F, C, and C, will be the point 
 fought. Euclidy Prop. XI. of the fecond book. 
 
 P R O P. II. B O O K VI. THEOREM. 
 
 If a right line Uy c, be drawn parallel to one of the fides A, C, of a 
 triangle A, B, C, (fig, 9.) it lhall cut the fides of the triangle propor¬ 
 tionally. See Euclid. 
 
 THE 
 
(*) 
 
 THE 
 
 PRACTICE 
 
 o F 
 
 PERSPECTIVE, &fc. 
 
 f g HIS treatlfe, being chiefly intended for thofc who are verfed 
 H in Defigning^ begins immediately with the praffice of perfpec- 
 
 M. tive j though the utmofl: care has been taken to render 
 
 every thing as clear, to any attentive reader, as the nature of the fub- 
 je6l will admit. With this view, as many of the known terms are 
 preferved, as poflible, that all may be readily underftood by thofe to 
 w'hom thefe terms are familiar; though others might have been in¬ 
 vented that would have been preferred as more fignificant, if the 
 author had intended to exhibit a theoretic fyftem. 
 
 His aim is to render the pradlice intelligible and eafy, to fuch as 
 above mentioned j for whofe fake, the terms and methods in common 
 ufe are employed, fo far as js confiftent v/ith the improvement pro- 
 pofed j and in thofe cafes where others become neceflary, they are in¬ 
 troduced and explained, and not before} by which means they will be 
 more readily underftood, and more ea'fily remembered. 
 
 The reader is fuppofed to be acquainted with fome of the firfl; 
 elements of Geometry, othervyife he wants the very language of the 
 fcience. 
 
 B 
 
 The 
 
2 
 
 rhe PRACTICE 
 
 The letter S, is every where ufed for the point commonly cdled 
 ^he point of fight ^ which is the fame point that Dr. Taylor (more pro¬ 
 perly) calls 'T’he center of the piSlurCy it being that wherein the pic- 
 ttire is interfe^led by a right line from the eye of the fpe6fator, per¬ 
 pendicular to the pidlure (or to its plane continued, if need be) which 
 line is the diftance of the pi6lure ; and that end of it, fuppofed to 
 be at the eye of the fpe6lator, is always marked D, whether placed 
 on the horizontal line, or elfewhere, and is called T^he point of 
 difance. 
 
 The point S, may very properly be confidered as the center of 
 the pi6lure, for if a circle be defcribed round it, with a radius 
 equal to the diftance, the point D, may be placed any where in the 
 circumference of that circle. 
 
 Fig, I. S, d, is the horizontal line, or vanifing line of the horizontal 
 plane, it being the interfe6lion (with the pi6lure) of a plane palling 
 from the eye, parallel to the horizon. S, the point of fight, or center 
 of the pi 5 lure. D, the point of difance j as are alfo d, and D, (in 
 the circumference of the fame circle.) G, H, is the ground line, or 
 mterfedlion (with the picture) of an original plane, which is here the 
 plane of the horizon. A, is an original point, on that plane, fup¬ 
 pofed to be beyond the picture, fo far as it is placed below the ground 
 line, that is, from A, to a j and therefore (once for all) it may be 
 proper to remark, that whatever is fo fituated, fhould be conceived to 
 be turned back, behind the ground line, and D, to be turned forwards, 
 on the point S, in fuch manner, that A, a, and D, S, be parallel to 
 each other, and (in the prefent cafe) both perpendicular to the pic¬ 
 ture ; then fuppofing the picture tranfparent, the point A, will be feen 
 through it at a, by an eye placed at D; or, to explain it otherwife, the 
 vifual ray from D, to A, (when in the fituation above) will interfe6l 
 the pi( 5 lure in a. For fuppofe a plane palling through the lines D, S, 
 and A, a,) when both are perpendicular to the pi6ture) that plane will 
 cut the picture in the line S, a. Now the point a, mull be fome- 
 where in the vifual ray, D, A, and it mull alfo be fomewhere in 
 the line S, a j therefore it mull be in thwr intcrfe6lion a, the only 
 
 point 
 
o/ PERSPECTIVE. 3 
 
 jpoint common'to both lines. And the fame line S, a, would be the 
 interfe61ion of a plane palling through D, S, and A, a, though thefc 
 two lines were not perpendicular, but in any other direftion (not 
 parallel to the picture) provided they were ftill parallel to each other ; 
 and therefore the fame point a^ will be as truly found in whatfoever 
 direction A, a, and D, S, are drawn, if ftill parallel to each other, as 
 here A, is tranfpofed to Ay on the ground linCy and D, to d, on the 
 horizontal line ; then drawing d. Ay and a, S, interfering it in a,, 
 that will be the fame perfperive reprefentation of A. 
 
 It is evident alfo that a, Uy is the perfpeBive of a. A, (an ori¬ 
 ginal line) and the whole line a, S, is the perfpeBive of the 
 fame original, continued infinitely, of which a, ay is a limited 
 part, and all lines terminating in S, reprefent originals pef- 
 pendicular to the pirure; for S, reprefents a point infinitely 
 diftant, to which all fuch lines tend, or (which is the fame 
 thing in perfi^erive) feem to tend j and is called their vanifj- 
 ing point. Hence it appears, that the perfpe6live reprefenta¬ 
 tion of every original right line, not parallel to the piBurCy is 
 included between its interfeBion with the pifture, and its 
 vanijhing point: —that is, having continued that original line 
 (whether perpendiculary or oblique) till it cuts the pi(5lure, as 
 here in a, and having drawn a parallel to the original line, 
 from the eye, cutting the pirure, as here in S, the line 
 drawn from a, the interfeBioUy to S, the vaniJJ.nng pointy will, 
 be the whole reprefentation of the original line j though that 
 line be infinitely continued beyond the picture. The repre¬ 
 fentation of the more diftant parts of which will approach 
 to Sj but the moft diftant point, fhort of infinite, will 
 not reach S j therefore that is very properly named the 
 vanifmig point of fuch line. 
 
 A, is here tranfpofed to Ay and a. Ay becomes by that means 
 parallel to »S, d j it is fo tranfpofed, becaufe in this, and moft cafes, it 
 is eafieft for the operation , but though it is neceflary that thefe two 
 
 B 2 lin^S 
 
The PRACTICE 
 
 4 
 
 lines fliould be parallel, yet they may be fo in any diredion j for fup- 
 pofe D, tranfpofed to £>, and A, to j, the vifual ray, D, j, will cut 
 a, S, in the fame point as is evident. 
 
 It is recommended to thofe readers who have not yet begun this 
 ftudy, to re-confider what has been faid, till they fully con¬ 
 ceive every part of it, before they proceed j and if they draw 
 the fchemes themfelves, they will apprehend the reafons of 
 the feveral operations much better, and even fave time by fo 
 doing. 
 
 In like manner may be found the perfpeBhes of any number of 
 points, and confequently of lines, and fuperficies: for inftance, 
 
 Fig. 2 . C, B, is a line in the fame original plane (whofe interfeflion e, 
 is found by continuing it to the ground line) the extremities of which, 
 being points, are fet off, (in the fame manner as was A) on the ground 
 line, to f and g; from each of which, by drawing a line to d, and 
 then drawing e, S, are found the points c, and b, the perfpeftives 
 of C, and B, and thus c, b, is the perfpedihe of the original line, 
 C, B. 
 
 Fig. 3 . E, F, is a line lying oblique to the ground llne^ whofe perfpeBhe 
 is found in the fame manner, viz. by drawing perpendiculars from E, 
 and F, to the ground Ime., and from each interfeftion drawing a line 
 to S ; then transferring the diftances of E, and F, to the fame line, 
 and thence feverally drawing to d, cutting the two lines (tending to S,) 
 in e, and f, and drawing e, f, this line becomes the perfpeSiive of the 
 original line, E, F. 
 
 Thus any right line, however fituated, may be reprefented, by 
 finding the perfpeBive of its two extremities. Hereafter a 
 fliorter, and better method will be fhewn of projedting any 
 oblique lines, but it was neceflary to begin with points. 
 
 A-> 5j 7 - It is obvious, that the perfpecfive reprefentations cf the 
 fquaie, and parallelograms are found the fame way, and that the reafon 
 why the fquare needs no pricked arch, is, that having all its fides equal, 
 the diagonal from d% determines the perfpedlive depth, without any 
 faither trouble. 
 
 Fig. 
 

 V1 
 
PERSPECTIVE. 5 
 
 Fig. 8. And for the fame rcafon, the eafieft way of defcribing the per- 
 fpe6:ive of a circle, is by including it in a fquare, and finding the 
 eight points marked i, 2, 3, 4, 5, 6, 7, 8. 
 
 Fig. 9. ’ The perfpedlive of any irregular plan, as A, B, C, &c. may be 
 found by the feveral points, as is evident. It is to be remarked, that 
 the pricked arches by which the diftances are fet off to the ground 
 
 line, fhould always be on the fide oppofite to d, (i. e.) when d, is to 
 
 the right of S, they fhould be transferred to the left, and fo vice 
 verfa. 
 
 Fig. 10. When a fquare is placed touching the ground line, in one point, 
 fo as to make, with that line, an angle of 45 degrees on each hand, 
 continue the fides, as A, B, and A, C, to the ground limy and draw 
 
 from the points b, and E, to d*, and from the- points c, and E, to d*, 
 
 which will give the perfpediive j (d*, and d*, being equally diftant 
 from S.) But in the fecond part, an univerfal rule will be given for 
 all fituations of original figures. 
 
 SOLIDS. 
 
 Fig. II, 12. ^ H E plans are firft reduced to perfpedtive, as here 
 
 JL of the cube, and parallelopiped, by the rules above, 
 then perpendiculars raifed on the ground line equal to their true, or 
 geometrical heights, and from the tops, lines drawn to S j then other 
 perpendiculars from the remaining angles of the perfpedlive plan 
 (meeting the lines drawn to S, from the firft perpendiculars) complete 
 the folids. 
 
 Fig. 13. This figure is aTufean pedeftal, whofe geometrical plan, and 
 elevation, are firft deferibed, then the plan in perfpedlive, which may be 
 either in its place, on the picture, or (as here,) below it, this being 
 chofen that it may not incumber the work above, and alfo that it may 
 be more diftindl, by being lefs crouded in fpace. This is performed as 
 the fquare at Fig. 4, and the inner fquares are determined by the dia¬ 
 gonals, crofting the rays drawn to S, from the loweft line taken from 
 
 the 
 
 2 
 
6 7^^ PRACTICE 
 
 the geometdcal plan with its divifions. After this operation, continue 
 the feveral parallel dines of the geometrical elevation, to the line of fec- 
 tion F, G, by pricked lines, and from all thofe interfe 61 :ions draw to S: 
 Then fet off from G, on the ground line, the divifions of the geometrical 
 ^lan, taken from the bafe of the pedeftal, [viz. r, 2, 3, 4—5, 6,7, 8,) 
 and from thefe divifions draw lines to d, which lines will cut the line 
 G, S : from which interfeftions, raife perpendiculars to the refpe6live 
 members of the pedeftal; thefe perpendiculars will complete the per- 
 fpedtive elevation marked E. 
 
 The perfpe( 3 :ive plan might have been made neat%r to S, or any¬ 
 where on, or below the ground line, beyond the numbers 
 
 I, 2,-8, fo as not to interfere with them, or on the other 
 
 fide of S, (e. g.) as far as Fig. 14 ; for the perfpeclive elevation 
 E, would have ferved for thatj by means of parallels. 
 
 Then the whole is completed, by raifing perpendiculars from the 
 feveral angles of the perfpeftive plan, and cutting them by parallels 
 from the correfponding angles of the perfpedlive elevation ; and laftly, 
 by tracing the figure thro’ thefe interfe(ftions j as for inftance, a per¬ 
 pendicular from 9, in the perfpe 61 :ive plan, Fig. 13, and a parallel from 
 9, in the perfpe 61 ;ive elevation, will meet at 9, in the finiftied pe¬ 
 deftal, and fo of the reft. 
 
 N. B. When the perfpeclive plan is, at once, reprefented in its 
 proper place, (i^e.J on or above the ground line, as at No. 14. 
 then parallels drawn from the feveral members of that 
 plan, will cut the loweft line G, 9, of the perfpeClive' eleva¬ 
 tion E, in the true points, from which the perpendiculars 
 are to be raifed to complete that elevation j but when it is found 
 more expedient to make the plan below the ground line (as 
 at No. 13.) it is neceflary to fet off the geometrical breadth of 
 the plan from G, on the ground line, with its divifions, 
 which muft be then drawn to d, as before direCled, and there¬ 
 by the perfpeClive figure completed j for perpendiculars from 
 this laft perfpeClive plan, tho’ below the ground line, will meet 
 the parallels from the perfpeClive elevation in the fame points. 
 
 Tkefe 
 
(9/ PERSPECTIVE. 7 
 
 Thefe feveral ways are explained, that the principles may be more 
 clearly underftood j but the beft of all methods to conceive them tho¬ 
 roughly, will be to perform thefe operations at the time of reading, and 
 not to pafs on to another figure, till all the former are fully compre¬ 
 hended : it is alfo recommended to fuch as are not pra£l:ifed in the art, 
 to perform this Fig. 13, in the feveral ways mentioned, before they 
 ■ read farther: they will then proceed with more facility and plea- 
 fure. 
 
 Fig, 14. This figure is projedted in the fame manner as the lafi:, except 
 that, inftead of the plan and elevation drawn geometrically, the breadths 
 only of the plan, and heights of the elevation, are marked with their fe¬ 
 veral divifions; all which are drawn to S, and a diagonal from d^, gives 
 the fquaresof the plan j then from the feveral divifions of this per- 
 fpe6liveplan, parallels are drawn to the loweft line of this fubftituted ele¬ 
 vation, and from thefe interfedtions, perpendiculars to the heights of 
 the feveral members : By means of this preparation, the whole is com¬ 
 pleted in its place j tho’, as hath been faid, the plan, and line of eleva¬ 
 tion, may be feparated, to avoid confufion. 
 
 Fig. 15. This example is of a rough pedeftal without mouldings. 
 
 After having made the geometrical elevation and plan, draw from 
 every angle of both to S, thro’ the line d. A, which is to be confidered as 
 the fe 61 :ion of the picture j with this difi:in6lion, that d, G, part of it, 
 is the perpendicular edge of the picture, and confequently will deter¬ 
 mine the heights of all the points, by means of parallels to the horizon¬ 
 tal line, drawn from the interfedfions of the rays, as i, 2, 3, but 
 from by to A, (inclufive) the intcrfecftions are fuppofed to be on the 
 bottom of the pidlure touching the ground j and are therefore to be 
 tranfpofed to G, H, the ground line, as at h, g, G, &c. —a, reprefent- 
 ing A J for fetting one foot of your compaifes at A, and extending the 
 other to h, on the line of fedtion, the whole is transferred to the ground 
 line, from a, to h, together with the intermediate divifions, from which 
 laft points, perpendiculars being drawn, will meet the refpedive 
 parallels in the true perfpe6live points, which being joined will form the 
 figure. 
 
 iV. B. In 
 
8 
 
 7%e F R a ‘c TICE 
 
 N. B. In the ground line, the point h, mufl: be placed exa 61 :ly 
 at the fame diftance from fy as h, is from d, on the line of 
 fe6lion ; otherwife the pedeftal will not be feen in the pi6lure, 
 as the fpeftator ftanding at S, fees the original. I’his method 
 of projeBiofi is Pozzo’i, in his fecond volumCy and is introduced 
 for reafons 'which 'will be explained hereafter. 
 
 Fig. i6. The next is without geometrical plan, or geometrical elevation.— 
 Having firft drawn the bafe line, a, by and divided it geometrically at 
 Cy and dy (for the body or trunc of the pedeftal) project the whole bafe 
 perfpectively, by means of a diagonal from D, then any where apart on 
 the ground line, as at k, ere6l a perpendicular, the height of the whole 
 pedeftal, and divide it geometrically at the heights of the feveral mem¬ 
 bers y and from thefe diviiions draw to any point in the horizon, as f: 
 after which, draw parallels from all the angles of the perfpeftivc plan to 
 k, fy (the lowed: line of the perfpedlive elevation,) and, from thefeinter- 
 feclions, cre6l perpendiculars, cutting the feveral lines drawn to f and, by 
 thefe lad interfedtions, form the perfpedtive elevation, then, by means of 
 parallels, from the feveral members (cutting perpendiculars, raifed from 
 the feveral angles of the perfpedtive plan) complete the figure. 
 
 Fig. 17. Here is added one object more, led any difficulty fhould arife 
 from fuch figures whofe fides are not fimiiar, but whoever has under- 
 dood thus far, will perceive how this is performed on infpedtion ; the 
 method being the fame as at Fig. 15, except that indead of drawing 
 all the lines of the geometrical to the point S, in the horizontal line. 
 In this fcheme, tliofe of the plan are drawn to another point below, as 
 T, to avoid confufion, but then it mud be remarked, that as T, f, is 
 equal to S, D, (the didance) fo R, by mud be equal to fy B, for the rea- 
 fon given above at Fig. 15, and here the line of divifions taken from 
 I B, ^c. tranfpofcd to 1 by was fet od' on the ground line, the contrary 
 w'ay to that of Fig. 15, (the rays being drawn to T, the contrary way 
 to S,) that the rays in the finiffied figure may run to S'*. 
 
 * The term geometrical and clfewhere, (when without a fubftantive) is iifed fubftantiv'ely 
 for the original objeft, as the term ferfgeBi’iie is frequently ufed for the reprefentation. 
 
 Fig. 
 
fi. 
 
 
 
 \ 
 
 
 a 
 
 '1? 
 
 
 ,"%.t . ., r-* ■ 
 
 ' •''•A . .; 
 
c/ PERSPECTIVE. 9 
 
 Fig. i8. Is performed in the fame manner as 14, but in this, the lines 
 which form the perfpe6live plan, are left vihble, that the operation may 
 more eafily be underftood. The meafures for the perfpe^live plan are 
 taken from the geometrical, and fet off, on o, p. From thence rays 
 are drawn to S’, and the diagonal from p, to D’, cuts the ray q. S’, in 
 the point z, which determines the perfpe6live fquare, that reprefents 
 the fquare B, Z, in the geometrical plan, and, by means of this, the 
 whole plan is put into perfpe6live. On the perpendicular o, 7, mark 
 the geometrical heights of the feveral members, and from thefe divifi- 
 ons draw rays to S’, and then draw parallels from all the angles of the 
 bafe, to o, S’, the lowed: ray; and from the feveral interfe 61 ions, raife 
 perpendiculars to the uppermoft, and fo form the perfpe6live elevation, 
 as at Fig. 14, but which is more apparent at Fig. 16, becaufe the elevation 
 is there feparated from the body of the pedejlaly tho' the method is the fame). 
 Now raife perpendiculars from all the angles of the plan, and, by 
 means of parallels from all the members of the elevation, meeting thefe 
 perpendiculars, complete the whole figure, in the fame manner as was 
 done at Fig. 14, and 16.— Particular care mufi be taken that each pa¬ 
 rallel, from the elevation, meet its correfpondent perpendicular from the plan, 
 to determine the fame member, and this is, perhaps, the eafieft, and fiiortefl 
 method of all: for the perfpe£live plan is made with as few lines, and 
 in as little time as the geometrical, which is unneceflary here j and 
 inftead of the whole geometrical elevation, the geometrical divifions, or 
 heights only (on the firfi: line q, 7) are necelTary : fo that the meafures 
 may be taken from a book of archite6lure, without drawing any thing 
 geometrically ; and if their meafures (in fuch book) be on a larger, or 
 fmaller fcale, it is eafy to fet them off in any proportion required for the 
 perfpe6live; as in this very figure, the meafure for the bafe is limited 
 to o, p, wherefore firfi: draw o, p, in its place -, but as the meafures 
 here, are equal to the original at B, and this expedient (for that reafon) 
 unnecefiary in the prefent figure, jt is more convenient to fliew it apart. 
 — Suppofe then, o, p, drawn in its place Fig. 18. (as direfted above) 
 draw from o, any other line, o, t, and on it mark all the geometrical 
 divifions of the plan from the book j then lay a parallel ruler ' from 
 
 C t, to 
 
10 
 
 rU PRACTICE 
 
 t, to p, drawing t, p, and parallel to it, all the reft of the divllions 
 from o, t, to o, p, and the line o, p, will thus be truly divided j 
 whether o, t, be longer or fliorter than o, p. 
 
 And alfo having drawn in its place a perpendicular to o, p, as o, 7, 
 for the elevation, draw any other line from o, as o, r, on which mark 
 the divifions of the geometrical elevations from the fame piece of archi- 
 te 61 :ure in the book, and (in order to preferve the proportion of the 
 bafe) fet oif the meafure of o, p, on the perpendicular o, 7, reaching 
 to V, and the meafure of o, t, on o, r, reaching to t'^, then lay the ruler 
 from v, to t*, drawing that line, parallel to it, transfer all the divifioiis 
 from o, r, to o, v, which will give them in the fame proportion as thofe 
 on o, p. 
 
 In this firft part, the feveral methods propofed by writers before Dr. 
 I’aylor are exhibited, any of which will anfwer the purpofe, when ob- 
 je6ls are placed diredlly in front, and on the horizontal plane j but 
 when objefls are in an oblique lituation, even on the horizontal plane, 
 and efpecially when they are on an oblique plane, or when the figures 
 to be reprefented on any plane are themfelves irregular, the new method 
 will appear preferable beyond all comparifon. 
 
 SECOND PART. 
 
 ^'' H E S E few examples are fiifficient for the firfi: part, that being 
 A intended only to exhibit the common methods, with fome im¬ 
 provements ; which methods, tho’ ufeful in many cafes, are no more 
 proper for fome, than the rule of addition, in arithmetic, is proper for 
 finding the produdl of a fum in multiplication ; and notwithftand- 
 ing a perfon, ignorant of multiplication, might find, by addition, how 
 much 300 times 278 makes ; yet, in order to afcertain it, he mufl fet 
 down 278 three hundred times, and add all together ; whereas, if he 
 underflood multiplication, he would do it in an infiant, and be much 
 iefs liable to miftake : It is not pretended that the cafes are exaftly pa¬ 
 rallel, but a few examples will fliew that this may not improperly ferve 
 
 for 
 
 2 
 

 
 tJ' 'v.. ■ 
 
 it. 
 
 ■ V •■ '■/-! 
 
 ■ 
 
 ■ 
 
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 •i; 
 
 
 
 - y.- -v^ ■ ■ ■.- ■ 
 
 
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 ••> i 
 
 
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 ■^i. -■ 
 
 y,"» 
 
 :=>: ' 
 
 . ' ■'f 
 
 ^n..: . ■ _ v^VAy 
 - . '.s _ .,, ,•■ ' 
 
 ' V ' ■ ' 
 
 V- 0 
 
 ■?' 
 
 ■■>*'*, 
 
 
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 V'. -.yV . 
 
 
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 '-"•.I 
 
 
 ■': 0 : 
 
 ■v‘,'-•;“ 
 
 
 ’ ^sV. 
 
 ’: ^ i' 
 
0 / PERSPECTIVE. ir 
 
 for illuftratlon, and that the common methods are attended with fuch 
 tedious operations, fuch a multitude of unnecelTary lines, and, in fome 
 fituations, with fuch perplexed and intricate fchemes, as require more 
 than human patience to execute, and, after all, render miftakes almoft 
 unavoidable, of which any one will be convinced who fliall examine 
 the plates of PozzOy even in his fecond volume, where he has publifhed 
 his Ihorter method, which he had promifed in his firft, as well as in 
 other authors; efpecially when they exhibit obje6ls in oblique pofitions, 
 not only on oblique planes, but even on that of the horizon. 
 
 In this fecond part, therefore, it is propofed to fhew the advantages 
 of the new method, by comparing it with the old, in feveral inftances: 
 and here it may be proper to obferve, that thofe readers, whofe leifure 
 or curiofity may not permit, or incline them to examine the feveral 
 comparifons propofed, may negle6l the examples of the old methods, 
 and go regularly thro’ thofe of the new, and fo arrive at the knowledge 
 of the pradfice the fliorteft way at once : thofe however who are already 
 acquainted with the old methods, will be better fatisfied on feeing the 
 different manners of operation, in the fame examples; and it is pre¬ 
 fumed that much the greater number of readers may be of this 
 clafs. 
 
 Fig. 19. A, B, C, E, a parallelogram, making a given angle with the 
 ground line G, H, reprefented, in perfpeclive, by the common method 
 before explained. 
 
 Fig. 20. The fame parallelogram by the new method. And here, in- 
 ftead of placing the dilbance on the horizontal line, it is proper to 
 raife it perpendicularly, as S, D ; then continue the fides of the plan C, 
 A, and C, E, till they cut the ground line in G, and H; from D, 
 draw D, a, parallel to A, B, and C, H, cutting the horizontal line in a, 
 and draw D, e, parallel to E, B, and C, G, cutting the horizontal line 
 in e ; then draw B, a, and H, a, and alfo B, e, and G, e, which com¬ 
 plete the perfpeftive reprefentation : the lines themfelves forming the 
 figure, without the trouble of finding points, or rifque of miflake, or 
 of inaccuracy in joining them when found. 
 
 C 2 
 
 N, B. A, 
 
12 
 
 The PRACTICE 
 
 jV* B. A, B, and C, E, H, being parallel, D, a, is parallel to 
 both j as is D, e, to E, B, and C, A, G j and it is an 
 univerfal rule, that all original lines parallel to each other, 
 (and not parallel to the picture) run to the fame point in per- 
 fpe6i:ive j which, when not the point of fight, is called by the' 
 old writers an accidental point, or more generally a point of 
 concourfe j but by Traylor^ a ^anljhing pointy whether it lies in 
 the horizontal line, or elfewhere: Thus a, is the vanifhing 
 point of A, B, and C, H, and B, and H, being their interfec- 
 tions, their perfpedlives are found between B, and a, and be¬ 
 tween H, and a j and fo, univerfally, the perfpedlives of all ori¬ 
 ginal lines (not parallel to the pi6lure) lie between their inter- 
 fe6tions with the pi( 5 bure, and their vanifhing points, as was 
 obferved before. 
 
 Fig. 21. A, B, C, E, the plan of a cube placed obliquely to the ground 
 line.—It is required to find the perfpe6live of the whole cube in that 
 fitiiation. 
 
 According to the old method: After having found the perfpe^live of 
 the plan, and raifed perpendiculars from all the angles, fet off the geo¬ 
 metrical height any whereon the ground line, as at and draw from 
 the extremities fy and to any point in the horizontal line h j 
 then draw parallels from all the angles of the perfpe6tive plan, to the 
 lower line h j and, from the interfe6tions, raife perpendiculars to the 
 upper line j and then, from thefe perpendiculars, draw back other pa¬ 
 rallels to the correfponding perpendiculars raifed from the angles of the 
 perfpedlive plan, which will complete the cube. 
 
 Fig. 22. To reprefent the fame according to the new method : After hav¬ 
 ing found the perfpeftive plan, (as at No. 20.) raife perpendiculars 
 from all the angles of the perfpedlive plan, and make c, i, (which 
 touches the ground line) of the true geometrical height j then from i, 
 the top of this line, draw to e, and k, (as before for the plan,) and 
 from 2, to K, and from 3, to e, interfedfing each other at 4, and 
 fo finifh the upper fquare, as the lower, which, completes the 
 figure. 
 
 After 
 
i4^-i I 
 
 '■ -w .fi Sf 
 
 i " ■' ■- 
 
 If- ■' 
 
 
 
c/ PERSPECTIVE. , 13 
 
 After what has been faid (at Fig. 15. and 17.) of Pozzo's fecond 
 method, it would not be neceffary here to add any thing 
 
 more in explanation of it, if it were not, that there will be 
 
 \ 
 
 feveral occafions for the fame kind of operation ; wherefore, 
 to render it as clear as poflible, the following example is 
 propofed. 
 
 Fig. 23. A, is an original parallelogram, fuppofed to be placed on the 
 horizontal plane, behind the picture, whofe interfe6lion with that plane 
 is G, h, the fpe6lator ftanding at d ; wherefore firft draw lines from 
 each angle to d, which will cut G, h, then transfer thofe interfe 61 :ions 
 to the proper ground line of the piffure G, H j begin by fetting one 
 foot of the compafTes in o, (which is the interfe6lion of d, o, with 
 G, h, perpendicular to it,) and fo transfer the feveral divifions from 
 the line G, h, to the line G, H, beginning at O, in this laft line j and 
 from I, 2, 3, 4, on G, H, raife perpendiculars. 
 
 After this operation, raife perpendiculars alfo from all the points of 
 the original figure A, to the line G, H, (continued behind G,) and 
 from their interfe^tions with this line, draw lines to S, which will cut 
 G, D, the fe6lion (or upright edge) of the pi6lure, and from thefe in- 
 terfeftions (viz. of the lines to S, with G, D,) draw parallels, which 
 meeting with the perpendiculars raifed from 1,2, 3, 4, determine all the 
 points of the perfpe6live ; but here care muft be taken that each pa¬ 
 rallel determines its correfponding perpendicular; as for inftance, the 
 perpendicular 3, correfponds with the lowefi: parallel, marked alfo 3, 
 and their interfe( 5 lion reprefents the nearefi: point of the obje6i: marked 
 3, and fo of the refi, which are all marked with their correfponding 
 numerical figures: this, and the samples before referred to, are per¬ 
 formed by the fliorter method of Pozzo^ exhibited in his fecond vo¬ 
 lume, which he propofed as the mofl expeditious manner of all ; and 
 for that reafon it has been thought proper here, and in fome following 
 examples, to make the comparifon between this method and the 
 new. 
 
 This example is the fame kind of objeff, and fituation, as 
 is above (hewn at Fig. 19, in the common method, and Fig. 20, in 
 
 the 
 
14 7 ^^ PRACTICE 
 
 the new, to which the reader is referred, who may compare them 
 together. 
 
 Plere is alfo added the fame objeft, according to the method of 
 A. Bojfe, (a famous French engraver and author, who wrote about 
 one hundred years ago) not only becaufe he aflerts his to be the 
 cafieft, fliorteft, and moft exa6t of any to that time, or that ever could 
 be invented but alfo becaufe it is ftill fo elleemed by fome moderns; 
 he propofes two methods little different from each other, both of them 
 are fliewn in this example. 
 
 Fig. 24. And firft at Fig. 24, where the original objeft is enclofed in 
 fquares of feet (or any known meafure) geometrically, the diftance is 
 fet off in the fame meafure, as,from p, to D, eight feet, and the height 
 of the eye, as from D, to O, five feet. 
 
 In order to put this in perfpe6tive, as below at Fig. 25, draw the 
 ground line G, H, and divide it into feet; draw S, S, for the horizontal 
 line, parallel to G, H, and the height of D, O, from it; and having 
 placed S, on that line perpendicularly over P, (which correfponds with 
 P, above,) draw rays from all the divifions to S ; then, in order to re¬ 
 duce the fquares into perfpe 61 :ive, (inflead of fetting off the diflancc 
 from S,) make a perfpe£l:ive fcale, or echelkfuyante^ (as he calls it,) by 
 marking from any point of the horizontal line eight parts of any 
 opening of the compaffes for the eight feet: as here from a, to b, and 
 take one of thefe parts from G, to d, draw d, a, and G, b, which 
 will cut d, a, in c, draw the parallel through c, and from the point 
 where that cuts G, a, as at e, draw again to b, and fo on, till you have 
 as many parallels as are wanted. G, a, d, is what he calls the fcale, the 
 peculiar advantage of which is, that you may always divide the fquares 
 perfpedlively within the pidlure, whatfoever diftance be taken, becaufe 
 any opening of the compaffes may anfwer to your foot, the truth of 
 
 * Hu n^vords are ,—laquelle maniere s'eft treuvee, fans contredit, la plus familiere, et abregce, 
 Jelle, & prccife qu’aucune qui ait encore parue, et j’ofc bien dire qui pareftra. Avertijfement. 
 
 H is book is intitled, “ Moyen univerfel de Pratiquer la PerfpeiUve fur les Tableaux ou 
 
 Surfaces irregulieres, &c. A Paris, MDCLIII. Par A. Boffe.” 
 
 And again Chap. I. J’ay dit que j’avois mis en lumiere un traite de perfpeftive que je crois, 
 avec plufieurs, etre le meilleur qui fe foie fait, et, fe fera, &c. 
 
 the 
 

 
 - • 
 
 1 
 
(?/ P E R S P E C T I V E. 15 
 
 the operation depends only on making G, d, equal to one fuch opening, 
 or divifion. Now find the feveral points, of the obje6f in thefe per- 
 fpeftive fquares, correfponding to the original in the geometrical plan, 
 join thefe points, which complete the work. 
 
 His other manner differs in nothing from this, except that inftead of 
 drawing the rays and the parallels quite through them. You need only 
 make the perfpedlive fcale, and divide the perpendicular S, P, by that 
 fcale, and fo meafure the depths of the feveral points by the line S, P, 
 and the breadths from the fame line on both fides, correfponding 
 to the original: but then, in order to fet off the parallel feet, it is 
 neceffary to add the line e, a, placing e, one real geometrical foot 
 diftant from G, which will determine the perfpedtive parallel feet, 
 all the way up. 
 
 The performance of all thefe particulars will convince any one of 
 the tedioufnefs, as well as uncertainty, of this manner of working ; it 
 will be found almoft impolTiblc to afcertain the exadl place of the feveral 
 points, even with the utmofl care ^ not to mention the neceflity of 
 making all that preparatory geometrical work, if not in fquares, yet 
 in divifions / 
 
 Fig. 26. Next follows the fame obje6t, according to the new method, in 
 order to be compared with thofe above, which having been before ex¬ 
 plained at Fig. 20, from which this differs only, in that the neareft 
 angle touches not the ground line. It is to be obferved, that the lines 
 here form the objedf, without the polfibility of miftaking, and with 
 the utmoft exactnefs, and in the tenth part of the time. The per¬ 
 pendicular S, D, is the diflance o, and d, the two vanifliing points, 
 found by drawing D, o, on one fide, and D, d, on the other fide 
 parallel to A, B, and E, B, refpedlively. 
 
 Fig. 27. This pedeftal is reprefented in perfpe6five by Pozzo's fliorter 
 method, as explained at Figures 15, and 17, to which nothing need 
 be added, except that the geometrical elevation mufl be formed by 
 
 * To do jufllce, however, to the author, h is acknowledged that this fcale is, in fome 
 eafes, a very ufeful expedient; though it will by no means jullify what he fays of his general 
 method. 
 
 perpendi- 
 
i6 P R A C T I C E 
 
 perpendiculars from all the angles of the plan, as placed obliquely, 
 which, in a complicated defign,'makes fometimes a very odd, and intri¬ 
 cate figure, fcarcely intelligible, as appears in feveral inftances in 
 Pozzo's fecond volume j whereas, according to the new method, this 
 never happens j but, on the contrary, how complicated foever the 
 original may be, the plans and elevations always make the fame kind 
 of figures as the original geometrical objedls. This will be fhewn 
 hereafter. 
 
 The operation is the fame as at Figure 23. And here, befides the 
 great number of lines, much time, patience, and care, are neceflary to 
 find the correfponding points, (after having drawn all the perpendicu¬ 
 lars, and parallels of the plan, and elevation,) which renders the work 
 very liable to errors. 
 
 Fig. 28. Here is alfo added the fame pedeftal, in the fame pofition, 
 according to A. Bojfes method, by means of fquares, (which is fhorter 
 than his other mentioned before, becaufe the meafuring is avoided, 
 which requires more time than making the fquares but, befides fo 
 many needlefs lines, there is great danger of miftaking the points by 
 the perfpeclive meafures, and much time is neceffary to complete the 
 figure with any exaclnefs. 
 
 N. B. The fmall plan above is, (in this method,) a neceffary 
 preparation, and the feveral points of the perfpeclive plan 
 are determined by marking them in the fame parts of the 
 perfpedlive fquares below, as in this, refpe 61 :ively. The per¬ 
 pendicular e, f, on the fide of the fmall plan above, is 
 divided geometrically for the height of the members, which 
 are to be transferred alfo to the perfpedtive j for inflance, the 
 top of the capital is four feet; therefore, on the ground 
 line take four feet, and with that meafpre turn the compaffes 
 from the neared angle 1, which touches the fame ground 
 
 line, to h.-Again, for the fame height at i, take four feet 
 
 on the parallel at'i, and turn the compailes up to kj and fo 
 for every other point 3 after which they mud all be joined. 
 
 Fig. 
 
Plate VI 
 

 s'"*' 
 
 i. 
 
 i 
 
 
 
 \ 
 
 m 
 
 / 
 
 I 
 
 4P 
 
 $ 
 
 ■y. 
 
Raw VII. 
 
s 
 
 PUte.VUL 
 
of P E R S P E C T I V:E. 17 
 
 Fig. 29. But in the reprefentation of the fame pedeftal, by the new me¬ 
 thod, the whole work is performed by means of three points only, 
 a, b, and c, to which all the lines are drawn, and thefe lines form the 
 figure itfelf; fo that having fixed one end of the ruler at a, the lines 
 of two fides, (i, e.) all that are parallel in the original, are drawn with¬ 
 out taking it off, and by placing it at c, the lines of the other two 
 fides are all drawn, without moving the end from thence, and, with 
 the utmoft exa 61 :nefs j b is the vanifliing point of one of the diagonals, 
 found by drawing D, b, parallel to d, b, in the geometrical plan. 
 
 If what has been hitherto faid of this method be underftood, 
 (efpecially at Fig. 22.) this will not need farther explanation; however, 
 to leave no difficulty j After having raifed perpendiculars from all the 
 inward and outward angles of the perfpedlive plan, the geometrical 
 meafures of the heights are marked on the perpendicular of the nearefl 
 outward angle, (which is pricked, or dotted •,) and, from thefe divi- 
 fions, lines drawn to b, cutting the perpendiculars of the nearefl and 
 farthefl angles of the die, or body of the pedeftal, determine the feve- 
 ral points of the die; and drawing from thefe interfe6lions to a, and c, 
 the reft of the die is completed, and fo of the mouldings. 
 
 Fig. 30. The next figure reprefents two bafes leaning one againft the 
 other, taken from the 28th of Pozzos fecond volume, both of them 
 raifed from the horizontal plane ^ for which reafon he fays, “ he could 
 “ not ajjign a point of fght^ and therefore was obliged to transfer all the 
 points one by one with his compaJfeSf that he might fnd the termination 
 “ and curvature of each line And although (in the plate referred 
 to) he has not left the lines by which thofe points were determined, 
 yet whoever underftands his method will perceive the neceffity of them, 
 and that in fuch oblique fituations, they muft be almoft innumerable, 
 as will appear throughout his book, on infpe6lion of the odd plans 
 
 I 
 
 * His'tvoreft are ,— defcendendo Cum linels occultls, ad perpendiculum, ab fingulis projefturis 
 limborum defcribuntur totidem circuli in veftigio, ut unufquirque apte collocetur, atquc ab 
 Btrifque iierunC bafes optice adumbrate : pro quibus cerium oculi punilum fiatuere non potui, e» 
 quod horizontales non Jint. Sed tranfiuli, (ircino, Jltfgillatim, puntia, ut finem, ac Jinuationem 
 cujufque tinea invenirem, &c. 
 
 D 
 
 he 
 
i8 
 
 rhe PRACTICE 
 
 he was obliged to make, which of themfelves are extremely difficult to 
 form, and intricate when formed. 
 
 The pains of fo tedious an operation, as this method requires, might 
 have been fpared j but that Pozzo's books (efpecially the fecond volume) 
 are not in every one’s pofTeffion j and that, of thofe who have them, 
 very few (if any) may have given themfelves the trouble to proje6l 
 thefe, or fubje6ts of the fame kind, by his rules j and, therefore, may 
 not be fenfible of the neceffity of ufing fo many lines. It was, there¬ 
 fore, thought expedient to proje6l thefe bafes in his way firft. 
 
 The manner of working is the fame as at Figures 15, 17, 23, and 27, 
 above explained. And firft the profiles A, and B, are geometrically 
 drawn, then the plans C, and D, by dropping perpendiculars from 
 every point of the profiles, and from the feveral points of the axes A, E, 
 and B, F, (which cut the members of the bafes) in order to find the 
 feveral centers on the line C, D, which line receives all the tranfverfe 
 diameters, as d, and its parallels of the bafe A, and likewife thofe of 
 the bafe B j but the perpendicular diameters are transferred from the 
 profiles geometrically, thus j C, reprefents the center A j d, reprefents 
 the point d\ and f, the point f-, all three found by the perpendiculars 5 
 then from C, upwards and downwards, the geometrical length A, dy 
 or A,/, is fet off from C, both ways, to h, and g, for the perpendicular 
 diameter g, h, which completes this circle ; the fame operation forms 
 each circle, After the profiles and plans are completed, lines muft be 
 drawn from every point of both to O, cutting the line i, 5, part of 
 which, viz, from 1, to 3, reprefents the interfedlion of the bottom of 
 the picture with the ground, and muft be transferred, with all its divi- 
 fions, to the proper ground line of the pi6lure, and perpendiculars 
 raifed from all thefe divifions. Another part of i, 5, viz. from 4, to 5, 
 reprefents the perpendicular, or upright fe^fion of the pidlure j and, 
 therefore, from all its divifions, parallels muft be drawn, meeting the 
 perpendiculars raifed from the divifions of the ground line, and thefe 
 interfering, will determine the points of the perfpe6live j but the 
 number and confufion of lines is fo great, that it will be neceffary to 
 fix every point with the compaffes, or (as Pozzo himfelf advifes) with 
 
 a pair 
 
Plate IX- 
 

 
 I 
 
of PERSPECTIVE. 
 
 a pair in each hand, as thus; place one foot of your compafles in i, 
 on the line of fe^lion, and extend the other to 6, which is the inter- 
 fe^Iion of the vifual ray from D, to O, and tranfpofe this meafure to 
 the piiSlure, fetting one foot there in i, and the other foot will reft on 
 the perpendicular marked 6, in which the center D, is to be found : at 
 the fame time, fet one foot of the other pair of compafles in 2, on the 
 line of feftion, and extend the other to the point where the ray B, O, 
 cuts that line, as at 7, and transfer that height to the perpendicular 6, 
 on the ground line, (found by the other compafles,) which will mark 
 7, on that perpendicular, the perlpeflive of the center fought; and this 
 double operation muft be repeated for every point, till all the points in 
 the perfpe6live are found, which muft afterwards be joined : in doing all 
 this, great care muft be ufed not to miftake ; and when completed, can 
 never be fo true as by the other method > becaufe here the feveral lines, 
 which fhould be drawn to the fame vanifhing point, muft be drawn 
 from point to point only. Here are ufed five points only for each 
 circle, viz, the center, and extremities of two diameters, to avoid 
 adding more lines. 
 
 As to the parallels and perpendiculars, which inclofe the perfpeftive, 
 they might have been omitted, if the two pair of compaflTes be made 
 ufe of i and efpecially if the perfon ufing them has got into the habit: 
 but thefe lines are left, that every thing may be clearly underftood ; 
 but as the parallels are in themfelves necelTary to mark the line of 
 feftion, they are only continued on to the bafes, and do not increafe 
 the number of lines. 
 
 All the other lines are abfolutely necefTary in Pozzo's fecond, or 
 fhorter method. The great number of lines, and the confufion arifing 
 from thence, has caufed even him to miftake, the lower bafe being falie 
 in his plate; for the lines reprefenting the thicknefs of the plinth, which 
 are perpendicular to the ground line, and parallel to each other, 
 ought to run towards a certain point, and fo be neither perpen¬ 
 dicular, nor parallel: if the fault be not his, it may be the engraver’s; 
 but whofefoever it be, the print is apparently wrong. 
 
to P R A C T I C E 
 
 Fig. 31. In order to reprefent this according to the new method, it was 
 neceffary to find the centers and diftance, both of the vanifliing line, 
 and picture, by continuing the two fides of the lower plinth till they 
 met in a point, (as here in C,) and then drawing C, £), parallel to the 
 ground line: Thus, C, D, becomes the vanifiiing line of the oblique 
 plane, which the lower bafe forms by being raifed, and C, its 
 center. 
 
 N. B. *^bis line is always to be ufed for obje 5 ts obliquely Jituafedy as 
 the horizontal line is ufed for objects on that plane. 
 
 The point of diftance jD, of this vanifliing line, was alfo found, by 
 drawing a line through the diagonal of the fquare of this plinth, from 
 the angle 3, to the vanifliing line C, D. ‘thus far y from Pozzo’j booky 
 for otherwifcy thefe are circumjiances always given. 
 
 Thefe points being found, half the meafure of Pozzo's was taken, 
 and fo the fame proportions were preferved. 
 
 The reft is all performed as before explained j but the fituation of 
 the objects being new, a more particular detail may be ufeful, and 
 will fliew the univerfality of the principles. 
 
 As the center of the pi6lure, found, by a line, perpendicular to it, 
 from the fpeftator’s eye, is that point to which all original lines, per¬ 
 pendicular to the pi6lure, tend; fo every vanifliing point being found 
 by fome line from the eye of the fpe6lator to the pi<fture, is alfo the 
 vanifliing point of all other lines parallel to that j and every line from 
 the fpe 61 :ator’s eye cutting the picture, (or plane of the picture how 
 far foever extended,.) makes fuch a vanifliing point j thus C is the 
 vanifliing point of the line i, 2, and of all original lines parallel 
 to it. 
 
 S, D, is the horizontal line, found by making the angle D, C, D, 
 equal to that which the plinth makes with the ground, and deferibing 
 an arc from D, to D, with an opening of the compafTes, or radius, 
 equal to C, D, (the diftance before found,) and drawing D, S, parallel 
 to £>, C ; then drawing through C, a line perpendicular to £), C, 
 cutting D, S, in S, that point S, becomes the center of the picture, 
 and D, S, the diftance of it,—~D, C, is the diftance of the vanifliing 
 2 point 
 
PlateX. 
 
21 
 
 of PERSPECTIVE. 
 
 point C, and of the vanifhing line Z), Cj and D, is to be confidered as 
 the eye of the fpe 61 :,ator j therefore (if D, C, finds the vanifhing point C,) 
 a line from D, perpendicular to D, C, muft find the vanifliing point of 
 lines perpendicular to i, 2, as D, finds </, cutting C, S, beyond 
 the limits of the paper, which will be the vanifhing point of the 
 line I, 3, and all others parallel to it, as are thofe at all the angles of 
 this plinth, which muft therefore be drawn to d, as the line D, d^ cuts 
 the line C, S, beyond the limits of the picture j but it is not necefiary 
 to ftop here to explain the manner of drawing lines to an inacceffible 
 point, (which is done in the fourth part :) then, for the thicknefs of 
 this plinth, draw a line through i, parallel to C, S, (which is the 
 vanifhing line of the planes i, 2, 3, and 4, 5, 7,) for that point i, is 
 fuppofed to touch the pi(fture, (where all obje6ls are of their true, or 
 geometrical fize,) and on that line, from i, downwards, mark the 
 geometrical thicknefs of the plinth, and having tranfpofed the diftance 
 of the vanifhing point dy from D, to dd, draw a line from d^, to 
 the point marked, which will cut 1, in the point 3 ; this deter¬ 
 mines the thicknefs of the plinth, by which it may be completed. 
 The reft of the members are determined exactly in the fame manner, 
 as if the bafe was on the horizontal plane, ufing C, Z), the vanifhing 
 line, as an horizontal line.—The heights, and breadths of the circles 
 are determined in fquares, as hath been taught in the firft part, and 
 the circles drawn through the eight points there fpecified. 
 
 For the other bafe, draw firft the pricked line 3, S, which 
 marks the ground or horizontal plane, perpendicularly under 
 3, k, C j then fet off from 3, to fy the geometrical diftance, 
 that the angle of the other plinth is from the point 3, and 
 from D, draw a line to fy cutting S, 3, in 4, and that interfeftion 
 will be the point fought, in which this plinth touches the ground; 
 and having drawn D, by making the angle S, D, by equal to 
 that which this plinth makes with the ground, (i. e.) 38 degrees, 
 by is the vanifhing point of the line 4, 5, and its parallels ; there¬ 
 fore draw 4, 5 J then to find the length of that line, make ufe of 
 the parallel to the vanifliing line C, S, before drawn, viz. 1, 3, e, 
 
 by 
 
 
22 
 
 The PRACTICE 
 
 by drawing firft a line from d^, (the diftance of D, b,) through 4, 
 to that line, cutting it in e i and from e, upwards, mark the geome¬ 
 trical length to g, and from g, draw back again to ^by which gives 
 the perfpe6tive length of 4, 5 j from which points 4, and 5, parallels 
 to the ground line are drawn ; and the length of the parallel at 4, is 
 found by drawing a line from S, to the point 6, cutting that parallel, 
 and drawing a line from by through this laft interfeftion, it will cut 
 the parallel 5, 8, in 8, and fo complete the bottom, or lower fquarc 
 of the plinth. 
 
 Or this fquare may be determined (without making ufe of the 
 line I, 3, e,) by firft finding the length of the parallel at 4, as laft 
 directed, and then drawing a line through the interfedlion which marks 
 that length, from D*, (the diftance of the vanifhing point by) to the 
 line by 4, cutting it in 5. This line D*, 5, gives the diagonal of the 
 fquare, by which it may be completed. 
 
 N. B. The diftance by D*, is brought down to D*, by fixing one 
 foot of the compaffes in by and the other in D, and defcribing 
 an arc, till by D®, is parallel to the ground line ; and fo it 
 becomes the vanifhing line of the plane, or fquare of this 
 bafe. 
 
 D, a, being drawn perpendicular to D, by finds the vanifhing 
 point of 4, 7, and its parallels j wherefore draw from 4, 5, and 
 8, to ay and having found the perfpe6live height of any one of them, 
 by the fame operation as for the other plinth, this is completed. And 
 to find that height, draw a parallel to S, C, ay from 4, upwards, and 
 'on that mark the proportional height or thicknefs, which is found 
 by drawing from S, through 4, to 4, the ground line, making another 
 parallel, as 3, i, there, of the geometrical height, and from i, the 
 top of that, a line drawn back to the fame point, S, will cut the 
 parallel drawn from 4, in the true proportion. Now, from d a, 
 (the diftance of <z,) draw a line to this laft interfe^tion, cutting 4, 7, 
 in 7, and from by through 7, a line cutting 5, ay and from that in- 
 terfeftion, a parallel to 5, 8, by which the plinth is completed. 
 The other members are determined, as thofe of the firft bafe. 
 
 Or 
 
c/ P E R S P E C T I V E. 23 
 
 Or (omitting the parallel to S, C, a, drawn from 4,) continue the 
 line 3, I, upwards, and then produce the line b, 4, till it cuts that 
 line (3, I, continued) as at 9, and there mark the true geometrical 
 height 9, 10, and draw 10, which will cut the line 4, a, in 7, and 
 fo finifh the bafe. By this, the trouble of finding the proportional 
 height at 4, is faved. 
 
 That the whole operation may be more eafily comprehended, it 
 is again reprefented apart from the picture, in pricked lines, 
 drawn from the fame points, and marked by the fame nume¬ 
 rical figures. 
 
 This explanation is lengthened by the neceflity of fhewing how the 
 fcheme was prepared from Pozzo, the intention being to reprefent thefe 
 bafes exadfly in the fame fituation, as he had placed them j for other- 
 wife, (i. e. without any reference to him,) the defcription would have 
 been more fimple j it is alfo very minutely particular, that nothing 
 material might be left unexplained, it having been faid at the begin¬ 
 ning, that thefe things fhould be referred to the occafions that might 
 require them, in order to avoid unneceflary definitions, ^c. Though 
 this method is lefs regular, yet it is much more eafy, becaufe the ex¬ 
 planation attends the ufe. 
 
 Thofe who may not readily comprehend every particular, at the firft 
 reading, are advifed to draw, in perfpe6live, the two plinths only, in 
 thefe, or the like fituations, and to place them at fuch angles with the 
 ground, that all the vanifhing points may fall within the limits of 
 the picture.——By this difpofition, they will better fee the reafon of 
 every operation; and the next figure is added to aflift them in it. 
 
 Fig. 32. Here the vanifhing points are all within the paper, and nothing 
 completed but.the two plinths, the lower of which is raifed higher 
 from the ground than in the preceding example, not only to bring 
 the vanifhing point dy within compafs, but alfo to fhew, evidently, 
 that the lines i, 3, and 6, 9, cannot pofTibly be perpendiculars to the 
 ground line, nor parallel to each other (as they are in Fozzo) when, 
 the bafe does not lie flat on the horizontal plane. 
 
 In 
 
2+ P R A C T I C E 
 
 In this fcherae 3, 6, touches the ground line ; and is therefore the 
 true geometrical length. C, Z), is the vanifhing line of the oblique 
 plane, to which the bafe is raifed, or, on which it may be fuppofed to 
 lie. The lower fquare is therefore projefted by means of C, the cen¬ 
 ter of that vanifliing line, and D, its diftance; as readily as, if it was 
 on the horizontal plane, it would be, by means of S, and D, the cen¬ 
 ter and diftance of that plane. 
 
 Now draw from (found as in the foregoing example) through 3, 
 and 6 ; then raife a perpendicular from either of them, as here from 6, 
 on which mark the geometrical height, or thicknefs of the plinth, 
 at e, and having fet off the diftance d, D, to dd, from thence draw 
 through e, which will cut d, 6, in 9, and 6, 9, will be the perfpedfive 
 height ; from 9, draw a parallel which will determine the point i, and 
 fo complete the plinth, by drawing i, C, and 9, C, and from draw 
 through the other two angles of the lower fquare, meeting i, C, 
 and 9, C. It is needlefs, here, to repeat what was before faid of the 
 other plinth in the laft fcheme. 
 
 The circles which are here barely traced, are done in the manner 
 explained at the beginning, juft as if they were on the horizontal 
 plane. 
 
 It is apparent how much work is faved by this method; neither 
 geometrical plan, nor profile are neceftary, if the meafures are but 
 known ; and if not, the plan and profile, in their common geometrical 
 fituation, will anfwer the purpofe of the moft oblique pofitions ; fo 
 that a printed book of the feveral orders may be referred to, with¬ 
 out the trouble of drawing the particular parts that may be occa- 
 fionally wanted. 
 
 Fig. 33. A figure in A. Bofes perfpedtive, for the rep^^fentation of 
 which, he makes ufe of feveral fchemes. Firft, that at No. r, where 
 in, e,—n, a, is, by him, defigned for the profile of the feat of the 
 objecl : z, e, is the inclination of the picture : o, the fpedlator’s eye ; 
 a, his ftation, or feet: o, z, the diftance : o, a, height of the eye. 
 If (fays he) the objed be only a plan, as b, c, d, on the geometrical 
 fquares at No. 2, find thofe feveral points in the correfponding perfpec- 
 
 tive 
 
■sC\- 
 
©/PERSPECTIVE. 25 
 
 tive fquares below at No. 3. (as has been taught before at Figure 25.) 
 But if the object be above the plane of the feat, as the Fig. f, r, s, (in 
 the faid geometrical fquares. No. 2.) then by means of the elevations 
 b, f,—c, r,—d, s, perpendicular to the feat b, c, d, draw' thofe eleva¬ 
 tions parallel to q, u, and from the ends of them b, c, d, drop perpen¬ 
 diculars b, I, —c, 2, —d, 6j and from their other ends f, r, s, draw 
 f, I, —r, 2,— -s, 6 , making (with them) angles b, f, i, —c, r, 2,—d, s, 6 , 
 equal to n, o, a, above at No. i, and draw i, 2, — r, 6, — 2 , 6, you 
 will have another feat i, 2, 6, and other elevations f, i, —r, 2, —s, 6. 
 Then find below, at No. 3, the perfpe< 51 :ive i, 2, 6,, of this other feat, as 
 B, D, C, was found, and make the perpendiculars i, F, 2, R, 6, S, 
 in proportion to their refpe6live plans; that is, on the perfpe6live 
 chequer take the fame meafures along thefeveral parallels of 1,6, and 2, 
 as they have above in the geometrical chequer; then join F, R, F, S, 
 R, S, by this means you will complete the perfpe6live of f, r, s ; and, 
 laftly, join F, B, S, D, and R, C, which finifh the whole prifm. 
 
 Fig. 34. Now, in order to rcprefent the fame obje6l in the fame fitua- 
 tion, nothing more is neceflary than to defcribe, either geometrically, 
 or in words, the form and fituation of the object, or to give one fide 
 of it, whofe form and fituation are known, or defcribed, and require 
 the reft ; as here, let E, F, be given, it is required to reprefent a 
 triangular prifm, whofe bafe is fimilar to the triangle Dtf, g, (above 
 this Fig. 34,) and whofe height is in proportion to the fide D, fy of 
 that triangle, and on a pidture inclined to the plane of the feat in 
 the angle o, a, n, Fig. 33, No. i. The pidture is as Bojfes, 
 
 Continue E, F, Fig. 34, till it cuts the vaniftiing line [C, D, Z, X,] 
 as in i, which will be its vaniftiing point; then from C, raife C, 
 equal to C, D, or [Z, X,] the diftance given; draw i, Dy and at D, 
 make the given triangle fy X), g, continue Dy g, to the vaniftiing line, 
 which finds k, the vaniftiing point of E, G ; therefore, draw k, E; 
 then in order to find the length E, G, which is geometrically equal 
 to E, F, bring down the diftance i, D, to the vaniftiing line at d, 
 draw E, b, parallel to the vaniftiing line, and draw d, F, which 
 
 E will 
 
26 PRACTICE 
 
 wili cut E, b, in b 5 then is E, b, the geometrical length fought, which 
 fet off from E, to a j bring down (in like manner) k, D, to e, and 
 draw e, a, which cuts E, k, in G; then draw G, F, which finifhes 
 the bafe. 
 
 And in order to complete the prifm in this fituation, having 
 defcribed the arc D, with the radius C, D, draw C, fo as to 
 make the angle D, C, equal to the inclination of the original 
 plane, with a plane perpendicular to the picture ; that is, to the angle 
 z, o, S, (Fig. 33, No. I j) and cutting the arc D, in d, draw 
 dj S, parallel to D, C 5 then S, will be the center of the picture, and 
 the angle C, d^ S, equal to D, C, draw dy N, perpendicular to C, dy 
 cutting C, S, (continued) in N, below, which will be the vanifhing 
 point of lines perpendicular to the original plane; fo that draw¬ 
 ing from N, through the points G, E, and F, the lines G, H, E, I, 
 and F, K, are got. And to determine their lengths, the diftance N, d, 
 is fet off on C, S, at N* and E, /, being made parallel to S, C, and 
 equal to E, b, (the geometrical height) draw N*, /, which cuts E, I, 
 in I, the length fought ^ whence drawing to i, and k, the points 
 H, and K, are determined, which, on joining H, K, completes the 
 whole figure. 
 
 N. B. Left: the reader ftiould not readily conceive the reafon of 
 this operation, there are reprefented on the profile of Bojfes 
 figure, No. i, the lines made ufe of in thisj for inftance, S, 
 is there the center of the picture (as formerly explained;) 
 and Oy S, properly the diftance of the pi£ture, Cy y, (being 
 parallel to o, S,) reprefents the profile of the plane, per¬ 
 pendicular to the pi6lurej and e, h, being the original 
 plane, on which the object is placed, hy Cy is the angle of 
 inclination of thefe two planes, equal to z, o, S, in the fame 
 fcheme, which is fuppofed to be given for working the pro¬ 
 blem ; and is the fame angle as D, C, dy and S, dy C, in 
 tlie pi(fture, Fig. 34. 
 
 Though 
 
(?/ PERSPECTIVE. 27 
 
 Though the work is fimple and fliort, the text may appear fome- 
 what long j but that is only becaufe the reafons of the operation arc 
 taught, and becaufe every particular is explained in the moft familiar 
 manner for the fake of learners. 
 
 It might have been remarked before, that this method of Boffe is 
 exceedingly operofe, and very uncertain j for in order to tranfpofe the 
 feveral points from the fmall geometrical chequer to the larger per- 
 fpe6live fquares, the rays Z, B, i, 8, (Fig. 33.) Z, Z), 6, and Z, C, 2,' 
 fiiould be drawn, and thefe eroded by lines from the point of diftance X, 
 to find each point, as Z, 8, and X, 7, are neceflary to find the point i, 
 only, (the diftance 8, 7, being the geometrical depth or diftance of 
 the point i, from the ground line ;) unlefs there were fo many fquares 
 in the geometrical plan as to crofs every point, which would not only 
 be exceflively tedious to perform, both there, and in the perfpeftive, 
 (where all muft be repeated ;) but the multitude of lines would necef- 
 farily produce confufion j and if the operation be performed without 
 thefe lines, then the feveral places of the points in the perfpedfive plan 
 can only be guelTed, by infpe6tion of thofe in the geometrical plan j 
 and whatever is done by guefs muft be uncertain. Whereas, in the 
 new method, there is not a line, or point, neceftary, more than are 
 here exhibited, and no polTibility of miftake, or occafion of uncer¬ 
 tainty, becaufe the lines form the figure of themfelves. 
 
 Fig. 35. In this figure, which is alfo BoJfe\ own, he propofes to reprefent 
 the prifm F, S, R, P, O, G, on a piifture, whofe profile z, e, above, 
 inclines forward, (as the laft did backwards;) the profile of the ori¬ 
 ginal plane is n, a, m, e ; the whole operation is the fame as his 
 laft, except that the pricked lines in his geometrical plan, on the 
 chequer, which in the former were drawn downwards, are here drawn 
 upwards, on account of the different kind of inclination : the corre- 
 fpondence of the perfpe6live plan with the geometrical, is evident, 
 therefore needs no farther explication. 
 
 Only it may be remarked, to what a confufed fcheme he was reduced, 
 by his limited principles. 
 
28 
 
 The PRACTICE 
 
 Fig. 36. And how fimple and eafy the other figure appears, whkh re- 
 prefents the fame objed, and is performed by exaiSbIy the fame opera¬ 
 tion as Fig. 34, with this only difference, that jD, N*, runs upwards j 
 whereas N, in the other, runs downwards, on account of the dif¬ 
 ferent inclination of the objedi: at 34, and 36, (N, in 34, and N% 
 in 36, are thevanifhing points of lines perpendicular to the original 
 planes of their refpeifive obje6ts j) and that this may, if poflible, be 
 more eafily conceived, the points S, N, and S, N*, are marked with the 
 fame letters in the geometrical fchemes. Fig. 33, No. i, and Fig. 35, No. i. 
 It is true Bojfe confines himfelf to the compafs of his picture, and un¬ 
 dertakes to reprefent all objects by means of lines terminating within it. 
 This is very well, if it be always the fhorteft, and fureft wayj but if there 
 are cafes which require more room j that is, if fome objects can be re- 
 prefented with greater certainty, by taking more fpace, or on a fmaller 
 fcale, and can then be transferred by the pidlure j and all this done in 
 lefs than a quarter of the time, that would be neceffary to produce 
 them by his method ; where is the advantage of confining the opera¬ 
 tion always to the picture ? Befides, the fcheme formed to be trans¬ 
 ferred, remains, and may be very ufeful another time. Though this 
 obje^l, however, might have been reprefented by an operation within 
 the pi6lure (with the addition of a few more lines,) on the new princi¬ 
 ples, as Jlmll be Jl:ewn hereafter. To all w'hich may be added, that 
 this very figure of Bojjes is falfe, though very neatly engraved by 
 his own hand, which muff be owing to what was hinted before, viz-. 
 that his method requires fo much guefling, as to make it almofi: im- 
 poffible not to fall into fome error from thence. And if, to avoid 
 thefe errors, a greater number of lines ftill are drawn, in order to 
 afeertain every point, the confufion would be fo much increafed, that 
 it would itfelf become a new caufe of miftake. 
 
 His error is in placing O, higher than P, G, in Fig. 35, as by the 
 pricked lines; and fo reprefenting P, O, G, as above the eye, (and 
 fecn on the under part) which cannot poflibly be, while it is below 
 Z, X, the vanifhing line. This will appear on infpediing his own 
 geometrical profile, Fig. 35, No, i, Vv'here, o, z, reprefents Z, X, 
 
 (Fig. 
 
A 
 
£/• P E R S P E C T I V E. 29 
 
 (Fig. 35, No. 3.) and fhews that if P, O, G, was even with that 
 
 line, it would be reduced to a lingle line, and muft be reprefented by it; 
 that if it were ever fo little under that line, as at 3, 4, (Fig. 35, No. i.) 
 then 4 would be feen (by the eye at o) higher on the picture z, e, 
 
 than 3, which is nearer; and laftly, that if it was above the line Z, X, 
 
 Fig. 35, No. 3. (and not otherwife) the obje 61 : would be feen as i, 2, 
 Fig. 35, No. I. (/. e.) 2, would be feen lower than i, which is 
 nearer: thus he has falfly reprefented P, O, G, />e//2g below the line Z, X, 
 haying given it the fame appearance as that in which it is truly 
 exlubited, p, o, g, Fig. 36, where it is abo’ve the line C, D, which is 
 the fame with the line Z, X, Fig. 35. P, O, G, Fig. 36. is alfo a true 
 reprefentation of the figure as it fhould have been given by Bojfe. 
 
 Fig. 37. Here is added another figure of Bojfe^ being a cube refting on one 
 of the folid angles, for which (by way of preparation) the geometrical 
 fcheme abo^'.^e is by him given, but not fufficiently explained. Some 
 lines therefore are added to render it more intelligible; ift, the fquare 
 g, n, o, p, is made, then the diagonal n, p, drawn, which is transferred 
 by the pricked arch p, b, to the point b, in the line n, g, and the 
 line b, a, drawn parallel and equal to g, p, and then n, a, is drawn, 
 and b, 7, perpendicular to, n, a, cutting it in 7, and from b, as a 
 center, with the interval, or radius, b, 7, a circle is deferibed, in which 
 two regular equilateral triangles 1,3,5, and 2, 4, 6, are inferibed ; thus 
 b, n, o, a, reprefents, geometrically, the cube Handing on the point n, 
 b, n, and a, o, being profiles of two oppofite faces feen angle wife; (that 
 is^ reprefenting diagonals of the cube,J b, a, and n, o, of two other 
 faces feen laterally; b, 7, and o, 8, two femidiameters (together) equal 
 to /, m, or (i, 4;) n, a, is the axis, to which are added the double line 
 /, n, m, and the two perpendiculars b, /, and o, m; /, n, m, reprefents 
 the profile or fedlion of the ground or plane on which the cube is fup- 
 pofed to reft, as on the point n; and n, b, (being equal to n, p, 
 the diagonal of a face) is the profile of the face B, P, C, F, (repre¬ 
 fented below;) B, reprefenting n, and /, b, (equal to 7, n,) is the per¬ 
 pendicular height of the point b, in the geometrical fcheme reprefented 
 *> 3> 5 j N, in the perfpedlivej as' o, m, equal to 7, a, 
 
 or 
 
rthe PRACTICE 
 
 30 
 
 or 8, n, the perpendicular height of the moft diftant point, and two 
 others reprefented by 4, I,—^2, P, and 6, F, in the perfpedlive ; for n, a, 
 is the greateft height, being the axis of the cube, reprefented in the 
 perfpedlive, by A, B j and thefe three are all the perfpedlive heights. 
 
 The manner of performing the perlpedtive, according to Boffcy has 
 been before explained; and it is to be carefully remarked, that;, in order 
 to find thefe perfpedlive perpendiculars, the meafure muft be firft taken 
 with the compalTes, above, on the geometrical j the compafies thus open 
 rnufl be applied to that chequer, to fee how many fquares, and parts 
 it contains, and then the fame proportion muft be taken along the 
 parallel fquares of tlie perfpedlive, even with that point in the per- 
 fpedlive plan, from which the perpendicular is required. For inftance, 
 to find the point c, in the perfpedlive, take the meafure /, b, with the 
 compafies, apply them to the geometrical fquares, where it appears, 
 that this line /, b, is equal to r, s, in the geometrical chequer (i. e.) 
 two fquares, and a part of another; then from the point i, in the 
 perfpedlive meafure, take two fquares and fuch part from i, to 10, and 
 make i, c, equal to it, by applying the compafies, thus open, from 
 I, perpendicularly to C, and fo for every height. 
 
 Fig. 38. No. I. Is the fame cube by the method fo often explained; and 
 here it is only necefiTary to require that a cube be reprefented perpen¬ 
 dicularly on its axis, and after the center, and diftance of the picture 
 are given, to give, alfo, the point B, the pole of the axis, on which 
 it ftands : C, D, is the diftance, wherefore make the angle C, D, W, 
 equal to b, n, /, (Fig. 37, No. i.) the inclination of the neareft face 
 of the cube (b, n,') with the ground, n, /, and draw D, O, perpendi¬ 
 cular to D, W; then draw B, O, and from B, draw B, i, parallel to 
 D, O, and (in order to find the proportional length of B, i,) draw 
 B, z, parallel to C, D; then make r, t, on the ground line, equal (geo¬ 
 metrically) to a fide of the cube, and draw t, Z, cutting z, B, in y; then 
 y, z, is the proportional length, at B, which fet off from B, to, i; now 
 draw D, i, cutting B, O, in I, then B, I, is one line determined. Fix 
 one foot of the compafies in W, and with the other, fet off tlie diftance 
 W, D, to D, and alfo the fame diftance on each fide of W, towards 
 
 a. 
 
c/ P E R S P E C T rv E. 31 
 
 a, and b, which will be the vanifliing points for the Tides of the neareft 
 (or front) face of the cube B, P, C, F, and its oppofite face A, G, 
 I, H j draw a, B, P, and b, B, F, and B, /, parallel to £), b, make 
 B,/, equal to B, i, draw Z),/, cutting B, F, in F, draw B, p, pa¬ 
 rallel to D, a, and equal to B, i, and draw D, p, cutting B, P, in P; 
 draw a, F, and b, P, meeting in C, then B, P, C, F, will be one 
 face determined; draw C, O, F, O, and P, O, and a, I, cutting P, O, 
 in G, and b, cutting F, O, in H, and draw a, H, and b, G, meet¬ 
 ing in A, which completes the whole cube. 
 
 Fig. 38. No. 2. The only difference between the operations to produce this 
 figure, and the laft, is, that here inflead of finding B, I, B, P, and B, F, 
 the diagonal B, C, is found, by drawing from W, its vanifiiing point, 
 through B, and the length of it, by drawing B, c, parallel to D, W, 
 and equal to the geometrical length of the diagonal n, p, (for this 
 cube is fuppofed to ftand on the ground line, and not within, as No. i.) 
 
 and drawing D, c, cutting W, B, in C-Befides which, the length 
 
 of B, I, is alfo found as before, fo that this whole reprefentation of the 
 cube will be produced, by finding the length of two lines only, and thefe 
 determine the lengths of all the reft, by means of the fame vaniftiing 
 points that ferved for the other. And in the fame manner, many 
 other cubes might be reprefented by the points already found. 
 
 N, B. The length of this diagonal B, C, No. 2, as well as of 
 the line B, I, in both the cubes, might be found by other 
 ways, which are fufficiently explained elfewhere 3 however, to 
 give an inftance, as W, D, is fet off to D, the line B, c, 
 might have been drawn parallel to W, D, and then drawing 
 from Z), through c, would find the point C 3 for it is fhewn, 
 in the beginning of this treatife, that the truth of thefe things 
 depends on the parallelifm of the original line with its line 
 of diftance, and not on their direction. 
 
 And at No. i, if the diftance O, D, had been turned up on the 
 point O, till that line became parallel to C, D,' then B, i, might have 
 been alfo drawn parallel to C, D, and if a, Z), had been turned down 
 
 on 
 
32 7 ^^ PRACTICE 
 
 on the point a, and b, JD, on the point b, both to the line a, W, b, 
 then B, p, and B, might alfo have been drawn parallel to C, D. 
 
 Fig. 39. Is alfo from Bojfe ^ the geometrical is above, which needs no 
 explication—And for the perfpe6live (according to him) the ho¬ 
 rizontal plane in the pifture muft be chequered, perfpe6tively, by- 
 means of the point of fight, or center of the picture X, and diftance 
 Z, the ground line E, V, being divided into fix equal parts by the 
 numerical figures, above that line, for feet, correfponding to the fame 
 number in the geometrical j then the line o, X, below, in the perfpec- 
 tive, which is the feat of the line 3, a, muft be divided perfpe6lively (to 
 reprefent the geometrical lines above, o, 9,) and marked with the fame 
 figures from o, to 9, inclufive: to effe6l which, fet it off geometrically 
 on the ground line from o, to V, dividing o, V, by the figures below 
 that line, and from the feveral divifions draw to Z, cutting the line 
 o, X, in I, 2, 3,-4, 5, 6,—7, 8, 9. Then from each of thofe divi¬ 
 fions ere 61 : perpendiculars, which muft be made (perfpeftively) equal to 
 their correfpondent perpendiculars in the geometrical fcheme above; 
 for inftance, the perpendicular 9, above, being taken by the com- 
 paffes, and one foot fet on E, V, at E, the other foot reaches to the 
 middle of the fquare between 4, and 5, (i^e.) four feet and a half; 
 wherefore, on the parallel at 9, in the perfpedlive line o, X, take 44 
 feet, and at 9, turn the compaffes up, with that opening, perpendi¬ 
 cularly over 9, which reaches to a, and determines the length of the 
 line 3, a, and 3, touching the ground, its place is thereby determined} 
 fo that drawing 3, finds that line, and the perpendicular at 7, is 
 found in the fame manner, (i. e.) by taking its meafure from the geo¬ 
 metrical above, which applied to E, V, appears to be 5 feet 4 i and 
 then on the parallel 7, in the perfpedlive, taking 54, and turning up the 
 compaffes (as before) the point b, is found} and thus is found the 
 point c, below} then by joining c, b,—b, a,—and 3, c, one fide of 
 the beam h, Cy 3, is found } then from b, and c, and alfo from 
 I, in the perfpe6live line o, x, draw parallels towards E, Z, and from 
 the parallel of the laft, raife a perpendicular to e, and draw e,y', and 
 raife another perpendicular from .the parallel of 7, {i. e.) at k, (in the 
 2 line 
 

 Plate xnn. 
 

(/PERSPECTIVE. 33 
 
 line E, f, X,) equal to 7, by which finds the point g j draw e, g, which 
 completes this beam: the parallels, found on this, being continued, will 
 fcrve for thofe of the other beam, and with correfpondent perpendicu¬ 
 lars, it may be completed.-Then for the crofs bar, perpendiculars 
 
 from 4, 5, and 6, in the perfpe 61 :ive line o, X, refpe6lively meafured, 
 firft in the geometrical, then on the perfpe6live parallels, will find all 
 the points neceflary, by which the whole is completed. 
 
 Fig. 40. Is the fame object reprefented by the method herein propofed. 
 Let the geometrical be either drawn as above, or only the form, and 
 meafures given in words, with the pofition, in confequence of which, 
 draw h, i, "(or any other known, or given line) j and having found S, 
 the center of the pi 61 :ure, and D, the diftance, by means of the 
 terms given, fet off the fame diflance, upwards, from S, to d, and 
 downwards from S, to b, then the angle d, D, b, will be a right angle, 
 and the angles S, D, d,—S, D, b, each of them 45 degrees, as will 
 
 alfo the angles D, b, S, and D, d, S, which is the angle of inclination 
 
 of the original object with the ground, (as well as with the perpendi¬ 
 cular 9, 10, in the geometrical above) wherefore d, is the vanifhing 
 point for i, k, and all its parallels, and b, for /, i, and all its paral¬ 
 lels. Draw m, p, on the fame line as h, i, equal to it, and at the dif¬ 
 tance given, from it; then draw i, d,—h, d,—m, d,—and p, d, all to 
 the fame point d. And for the length of them, firft draw from D, 
 
 through m, to M, in the ground line, then draw M, a, (which is to 
 
 be confidered as an original line) parallel to D, d, and equal to the 
 geometrical length of the originals, which are all equal. Now divide 
 M, a, as 3, 10, (the original above) is divided, (/. e.) in z, and q^ and 
 draw z, D,—q, D, and a, D, cutting m, d, in Z, Q^and n, which 
 will be the perfpe6live points anfwering to z, q, and 10, in the geo¬ 
 metrical. Through Z, and Q, draw from b, two lines Z, r, and Q, s, 
 and then cut thefe laft lines from the perfpe6five divifions of 4, m, 
 found by like means, (explained a few lines lower) ; and thus the four 
 points, for the crofs beam, are determined, in the plane 4, m, n. 
 
 N. B. o, S, is the feat of A, d, on the plane of the horizoUy u'hich 
 two lines cut each other in angles of 45 degrees perfpeBi^eljy as 
 their originals do geometrically. 
 
 F 
 
 For 
 
34 7?^ P R A C T I C E 
 
 For thefquares at the bottoms, and tops, draw b, i,—b, h,—b, m, 
 and b, p, alfo b, k,—b, n j then from g, draw g, h, cutting b, i, in 
 /, and g, p, cutting b, m, in 4; drawing alfo from g, through the 
 divifions of p, m, are got thofe of 4, m j for g, is the diftance of the 
 vanijhlng point b, (equal to D, b,) and therefore is the 'uanijhing poi?it 
 of the diagonals p, 4, on b, g, the •danifing line of the plane 
 
 b, /, i, — p, 4, m, and their parallels. Now draw /, d, and 4, d, &c. 
 Laftly, the crofs beam is finiftied, by drawing its parallels, having before 
 found the feveral points by means of the vanifliing points b, and d j 
 and this completes the whole. 
 
 Fig. 41. At the end of this treatife of Boffe a figure is propofed, which 
 he calls a cage, and which is alfo inferted in the Jefuit’s perfpeflive 
 borrowed from this j and, in both, faid to be by the univerfal method 
 of Monf. Defargues. Its excellence confifts in this, that by it the obje6l 
 may be projefled as large as the picture, by lines and points all with¬ 
 in thecompafs of itj-—now, befides that it might be more accurately 
 done apart, and transferred to the pi6l;ure with much lefs than one 
 quarter of the work, and in lefs than one quarter of the time, it may 
 alfo be done within the fame compafs, in a fliorter, and lefs com¬ 
 plicated way, and with fewer lines, as follows. 
 
 The fame circumftances, viz. fhape, fize, and particular meafures, 
 are given, as are given by Bofe, with its fituation, height of the ho¬ 
 rizon, diftance, and vanifhing points 3 all which are exprelTed geome¬ 
 trically on the fide, in a fmall fcale from him. The fame letters are 
 alfo ufed throughout, as many at leaft as are neceftary here, that the 
 two fchemes may be more readily compared. Having marked G, the 
 point of fight, or center of the picture, take fo much of the true 
 diftance as comes conveniently into the picture j for inftance one fourth, 
 which is 6 feet (the diftance given being 24) fet it off from G, to Z, 
 on the horizontal line. 
 
 A, B, is the ground line divided into 12 feet, draw A, G, and 
 B, G, and rays to G, from 12, to 7, inclufive. 
 
 Now in order to find any point, as M, which is 17 feet within the 
 piGure, and 1B from A, G, towards B, G, as appears by the fmall 
 2 geometrical 
 
( 9 / PERSPECTIVE. 35 
 
 geometrical fcheme above, take from A, towards B, ^ of 17, that is, 4'- 
 feet, and thence draw to Z, cutting A, G, in R, which will be 17 feet 
 within the pi6lure, (for if Z, was 24 feet from G, and 17 feet had 
 been taken from A, totvards B, a line drawn from 'Z,fo placed to 17, 
 would have cut A, G, in the fame point R, the proportion being 
 exa6lly the fame. Then through R, draw a parallel to A, B, and, 
 on that parallel, meafure i foot t between 7, and 12, (which is a 
 perfpedtive fcale j) transfer that meafure to R, and fet it off from R, to 
 M; thus the point M, is found. 
 
 K, is 29 feet within the pidlure, and yr behind A, G ; therefore 
 from A, towards B, take ^ of 29, which is y^, and draw from thence 
 to Z, cutting A, G, in e, which will be 29 feet witbinj—through e, 
 draw a parallel, and on that parallel meafure 7 feet t, which meafure 
 carry to e, and fet it off to K j—then for L, which is 26 feet deep, 
 draw from 6 t (on A, B,) being one quarter of 26, to Z, which finds 
 a point in A, G, 26 feet within i and on the parallel drawn through 
 that point, fet off 13 feet t, being its diftance from A, G, meafured 
 as before; that is, on this parallel take the whole line from e, to its 
 interfedlion with G, B, which is 12 feet, and add if of the fame mea¬ 
 fure, which will determine the point L. 
 
 In like manner for the point I, which is 38 feet within, draw from 
 9f, the middle point between 9, and 10, to Z, which will cut A, G, 
 38 feet deep, and from this laft interfedtion fet off on the parallel in 
 which it lies 4f perfpeftively, as before for the others ; join M, K,— 
 M, L,—L> I, and I, K, which completes the lower fquare of the 
 cage. 
 
 The perpendicular fides are all 17 feet high, wherefore take 8 feet f, 
 (the half of 17) on the parallel of each pointy M, K, L, ^^nd I, and 
 doubling them over each point, refpedtively, the four corners, above, 
 are determined, and by a like means the apex is found; (i. e.) after hav¬ 
 ing drawn the diagonals of both fquares, di^aw an indefinite perpen¬ 
 dicular through both centers upwards; then take, on the parallel of the 
 center of the lower fquare, 13 feet f, being the height of the apex, 
 above the upper fquare, and mark it on the perpendicular drawn, to 
 
 F 2 which 
 
rhe PRACTICE 
 
 36 
 
 which draw lines from the four corners, this completes the whole oh- 
 je6l.—The work is apparently lefs than his, for befides that he makes 
 ufe of a double operation for three of the four angles, merely to find 
 their depth, which are found here by a fingle one, and all the four 
 by the fame method j this double feries of numerical figures, which in 
 his method is neceffary, is apt to confound, and it requires much time, 
 and great care, to divide the lower feries wdth exadlnefs, which is wholly 
 unnecefTary in the method here ufed. 
 
 Fig. 42. In the fecond Volume of Pozzo^, Plate 9, thefe eight pilaflers 
 placed circularly are reprefented by his fliorter method, which has been 
 before fufhciently explained. The lines are all left, that the quantity 
 of work may be feen, and none are drawn, but fuch as are neceffary, 
 thofe tending to O, are. drawn only on one fide of Bj—becaufe the 
 other fide exadlly correfponds j fo that having placed one foot of the 
 compaffes on the point B, the other is to be extended to the feveral di- 
 vifions, and to be transferred each twice, that is, on both fides of the 
 point C, as the objedls are placed at equal diflances from it on either 
 fide j for inflance, B, i, is fet off from C, both ways, and fo of the reft. 
 
 Fig. 43. The fame fubjedl according to the new method. And here the 
 double circle is firft made perfpedlively, as has been taught, then at 
 the point of diftance D^y a geometrical double circle is drawn with 
 one fquare. A, B, in its plane, as a plan of one pilaftcr, and A, 
 B, drawn, which find a, and b, on the vanifhing, or horizontal 
 line ; thefe would be the true vanifhing points, if Ds was the true 
 diftance, but it being only half, the diftance C, a, is doubled to a*, 
 and C, b, to P, which become the vanifhing points, (for the tri¬ 
 angle C, a, being half of, and fimilar to, what the true diftance 
 v.^ould produce, c, a*, is the bafe of that triangle); wherefore drawing 
 a"", S, and b'^y S, the perfpe^tive plans of the pilaflers i, and 5, are 
 found, and fetting off the fame meafures from C, on the other fide, 
 and thence drawing through S, as before, the plans of 4, and 8, two 
 more of the pilafters, are alfo found ; but there not being room for the 
 vanifhing points of E, F, the next pilafter in the geometrical plan, 
 (which would complete the whole) another operatioju becomes ne¬ 
 ceffary. 
 
T 
 
 Plate W. 
 
(?/ P E R S P E C T I V E. 37 
 
 ccffary, viz, the geometrical place of that pilafter, or its oppofite, muil: 
 be found below, as at G, and reprcfented, as was explained in the 
 firft plate of this treatife; the plan of No. 2, will then be found, the 
 tides of which being continued through S, finds alfo the plan of 6; the 
 remaining two are found by parallel lines from thofe already done ; 
 for 3 is parallel to 6, and 7 to 2 j then the geometrical height is fet 
 perpendicularly from e, to f, and lines drawn from both to C, be¬ 
 tween which all the feveral heights are found by means of parallels 
 drawn from the bafes or plans cutting e, C j and perpendiculars from 
 thofe interfedlions to f, C, and parallels drawn back from the inter- 
 fe6tions of f, C, complete the whole. ^ 
 
 Fig. 44. This is another reprefentation of the fame pilafler, added merely 
 to thew how little work is neceflary, where room is not wanting for 
 all the vanithing points j and this reprefentation is fo eafy to be under¬ 
 flood, merely by infpedlion, after what has been faid above, that no ex¬ 
 planation is neceffary. 
 
 Indeed this alone might have been fufficient to have fhewn the prac¬ 
 tice; but then it might have been obje6led, that the Fig. 43. was 
 avoided, to conceal the difficulty of this method, when the fpace 
 allotted is too fmall to receive all the vanifhing points neceffary; 
 but for the future, fuch a diflance will be taken, as may admit of all, 
 or mofl of the vanifhing points ; both becaufe the work will be clearer, 
 and fhorter, and alfo becaufe a proper place is referved in the 
 fourth part, for the explanation of feveral expedients that may be ufed, 
 in cafes that fhall require them. 
 
 The 
 
The PRACTICE 
 
 The THIRD PART. 
 
 H E reader is fuppofed, by this time, to be fufficiently con-. 
 
 JL vinced, that the new method is preferable to any former \ there¬ 
 fore no more comparifons will be made, but the new principles 
 conftantly recommended in this treatife will be regularly purfued. 
 
 Perfpe6live is principally exercifed in proje 61 :ing points and lines, 
 and planes compofed of lines j for folid bodies, of all kinds, are to be 
 projefled either by two planes perpendicular to each other, as the ich- 
 nography, and orthography, or by that number of planes which com- 
 pofe fuch bodies refpe6i:ively j in either cafe it is necelfary, after 
 having found the vanilhing line of each plane, (the center and diftance 
 of the pi6lure, on which it depends, being always given) to projefl 
 the feveral lines which form fuch plane. When the whole number of 
 planes are projected, the body or figure is completed by fuch proje6lion, 
 without any further operation j but when only the two planes of the 
 ichnography, and orthography are chofen to be proje6led, it is necef- 
 fary, afterwards, to join their correfponding points, by perpendicijlars 
 drawn from them refpeflively. 
 
 It is apprehended, that the following examples will be fufficient to 
 explain, and illuftrate thefe two manners of projedling obje6ls, per- 
 fpe6lively. 
 
 Here it may be proper, more explicitly, to deferibe the nature of va- 
 niflaing points, and lines, (hitherto occafionally explained) and to fhew 
 how they are generated. 
 
 A vanifliing point, is that point, wherein a line, paffing from the ^ 
 eye, parallel to an original line, cuts or interfeffs the piflure; and a 
 vanifhing line, is that line wherein a plane, palling from the eye, pa¬ 
 rallel to an original plane, cuts or interfedfs the pidture. Thus the 
 point, commonly called the point of fight, or center of the pidlure, 
 being determined by a line paffing from the eye, at right angles, or per¬ 
 pendicular to the plane of the pidlure, is the vanilhing point of all 
 
 original 
 
f 
 
PERSPECTIVE. 39 
 
 original lines, making right angles with, or which are perpendicular to, 
 the plane of the pidlure. And when the picture is perpendicular to the 
 plane of the horizon, which is the moft ordinary fituation, the line, 
 commonly called the horizontal line, being formed by a plane palTing 
 from the eye, at right angles, or perpendicular to the ’pidlure, is the 
 vanifhing line of the horizon, as well as of all other planes parallel to 
 the horizon j but when the pi6ture is inclined to the horizontal plane, 
 in any other than a right angle, then the vanifhing line of the horizon 
 will be higher or lower than tlie vanifhing line of a plane per¬ 
 pendicular to the pidlure, according as the pidture is inclined back¬ 
 wards or forwards, as fhall be explained hereafter. 
 
 And thus, in general, the vanifliing line of any original plane, is 
 that line in which the parallel of fuch original plane (pafling from the 
 eye) cuts the pidlure. 
 
 Fig. 45. Let it be required to reprefent a cube {landing on a plane, 
 making a given angle with the horizon, (fuppofe thirty degrees) the 
 point of light, or center of the pidlure, and the diflance being given. 
 Firfl mark the center of the pidlure S, and draw S, D, parallel to the 
 
 interfedlion of the original plane with the pidture, and equal to the 
 
 diftance given; through S, draw S, P, perpendicular to S, D, and 
 from D, draw D, C, making the angle S, D, C, 30 degrees; and 
 cutting S, P, in C, draw d, C, parallel to S, D, which will be the 
 
 vanifhing line of the plane on which the cube Hands j draw D, P, 
 
 perpendicular to D, C, cutting S, P, in P, and bifedl the angle C, D, P, 
 to X, and fet off the diflance (D, C,) of the vanifliing line, d, C, 
 from C, to d, then will the points C, P, X, and d, be all the vanifli- 
 ing points requifite for projedling the cubes No. i. and 2, and as many 
 more as may be required, with a fituation, diredl, on the plane, v^^hofe 
 vanifhing line is d, C j for draw at pleafure e, f, parallel to d, C, 
 then e, C, and f, C, and d, f, cutting e, C, in g, and g, h, parallel 
 to e, f, which finiflies the lower fquare, then P, e,—P, f,*—P, g, and 
 P, h, after which draw X—h, cutting P, f, in /, and the diagonal 
 h, /, will determine the length of f, / j then draw /, C, and /, u, 
 parallel to e, f, then u, C, and the remaining parallel, by which the 
 cube is completed. The 
 
40 7 :^^ P R A C T I C E 
 
 The fame points' and operation are fufficient for No. 2, or any others 
 in a like fitiiation. 
 
 For No. 3, the fame lines, with very little addition, (on account of 
 the different pofition of it) are fufficient. This cube is in an oblique fitu- 
 ation on the fame plane; m, 6, is firft drawn at pleafure, and continued 
 to a, its vanifhing point, then the diftance of the vanifliing line, ‘utz. 
 D, C, is fet off from C, to D, then a, D, is drawn, and D, at 
 right angles to it, and fo becomes the vanifliing point of m, q, 
 audits parallels; the length of m, q, is taken at pleafure (as was e, f, 
 of No. I.) but by that all the other lines are determined. Draw q, a, and 
 to find the length, q, 5, bife6t the angle a, £), b, to X*; draw Xh m, 
 cutting q, a, in 5; draw 5, cutting m, a, in 6, which finifhes the 
 lower fquare; draw P, m,—P, q, and P, 6, and to find their lengths, 
 the diftance of P, b, the vanifliing line (of the plane of m, q, n,) muff 
 be found ; therefore draw through S, a perpendicular to P, b^ cutting 
 it ill C'^, which will be the center of that line, and on P, by as a dia¬ 
 meter, defcribe a femicircle, cutting that perpendicular in d'^, then 
 F, b, will be a right angle (by 31. III. Fuel.) and confequently 
 
 C% the diffance, of that vanifhing line; or draw S, D®, parallel 
 to the vanifliing line P, b, and equal to S, D, (the diffance of the 
 piclure) and draw from Z)*, to G®, (the center of P, b,) and fet off 
 to d^, by placing one foot of the compaffes on and with the other 
 foot deferibing the arc D '^,—Zh at d"^, bifedl the right angle P, d^, by 
 to Xh and draw Xh q> which will find the diagonal q, n, of the 
 fquare m, q, n ; draw n, a, and n, by &c. and fo complete the 
 cube. 
 
 No. 4. Is another cube reprefented by means of the fame vanifliing 
 points, as No. 3, without the addition of one other point or line. But 
 becaufe in fome cafes it may not be fo convenient to make ufe of the 
 diagonals to determine the lengths of lines, the following method 
 is added, which is univerfal. Having drawn from k, to the three vanifh¬ 
 ing points a, b, and P, in order to determine the length of any line, as 
 for iiiffance k, i, (whofe vanifliing point is b,) fet off b, D, the dif¬ 
 tance of that vanifliing point, on its proper vanifhing line a, C, b, 
 
 from 
 
c/ PERSPECTIVE. 4 £ 
 
 from jD, to the point o, and parallel to that vanllhing line draw k, 
 equal to the original of k, i, in that place : Draw o, r, cutting k, b, 
 in i, which will be its perfpe<^live length i make k, t, on the other fide 
 equal to k, r, and, in like manner, fet off the diftance of a, D, to u, on 
 
 the fame vanifliing line j draw u, t, cutting k, a, in w_The fame is 
 
 repeated at z, (i-e.J the diftance P, is fet off to z, on the vanilh- 
 ing line b, P, to determine one line in that plane, viz. k, 7, for which 
 purpofe the fame geometrical length is placed from k, to 8, parallel to 
 by P, and drawing z, 8, cuts it in 7, the perfpeftive length. 
 
 If it were required to find the plan of one of thefe cubes on the hori¬ 
 zontal plane, this might be done by dropping perpendiculars from every 
 point of the cube, as at No. 4, and cutting thofe perpendiculars by 
 lines drawn from vanifhing points found in the horizontal line S, D, 
 by means of perpendiculars from the correfponding vanifhing points 
 of the feveral lines of the proje 61 :ed figure ; as for inftance, S, is the 
 vanifiiing point P, brought up to the horizontal line, wherefore draw 
 from S, through 7, the lower angle of the cube (fuppofed to touch 
 the ground) cutting the perpendicular from k, in the point a, which 
 is the feat of the point k, on the horizon j from a, draw to the point 
 perpendicularly (under by) in the horizontal line, which determines the 
 plan or feat of i j from a, draw to the point under which finds the 
 feat of w J and from thefe two lafi; found points, the feats of i, and 
 w, draw to the fame vanifhing points, of the horizontal plane, which 
 completes the plan of the upper face, of the cube; and in the fame 
 manner is the plan of the lower face found, and by joining the two 
 extreme points on each fide, the plan of the whole cube is completed. 
 
 That of No. 2. is alfo found in the fame manner, but as one whole 
 fide of this cube touches the horizontal plane, it being placed parallel 
 to the horizontal line, and only the point S, ufed, the plan is more fiin-* 
 pie. And if it were required to find the plan at any given diftance be¬ 
 low, on fuppofition of the object being above, and not touching the 
 horizontal plane, the fame method will anfwer the purpofe; thus, at 
 'No. 3, ‘drop a perpendicular as low as required, (e. g.) from N, to M, 
 and proceed as at No. 4, beginning with M. 
 
 G 
 
 N.B. 
 
42 
 
 The PRACTICE 
 
 N. B. ^hefinding plans of ohjeUs already projediedy is by no meam 
 an ufelefs curiofityj but in fome cafes abfolutely neceffary^ and par~ 
 ticularly^ in order to the projection of their Jhadows, 
 
 Suppofe it required to reprefent an obje6t Branding on a plane in¬ 
 clined to the horizon, in any given angle, as (e.g.) in an angle of 20 
 degrees, and on a pifture perpendicular to the horizon. 
 
 Fig, 45. No. 5. Let S, be the center of the pidlure ? S, D, the diftance, 
 marked on the horizontal line; S, C, drawn perpendicular to S, D j 
 and D, C, drawn fo, as to make with S, D, an angle of 20, (the required 
 inclination of the original plane, with that of the horizon,) and cutting 
 S, C, in C. Then the angle at C, will be 70, the complement of 20, 
 to a right angle, and equal to that which the original plane makes 
 with the pidirure. 
 
 By filch original plane is to be underftood a plane whofe interfedfion, 
 with the pidlure, is parallel to the horizontal line, in which cafe its 
 vanifhing line will neceffarily be parallel alfo ; wherefore C, D*, drawn 
 through C, parallel to S, D, is that vanifhing line, and C, D, its dif¬ 
 tance, which may be raifed up to its vanifhing line C, D®. 
 
 The objedf, or wedge A, is an example, Ihewing the ufe of fuch 
 vanifhing line, the bafe of it is a fquare on the horizontal plane, projedled 
 by means of the horizontal line F, S, D. But the upper face of it in¬ 
 clines to the horizon in an angle of 20, and is, therefore, projedled 
 by means of the vanifhing line C, D®, as appears by the lines of the 
 operation, in the diagram. 
 
 But when the original plane (though with the fame inclination to 
 the horizontal plane) is oblique to the pidlure, then the vanifhing line 
 of that plane will be oblique alfo, and will interfedt the horizontal line, 
 as E, D, the vanifhing line ef the upper face of the wedge B, which 
 objedf is, in all refpedls, fimilar to A, its pofition only being different, 
 and is projedfed on its proper vanifhing line E, D, by means of points 
 exadfly correfponding to thofe on the vanifliing line C, D“, for the 
 upper face of the wedge A. 
 
 ^Now, here, though the inclination of the upper face of B, to the 
 plane of the horizon, is ffill the fame, yet, its inclination to the plane 
 3 of 
 
PlateXK 
 
•* 
 
 1 
 
 r 
 
 i 
 
 rr- 
 
 % 
 
 I 
 
 i 
 
 ■'•■Jr'i ../ ■ 
 
 r;«i 
 
 
 
 
 1 .^* 
 
 
 1 
 
 
 
 
 
 
 v.t 
 
 ,T., - 
 
 , • t'!. •>.' .' 
 
 -'r.- 
 
 4 ‘• 
 
 ■ '; . ■' ';’ . ■ 
 
 
 ^ ;/h - 
 
 v-^iS 
 
 
 
 V. 
 
 ;-V 
 
 
 
 
c/ P E R S P E C T I V E. 43 
 
 of the pi^ure is altered hy the obliquity of its pofition; and as tiic 
 inclination of two planes is always meafured on a third plane perpen¬ 
 dicular to both (or which is the fame thing) perpendicular to their 
 common interfeflion; fo it appears in the diagram, that the angle 
 E, d, F, which meafures the inclination of the upper face of the ob- 
 je6t B, with the horizon, is equal to C, D, S, which meafures that of 
 the upper face of A.—But that </, C®, S, which meafures the inclina¬ 
 tion of the upper face of B, ‘wkh the piBtirey is larger than D, C, S, 
 which meafures the inclination of the upper face of A, with the piBure, 
 
 And as the angle C*, S, is larger than d, E, F, fo the angle at 
 dy (the complement of dy C®, S,) is neceflarily lefs than that at d, (the 
 complement of d, E, F,) which angle at dy is the inclination of a 
 plane, to another plane whofe vanifhing line would be dy S, continued, 
 (i, e.) parallel to D, E, and not to the plane of the horizon. 
 
 The vanifhing line E, D, for the upper face of B, is found by bring¬ 
 ing down F, D’, (the diftance of the vanifhing line E, F,) to d, on the 
 horizontal line, and drawing d, E, making F, d, E, an angle of 20, 
 and then drawing E, D, which is the vanifhing line required. 
 
 For if the triangle F, D, be raifed up on the line F, 5 , D, fo 
 
 as to become perpendicular to the pidtiure, and the triangle E, d, F, 
 raifed up, with it, on the line E, F, till d, coincide with (in that 
 perpendicular fituation,) then it will be evident that the plane E, d, F, 
 will be perpendicular both to the plane of the horizon, and to the plane 
 E, d, D, (when in fuch fituation) which plane, being parallel to the 
 original, oblique plane, and cutting the pifture in E, D, that line 
 becomes its vanifliing line. 
 
 As this part, relating to oblique planes, and efpecially when oblique¬ 
 ly fituated with refpefl to the pi6ture, is fomewhat difficult, par¬ 
 ticular attention has been employed to render it as clear, as poffible, 
 and for that purpofe the two principal parts of the laft diagram, are 
 again feparately reprefented, and difl:in6lly confidered, in the two fol¬ 
 lowing fchemes, and with a greater angle of inclination to the horizon. 
 
 Fig. 45. No. 6. F, D, is the horizontal line j S, the center of the pifture j S, D, 
 the diflance of the pi( 5 lure j E, D% the vanifhing line of a plane in- 
 
 G 2 dined 
 
44 P R A C T I C E 
 
 dined to the horizon in an angle of 40 degrees, and confequently to the 
 plane of the picture in 50, the complement of 40 j C, the center of 
 that vanifhing line j C, D, its diftance j C, D^, the fame diftance raifed 
 up to its proper vanifhing line. This wants no farther explanation. 
 
 Fig. 45. No. 7. F, the horizontalline; S, the center of the pi< 5 tu re; S,D, 
 the diftance of the picture j E, D®, the vanifhing line of a plane inclined to 
 the plane of the horizon in an angle of 40 degrees, (as is E, jD*, No. 6 .) 
 but in a different diredtion, (i. e.) obliquely, with refpedt to the pidure. 
 For it is inclined to the pidure in an angle of 55, (and not of 50, as 
 No. 6.) 
 
 Now to explain the reafon of this difference, it is to be confidered, 
 that, in this fcheme, No. 7, the circumftances required are, to find the 
 vanifhing line of a plane inclined to the horizon, in an angle of 40, 
 but with a certain given diredion, (i> e.) fo as to interfed the horizon¬ 
 tal line in a given point, as D*.—In order to effed which, the firfl 
 flep to be taken is to find the vanifhing line of a plane perpendicular 
 to the line whofe vanifliing point is D*, (becaufe on fuch plane the incli¬ 
 nation is to be meafiired^ as has been before mentioned,) therefore from S, 
 (the center of the pidure) raife a perpendicular S, D’, equal to the 
 diftance of the pidure, and draw the line D®, Dh and then, perpendi¬ 
 cular to it, draw Dh F, cutting the horizontal line in F, at which 
 point raife the perpendicular F, E, and this will be the vanifhing line 
 (fought) of a plane perpendicular to the vanifhing point firft given. 
 
 F, is the center of this vanifliing line, F, Dh its diftance j where¬ 
 fore bring down that diftance to d, on the horizontal line, and there 
 make the angle of inclination required, by drawing d, E ; and laftly, 
 draw D^ E, which is the vanifhing line of the oblique plane required. 
 And this plane inclines to that of the horizon in an angle of 40 j for 
 if the triangle F, Dh D*, be raifed up on the horizontal line, till it is 
 perpendicular to the pidure, and the triangle F, d, E, be raifed up 
 at the fame time with it on F, E, till d, and Dh coincide j then the 
 plane F, d, E, will be perpendicular both to the horizontal plane, and 
 to the plane whofe vanifhing line is E, (for in the fituation defcribed, 
 
 E, D^, ly, or which is the fame, E, d, D^, will be that plane) and 
 
 will 
 
Hate XX. 
 
 C 
 
If,. 
 
 ’ L » 
 
 
 ) 
 
 9 ^ 
 
 
 I 
 
 / 
 
 \ 
 
 \ r 
 
 
 
 h- 
 
 / 
 
 . -I 
 
 5 
 
 
 i-’’ 
 
 r • 
 
 - ' ^ ' 
 
 ^tt<4 
 
 
 (. 
 
 Ml 
 
o/ P E R S P E C T I V E. 45 
 
 will truly meafure their inclination j therefore thofe two planes are in¬ 
 clined in an angle of 40, as required. 
 
 But as the plane F,.d, E, is not perpendicular to the picture alfo, 
 it cannot meafure the inclination of the oblique plane with that of the 
 pi6lure. 
 
 And as this is to be found by means of a plane perpendicular to 
 both, draw S, C, perpendicular to that - vanilhing line, and S, D, 
 parallel to it, and equal to the diftance of the pidlure, and draw D, C ; 
 then imagine the triangle S, D, C, raifed up perpendicularly on the 
 line S, C, and the other planes raifed up with it (as before), and, in 
 this fituation, the plane S, D, C, will be perpendicular to both the pi6lure, 
 and the oblique plane, (for D, njoill then coincide with D^, and d, per- 
 pe 7 tdicidarly over S,) and will therefore truly meafure their inclination, 
 which, thus, is found to be an angle of 55, and its complement 35, 
 is the angle of inclination of the oblique plane, with a plane whofe 
 vanifhing line would be D, S, continued, (and not the horizontal 
 plane) but which would be inclined to the horizon in the angle 
 D, S, D* j for palling thro’ the center, it is the fame as the real, ori¬ 
 ginal, or geometrical angle of inclination. From the two laft dia¬ 
 grams, appears the difference between the relations of an oblique plane, 
 whofe vanifliing line is parallel to the horizontal line, and an oblique 
 plane, whofe vanifhing line interfe6ls the horizontal line. 
 
 For at No. 6, the plane S, D, C, (when raifed perpendicularly on 
 the line C, S,) is perpendicular to all the three planes, viz, of the hori¬ 
 zon, of the picture, and of the oblique plane \ and therefore meafures. 
 the inclination of any two of them. 
 
 But at No. 7, the plane F, d, E, when raifed, fo that d, be per-» 
 pendicularly over S, is perpendicular to two of them only^ viz, to that 
 of the horizon, and to the oblique plane, but not to the pidlure. And 
 alfo, that the plane S, D, C, (when raifed perpendicularly over the line 
 S, C,) is perpendicular to two only of the three before mentioned j viz, 
 to the pidure, and to the oblique plane ; but not to that of the ho¬ 
 rizon. 
 
 N. B. This 
 
46 
 
 m PRACTICE 
 
 iV. B. This laft is alfo perpendicular to a third plane, (though 
 not one of the three here required) viz, to a plane perpendicu¬ 
 lar to the pi6hire, whofe vanilhing line would be D, S, con¬ 
 tinued, as was before obferved. 
 
 Fig. 46. Here are fome circumftances explained, which were referved for 
 this place, both that the learner might be prepared by what has pre¬ 
 ceded, and alfo that he might not be embarraffed by too many lines 
 in one diagram. S, is the center of the pidlure; S, D, the diftance 
 drawn in the diredlion of an original plane, with refpedl to the pic¬ 
 ture, or in which a vaniHiing line is required j S, C, drawn perpen¬ 
 dicular to S, D ; and D, C, drawn parallel to the inclination of the 
 original plane, (i,e.) making with D, S, a certain given angle g.) 
 an angle of 24, and cutting S, C, in C; then the angle at G, will be 
 66, the complement of 24, and equal to the angle fuch original plane 
 makes with the pi6lure. 
 
 Now thro^ C, draw <2, C, parallel to S, D, which will be the va- 
 nifhing line of the original plane,' and on which feveral cubes are fup- 
 pofed to be placed. Let C, S, be continued downwards j draw D, P, 
 perpendicular to C, D, cutting C, S, P, inP; now fuppofing the plane 
 C, D, P, to be raifed up fo, as that the point D, (which reprefents al¬ 
 ways the eye of the fpedlator) be perpendicular over S, then that plane 
 C, D, P, becomes perpendicular to the picture, and P, the vanilhing 
 point of lines, perpendicular to the planes whofe vanifliing line is 
 £ 1 , 0 ,^ b \ therefore any line, as a, P, palTing thro’ P, and cutting a, C, b^ 
 will be the vanifliing line of a plane perpendicular to a, C, b but in 
 order to find a third vanifliing line perpendicular to both thefe vanifli- 
 -ing lines already found, the diftance C, D, of C, b^ is fet off on 
 C, P, at d’ j then a, d‘, is drawn, and d‘, b^ perpendicular to it j and 
 thus P, being drawn, becomes the third vanifhing line of planes 
 perpendicular to both the others j for (as was remarked before) any line 
 pairing through P, meeting the vanifliing line a, by will be the vanifli¬ 
 ing line of a plane perpendicular to it; therefore P, by is a vanifliing 
 .line perpendicular to ay by and it is perpendicular to P, byconflruc- 
 tion, Uy d\ by being made a right angle. This vanifliing line P, by 
 
 might 
 
P E R S P E C T I V R . 47 
 
 might have been found, by drawing a, S, and S, perpendicular to 
 it, equal to S, D, (the diftance of the pi< 51 :ure) and drawing a, £)^ and 
 X)^, C^, perpendicular to it, cutting a, S, in ; then drawing 
 P, cutting by in by as is evident} and fo of the reft, 
 
 C, is the center of the vanilhing line a, by found by drawing 
 a line through S, perpendicular to it, and the point fo found in 
 every vanifhing line, is always callechits center, which is to be ufed on 
 fuch vanifhing line in the fame manner, and for the fame purpofes with 
 refpedl to the plane which it reprefents, as the center of the horizon-- 
 tal line with refpeft to the horizontal plane ; and in the fame manner 
 are found the center of a, P, and Ch the center of by P.—If a 
 circle be defcribed round S, the center of the pi6hire, with the radius 
 S, D, which is the diftance of the pifture, as all the radii are necefla- 
 rily equal, any line from S, to the circumference will be equal to, or 
 will be properly, the diftance of the pi6ture ; therefore drawing S, Z)®, , 
 at right angles, on the perpendicular by C®, of the vanifhing line a, P, 
 and drawing C®, D®, it will be the diftance of that vaniftiing line. 
 
 In the fame manner drawing S, Z)h at right angles, on the per-* 
 pendicular <?, Ch of the vanifhing line by P, and drawing C^, Z>h it 
 will be the diftance of that vaniftiing line j S, D,—S, Z)®,—S, Z)^, 
 will all be feverally parallel to their refpedfive vanifhing lines ay by”—- 
 a, P,—and b, p. Now if D, S, be raifed up perpendicularly over ' 
 
 C, P,—Z)*, S, over by C®,—and Z)^, S, over C^, thefe three points 
 
 D, D®, and Z)h will coincide over S.—Again, if C, D, (which is the 
 diftance of <2, b,J be transferred to d', on C, P, the perpendicular of 
 the faid vaniftiing line a, by and C®, D®, the diftance of a, P, to d®, 
 on its perpendicular by C®, and alfo Ch Z)h the diftance of by P,.. 
 to d^, on its perpendicular a, Ch the three points d^, d®, dh being 
 raifed on their refpe6live vanifhing lines a, by —a, P,—^and b, P, fo 
 far as that each be perpendicular over S j thefe three points will all '’ 
 coincide, not only with each other, but alfo with the three firft named ^ 
 D, Z)®, DK 
 
 The learner isadvifed to make all thefe points, and lines, as familiar^ 
 to himfelf as pofiible, by drawing vaniftiing lines in feveral direcftions, , 
 
 and 
 
The PRACTICE 
 
 and efpeclally three, to reprefent planes at right angles to each other, 
 as in this fcheme, for the proje 61 :ion of cubes, and cubical forms, in 
 all pofitions j becaufe nothing is more neceflary in the pradlice of 
 perfpedlive: and if he has underftood all the preceding part, it is ap¬ 
 prehended this will not be difficult to him. In order to affift the ima¬ 
 gination a little, let him confider the two cubes. A, and B, one of 
 which, A, is feen dire£l, the blank parts to be fiippofed on the other fide 
 of the pi6lure, and the lines n, /,—/, m, and m, n, to interfcdf the 
 pi 61 :ure ; thefe three lines may be conceived to be the vanifhing lines of 
 the three planes which form the folid angle of a cube, and m, n, /, 
 the vaniffiing points of its Tides, or legs. The fame thing is repre- 
 fented in B, with this only difference, that it is oblique, as the large 
 fcheme, juff explained ; for which reafon, two of the legs cut 
 the pi6lure on this fide of the angles of the cube, and the lines a, — 
 
 p, and p, a, reprefent thole three vanifhing lines, in the large 
 fcheme, to which they are refpectively parallel, and are, in both, the 
 vanifhing lines of the folid angle of the cube. 
 
 On one fide is a cube E, proje 61 ;ed as in the former, but removed 
 out of its place of projedlion, that the lines might not be confounded 
 with thofe of the fcheme ; the reader is to refer it to g, within, 
 which point (correfponding with g, on the figure) the whole is fup- 
 pofed to be performed. 
 
 Here is a circumftance determined which before was not fuppofed 
 to be required, mz. that it fliould touch the pidlure in the point g, the 
 body of the figure being behind the picture; to effe 61 ; which, from the 
 point D, on the line C, D, raife a perpendicular D, G, equal to one 
 Ede, or leg of the cube: draw G, F, parallel to C, P, and confequently 
 to the picture. In proje6ting the cube E, after having drawn g, b, 
 and g, a, fi. e.) fuppofe from g, in the original fcheme, draw g,f, pa¬ 
 rallel, and equal, to G, F, (which was made parallel to C, P,) then 
 draw f, C, tending to C, (i. e.) to the fame vanifhing point, as F, C, 
 and reprefen ting a, parallel to it j draw alfo g, P, cutting/, C, in 
 h, then the line g, h, will reprefent G, D, and be (perfpe6tively) equal 
 tio it, and g, vdll touch the pidture. The reft is performed as has been 
 
 fhewn 
 
®/ P E R S P E C T I V E. 49 
 
 fhewn before. All this operation is fuppofed within the fcheme or 
 
 diagram (as is faid above) beginning at g, and then tranfpofed to 
 
 E, only to avoid confufion of lines in the great fcheme. The other 
 cube K, is fuppofed to interfefl the pi6lure in the lines 5, 6,—6, 7, 
 and 7, 8, the reft being fuppofed behind j to reprefent which, another 
 perpendicular H, M, is drawn on C, D, F, fo much before the line 
 G, F, as the cube is fuppofed to be before the pi6lure, the reft be¬ 
 hind j and the fame method is ufed as for the other E, only the trian¬ 
 gle D, G, F, is here reprefented by dy gy fy within the cube; and, 
 in order to make it advance before the picture, in the proportion re¬ 
 quired, a line g, t, equal to G, H, is fet off from g, parallel to the 
 
 vaniftiing line <z, and the diftance C, D, fet off from C, to q: 
 
 then q, t, is drawn, cutting c, g, in r j and r, P, cutting C, dy /, 
 in 5, determines one line of the advanced part of this cube, from which 
 the reft is finifhed j and when completed, the lines forming the triangu¬ 
 lar interfeiSlion, are refpe6lively parallel to the three feveral vaniftiing 
 lines; the uppermoft paffes through g, determining the other two. 
 This alfo is performed within the fcheme, the point r, there, anfwering 
 to that on the cube, and the whole being tranfpofed to K. 
 
 It appears, by thefe proje^iions, that if the meafures, and forms of 
 objefts, are known, they may be reprefented without geometrical plans, 
 or elevations, which faves much time and trouble. And if it be re¬ 
 quired to alfign the geometrical fize, and fituation of any figure already 
 proje 6 led; as, for inftance, the cube E , firft draw through fy a line 
 u, w, parallel to the vaniftiing line by which line is therefore the 
 interfe<Sl:ion of the original plane with the pi 6 lure (or what is called 
 the ground line;) for it was before mentioned, that gy fy touches the 
 pifture; continue the lines i, h, and /, h, till they cut u, w, in n, 
 and o; continue alfo m, /, and m, i, to u, and w: then having 
 meafured the angle d*, a, by in the large fcheme, make u, o, L, 
 equal to it, for a, is the vaniftiing point of h, / j draw w, I, pa¬ 
 rallel to o, L: in like manner meafure the angle d*, by <7, and make 
 w, n, I, equal to it j draw u, L, parallel to n, I, which will 
 complete the figure, or plan, L, I, in its geometrical fituation, and 
 
 H pro- 
 
50 7^^ PRACTICE 
 
 proportion.—Or, to avoid the trouble of meafuring, and transferring 
 the feveral angles, continue d*, C, upwards, till c, £)♦, is equal to 
 C, d*, (which was made equal to C, D,) and make o,.L, and w, I, 
 parallel to a j and u, L,—^n, I, parallel to D^, b, which will 
 anfwer the fame purpofe. 
 
 The like operation is repeated for the cube K, on the other lide, tranf- 
 pofed from r, (within the fchemej) to which r,on the cube, correfponds, 
 only this cube advancing, in part, before the pi6lure, (as appears by the 
 ground-line v, w,) is larger on that account j and, for the fame reafon, 
 the original geometrical fquare, or plan, is neceffarily cut by the ground¬ 
 line j in confequence of which the points n, o, are, in the interfeflion 
 of the original fquare, with the reprefentation. The reft needs no 
 explanation, as the two figures are intirely fimilar.—If this large fcheme 
 appears, at firft fight, overcharged with lines, the reader, who has 
 underftood the preceding rules, will readily perceive that very few of 
 them (only) are neceflary to the proje6tion of the obje6ls reprefented; 
 and that the others are added, partly to exhibit different manners of 
 projecting the fame objeCls, but principally to fhew the corre- 
 fpondence, and relation that feveral fyftems of lines have with each 
 other, which, thus, are more evident than if drawn in feparate 
 diagrams, and more effectually illuftrate the nature and ufe of 
 vaniftiing lines. 
 
 iV. B. The whole profile of this fituation of the cube K, is 
 geometrically ereCted on the line C, D, M, which is taken from 
 d*, where the original plan is defcribed, in the pofition feen 
 by the fpeCtator, and reprefented in the perfpeCtive, where 
 C, 5, anfivers to C, d*, in the geometrical plan, and t 5, 
 to r dh &c. The lines crofting from the angles (in the 
 geometrical plan) are parallel to the vanifhing line b, and 
 confequently to the interfeCtion j and N, M, anfwers to 
 Ny M, the feCtion or profile of the plan. 
 
 Fig. 47. In this fcheme the vanifhing line is ftill more oblique, 
 
 and croffes that of the horizon, in order to fhew that even in fuch 
 fituation, the manner of finding a plan on the horizontal plane is the 
 2 fame 
 
.1 
 
 t 
 
 i 
 
 
 
 ■ t- 
 
 ■ A> 
 
 f. 
 
 i 
 
 *v 
 
 V 
 
 
 « 
 
 >. ' ' '• ., A ‘ '• ' ‘ 
 
 ri 
 
 
 
 
 \ 
 
 
 ..ij? 
 
 ' .<* 
 
 " ‘ '’/li® 
 
 V 
 
, ./perspective. 5r 
 
 fame as before exhibited, notwithftanding a feeming difficulty arifing 
 from the vanifhing point being below the horizon.-—The cube A, 
 being proje6led, as has been before taught, in order to find its plan 
 perpendicularly on the plane of the horizon, firft transfer the three 
 vanifliing points tf, and P, perpendicularly to the horizontal line, 
 viz. downwards; <7, and P, upwards, to b, a, and p; then draw, 
 from the point i, which touches the ground, to a, and to b ; drop 
 a perpendicular from II, to 2, and raife one from III, to 3 j then 
 draw from 2, to a, and from 3, to b, which will complete the plan 
 of the lower fquare or face I, II, III, IV; for the interfe6lion of 
 2, a, and 3, b, will mark the point 4, perpendicularly, under IV; 
 now draw from p, through I, and drop a perpendicular from V, in- 
 terfe^Iing p, I, in 5, and draw 5, a,—5, b; drop a perpendicular 
 from VI, to 5, a, cutting it in 6 j and from VIII, to 5, b, cutting 
 it in 8 J draw from 8, to a, and from 6, to b, which will complete 
 the plan of the upper face V, VI, VII, VIII j for the point 7, will 
 be found in the interfedlion of 8, a, and 6, b, perpendicularly under 
 VII; after which, joining I, 5, which is the plan or feat of the line 
 I, V, and 7, 4, the feat of VII, IV, the ichnography of the whole 
 figure is determined on the plane of the horizon, and it will be found, 
 on infpeclion, that every line, and, confequently, every face, is planned, 
 which is eafily examined by the correfponding figures. 
 
 The figure B, is not a cube, but a right angled folid, or parallelo- 
 piped, four of whofe faces are parallelograms, as in the geometrical 
 D, i, g, h, on the line C, D, and whofe upper and lower faces are 
 fquares, as appears by the diagonal drawn from x', which is a bi- 
 fe^tion of the right angle D, and whofe fides are equal to 
 h, i; but the depth of the whole figure is only equal to h, g \ and, 
 in order to determine that depth, either the angle h, D, g, may be 
 transferred to the vanifhing line of one of the faces, as here to a, P, 
 by making a, d*, f, equal to it, (d*, c*, being the diftance of that 
 vanifhing line,) and drawing /, cutting m, P, in n; and then 
 n, /, will be a diagonal, reprefenting D, h, by means of which 
 the figure may be completed.-—Or otherwife, thus: drawing, /, o, 
 
 H 2 parallel 
 
5 2 P R A C T I C E 
 
 parallel to P, and equal to g, h j and transferring the diftance 
 P, d*, to e, on the vaniftilng line P, a j and drawing e, o, cut¬ 
 ting /, p, in q, the length /, q, is determined, by which the reft 
 may be finifhed. 
 
 There is another kind of plan which may be projetfted on the plane 
 of the horizon, by lines perpendicular to the oblique plane, on which 
 the object is fuppofed to ftand j for this the reader is referred back to 
 Fig. 46, at the cube E, where the vanifhing points ufed are S, and 
 t, on the horizontal line in the large fcheme, in which the vanifliing 
 points and are transferred to that line, by P, and P, repre- 
 fenting perpendiculars to the vanilhing line of the original ob¬ 
 lique plane. Now at the cube E, draw h, s, and h, t, then i, P, and 
 /, P, interfering them, in 5, and 6; draw 5, t, and 6, s, which com¬ 
 pletes this plan i for as it is formed by the continuation of the lides of 
 the figure which are perpendicular to the plane on which it ftands, 
 the plan of one face is (neceflarily) that of the whole cube; the point 
 h, being fuppofed to touch the horizontal plane. 
 
 As cubes, and cubical forms, are apprehended to be more ufeful 
 than any others, as approaching nearer to thofe of buildings, and moft 
 common objc6ls, they have therefore been confidered in many various 
 fituations: the plans projefted after the figures themfelves, if not al¬ 
 ways necelTary, are fometimes fo, as was before obferved, and were exhi¬ 
 bited on account of difficulties that had been fuppofed relating to fuch 
 piojerions. Thus having fufficiently treated of the cube, and there 
 being no more than five regular bodies, or folids, it might be deemed 
 an omiffion wholly to negledl the other four; wherefore here are given 
 reprefentations of them all, by the vaniftiing lines of their faces, and, 
 of feveral, by means of the ichnography, and orthography, alfo, to fhew 
 the different ways of proceeding. On the fide of Fig. 48. is a geo¬ 
 metrical defeription of the feveral angles, as well of the fedlion as fa¬ 
 ces, of a tetraedron, which muft be underftood before a perfpe6live re- 
 prefentation can be made.— G, H, I, is an equilateral triangle (whofe 
 angles are each 60 degrees) the bafe of a tetraedron.—I, N, K, is the 
 feftion fuppofed to be raifed up, perpendicularly, on the line I, K; 
 
 in 
 
riateXXl 
 
e/ P E R S P E C T I V E. 53 
 
 in which fituatlon the angle I, K, N, reprefents the inclination of 
 two of the planes or faces j the other two, viz. I, N, K, and 
 K, I, N, reprefent the angle made by one fide with a plane or face ; 
 I, K, is the diameter j and O, N, the axis. 
 
 N. B. By referring to this, occafionally, the reafons of the fol¬ 
 lowing operations will be better underftood. 
 
 Fig. 48. No. I. The tetraedron A, B, F, E, is thus proje 61 :ed. Firft 
 draw, at pleafure, the vanifhing line S, rj from S, raife a perpendi¬ 
 cular to D, the diftance given; let the fide B, A, be alfo given, 
 which continue to its vanifhing point b; draw D, b: Then make an 
 angle of 60 degrees on each fide D, b, to and c ; draw B, r, 
 and A, cutting B, r, in F; then the bafe, or one face, is 
 finifhed on the plane of a, S, c. —Now find the vanifliing line, c, h, f\ 
 of planes, inclining to the face A, B, F, in the angle of in¬ 
 clination of two of the faces; that is, in the angle I, K, N; on 
 the fide B, F; that is, on its vanifhing point c ; and in it find the 
 vanifhing points e, and (r, being already found) and then drawing 
 e, B, and f, F, the point E, is determined by their interfe^fion; 
 wherefore joining E, A, the tetraedron is completed. 
 
 N. B. The angle of inclination of two planes is always mea- 
 fured, as hath been already faid, on a plane perpendicular 
 to both of them; that is, perpendicular to their common 
 interfedlion. Now D, r, (if turned forwards with D, S, 
 till D, S, is perpendicular to the pifture) is that inter- 
 fedlion; therefore draw D, g, at right angles to D, r, and 
 from g, ere( 5 l a perpendicular (to a, S, c,) as g, h, which 
 will be the vanifhing line of planes, perpendicular to D, c, 
 the common interfedlion ; and D, g^ being the diftance of that 
 perpendicular vanifhing line, fet it off" on either fide of g^ as at 
 d ; there make the angle of inclination, g^ h', then draw r, h, 
 which will be the vanifliing line of the face B, F, E : Find 
 the diftance of this vanifhing line (/. e.) draw from S, a 
 perpendicular to r, h, cutting it in C, which is its center; 
 fet off S, D, to d, parallel to r, h, then d, C, will be the 
 
 diftance 
 
54 
 
 7lje PRACTICE 
 
 diftance of that vanifhing line, which transfer to C, D, 
 perpendicular on it; draw c, £>, and make c, D, e, and 
 e, jD, fy both 6o degrees. If any difficulty remain con¬ 
 cerning the diftance of the vanifhing line r, h, /, let 
 it be conceived that d, and D, are both brought forward, 
 fo as to be perpendicular to the pidlure over S, then they 
 will coincide, and be the place of the eye; whence it will 
 be evident, that d, C, muft be the diftance of the vanifhing 
 line r, h, f, ’ 
 
 The vanifhing line of a plane, perpendicular to another plane, is 
 determined, by finding only the vanifhing point of lines perpendicular 
 to fuch given vanifhing line, becaufe any line, perpendicular to a 
 plane, makes an angle of- 90 degrees with that plane every way: 
 whereas a line, cutting a plane in any other angle, (for inflance 30) 
 makes that angle but one way on that plane, wherefore it is neceffary, 
 in order to find a plane at 30 degrees, to take another method: and, 
 for the fame reafon, a plane palling through a line perpendicular to 
 another plane, will continue always perpendicular, though turned 
 round fuch line every way. But if a plane were to be turned round 
 a line making an angle of 30, &c. the angle would vary continually, 
 fo as to make every other angle between 30, and its complement 150, 
 (/. e, to 180, or two right angles;) for this reafon, it becomes neceffary, 
 in order to find the vanifhing line of a plane interfe 61 :ing another 
 plane at 30, (or any other angle except 90,) to find, firll, the va- 
 nifliing line of a plane, perpendicular to the interfedlion of the two 
 planes, whofe inclination is fought, on which to meafure that angle of 
 inclination, otherwife it cannot be truely found. 
 
 Fig. 48. No. 2. N. B. In this fcheme, B, C, G, is the vanifhing 
 line of the plane perpendicular to E, which is the 
 vanifhing point of the interfedlion of the planes B, E, 
 and E, C, inclined to each other in 30 degrees; B, A, C, 
 the geometrical angle of 30 degrees; G, A, (equal to 
 G, D,) being the diflance of the vanifhing line B, C, G ; 
 fo that if D, S, be raifed up perpendicularly over S, and 
 
 the 
 
X.Ml 
 
« 
 
 t 
 
 i 
 
 
c/ PERSPECTIVE. 55 
 
 the arc A, D, together with it } and alfo if the triangle 
 B, A, C, be raifed on the line B, Cj the point A, will • 
 move along the arc A, D, till A, coincide with D, 
 which is the true lituationj then will B, A, C, be a plane 
 perpendicular to D, E, the interfe6lion of the two planes 
 F, E, and C, E. 
 
 No. 7. of Fig. 45.—S, is the center of the pi6l:ure j S, D, the diftance; 
 the line A, S, D, is a vanilhing line of planes perpendicular to 
 the picture j and E, D% another vanilhing line, parallel to A, S, D; 
 but of planes inclining to the planes, whofe vanilhing line is 
 A, S, D, in the angle S, D, C.—F, D% is the horizontal line. 
 
 It is required to find the angle of inclination, of the plane of the 
 horizon, with the planes whofe vanilhing line is E, D% and the 
 difference of the angles of inclination, between that of A, S, D, 
 to E, D*, and that of F, D*, to the fame E, D*. 
 
 Set off the dillance of the pi6lure S, D, to Dh perpendicular to 
 the horizontal linej draw D*, Dh and Dh F, perpendicular to it; 
 at F, raife the perpendicular F, E, cutting D*, E, in E; then 
 E, F, will be the vanilhing line of planes, perpendicular to both 
 the planes of E, D*, and F, D*; from F, fet off F, Dh (the di- 
 ftance of the vanilhing line E, F,) to d, on the horizontal line. 
 Now draw d, E, and E, d, F, will be geometrically the angle of 
 inclination fought j (/. e.) of the plane of the horizon, with the 
 planes whofe vanilhing line is D*, E, 'which was the jirji thing 
 required. 
 
 And this angle E, d, F, is larger than S, D, C, by 5 degrees, 
 •which was the fecond thing required. 
 
 N. B. The angle of inclination of the planes of A, S, D, and 
 F, D®, is A, S, F, or D, S, d, the real geometrical 
 angle, made by their interfe6tion, on the picture; becaufe 
 they both pafs through S, and are therefore both perpen¬ 
 dicular to the pi6ture. 
 
 The reprefentation. Fig. 48, No. i, was formed entirely by vanifh- 
 ing lines j but the principles are fo general, that many other methods 
 
 may 
 
56 P R A C T I C E 
 
 may be ufed, fome of which are ftill fhorter in particular cafes j as an 
 inftance, here is added one other projection of the fame objeCt, with 
 one vanifhing line only. 
 
 Fi^. 48. No. 3. After having found the face Ay By jP, No. 2, as 
 
 before at No. i, draw B, ay and c, A, meeting in i, and F, b, 
 and A, c, meeting in 2; 'draw i, F, and B, 2, 3, whofe inter- 
 feCtion o, will be the center of the face A, B, F j ereCl a perpen¬ 
 dicular at o, and, at 3, raife the perpendicular 3, 4; fet off 3, D, 
 (the diftance of the vanilhing point 3,) of either fide, on the 
 vanilhing line dy S, c, as at e; draw e, 4, making the angle 
 3, e, 4, equal to O, I, N, and cutting the perpendicular 3, 4, 
 in 4, which will be the vanilhing point of the fide B, 5; and 
 the line 4, B, will cut the perpendicular o, 5, in the apex; from 
 whence draw to A, and to F, by which the whole is completed. 
 
 N. B. e, 4, fy reprefents the feCtion I, N, K, in the geo- 
 
 metiical. 
 
 Fig- 49. For the oClaedron, make an equilateral triangle R, F, Gj 
 draw its diameter F, L; on R, G, defcribe a fquare; draw 
 the diagonal R, H, and from H, and R, with the radius F, L, 
 defcribe two arches interfering each other in Ij then the angle 
 
 R, I, H, will be the angle made by two planes, or faces of the oClae- 
 dron on the infide ; and the angle K, I, H, will be the angle 
 on the outfide, or (properly) the angle of inclination, and to be 
 ufed in projecting this figure; for R, H, is the axis of the folid, 
 and R, I,—H, I, the diameters of two faces meeting in I. 
 
 Fig. 49. No. I. To projeCl the oCfaedron, perfpeClively, ay S, c, is 
 the given vanilhing line of the plane on which it rells; A, B, a 
 given fide of the figure, continued to its vanilhing point a ; make 
 
 S, D, equal to the diftance given ; draw the lines dy D, and D, b, 
 making with dy D, an angle of 60 degrees; and D, c, making the fame 
 angle with D, b; then draw A, c, and B, b, cutting A, c, in E, 
 which finillres the face on which the folid refts; then find the va¬ 
 nilhing line of one other face, (which will be all that is necelTary;) 
 and, in order to it, find k, the yaniftiing point of lines perpendicular 
 
 to 
 
 2 
 
^/PERSPECTIVE. 57 
 
 to the lines whofe vanifliing point is a\ (/. e.) draw D, k, perpendicular 
 to D, cutting S, c, in k; draw k, /, perpendicular to 
 S, c; find its diftance k, m, by fetting off K, D, to m; from 
 m, draw a line downwards to /, .making an angle k, m, /, equal to 
 K, I, H, (in the geometrical) with the line <?, S, c; which line /, m, if 
 continued upwards^ would make an angle with the fame line a, S, c, equal to 
 the inner angle of the inclination of two faces ; then draw /, a^ which 
 IS the vaniftiing line'fought; find its diftance C, D, (i. e.) draw 
 S, C, jD, perpendicular to /, a\ fet off the diftance S, D, to d, pa¬ 
 rallel to /, a \ then fet off the diftance d, C, from C, to D;—/, 
 thus found, is the vanUhing line of the face A, B, h, and its oppofite 
 
 g, E, i. Now find the vanifliing points, as dire6led above in the laft 
 figure; then draw e. A, and B, f cutting it in h; then h, b, and 
 
 h, c, and e, E, cutting h, c, in i; draw i, cutting h, b, in 
 g ; draw g^ A,— g^ E, and i, B, which will complete the whole. 
 
 Fig. 49. No. 2, Is a repfefentation of the fame figure ftanding on one of 
 its points, or folid angles, with very few lines. For this prcjedlion, firft 
 draw any one given fide i, 3, to its vaniftiing point a ; find b, the 
 vaniftiing point of lines perpendicular to thofe, whofe vaniftiing point 
 is and draw i, b; then draw 3, b; bife6l the angle a^ D, b, to 
 o; draw o, i, cutting 3, b, in 4; draw 4, cutting i, b, in 
 
 2, which finifties the fquare i, 2, 3, 4; at o, drop a perpendicular j 
 
 fet off the diftance o, D, to e; thence draw e, p, making with o, e, 
 
 an angle of 45 ; and, having found the center of the fquai-e 
 I, 2, 3, 4, and drawn a perpendicular through it, draw p, i, cut¬ 
 ting that perpendicular below in 6, and p, 4, cutting it above in 
 5 ; from thefe two extreme points of the axis, draw to i, *2, 3, 4, 
 which will complete the whole. For i, 5, 4, 6, reprelents a fquare 
 
 (as v.^ell as i, 2, 3, 4 ;) and o, e, p, boing half a fquare, the angles 
 
 are rightly found. 
 
 Though the oftaedron may be projected in this pofition with 
 fo few lines, yet as projections, by means of the ichnography 
 and orthography, are proper in many cafes, the manner of con- 
 ftruCling and ufing them is here explained in the fame example. 
 
 I Suppofe 
 
58 P R A C T I C E ^ 
 
 Suppofe then, No. 3, the fide i, 3, only given- draw from its 
 vanifhing point any line, where there is convenient fpace, as 
 7, 8 } and from b, the vanifhing point of lines perpendicular 
 to thofe whofe vanifliing point is Oy draw through 3, and i, of the 
 line given, cutting the line a, 7, 8, in 7, and 8; then, through a, draw 
 fy gy perpendicular to tiy S, b; fet off the diftance D, to e, 
 and draw e, fy upwards, making (with e, Cy) half the angle 
 R, I, H, (in the geometrical) and e, gy making the fame with e, <7, 
 downwards, fo that the angle f e, gy be equal to the whole inner 
 angle R, I, H. Now draw fy 8, and gy 7, meeting in 9; and 
 fy 7, and gy S, meting in 10 j which will form the perfpeflive of the 
 profile, or orthography. The plan or ichnography is fo eafy to be 
 underftood by infpe6lion, being only the reprefentation of a fquare, 
 that it needs no defcription in words. 
 
 Now perpendiculars, from the correfponding points of ichnography 
 and orthography, will meet in the feveral points which form the 
 figure; and, by joining thofe points, the figure is completed j (?. g.) 
 drawing from the point 8, to b, and raifmg perpendiculars from i, 
 and 2, of the plan, meeting i, b, in i, and 2, the points j, and 
 2, in the figure itfelf, are found; and fo of 3, 4; and by raifmg 
 a perpendicular from the center of the plan, and cutting it by lines 
 (tending to b,) from 9, and 10, of the profile, the points 5, and 6, 
 in the figure, will be found alfo, by which it is completed. 
 
 N. B. From ichnography, the perpendiculars are geometrically 
 fuch ; but from orthography, perfpe6lively fuch. 
 
 This manner Of projecting the orthography, is one of the great 
 advantages of the new principles; for, according to the old, the 
 geometrical orthography of this figure would not have been fo fimple; 
 nor, indeed, would it have been any regular figure; and in archi¬ 
 tecture (where many members are to be reprefented) the orthographic 
 projections for oblique fituations are fo confufed, as to be fcarce in¬ 
 telligible, and give no idea of the thing intended to be reprefented; for 
 proof of which the reader is referred to Pozzo's fecond Volume.—The 
 geometrical orthography is drawn above, by which it is evident -that 
 2 the 
 
PlateBaH 
 
if,', 
 
 « ■ 
 
 r 
 
 4 
 
 * s 
 \ 
 
 / 
 
 
 
 Vv 
 
 
 -©a 
 
 
 i ■■ _ 
 
 V 
 
 t 
 
 '■a 
 
 .) 
 
 s: ■ ' 
 
 1 
 
 -i 
 
 . i 
 
 
 ( 
 
 >■- 
 
 
 
 
P E R S P E C T I V E. 59 
 
 the profile, here made ufe of, is the fame kind of figure, having the 
 fame lines and angles perfpe6lively. 
 
 And, on thefe principles, the orthography may, in all poffihle fitua- 
 tions, be fuch plain, fimple, and regular draughts, as an architect 
 would make, for an elevation, with no other change than what the 
 perfpe6tive necefiarily produces, which never exhibits the figure of 
 any object, otherwife than as feen in nature, fuppofing always the 
 fame fituation of object, and fpe6tator, as required in the pi6ture. 
 
 I To fhew what expedients this general method affords, and how 
 extenfive the principles are, there is added another orthography of 
 this figure, which reprefents a perfedt fquare, with the two diagonals. 
 In fome cafes this orthography, and - in fome the other, may be 
 moft convenient, according to the diflances of the vanifhing points. 
 To projedt the figure by this orthography, let a diagonal 4, i, 
 (No. 4.) be given, whofe vanifhing point is o. Draw any where 
 from o, at a convenient diffance; for the orthography, another line 
 o, 4, I; find h, the vanifhing point of lines perpendicular to thofe, 
 whofe vanifliing point is o; draw h, 4, 4, and h, i, i, this deter¬ 
 mines the diagonal of the orthography j then find the vanifhing point 
 of lines, making 45 degrees with this diagonal; (i. e.) find the center 
 c, of p, q, the vanifhing line of the orthography, and its diftance 
 c, X)*; draw D®, o, and then make an angle of 45 above, and be¬ 
 low the line o, i)*, to q, and rj for o, is the vanifhing point of the 
 diagonal, and o, Z)*, the diftance of that vanifliing point; and from 
 thefe two points q, and r, finifh the orthography (as was done from 
 and g, in the laft figure); then draw from a, to i, and from b, 
 through 4, of the firft given diagonal, which will determine the fide 
 3, I, in its place; then draw from 3, and from i, to p, and another 
 line, a, 3, i, below for the ichnography, which complete, by drawing 
 3, b, and i,,b, and h, 3, cutting i, b, in 2, and a, 2, cutting 3, b, 
 in 4. Now through this ichnography, drawing from p, perfpedtive 
 perpendiculars, thefe will determine all the points, as the geometrical 
 perpendiculars did in the former figure. Laftly, lines from h, to the 
 orthography, and from p, through the ichnography, will cut each 
 
 I 2 other, 
 
6o 72^ PRACTICE 
 
 other, refpe£tlvely, in the correfponding points, to complete the 
 figure. 
 
 This is again reprefented, under all the fame circumftahces, with a 
 little difference in the pofition, only, at No. 5; in which h, o, b, is the 
 vanifhing line of the plane i, 2, 3, 4,—o, p, of the plane i, 5, 4, 6, 
 and p, h, of the plane 3, 2, 6, 5, which are perpendicular to each 
 other j as are the planes that form the folid angle of a cube. 
 
 Thefe are on an oblique plane (the center of the pi6lure being S,) 
 which is the reafon that the perpendiculars to the fquare i, 2, 3, 4, tend 
 to p; whereas in that, at No. 3, thofe perpendiculars were parallel to 
 each other, the figure there being on the horizontal plane; but if in 
 that the oftaedron had been feen in front, fo that the lines i, 2, and 
 3, 4, had been parallel to the horizontal line a, b, the perpendiculars 
 from the orthography would alfo have been all parallel to that line, 
 
 and to each other-The orthography of No. 3, is a fe6fion through 
 
 the axis, parallel to one fide 2, i, or 4, 3; that of the laft, No. 4, through 
 the axis, and diagonal; fo that here both plan and profile are fquares. 
 
 N. B. Though No. 4. might have been proje6led, in its 
 fituation, without either orthography, or ichnography (by 
 means of the feveral vanifhing points) yet as, in more com¬ 
 plicated obie 61 :s, this method is very expedient, it was thought 
 proper to fhew it, firft, in fo fimple a figure, that it might 
 be the more eafily comprehended. 
 
 Fig. 50. For the reprefentation of a dodecaedron, by vanifhing lines only. 
 Having the center of the pidlure S, and diffance S, D, given, draw at 
 pleafure the vanifliing line of the plane of one face; and having found 
 the center, and diflance of that vanifhing line, ere6l the diftance per¬ 
 pendicularly to that line from that center, and there find the feveral 
 
 vanifliing points of the face propofed: this is the general method.- 
 
 Here the face a, b, c, d, e, is chofen, which being fuppofed to lie on the 
 Iiorizontal plane, the vanifhing line paffes through S, and the diflance of 
 the picture S, D, is (in this cafe) the diftance of the vanifhing line; drav/ 
 at pleafure b, c, for one fide of the face propofed, which if it be not 
 parallel to the vanifliing line, continue till it cuts that line, and from 
 
 fuch 
 

 
./PERSPECTIVE. 6i 
 
 fuch Interfeftion draw to D, and thence find the vanifhing points j but 
 as b, Cy here, is parallel to the vanifliing line, draw through D, another 
 parallel, and make the feveral angles necelTary to produce the vanifhing 
 points j (i. e.) defcribe a pentagon at D, in the pofition required, and 
 divide it into triangles; continue DE, DC, DB, and DA, to the vanifh¬ 
 ing line, cutting it in i, 2, 3, and 4, which are all the vanifhing 
 points necefTary: then draw 2, by for D, C, 2, is parallel to B, A; 
 and draw 3, r, for D, B, 3, is parallel to E, C : then draw i, Cy 
 
 cutting 2, by in Cy for D, E, i, is parallel to C, A; and draw 4, by 
 
 cutting 3, Cy in Cy for DA, 4, is parallel to BE: laftly, draw i, 
 
 and 4, ay meeting in dy which finifhes the face by Cy Cy dy a. 
 
 For the next face, find the vanifhing line of planes inclined to that 
 of the face already defcribed in the fame angle as the faces of a dode- 
 caedron are to each other, (i. e.) 63 deg. 30 min. externally (the 
 angle within being 116 deg. 30^min. the complement of 63 deg. 30 min. 
 to two right angles *. Now draw D, x, perpendicular to 4, D j from x, 
 drop a perpendicular (to the vanifliing line 4, S, i, before found); 
 fet off the diflance x, D, to d; draw d, p, making with x, d, an angle 
 of 63 deg. 30 min. and cutting x, P, in P, which line will confequently 
 form an angle of 116 deg. 30 min. with the horizontal line, if continued 
 upwards; then draw 4, P, which is the vanifhing line fought; wherefore 
 find its center Ch by drawing through S, perpendicular to 4, P; find 
 its diflance Ch Dh (by taking the diflance S, D, in the compafles); 
 then fetting one foot in Ch and the other at Y, in the fame line, and 
 from Y, to S,' will be the diflance, which tranfer to Ch D ^); draw 
 4, Dh and find the feveral vanifhing points on this vanifhing line, as 
 on 4, S, I, by making the fame angles at Dh as at D, (i. e.) of 36 
 
 Fig. 50. No. 2. * The angle of inclination of two faces of the dodecaedron is found, by making a regular 
 pentagon, and drawing any diagonal, as a, a -, then bifedting that diagonal in A, and railing a per¬ 
 pendicular A, B, to the oppofite fide, bifeding that fide in B; and then with the dillance A, B, as 
 radius, defcribing the arcs a, b, and a, b, meet.ng in b, and producing a, b, to E, the external 
 angle E, b, a, is the angle of inclination fought. 
 
 For if on the real folid any two parallel diagonals are drawn on two adjoining faces, and thefe 
 diagonals are bifeded, and perpendiculars drawn from each point of bifedion to the fide of con- 
 tad, thefe perpendiculars will form the inner angle of 116 deg. 30 min. (very nea’"/) : and the 
 external angle of 63 deg. 30. min. is the angle of inclination. 
 
 degrees 
 
63 
 
 rhe PRACTICE 
 
 degrees each, and proceed as before, for the firft face; that is, taking 
 a-, d, here, for the given fide, becaufe 4, is its vanifhing point, in the 
 plane of 4, P, as well as in that of 4, S, i, the point 4, being the 
 interfe(5tion of the vanidiing lines of thofe two planes; draw 6, d, and 
 7, a-, then 5, a, cutting 6, d, in /, and 4, /, cutting 7, in 
 but 8, being too far diftant to come within the pidlure, (inftead of 
 drawing 8, and 5, g, meeting in h, as in the firft face) draw 5, g^. 
 and 6, a, cutting it in h: laftly, join /, which finifties this face. 
 
 Now draw a line through 2, 7, which will be the vanifhing line 
 of the face a, b, g, /, 5, for 2, is the vanifhing point of ay by and 
 7, of gy tty both of thefe being fides of this face, (and two points in 
 any right line being given, the line is thereby given); and having found 
 the center C, and diftance D*, of that vanifhing line, with its vanifhing 
 points 9, and 10, by the fame operation as the laft, drawg, 10, and 
 7, by cutting it in i ; draw /, 9, and gy 2, cutting it in r; laftly, join 
 Sy by which finiflies this face. 
 
 If it had been necejfary to have found this vanijhing line (for want of a 
 fecond pomt) the fame method muf have been ufed as for the othery except¬ 
 ing that the perpendicular mujl have been drawn upwardsy and the angle 
 taken on the upper fide of the vanifing linCy 4, S, i, and from thence a line 
 mufl have been drawn cutting the faid perpendiculary becaufe this vanijhing 
 line (by the filiation of its original plane) muf necejfarily make its acute 
 anglcy or angle of inclinationy above the horizontal line. 
 
 Now draw 10, 3, 5, which will be the vanifliing line of the face 
 gy hy iy ky /, foi' ^y Is thc vauifiiing point of g, hy and 10, of g, /, 
 and find the other vanifhing point ii, of that vanifhing line (for 3, 
 through which it paffes, was before found); draw ii, i then draw 
 II, g, and 5, I, cutting it in /, and/, 3, cutting ii, /, in K; laftly, 
 join /, hy which finiflies this face. 
 
 Thus having got two lines, /, ky and iy r, (of the next face 
 /, ky niy ty Sy) wlioft vaiiiftilng points are ii, and 9, draw through 
 thofe two points the vanifhing line of this face, which will alfo pafs 
 through Iy and 6, the remaining vanifliing points; draw ky i; then 
 draw ky 9, and /, i, meeting in draw 6, t, cutting ky i, in m-y 
 l^iftly, join Sy ty which finiflies this face. The 
 
c/ PERSPECTIVE. 63 
 
 The uppermoft face /, m, n, 0, being parallel to c, d, e, 
 has, of confequence, the fame vanifliing line, of which face the lines 
 
 /, and niy being already drawn, draw 4, w, and /, i, cutting 
 
 it in «, and draw 2, ni laftly, draw /, 0, parallel to by c, which finilhes 
 this face. 
 
 The remaining four faces are oppofite, and parallel to four, on the 
 ' other fide, and are therefore drawn by means of their refpeflive vanifli¬ 
 ing lines; the face e, fy r, y, is oppofite and parallel to /, ky niy ty s-, 
 the face c, y, Uy Wy is oppofite and parallel to g, by i, ky /; the face 
 r, y, Wy fly Oy Is oppofite and parallel to ^ , by j , /, g ; and the face 
 
 ty niy Uy Wy Uy Is oppofitc aod parallel to d, fy by g: the fame vanifli¬ 
 
 ing points, and the fame manner of proceeding, determines all the points, 
 and lines of thefe four, as of their oppofites, though in contrary pofi- 
 tions, and thefe compleat the dodecaedron. 
 
 There is one other dodecaedron on the fide, which is proje 61 ;ed by the 
 fame vaniflbing points. 
 
 Fig. 51. No. I. In order to reprefent an icofaedron, the fifth and laft of the 
 regular folids, which is compofed of 20 equilateral triangles, let one 
 fide 3, 4, be given, with its vanifliing line by c, and diflance S, D. 
 Continue the given fide to its vanifliing point a j draw Uy D, and find 
 the vanifliing points b and c, by making Uy D, by and by D, Cy angles 
 of 60 degreesj draw 3, by and 4, r, cutting 3, by in 2, which deter¬ 
 mines one face 4, 3, 2 j then through find the vanifliing line of 
 planes, inclining to that of Uy b'y Cy in the fame angle as the faces of an 
 icofaedron to each other (viz. 42 degrees) being the acute angle v/ithout, 
 (the complement of 138, the obtufe angle within *,) by drawing D, y, 
 perpendicular to D, and y, />, to by c, which will be the vanifli- 
 
 Fig. 51. No. 2. • The angle of indination of two faces of the icofaedron is found, by making a regular pen¬ 
 tagon, and on one fide deferibing an equilateral triangle, and having drawn a diagonal of the pentagon; 
 a, a, and the diameter of the triangle, as A, Bj then taking A, B, for radius, and deferibing 
 the arcs a, b, and a, b, meeting in b, and producing a, b, to E, the angle E, b, a, is the 
 angle of inclination fought. 
 
 For, on the real folid, five equilateral triangles form a pyramid, whofe bafe is a pentagon 
 therefore a diagonal of that pentagon will be the bafe of a triangle, whofe legs (being the diameters 
 of two of thefe equilateral triangles) will form the internal angle of 138 deg. (nearly) ; and the 
 external angle of 42 deg. is the angle of inclination. 
 
The PRACTICE 
 
 ing line of planes perpendicular to the lines, whofe vanifhing point is a\ 
 then fetting off the diftance of that vanifhing line (which is j-, D,) to 
 r, and from r, drawing r, making with r, an angle of 42, and 
 then drawing a, p, that will be the vaniHiing line fought. To find the 
 center C, and diftance C, D*, of this vanifhing line, draw ay Z)*, and 
 find the other vanifhing points of an equilateral triangle, as was done 
 on the firft vanifhing line, b, c, viz. here, n, and o; now draw 
 f?, 4, and 0, 3, cutting n, 4, in 5, which determines the face 3, 4, 5. 
 V Then find, in the fame manner, the vanifhing line c, ky /, for the 
 face 4, 2, II, with this difference, that, as the former «, 0, was 
 taken below the firft vanifhing line Cy by c, becaufe the face 3, 4, 5, 
 comes forward, with refpe6l to 3,4, 2, this falls backward, with re- 
 fpe6l to the fame face, and muft therefore be drawn above Cy by c; and 
 having found the vanifhing points, as in the others, draw 2, /, and 
 4, by cutting 2, /, in ii, which finifhes this face. 
 
 Then through by the vanifhing point of the fide 3, 2, find another 
 vanifhing line e, by fy for the face 3, 2, 16, and in it the points c, 
 and fy as in the former vanifhing lines, and draw c, 2, and 3, fy 
 cutting it in 10, which finifhes the face 3, 2, 10, as the line D, a, (per¬ 
 pendicular to by D,) goes beyond the pidture, take S, d, a fourth of S, Z), 
 and draw a parallel to D, a, and proceed as if this 4th was the whole di¬ 
 ftance, till you find the vanifhing line j then from b draw a parallel to the 
 vanifhing line, which parallel will be that fought. The perpendicular 
 is taken downwards, viz. Uy f. 
 
 Through €y the vanifhing point of 2, 10 ; find another vanifhing line j 
 but as €y by fy (111 whlch is the point c,) does not pafs through S, 
 draw Cy S, and proceed on it as before, on the vanifhing line ay by c, 
 which will produce the vanifhing line c, g, hy for the face 2, 10, 12, 
 %hich is determined, by drawing 10, g, and 2, hy cutting 10, g, in 
 12; and now join 12, ii, which finifhes another face, 2, 12, ii. 
 
 Then through fy the vanifhing point of the fide 3, 10, find one more 
 vanifhing line, yj /, for the face 3, 10, i, by the fame procefs, as 
 the laft; and, having found the vanifhing points, draw m, 3, ^nd /, jo, 
 cutting hiiy 3, in i, which finifties that face. 
 
 Then 
 
Plate XXVI 
 
 O 
 
( 9 / P E R S P E C T I V E. 6? 
 
 Then draw from /, through 11, for the fide 11, 6, is parallel to t, lo ^ 
 
 draw 5, g, cutting ii, 6, in 6; then 6, and 5, /^, cutting <?, 6, in 7, 
 which finillies the face 5, 6, 7, oppohte, and parallel to 2, 10, 12 j 
 join 6, 4, this determines the face 4, 5, 6, (which, here, happens to 
 fall in one line) and alfo the face 4, 6, 11. 
 
 Draw 6, and 7, cutting 6, fy in 8, which finilhes the face 
 6, 7, 8, oppolite and parallel to 2, 3, 10. 
 
 Draw Oy 8, and 7, Cy cutting a, 8, in 9, which determines the 
 
 upper face oppofite and parallel to 2, 3, 4; join 9, 10,—9, i,—. 
 
 9, 12,—8, 11, and 8, 12,—5, i, and 7, i, which finilhes the for¬ 
 warded face, I, 5, 7, and completes the whole figure. 
 
 2, 3, 4, and 7, 8, 9, being oppofite and parallel, their vanifhing 
 
 line is the fame, viz. by c. 
 
 3, 4, 5, and 8, 9, 12, being oppofite and parallel, their common 
 
 vanifhing line is ay rty 0. 
 
 2,4, II, and I, 7, 9, being oppofite and parallel, their common 
 vanifhing line is Cy k, i, 
 
 2,3, 10, and 6,7, 8, being oppofite and parallel, their common 
 vanifhing line is r, by f, 
 
 2, ic, 12, and 5, 6, 7, being oppofite and parallel, their common 
 
 vanifhing line is Cy g, h. 
 
 3, i, 10, and 6, 8, ii, being oppofite and parallel, their common 
 
 vanifhing line is fy /, m. 
 
 Fig. I. Below is a reprefentation of the fame folid, proje 61 ;ed without any 
 other vanifhing line than that of the horizon, and alfo without ichno- 
 graphy, or orthography, any farther than the geometrical draught of a 
 decagon, or double pentagon A, and the axis of the folid marked 
 Oy I, 2, 3, which axis is divided by taking 0, i, equal to a fide of the 
 decagon, i, 2, equal to radius (or a fide of the hexagon), and 2, 3, 
 equal to Oy i, in which divifion the middle part i, 2, with either end, 
 will be in extreme and mean proportion. Prop. 9. Book 13. Euclid. 
 
 N .B. A line is faid to be cut in extreme and'mean proportion, 
 when the whole is to the greater fegment, as the greater feg- 
 ment is to lefs, Def. 3. Book 6. Euclid. 
 
 K 
 
 Firlh 
 
66 7 ^ 5 ^ PRACTICE 
 
 Firft draw the axis perpendicular to the horizon, and mark the divi- 
 fions; then through i, and 2, draw tranfverfe lines, (i, e.) parallel to the 
 horizon, and on them mark the length c, on one fide, and r, 
 on the other, but contrariwife; then fet off the diftance on the hori¬ 
 zontal line, from S, the center of it, to J, and from d, draw through 
 each extremity of the tranfverfe lines, cutting the axis, and the points 
 in which that is cut will determine the depth of each pentagon; the 
 pricked line pafling through 2, is fo divided. The fide 2, towards dy 
 being equal to by r, in the plan, and the fide 2, a, equal to r, and 
 the line drawn through i, contrariwife, (i. e.) r, towards dy and 
 Cy by on the oppofite fide; draw dy ay cutting the axis in g, and dy by 
 cutting it in after which, find the vanifhing points of the fides of the 
 pentagon, 4, 5, 6, 7, by drawing from Z), (below) Z), 4,—Z), 5, &c. 
 making 4, Z), 5, an angle of 36 degrees, and the fame angle with 
 5, Dy 6, and 6, Z), 7, which are all the vanifliing points necefiary: 
 then, parallel to the horizontal line, draw /?, ?, through ky the point 
 in v/hich d, by interfedts the axis, and draw 6, gy and 5, gy cutting 
 by iy in hy and / ; then by 5, and /, 6: laftly, 4,^, and 7,^, cutting 
 iy 6, in by and by 5, in which finifhes the lower pentagon. The 
 upper pentagon is determined in the fame manner, and by the fame 
 points, with this only difference, that the operation is reverfed, in order 
 to produce a contrary pofition. After which the whole icofaedron is 
 completed, by joining the refpedlive points of the two pentagons, 
 and drawing from the fame points to the two poles of the axis. All 
 which may be performed with lefs trouble, and in lefs time, than is 
 requifite to form the geometrical ichnography, and orthography. 
 Here is added another on the fide, projedled by the fame points, only 
 infiead of drawing d, by cutting the axis (as in the former) a pricked 
 line is drawn through the axis from S, which being cut, from dy (the 
 diftance) through by finds ky the middle of by /, and from dy to a, finds 
 gy and fo for the upper pentagon. 
 
 Fig. K. In order to reprefent this figure, by the fame method, on an 
 inclined plane ; Draw the vanifhing line of fuch plane C, E, and from 
 S, draw S, D, parallel to it; join C, D, and make D, P, perpendicular 
 
 to 
 

 JO 
 
 Watc xmi 
 
 4 
 
riF 
 
^/PERSPECTIVE. 67 
 
 to It: draw at pleafure P, G, B; draw B, /, parallel to D, P, and 
 equal to 0, 3, the axis at figure A, which divide in i, and 2, according 
 to the geometrical proportion j then from D, draw to thdfe divifions, 
 cutting B, G} thefe interfedlions determine the axis according to the 
 perfpedtive- proportion. Draw from C, lines through the two interme¬ 
 diate points of B, G; and, in order to find their refpeftive lengths, fet 
 them off geometrically on C, D, from D, to and to a j that is, make 
 D, equal to c, b, and D, a, equal to c, a \ and on D, P, fet off 
 
 D, hy equal to 0, i j then drawing a, and D, a, parallel to it, 
 
 a, will be a vanifhing point (in the vanifhing line P, S, C,) from 
 which the axis, and tranfverfe line, will be divided in the proportion, 
 
 of D, h, to D, a-, w'herefore, by drawing a, G, and a, B, the two 
 
 tranfverfe lines will be cut, the lower backwards,‘the upper forwards \ 
 in that proportion. D, g, is equal to Z), h fo that drawing g, b; 
 and D, parallel to it, m.^ will be the vanifliing point dividing the 
 fame axis, and tranfverfe lines, in the proportion of D, g, to D, ; 
 wherefore draw G, and B, which will cut the tranfverfe 
 lines on the oppofite fides in this laff proportion. 
 
 This is the preparatory work for the figure K, which is pro]e6led 
 upon it, as on a fkeleton, all the lines correfponding, as appears on 
 infpecffion j and all the reft of the operation is the fame as at figure 1. 
 
 Fig. L. This method is an univerfal one; (i. e.) the folid in any fitua- 
 tion may be projedfed by it, and is perhaps the fhorteft of all.—^For 
 let any face be given, 2, 3, 4, and through any angle of that face, 
 find the axis of the folid; divide that axis, perfpedtively, in the two 
 points, ferving for centers of the two pentagons ; complete thofe 
 pentagons in the manner before taught, which, with the two poles 
 of the axis, are all the points; and thefe being joined, the whole 
 figure is formed. 
 
 It will be proper to draw (fomewhere apart) the geometrical pro¬ 
 portion of the axis, with its divifions, and the angles made by the 
 planes ; (e. g.) draw g, for the axis; divide it in y, and 2:, the 
 centers of the two pentagons; and through draw, at right angles, 
 Zi make z, k, equal to the fliorter fide of the diameter of a 
 
 K 2 pentagon ; 
 
68 
 
 rhe PRACTICE 
 
 pentagon j draw g j then 2:, ky gy will be the angle made by the 
 interfe6lion of the plane of the lower pentagon, with a face of the 
 folid, terminating at the lower pole gy and confequently Zy gy ky will 
 be the angle made by the axis with that face, gy k. 
 
 Having given the face 2, 3, 4, with -its vanilhing line, diftance, 
 and points, find the vanifhing line hy />, of planes perpendicular to 
 Cy the vanifhing point of 4, 2, (one fide of that face) j and at x, the 
 diftance of this vanifhing line hy py draw x, py making the angle 
 Cy Xy py cqual to Zy ky g-y di'aw Cy py vdilch will be the vaniflring 
 line of the planes of the lower and upper pentagon (2, 4, 5, i, 10, 
 being the lower) of which one fide is 4, 2-, and by means of that 
 fide, with the vanifhing line c, py that pentagon is finifhed. Now 
 find hy the vanifliing point of lines perpendicular to the vanilhing 
 line Cy py and draw 3, h, which will be the indefinite axis of the 
 ieofaedronj bifecl the angle Cy d, f, to /> '^j draw py i, cutting 4, 2, 
 in ly and I, i, will be the diameter of this pentagon, whofe center is 
 determined by the interfedlion of its diameter with the axis, and from 
 that center to 3, will be the perfpedlive reprefentation of z, g ^ 
 wherefore from 3, draw a line in any convenient direiSlion, fo as 
 not to interfere with the figure, as 3, o, and parallel to it, hy d, 
 equal to hy D^, the dillance of h j then draw from dy through the 
 center, cutting 3, 0, in k-y this interfedlion will mark the place of 
 the center of this lower pentagon, geometrically on 3, Oy by which 
 meafure the reft of the geometrical axis is divided, as appears above, 
 at y, g, by means of parallels from Zy and g. Now draw from the 
 divifions of 3, o, to dy cutting the axis in the upper center and polej 
 from py draw through the upper center, and make that part of the 
 upper diameter, from the center to 11, equal to the lower, from 
 
 * N. B. I'his bifedtion is made to find the vanifhing point p, from which a line (drawn to i,) 
 will divide 4, 2, in half, and fo become the diameter of the lower pentagon; for e, is the 
 vanifhing point of 4, i, which is parallel to 2, 10 ; and f, of 2, i, which is parallel to 
 4, 5. The fame point p, might alfo have been found, by drawing d, p, perpendicular to d, t, 
 for c is the vaniflung point of the line 4, 2. Or as p, is in the plane of the vanifhing line 
 (, €, and alfo in that of h, p, it muft be in their interfedion, and therefore is found, as here, 
 in the interfedtion of thefe two vanilhing lines. 
 
 the 
 
c/ P E R S P E C T I V E. 69 
 
 the center to i, by drawing from i, through the upper center to p, (the 
 vanifhing line of the plane palling through the axis,) cutting it in i ; 
 then drawing from /, to the lower center, cutting the upper diameter inn, 
 which will be the angle of the upper pentagon over the middle of the fide 
 4, 2, of the lower pentagon ; and, in the fame manner, find the other 
 part of the upper diameter j that is, from the upper center, through 
 /, the middle of 4, 2, draw to the fame vanifhing line hy y>, and 
 from the point of interfeftion draw through the lower center, 
 cutting the upper diameter in r, which will be the middle of 7, 9, 
 the line of the upper pentagon, over the point i, of the lower 
 pentagon; draw from the vanifhing point c, through r, which will 
 produce the line 9, 7, indefinitely3 draw from through ii, cutting 
 that line in 9, and from fy through the fame point ii, cutting it 
 in. 7, which determines the length of 7, 93 draw g, ii, and f, 7, 
 which finds 63 then fy 9, and c, 6, cutting it in 123 draw ii, 12, 
 which finifhes this upper pentagon. Now draw from 6, 7, 9, ii, 
 and 12, to 8, the upper pole 3 then join the correfponding points 
 of the two pentagons, which completes the icofaedron. 
 
 * N. B. The line here direfted to be drawn, is fo nearly parallel to h, p, that the point of 
 interfeftion falls at too great a diftance to be conveniently ufed; yet being the fame method by 
 which the laft point ii, (of the fame diameter) was found, it was proper to diredl it, as beft, 
 when the points fall within reach. 
 
 But an expedient may be ufed to find r, the other extremity of this upper diameter. From /, 
 (in the line h, p,) draw one line through the center of that diameter, indefinitely, and another 
 through II, (the extremity already found); and, at any convenient diftance, draw a line through 
 them both, parallel to h, p, as L, ‘iv, I, at, cutting thefe lines in I, and w: now as /, 11, necefiarily 
 pafles through the center of the lower pentagon, and i, i, through one extremity of its diameter; 
 draw alfo /, /, through the other extremity, cutting the fame line L, w, I, .t, in L ; by this 
 means the proportion of the two parts of the diameter is found geometrically; therefore, in the 
 line L, w, I, x, make I, x, equal to L, w, and draw ;, x, which will cut the upper dia.- 
 meter in the point r.-^This is thus particularly explained for its general ufe. 
 
 The 
 
rhe PRACTICE 
 
 The FOURTH PART. 
 
 Fig. 52.TN this part it is propofed to exhibit feveral expedients to fa- 
 No. cilitate the pra6licej and firft to divide a perfpe6tive line in 
 any proportion. 
 
 Let it be required to divide A, B, whofe vanijQiing point is V, into 
 four equal parts. Draw V, d, in any direftion, and of any length, 
 and Aj yi parallel to it 5 draw d, B, cutting A, in f-, divide 
 A, yi into four equal parts, at c, y and e ; draw d, r,—d, 
 and d, which will cut A, B, in C, D, and E, the points 
 required. 
 
 If room be wanting, any nearer diftance will anfwer the fame 
 purpofe, as D: in this cafe, draw D, B, cutting A, y, in 4; and 
 divide A, 4, in the fame number of equal parts, at i, 2, and 3; 
 and draw D, i,-—D, 2, and D, 3, which will find the fame 
 points. 
 
 Again, at No. 2, V, D, is drawn in another diredlion; for it 
 may be in any, at pleafure, provided A, y be drawn parallel to 
 it j and here it is required to divide A, B, into tv/o equal parts onlyj 
 therefore bifedt A, y in r, and draw D, cutting A, B, in E, 
 the point fought. Or, if it be more convenient, draw V, d, and 
 A, f, parallel to itj bifedf A, f, in 5 , and draw d, which 
 will cut A, B, in the fame point Ej for the truth of the operation 
 depends on the parallelifm of the two lines, V, d, and A, f. And 
 in all thefe cafes, the lines from D, or d, to the feveral divifions 
 of A, y, or A, f, reprefent parallels, therefore the lines A, B, 
 inuft be divided as the originals. 
 
 If V, d, No. I, be the true diflance of the vaniffiing point V, and 
 A, the interfedlion of the pidfure, by the original line; then A, r, 
 y r, y, are the true originals, or geometrical proportions; and there¬ 
 fore when it is required to find the original proportions of a per- 
 ipedlive line, already divided, the true diftance and interfedfion muft 
 
 be 
 
 2 
 
0/ PERSPECTIVE. 71 
 
 be taken.—But in order to find the perfpe6i:ive divifions on a line 
 already projected, the above operations are equally true, whatever 
 diftance be taken j and although A, be not the interfe6tion j for 
 A, B, any part of a perfpe6tive line^ will be truly divided by this 
 1 method. 
 
 At No. 3. it is required to make a fegment on the perfpe6tive 
 
 line A, V, from the point C, towards V, equal to A, B, on the 
 
 ' fame line. Draw D, C, till it cuts A, in c; make c, equal 
 to A, and draw e, D, cutting A, V, in E ) then will C, E, 
 
 I be, perfpeflively, equal to A, B. 
 
 >i ^ Or, if it be required to make the perfpeftive of a part equal to 
 l>, A, (on the original line;) at the diftance of c, from divide the 
 original line A, e, in the manner required, and draw c, D, and e, D, 
 
 ' which will determine the part C, E; and fo of any other pro¬ 
 portions. 
 
 Fig. 53. No. I. Here is an original plan, in its geometrical proportion, 
 
 I placed obliquely below the ground line; it is required to project it in 
 perfpedlive. Continue the feveral divifions to that line, and having 
 found the two vanifhing points a, and draw from thofe points to 
 the feveral inteife6lions, which will form the perfpe6tive reprefenta- 
 tion; but as it often happens, that on one fide there may not be 
 I room for many interfeftions, becaufe they run much wider than on 
 the other hde, after having drawn one only, as to r, and drawn from 
 thence to which finds the point i, take any other point between 
 and S, as B, and draw from thence through i, cutting the 
 ground Fine in C, and make ufe of the diftance o, C, (inftead of 
 0, c,) fetting that off, from C, to f, and from to g, &c. as 
 often as neceffary; and drawing from to B, and from to B, 
 
 ' &c. the fame points 2, 3, 4, &c. will be found, as if there had 
 
 been fpace to repeat the diftance c, 0, as many times; then draw 
 i I, b, —2, b, —3, b, —&c. which will complete the work. 
 
 Fig. 53. No. 2. The fame thing is done without a geometrical 
 plan: and here the meafures of the original fquares are fet off equally 
 on each fide of <?, as i, 2, 3, 4, and the diftances D, and a, D, 
 
 brought 
 
72 P R A C T I C E 
 
 brought down to the vanifliing line at and d j and, after having 
 drawn the two extreme lines <7, 0^ and 0, then drawing, refpedtively, 
 3 -, I, — S', 2,— S, 3,— S, 4, and d, i,— d, 2,— d^ 3,— d, 4, the points 
 of the extreme lines b, 0, and a, 0, are found j from which points, 
 
 on one fide, drawing to and, on the other, to b, the plan is com¬ 
 
 pleted. And if room were wanting for S, D, above, the points 
 h, a, and S, d, might be found by means of d, below, making 
 S, d, equal to the diftance S, D. 
 
 Fig. 54. If it be required to find a fegment of F, from F, towards 
 b, equal to A, B, of the line A, bring down D, the diftance 
 of the vanifhing point a, to d; draw d, B, cutting the ground line 
 in e j bring down alfo b, D, the diftance of the vanifhing point 
 
 to d ; draw d, F, cutting the ground line in f-, make g, equal 
 
 to A, e, and draw d, g, cutting F, b, in E; then will E, F, repre- 
 fent an original line, equal to the original of A, B, and therefore 
 will be equal, perfpedtively, to A, B, which is all that is neceffary to find 
 the length F, E, perfpedtively, equal to A, B. But as it is indif¬ 
 ferent whether the ground line A, g, be ufed for this purpofe, or 
 any other parallel, and as fome other is often more convenient, take 
 any line parallel to the vanifhing line, as B, 6, and draw d. A, 
 cutting it in 4, and d, F, cutting it in 6; then make 6, 5, equal 
 
 to 4, B, and draw d, 5, which will cut F, b, in E. Or take 
 
 any other parallel, as i, Fj make F, 3, equal to i, 2, and draw 
 
 d, 3, which finds the fame point E ; for d, A, e, and d^ g^ 
 
 are equal triangles, being between the fame parallels, and having 
 equal bafes; [Euclid, Book 3. Prop. 38 ] and B, 6, or i, F, being 
 parallel to the fame vanifiiing line a, d, will cut off equal triangles 
 from e, d. A, and d, f‘, therefore, &c. 
 
 Fig. 55. Another way of dividing perfpedlive lines, is by mark¬ 
 ing the geometrical proportions on the rays, from D, as on D, a ,— 
 D, b^ the rays of the vanifhing points and b, of A, B, and 
 A, G, F. Make D, i, and D, 2, equal, and draw D, 6, paral¬ 
 lel to I, 2 j then a line from 6, cutting A, b, and A, a, in any 
 
 parts, will divide thofe two lines equally, as 6, G, B, makes 
 
 A, G, and A, B, perfpedlively, equal. If 
 
Plate XX\^. 
 
PlateXXIX 
 
 id 
 
o/ P E R S P E C T r V E, 73 
 
 ]f any other proportion be required, mark it on the ray of the 
 line ; for inftance, D, 3, is equal to D, i, and r. Draw 
 
 D, 5, parallel to 1,3; and draw 5, B, cutting A, by in F; 
 then A, F, is in the fame proportion to A, B, as D, 3, is to 
 D, I. 
 
 A, Cy is another perfpe6live line j D, c, its ray j and Cy its 
 
 vanifliing pointj and i, 4, divides D, dy and D, r, unequally; 
 and D, 6, being parallel to i, 4, the line 6, E, B, divides' A, c, 
 and A, a, in E, and B, perfpe6lively, in the fame proportion { 
 A, E, being to A, B, perfpedtively, as D, 4, to D, i, geome¬ 
 
 trically. 
 
 Fig, 55. No. 2. This method may alfo be ufed in cafes like that of 
 54. Let E, F, be a perfpedlive line, 'vLofe vanifhing point is 
 
 dy it is required to make A, B, (of another line) equal to E, F. 
 Draw from A, to dy the vanifliing point of E, F; and draw A, E, 
 cutting the vanifhing line in e j draw <?, F, cutting A, a, in C; 
 then A, E, F, C, will reprefent a parallelogram, whofe oppofite 
 fides being parallel and equal. A, C, mufl be equal to E, F. Now 
 .make D, i, and D, 2, equal; and draw D, /, parallel to i, 2; 
 
 draw fy C, cutting A, by in B; then will A, B, be equal to 
 
 A, C, (as by the lafl Figure;) but A, C, is equal to E, F ; 
 therefore A, B, is alfo equal to E, F ; which was to be done. 
 
 Fig. 55. No. 3. B, A, is given, it is required on E, A, to make 
 
 A, C, equal to B, A. Draw 4, i, making D, 4 and D, i, equal; 
 
 draw D, 5, parallel to 4, i ; then draw 5, B, cutting E, A, in C, 
 
 and C, A, will be equal to B, A; which was to be done. 
 
 But if 5, goes beyond the pidture, divide D, 4, in half, at 2; 
 and draw D, 3, parallel to 2, i ; then (having divided A, B, 
 perfpeclively, in half at 6,) draw 3, 6, which will cut E, A, in 
 the fame point C. 
 
 N. B. The manner of dividing a perfpedlive line In half, is 
 defcribed at Fig. 52, No. 2. 
 
 Fig. 55. No. 4. Let the fame things be given as before; but the di- 
 flance D, being beyond the pidure, take a fourth (or any) part of 
 
 L the 
 
74 755^ P R A C T’ I C E 
 
 the diftance at D ; then take a fourth of S, V, at E, and draw 
 Z), E, which will be parallel to V, D, which tends to the true di- 
 ftarice; therefore the angle at Z), will be the fame as at D.—Draw 
 c, d, making Z), c, and D, d, in any proportion required, and 
 Z), parallel to c, d fet off S, e, four times to F, and draw 
 
 F, B, cutting S, A, in C : then will A, C, be to A, B, as Z), d^ 
 is to D, Cy (/. e.) in the proportion required. 
 
 N. B. Any line drawn from F, will cut the fame lines in the 
 fame proportions, as F, If; for A, 1 ?^ is to A, f\ as 
 A, B, is to A, Cj (/. e,) as D, r, is to D, d. 
 
 Fig. 56. As this is the place allotted for expedients, the reader is referred 
 back to Fig. 45, where the manner of finding, on the horizontal plane, 
 the plans, or ichnographies, of cubes projected on oblique planes, is de- 
 fcribed. The letters, conftantly ufed, fhew the fituation in general j but 
 here is a difficulty peculiar to this pofition, which is, that the fide a, by is 
 exactly in front, and, for that reafon, it is impoffible to find the feat, 
 or ichnography of the point a, without fome expedient j therefore 
 draw a, ay and b, b, both parallel to the horizontal linej draw at 
 pleafure from S, (which is the vanifhing point P, brought up to the 
 horizontal line), cutting b, by m b-y draw P, by cutting a, in 
 a ; draw C, perpendicular to the horizontal line, cutting S, by 
 in Cj draw C, A, parallel to by b; then A, will be the point 
 fought, viz. the feat of a, on the horizontal plane, by which the 
 feat, or plan of the upper fquare is found on that plane. 
 
 57* I* Fig. 31, are feveral lines tending to a point beyond 
 the limits of the pidlure, and there, as well as elfewhere, the reader 
 is referred to this place for expedients in fuch cafes; which frequently 
 happen, efpecially when the diftance of the picture is confiderable. 
 The prefent figure fliews the manner of drawing lines to an inacc.effible 
 point. Suppofe A, B, and C, D, two lines, tending to fuch point, 
 
 beyond the pidlure; it is required to draw from g, a line, tending 
 
 to the fame point. Draw N, O, through gy in any diredlion, and 
 B, D, parallel to it, at any convenient diftance, within the pifture j 
 
 at D, with the compafTes, fet off the length of N, O, as D, E; 
 
 fo 
 
rJalcXXX. 
 
 \ 
 
c/ PERSPECTIVE. 75 
 
 fo that E, interfe6ls the line A, B j divide D, E, in F, making 
 D, F, equal to O, g; draw F,parallel to A, B, cutting B, D, 
 in f ; then draw g, f, which will tend to the fame point with A, B, 
 and C, D3 for B, D, is divided in in the fame proportion as 
 D, E, is divided in F, (i. e.) as N, O, in (Prop. 2. Book 6. Eucl.) 
 Let there be another point, from which it is alfo required to draw 
 to the fame inacceffible point, whether between the lines A, B, and 
 
 C, D, or without, as here at h 3 which, if it be fo fituated, as that 
 
 a line N, O, may conveniently be drawn to it through gi then 
 draw N, g, O, h, and take the diftance O, and add it to 
 the line E, D, from D, to 1 5 and draw I, /, parallel to f, F, 
 cutting B, D, in /, and drawing /, it will tend to the 
 
 fame point 3 for the triangle D, I, /, is fimilar to D, F, y'3 and 
 therefore D, i, is to D, I, as D, /*, to D, F 3 and, confequently, 
 
 D, /, to O, h, as D, /, to O, g. 
 
 Let it alfo be required to draw from to the fame point. Draw 
 
 I, 2, through k, parallel to B, D 3 fet off i, 2, from/i to 33 
 
 divide yi 3, in 4, as i, 2, in ^3 draw 4, 5, parallel to A, B3 
 
 and then drawing k, 5, it -will tend to the fame point.—Here g, J\ 
 is ufed inftead of C, D, as more convenient 3 for any two lines 
 tending to the fame point, will anfwer the purpofe. 
 
 Fig. 57. No. 2. Another method is propofed: let A, B, and C, D, be 
 two lines tending to a point, beyond the pidlure 3 now, in order to 
 draw from c, to the fame point, draw any line through e, cutting 
 thefe two lines in A, and C3 and at any convenient diflance B, D, 
 parallel to A, C, draw A, D, and (parallel to D, C,) draw B, 
 cutting A, D, in b 3 then draw parallel to B, D, and con¬ 
 fequently equal to it 3 draw c, D, cutting in F, which di¬ 
 vides b^ d, in the fame proportion as A, C, is divided by e-, 
 
 wherefore fet off d^ F, to D, f, or (which is the fame thing) draw 
 F, /, parallel to d^ D, and drawing c, f, it will tend to the point 
 required. 
 
 By the fame procefs, p, is found, tending to the fame point, 
 -viz. draw m, f/, parallel to A, C 3 draw 72, D, cutting B, by m s-, 
 
 L 2 draw 
 
The PRACTICE 
 
 76 
 
 draw r, t, parallel to B, Dj draw D, cutting it in t then 
 let off 5, t, from B, to / j and drawing m, p, it will tend to the 
 fame point.—The only requifite is to find the fame proportions on 
 the line B, D, as on A, C, or on B, p, as on 7/, 7 fi. 
 
 Fig. 57. No. 3. Now, on thefe principles, fuppofe S, /, the horizontal 
 line, (or the vanifhing line of any other plane,) S, D, the diftance, 
 D, bt a parallel to an original line, tending to a vanifhing point 
 beyond the pidlure j it is required to draw from to the fame 
 vanifliing point.—-Draw and e, both parallel to S, D; 
 
 continue b^ D, till it cuts J, in ^j draw b^ parallel to S, /"j 
 
 draw dy g, cutting S, fy in /; draw /, cutting D, S, (conti¬ 
 nued) in h i fet off g, hy from by to f, (or S, hy from fy to e,) 
 or draw hy e, parallel to S, then drawing e, it will tend to 
 
 the vanifhing point required.—It is evident, that by fy might have 
 been fet off from S, to gy inflead of drawing the parallel by g 
 or that a parallel to S, fy from h, would have determined as 
 well as fetting off S, hy at f e, 6cc. 
 
 F'g. 57. No. 4. Let ay by and C, dy be two lines, tending to a va¬ 
 nifhing point i if there are any number to be drawn to the far!:ie 
 point in the fame line, as 1, 2, and 3, draw a, c, parallel to 
 ay Cy at any convenient diftance j then draw from C, i, 2, and 3, 
 parallels to a, cutting a, c ; then from c, and the other inter- 
 
 feftions of a, draw to O, any point in ay by and through 
 
 7, the interfedlion of r, O, with C, dy draw 7, 8, another pa¬ 
 rallel to a, c then from i, 2, and 3, draw through the refpedlive 
 interfedlions of the line 7, 8, viz. i, 4,-2, 5, and 3, 6, which 
 ' will be the lines required, all tending to the fame vanifhing point. 
 
 Fig. 58. No. I. Suppofe A, B, a perfpedlive line, to be divided in 
 any proportion, but its vanifliing point to be beyond the picture 3 
 continue D, S, to A, B, cutting it in e -y draw at pleafure Cy i ; 
 fet off B, O', from r, to h ■, draw S, hy and D, /, parallel to it j 
 fet off h, iy from g, to /, upwards; draw D, /, and A, M, parallel 
 to it. (By this means, e, /, will be divided proportionally to D, ey 
 
 (l. e.) Cy hy to Cy Sy US hy ty tO S, Dj aUd gy ly Will be to gy B,. 
 
 as 
 
Plate XXXI . 
 
■if 
 
«/ P E R S P E C T I V E. 77 
 
 fis S, D, to S, p, which is the proportion wanted.) Now A, M, 
 
 will be the original of A, B j therefore, on it, make the geometrical 
 divilions required; draw from thefe divifions to D, which will cut 
 A, B, perfpe^fively. 
 
 N. B. If D, /, had been already drawn (as well as A, B,) 
 then only draw A, M, parallel to it, and proceed as above. 
 
 Fig. 58. No. 2. After e, /, is divided, if room be deficient below 
 for A, M, draw «, p, (from p, the extremity of the pidlure) pa¬ 
 rallel to D, /, and bring down to q-, (i, e.) make p, y, equal to 
 
 n then draw from D, parallel to n, y, which finds r, the 
 
 diftance of the vanifhing point of A, B; then make the geometrical 
 divifion on A, c, as ^7, r, from which points draw to r, and the 
 lines r, r, and p, r, will cut A, B, in the fame points. 
 
 Fig. 59. No. I. Let E, F, be given, and the direction of E, G, drawn 
 at pleafure, in order to form either a cube, or any other cubical 
 figure, of known meafures, (the horizontal line, with its diftance, being 
 always fuppofed to be given, or known). Continue the fide E, G, 
 till it meets the horizontal line in which will be its vanifiiing 
 point; draw a, D, and D, perpendicular to it, cutting the hori¬ 
 zontal line in b j draw E, b ^ then E, parallel to D, a, and 
 E, h, parallel to D, and make thefe two lines of the geometrical 
 lengths required (as here they are both equal to E, F): draw g, D, 
 cutting E, in G, and i?, D, cutting E, b, in FI; by which 
 operation thofe lengths are determined : draw G, b^ and H, a, 
 which finifii the lower fquare or plan: raife perpendiculars at G, 
 and H, and the remaining angle; and draw F, a,- —^F, b^ cutting 
 the perpendiculars at G, and H, in I, and K r laftly, draw I, b^ 
 and K, which completes the cube. 
 
 Fig. 59. No. 2. If room be wanting, below, for the lines E, g^ and E, 
 they may be drawn parallel to the horizontal line, as in this fcheme; 
 but then the diftance b^ D, muft be brought down to d, and draw¬ 
 ing d, g, will find the fame point G, in the line E, b^ as by the 
 above operation ; and the diftance a, D, muft alfo be brought 
 down to d, from whence, drawing to /j, the point H, will be found. 
 The reft is as No i. Fig. 
 
ne PRACTICE 
 
 78 
 
 Fig. 59. Ko. 3. Is an expedient, in cafe the point a, Is beyond the 
 picture. Continue tlie line G, E, whofe diredlion is given as be¬ 
 fore, indefinitely towards D ; and fet off the diftance of the pidlure 
 S, D, from S, to D, the point where that diftance cuts G, E, 
 continued. From O, in the horizontal line, at the extremity of the 
 pidlure, draw a parallel to G, E, D, cutting S, D, in d j tranf- 
 
 pofe S, d, to d, on the line S, D; draw d, O, and D, a, pa¬ 
 
 rallel to it, which will tend to the true vanifliing point a, beyond 
 the pidfure. Now proceed, as at No. i, or No. 2; and having 
 drawn E, and found the points G, and H, and drawn G, 
 bifedt the angle D, to <7; draw c, E, cutting G, in L; 
 draw L, Hj then raife the feveral perpendiculars at G, L, and Hj 
 draw F, /?, cutting H, K, in K, and c, F, cutting L, M, in Mj 
 draw b^ M,. cutting G, I, in I; join F, I, and M, K, which 
 completes the whole. If a parallel to S, D, be drawn from O, 
 cutting the line D, E, G, in a, then O, a, will be equal to O, 
 
 (fo that if O, and a, O, were both raifed up perpendicular to 
 the pidture, and alfo D, S, and Z), S, on S ; then would coincide 
 
 with a, and D, with Z);) by which means the lines O, d, and 
 
 O, d, may both be fpared. 
 
 The fame figure is repeated feveral times, on purpofe to fhew by 
 what various means the fame effedt may be produced j but if the 
 lines of the three operations were crowded together in cue fcheme, 
 they could fcarce be feparated by the eye or underftanding, fo as to 
 difcover what was peculiarly, and diftindlly eflential to each. 
 
 If they appear to be fomewhat intricate, it muft be confidered, 
 that they are to be ufed on extraordinary occafions only, and may 
 be always avoided, if the artift chufes rather to take fufftcient room 
 on another fcale, than confine himfelf to the fpace which happens to 
 be left in his pidfure. 
 
 Fig. 60. Let A, C, be a perfpedlive line j it is required to find the 
 geometrical length of a part, as A, B. If its diftance C, D, be 
 within the pidture, and room to fet it off on the vanifliing line, 
 tranfpofe that diftance from C, D, to E} draw E, B, cutting the 
 
 ground 
 

 7^a 
 
 Plate XXXll. 
 
e/ PERSPECTIVE. 79 
 
 ground line in F j then A, F, is the geometrical length fought.—But 
 if D be out of the piflure, take any portion of S, D, as here 
 S, d, a fourth, and S, c, a fourth of S, C j draw c, d, which 
 will be a fourth of C, Dj tranfpofe c, d^ from C, to e, and draw 
 B, cutting A, F, in y' j then A, will be a fourth of A, F, 
 the geometrical length fought.—It is evident, that the fame expedient 
 will ferve to find any proportion of A, C; for fuppofe it had been 
 required to cut off the geometrical length A, F, thereon, this is 
 only, reverfing the operation, (i. e.J drawing E, F, cutting A, C, in B; 
 or, if room be wanting, drawing e, which finds the fame point B. 
 
 Fig. 6i. No. I. (In order to find the vanifhing line of a plane making a given 
 angle with another vanifhing line, it has been taught to find, firft, 
 the vanifhing line of planes at right angles with both, on which to 
 meafure the angle of inclination. Now fuppofe Q, S, B, the vanifh¬ 
 ing line given J it is required to find the vanifiiing line of planes, 
 making a certain angle therewith, and paffing through B, their 
 common interfedlion. To this end, find Q, the vanifhing point 
 
 of lines perpendicular to B; draw Q^P, perpendicular to Q^B, which 
 will be the vanifhing line, perpendicular to both planes j fet off its 
 difiance to d ; draw d^ P, making Q, d^ P, the angle of in¬ 
 clination 5 and, lafily, draw B, P, which will be the vanifliing line 
 fought. But if room be wanting, above, for the difiance S, D, 
 
 divide S, B, in half, at b, and take S, Z), half of S, D, and find 
 y, the vanifliing point of perpendiculars to b ; draw y, p, perpen¬ 
 dicular to y, B J fet off the difiance of y, to r; and draw r, y, 
 making y, r, the angle of inclination : draw b, y>, which would 
 be. the vanifhing line fought, if half the difiance was the true difiance. 
 Now, therefore, from B, draw a parallel to by y, which parallel 
 will be the vanifliing line fought. 
 
 Fig. 61. No. 2. Let. it be required to draw a line through B, S, at 
 the point A, in any angle, perfpedlively, as (e. g.) 45 degrees; this 
 is done by making the fame angle at D, drawdng D, d, and then 
 d. A, which is the line fought. But if room be wanting, take S, d, the 
 half of the true diftance (or any lefs proportion, as may be neceflary;) 
 
 make 
 
8o 
 
 rhe PRACTICE 
 
 make the fame angle at d ; draw D ; then divide S, A, in half, at 
 a ; draw D, a j and, laftly, draw through A, a, parallel to Z), a, 
 which will be the line required. 
 
 Fig. 62. When the parallel D, C, of any line b, a, runs out of the pi< 51 ;ure, 
 before it reaches the vanifhing line, any other line, within the pic- 
 ture, will anfwer the purpofe, as D, c, by drawing from by and 
 yi parallels to it, cutting the ground line in Cy g, and b, from which, 
 feverally, drawing to c, the perfpedtive points of a, by and fy are 
 found in the pofition required. 
 
 Fig. 63. No. I, The center and diflance of the pidlure being given, 
 let B, A, be an original line, in any direction {e, g.) inclined to 
 the picture in the angle B, A, and, cutting it in A, (A, By being 
 its feat, or orthographic projedlion on the pi6lure) it is required to 
 find the perfpe6live length of A, B. Draw S, V, parallel to that feat, 
 and S, D, perpendicular to it, and equal to the diftance; draw 
 D, V, parallel to the original A, B ; draw A, V j then V, is the 
 vanifliing point, and A, V, the indefinite perfpedlive reprefentation. 
 And the length of A, B, is determined, by drawing B, D, cutting 
 A, V, in b. Or fetting off the diftance V, D, to dy and A, B, to 
 By and drawing By dy finds the fame point b. 
 
 This is the moft general fcheme for the purpofe, becaufe the angle 
 of incidence is, at once, referred to the picture, without regard to any 
 other plane, and fo the original line may have any inclination, 
 without making the leaft difference in the operation, on account of 
 the pofition of the pidfure. 
 
 But another example or two, with additional circumftances, may 
 farther illuftrate this kind of operation. 
 
 Fig. 63. No. 2. A, B, is an original line; A, ay its feat on the pic¬ 
 ture; A, by the perfpedtive of A, B, found as in the former; B, /7, 
 (parallel to S, D,) its feat on the ground; <7, the perfpedfive of 
 that feat, found by drawing <7, S ; for if D, S, be turned forward on 
 the point S, and B, turned backward on the point /?, till both 
 are perpendicular to the pidlure, it is evident that by will be the 
 perfpedlive of B, and, confequently, dy by of dy B, 
 
 Fig. 
 
 2 
 
Rate XXXin. 
 

of PERSPECTIVE. 
 
 Fig. 63. No. 3. A, B, is an original line; A, its interfection wit^i 
 the pi6lure ; A, its feat on the pi6lure; V, its vanifliing point, 
 found as at No. i ; and A, b, its perfpeclive, which is all that 
 is necefiary: but befides, let it be required to find its feat on the 
 ground, in the diredlion of the original line; (i. e.) fuppofing a 
 plane palling through it, and its feat on the pi 61 ure. Draw B, 
 parallel to S, D, which will be that feat; for, turning the triangle 
 S, D, V, forwards on S, V, and the triangle A, B, backwards 
 on A, a, till both are perpendicular to the picture, the plane A, B, g, 
 will cut the pi6ture in the line A, a-, and the ground, in the line B, a. 
 
 And to explain it Itill farther, B, is turned round on to 
 a, By fuppofed perpendicular to the pidture, and lying on the ho¬ 
 rizon ; A, a, drawn on the pidture perpendicular to the horizon; 
 and to thefe is alfo joined the line B, a, which is the true geo¬ 
 metrical feat of the original A, B, (perpendicularly) on the ground; 
 as B, <r?, is its feat in the diredlion of the original line, a, and being 
 their interfedlions with the pidture. The perfpedlive of a, B, is by 
 for S, muJd be its vanifliing point by conftrudfion, D, S, being pa¬ 
 rallel to B, a. And the perfpedlive of a, B, is a, by drawn to Vy 
 its vanilhing point, which is found by drawing a perpendicular from 
 V, to the horizontal line: for the geometrical line a, B, being on the 
 ground, perpendicularly under the original line A, B, its vanifliing 
 point mufl neceffarily be on the horizontal line, perpendicular to V, 
 the vanifhing point of A, B ; wherefore a, by drawn towards ‘Vy is 
 ■the perfpedlive of a, B. To make this (if poflible) more clear, S, 
 is drawn perpendicular to the horizon, and equal to S, D, (the 
 diftance); and -u, drawn parallel to a, B, which finds the fame 
 vanifhing point -u. 
 
 Thefe circumflances are thus minutely explained on this, and other 
 occafions, that the univerfality of the principles may appear in their 
 application to the various cafes that occur, or may be required. 
 
 Fig. 64. Let A, B, be given as the fide of a fquare to be projedled, 
 fpace being deficient every way; draw A, S, and B, S; and from 
 any convenient portion of A, S, or B, S, (as here a 4th,) draw 
 
 M fy 
 
S2 7?^ P R A C T I C E 
 
 /, parallel to A, B j then take the fame portion of the diflance^ 
 ('L e.) a 4th, as S, D j draw /, D, and D, g, at right angles to 
 it j draw g, b, and fo finifli the fmall fquare, h, e, Cy by the 
 ufual method —then the large fquare is completed, by drawing pa¬ 
 rallels to the fides of the former, as B, E, parallel to by e ; A, C, 
 parallel to c\ B, C, parallel to 0, by for the diagonal j and 
 lallly, C, E, parallel to r, e. 
 
 Or, inftead of B, C, parallel to b, c,—-S, c, continued, will find 
 the point C ; or S, e, continued, will find the point £5 either of 
 which is fufficient for the purpofe. 
 
 This may not be an improper place to decide a queftion much 
 debated, viz. Whether the reprefentation of a long wall, on a pi6lure 
 parallel to it, fhould be made of the fame height, at the utmoft 
 extent, as dire^lly oppofite to the eye, fince it appears of lefs height 
 the farther it is extended ? To which queflion, the anfwer is, that 
 the wall fhould be drawn of equal height, how far foever extended j 
 becaufe the reprefentation will appear as much lefs, in proportion, at 
 the extent, as the original appears. 
 
 Fig. 65. Let A, —-C, E, be the original wall; I>, the eye of the 
 
 fpe^lator; and, confequently. A, D,—B, D,—C, D, and E, D, 
 vifual rays; and a, h,. c, the reprefentation on a parallel plane;, 
 the triangles A, D, B, and D, by are fimilar, as are the tri¬ 
 angles C, D, E, and r, D, <?; and the lines A, C,-—B, E, and 
 iiy Cy—by Cy aTC aU parallel, as are alfo by and A, B, and c, 
 to C, E : therefore c, Cy muft be equal to Uy by which was to be 
 froved', and in like manner, and for the fame reafons, 2, b, and 
 i*, /\y are equal, being the reprefentations of the equal lines i, B„ 
 and E, 3, made equal to A, B, and C, E. See Euclid, Book I. 
 
 37 > 39 > 
 
 Of the fame nature is that other queflion. Whether, in reprefenting a 
 row of columns, flanding on a line parallel to the picture, thofe, which 
 are more diftant from the center of fuch picture, fhould be made equal to, 
 or lefs, or bigger than the nearer ? It is allowed they appear lefs; but 
 the anfwer to this queflion is, that they ought (in this lituation of the 
 
 picture) 
 

 
 HJKXXUU. 
 
 
4 
 
^/perspective. 8^ 
 
 pi( 5 ture) to be made bigger j and, though £o painted, will really appear 
 as much lefs, in the painting-, as they appear in nature. 
 
 Tig. 66. Let A, B, and C, be the plans of three columns, either 
 fquare or round} and firft fuppofe them fquare } it is evident, that 
 the reprefentation of them will take up the fpace marked by the vifual 
 Tays, from the extreme angles to D, the fpedator’s eye, on the parallel 
 pidlure, whofe fedlion is 5 , ^} (L e.) the reprefentation of A, will 
 fill the fpace S, /} that of B, will fill the fpace g, and that 
 of C, the fpace /, k. 
 
 If the columns are round, the feveral fpaoes, which are filled by 
 their reprefentations, will he determined by the pricked rays, cutting 
 the line S, i, which fpaces are marked by fmall arches. 
 
 But if the pidlure be placed on the line S, 2, the reprefentations of the 
 round columns will be equal to each other, or nearly fo. And if on the line 
 S, 3, or any other between 2, and D, (the end S, remaining unmoved) 
 the reprefentations of the more diftant columns will then, indeed, be 
 in lefs fpaces of the pidlure, by certain proportions, according to their 
 ieveral diftances.—-But on all thefe pidlures, they will be truly repre¬ 
 sented, and will equally exhibit the images of the originals to the eye 
 of the fpedtator at D, who will neceflarily form the fame ideas of 
 the proportions, and diftances of the objects, from any one of thefe 
 pidlures, as from any other of them j which may all be confidered as 
 tranfparent planes, or as one fiich plane, moveable on a hinge at S, 
 from ky to 2, or 35 which plane no more hinders the fpe6lator 
 from difeerning the original objects, than the common medium of 
 air} and as all the vifual rays are neceflarily right lines, the pidlure, 
 or medium, makes no alteration in their diredtions, which are con¬ 
 tinued, without interruption, from the feveral parts of the originals, 
 to D, through any one of thefe tranfparent planes, and whichfoever 
 be chofen, the reprefentations can be determined bynothlng but the 
 interfedtions of thofe vifual rays with fuch plane, and cannot polTibly 
 "be falfc, if thefe interfe^lions are truly found, 
 
 N. B. The rays for the round columns are determined, by 
 making tangents to the feveral circles from D j and the 
 
 M 2 points*, 
 
84 ^ 7?^ P R A C T I C E 
 
 points, in which they touch, are found, by bifeffing the iinc? 
 from D, to the center of each circle, as D, 5, for the circle 
 Cj and with the length 4—5, as radius, making an arc 
 tlirough the center, cutting the circumference in the points- 
 fought. 
 
 If the circles were nearer each other, and D at a greater diftance,. 
 the difference would be proportionally lefs, and at a fufficient diftance,. 
 not at all offenfivej as indeed nothing, that is truly reprefented, cam 
 be ; but even at this, or any diftance, the rule (being demonftrably. 
 juft) cannot vary, and therefore muft be univerfal. 
 
 Fig. 67. No. 1. The ufual points and lines being given, it is required^ 
 to reprefent a door open at any angle. Let /, r, be the fide given,, 
 on v/hich it is fuppofed to turn; S, .6, the fide of the room on the 
 floor j make S,. D, a, equal to the angle required; draw Cy. 
 cutting the ground line in then f,. r, /j, will reprefent the fame 
 angle: and, for the breadth of the door,, draw /, parallel to 
 D, a, and D, c, cutting it in gi from g, to fet off the geometrical 
 breadth, and draw D, cutting r, /, in 0-’,. then will c, e, be the per- 
 fpedlive breadth fought, equal to g, k. —Now, for the thicknefs, drav/* 
 D, b, perpendicular to a, D j, and draw b, e, to /;, which will- 
 be the direction of the edge; draw h, k, which, will be parallel to 
 D, b\ and make j\ of the thicknefs required; . draw f, D, which 
 will determine the thicknefs, perfpe 61 ively ; or, inftead of drawing 
 hy k, and D, continue g, to /, on the ground line, and a 
 parallel from f, to the fame; and, from thofe interfeclions, draw to 
 i/, which v;ill give the thicknefs of the door; draw f, parallel to 
 r, /, and n, /, cutting it in ?n and draw w, which finiflies- 
 the door. Or, as 
 
 Fig. 67. No. 2. Inftead of drawing /, g, and k, below the ground 
 line (in order to determine the breadth and thicknefs of the door), 
 bring down the diftance a, D, to d, and draw d, r, cutting the 
 ground line in and make g, ky equal (geometrically) to the 
 breadth ; draw d, i, cutting c, in which finds the perfpective 
 breadth: then, for the thicknefs of the edge, bring down, in like 
 
 manner^ 
 
Fl.'SSW i 
 
 A 
 
 3 ^ 
 
 
 "■'■-'-if) 
 
^/PERSPECTIVE. 85 
 
 manner, D, to d, and draw d, e, cutting the ground line in 0 ; 
 make 0, equal to the thickncfs, and draw d\ all the reft is* 
 
 as the former. 
 
 N.B. If the door be Ihut, the point (at No. i.) will'"* 
 
 touch E j and w, will touch M. Or again, 
 
 Fig. 67. No. 3. For the breadth, draw the pricked line c, 4, parallel to 
 the horizontal line, and equal to one third of c, /, the heighth (which 
 is here the geometrical breadth,) and D, i, parallel to it, of any 
 length; draw i, 2, cutting D, i, and D, <?, equally, in i, and 
 2, and D, 3, parallel to i, 2; and draw 3, 4,' which will cur 
 <r, in e then f, c, will be the perfpedtive breadth. This me¬ 
 thod has been before explained at Fig. 55, No. i, and 2, and is, iir 
 many cafes, very expedient. The thicknefs is found as in the two former, 
 and the'perfpe 61 :ive dire6tion is determined by the lines e, and b^ m. 
 
 N. B, The point which determines the breadth of the door, 
 may be any where in the pricked femicircle, if the angle of 
 the aperture be not particularly required. 
 
 Fig. 6’8. Having' a fquare e, b, g, J] given, to make, on it; an octagon, 
 draw a, b, through the center; perfpe6tively parallel to the fides 
 e, g, and h, f', fi. e.) from their vanifliing point; and c, d, alfo,' 
 
 through the center, at right angles to a, perfpeftively; then draw 
 
 D, JE, making an angle of 67 r degrees witli S, D, (which is' 
 
 the geometrical angle that e, makes with b^\ and, then, from' 
 draw to r, cutting b, in a, which is one of the angles; 
 and from the fame point through h, cutting d^ r, in r, another 
 angle; and again to g, cutting- r, d, in d ; and alfo through 
 
 cutting b, in b, which determines the laft angle; e, —c, - 
 
 g^ d ,—and b, f being all parallel; and, laftly, join <r, f,—r, g,—— 
 J, b, —3/1 <-/,— d, h, and b, n, which completes the o6lagon. 
 
 This is done in lefs time than defcribed, for all the four points’ 
 wanting, are. found without once moving the end of the ruler 
 
 from .ds. 
 
 If the piffure will not admit the length S, ./E, then, inftead of* 
 the angle S, D, of ’67^, make S, D, B, an angle of 
 
 wliich-:. 
 
-86 PRACTICE 
 
 •which (being the complement of 67-^ to 90) is the geo¬ 
 metrical angle that c, makes with e, h ; and from B, draw 
 through e, which will cut c, in c ; and from the fame point B, 
 to yi cutting c, d, in d and through h, cutting a, b, in b-, and, 
 laflly, through <7, to cutting b, in a, by which means all the 
 fame points are found, and the odlagon completed, by joining the reft. 
 
 The other o6lagon (of pricked lines) is formed by the fame va- 
 niftiing points. 
 
 Tig. 69. No. I. In order to reprefent a cornice, firft draw the geo¬ 
 metrical elevation only, as here for the Doric, in pricked lines then 
 draw lines from S, through every angle of the projection, as i, 2, 3, 
 ,&:c. and draw from D, through A, meeting the line S, i, in a.\, 
 and fo on, from D, through every interfeCtion of the line A, B, 
 as D, 4, meeting the line S, 2, in c, and D, 5, meeting the line 
 
 5, 3, in y>, &c. by which operation the whole cornice is completed 
 ^without any geometrical plan. 
 
 The reafon of this proceeding will appear on infpeCtion of the fquare 
 
 6, I, A, which is the fquare of the whole projection, and of 
 which A, is tlie diagonal, iffuing from the corner, or angle of 
 :.the wall. 
 
 And when it is neceflary to determine an outer angle at the extre¬ 
 mity, make there a fquare correfponding to the above, as ii, 7, 
 8, 9, which is eafdy done, by means of the lines already found j and 
 to determine the mouldings of this angle, perpendiculars muft be 
 raifed from thofe of the firft, already completed, to the diagonal A, a.y 
 and from the feveral interfeCtions, lines drawn to S, will cut the 
 diagonal 11, 8, of this fquare j from which interfeCtions, dropping 
 perpendiculars, thefe, meeting the feveral members, will determine 
 the mouldings of this laft angle. For inftance, from the point 
 raife a perpendicular to y, in the diagonal A, a ; and from y, draw 
 to S, cutting the diagonal j i, 8, in r; from r, drop a perpen¬ 
 dicular, meeting the ray /», S, in / j which is the point fought^ 
 ,and fo cof the .reft. 
 
 The 
 
^'1 
 . Iff* 
 
0/ P E R S P E C T I V E. 87 
 
 N. B. The fecond diagonal I'l, 8, is that ifTuing from the 
 
 , corner of the wall, in this place; and is parallel (in the 
 geometrical) to 6, i, in the former fquarej and the reafon 
 
 for ufing thefe different diagonals, is that A, proje^s 
 
 obliquely forwards, and i r, 8, projects at right angles to^ 
 it, or obliquely backwards. 
 
 Fig. 69. No. 2. The operation is, here, for an inner angle, exa^^ly the 
 fame as in the former, for an outer angle, to the determination of the 
 outlines of the mouldings inclufive; after which, the difference is, that 
 from and the reft of the pro}e6lians (in the former) the line a, 6, 
 with thofe under it, are parallels j whereas, in this latter, they are all 
 rays from S; and the line 8, with thofe under it, in the 
 former are rays, but in this latter are parallels. 
 
 Fig. 69. No. 3. Omitting the geometrical elevation, only knowing the 
 meafures; let it be required to proje6t the cornice (for inftance, of the 
 Ionic order) immediately on a given part of the picture, without 
 raifing it higher (as in the former example). Draw A, B, for the 
 uppermoft line; and from B, to A, (the geometrical projedfion of the 
 whole cornice) fet off the feveral parts of that projedfion j draw from 
 B, to d, the diftance, brought down to the horizontal line, and from 
 A, to C, the center, cutting B, d, in then B, will be the 
 perfpedfive diagonal of the cornice. Now draw from all the divifions 
 betVv’een A, and B, to C, cutting the diagonal B; and, from all 
 thefe interfedlions, drop perpendiculars, and another perpendicular 
 alfo from B; and, on this laft, mark the feveral geometrical heights 
 of the members; and from thefe points, draw to d^ cutting all the 
 perpendiculars, and their refpedlive interfedfions will determine the 
 perfpedfive projedtions of all the members, by which the cornice will" 
 be completed 3 gO C, cuts the diagonal in f, and c, d, cuts 
 the perpendicular /*, E, in E, which determines the projedlion of 
 that member, and fo of the reft.—-The points of the projedlion being 
 thus found, may ferve either for an outer angle, (as here,) or for an 
 inner angle, (as in the laft example,) the perfpedfive extremities re-- 
 maining the fame. And if an inner angle be req\tired from any other 
 3 point 
 
cpoint in B, C, as G, draw G, d, cutting A, C, in k‘, then G, will 
 be (perfpedllvely) parallel to B, a, and, confequently, will reprefent 
 ,the diagonal, by which fuch inner angle may be completed, as the 
 outer was by means of B, a. 
 
 ^•.Fig. 69. No. 4. When it is required to projefl a cornice (as here of the' 
 Corinthian order) not parallel to the picture, from a point given, as B ; 
 draw firfl the geometrical diagonal of the projection B, A, parallel to 
 the horizontal line, and mark on it the angular projections of the 
 feveral members; and having bifeCted the right angle d, D, d, and 
 continued the line of bifeCtion to c, in the horizontal line, and 
 brought down the diftance 0^ D, from D, to Z)f, draw B, 0^ and 
 A, £>£-, cuttmg it in a, then B, will be the perfpeCtive diagonal. 
 Now draw from all the divifions of A, B, to Dr, cutting Bj a, 
 in the feveral perfpeCtive points of the diagonal, from which drop 
 perpendiculars, as alfo one from B j and, on this latt, mark the geome¬ 
 trical heights of the feveral members j .and, from all tliefe points, 
 draw to o, cutting their ^orrefpondent perpendiculars, which inter- 
 feCtions will determine.the angular points of the cornices and draw¬ 
 ing lines from every one of thefe angular points, to the vanifliing 
 ,points -d, and dy the cornice is, thus far, -completed. ^ 
 
 And, for the inner angle H, draw H, 0, and a, -dy cutting it in 
 hy which gives this diagonal; and divide it, by drawing from the 
 feveral divifions on B, ay io d-, then dropping perpendiculars, from 
 the points thus found, in FI, hy they will meet their refpeCtive cor- 
 refponding lines (already drawn) from the perfpeCtive angular points 
 of the outer angle B, to d, and thefe laft interfeCtions will determine 
 this inner angle. 
 
 The fame operation determines the outer angle G, with thefe only 
 differences, that the diagonal of this laft is not parallel, but per¬ 
 pendicular to the two former, (as was particularly explained at tl^e 
 N. B. of No. I, with refpeSl to that Doric cornice) ; and the lines 
 ;run to the oppofite vanifliing point d. 
 
 N. B. ^he manner of finding the Jloadows of thefe .cornices is ex-- 
 plained in the Supplement. 
 
P1.5XSVIT, 
 
<?/ P E R S P E C T I V E. 89 
 
 Fig. 69. No. 5. In this, and the following figures, the fkeleton, or cafe 
 only, of the whole cornice, is proje6led in ftraight lines, that the reafon 
 of the operation may appear more limply and clearly, unembarrafled 
 with that number of points, and lines, which are neceffary in deter¬ 
 mining minutely each member. 
 
 The center and diftance being given, as ufual, draw firfl; the fquare 
 
 A, B, E, F, for the breadth of the folid j then draw A, C, and D, E, 
 cutting it in G; draw G, H, parallel to A, E, and draw E, C, cutting 
 it in H, which will give the cubical perfpedtive of the folid; produce 
 
 B, A, to I, making A, I, equal to A, E, for the proje£fion of the whole 
 cornice, (which, in four of the orders, is equal to the height, ac¬ 
 cording to mod: archite6ls; (i. e.) in all but the Doric); draw C, I, 
 and 3, E, cutting it in K ; and i, H, cutting the fame line C, I, in 
 L ; then draw 2, F; and laftly draw K, M, (parallel to A, B,) 
 cutting 2, F, in M, which completes the figure. 
 
 N. B. A, E, is the height of the cornice; and A, K, is the 
 diagonal of its projeftion. 
 
 Befides that the pricked lines, within, fliew the conformity of this 
 operation to the former figures, let it be confidered that the triangle 
 K, A, E, reprefents the angular proje6tion of the whole cornice; 
 for 3, 4, is the vanifliing line of the plane of that triangle, and 
 
 3, the vanifliing point of K, E; as the triangle L, H, G, repre¬ 
 fents another proje6lion of the cornice (the plane of which is per¬ 
 pendicular to K, A, E,) whofe vanifliing line is therefore i, 2, and 
 the vanifliing point of H, L, is i. The other angles are determined 
 in the fame manner; for i, 2, is alfo the vanifhing line of the 
 triangle M, B, F, it being in the fame plane as L, G, FI, and 2, 
 the vanifhing point of F, M; fo is alfo 4, 3, the vanifliing line of 
 P, N, O, this triangle being in the fame plane with K, A, E; and 
 
 4, is the vanifhing point of O, P. 
 
 I, C, 3, is alfo the vanifhing line of the plane K, L, H, E ; 
 for I, 3, and C, are all vanifhing points in this line: 2, 3, is the 
 vanifhing line of the plane K, E, F, M; and i, 4, is the vanifliing 
 line of the plane L, FI, P, O; and 4, C, 2, is the vanifhing line 
 
 N of 
 
go P R A C T I c E 
 
 of the plane M, F, O j for 4, 2, and C, are all in this line, 
 
 Fig. 69. No. 6, If the cornice be Doric, as that projefts more, in pro¬ 
 portion to its height, than the other orders, all the difference is 
 in taking A, e, inftead of A, E, with its correfpondent meafures, &c, 
 which will neceffarily give 3, 2, in this fcheme, for a vanifliing line, 
 inftead of 3, 2, (at No. 5.) and i, 4, inftead of i, 4; the reft of the 
 operation being the fame as above. 
 
 N. B. In this figure, the plane M, />, jf, 0, happens to fall in 
 the vanifliing line 4, C,. 2. 
 
 Fig. 69. No. 7. This figure (which is feen by the angle) cannot need 
 much explanation, after what has been already faid; the lines e, f, and 
 gy by being (with g, and fy h,) in a plane parallel to the picture, have 
 no vanifliing points; a, b, has 0, for its vanifhing point; and ky 
 has p: for as this vanifliing line Oy p, pafTes through the center of 
 the pidlure, the diftances C, o, and C, p, are to C, dy (the diftance 
 of this vanifliing line) as the fide of a fquare to its diagonal j by 
 which means the height and projedtion of the cornice keep their 
 proportions, the angular projedlion being to the height, as the dia¬ 
 gonal of a fquare to its fide, in the four orders, which have their 
 heights and projedlions equal. Hence it is evident, on infpeclion, 
 that the perfpedldve angle by Cy q, (or C, ay 0.) reprefents the geo¬ 
 metrical angle C, dy o, as r, k, i, (or C, k, pd) does C, dy p. 
 
 Fig. 69. No. 8. This figure differs from the laft, only, in its being 
 obliquely fituated to the plane of the picfure in every refpedl:, the laft 
 having the lines c, g, and jf by parallel to it. 
 
 The fcheme fufficiently ftiews the operation, on the principles fo- 
 often explained ; the letters a, c, g, and d, mark the angles of the 
 cornice, as in the former j and 0, is the vanifliing point of a, by 
 found by making c>, as the ftde of a fquare, to y, the diagonal, 
 (which -1, y, is equal to q, D, the bifedlion of the angle E, D, F,) 
 for this is the proportion of y, by to y, a. E, <?, is the vanilhing 
 
 line of the plane ay Cy b j and F, c, the vanilhing line of the 
 
 plan; U, gy by k. 
 
 4 
 
Pl.XXXMU 
 

 , 1 ,: 
 
 1 
 
 
 I 
 
 { 
 
 ■ -i! 
 
 ‘"^ ■■ :.1 
 
 ■ rJ-.v 
 
 
 
 f 
 
 ■'\ ■ 
 
 '•. ■< 
 
of PERSPECTIVE. 
 
 9 ^. 
 
 The FIFTH PART. 
 
 OF SHADOWS. 
 
 T hat part of perfpe6llve, whLch relates to the proje6lion of 
 fhadows, is lefs neceflary, than any of the preceding parts, and 
 is wholly omitted by Pozzo, in his treatifes on the fobjedt, though 
 one of the greatell mailers in the executive part, and who feems to 
 have done every thing elfe by rule. 
 
 However, it Was thought proper to give a few examples, not only 
 of cafes that moft commonly occur, in the courfe of pradlice, but alfo 
 of fome others lefs ufual, and more difficult, to fhew the application 
 of thefe principles to this purpofe, as well as to corredl the miftakes 
 of fome writers on the fubjedl. 
 
 Fig. 7c. No. I. In this fcheme, the light is fuppofed to come 
 from the fun, which being confidered as at an infinite diftance, the 
 rays are treated as parallel; but it does not follow, that therefore the 
 fhadows muft be reprefented parallel, except in one cafe, (and that 
 very rare,) which is, when the rays are parallel to the pidlurej an 
 inflance or two of which fhall be firfi; given. 
 
 The rays of light are here fuppofed to come from the fun, and arc 
 not only parallel among themfelves, but alfo to the pidture, and 
 therefore the moft fimple, and moft eafy to projedl. Any one ray, as 
 yi being drawn in the diredlion required, all the reft muft be 
 parallel to it. Draw from F, the bottom of the line y, F, a pa¬ 
 rallel to the horizontal line, meeting y, g, in g, which determines 
 that fliadow j for g, is the ftiadow of y on the ground, and F, 
 of the perpendicular line F, y; and fo for each perpendrcuiar line, 
 as, for z, draw H, parallel to y g-, and i, H, parallel to the 
 horizontal line, meeting in H; and K, parallel to y, g j and, 
 laftly, /, K, parallel to the horizontal line, meeting in Kj and the 
 fame operation for the open door, wffiich completes the whole. 
 
 N - N. B, If 
 
^ 2 7 : 5 ^ P R A C T I C E 
 
 N. B. If one only of the points g, or H, had been found, 
 the other would be determined, by drawing through that, 
 already found, to S, the vanifliing point of /, F, which is- 
 perfpe6tively parallel to H, gy &c.—for H, g, the Ihadow 
 of by fy is parallel to it in the original, and therefore 
 both run to the fame vanifhing point S. 
 
 Fig. 70. No. 2. This figure, though two walls only, is exhibited, to fhew 
 the manner of determining the fhadow on a plane Handing obliquely 
 to the horizontal line. Having drawn B, E, parallel to A, by the direc¬ 
 tion of the rays, as in the former, and G, E, parallel to the horizontal 
 line, meeting it in E, G, E, will be the fhadow of G, B; but as it 
 is interrupted by the wall C, L, raife a perpendicular at L, (where 
 G, E, cuts the bottom of that wall) and draw A, C, cutting it in 
 e y then C, e, will be the fhadow of the line B, C, on the plane 
 C, L i for, all the rays are parallel to A, b, and by is the vanifhing 
 point of B, C, therefore A, by is the vanifliing line of the plane made 
 by the rays which pafs through the line B, C; and (ay A, being 
 the vanifhing line of the wall C, L,) A, is the vanifliing point of 
 the common interfe 61 ion of the two planes, whofe vanifliing lines are 
 by A, and A; therefore A, is the vanifliing point of the fhadow 
 C, ey of the line B, C, on the wall C, L. 
 
 If the reader fliids any difficulty in conceiving this, let him imagine 
 the wall C, L, continued to its vanifliing line A, and the plane 
 of rays to its vanifliing line by A, thefe planes-will then interfedl at 
 A j and as both planes pafs through the eye, fb alfo muft their in- 
 terfedlion, which will be a line from the eye terminating in A.—It 
 cannot be forgot, that all lines, drawn from the fame vanifliing point, 
 are, perfpeclively, parallels j (/. e. reprefent lines geometrically parallel;) 
 wherefore if D, were brought forwards on S, perpendicular to the 
 pidlure, it would reprefent the eye, and D, A, the interfedfion fought. 
 Now A, C, c, reprefents a parallel to D, A, in this fltuation of D. 
 
 The point c, however, might have been determined by the 
 line B, E, interfering the perpendicular from L, in tliis particular 
 cafe, as the wall breaks the fliadow G, E, in L ; but, otherwife, the 
 general rule is as above. Fig. 
 
c/ P E R S P E C T I V E. 93 
 
 Fig. 70. No^ 3. Is the fame fabjeft, with this only difference, that the 
 wall C, L, is fhorter, on which account, the fhadow G, E, of B, G, 
 is thrown,, wholly, on this fide of it, fo that the line G, L, E, cannot 
 interfe6t the bottom; which difpofition is chofen, on purpofe to fhew 
 the ufe of the vanifhing point A, in determining the dire6lion of the 
 fhadow on the wall C, L, as explained above. 
 
 Though this alfo might have been found, by continuing the line 
 N, L, till it cuts G, E, in L, and there raifing a perpendicular, 
 meeting the ray B, E, in e, and then drawing C, e .—But in- 
 flead of all this work for the direction of one line, nothing more is 
 neceffary, than drawing from A, through C, as has been explained. 
 
 If the line N, L,. be continued, and the ray A, r, till they meet 
 in E ; and the parallel E, G, be drawn, and alfo the line N, G, till 
 it meets E, G, in Gj and the perpendicular G, E, be raifed, and b, c, E, 
 continued, till it meets Ej and the ray E, E, be drawn parallel to 
 A, b j then the fhadows (reprefented by r, r, and G, E,) brought 
 down to G, E, will unite in E, and, by that means, illuftrate the 
 whole operation, 
 
 Fig. 71. No. I. Here, though the fhadows are geometrically parallel 
 among themfelves, they are, neverthelefs, oblique with refpe6l to the 
 picture (as is generally the cafe) j and, for this reafon, the fhadows 
 will all tend to the fame vanifhing point in the horizontal line. 
 
 U, is the vanifhing point of the rays of light, found by drawing 
 from the eye, to the pi6lure, in the direction chofen, or given, for 
 that purpofe. To conceive this, fuppofe D, S, raifed up, on S, till 
 perpendicular to the picture : in that fituation, E), U, is the parallel 
 of the rays, cutting, the pidlure in U, which is therefore the vanifhing 
 point of them. 
 
 And ralfing a perpendicular from U, to the horizontal line, cut¬ 
 ting it in V, that will be the vanifliing point of all the fhadows, caff 
 on the plane of the horizon, by all objeiffs perpendicular to that plane j fo 
 that the fhadow of any point is found, by drawing firfl from fuch point 
 to U j and then from the feat of the fame point to V ; for inflance^ 
 from the point b, (of the objedt A,) draw to U y and again from B, 
 
 til®. 
 
94 77.^ P R A C T I C E 
 
 the feat of b, draw to V, cutting b, V, mb-, this will be the 
 fliadow of b, on the ground, and B, b, will be the fliadow of the 
 whole line b, B : in like manner, dravhng c, U, and C, V, cutting 
 it in e, that is the fliadow of c ; and thus is alfo found /, the 
 fliadow of f: after which, by joining all thefe points, is completed 
 the whole fliadow of the objedl A, on the ground. 
 
 For the fliadow of the odtaedron, the feveral points of it marked 
 I, 2 , 3, 4, are all found in the fame manner, viz. by drawing from 
 the feveral angles to U, and from their feats refpedtively to V, each 
 sngle, feat, and fliadow, being marked by the fame charadter; for 
 infiance, 2, the angle on the body of the ocfaedron; 2, its feat 
 perpendicularly under it j and 2, its fliadow; and fo of the refl. 
 Thus alfo may be found the fliadow of any obje( 5 l not perpendicular, 
 by finding the perpendicular feats of the principal points, and ufing 
 fuch feats, and points, as perpendicular lines j (e. g.) having found g, 
 the feat of E, draw E, U, and g, V, interfedliiig in b, that will 
 be the fhadow of E ; and drawing h, G, is determined the whole 
 fhadow, on the ground, of the pole G, E. 
 
 But for thofe fliadows, which fall on planes not horizontal, fome 
 other expedients are to be ufed, as that of A, on the parallelopiped 
 H. Having firfl: found the fhadow on the horizontal plane, raife two 
 perpendiculars where that fliadow touches the edge, (as at /, and k,) 
 to /, and 771 and fince the upper fide is parallel to the horizon, 
 draw from /, and 777, to the fame vanifliing point V, which de¬ 
 termines it: fo alfo is found the fhadow of the cube on the bottom 
 of the parallelopiped A, firfl drawing a perpendicular from 7i, up¬ 
 wards (where the fhadow on the ground cuts the line 71, B,) and 
 determining the top of that perpendicular w, by the line 0, U then 
 for the point p, continue V, f backwards, till it cuts q, S, in r ; 
 at r, raife a perpendicular to t draw t, U, which finds the point 
 p i and, laftly, draw p, w, which finifiies that fhadow. Or if B, 77, 
 be continued backwards, till it meets q, S, as in x*, and a perpen¬ 
 dicular be there raifed, cutting the line 0, S, in y, a line drawn 
 from y, to w, will find the fliadow p, ‘w. 
 
 N. B. This 
 
I'L XXXIX 
 
■A. 
 
PERSPECTIVE. 95 
 
 N. B. This line y, w, runs not to any vanifhing point, 
 becaufe it is parallel to the pi6ture. 
 
 The fhadow, on the ground, of the plank, that refts upon the 
 cube, is found by dropping a perpendicular from the point, 
 where it touches the edge, to the ground j and drawing from the top 
 to U, and from the bottom to V,- is found one point on the ground, 
 which is fufhcient. For, drawing from N, through that point, ta the 
 horizontal line, is found the vanifhing point M, of the fhadow on the 
 ground; to which vanifliing point, draw from the other corner of 
 the plank, which refts on the ground, and one line more (from the 
 top of the plank to U,) completes this fliadow, fuppohng the whole 
 of it on the ground : that part of it, on the upper face of the 
 cube, is found, by drawing from the points, where the plank touches 
 it, to the fame vanifliing point M j and that other part, on the front 
 of the cube, by continuing the line, of the bottom of the cube, from 
 y, through the lhadow on the ground to ; then drawing from the 
 points in which the plank touches to 2:, and •f*. The reafon of this 
 operation is obvious j for, fuppofing the front of the cube to extend 
 beyond 2;, the fhadow of the ground would there meet it, and,that 
 part of the fhadow, on this face, muft be between the points in which 
 the plank touches the cube, and the ground at -f*, 2;. 
 
 Here are added four perpendicular polls, on the fame line, to fliew, 
 that though the fliadows of them are geometrically parallel, yet as 
 they are not call in a diredlion parallel to the pi6lure, (in which 
 cafe they would be parallel, and equal, in perfpedlive,) they muff, in 
 this direction of the light, have all the fame vanifhing point V, and, 
 therefore, cannot be parallel in their reprefentations, nor of equal 
 lengths, though they are all of the fame height, or depth, on the 
 pidure, from the front line on which the pofts (land. 
 
 The error of making the fliadows parallel, in this cafe, is to be 
 obfcrved in the Jefuit’s Perfpedlive. The author might pofhbly be 
 mifled, by confidering, that as. the fun is fo large a body, and fo 
 diflant, the rays of light from thence defcend in parallel lines, or 
 (which is the fame thing) in lines not to be diiLingulfhed from 
 
 parallel 
 
The PRACTICE 
 
 96 
 
 parallel; but he fliould alfo have confidered, that all parallel lines, 
 not parallel to the pi 61 :ure, will, on the pidture, have a common 
 vanhhing point, to which they mutually tend, as the prefent cafe 
 requires, and which makes fo great a difference in the reprefentation. 
 For all lines, in perfpective, are fuppofed to pafs through the eye of 
 the fpeflator, and to meet the plane of the pidture fomewhere (except 
 thofe which are parallel to that plane); and the point wherein any line, 
 fo pafling through the eye, interfefts the plane of the pidlure, will 
 be the vanifhing point of all other lines parallel to that line. 
 
 This is the very con{l:ru6lion of a vanifhing point, on which almoft 
 the whole pradfice of perfpe^live depends. 
 
 iV. B. In the Jefuit’s Perfpe6five, the 3d edition, tranflated by 
 Chambers^ i 743 j P^ge 132, the fecond example is falfe, as 
 the extreme lines of the fhadow (though calf forwards) 
 are made geometrically parallel, which fhould run to a va- 
 nifliing point in the horizontal line, and would then repre- 
 fent parallels j and, in the text, the reader is inflrudled in 
 this falfe method. Page 133, falfe, inafmuch as all the 
 lines, at the feet of objedls, (i. e.) the direflion of the fhadow 
 which ought to run to a vanifhing point to reprefent pa¬ 
 rallels, are made geometrically parallel to H, K, L, and 
 the reader is inflru6fed fo to make them. Page 134, is right, 
 where the fun is direftly behind j and he is right alfo, 
 where the fun is fuppofed to call the fhadows parallel to the 
 pi 61 :urej but he is falfe in every oblique dire6tion, which 
 diredions are much more frequent than any others. Thefe 
 errors are plainly owing to his not underftanding (or not con- 
 fidering) the neceflity, and ufe of vanifhing points j perhaps 
 he was fenfible of the difficulty of oblique dire 61 :ions, which 
 feems to be the reafon that, for five pages together, all the fha¬ 
 dows are caft in the fame diredion, (ue.) all of them parallel 
 to the pidure j fo that they afford no variety, except of the 
 objeds j that is, no variety of inflrudion, or any cafes that 
 
 need 
 
»/ P E R S P E C T I V E. 97 
 
 need it. His fliadows, from artificial lights, are tiMe j but no 
 difficult or intricate cafes are given, either of objedls placed ob¬ 
 liquely, or fhadows thrown on planes oblique to each other; 
 but all on planes parallel, or at right angles to each other. 
 And moft of the miftakes into which many painters, and wTiters 
 on this fubje^f, have fallen, (who perhaps may not have been altogether 
 deficient in fcience) are owing to the attention to obje 61 :s, as they 
 appear generally in nature, without referring their appearances to' 
 the eye of a fpe6tator fixed to a point, or without fufficiently confider- 
 ing what kind of images fuch appearances, in nature, muft neceffarily 
 form, on a tranfparent plane, between the eye and objedl:, in every 
 different dire 61 :ion. 
 
 It was faid above, that by drawing a perpendicular from U, to V, 
 this laft would be the vanifliing point of the fhadows of all lines per¬ 
 pendicular to the horizontal plane; and alfo, that the fliadows of any 
 other lines (not perpendicular) might be determined, by finding the 
 feats of any points on the ground, and drawing from fuch points to U, 
 and from their feats to V j and thus was found /», the fhadow of E, 
 and the whole fhadow of the oflaedron, and other obje 61 :s. 
 
 But to fliew the extenfive ufe of vanifliing points, here is added 
 another method for oblique lines : 7, 8, is a pofl: inclined to the 
 horizon, but parallel to the pi6lure. Draw from U, a parallel to 7, S, 
 cutting the horizontal line in which will be the vanifhing point of 
 the fhadow of 7, 8, on the ground; draw 8, U, and 7, «, cutting 
 it in 9; then 7, 9, is that fhadow, without farther operation: and 
 if there were many lines in the fame diredlion, that is, parallel to 
 7, 8, it might be worth while to find their common vanifhing point 
 to which all their fliadows would tend; but, otherwife, it may be 
 determined by the vanifhing point V; for dropping a perpendicular 
 from 8, to 10, this will be its feat; wherefore, having drawn 8, U, 
 as before, draw i o, V, which finds the fame point 9; then drawing 
 7, 9, that will find the fame fhadow. 
 
 The reader will obferve, that the pofl 7, 8, though not perpen¬ 
 dicular to the horizontal plane, is, however, parallel to the picture; 
 
 O and 
 
PRACTICE 
 
 and in cafe of any obliquity in an objed, if flill pirallel to tKe- 
 picture, a line, as U, u, drawn parallel to fuch obliquity, will al¬ 
 ways truly find Uy the vanifliing point of the fliadow. For the tri¬ 
 angles Uy 9, U, and 8, 9, 7, are firnilar, as well as the triangles 
 V, 9, U, and 8, 9, i o j and the line 8, 9, U, is common to both 
 pair of triangles; therefore it muft be divided in the fame point 9, 
 by either of the correfpondent lines 7, «, or i o, V. 
 
 But when the original is not only oblique to the horizon, but alfo 
 to the picture (as the pofl ii, 12) j then the vanifhing point of fuch 
 oblique line muft be found by means of a fcheme like that at 
 Fig. 63, which (if underftood) will render this intelligible. Or the 
 vanifhing point of 11, 12, for inftance, which is fuch an oblique 
 line, and whofe feat is 13, 12, may be found, by continuing the 
 line 13, 12, to its vanifhing point S; thence dropping a perpendi¬ 
 cular j and, laftly, continuing the projedled line 11, 12, till it cuts 
 that perpendicular in ©, which is the vanifliing point fought. This 
 poft leans forwards, in an angle of 65, marked at Dj (i. e.) the 
 angle S, D, ©, and whofe vanifliing point being ©, below; thence 
 drav/ through U, till it cuts the horizontal line, which iiiterfedfion will 
 be the vanifhing point of the fliadow; and having drawn ii, U, (as 
 in ail the former cafes) draw from 12, to that vanifliing point, cutting 
 II, U, in 14, and then 12, 14, will be the fliadow fought. To fliew 
 that this fhadow is truly found, draw from 13, the feat of 11, to 
 V, which will cut ii, F', in the fime point 14, and is a proof. 
 
 N. B. When the vanifliing point is tirft given, then the feat is 
 found by drawing a perpendicular from 0, to the horizontal 
 line, cutting it, as in S ; then drawing S, 12, and a per¬ 
 pendicular from ij, cutting it in 13. 
 
 And this will be general, for any line, viz. to draw from its va- 
 nifiiing point, through the vanilliing point of the ray of light, to 
 the vaiiilhing line of the plane on which the fliadov/ is to be p'ojecfted, 
 whether it be the horizontal plane, or any other; and this interfe^lion, 
 with the vanilliing line of the plane on which the fliadow is caft, will 
 be the vanifliing point of the fhadow. For (in this fcheme) imagine 
 4 
 
^/PERSPECTIVE. 99 
 
 the plane D, S, 0, raifed on S, 0, till D, S, be perpendicular to 
 tlie pidture j then a line D, U, determines the vanilhing point of 
 the rays; and, confequently, a line through 0, and U, to the 
 vanifliing line of the horizon, will give the vanifliing line of the 
 plane of rays paffing over the whole line 0, 12, ii, and therefore, 
 alfo, the vanilhing point of the lliadow 12, 14. See this fa 7 'ther ex¬ 
 plained in the Supplement. 
 
 Fig. 71. No. 2. Is a cube on an inclined plane; a, S, is the vanilhing 
 line of that plane, W, is the vanifliing point of the lines i, 2,— 
 3, 4,—5, 6, and 7, 8; wherefore, having given U^, for the vanifliing 
 point of the rays, draw from W, through U% to V®, (in the vanifli¬ 
 ing line S, b \) and then, drawing from 3, 5, and 7, to U*, and 
 from 4, 6, and 8, to V% the fliadows of 3, 5, and 7, are determined, 
 which are all the points necelTary for completing the whole fliadow. 
 
 Fig. 71. No. 3. In this fcheme the obje6l is (as the laft) on an inclined 
 plane, and the fun obliquely behind the objeft; on Vv^hicli account U, 
 the vanifliing point of the rays, will be above V, the vanifliing point 
 of the fliadows; for it is to be always remembered, that every vanifli¬ 
 ing point is formed by a line pafling through the eye parallel to the 
 original; and wherefoever fuch line cuts the plane of the picture, that 
 interfedtion is the point fought. Now'if the fun be beyond the obje6t, 
 a parallel to its rays will, neceflarily, cut the pidture fomewhere above; 
 and as S, is the center of the pidlure, and S, D, its diflaiice, ima¬ 
 gine D, S, raifed on S, perpendicular to the pi6ture; in that fitua- 
 tion, D, U, is the parallel to the rays (whofe direftion is fuppofed to 
 be given) ; therefore U, is the vanifliing point of thofe rays; and 
 as P, is the vanifliing point (which will be perfpeclively perpendicular 
 to the vanifliing line a, C, b, and) of i, P, and all its parallels, draw 
 U, P, cutting the line C, b^ in V, and that will be the vanifliing 
 point of all the fliadows; wherefore, drawing U, i, U, 2, U, 3, 
 and then from V, through the feat of each line, the points i, 2, 
 and 3, of the fliadow, are found. 
 
 Fig. 71. No. 4. In order to fliew the conformity of operation, here is 
 added the fame figure, with fimilar circumftances, on the plane of the 
 
 O 2 horizon. 
 
lOO 
 
 rhe PRACTICE 
 
 horizon. U, V, is, in this fcheme, therefore, geometrically perpendi¬ 
 cular to the vanilhing line a, S, 3 , and parallel to the lines i, p ,—« 
 2, py —'3, p, whofe fhadows are fought. The reft needs no- 
 
 explanation. 
 
 N. B. In this, and all the former, after having drawn U, r,, 
 or any one of the rays, and V, P, interfedling it in i, all 
 the other points of the ftiadow may be found, by only draw¬ 
 ing through the bottoms, or feats, of the remaining lines 
 from V, and then drawing to the vanifhing points a, and< 
 3 , of the figure, which will find the points 2, and 3, as 
 1, ay finds 2, and 2, finds 3, &c. 
 
 Fig. yi. No. Here is once more the fame objedt, with the fun 
 behind itj V, the vaniihing point of the fliadows coinciding with Sj 
 U,. the vanifhing point of the rays perpendicularly above it. Having 
 drawn U, i, and V, 0, cutting it in 2, the reft is determined as 
 in the N. B. of the laft, viz. by means of tlie vanifhing points 
 and b \ and as the cube is a little oblique before S, the ray U, i, 2, 
 determines the plane of 2,. by which the whole is completed, as has 
 been explained. But if the cube had been diredlly before S, it woul(h 
 have been very difficult j or if the objedl had been a fingle line, as 
 4, 7, it would have been impoffible, without an expedient j becaufe 
 the ray U, 6, and the line of the fhadow V, 7, coincide; in which 
 cafe, draw V, 8, at pleafure, and from 7, draw a parallel to S, by 
 cutting it in 8 ; draw 8, 9, parallel to 7, 4, and equal to it; drav/ 
 U, 9, cutting V, 8, in 3; laftly, draw 3, 6, parallel to 8, 7, which 
 determines the point 6. 
 
 Fig. 72. No. I. In this fcheme, the light O, is fuppofed that of a* 
 candle, or other luminous point, whence it is diffufed every way as^ 
 from a center; its foot, on the ground, is F. In order to find the 
 lhadow of the door by c, by draw F, by cutting S, H, in g ; there 
 raife a perpendicular, and draw O, c, cutting it in c; then by g, 
 will be the fliadow on the ground, and g, c, on the fide of the 
 room ; and to find the diredlion of c, f, as a is the vanifhing 
 point of the top and bottom of the door, draw a, by H, cutting, 
 
 S, H, 
 
lOI 
 
 of PERSPECTIVE. 
 
 S, H, in H; and then H, will be the mterfe£lion of the plane of 
 the door with the plane on which this part of the lhadow is call 5 
 draw H, parallel to e, and by e, cutting it in which will 
 be the point wherein the line b, Cy meets the fame plane j wherefore 
 draw iy c, which will be the dire6lion, fought, of the fhadow r, /*j 
 laftly, draw fy by which completes the whole fhadow. 
 
 It is evident, that if the plane of the door was continued, it would 
 meet the fide of the room in H, / j and the point r, having been 
 found by the perpendicular gy c, the direftion of the fhadow, paffing 
 through Cy muft be /, r, f. 
 
 Or the fhadow by fy c, might be found by a method more geo¬ 
 metrical, thus. 
 
 Fig. 72. No. 2. Having drawn F, by gy (as before) and the perpen¬ 
 dicular gy c'y and’ alfo having drawn by H, and the perpen¬ 
 dicular H, iy and dy by Cy cutting it in i-, and found the vanifh- 
 ing line dy ky of the plane of rays palling through the line 
 Cy by By i -y from ky the interfedlion of this vanifhing line with S, ky 
 the vanifhing line of the fide of the room (on which part of the 
 fhadow is caff) draw ky iy cutting gy c, in c, and the angle of the 
 room in f j and draw b, fy which finifhes the fliadow. 
 
 N. B. The vanifhing line dy ky is found, by drawing a line 
 through dy parallel to 0, ni ; for 0, my reprefents a ray 
 parallel tO’ the pidfure, found by drawing F, /, parallel to 
 the horizontal line, and /, perpendicular to it, cutting 
 dy by Cy lu 171 'y aud thco, drawing 0, m. 
 
 Or (without ufing the vanifliing line dy ky) draw 0 , dy and 
 F, dy raifing a perpendicular at the interfe6lion of it, with 
 the edge of the room) cutting 0, dy in c>, which will be the 
 foot of the light on the fide of the room ; and 0, Oy will 
 be perfpedlively in the direction of (or parallel to) Cy ^ j. 
 wherefore draw o, by fy &c. 
 
 This is farther explained in the feveral manners following, becaufe 
 many fuch cafes happen, and the underftanding this, fully, may be 
 of great ufe. dy by continued, cuts the fide of the room in H, and- 
 
 a per^ 
 
102 
 
 I'he PRACTICE 
 
 a perpendicular being raifed at H, and continued, they 
 
 meet in —So that if the plane of the door was continued, 
 H, /, would be the extreme edge, touching both that fide, and the 
 floor of the room, and could have no fliadow, either on the fide, or 
 below but, in that cafe, there would be a fliadow above becaufe /i 
 is the angle of the room, and b, f, being in a plane parallel to the 
 pidlure, and J\ i, in a plane perpendicular to it, the whole fhadovv- 
 would be the triangle b,f, i ; for 0, F, /, is a plane parallel to the 
 pidfure, and w, e, b^ the edge of the top of the door, continued 
 to the horizon at a-, therefore 0, (being parallel to the pidfure) 
 may be confidered as the interfedtion of the plane of rays, paffing 
 over the top of the door; and, confequently, ^7, (parallel to it, 
 palling through the eye, and cutting the pidlure) is the vanifiiing line 
 of that plane j and, cutting S, the vanifiiing line of the fide of 
 the room, on v/hich part of the fliadow is caff, d, will be the vanifh- 
 ing point of the interfedtion of thofe two planes, viz. of the rays, and 
 fide of the room ; wherefore, drawing /, this line determines the 
 lliadaw c, yi and drawing b, the whole is determined. 
 
 Or drawing b^ parallel to meeting the angle of the room, 
 
 that will determine the point by which c, is found; for the 
 plane of the rays is parallel to /<:, and b^ is parallel to the 
 picture, and is interfedted by the plane which generates the vanifiiing 
 line k', as a, k, is the vanifiiing line of the plane of rays paffing 
 through the line b, e, i-, and o, b, being the interfedtion of that 
 plane, with the plane on which that part of the fliadow is caft, b^ y, 
 muft be parallel to a, k. Again, o, is the feat of the light on the 
 plane of the fliadow, and is the feat of e, b^ on the fame plane j 
 therefore b^ the fliadow of b, muft be the continuation of the 
 line Oy b ; and that line is neceflarily parallel to k, (by conftrudtion) 
 becaufe the plane of this part of the fliadow is parallel to the pidturej 
 and the plane 0, ?n, /, k, a, cuts both the plane of the pidture in 
 a, d, and that which receives the fliadow in <?, f, which two latter 
 are parallel planes. 
 
 The other door /, y, k, in No. i, is opened at right angles, 
 
 and. 
 
(?/ P E R S P E C T I y E. rc3 
 
 and, confequently, S, is the vanifhing point of the top and bottom. 
 Draw F, cutting the fide of the room in m ; raife a perpendicular 
 
 m, n j draw O, /, cutting w, in n j then draw S, cutting 
 
 the angle of the room in p ; and join /, y, which finifhes the Iha- 
 dow. Or y, might have been firft found thus: Draw O, S, and, 
 in that line, find by a perpendicular from the interfe 61 :ion of F, S, 
 with the bottom of the room, which point is the feat of O, on 
 the plane p, y, that is, the farther fide of the room j and drawing 
 /, y, py that will be the diredlion of the fhadow, on this plane j and 
 draw S, and O, /, meeting in n. 
 
 The two fquare blocks, A, and B, againft the wall, are introduced 
 to fliew the courfe of the fliadows, which need little explanation j only 
 it ia to be obferved, that being the foot of the light on the wall, is 
 
 to be ufed, on that plane, as F, on the floor. Thus draw from /, 
 
 through all the points which touch the wall, to the beginning of the 
 cieiing, and through thofe interfedlions (with the edge, or angle of the 
 deling) draw from S ; then from O, draw through the projeding 
 angles, of each block, lines meeting all thofe rays from S, and the 
 feveral points of the fliadow, on the cieiing, are terminated. 
 
 The fhadow of the hollow fquare C, is found in the fame manner, 
 only remembering, that as t\ reprefents the foot (of the light) on the 
 wall, and as the depth of the hollow is to be regarded in calling the 
 fnadow, fo the foot mull be as low as that depth, which is found by 
 drawing from the top of one of the lines, viz. i , to /, and a parallel 
 to it from the bottom of the fame line 2, cutting O, S, in V, which 
 will be the foot of the light for this hollow j then drav/ing V, 2, and 
 O, I, meeting in 3, draw from 3, a line parallel to i, 2;, which will 
 give the indefinite fliadow of i, in the bottom; and draw from 
 I, meeting that line where it interfecls the angle of the upper fide.; 
 and one more line from O, through 2;, cutting the parallel from 
 3, which gives the determinate fliadow of and completes 
 the whole. Or the fliadow of the line i, (on the upper 
 plane of the fquare hollow) may be found, by means of the foot of 
 the light on that plane, thus. Continue the line from j, parallel to 
 
 ther 
 
S 04 7%^ P R A C T I C E 
 
 the horizontal line, and continue the perpendicular from /, till that 
 cuts it; then draw a line from S, through that interfedlion, and a 
 perpendicular from O, cutting it; and from this lafl interfedlion (which 
 is the foot of the light) through i, draw the fhadow fought, and a line 
 from O, through Zy determines the length of it. 
 
 Then, for the lhadow of the ladder, draw fi*om its vanilhing point 
 4, through O, and from S, through F, meeting in G; draw from 
 G, through each foot of the ladder, to the interfedlion of the floor 
 with the farther fide of the room continued, cutting it in 5, and 6; 
 then, where G, F, S, cuts the fame line of interfedlion (as at 7,) 
 laife a perpendicular, cutting G, O, 4, in A, and draw A, 5, and 
 A, 6, which lines will cut the top of the ladder, and (having before 
 drawn G, 5, and G, 6,) by thefe means the whole fhadow is found 
 from G, to 5, and 6, on the floor, and* from A, to 5, and 6, on 
 the wall, fuppofmg no other objedl to intervene. 
 
 For G, O, 4, is the line in which all the plane of rays mufl: 
 pafs to the two fldes of the ladder G, 4, reprefenting a line parallel 
 to them, pafTing through the light O, and touching the ground in 
 G, and the wall in A. 
 
 But as G, 5, and G, 6, meet the other fide of the room in 9, and 10, 
 draw from thefe lafl: points to 4, which will determine that part of the 
 fhadow, and would meet A, 6, and A, 5, in their interfe6lions with 
 the angle of the room, if there were no door, or if the door were fhut. 
 Again, as the door will alfo receive part of the fliadow, draw from 
 II, and 12, (where G, 5, and G, 6, meet a, H, the interfedtion of 
 the plane of the door) to 13, and 14, where A, 6, and A, 5, interfeft 
 the edge of the door, which completes the whole fhadow. Or the 
 vanifhing points of thefe two lines ii, 13, and 12, 14, may be found, 
 by raifing a perpendicular at which perpendicular will be the vanifh- 
 ing line of the plane of the door; and from F, and G, (both in the 
 horizontal line) drawing F, 4, and G, 4, cutting that vanifhing line, 
 in their refpedlive vaniihing points J', and g ; and then drawing 
 14, and g, 13, is found the fliadow on the door. 
 
 N. B. Fhis IS farther explained in the Supplement. 
 
 The 
 
<?/ P E R S P E C T I V E. 105 
 
 The fhadow of the table on the ground is determined, by drawing 
 lines from F, the foot of the light, through each angle of the table, 
 on the ground j and then drawing from the light, itfelf, through all 
 the upper angles, meeting thofe lines from F, refpe6tively. And for 
 that part of the fhadow on the door, continue the line of the bot¬ 
 tom of the door w, till it meets F, as in y; at raife a per¬ 
 pendicular, cutting O, jic, in L; draw L, S, which determines the 
 fhadow on the door, and completes the whok. 
 
 The fliadows of the windows are found, in a like manner; for in- 
 flance, the window E. Find the foot of the light on the plane E, 
 which is to receive the fhadow, thus: Draw a parallel from F, cut¬ 
 ting the fide of the room, 15, continuing the line F, 15, and find, 
 there, the thicknefs or depth of the window 15, 16 j draw O, 17, 
 parallel to F, 16, and 16, 17, parallel to F, O ■, then 17, will be the 
 foot of the light fought: draw 17, 18, and O, 19, cutting it in 20; 
 from 20, raife a perpendicular, cutting the inner lines of the window, 
 which determines the fliadowj and fo of the reft.—For the round 
 window, the like method is ufed, the fame point 17, being the foot 
 of the light, for that whole plane j take tv/o or three lines from the 
 outer, to the inner circle, parallels to O, 17, and draw from 17, 
 through the bottoms and from O, through the tops, and their feveral 
 interfedlions will give the points of the fhadows, &c. on the inner 
 plane. 
 
 N. B. Fig. 72, No. 3, anJ 4, in the Supplement, is the 
 continuation^ and conclufion of what relates to fadows. 
 
 Of the images or refleBions of obje&s m refleBmg planes. 
 
 73* T ^ fcheme are reprefented the reflections of feveral ob- 
 ^ jeCts, in the water. Every objeCt is fecn as far, or deep 
 within the reflecting plane, as it is placed without it. Thus, to find 
 the reflexion of the pile A, part of which is in the water, and the 
 refl: above the furface of it, meafure from the top to that furface, 
 and fet off the fame meafure downwards to a. 
 
 P 
 
 But 
 
the PRACTICE 
 
 io6 
 
 Bat for the pile B, which inclines, the perpendicular height from the 
 furface of the water muft be taken. If it inclined fo, as to continue 
 ftill parallel to the plane of the picture, then a perpendicular from 
 the top would touch the water at 2, in which cafe, B, .2, would be 
 the meafure to be taken downwards, and then the reflexion would be 
 equal to the reprefentation of the objedt; and if it inclined inwards, fo 
 as that the top was perpendicularly over i, then B, i, would be that 
 meafure (fliorter than the reprefentation of the obje6t); if forwards, 
 fo as to be perpendicularly over 3, then B, 3, is the meafure, which is 
 the meafure here taken, and 3, 3, is the perpendicular depth of the 
 reflection, which therefore makes the reflection longer than the repre¬ 
 fentation of the object. 
 
 The principal difficulty, in molt reflections, is to find the feat of 
 the objeCt on the furface of the reflecting plane; for when that is 
 found, the reflection is made, by repeating that diftance either geo¬ 
 metrically, or perfpeCtively, as the cafe may require, in, or on the re¬ 
 flecting plane: the block C, hangs over the water, and projects from 
 the wharf, touching the edge of it, at the point 4; therefore meafure 
 from 4, to the furface of the water 5 ; then draw from S, (the va- 
 nifliing point of the four fides of the block) through 5, and drop a 
 perpendicular from its extreme angle, cutting S, 5, in 6, which is its 
 feat on the furface of the water ; and this meafure repeated downwards, 
 finds the reflection of that angle, by means of which the reft is eafily 
 determined. Or having firft found the reflection of the edge of the wharf, 
 by repeating the perpendicular 4, 5, downwards, to 7, draw a line 
 from S, through 7, and a perpendicular, from the corner of the block, 
 cutting that line, will find the fame corner, or angle fought. The re¬ 
 flection C, is fimilar to its original, and all the rays run to S, as in 
 the original. 
 
 The reflection of the block E, is found in the fame manner; the 
 vanifning point of E, is F, but the vanifhing point of its tranverfe 
 fides is beyond the picture, to which the correfponding fides of the 
 reflection, or image, js dravv^n, as well as thofe of the block itfelf: the 
 piece E, hangs over the piece C, but makes fo fmall an angle (with a 
 
 per- 
 
<?/ P E R S P E C T I V E. 107 
 
 perpendicular from its interfedion e,) that it becomes neceflary to ufe 
 an expedient; therefore draw (from e,) a line parallel to D, S, as e,f, 
 and another from the edge, or corner, of the block, to draw 
 Fj/j S* drop perpendiculars from f, and g ; then meafure from 
 y', downwards, to the furface of the water b, and repeat that meafure 
 to /; draw F, /, cutting the perpendicular from g, in ky and transfer 
 i, ky backwards, by parallel lines, perpendicularly under E, which will 
 determine the refledtion I, K, by which the reft is completed j (/. e.) 
 .dropping perpendiculars from ey and the corner of E, the parallels 
 from /, and ky will meet them in I, and K, and the line K, I, will 
 terminate in F, the vanifhing point of its original. 
 
 The refledlion of G, is found on the fame principles. In order to 
 find its feat, draw from its lower angle a line parallel to D, S, and a 
 ray from ii, the angle of the wharf, to S, cutting that parallel in 8 ; 
 drop a perpendicular from 8, and cut it in 9, by another ray from the 
 furface of the water, under 11; drop a perpendicular from the lower 
 angle of G, and draw 9, 10, parallel alfo to S, D, cutting that per¬ 
 pendicular in 10, which will be the furface of the water; meafure 
 therefore from the top of G, to lo, and repeat that to 12, which will 
 be the reflexion fought. 
 
 Here is introduced a row of trees, to fhew in what manner they are 
 to be refledted, and how many of them would be feen. Continue the 
 ray, or line, on which they ftand, forwards to the margin, or edge, 
 of the wharf; from thence take the diftance to the furface of the 
 water, and repeat it downw'ards by a perpendicular; and, from the 
 lower extremity of that perpendicular, draw a line to S, and then to that 
 line drop a perpendicular from the firft tree ; and from this laft 
 point of interfedlion, meafure the tree downwards, and mark the top, 
 and ftem, of that inverted tree; thence draw lines to S, which 
 lines will receive perpendiculars from each tree above, whofe inter- 
 fedlions will give the places of the refledted trees, refpedfively. 
 
 And it appears that no more than parts of the three forwardeft 
 trees will be vifible, in the water, by refledtion. 
 
 P 2 
 
 Here, 
 
io8 7^^ P R A C T I C E 
 
 Here, and every where, fuch objedls, and fuch aflemblage of obj^ 
 jedls, only are reprefented, as feem beft calculated to exhibit 
 examples for the inftrudlion intended, and not any regular 
 figures, or agreeable pidlures. 
 
 l^ig. 74. No. I. The reflection, or image, of the figure A, fuppofed to ftand 
 before a looking-glafs, is found on the fame principles as refledtions 
 in the water, with this difference, in the procefs, that the feat on the 
 refledting plane, and the image beyond, or within it, are both per- 
 fpcBhely found in this and every vertical fituation, except when the 
 object, and its refledted image, are in a line parallel to the pidture-; 
 whereas in the horizontal, (i. e.) in water, they are only, and always 
 found geometrically j therefore, having found the diftance from A, 
 the foot of the figure, to the plane of the looking-glafs, on the 
 vifual ray A, S, repeat the fame diftance, on the fame ray, onwards, 
 that is, within the glafs, to a, which will determine the foot of the 
 image j there raife a perpendicular, and draw another ray from the 
 head of the figure to S, cutting that perpendicular, which interfedticn 
 determJnes the height of the image j' and in like manner any points of 
 the original figure may be found, by drawing rays to S, from fuch 
 points through correfponding perpendiculars. 
 
 And in the fame manner is found the image of the figure B, in 
 the glafs, before it; but this figure is again refledled in the glafs, be¬ 
 hind it (which is over the chimney,) by geometrical meafures, taking 
 firfl the diflance to the plane of the glafs from B, to /», and again 
 the fame diflance to b, within the glafs j which is an operation fo fully 
 explained in the preceding figures (refiedled in water) as to need no¬ 
 thing more here : it will be obferved, that, in this latter cafe, the 
 image is always equal to its original, whatever be the diftance; where¬ 
 as, when perfp^dfive meafures are neceftary, the images become lefs, in 
 proportion to their diftance. 
 
 As thefe reprcfentations are made purely for inftruiftion, it was ne- 
 cefi’ary to place the figures where the images could be refiedled without 
 grouping, or giving them any relation to each other; and for tbse 
 fame reafon here are two of the glafles inclining forwards, in order to 
 % ftiew 
 

r 
 
r/ P E R S P E C T I V E. 109 
 
 fhew the manner of finding reflexions, in fuch fituation j they both 
 incline in an angle, of 18 degrees, -with the fide of the room; as that 
 oppofite to the figure F, whofe image is found by the fame general 
 law, but the particular circumftances require explanation ; and firfi: 
 the vanifhing line of the plane of the glafs runs through E, below, 
 which is the vanifhing point of the fides of it, and the bottom touches 
 the wainfcot in the points c, and d. Wherefore, having drawn E, r, 
 and E, and dropped a perpendicular from r, or d, to the floor, 
 as here from r, to 5, draw S, 5, cutting E, r, in 6, through which 
 a parallel to S, D, (as 6 ,fd) will be the bottom of the glafs on the floor 
 (fuppofed beyond the wainfcot). Now drawing F, S, cutting 6, yi in 
 f\ and drawing E,y', this line will give the indefinite feat of the axis of 
 the figure on the glafs, and G, being the vanifhing point of perpen¬ 
 diculars to the plane of the glafs, draw F, G, cutting E,yi in g, 
 which is the feat of the foot of the figure; wherefore double, or re¬ 
 peat F, g, within the glafs perfpeXively (as has been frequently taught) 
 to /6, which will be the foot of the image, or refleXion in the glafs, 
 and alfo drawing from the head of the figure to G, cutting E, 
 in 7, that will be the feat of the head; this diflance from the head to 7, 
 being repeated perfpeXively, finds the head of the image, which is now 
 eafily completed, as the others. 
 
 The axis of the image might be found, by continuing the axis of 
 the original figure, perpendicularly from F, upwards, till it meets the 
 axis of the feat E, /i 7, in H, and drawing H, I, which will be 
 the axis of the image ; for this point I, is the vanifhing point of the 
 image, found, by making E, D, I, an angle of 18 degrees. 
 
 If the reader finds any difficulty to conceive this operation, or the 
 reafon of it, he is referred to the fcheme below, 
 
 ‘Fig. 74. No. 2. which is a geometrical reprefentation of it, with the fame 
 charaXers. F, 0, H, is the axis of the original figure; /, g, 7, H, the 
 feXion of the glafs with the fame inclination as in the piXure; g, 7, 
 is therefore the feat on the glafs, and /j, 0, the image, which laft meets 
 F, 0, in H, as above. F,/, in the geometrical fcheme, is the floor ; /, /&•, 
 the fame refleXed (the angle of refleXion being equal to that of in¬ 
 cidence) 
 
no 
 
 I'he PRACTICE 
 
 cidence) y and hence appears the reafon of drawing D, I, to find the 
 vanifhing point of the image, for this line reprefents H, h, in the geo¬ 
 metrical, as D, L, perpendicular to D, I, reprefents f, h, in the geo¬ 
 metrical 3 H, being alfo a right angle, and therefore L, is the va- 
 nilhing point of all the radials in the refiedled fioor. It may be worth 
 while to examine the correfpondence of thefe lines in the geometri¬ 
 cal fcheme, and the perfpeftive. The floor refledled in the glafs, 
 is reprefented at ii, 10, above, which are the images of 11, and 
 1C, below, on the firft line of the floor, found by drawing lines from 
 the originals to G, (the vanifiiing point of perpendiculars to the glafs,) 
 and cutting them by other lines from the interfedlion of the glafs with 
 the floor 5 as from P, to L, the vanifhing point of the radials on the 
 refledted floor, and (therefore) of perpendiculars to the image. Since 
 therefore the refledled image of 11, muft be in each of thefe lines, it 
 , muft be in their common interfeclion. , Here are three fets, or pairs of 
 lines, perpendicular to each other ; the firfl: pair are D, S, the common 
 diftance, and geometrical perpendiculars to it, for the reprefentatioii 
 of the room, &c. the fecond pair D, E, and D, G, for the reprefenta- 
 tion of the glafs, and the feats of objedls on it j and the third pair 
 D, I, and D, L, for the reprefentation of the image, or refledlion, of 
 the man, and of the floor, at right angles to the man. 
 
 The image of the objedl; K, in the glafs M, is found by lines drawn 
 from each point, geometrically perpendicular to the plane of the look- 
 ing-glafs, (which is inclining in the fame angle from the wall, as the 
 laft, viz. 18 degrees); thefe points, on the furface of the glafs, are feats 
 of the originals, from each of which, an equal diftance is taken with¬ 
 in, or beyond the furface, which laft fet of points being joined by 
 right lines, become the image fought. The feveral feats on this glafs are 
 found, by drawing lines, from each original point on the floor, parallel 
 Ao the horizontal line, cutting the fedlion of the glafs on the floor, and 
 thence drawing a parallel to the fide of the glafs, and drawing a per¬ 
 pendicular to that parallel from the original point. And for any ori¬ 
 ginal point above the floor, find its feat firft on the floor, and proceed 
 as if that was the point, and then draw a perpendicular from the real 
 
 original 
 
£/* PERSPECTIVE. lii 
 
 original point above, to the parallel on the glafs, as before directed. 
 Or as at 
 
 Fig. 74. No. 3. Firft fuppofe the glafs to be clofe to the wall, (/. e.) to 
 coincide with the line f, P, and confequently perpendicular to the floor, 
 both before and behind. 
 
 Then (as the glafs inclines forwards) the floor, behind, rifes in pro¬ 
 portion, (/. e.) to the pricked lineyi O, (18 degrees) ; yet ftill 90 de¬ 
 grees will be left, between that line, and the back of the glafs, from 
 which there miift now be taken 18 more by the line/i to reduce the 
 angle, behind, to 72, equal to that before, which has lofl 18, by its in¬ 
 clination forwards. 
 
 And this is the reafon of the two angles of 18, behind the glafs, 
 between yi N, and g. 
 
 Draw A, B, and its parallels, (from the interfedlioris of b. A, and 
 its parallels) all parallel to f, g. 
 
 Transfer the feveral divifions of the line e, /i on g, and from 
 thefe lafl, draw to S, crofling all the parallels of A, B, which finiflies 
 the refledfion of the floor, in the glafs. 
 
 Every particular has been fo repeatedly explained, in relation to the 
 former objedls, that nothing need be here added. If any poflible dif¬ 
 ficulty arife, a careful infpection will remove it.—The reader mull ob- 
 ferve, that though the glafs receives but part of the image, yet the 
 whole is defcribed by pricked lines, that the procefs may be entirely 
 comprehended. 
 
 The feveral numerical figures of the image correfpond with thofe 
 of the originals, refpedlively. 
 
 N. B. I, 3,—2, 4, at the bottom, and top of the image. No. i. 
 run to the fame vanifhing point S, as 1,3, and 2, 4, &c. of 
 the original, with which they correfpond j and i, 2,—3, 4, 
 &c. are, alfo, parallel to each other, as in the original, being 
 parallel to the pidlure. 
 
 The floor is chequered on purpofe, to give occafion for fliewing its 
 image, or reflection in the two inclined glafies; and in that marked, 
 M, there is alfo the image of fo much of the window neareft to it, 
 
 as 
 
I'l 2 
 
 The P R A C 'r I C E 
 
 as could be feen, by refiedlion, in it j as alfo in the oppofite glafs over 
 the chimney is the refiedlion of part of the neareft perpendicular glafs, 
 and of the window, and, if the chimney glafs had been wider, the 
 very image of B, in the glafs before it, would have been again refiedled 
 in that of the chimney, where the pricked touches are made, that being 
 the place in which the diftance from the image, to the fide of the 
 room, is doubled, fuppofing that fide continued on, as far (back¬ 
 wards) as the image is cad. 
 
 Though very few defigners will, perhaps, take the pains to projedf, 
 by rule, every reflected objedf; and though it feldom happens that 
 very intricate difpohtions of fuch objedls occur, yet it may not be ufe- 
 lefs to add two or three cafes, in which the principles herein explain¬ 
 ed, and the methods founded on them, will appear peculiarly advan¬ 
 tageous : thofe who are curious in fpeculation, and thofe who defire 
 to be correct in the execution, will be gratified; and all will be better 
 able to judge of what they fee, as well as the practitioners will be 
 better able to perform, (even though it be by guefs) after knowing, and 
 confidering the rules, than without fuch knowledge, and confideration. 
 
 Fig. 75. No. I. P, R, T, is fuppofed to be a looking-glafs ftand- 
 ing, perpendicularly, on the horizontal plane; A, B, E, F, an objeCl to 
 be reflected, which is defignedly made plain, and Ample, to avoid con- 
 fufion of lines; draw A, a,—E, e,—B, b, and F, f, all parallel to 
 the horizon, (which lines will be perpendicular to the plane of the glafs,) 
 and in thefe lines will be found both the feat of the objeCt, on the 
 glafs, -and alfo its reflection, in it. S, is the vanifhing line of the 
 glafs, and C, of the objeCt to be reflected; but as C, (the vanilbing 
 point of A, B, and E. F,) is above the horizon, continue ;;;, C, till 
 it cuts the horizontal line in r, and draw E, cutting P, Q, S, in 
 11, which will be the point where the two planes, of the glafs and ob - 
 •jeCt, meet on the ground; and as their vaniflring lines meet, above, 
 in w,—draw w, «, which will be their common interfeCtion, and of 
 which, m, is the vanifliing point. 
 
 Draw C, L, perpendicular to ni, S, the vanifliing line of the glafs, 
 .cutting it in K, then K, will be the vanifliing point of the feats of 
 
 A, B. 
 
■ TTTV' 
 

 
0/ P E R S P E C T I V E. 113 
 
 A, B, and E, F, on the plane of the glafs; for C, is their original va- 
 nifhing point, and K, is perpendicular to it, on the vanifliing line of 
 the glafs j and w, being the interfedtion of thefe two planes, and 
 
 A, B, C, cutting that interfeftion in 0, draw K, <?, cutting A, a, in 
 
 I, and B, b, in 2, then i, 2, K, will be the feat of A, B, C j and 
 
 as E, F, C, alfo cuts the fame line /«, «, in V, draw K, V, cutting 
 
 E, e, in 3, and F, f, in 4; then 3, 4, K, will be the feat of E, F, C, 
 and drawing 1, 3, and 2, 4, the whole feat is completed j and doub¬ 
 ling the four parallels A, i, to a; B, 2, to b; E, 3, to e; and F, 4, 
 to f; the image is completed, by joining a, b, e, f. 
 
 Or, fince the image or refle< 5 tion is juft as far behind the feat, as 
 the original is before it, in C, K, L, make K, L, equal to C, K, and 
 then L, will be the vanifhing point of the image; wherefore draw 
 L, c, cutting A, a, in a, and B, b, in b, and draw alfo L, V, cutting 
 E, e, in e, and F, f, in f, and joining a, e, and b, f, the image w 
 completed; and by this method it may be found, even without the 
 trouble of firft finding the feat. 
 
 Or, inftead of drawing four parallels, two will fuffice. A, a, and 
 
 B, b; and having drawn L, 0, cutting thofe two parallels in b, and 
 
 a, and continued the fides of the original A, E, and B, F, till they 
 
 cut the line w, in y», and y, draw y, a, and />> h, cutting L, V, in 
 
 ‘ e, and f, this will complete the image alfo, without finding the feat. 
 
 Fig. 75- No. 2. Proceed as was fhewn at large in the former figure, 
 which is the univerfal method recommended. The fame letters, and 
 numerical figures, are ufed in this, as in that, to fliew the correfpon- 
 dence. The only difference is, that, as this glafs does not ftand per¬ 
 pendicularly on the horizontal plane, fo the parallels A, a,—B, b, 
 &c. are not parallel to the horizon, but are here^ as they muft be, 
 always, perpendicular to the reflefting plane. 
 
 Fig* 75 - No. 3. In this fcherae is the fame general method ufed, as in 
 the two preceding; however, that nothing may be left unexplained, it 
 is to be obferved, that the glafs here is oblique, not (as the laft) on 
 the horizontal plane, but above it, whofe vanilhing line is m, K, r, and 
 the vaniftiing point of the fides R, T, and P, Q, being K, the lines 
 
 Q per^ 
 
He PRACTICE 
 
 114 
 
 perpendicular to this plane, are drawn peifpeflively (and not geome¬ 
 trically, as in the two former); that is, the vanifhing point W, of lines 
 perpendicular to it, is found, by the rules heretofore taught, and 
 A, W,—”B, W,'—-E, W, and F, W, drawn, reprefenting perpendicu¬ 
 lars, and confequently K, L, is made perfpeftively (not geometrically) 
 equal to K, S : after which the image of the objefl is found by means 
 of L, the vanifhing point of its fidesi L, being its vaniflring line^ and 
 the feat (though not necelTary) is found (as before) to fhew the confor¬ 
 mity of the lines, and of the procefs, with the preceding fchemes. 
 
 N. B. As the fame letters fiiand for the fame points, it is needlefs 
 to enter into the explanation over again, except that here, it 
 became neceflary to vary two or three j as K, for inllance, at the 
 fame time that it is the vanifhing point of the fides of the 
 glafs, is alfo the vanifhing point of the feats of A, B, and E, F,. 
 and that the vanifhing point of the fame original lines A, B, 
 and E, F, which was marked C, in the former fchemes, is 
 here marked S, becaufe it coincides with the center of the 
 picture, which is always diflinguifiied by the fame letter, &c. 
 n, and V, alfo coincide in this fchemej S, D, is the diftance 
 of the pidliire; and the point L, is found by drawing S, 5, 6 , 
 parallel to W, D, drawing D, K, cutting it in 5, making 
 5, 6, equal to S, 5, and then drawing D, 6, catting S, K, L> 
 in L. 
 
 . 75. No. 4. This laft fcheme is ftill by the fame method j but that 
 no difficulty might be avoided, the center of the pidfure is not the 
 vanifhing point of either the objedt, or glafs, both which are placed 
 obliquely, the one above, the other below the horizontal linej and as 
 E, is the only point of the objedt that touches the ground (E, C, and 
 confequently F, being under it) E, r, is drawn to the interfedlion of 
 ks vanifhing line with the horizontal line, and alfo P, being the only 
 point of the glafs that touches the ground, P, r®, is drawn to the 
 mterfedlion of its vanifhing line with the horizontal line, and E, r,, 
 cutting P, rh finds ji, through which, from my (the interfedlion of 
 two vanifiiing lines) viz. of the glafs, and objedf, is drawn n, the 
 
 mter- 
 

 w 
 
 
 ■ii' 
 
 ■*<< 
 
 *: . : 
 
 i 
 
 P'*"' 
 
 •f «S(f: i '-V; ■ 
 
 • ' t’■ it, '.v^ 
 
 
 1 '4 
 
 u ^ 
 
 : J- . 
 
 '^.v. 
 
 ' -■ s 
 
 s Tk 
 
 ■ . . - 
 
 ■^Wsui '-IM 
 
 -Ji 
 
 
 
 V* 
 
 
 Tg? 
 
 ■4f4i|^ __ _ -| 
 
 i ■;>;> ' ■•? ■( 
 
 ■% 
 
 . v'w ■ -m-.f 
 
interfe6lion of their two planes. The reft is all as the former, only 
 one letter (viz. Y,) is added here j for S, ferved in all the formef 
 cafes as the vanifhing point, either of the glafs, or of the obje 61 :, one 
 of which coincided with the center of the pidfure, but Y, is (in this 
 fcheme) the vanifhing point of R, T, and P, Q, and Y, S, W, is the 
 vanifhing line of a plane perpendicular to Y, (the vanifhing line of 
 the glafs) on which Y, D, W, is made a right angle to find W, the 
 vanifhing point of perpendiculars. 
 
 The four laft fchemes, Fig. 75. No. i, 2, 3, and 4, are defigned to 
 explain, on the principles of Brook T'aylor^ the method of finding re- 
 fle6ted objects in mirrors, and have a more particular reference to the 
 iaft fcheme in his book, where he reprefents the image of a pidiure as 
 refiedled in a glafs ftanding obliquely on a table. They are exhibited 
 Vv^ith all poffible fimplicity, and without any ornament, that fo no lines 
 may enter into the diagrams, but fuch as are abfolutely neceffary to 
 the projedtion of the image propoled. 
 
 This is one of the parts of that work, which is mentioned (in the 
 Preface of this Treatife) as attended with difficulty. No. 3, is nearly 
 Brook Traylor own example, but with all the neceffary lines; No. i, and 
 2, are preparatory, and explanatory of the principles; and No. 4. a 
 cafe ftill more difficult, but all on the fame principles. 
 
 CONCLUSION. 
 
 K E author has, in this work, endeavoured to exprefs himfelf 
 A with all the perfpicuity that the nature of the fubjedl will ad¬ 
 mit, and has been lefs folicitous to avoid repetition, than to avoid ob- 
 fcurity. That over fcrupulous exadnefs, which permits not to re¬ 
 peat an inftrudtion (once delivered) though at the diftance of many 
 pages, makes references backwards continually neceffary, and not 
 only perplexes and wearies the reader, but difgufts him more than, 
 now and then, a feafonable repetition; and the getting by heart a 
 great number of definitions, before their life can be known, efpecially 
 when moft of them will afterwards appear to be unneceflary, is burthen- 
 
 Q 2 fome 
 
Ii6 P R A C T I C E ^c. 
 
 fometo the memory, and tedious even to patience itfelfj yet everyone 
 of thefe muft be diftindlly remembered, or the reader muft be con¬ 
 tinually turning back to analyze them, which interrupts him beyond 
 meafure. If, on the contrary, it were thought fufficient to call a 
 fhadow, a lhadow, to call the ground, the ground, and to give the 
 common names to common things, and to treat this fubje^l in a more 
 familiar way, it might, undoubtedly, be thereby more accommodated 
 to the apprehenfions of the generality of thofe, whofe profeffions re¬ 
 quire a knowledge of perfpedlive. And this is what the author has 
 endeavoured to execute. 
 
 He is far from faying, or thinking (as the Jefuit in his Preface) 
 “ That perfpe6tive is the very foul of painting,' and which, alone, can 
 “ make the painter a mafterf’ or as fome others, who may have fet 
 it too high among the requilites, in forming a painter fince many 
 very great mafters have been deficient in it, fome egregioufly, who 
 have, notwithfianding, pofiefled the other, and more excellent parts 
 in a high degree; as invention, compofition, expreflion, corredlnefs 
 of defign, and colouring; which will produce finepidlures, though the 
 perfpedtive be, in fome refpedfs, faulty, and much finer, than any, in 
 which the perfpedlive may be abfolutely true, and thefe other parts 
 but in a low degree. 
 
 It is certain, however, that perfpedfive is an efiential, and that what¬ 
 ever is erroneous in this refpedl, docs not truly reprefent the thing in¬ 
 tended ; that it is abfolutely neceffary to the perfedlion of painting j 
 and that fome fubjedls, particularly architedlure, cannot be reprefented 
 without it. It is alfo certain that a man will invent, and cqmpofe 
 with more facility, and precifion, who underfiands it well, than he who 
 iindei ftands it but imperfedlly, fuppofing other qualifications equal j. 
 that great errors in it are monftrous, and ftiocking, and that a total 
 ignorance of it is unpardonable in a painter, or defigner. 
 
 end. 
 
Plate'XUn. 
 
1 
 
 Y't V.: 
 
 I 
 
 A 
 
 I 
 
 
 / 
 
 J 
 
 % 
 
THE 
 
 SUPPLEMENT. 
 
THE 
 
 SUPPLEMENT; 
 
 Added to illuflrate and explain fome of the more difficult 
 parts of the foregoing Treatife, which, in their feveral places, 
 were neceffiarily complicated ; but are here feparated, in 
 order to their being iingly, and diftindly confidered. For 
 this piirpofe it has been thought moft convenient to repeat 
 the fame figures and numbers as in the body of the work, 
 where the fame fubjeds are treated (with the addition, only, 
 of capital letters) that reference may eafily be had to fuch 
 places. 
 
 AND FIRST, 
 
 T H E reader is referred back, from this place, to Fig. 71. No. lo 
 where he will find, that the vanifiiing point O, of the poll: 
 II, 12, is at a confiderable difiance below, among other objeds, 
 and (on that account) not fo readily diftinguifhed j and, alfo, that the 
 vanifiiing point of its fiiadow is beyond the limits of the plate. For 
 thefe reafons it has been thought proper to repeat this diagram by itfelf,, 
 and in a narrower compafs, that all the points may be feen at once, and 
 fo their relation more difiindly appear, efpecially as this is a matter 
 of fome difficulty, and of great ufe. 
 
 Fig. 71. No. 1. A.— 11, 12, is aline fiandlng obliquely on the horizontal 
 plane, whofe feat is 13, 12, and the vanifiiing point of that feat is 
 S.—U, is the given vanifiiing point of the fun’s rays. Firfi: find the 
 vanifiiing point of 11, 12, by dropping a perpendicular from S, and 
 continuing ii, 12, till it cuts that perpendicular in O, which will be 
 its vaniffiing point; then draw from O, through U, to the horizon¬ 
 tal line, cutting it in X, which will be the vanifiiing point of the 
 fliadbw; then draw ii, U, and 12, X, cutting it in 14; then 12, 14, 
 is the fhadow fought. For U, V, cutting the horizontal line (perpendi¬ 
 cularly over U,) in V, this will be the vanifiiing point of the jfiiadow 
 
 of 
 
E 20 
 
 The SUPPLEMENT. 
 
 oF any line {landing perpendicularly on the ground. Now if fuch a 
 perpendicular line ii, 13, be drawn, cutting the feat in 13, the (ha- 
 dow of that perpendicular will be 13, 14, and 14, will be (in that 
 cafe, alfo,) the fhadcrw of the point ii ; therefore 12, 14, mufl be 
 the fliadow of the whole line ii, 12, which is a proof, that the 
 6rft operation was true. 
 
 All the references are the fame as in the large fcheme, and there is 
 no difference in any circumflance, except that this poll leans forwards 
 in an angle of 58, and the former in 65; which change was made, only 
 to avoid the too great diflance of the vanin:iing points O, and X. 
 
 So that if the text, relating to the former, be read with this fcheme, 
 it will anfwer throughout. 
 
 And this will be general, for any line, viz. to draw from its va- 
 nifliing point, through the vaniihing point of the ray of light, to 
 the vaniming line of the plane on which the fnadow is to be projected, 
 wdicther it be the horizontal plane, or any other; and this interfedlion, 
 with the vanifliing line of the plane on wdiich the fhadow is caif, will 
 be the vanifliing point of the fliadow. For, (in this fcheme,) imagine 
 the plane D, S, O, railed on S, O, till D, S, be perpendicular to 
 thepidlure; then a line from D, to U, determines the vanifliing point 
 of the rays; and, confequently, a line through O, and U, to the 
 vanifliing line of the horizon, will give the vanifliing line of the 
 plane of rays pafling over the whole line O, 12, 11, and therefore, 
 alfo, the vanifliing point of the fliadow 12, 14. 
 
 Fig. yj. No. 3. E. Is an example of the fame kind on an oblique plane. 
 Here C, q, is the vanifliing line of fuch plane; Q, P, a vanifning line 
 of planes perpendicular to it; ^7, B, a line (landing perpendiculaily on 
 the plane C, y;—B, its feat, on that plane, and P, its vanifhing point; 
 U, the given vanifliing point of the rays of the fun, and V, the vanifli¬ 
 ing point of the fliadow, found as X, in the lafl; that is, by drawing 
 from P, the vanifliing point of the line a, B, through U, the vanifli¬ 
 ing point of the rays, to the vanifhing line of the plane on which the 
 fhadow is cafl; draw U, and B, V, cutting it in r, then e, is the 
 fliadow of and B, c, of B, a., on the plane C, q. 
 
 And 
 
121 
 
 The SUPPLEMENT. 
 
 And if any other line, as be given, on the fame plane C, y, 
 ftanding obliquely on it (yet being parallel to the plane Q, P,) con¬ 
 tinue that line by to its vanifhing point Q, and draw Q, U, cutting 
 C, y, in y, then y, will be the vanifhing point of its fhadow on the 
 plane C, y. Wherefore, 
 
 Draw by y, cutting the ray U, in and by ey will be its fhadow. 
 
 If dy B, were continued to gy then gy by would be the fhadow of 
 
 And if dy by were continued to fy (or any length) the fame ope¬ 
 ration finds the fhadow; thus fy by is the fhadow of a, fy &c. 
 
 Fig. 71. No. I. F. Here is one more example for the fhadows of ob¬ 
 lique lines. Thefe incline inwards, and therefore have their vanifhing 
 points above the horizontal line. 
 
 O, H, the horizontal line; Y, the vanifhing point of the four lines 
 A, B ;■—U, the vanilliing point of the rays of light, and confequently 
 U, Y, is the vanifhing line of the rays which pafs over thefe four 
 lines; and W, (being the point, in that vanifhing line, which cuts 
 the horizontal line) is the vanilliing point of their fhadows on the ho¬ 
 rizontal plane, (/. e.) on the ground. The line A, L,'being in a diffe¬ 
 rent diredtion, has another vanifhing point, viz. jy, and therefore the 
 rays paffing over it will produce another vanifhing line, as U, which 
 cutting the horizontal line in Wy that becomes the vanifhing point of 
 the fhadow on the ground of this line A, L. 
 
 The reft needs no explanation, only as H, is the perpendicular feat of 
 Y, on the horizontal line, and by ofy; if the perpendicular feats of A, are 
 found, then, by means of thefe feats, the fame fhadows might be de¬ 
 termined, as in the cafe of perpendicular lines; for the fhadows of the 
 point A, would exadtly coincide with thefe, here, found, and then the 
 point Oy perpendicular to U, muft be ufed as the vanifhing point of 
 thefe fhadows on the ground. 
 
 The feat of any of the points A, is found by drawing a line 
 from B, to H, then dropping a perpendicular from A, to the line B, H, 
 which line is the feat of the line A, B, on the ground, as a^ A, i, 
 and at A, 2, the feat is a ; and then drawing Oy and A, U, inter- 
 
 R fefting 
 
122 
 
 7%e SUPPLEMENT. 
 
 feeling it in a, this point is the fhadow of A, which coincides with that 
 already founds and is a proof, that the method, propofed, is univer- 
 fally true. 
 
 N. B. The center and diftance are not given, being no way 
 concerned in this diagram, for wherever the center is placed, 
 in the horizontal line, or whatever be the diftance, all the 
 feveral relations of thefe lines remain the fame. 
 
 It is alfo evident, that two lines, only, are neceflary to the determi¬ 
 ning any fliadow, as appears at A, B, 4, viz. one from the top A, 
 to the vaniftiing point of the rays U, and another from the bottom B, 
 to the vanilhing point of the fhadow W j the additional lines at i, and 
 2, are merely for illuftration, or proofs and at 3, there is another line 
 A, L, whofe vanifhing point is and the vanilliing point of its ftia- 
 dow w. 
 
 The foregoing fchemes have been introduced, and explained, in or¬ 
 der to facilitate the pradlice, in the perfpedfive of fhadows, and prin¬ 
 cipally with refpedl to the members of architecture j for though, hi¬ 
 therto, in hmple lines, only, (that they may be more eafily compre¬ 
 hended,) yet their application, and utility will appear by thofe which 
 follow. 
 
 Fig. 69. No. I. (in the foregoing treatife) is the reprefentation of a Dork 
 cornice; now to find the fhadows of the projedting members, draw 
 from S, (the center of the pidture) a line to the extremity of any 
 member, {e. ^.) tojf, the extreme angle of the modillion; which line 
 will be (perfpedtively) perpendicular to the plane of the pidture, and 
 find the point g, in which fuch member touches the naked, or folid, of 
 the building, (i. e.) the plane on which the fhadow is to be caftj and 
 from that point draw a line g, h, parallel to S, R, (R, being the 
 given vanifhing point of the rays of light); then dravvyi R, cutting 
 gy h, in which will be the fhadow of f-, and fo of the reft. 
 
 Fig. 69. No. I. A. But to explain this operation, unembarraffed with 
 other lines, is the following fcheme. Here S, is the center of the pic¬ 
 ture 5 S, D, the diftance; R, the vaniftiing point of the rays of the 
 fun j yi g^ a line perpendicular to the pidture,, and alfo to the plane 
 
 on 
 
rhe SUPPLEMENT. 123 
 
 on which the fhadow is to be cad, which plane is parallel to the 
 picture. Now, being the leat of on the parallel plane, 
 
 or the point in which it touches that plane, draw g, h, parallel to 
 S, R, and draw fy R, cutting it in h ; then g, /?, will be the fhadow 
 
 pf<g-,/. 
 
 For as all lines tending to S, reprefent lines parallel to D, S, when 
 raifed up on S, (/. e.) perpendicular to the pi6lure j fo all lines tending 
 to R, reprefent lines parallel to D, R j therefore S, is the vanifhing 
 point of all the lines/*, g, and R, of all the rays pafling over them, 
 which rays (though parallel among themfelves) being oblique to the 
 pidlure, mud have a common vanifhing point. And as the plane, on 
 which thefe diadows are cad, is parallel to the pidlure; and the objedfs, 
 wdiofe fhadows are fought, all perpendicular to that plane j the fhadows 
 mud neceffarily be all parallel to each other, and to the feat of the rays 
 on that plane, which is S, R, and therefore can have no vanifliing 
 point; and for the fame reafons, all perpendicular objedls, that are of 
 equal length, will projedb fhadows, on this parallel plane, of equal 
 length alfo. 
 
 Fig. 69. No. I. E. As for the fide plane of the fame objedt, which plane 
 is perpendicular to the picture, the fhadows will have a vanifhing 
 point, as Z, perpendicularly under S, for S, Z, is the vanidiing line 
 of that plane, and R, Z, drawn from R, perpendicular to S, Z, will be 
 parallel to the lines, whofe fhadows are fought, on this plane, which 
 lines are all parallel to the pi6lure. Thus, draw a line from /, (the 
 point where /, /, touches the fide or profile of the building) to Z, 
 and then draw another line, /, R, cutting it, in w, which will be the 
 point fought, that is, the fhadow of /, on the folid of the building, 
 exadlly as on the horizontal plane in finding the fhadow of a line 
 danding perpendicularly on it j for (turning the pidture) the whole 
 correfponds to that; and having found my the fhadow of any fuch 
 point /, draw S, /w, which will determine the whole fhadow of fuch 
 projecting member; for S, w, reprefents a line parallel to /, /, and 
 4 4 
 
 R a 
 
 To 
 
,24 SUPPLEMENT. 
 
 To find the fliadows on a plane oblique to the pidture, fuch as at 
 Fig. 69. No. 4. (in the foregoing treatife) a line muft be drawn (per- 
 fpe 61 :ively) perpendicular to that plane, {i. e. as the modillions are, on 
 this face of the building,) and then the operation will be like the 
 I aft; but as there, fpace is wanting to introduce the vanifiiing points 
 of the rays, and of the fhadow, fee 
 
 Fig. 69. No. 4. G.—Draw if. A, to its vanifiiing point and from g, draw 
 through R, the given vanifiiing point of the rays, to S, e, (the vanifii¬ 
 ing line of this face of the building,) cutting it in e, which will be the 
 vanifiiing point of the fhadow. Now draw if, R, and A, e, cutting it 
 in f, then A,/*, is the fhadow of A, b, on this plane. Tliefe are all the 
 lines that are neceffary for the purpofe; and this is the fhortefi; method. 
 
 But the fiiadow f, of the point, b, might be found by other lines, 
 which are here added merely for illuftration, and to fiiew a kind of 
 correfpondence. 
 
 For, in every method, the truth of the operation depends upon the 
 fimilarity of triangles, either geometrically, or perfpedfively, and in 
 this already explained, the triangles b. A, f, and R, e,f, are perfpec- 
 tively fimilar, that is, reprefent fimilar triangles; for g, b, and g, e, re- 
 prefent parallel lines, as having the fame common vanifiiing point g. 
 
 And if R, L, be drawn parallel to S, g, (the horizontal line) cutting 
 the fame vanifiiing line S, e, in L, and b, a, be drawn parallel to it, 
 then draw a, L, which cutting b, R, finds the fame point f, for the 
 fhadow of b, and here the triangles b, a, f, and R, L, f, are, geome¬ 
 trically, fimilar. 
 
 Again, if a perpendicular from R, be drawn cutting S, g, in r, draw 
 r, b, cutting S, A, in a, and draw a, f, parallel to r, R, this alfo 
 will cut the ray b, R, in the fame point f, and here the triangles b, a, f, 
 and b, r, R, are fimilar. 
 
 And having found f, the fiiadow of b, draw S,yi which will de¬ 
 termine the whole fiiadow of the projedling member; as in the laji 
 example. For %,f, reprefents a line parallel to S, A, and S, b. 
 
 •fig. 72. No. 1. A. This fcheme is introduced to explain fome particu¬ 
 lars relating to the fiiadow of the ladder at Fig. 72. No. i. and there¬ 
 fore 
 
HateXLVI 
 
\ • 
 
 f] 
 
 L * 
 
 • • 
 
rhe SUPPLEMENT. 125 
 
 fore has the fame letters, and numerical figures of reference j only in- 
 ftead of that object, here is a plank, in order to fhew the operation 
 and efFe6l more diftinflly. 
 
 G, 0, 4, is a ray parallel to the plank, and the triangle G, 4, S, is 
 a plane of rays continued to the horizon.—A, 7, is the interfedlion of 
 that plane, by the plane of the wall againft which the plank leans: 
 therefore from the point A, (the top of that interfedion) drawing A, 5, 
 and A, 6, through the top of the plank, thefe lines will give the fha- 
 dow of it on the wall, but being interrupted by the door, at 13, 14, 
 the fiiadow will thence take another direction. Now fince the plane 
 of the door (whofe vanifhing point is a^) cuts the triangular plane of 
 rays in B, 8, {as the plane of the wall does in A, 7,) therefore from the 
 point B, draw through 13, 14, which will give the direflion of the 
 fhadow on the door, which fhadow will meet that on the ground 
 (from the bottom of the plank) in ii, 12. The triangle B, 11, 12, 
 correfponding to the plane of the door, exadtly as the triangle A, 5, 6, 
 does to the plane of the wall, and as the triangle 4, F, G, does to that 
 of the utmofi: diftance, which, being parallel to the pidlure, and to 
 the wall, the line 4, G, is parallel to A, 6, and 4, E, to A, 5, which . 
 fliews the correfpondence of the operation, and proves the truth of it» 
 And as is the vanifhing point of the top, and bottom- oPthe 
 door, fo a, g, f will be the vanifhing line of its plane; wherefore, con» 
 tinuing 11, B, to^, and 12, B, tof thefe will be the vanifiiing points of 
 thofe lines;—and as a,g,f is at the utmofi: extent, or on the hori¬ 
 zontal line, as well as F, 4, and G, 4, fo, by continuing F, 4, to f and 
 G, 4, to g, thefe fame points f and will alfo. be the interfedtions 
 
 of the vanifhing lines F, 4, and G, 4, with the vanifhing line a, g, f 
 which is a farther illuffration of the whole. 
 
 N. B. It feems hardly neceffary to add, that G, (being the point' 
 in which the fame ray 4, ©, G, touches the ground) anfwers 
 the fame purpofe for the plane of the ground, as Aj and B, 
 for the planes of the wall, and the door, and that the triangle 
 G, II, 12, on the ground (therefore) correfponds to that 
 planCj as A, 13, 14, and B^ 11, 12, to their refpe^live planes. 
 
 As 
 
126 S U P P L E M E N T. 
 
 As a farther illuftration of the two laft fchemes, and to render this 
 kind of operation (which is of great ufe) hill more clear, here is added 
 a third. 
 
 Fig. 72. No. I. B. Let L, 4, be confidered as one leg of the ladder, ora 
 pole, continued to its vanifhing point 4—L, S, the feat of the pole, on 
 the ground, continued to its vanifhing point S; fo that L, 4, S, may 
 be confidered as a triangular plane, whofe vanifliing line is S, 4. Draw 
 from 4, through 0, the light, and from S, through F, the foot of that 
 light, meeting in G, which will be the foot of the light, for the pole, 
 becaufe 4, G, reprefents a parallel to 4, L. 
 
 Draw G, L, cutting a, ii, (which is the interfecfion of the plane 
 of the door with the ground) in b, and from 8, (the interfeftion of 
 L, S, with a, 11,) raife a perpendicular, cutting L, 4, in d-, draw d, 
 which will be the fhadow of L, d, on the plane of the door j con¬ 
 tinue b, d^ till it meets <?, (the vanifhing line of the plane of the 
 
 door) in which will be the vanifliing point of b, d. 
 
 Or that vanifhing point may be found, by continuing G, L, b^ to 
 E, in the horizontal line, and drawing F, 4, which will find the fame 
 pointy) and then drawing b^ y, which will cut L, 4, in d. 
 
 And this lafir method will be true for any plane, whofe vanifliing 
 line is a^f .—For fuppofe a plane whofe interfedfion with the ground 
 is 7, and which is cut by L, S, in h-, raife, at by a perpendi¬ 
 cular up to L, 4, cutting it in k, and draw 7, ky which will tend to 
 
 the fame point f ; or draw 7, /) which will cut L, 4, in ky and 7, k, 
 will be the fhadow of L, ky on that plane. 
 
 And fo univerfally of any plane, whofe vanifhing line is dyfy from 
 
 I I, to ^7, F. 
 
 Again, fuppofe a line drawn through L, parallel to the horizontal 
 line dy Fy cutting 4, ©, G, in g, and ii, in ii, and S, cutting 
 0, F, in f, then f, will be the foot of the light (removed farther with¬ 
 in the plane of the ground) j and drawing 4, U, parallel tog, L, 11, 
 cutting the vanifhing line dy /, in U, then U, will be the vanifhing 
 point of the fhadow of L, 4, (or of L, dy) on the plane of the door; 
 and drawing ii, U, cutting L, 4, in dy —ii,y will be that fhadow; 
 
 remenok 
 
FLateXLATI. 
 
2?^ S U P P L E M E N T. 127 
 
 remembering, that in this laft cafe, the foot of the light is fuppofed to be 
 
 f, (farther removed within the pi6ture) which occafions the fhadowfrom 
 
 g, to be fo much longer, than that from G, produced by the foot F, 
 which is nearer; and that, in either cafe, 0, (the light) is juft as far 
 within the pi6lure, or the ground, as its foot, whether it be F, or f. 
 
 Fig. 72. No. 3. Here the rays of light, are thofe of the fun, fuppofed 
 to be parallel to the pidlure, and to H, I; it is required to find the 
 ftiadow of the fquare projedlion a, b, e, of this building on the parts 
 contiguous to it. 
 
 Firft, draw c, y, parallel to K, I, the horizontal line, this gives that 
 part of the fliadow which is caft on the ground j at y, raife a perpen¬ 
 dicular to r, the edge of the roof j then draw r, parallel to K, L, 
 the vanifhing line of the plane of the roof, and draw n, parallel 
 to H, I, cutting r, n, in which is the fhadow oi b fo that y, r, w, 
 
 is the whole fhadow of the line b, and is, evidently, in a plane of 
 
 rays parallel to the pi6lure, and to H, I. 
 
 For c, y, is on the horizontal plane, and parallel to the horizon¬ 
 tal line, and y, r, is parallel to c, b, and in the fame plane with that, 
 and c, yj and r, (joining y, r,) is on the roof, and parallel to its 
 vani filing line L, K; therefore, &c. 
 
 Now, in order to find the fliadow of b^ a, whofe vanifhing point 
 is I, let it be confidered that I, FI, is the vanifhing line of the plane 
 of rays which pafs over b, and L, K, is the vanifhing line of the 
 roof, and, therefore, that the interledlion of thefe two vanifhing lines E, 
 muft be the vanifliing point of the fhadow of b^, on the roof, for the 
 fadow itjelf is ibe interfeBicn of thofe t^wo planes. Therefore draw 
 E, n, which will be the indefinite fliadoiv of b^ on the roof, and 
 drawing o, parallel to H, I, cutting E, n, in Oy this would determine 
 0, Uy the fliadow of ay by (for o, would be the fliadow of j, on the 
 roof) if the roof were continued fo high; but as this is interrupted by 
 the perpendicular plane /, o, whofe vanifhing line is K, H, which 
 interfedfing I, H, (the vanifliing line of the rays) in H, this inter- 
 fedlion will be the vanifhing point of the fliadow on that plane, for 
 this fhadow is the interfeBion of the raps with that plane. There¬ 
 fore 
 
5 28 The SUPPLEMENT. 
 
 fore draw H, cutting «, o, in p, and then a, p, is that part of the 
 fhadow of a, b, which falls on the wall, and p, n, the reft of it, 
 which falls on the roof \ fo that a, p, is the whole Ihadow of a, h, 
 produced by rays all parallel to the pidlure, and to H, I. 
 
 And to prove the truth of all this, draw from c, the feat of b, 
 on the ground, and from the feat of a, two lines, c, g, and e, j\ pa¬ 
 rallel to the horizontal line; then draw b, g, and a, j\ both parallel 
 to H, I, cutting them in g, and f j then g^ will be the ftiadow of b, 
 and j\ the ftiadow of on the ground. And, having continued 
 I, c, and L, h, till they meet in /, draw K, i, and I, /, g^ meeting 
 in X ; draw X, L; now raife perpendiculars from g^ and f, cutting 
 X, L, in /, and;^; draw k, /6, and /, /;/, which will be perpendicularly 
 overyi and^, c, and parallel to each other, and to the vanifhing line 
 L, Kj and therefore will be the fliadows of a, h, and b^ m, on the roof; 
 wdiich lines meeting with the rays b, g, and a, in n, and o, thefe 
 rays determine the lengths of the fhadows j that is, «, is the fhadow 
 of bj and o, would be the ftiadow of a, if the roof reached fo high ; in 
 which cale, o, would be the fhadow oi a, b but as the roof is in¬ 
 terrupted by the perpendicular plane above, (which touches the line 
 hy y>, in ph) therefore a, y>, will be fo much of the fhadow of a, b, as 
 falls on that plane, and the reft of it is /, n, on the roof. 
 
 Fig. 72. No. 4. Is a cylinder, lying on the ground, whofe bafe is pa¬ 
 rallel to the picture. 
 
 S, is the center of the pidlure; R, the vanilhing point of the rays of 
 light, which are fuppofed to come from the fun j L, the vanifliing 
 point of the fhadow, found by railing a perpendicular from R, up to 
 the horizontal linej it is required to find the fhadow of any point, 
 or points, of the circumference of the bafe of the cylinder, on the 
 inner furface of it. 
 
 The fhadow of A, is found on the inner furface, by drawing A, a, 
 parallel to S, R, and drawing a, S, and, laftly. A, R, cutting a, S, 
 in a. 
 
 For the fhadow of A, muft be determined by the ray, pafiing over 
 that point, to the vanifhing point R, and it muft be in the line a, S, 
 
 in 
 

 I 
 
 
 A’ ' 
 'i 
 
 J 
 
S U P P L E M E N T. 129 
 
 in which that ray cuts the inner furface, and alfo it muft be in the 
 point (of a, S,) in which A, R, cuts that line; therefore it mufl be 
 the point a. 
 
 And fo for any other point, or points of the circumference: by 
 which operation a number of points in the inner furface may be 
 found fufficient to trace the lhadow of the circumference, within the 
 hollow of the cylinder. 
 
 The fliadow within the other cylinder is found in the fame man¬ 
 ner j but that is introduced, principally, to fliew the method of finding 
 the fliadow call: on the outer furface, by any obje6l, as B, interpofed 
 between it, and the light. In order to which j firfl find the fliadow of 
 that objedl on the ground, then mark any point on the bafe of the 
 cylinder, as b, and find its feat c, on the ground : draw c, S, cutting 
 the fhadow of B,' in j at raife a perpendicular, and draw b, S, 
 cutting that perpendicular in c, which will be the point of fhadow 
 fought. And repeating the fame operation for as many points as fhall 
 be neceffary, the whole fhadow of the obje6l B, may be found on 
 the outer furface. 
 
 For, b, c, may be confidered as a perpendicular plane touch¬ 
 ing the cylinder in the line b, e i and d^ c, would be the fliadow of B, 
 on fuch plane; but, as the cylinder is circular, the plane of the 
 fhadow touches it only in the point r, which is the reafon that other 
 points mufl be found, by the fame method j that is, by marking feveral 
 points on the bafe of the cylinder, finding their feats on the ground, 
 then drawing lines from thofe feats to S, cutting the fhadow of B, on 
 the ground, and thence railing perpendiculars; and laflly, drawing 
 lines from the feveral points (marked on the bafe) to S, meeting their 
 refpeclive perpendiculars in the points of fliadow. 
 
 The End of thv Supplement. 
 
 S 
 
DIRECTIONS to the BOOKBINDER 
 
 The two unnumbered Plates to be placed in the Introdudion as paged, to- 
 wit, X and xii, and thofe numbered as follow. 
 
 Plate I. - - - to front page 4 
 II.. 8 
 
 in. 10 
 
 IV.. 
 
 V. ------ - 14 
 
 VI. .16 
 
 VII. 16 
 
 VIII. . 16 
 
 IX. ------ - 18 
 
 X. ------- 20 
 
 XI. - 22 
 
 XII. ------ 24 
 
 XIII. .28 
 
 XIV. ------ 32 
 
 XV. ----- - 36 
 
 XVI. ------ 38 
 
 xvn. ------ 36 
 
 XVIIL 50 
 
 XIX. ------ 42 
 
 XX. 41 
 
 XXI. ------ 52 
 
 XXIT. ------ 54 
 
 XXIIT, ------ 58 
 
 XXIV. ------- 60 
 
 Plate XXV. - - to front page 72 
 
 XXVI..64 
 
 XXVII. - - - - ’ - - 66 
 
 XXVIII. ----- 70 
 
 XXIX. ------ 72 
 
 XXX. ------ 7^ 
 
 XXXI. .76 
 
 XXXII. ------ 78 
 
 XXXIII. ----- 80 
 
 XXXIV..82 
 
 XXXV..84 
 
 XXXVI. - - . - - -86 
 
 XXXVII. ----- 88 
 
 XXXVIII. ----- 90 
 XXXIX. ----- 94 
 
 XL. ------ 100 
 
 XLI. -.108 
 
 XLII. - - - - - - 112 
 
 XLIII..- 114 
 
 XLiV. ------ 116 
 
 XLV. ------ 122 
 
 XLVI. ------ 124 
 
 XLVII. ----- 126 
 
 XL VII I. - - after page 129 
 
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