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T R E 4„ T I S E 
 
 O P T H E 
 
 THEORY AND PRACTICE 
 
 O F 
 
 PERSPECTIVE. 
 
 WHEREIN 
 
 The Principles of that mofl ufeful Art, as laid down 'by Dr. Brook 
 Taylor, are fully and clearly explained, by Means of moveable 
 Schemes, properly adapted for that Purpofe. 
 
 The whole being defigned as 
 
 An Eafy Introduction to the Art of Drawing in Per spec tive, 
 
 AND 
 
 Illuftrated by a great Variety of Curious and Inflrufling EXAMPLES. 
 
 A Figure in a PIfture, which is not drawn according to the Rules of Perfpe£tive, does not repre- 
 fent what is intended, but fomething elfe : fo that it feems to me, that a Pifture which is faulty 
 in this Particular, is as blameable, or more fo, than any Compofition in Writing which is faulty 
 in Point of Orthography or Gram mar. 
 
 Dr. Broojc Taylor's Preface. 
 
 By DANIEL FOURNIER, 
 Drawing- Mafter, and Teacher of Perfpedive. 
 
 LONDON: 
 
 Printed for the AUTHOR, at his Houfe in Wilde Court, near Great Q^ieen-ftreet, 
 
 Lincoln's-Inn-Fields : 
 
 And Sold by Mr. Nourse, facing Catharine-Street, in the Strand; and by Mr. Lacey. the 
 •Corner of St. Martin's Court, St, Martin's Lane. 1761, 
 
t o 
 
 Mr. M A R 0 U O I S, 
 
 Mafter of the Military Academy at Noreland, 
 near Kenfington Gravel-Pits, 
 
 SIR, 
 
 r I "1 H E following Treatife of PerfpecSti ve being 
 JL drawn up for the ufe of the noblemen and 
 gentlemen, whofe military education you fuperin* 
 tend fo fuccesfully, it is, I prefume, with the 
 ftridleft propriety that I infcribe it with your name* 
 
 It aclcl^5 greatly to yuui iCpUtiltion, that yOU 
 
 are the firfl: man who opened, in a private way, 
 an academy of this kind. How very important 
 fuch inftitutions are judged by the government, 
 appears from the Royal Academies of Wool-* 
 wich and Portfmouth, founded under the aufpices 
 of his late Majefty King George II. 
 
 The valour of the Englifli has been ever ac- 
 knowledged, as the French have often experi- 
 enced to their coft : but it was long obferved, that 
 
 a z the 
 
the miiitary fkill of the French officers v/as far 
 
 It 
 
 fuperior to that of theEnghfli, and this through ' 
 want of military academies, where our noble and i 
 p-enerous youth might be taught thofe arts, which 
 only can enable them to ihine both in the field 
 and on the ocean. 
 
 How ufeful a membjer. Sir, you are to fociety, 
 appears from the excellent pupils you are daily 
 forming : 'tis to your friendihip that I owe the 
 great honour I have to inftrudl them in the art 
 of drawing and perlpeilive. For this, and many 
 other favours from you, gratitude obliges me to 
 wilh all imaginable fuccefs to your laudable en- 
 deavours 5 and to aflure you, that no one can be 
 with more perfect efteem, 
 
 SIR^ 
 
 Your moft humble, 
 
 and moft obedient fervant. 
 
 DANIEL FOURNIER. 
 
[ V ] 
 
 T 11 E 
 
 P R E F A 
 
 As Dr. Brook Taylor's Perfpeaive is undoubtedly the moH: excel- 
 lent performance of its kind hitherto made public, I chofe rather to 
 attempt arr explanation of his principles, than to offer any thing of my own, 
 with regard to the theory of this mofl- ufeful art ; and chiefly bccaufe the 
 generahty of thofe who apply themfelves to the ftudy of that treatife, with- 
 out being firfl acquainted with geometry, find great difficulty in conceiving 
 the author's defign, as all his fchemes are drawn in piano : I have, to obviate 
 this difficulty, introduced moveable fchemes in the following work, fo con- 
 trived, as to be raifsxl vip onr? pla<.»-d awwrdiu^ aff the nature of the problem 
 
 may require. 
 
 In the pradlical part, which begins at page 32, I have defined and made 
 ufe of the terms, ground line, horizontal line, Sec. This I did for thecon- 
 veniency of thofe who may have already learned the theory of perfpeflive, 
 from fuch authors as have treated thereon agreeable to thefe definitions, and 
 becaufe painters commonly ufe the horizontal line as the boundary of the pic- 
 ture. 
 
 As the art of perfpedive confifls in the knowledge of truly reprefenting an 
 objedt according to its natural appearance, fo whatever is not drawn agreeable 
 the rules thereof, cannot poffibly repreient what is intended, but fomething 
 elfe : it is true, that as the art of painting confifls of two parts, the inventive 
 and the executive, the former may indeed be varied according to the taffe 
 
 and 
 
and genius of the artift, in the difpofition of the parts of his fubjed j but the 
 latter is wholly confined and ftridly tied to the rules of art, and therefore in 
 this the artlft is not to take any liberties whatfoever ^ for what is perfe£lly jaft 
 in the real original objeds themfelves, can never appear defective in a pi<flure, 
 v/htre thofe objedts are reprefented according to the true rules of perfpedtive. 
 
 Notwithftanding the principles upon which Dr. Taylor has founded his 
 treatife fecm fo very obvious as not to admit of any objedtion ; yet fome 
 modern writers upon this fubjed: have fo far deviated from his theory, as to 
 iiiform us, that a circle, when viewed in an oblique pofition, cannot poflibly 
 tal^e the appearance of a true mathematical ellipfis. I (hall not at prefent en- 
 ter into a confutation of the reafons they give to fupport the truth of this ex- 
 ti-aordinary affertion, but only point out wherein (I think) they are miftaken. 
 That the oblique fedion of a cone may be an ellipfls, is certainly true : I fay 
 may be an ellipfis, becaufe in fome cafes it may be an hyperbola; wherefore, 
 if the cutting plane be fuppofed to be the plane of the picture, the reprefen- 
 tation of a circle thereon muff be elth^^r a rircle, ellipHs, or a right line, and 
 that according to the different fituations of the original circle with regard to 
 the plane of the pidure ; if after having found the elliptical reprefentation of 
 an original circle, we find that of its center, it will not coincide with the 
 center of the ellipfis ; and this I apprehend may have deceived thofe who 
 have imagined the oval reprefentation of an oblique circle would not be of 
 equal curviture in oppofite parts of its circumference. 
 
 A 
 
A LIST of S 
 
 A. 
 
 CAptain Allwright 
 Mr. John Allen 
 Mr. William Atkins 
 
 B. 
 
 William Boothby, Efq; 
 
 Mr. Brewer, Miniature Painter 
 
 Mr. Sampfon Bifhop 
 
 Mr. Jacob Bonneau, Drawing-Mafler 
 
 Mr. John Barrell, Statuary 
 
 Mr. William Burton 
 
 Mr. Francis Bowry 
 
 Mr. Beauvais, Miniature Painter 
 
 Mifs Carolina Boucher 
 
 C. 
 
 Mr, Robert Clee, Engraver 
 
 Mr. James Caton, Painter 
 
 Mr. Charks Cock 
 
 Mr. Thomas Carter, Statuary 
 
 Mr. Thomas Chambers, Engraver 
 
 Richard Cambridge, Efqj 
 
 Mr. John Caffin 
 
 Mr. John Crompton 
 
 Mr. Patrick Craufurd 
 
 D. 
 
 Mr. Thomas Dade, Junior 
 Mr. Davers 
 
 Mr. James Durno, Junior 
 Mr. John Dibb 
 Mr. Henry Downing 
 Mr. Thomas Davis 
 
 E. 
 
 The Right Honourable the Earl of 
 
 Effingham 
 Mr. John Ellis 
 
 J B S C R I B E R & 
 
 Mr. Thomas Elliot 
 Mr. John Edwin 
 
 F. 
 
 William Windfor Fitzdiomas, 
 
 Patrick Fergufon, Efq-, 
 
 Mr. Thomas Filewood 
 
 Mr. William Franks 
 
 Mr. Gabriel Fournier, Upholder 
 
 Mr. John Farmer 
 
 G. 
 
 Capt. Grenvllle 
 
 Thomas Garth, Efq; 
 
 Mr. Thomas Grignion, Junior 
 
 Mr. Gibfon 
 
 Mr. James Gandon, Architect 
 Mr. Francis Gould 
 
 H. 
 
 Right Honourable the Lord Howard 
 
 John Hamilton, Efq; 
 
 Mr. William Henry Higden 
 
 Mr. Charles Hue, Painter 
 
 Mr. John Henderfon 
 
 Mr. George Richard Hoare 
 
 Mr. William Hoilingworth, Upholdei" 
 
 Mr. Charles Hall, Engraver 
 
 Edward Havvke, Efq; 
 
 Mr. Thomas Haines 
 
 I. 
 
 Mr. John Jackfon, Junior 
 Mr. Thomas Jarvis 
 Mr. William Jones 
 
 K. 
 
 Mr. John Keller 
 Mr. Thomas Knipe 
 Mr. Thomas Knowies 
 
I viii ] 
 
 L. 
 
 Mr. John Lacey 
 
 Mr. Thomas Loveday 
 
 Mr. William Lowe 
 
 Mr. David Lowns 
 
 Mr. Thomas Landfdowne 
 
 M. 
 
 John Muller, Efq; Profeffor of Forti- 
 fication, Artillery, 8cc. and Chief 
 Mafter at the Royal Academy at 
 Woolwich 
 
 Lieut. Muller 
 
 Thomas Marquois, Efq; 
 
 Mr. Merger, Miniature Painter 
 
 Mr. Charles Mackenzie 
 
 Mifs Sibela Caroline Mackenzie 
 
 Mr. James Mack ay, Upholder 
 
 Mr. Morrier, Houfe Painter 
 
 Mr. Maltby 
 
 Mifs Barbara Marfden 
 
 N. 
 
 Alexander Napier, Efq; 
 Thomas Nefbit, Efq; 
 Mr. Noverre 
 Mr. John Nevil 
 Mr. Francis Newton 
 Mr. Frederick Newholt 
 
 O. 
 
 The Right Rev. Lord Bifhop of OfTory 
 Mr. John Oliphant 
 Mr. William Oliver 
 Mr. Thomas Overton 
 
 P. 
 
 The Honourable Edward Perci?al 
 
 Mr. John Potter 
 
 Mr. Louis Pingo Medalift 
 
 Mr. Probyn, Engraver 
 Mr.. Peacock, Surveyor 
 Mr. James Perkins, Engraver 
 Mr. Robert Pringle, Engineer 
 Mr. Charles Pinckney 
 
 R. 
 
 Major Rooke 
 
 Mr. Nicholas Read, Statuary 
 Mr. John Roberts, Upholder 
 Mr. John Ringe 
 Mr. Reynous 
 
 S. 
 
 Samuel Strod, Efq; 
 
 Mr. Edward Stevens, Archlted 
 
 Mr. Anthony Swift 
 
 Mr Richard Sharp, Carver 
 
 William Stuar;, Efq; 
 
 Mr. R'.ciiard Synge, Upholder 
 
 Charles Smith, Efq; 
 
 T. 
 
 William Thompfon, Efq; 
 William Temple, Efq; 
 Mr. Pe er Tri :)uet 
 Mr. William Twifs 
 
 ■ y. 
 
 Mr. Ferdinand Vigne 
 Mr. Robert Venable 
 Mr. John Vincent 
 
 W. 
 
 Mr. John Wolf Sui-veyor 
 ?vlr. Francis Wh.atly 
 Mr. Williams, Painter 
 Mr. Thomas Weft 
 Thomas Whitmore, Efq; 
 
O F 
 
 LINEAR PERSPECTIVE. 
 
 IN order that the learner may form a clear idea of the inveftigations 
 relating to the theory of PerfpeBtve contained in the enfuing pages, 
 it may be neceflary to premife the following geometrical Definitions and 
 Problems. 
 
 ^^^^^^ Definitions. 
 
 1 . Magnitude is whatever is extended, and therefore may be conceived 
 to be contained or limited by fome certain term or terms. 
 
 2. h point is that which has no parts, confequently of itfelf no mag- 
 nitude, and may be confidered as indivifible. . 
 
 3. A right line is the neareft diftance between two points. 
 
 4. If two right lines inclined to each other be produced till they meet, 
 they will form a plane or right lined angle. Thus the angle formed by 
 the lines BD, CE, Plate I. Fig. i. produced, is denoted by BAC. 
 
 N. B. An angle being generally denoted by three letters, the middle 
 one always reprefents the angular point. 
 
 5. When a ftrait line ftands on another, fo that it does not incline to 
 one end more than the other, but makes the angles on both fides equal ; 
 then are thefe called right angles : and the lines are faid to be perpendicular 
 to each other j as BD Plate I. Fig. 2. is perpendicular to FE. 
 
 ^ 7. A 
 
2 LINEARPERSPECTIVE. 
 
 6. Every angle greater than a right one, is called an obtufe angle 'y and 
 every angle lefs than a right angle, is called an ^c^ite^angle. ; 
 
 7. A plane furface terminated by three right lines, is called a plane 
 triangle. 
 
 8. When the three fides of a triangle are equal, it is called an equi' 
 lateral triangle. 
 
 « 
 
 9. When only two fides are equal, an ifofceles triangle. 
 
 10. When the three fides are unequal, a fcalene triangle. 
 
 1 1 . When one angle of a triangle is a right angle, it is called a right 
 angled triangle. 
 
 12. Any one fide of a triangle may be called the bafe^ and the angular 
 point oppofite to the bafe is called the 'vertex. 
 
 13. In a right-angled triangle, the fide oppofite to the right angle is 
 called the hypothenufe. 
 
 14. A plane contained by four right lines, is called in general a qua~ 
 trilateral figure ; and when the four fides arc equal as well as the four 
 angles, it is called a fqiiare. 
 
 15. If the four angles are equal, and only the oppofite fides, a re5i' 
 angle. 
 
 16. If the oppofite angles are only equal, and the oppofite fides, a 
 parallellogram. 
 
 17. A plane figure of above four fides is called, in general, ^polygon. 
 
 18. A circle is a plane figure bounded by one continued line called 
 the circumference^ which is every where equally diflant from a point 
 within the circle, called the center. 
 
 19. Any line drawn from the center to the circumference is called a 
 radius 3 the double of which is the -diameter of the circle. 
 
 20. The meafure of an angle is the arc defcribed from the angular 
 point as center with any radius, and terminated by the lines which form 
 the angle. 2 21. A 
 
LINEAR PERSPECTIVE. 3 
 
 21. A line which touches a circle in one point only, is called a tangent i " 
 and the point where it touches, the point of contaSi. 
 
 22. A polygon is called regular ^ when all its fides are equal as well 
 as all its angles ; and irregular y when they are not equal. 
 
 23. A polygon is faid to be infcribed in a circle, when all its angles 
 touch the circumference; and circumfcribed about a circle, when all 
 its fides touch the circle. 
 
 Polygons are diftinguifhed by the number of their fides, viz. 
 
 5 Pentagon, 9 Enneagon, 
 
 6 Exagon, 10 Decagon, 
 
 7 Eptagon, 1 1 Ondecagon, 
 
 8 Odagon, 12 Dodecagon, 
 
 Problem i. 
 I0 divide a given right line into two equal parts. 
 
 Let the line given be AB, Plate I. Fig. 3. Set one foot of the com- 
 pajfes in A, and with any convenient diftance fweep two arches one above 
 and the other below the given line ; then fet one foot of the compaffes in B, 
 and with the fame diflance as before crofs the aforefaid arches, join the 
 points of interfedtion with a right line, and it will cut AB in two equal 
 parts. 
 
 Problem 2. 
 To raife a perpendicular from any point propofedy in a given right line. 
 
 Let the given line be AB, Plate I. Fig. 4, and the perpendicular to be 
 raifed from the point C ; to do which, fet off two equal diflances from 
 the point C, as CR, CS : then the compaffes being opened to any conve- 
 nient diftance greater than CR or CS, with one foot of the compajfes in 
 the point R defcribe the arch DE : then with the fame extent, and one 
 foot of the compafes placed in the point S, defcribe the arch FG ; then 
 
 B 2 draw 
 
4- LINEAR PERSPECTIVE. 
 
 draw from C a right line through the interfeftion of the two arches FG, 
 DE, and it will he the perpendicular required. 
 
 •PR0BLEM3. 
 I'o let fall a perpendicular from a point ajjigned^ to agi^ven right line. 
 
 Let AB, Plate I. Fig. 5. be a right line given, and C the point from 
 whence the perpendicular muft fall. 
 
 Set one foot of the compajjes in C, and opening them to a proper 
 diftance defcribe the arcliRS ; then by Prob. i, divide RS into two equal 
 parts in D, join the points C, D, and CD will be the perpendicular required. 
 
 Problem 4. 
 
 To draw a right line parallel to a given right line, and alfo through a point given. 
 
 Let AB, Plate I. Fig. 6, be the given line, and C the point given. 
 From C let fall, by the lafl Problem, the perpendicular CD upon the line 
 AB. Take any point, as E in AB, in which fet one foot of the coinpafes, . 
 and with an opening equal to CD, defcribe an arch FG ; lay a ruler, fo as 
 jufl to touch this arch and the point C, then draw the line CH, it will 
 be the parallel required, 
 
 Problem 5. 
 T 0 divide a given right line into any number of equal parts. 
 
 Let AB, Eig. I. ^^fe 7, be the given line. 
 
 From the point A draw a right line AC at pleafure ; from the point 
 B draw a right line BD parallel to AC, fet off along AC and BD from 
 the points A and B, as many equal parts AE, EF, FG, &c. and BK, KL, 
 LM, MN, &c. towards C and D refpedively, as are one lefs than the 
 number into which you intend the given line to be divided ; join the laft 
 point in AC with the firft in BD, and alfo join the other points I, L, G, 
 N, F, O, &c. and the right lines KK, XL, HM, GN, &c. will divide AB 
 into the defired number of equal parts at the points k, i, h, g, &c. 
 
 c Problem 
 
LINEAR PERSPECTIVE. 5 
 
 i 
 
 Problem 6. 
 5*0 divide a given angle ACB, Plate 1. Fig. 8, into two equal parts. 
 
 From the angular point C as center, defcribe an arch with any radius, 
 meeting the legs of the angle at A, B : from thefe points, as centers, de- 
 fcribe two arches with the fame radius, fo as to interfeft each other at D ; 
 then the line drawn through this interfedion and the angular point C, 
 will bifed the given angle. 
 
 Problem 7. 
 
 To draw a line CA, fo as to make an angle ACB, with a given line CB, equat 
 to a given a?jgle a c b, Plate I. Fig. 9. 
 
 From the points C, c, as centers, defcribe two arches with any radius 
 at pleafure : make the arch AB equal to the arch a b ; and the line drawn 
 through the points C, A, will form the angle required. 
 
 Note, The circumference of every circle, whether great or fmall, is 
 fuppofed divided into 360 equal parts, called degrees, each degree into 60 
 equal parts, called minutes. Hence a quadrant, or the meafure of a 
 right angle, is 90 degrees, being the fourth part of the circumference 3 
 a femicircle, or the meafure of two right angles, is 180 degrees, being the. 
 one half of the circumference. 
 
 Problems. 
 
 T 7 defcribe the circumference of a circle, through three given points A, B, C, 
 
 Plate I. Fig. 10. 
 
 Join thefe points by two lines AB, BC, bife6t them with the perpen- 
 diculars DO, EO ; their point of interfedion O, will be the center of 
 the circle defired. 
 
 Problem 
 
6 
 
 LINEAR PERSPECTIVE, 
 
 Problem 9. 
 To infcribe a fquare in a circle, Plate I. Fig. 1 1 . 
 
 Draw the diameter AB, to which draw the diameter CD perpendicular j 
 then the lines joining the extremities of thefe diameters will be the 
 fquare i\.BCD required. 
 
 Problem 10. 
 To infcribe a regular polygon of a given number of fides in a circle. 
 
 Draw two radii from the center of the propofed circle, making an 
 angle equal to the angle at the center of a polygon anfwering to the afligned 
 number of fides (as in the table annexed) : join the two points in the cir- 
 cumference, where the radii interfed it, add that line or chord will be 
 the fide of the required polygon. 
 
 A TABLE of regular Polygons. 
 
 
 Anglp ill- 
 
 Angle 
 
 Polygon. 
 
 the 
 Center. 
 
 macle hy 
 
 the 
 Sides. 
 
 5 
 
 72 
 
 108 
 
 6 
 
 60 
 
 120 
 
 7 
 
 
 1284 
 
 8 
 
 45 
 
 135 
 
 9 
 
 40 
 
 140 
 
 10 
 
 36 
 
 144 
 
 1 1 
 
 32- 
 
 147 A 
 
 12 
 
 30 
 
 150 
 
 They 
 
LINEAR PERSPECTIVE. 7 
 
 The angle at the center of a regular polygon is found by dividmg 360 
 degrees by the nunnber of the lides : thus 360 divided by 5 gives 72 de- 
 grees for the angle at the center of a Pentagon ; and 360 divided by 6 
 gives 60 degrees for the angle at the center of an Exagon. 
 
 But the angle made by two adjacent fides of a polygon is found by fub- 
 trading the angle of the center from 180 degrees : thus from 180 de- 
 grees take 72, there remains 108 degrees, which is the angle made by 
 the fides of a Pentagon ; and if from 180 we take 60, (the angle at the 
 center of an Exagon) there will remain 120, the angle made by the fides 
 of an Exagon : in the fame manner may the remaining numbers in the 
 third column of the foregoing table, be found. 
 
 Having furniflied the reader with the moft neceflary geometrical Pro- 
 blems, for the underftanding of Perfpedlive, we fliall next proceed to the 
 Definitions thereof. 
 
 Definition i. 
 In order to have a clear idea of the principles of this art, we are to 
 confider that a picfture drawn perfectly true, and placed in a proper po- 
 fition, ought fo to appear to the fpedator, that he fhould not be able to 
 diftinguifh the reprefentation from the real original objeds adlually placed 
 where they are reprefented to be. To produce this efted, it is necef- 
 fary that the rays of light ought to come from the feveral parts of the 
 pidlure to the fpedator's eye with the fame circumftances of diredion, 
 flrength of light and fhadow, and colour, as they would do from the 
 correfponding parts of the real objedts feen in their proper places. Thus, 
 Plate II. Fig. i. let O be the fpedator's eye, FH the plane of the pic- 
 ture, which, for more eafy conception, we may fuppofe to be tranfparent, 
 ABCDE an original cube i then if we imagine lines to be drawn from O 
 to every part (in view) of the folid ABCDE, their interfe<ftion with the 
 plane FH will mark thereon the reprefentation abcde of the original 
 figure ABCDE. 
 
8 LINEAR PERSPECTIVE. 
 
 2. Cone of rays. Lines drawn from the feveral parts of any figure, 
 and continued fo as to pafs through one and the fame point, conftitute 
 a cone of rays ; and when that point is confidered as the eye of a fpec- 
 tator, it is called the optic cone. 
 
 3. Ichncgraphy. When a fyftem of rays (parallel to each other and 
 perpendicular to the horizon) coming from the feveral points in any figure, 
 is cut by a plane parallel to the horizon, the projecftion thereon is called 
 the ichnography of the faid figure. 
 
 4. Orthography. When a fyftem of rays (parallel to each other and 
 to the horizon) coming from the feveral points in any figure, is cut by a 
 plane perpendicular to the horizon, the projedion thereon is called the 
 orthography of the faid figure. 
 
 Thefe are the common definitions of the terms ichnography and orthography ; 
 but in the following pages we fhall ufe them to fignify any two projec- 
 tions that are made by fyftems of parallel rays, when thofe fyftems are 
 perpendicular to each other, and to the planes on which the projedions 
 are made, without having any regard to their fituation with relpeft to the 
 horizon. 
 
 5. Schenography. When the optic cone is cut by a plane, the repre- 
 fentation of the propofed figure thereon is called the fchenography thereof. 
 It is evident that the fhadows of figures are this projedlion, when the 
 light is confidered as a fmgle point: though in the cafe of the fun or 
 moon, tha«- point being at an infinite diftance (as to all fenfe), the pro- 
 'leding rays are parallel to each other. 
 
 6. Point of fight is the vertex of the optic cone, or it is that point 
 where the fpedlator's eye fliould be placed to look upon the picture. 
 
 ■7. Di fiance of the picture. The point in which a perpendicular let 
 fall from the point of fight upon the pidure cuts it, is called the center of 
 the pidure 5 the diftance between the center and poiitt of fight is called 
 the difiance of the pi5lure. 
 
LINEAR PERSPECTIVE. 9 
 
 8. Dirediing plane is that plane which pafTes through the point of fght 
 parallel to the pi<5lure. 
 
 9. Original ohjedi the real objecft (whether it be point, line, furface, 
 or folid,) placed in the fituation it is reprefented to have by the pidture. 
 
 10. Original plane. The plane wherein is fituated any original point, 
 line, or plane figure, is called the original plane. 
 
 1 1 . Interfedlion of an original line is that point wherein the faid line 
 (continued if need be) cuts the picture. 
 
 12. InterfeSlion of an origiiial plane is that line wherein the faid plane 
 cuts the pidture. 
 
 13. DireBing point of an original line is that point wherein the faid line 
 cuts the dirediing pla7ie. And a right line joining the dirediing point and 
 point of fight, is called the dire5lor of that original line. 
 
 14. DireSling line of an original plane is that line wherein an original 
 plane cuts the dirediing plane. 
 
 15. 'Parallel of an original line is a right line drawn through the point 
 of fight parallel to the faid original line. 
 
 16. Parallel of an original plane is a plane pafling through the point 
 of fight parallel to the faid original plane. 
 
 17. VaniJJ:ing point is the point where the parallel of any original line 
 meets the pidlure, and is called the 'vanifloing point of that line ; the dif- 
 tance between the vanifiiing point and the point of fight is called the dif- 
 tance of that vanifhing point. 
 
 18. Fanifoing line is that line wherein the parallel of any original plane 
 interfedts the pidlure. From the point of fight let fall a perpendicular to 
 the vanifhing hne j the point where that vanifliing line is fo cut is called 
 the center of it, and the difiiance between the center and point of fight is 
 called the dijlance of that vanifhing line. 
 
 C 
 
 19. Repre- 
 
lo LINEAR PERSPECTIVE. 
 
 19. Ucprejcntationcif <J?7)'^^^rr is the projection of that figure. In order 
 to comprehend the fenfe of thefc definitions more fully, let the planes 
 ABIC, DEO,. Plate III. Fig. i. be fo raifed upas toftand parallel to each 
 other, and bring the plane ACO par>dlel to the plane FGfl: this dene, 
 imagine the plane ABIC to be the furface of the picture, O to be the point 
 of fight, (Def 6. Perfpe(!live) the plane DEO to be the directing plane, 
 (Def. 3.) FG to be an original line (Def. 9.) in the original plane FGH, 
 (Dcf. 10 ) cutting the pi^ilurc in BI and the direfting plane in DE : and 
 let ACO cut the pidare in the line AC, and fuppole OV to be drawn 
 parallel to the original line FG, and cut the pivTture in V : and let FG 
 (produced) cut the pitlure in B, and the directing plane in D ; then is 
 B the interfedtion of the original line FG, (Def. 1 1 .) D its direding point, 
 (Def. 13.) and V its vanilhing point, (Def. 17.) and OD its diredor, 
 (Def, 13.) and OV is the didance of the vanifliing point V (Def. 17.) ; 
 BI is tiie interfe(5lion, (Dcf. 12.) DE the diredting hne, (Def 14.) ACO 
 the parallel, (Def. 16.) and AC the vanifhing line (Def. 18.) of the ori- 
 ginal plane FGH. Froin O let fall a line perpendicular upon the vanilh- 
 ing line AC, and the point wherein the faid perpendicular meets AC, as 
 S, will be the center oi the picture, (Def. 18.) and SO will be the diftance 
 of the vanifliing line AC (Def. 18.). 
 
 A X I o M I ^ 
 
 The common interfedion of the two planes is a flraight line. 
 
 Axiom 2 . 
 
 If two ftraight lines meet in a point, or arc parallel to one another, 
 there may be a plane pafilng through them both.. 
 
 Axiom 3. 
 
 If two parallel ftraight lines are cut by a third line, they will all three 
 be in the fame plane j that is, a plane paffing through any two of them 
 will alfo pafs through a third. Axiom 
 
LINEAR PERSPECTIVE. 
 
 rt 
 
 Axiom 4. 
 
 Every point in any flraight line is in any plane that line is in. 
 
 Theorem i. 
 
 A line drawn from the center of the pidure to the center of a va nifl:- 
 ing line, is perpendicular to that vanifliing hne. 
 
 Demonstration. 
 Raife up the plane AC (Plate III. Fig. 2.) perpendicular to the plane 
 of the pidure AFG ; draw OC perpendicular to AF, and move the tri- 
 angle OSC about OC as an axis, until its bafe CS is perpendicular to a 
 given vanlfliing line, as AB ; then will OS be perpendicular to ASB, and 
 confequently (Def 18.) S is the center of the vanifhing line propofed. 
 
 Corollary. 
 The dillance OS of any vanifliing line ASB, is the hypothenufe of a 
 right-angled triangle, whofe legs are the diftance of the pi6ture OC, 
 and the diftance CS between the center of that vanifliing line and the cen- 
 ter of the picture. 
 
 T H E o R E M 2. 
 
 The perfpe£live reprefentatlon, or projeiflion of any objeft, is the fame 
 as the ichnographic projedtion of it on the plane of the pi6turc, the point 
 of fight being the vertex of the optic cone. 
 
 Demonstration. 
 For, by the explanation of the principles in Definition i . the light mufl 
 come to the fpedator's eye O, (Plate II. Fig. i.) in the fame diredion 
 from any point a of the projection as it would do from the correfponding 
 point A of the original objedit ; therefore it is evident that the rays ^O and 
 AO are in one and the fame flraight line. Wlience it follows that the 
 projection a is the interfcCtion of the pidlurs with the ray AO, and the 
 
 C 2 wliole 
 
1^ LINEAR PERSPECTIVE. 
 
 whole prqjedllon ahcde is the fchenography of the original figure ABCDE, 
 made by the optic cone OABCDE, whofe vertex is the point of fight O. 
 
 Corollary i. 
 
 The projedtlon of a ftralght line is a ftraight line. For conceive a plane 
 paffing through the point of fight, and the original line to interfecft the 
 picture, it is then evident that the interfe<5lion can only be a ftraight line. 
 As ^ is the interfedlion of the pidlure with the triangle ODE, and confe- 
 quently a ftraight line (by Ax. i.). 
 
 Corollary 2. 
 
 The original of a projedion may be any objedt that will produce the. 
 fame cone of rays. Thus the original of the projedlion de may be any 
 line^j^, which produces the optic cone ODE, as well as the line DE. 
 
 This being fo, it may reafonably be afked, whence it happens that 
 figures drawn on a pidture appear to be what they are defigned to repre- 
 fent. The reafon is, becaufe the mind has acquired a habit of judging 
 objeds, that are fo and fo related, have fuch and fuch colours, and are fb 
 and fo enlightened and fhaded, to be of fuch a fliape and alike fituated. 
 Thefe circumftances are all of them neceflary to make a picture complete, 
 though the fimple drawing is fometimes almoft fufficient, upon account 
 of the relation of the parts 5 as in a pavement, where all the ftones ap- 
 pear to be fquare, though they are reprefented by very irregular figures. 
 I fay, it is the relation of the parts which produces this effed: j for the 
 reprefentation of any one of the fingle fquares would hardly appear to be 
 fquare> were there no other objedls to biafs the judgment by their rela*- 
 tion to it. 
 
 Theorem 3. 
 
 The projedlion of a ftraight line not parallel to the pi<5ture, paffes thro' 
 both its interfeftion and vanilhing point, 
 
 PEMON- 
 
LINEAR PERSPECTIVE. 
 
 13 
 
 Demonstration. 
 Imagine an indefinite plane (paffing through the point of fight and the 
 original line) to cut the picture, then it is very obvious that in the line 
 which forms their interfedion, that of the original line with the pidlure 
 will always be found : but by Def. 17. the line determining the vanifhing 
 point will alfo be in this plane, therefore the projedlion of an original line 
 FG (Plate III. Fig. 1.) pafTes through its interfecflion B and vanifliing 
 point V. 
 
 This Theorem being the principal foundation of all the praftice of per- 
 fpediive, the learner would do well to make it very familiar to him. In 
 order to which I have again reprefented the meaning of it in Plate II. 
 Fig. I. where the projection be meets the original line BC in its inter- 
 fedlion K, and pafTes alfo through its vanifhing point V, which is pro^ 
 duced by its parallel O V. 
 
 N. B. When the original line itfelf pafTes through its vanifhing point, 
 the whole projection of it will be that point ; fo that in that cafe the 
 line may be faid to vanifh-: this is one reafon for ufing that term. An- 
 other reafon is, that the further any objed: is off upon any line, the 
 fmaller is its projedlion, and, at the fame time, the nearer to this point, 
 and when it comes inta this point, its magnitude will entirely vanifh, be- 
 caufe the original obje<ft is at an infinite diftance. This is eafily conceived,, 
 by imagining a perfon going from you in a long walk, who appears to be. 
 fmaller and fmaller the further he goes. Tlie reafon of this diminution, 
 will appear from the following Corollaries. 
 
 Corollary i» 
 Projedions of original lines that are parallel to each other, but not to the 
 picture, pafs through the fame vanifhing point: for they have but one- 
 parallel common to them all, and confeq[uemly but one vanifhing points. 
 
H LINEARPERSPECTIVE. 
 
 As in Plate II. Fig. 1. where theprojedlon da and cb of the parallel lines 
 DA and CB meet in their common vanifliing point V. 
 
 Corollary 2. 
 The center of the picture is the vanifhing point of lines perpendicular 
 to the pidlure. 
 
 T H E o R E M 4. 
 
 The proje<5lion of an original line parallel to the pi6ture is alfo parallel 
 to that original line. 
 
 Demonstration. 
 Imagine a triangular plane, whofe vertex is at the fpedtator's eye, and 
 its bafe the propofed original line, to be cut by the pidlure. It is evident 
 tlic reprefentation upon the picfture will be parallel to the bafe of this tri- 
 angular plane, becaufe the picture is fuppofed to be fo ; therefore if EF, 
 Plate III. Fig. 3. be the pid:ure, AB an original line parallel to it, then 
 ^vill its projection ab be parallel to the original AB, O being the point of 
 fight. 
 
 Corollary i. 
 Lines parallel to one another, and to the pidure, are projed:ed into 
 lines parallel among themfelves. Thus ab and dc are parallel to each 
 other, and to their originals AB and CD. 
 
 Corollary 2. 
 The projedlion abed of any plane figure ABCD parallel to the pidure, 
 is fimilar to its original. For draw AC and ac^ then v/ill the triangles 
 ACB, acbi be fimilar, as will alfo the triangles ADC, adc, confequently 
 ABCD and abed are fimilar. 
 
 Corollary 3. 
 AB -.ab diftance of the pidure : diftance between the point of fight 
 and the plane of the original figure j that is, {OgG being drawn perpen- 
 dicular to thofe two planes) AB : : : : OG. 
 
 Theorem 
 
LINEAR PERSPECTIVE* 
 Theorem 5 . 
 
 The projection of a line is parallel to its dired;or. For the lines OF, 
 OG, OD,fg. (Plate III. Fig. i.) are all in the fame plane ; but the di- 
 re6ling plane ODE is parallel to the plane of the piiSlure ABIC (Def. 8.) ; 
 therefore the director OD is parallel to the projeiftlon fg. Eu. n. 16. 
 
 Corollary i. 
 The projedlions of lines that have the fame director are parallel to each 
 other. 
 
 Corollary 2. 
 If the original line is parallel to the pidlure, its director is fo too, and 
 confequently is in the parallel of any plane paffing through that original 
 line ; and therefore the vanifhing line of that plane, and the projecftion 
 of the line are parallel to one another. 
 
 Theorem 6. 
 
 The vanlllilng line, intcrfed:ion, and diredling line of any original 
 plane, are parallel to each other. 
 
 For the planes OVC, DFH, (fee the laft Figure) are parallel, (Def. 
 16.) as alfo ODE and CAB (Def. 8.) therefore the vanifhing Hnc CV, 
 interfcdtion IB, diredling line ED, are parallel to each other. 
 
 Corollary. 
 
 The triangles O/V, DOF, are fimilar, and D0= BV, therefore 
 /V : VO : : BV : DF. 
 
 T H E O R E M 7. 
 
 All lines in any original plane have their vanifhing points in the vanidi- 
 ing line of that plane. 
 
 Demonstration. 
 For as all the original lines are in the fame plane, the lines which de- 
 termine their vanifl^iing points, will interfedt the picture in the vanifhing 
 line of the original plane. 
 
 Corollary 
 
i6 
 
 LINEAR PERSPECTIVE. 
 
 Corollary I. 
 Original planes parallel to each other have the fame vanlfhing line. 
 
 Corollary 2. 
 The vanifhing point of the common interfe<ftion of two original planes, 
 is the common interfetftion of their vanifliing lines. 
 
 COROLXARY 3. 
 
 The vanifhing line of a plane perpendicular to the pidiure, pa0es 
 through the center of the pidure. 
 
 Theorem 8. 
 
 Interfedions of all lines in the fame original plane, and alfo the line 
 which determines the interfedlion of that original plane with the pidure, 
 are in that plane. 
 
 This needs no Demonstration. 
 
 Corollary i. 
 The interfedion of two original planes, which is not parallel to the 
 pidure, being produced, will interfedl the pidture in that point which is 
 the common interfedion of the two planes therewith. 
 
 Corollary 2. 
 If the common interfedion of feveral planes is parallel to the pidure, 
 then thofe planes have parallel interfedion therewith, and have alfo pa- 
 rallel vanifliing lines. 
 
 Problem i. 
 
 Having given the center and diftance of the pidure, to find the pro- 
 jedion of a point, whofe feat * on the pidure, with its diftance from it, 
 are given. 
 
 Let O be the fpedator s eye, Plate III. Fig 4 S the center of the pic- 
 ture, and ^ the given feat of the original point A ; then if we imagine a 
 
 * By feat cn the piSittre^ we mean that point wherein a perpendicular faUing on the pi(flure 
 from the point to be projefted, interfecls it. 
 
 r line 
 
LINEAR PERSPECTIVE. a-7 
 
 line drawn from O to A, it will cut the plane of the picture fomewhere 
 in that line which joins the center of the picture and feat of the original 
 line, as at a, which is therefore the reprefentation of the point A, by 
 Theo. 2. The triangles hah O^S are fimilar; whence :ab SO : Kb. 
 
 Corollary i. 
 The point A may be found by a fcale and compalTes, dividing ilie line 
 Si^ in J, fo that may be to aby as the diftance of the pi(flure OS, is to 
 the diftance h.b of the original point from its feat b. 
 
 Corollary 2. 
 By this Problem the projeftion of any right line may be found ; for it 
 is no more than finding the projedion of any two points therein, and 
 drawing a right line through thofe two projections. 
 
 Problem II. 
 
 To find the projeflion of a line, its vanifhing point and diftance ; hav- 
 ing given its feat, interfedlion, and the angle it makes with its feat, and 
 the center and diftance of the pidture. 
 
 Let W (Plate IV. Fig. i.) be a point in the given line, E its feat on 
 the pidlure, and D its interfedion. 
 
 Make SO equal to the diftance of the pidure, draw SV parallel to 
 DE and perpendicular to SO, join the center of the pidture and feat of 
 the point W by drawing ES ; then it is evident that the projection of W 
 will be in this line : but to determine the exaCl place, we muft divide ES 
 in the proportion of WE to SO, which may be done thus : MakeET^EW, 
 SM=SO J draw TM, and the point of interfedlion iv will be the place 
 required : join the points D, w, and Dw will be the indefinite reprefent- 
 ation of the line propofed. If from the point of fight, a right line be 
 drawn parallel to DW, it will interfedt the pidure in the vaniftiing point 
 thereof, and its diftance will be equal to OV, 
 
 D Corollary 
 
LINEAR PERSPECTIVE. 
 
 Corollary 2. 
 If the vanifhing point and interfeftion of the original line DW be 
 joined by a right line, it will cut ES in w, the projedlion of the original 
 point W. 
 
 Problem III. 
 Having given the projedion of a line, and its vanishing point, to find 
 the projed:ion of the point that divides the original line in any given 
 proportion. 
 
 Let AB (Plate IV. Fig. 2.) be the given projedion of the line to be di- 
 vided, and V its vanifhing point. Draw at pleafure VO and ba parallel 
 to it; and through any point Vof the line VO, draw OA and OB cutt- 
 ing ba in a and b. Divide ab in c in the proportion given, and draw O^ 
 cutting AB in C. Then fhall C be the projection fought, the original of 
 BC being to the original of CA,. as be is to ca. 
 
 For OV being parallel to ba,. ba may be confidered as the original line, 
 and OV as its parallel, and confequently O as the point of fight, and 
 ^O, cOj as vifual rays projeding the points A, B, C. 
 
 Corollary. 
 CAxBV:CBxAV::fiz:c^; for draw ef through C parallel to ab, 
 then we fhall have 
 
 AC : AV : : eC : OV 1 
 
 whence ACxBV : AVxCB ::eC: C/: : ca : cb. 
 
 Wherefore the point C may be found by a fcale and compafTes, making 
 CA:CB: : AV x^-^ : BV x^^. 
 
 Problem IV. 
 Having the projedion of a line, and its vanifliing point, from a given 
 point in that projedlion, to cut ofFa fegment, that (hall be the projection 
 of a given part of the origioal of the projection given. 
 
 Let 
 
 BV : CB : : OV : C/ 
 
LINEAR PERSPECTIVE. 19 
 
 Let AB (Plate IV. Fig.f3.) be theprojed:ion given, V its vanifhing point, 
 and C the point from whence is to be cut off the fegment. Draw at plea- 
 fure VO, and abc parallel to it, and from any point O in VO, draw OA, 
 OB, OC, cutting ^ ^ in ^z, c. Make cd to aby as the given part is to 
 the original of AB, and draw O^ cutting AB in D : then will CD be the 
 fegment fought. 
 
 The Demonftration of this problem is obvious from that of the fore- 
 going one. 
 
 Corollary 
 
 The point D may be found by a fcale and compaffes, making DC : DV 
 : : X AB X CV : «3 X AV ^ BV. For, per laft Prob. 
 
 ABxBV: AVxBC wab-.bc. 
 BCxDV :BVxCD: : bc:dc. 
 
 and therefore DC :DV : : ABxCV : ^z^x AVxBV. 
 
 N. B. The foregoing Problem may be confidered as a particular cafe of 
 this, wz. when the point C in this coincides with one of the points 
 A or B. 
 
 Problem V. 
 
 Having given the center and diftance of the pidure, to find the vanifli- 
 ing line (with its center and diftance) of a plane, whofe interfeftion is 
 given, with the angle of its inclination to the picture. 
 
 Let C (Plate III. Fig. 2.) be the center of the pidure. Draw CG pa- 
 rallel to the interfedion of the original plane with the picture, and equal 
 to the diftance of the pidlure, and from C ere(£t the indefinite right line 
 CS perpendicular to CG ; draw GS, making- an angle SGC equal to the 
 complement of the angle of inclination of the original plane to that of the 
 pidlure ; draw through S the right line ASB. Then will ASB be the 
 vanifliing line fought, S its center, and GS its diftance. 
 
 Becaufe GC=CO, and the angle GSC equal to the given inclination of 
 
 D 2 the 
 
 I'whence AB x C V x D V x = AV x B V x CD x 
 
20 LINEAR PERSPECTIVE. 
 
 the original plane and pifture (by conftrudtion), it is evident that the right 
 angled triangle GCS is equal to the moveable right-angled triangle COS ; 
 therefore, by Defin. 18. ASB is the vanifliing line, and S its center. 
 
 N. B. Taking GC for radius, CS is the cotangent, and OS the cofecant 
 of the inclination of the original plane to the pidure. 
 
 Problem VI. 
 
 Having given the interfed:ion of an original plane, with its vaniiliing 
 line, its center and diftance ; to find the projeftion of any line in the ori- 
 ginal plane, having the original figures drawn out in their juil proportions. 
 
 Let DF (Plate IV. Fig. 4.) be the interfedion given, HG the vaniiliing 
 line, and S its center. Draw SO perpendicular to GH, and equal to the 
 diftance of the vanifliing line GH, and let the fpace X be the original 
 plane, feen on the reverfe, as objedls appear in a looking-glafs ; the fpace 
 Y being the parallel plane in the fame manner folded down on the pidbure : 
 and let AB be the original line whofe projedtion is fought. Let AB cut 
 the interfe(flion in D, and draw OG parallel to AB, cutting the vanifli- 
 ing line in G. Draw DG, which will be the indefinite projection of AB. 
 Through A and B draw at pleafure AC and BC meeting in C, and in the 
 fame manner find their indefinite projedions FI, and EH, cutting DG in 
 a and Then will be the determinate projedlion of AB, a being the 
 projedion of the extremity A, and ^ the projedtion of the extremity B. 
 
 Otherwise, 
 
 Suppofe KL to be the original line given. Having found its indefinite 
 projedtion QG, as before, draw OK and OL, cutting it in k and /, which 
 will be the projedlions of the extremities K and L. 
 
 Otherwife, by the Directors, 
 Let DF (Plate IV. Fig, 5.) be the interfedtion given. And let the origi- 
 nal plane be folded down on the pidture, fo as to bring the diredling line 
 into the plane HI,the diftance between AF and HI being equal to the 
 
 diftance 
 
L I N E A R P E R S P E C T I V E. 21 
 
 diflance of the vanifliing line given. Let alfo O be the point of fight 
 brought into the pidure at the fanae time along with the directing plans 
 HIO. To find the indefinite proje6tion of any original line AB, conti- 
 nue it till it cuts EF in E, and HI in G ; then draw OG, and ¥a drawn 
 parallel to it will be the indefinite projedion fought. Then finding in the 
 fame manner the indefinite projeftion E^/ of any other line, AD paffing 
 through A, by its interfedion with Ya is got the projedion a of the ex- 
 tremity A. And in the fame manner is got the other extremity b ; or 
 thofe extremitie-s might be found by drawing lines from A, and from B 
 to O, as in the foregoing confi;ru£tion. 
 
 To prove the truth of thefe operations, imagine the figures (Plate IV"» 
 Fig. 4.) to be folded in DF and HG, and (Plate IV. Fig. 5.) in EF and HI, 
 till the original plane, its parallel, and the directing plane, and along 
 with them the point of fight O, come into their proper places. Then you 
 willfind, thatD (Plate IV. Fig. 4.) and F (Plate IV. Fig. 5.) will be thein- 
 terfe<f;tion of AB, and G (Plate IV. Fig. 5.) will be its direfting point. But 
 OG (Fig. 4.) is fiill parallel to AB, wherefore G is its vanifhing point, 
 and DG its indefinite proje6lion, (by Theo. 3.) and F^ (in Fig. 5.) is flill 
 parallel to OG, which is the diredor of AB ; wherefore F^ is the inde- 
 finite projection of AB (Theo,. 5.) That a found by the interfeftion of 
 FI with DG (in Fig. 4.), and of Ed with ¥a (in Fig. 5.), is the projec- 
 tion of the interfedtion of the original lines AB and AC (Fig. 4.), and of 
 AB and AD (Fig. 5.) is obvious. The other conftrudion by the lines AO 
 is the fame as by the lines AO and CO for finding the points a and 
 Prob. II. 
 
 N. B. In Fig. 5. the projedions ad and kl are parallel, their originals^ 
 having both of them the fame diredtor OH, according to Cor. i . Theor. 
 5. The fame may be obferved in Im and cd, which have the fame di- 
 redor OI. 
 
 P R O B L E ?A 
 
22 
 
 LINEAR PERSPECTIVE. 
 
 Pr O B L E M VII. 
 
 Having given the fame things as in the foregoing problem, to find the 
 projedlion of any figure in the original plane. 
 
 This is done by finding the projedlions of the feveral parts of the figure 
 given, by the foregoing problem. 
 
 For example, the projection i^/w;^^ (Plate V. Fig. i.) of the pentagon 
 KLMNP is found thus. Drawing OG, OH, OI, OV, parallel to KL, 
 LM, MN, KP refpediively, the points G, H, I, and V, are vanifhing 
 points : and KL, LM, MN, being continued, cut the interfedtions in 
 their interfedtions Q^R, T whence drawing QG, RH, TI, are got the 
 projedlions /, m, of the points L and M, by their mutual interfedlions. 
 Then drawing OK and ON, are got the points k and n. Then drawing 
 ^V to the vaniQiing point V of KP, is got the indefinite projedlion of KP. 
 Laftly, drawing OP is got the point p. 
 
 The projections of curve-lined figures are to be got by finding the pro- 
 jections of feveral of their points, and afterwards joining them neatly by 
 hand. Thus (Plate V. Fig. 2.) DE being the interfeCtion, and VFthe va** 
 nifhing line, and O the point of fight, and ABC an original circle, placed 
 as in the foregoing problem, the projection a of any point A may be found 
 by drawing at pleafure AD, and OV parallel to it ; then drawing DV 
 and OA meeting in the point fought a, according to the conftruCtion in 
 Problem VI. D being the interfeCtion, and V the vanifhing point of the 
 line AD, and the feveral lines AD being drawn parallel to one ano- 
 ther, the fame vaniOiing point V may ferve for them all. 
 
 Or VF (Plate V. Fig. 3.) being the directing line brought into the pic- 
 ture, the reft remaining as before, drawing at pleafure AD cutting DE 
 and VE in D and V ; then drawing O V and D^ parallel to it, the projec- 
 tion a is got by drawing OA cutting Da in a. And the fame point V be- 
 ing ufed for all the points A, all the lines Da will be parallel to one an- 
 cther, and to the lame line OV^ 
 
LINEAR PERSPECTIVE. 
 
 Problem VIH. 
 
 To find the proje^lion of any figure in a plane parallel to the pidlure. 
 
 The projedion being fimilar to its original, (by Cor. 2. Theo. 4.) this 
 IS done by making an exadt copy of the original figure ; making the ho- 
 mologous fides in the proportion explained in Cor. 3. of the fame Theorem^ 
 
 Problem IX. 
 
 Having given the vanifliing line of a plane, its center and diftance, and 
 the projeiftion of a line in that plane; to find the proje<5lion of another 
 line in that plane, making a given angle with the former. 
 
 Let the plane HFIE (Plate V. Fig. 4.) reprefent the pidlure, which raife 
 up and place the plane EIO parallel to the original plane HBFH. Now 
 if we fuppofe O to be the point of fight, EI will reprefent the vanifliing 
 line. Let ab be the given projection ; it is required to draw ac, fo that 
 the original of the angle bac may be equal to a given angle. Continue ab 
 to its vanifhing point G, draw GO and OI, making GOI equal to the 
 given angle, and cutting the vanifhing line in I i then draw lea, which 
 will be the line fought. 
 
 The reafon of this conftrudtion is very eafy to be underftood ; for the 
 plane EOI being parallel to the original plane, it is evident that any two 
 lines, as EO, GO, drawn from the vaniflbing points E, G, to the point 
 of fight O, will form an angle at that point equal to the angle contained 
 by thofe original lines, of which E and G are the vanifliing points, 
 
 N. B. If it had been required to make abc to reprefent the angle ABC,- 
 the angle EOI mufl have been made equal to the complement of the angle 
 ABC to tvv^o right angles. 
 
 Problem X. 
 Having given the vanifliing line of a plane, its center and diflance, and 
 the projection of one fide of a triangle of a given fpecies in that plane j to 
 find the projection of the whole triangle. 
 
 Ther 
 
24 LINEAR PERSPECTIVE. 
 
 The proje<5lions of the fides wanting are to be found by the foregoing 
 problem, the angles of the triangle being given. Thus having given the 
 projedion aby (fee the laft Figure) of the fide AB of the triangle ABC, 
 the vanlfhing point of the fide ac is found by making the angle JOG equal 
 to the angle CAB, and the vanifhing point of the fide be is found by mak- 
 ing the angle JOE equal to the complement of the angle CBA to two 
 right angles. 
 
 N. B. If the vaniil:iing point of the line given ab, is out of reach, you 
 may proceed thus : Taking any line DR, Plate VI. Fig. i . (parallel to the 
 vanifliing line HG by Theorem 6.) for the interfecftion, by means of two 
 two lines, Hi^E, la¥, drawn at pleafure through b and find the origi- 
 nals A and B of the points a and b ; (by the converfe of Problem I.) and 
 draw AB. Then on the fide AB complete the original triangle, and find 
 the projection of the fides wanting by Problem VII. 
 
 Problem XI. 
 
 Having given the vanifhing line of a plane, its center and difi;ance, and 
 the projedlion of one fide of any figure in that plane, to find the projection 
 of the whole figure. 
 
 Divide the whole figure given into triangles, by means of diagonals, 
 and find the projections of thefe triangles one after the other (by the lafl 
 Problem), beginning with thofe that have the line given forDnc of their 
 fides. 
 
 Problem XII. 
 Having given the center and diftance of the picture, and the vanifhing 
 line of a plane, to find the vanifhing point of fines perpendicular to that 
 plane. 
 
 Let AB (Plate VI. Fig. 2,) be the vanifhing line given, and C the cen- 
 ter of the picture. Draw CA perpendicular to AB, and CO parallel to 
 it, and equal to the diftance of the picture. Draw AO and OD perpen- 
 dicular 
 
LINEAR PERSPECTIVE. 25 
 
 dicukr to it, cutting CA in D, which will be the vanifhing point fought; 
 for raife up the triangle AOD perpendicular to the plane, of the fcheme, 
 (which here reprefents the'jpidture) then a plane pafling though the point 
 of fight O, and the line AB will be the parallel of the original plane, and 
 the line OD will be perpendicular to it, and confequently will be the 
 parallel of lines perpendicular to that original plane. Wherefore D is 
 the vanifliing point of thofe perpendiculars, (by Def. 17.) 
 
 N. B. I. When the vanifliing line AB paffes through the center of the 
 pidlure, that is, when the original plane is perpendicular to the pidurc, 
 the point D will be infinitely diflant, the line OD being parallel to AD, 
 and the projedtions of the lines perpendicular to the plane propofed, will 
 all of them be perpendicular to AB, they being to meet the line AC, 
 which is perpendicular to it at an infinite dillance, and confequently they 
 will be parallel to one another; which they ought to be upon another 
 account, their originals being all parallel to the pidlure. 
 
 N. B. 2. But when the original plane is parallel to the picture, the 
 diftance CA will be infinite, and confequently OA will be parallel to 
 CA, and OD will coincide with OC, making the point D to fall into 
 the center of the picture C, agreeable to Cor. 2. Theo. 3. 
 
 N. B. 3. CD is a third proportional to AC and CO, as alfo is AD to 
 AC and AO. 
 
 N. B. 4. OD is the diftance of the vanifliing point D, 
 
 Problem XIII. 
 Having given the center and diftance of the picture, to find the vanifli- 
 ing Une, its center and diftance, of planes that are perpendicular to thofe 
 lines that have a certain vanifliing point. 
 
 The folution of this problem eafily follows from that of the laft, for 
 let C be the center of the pidlure, and D the vanifliing point propofed, 
 (fee the laft figure) then it is very evident that in order to determine 
 
 E the 
 
'26 LINEARPERSPECTIVE. 
 
 the point A, we muft join the points D C, and draw CO perpendicular 
 to DC, and equal to the diflance of the pidure ; then draw DO, and 
 mak'e OA perpendicular to itj draw AD perpendicular to the right line 
 DCA, and ir will be the vanilliing line required, A being its center, (by 
 Theo. I.) and OA its diftance. 
 
 Problem XIV. 
 
 Having given the center and diflance of the pidure, through a given 
 point to draw the vanifliing line of a plane that is perpendicular to ano- 
 ther plane, whofe vanifliing line is given, and to find the center and 
 diftance of that vanifhing line. 
 
 Let AB (the fame Fig. as before) be the vanifliing line given, and C 
 the center of the pidure, and let E be the point given. Find the vanifh- 
 ing point D of lines perpendicular to the original planes of AB (by Prob. 
 XII.) : Draw DE, which will be the vanifhing line fought j draw CF 
 cutting DE at right angles in F, and F will be the center of the vanifh- 
 ing line DE (by Theo. i.) Make aright-angled triangle, whofe bafe is 
 CF, and its perpendicular is equal to the diflance of the pidure, and its 
 hypothenufe will be the diftance of the vanifhing line DE (by the 
 Cor. Th. I.) 
 
 Now becaufe the plane, whofe vanifhing line is fought, is perpendi- 
 cular to the other plane, its vanifhing line muft pafs through the vanifli- 
 ing point D of lines perpendicular to that other plane, becaufe fome 
 of thofe lines are in the plane fought. Therefore DE is the vanifhing 
 line fought. 
 
 In the above conftrudion, where DE is faid to be the vanifhing line 
 fought, it is to be underftood as the vaniftiing line of a plane perpendi- 
 cular to that whofe vanifhing line AB is given, without having regard to 
 its fituation with refped to the pidure, only that it fhall be perpendicu- 
 lar to the plane of which AB is the vanifhing line j for were the propofed 
 
 perpendicular 
 
LINEAR PERSPECTIVE. 
 
 perpendicular plane, when produced, to interfedt the pidure in a given 
 angle, fuch reftridion might render it impoffible that the plane producing 
 its vanifliing line, fliould pafs through both the points D, E. 
 
 Corollary i. 
 If PC be continued till it cuts the vanishing Hne given in B, B will be 
 the vanifhing point of lines perpendicular to the original plane of the va- 
 nifliing line DE : For that vanifhing point is in the line EC, by the con- 
 ftru(Stion of Problem XII. and it is in the vanifliing line given by what 
 has been proved in the prefent Problem. 
 
 Corollary 2. 
 And therefore, if the vanifhing lines AB and DE meet in G, the points 
 B, D, and G, will be the vanifhing points of thethree legs of the folid 
 angle of a "cube, which are perpendicular to one another. And drawing 
 DB, BG, GD, and DB will be the vanifhing lines of the three planes 
 that contain that folid angle. 
 
 Corollary 3. 
 The diflance of the vanifhing line DG is equal to the line FP, the 
 point P being the interfedion of the line FC, with a circle defcribed on 
 the diameter DG. 
 
 Problem XV. 
 
 Having given the center and diftance of the pidure, and the vanifliing 
 point of the common interfedlion of two planes that are inclined to one 
 another in a given angle, and the vanifhing line of one of them ; to find 
 the vanifhing line of the other of them. 
 
 Let C (PL VI. Fig. 3.) be the center of the pidure, BG the given 
 vanifhing line of one of the planes, and B the vanifhing point of their 
 common interfedion, and H the angle of their inclination to one another. 
 Find the vanifhing line GD, of planes perpendicular to the lines whofe 
 
 E 2 vanilhing 
 
2^ LINEAR PERSPECrrVE. 
 
 vanifliing point is B (by Prob. XIII.) ; let that vanifliing line cut the va- 
 nlfhing line given in G. In GD find the vanifliing point E of lines 
 making the given angle H, with the lines whofe vanifliing point is G 
 (by Prob. IX.) ; that is, in BCF perpendicular to GFD ; take FP equal 
 to the diftance of the vanifliing line GD, (found by Prob. XIII.) and 
 draw PG, and PE, making the angle EPG equal to H ; draw BE, which 
 will be the vanifliing line fought. 
 
 For a demonflration of this conflrudion, raife up the planes EPG, 
 BPG, until they meet in P, the point of fight perpendicular over the 
 center of the pidlure C ; then if we imagine a plane to pafs through P, 
 and making the vanifliting line BD, the planes BPG, GPD, DPB, will 
 be the parallels of three original planes, whofe vanifliing lines are BG, 
 GD, DB 5 that whofe vanifliing line is DG, being perpendicular to the 
 other two (by the confl:rudlion, becaufe it is perpendicular to their com- 
 mon interfedion, whofe vanifliing point is B.) Therefore the original 
 planes, whofe vanifliing lines are BG and BD, are inclined to one ano- 
 ther in the angle EPG, or H ; (for the inclination of two planes is al- 
 ways meafured in a plane perpendicular to their common interfedion) 
 therefore BG being the vanifliing line given, BE is the vanifliing line 
 fought. 
 
 N. B. The center of the vanifliing line BE is found by drawing a line 
 perpendicular to it from C, (by Theo. i.) and then its diftance is 
 found, as was found the diftance PF in Prob. XIV. 
 
 Otherwise, 
 
 Having drawn the indefinite right line BCF, make CF equal to a third 
 proportional to BC, and the diftance of the pidlure; draw DFG perpen- 
 dicular to BF ; then take FP equal to the quotient arifing by dividing 
 the fum of the two fquares of BC, and the difl:ance of the pidiure, by 
 the right line BF ; draw GP, and from the point P draw P E, making 
 
 an 
 
LINEAR PERSPECTIVE, 29 
 
 , an angle GPE equal to the given angle H j which being done, draw BE, 
 and it will be the vanifliing line required. 
 
 Problem XVI. 
 
 Having given the center and diftance of the pidure, and vanifhing 
 line of one face of any folid figure propofed, and the projedlion of one 
 line in that face ; to find the projedlion of the whole figure. 
 
 By means of the projection given, find the projection of the ichno- 
 graphy of the figure propofed on the plane of that face, whofe va- 
 nifhing line is given (by Prob. XI.) ; then (by Prob. XIV.) find the va- 
 nifhing line of the plane of the orthography, and defcribe the projection 
 of the orthography, by help of the lines already given in the ichnogra- 
 phy (by Prob. XI.). L'uftly, by the interfedions of the projections of 
 perpendiculars to the ichnography and orthography, will be found the 
 feveral points of the projection required. 
 
 Otherwise, 
 
 Having found the projection of the face whofe vanifhing line is given, 
 by means of the projection of the line given, find the vani/hing lines of 
 the adjacent faces, (by the lafi: Problem) and defcribe their projections 
 by help of the lines given in the projection of the firft face, and fo on, 
 till the whole projection fought is completed. 
 
 This is a general defcription of the method to be ufed in putting any 
 figures propofed into PerfpeCtive, and by the rules herein delivered, the 
 following examples were aCtually delineated. Having in thefe few pages 
 done my beft endeavour to render Dr. Brook Taylor's PerfpeCtive eafy to 
 be underflood, I fhould have here ended ; but as feveral confiderable 
 writers upon this fubjeCt have objeCted againfi: the univerfality of thefe 
 rules, and their holding equally good, when applied to putting of fome 
 particular forts of objeCts, fuch as round pillars, &;c. into Perfpec- 
 
 tive. 
 
30 LINEAR PERSPECTIVE. 
 
 tive, I hope the following attempt to fet this matter in a clear light, will 
 not be difagreeable to the reader. 
 
 The appearance of an objedl and its reprefentatlon are two very diffe- 
 rent things, the former being determined by the angle (at the eye) under 
 which it is feen ; but the latter is the adual figure formed upon the 
 pidture, by the pencil, of rays paffing from every point (in view) of the 
 obje6t to the fpe(flator's eye ; confequently, in viewing a pidlure, if the 
 eye be fuppofed to move about from the proper point of fight defigned by 
 the painter, it is no wonder that feme remote parts in the picture fliould 
 appear diftorted ; and for a very obvious reafon, namely, becaufe it is 
 very poflible that the reprefentation of a right line may, upon account of 
 its fituation with regard to the pidtare, be much longer than the original 
 line itfelf : but that a right line placed in any pofition whatfoever, can ap- 
 pear longer than itfelf, is abfolutely impofilble, and contradidlory to the 
 very nature of the fcience of Perfpedive. But to render this fiill plainer 
 by an example, let AB, PI. 39. Fig. i. be the plane of the pidure, CD 
 an original right line, O the point of fight j draw OC, OD, and fup- 
 pofe CD fo fituated, that the angle ODC may be a right one : then it is 
 very evident that is the reprefentation of CD; but becaufe C^/ is 
 the hypothenufe of the right angled triangle CD^, it follows that C^ 
 mufl be longer than CD, that is, the reprefentation is longer than the 
 original line ; but notwithftanding this, it is certain, that to the fpedator's 
 eye at O, the reprefentation Cd can only appear as the tangent DC (OD 
 being radius) of the angle DOC, formed by the rays OD, OC, pafiing 
 from the points C, D, to the eye placed at O. For a fecond example of 
 this kind, let it be required to draw the reprefentation of a range of 
 columns parallel to the pidture. If they are drawn according to the fi:ridl 
 rules of Perfpedive, then that column which is in the center of the pic- 
 ture will be the leaft, and confequently thofe on each fide of it will be 
 
 larger 
 
LINEAR PERSPECTIVE. 31 
 
 larger and larger continually, the farther they are removed from the center 
 of the pidure, (if the original columns are all of the fame diameter.) 
 But to explain this more fully, let PP reprefent the plane of the pidure, 
 PI. 39. Fig. 2. H, G, and I, the centers of the feclions of three columns, 
 whofe reprefentations we are to find upon the pidure ; and let E be the 
 place of the fpedator's eye : draw the tangent E//, E^, EA, EB, E/, and 
 E/^; then will their interfedions with PP determine the reprefentations 
 cd, ab, ef. Now it is very evident that cd and ef are much longer than 
 ab. From whence we may conceive, that the farther any column is re- 
 moved from the center of the pidure, the longer will its reprefentation 
 be; and we may moreover conceive, that this increafe of the reprefenta- 
 tions of the magnitude of the columns, is owing to the obliquity of the 
 chords hg, ik, with the pidure ,• which chords will meafure the appa- 
 rent widths of the columns, if viewed from the proper point of figlit, 
 and will appear lefs and lefs, the farther they are removed from the per- 
 pendicular EG. But if we were to draw the appearances of thefe co- 
 lumns upon the pidure, without having regard to any particular point of 
 fight, but only that the eye be placed at a due diftance from the pidure j 
 then with the perpendicular diflance of the eye from the plane of the 
 pidure, defcribe an arch of a circle cutting the tangent E//, E^, E^, 
 &c. and the chords of the arches fo made by thefe tangents, will be the 
 apparent widths of the columns to be reprefented in Perfpedive. Thus 
 if "Ea = 'Eb be the radius, then will rs and tw reprefent the perfpedive 
 appearances of the widths of the columns I and H refpedively. 
 
 E N D of the T H E O R Y. 
 
( 3« ) 
 
 PRACTICAL PERSPECTIVE. 
 
 IN this pracflice of Perfpecftive, I have thought it convenient, that 
 the vanifliing line to the ground plane, fliould be called the horizontal 
 line, and the interfeSlion of the piBure with that plane, the ground line. 
 In order to underftand this well, I have fet down the following general 
 definitions. 
 
 Plate XXXVIII. F i g. I. 
 
 ' LH, ED, the pi^ure ; FM, GK, the ground phne. 
 O, the point of fight; LVH„ the horizontal line. 
 V, the center \ the line OV, the dijlance. 
 
 CI, LH, the horizontal plane ; CI, FM, the direBlng plane : AB, 
 an original line j ab the perfpecStive reprefentation of it j P the inter- 
 feding point j and Z the direding point. 
 
 CHAP. I. - 
 
 Shewing in what manner any fuperficialfigure may be put in 
 
 Perfpedive. 
 
 The diftance the fpedlator fhould ftand from the p'lBure (or what 
 is meant by the point of dijlance) is very efTential j for on a right choice 
 of it depends the agreeable appearance of the objects. If it is placed 
 very near, the reprefentation of fome objeds will be greater than their 
 6 originals. 
 
PRACTICAL PERSPECTIVE. 33 
 
 originals, as a, ^,{lnFig. 2. Plate 38.) but this muft be left to the dif- 
 cretion of the artift, which experience will improve. 
 
 EXAMPLE I. 
 Plate VII. Fig. i. 
 (Is a method made ufe of by an ingenious author, as the nearefl diflance.) 
 
 ABCD, is the piSiure^ G, the centre, CD, the interfetling or ground 
 line. Through the center, draw HL parallel to CD, for the horizontal 
 line, on G raife the perpendicular GO, equal to GD, or GC, let O be 
 the point the Ipedtator's eye is fuppofed to be. 
 
 EXAMPLE II. 
 Plate VII. Fig. 2. 
 To find the reprcfentation of a line given. 
 
 The pidure being prepared as above, HL, for the horizontal line, C, 
 the center, O, the point of fight, DB, the ground line, and AB, the 
 given line ; touching the ground line in B, make OH parallel to AB, 
 interfering the horizontal line at H, for the vanifhing point of AB ; draw 
 BH, and from A draw the ray OA, interfering HB at a ; join aB for the 
 Perfpedive reprcfentation of the original AB. 
 
 Fig- 3- 
 
 The original AB is continued to the ground line at d, the vanifliing 
 point H, is found as in the foregoing Ex. The rays BO, AO, cutting 
 the line dH in ba, gives ab for the reprcfentation of AB, and as deep in 
 the pi^ure as AB is from the ground line. 
 
 EXAMPLE III. 
 Plate VIII. Fig. i. 
 
 In this example the original line DB is perpendicular to the piSlurCy 
 HL, the horizontal line, C, the cer:ter, GF, the ground line, and OC, the 
 
 F d fiance j 
 
34 PRACTICAL PERSPECTIVE. 
 
 dijiance ; continue DB to the ground line at e, and draw at pleafure two 
 lines, B2, Bi, parallel to each other, cutting GF at i and 2, make OE 
 parallel to any one of them, draw 2E, lE, then draw ^C, interfering 
 them at hd^ join bd for the reprefentation of DB. For the line A, make 
 OC parallel to it, then C is the vanifhing point of A, and DB. 
 
 EXAMPLE IV. 
 
 7(? find the reprefentation of a given Triangle. 
 
 Let BAD be an original triangle, HL, the borizo?ital line, GF, the 
 ground line, OC, the diflance, produce the fides AB, BD, to the ground 
 line at 1 and 2. Make OH parallel to AB, and OL parallel to BD, 
 cutting the horizontal line at H and L the 'vaniping points, for AB, and 
 BD, draw iL, 2H, interfe£ling each other in by draw the rays AO, 
 DO, cutting the lines iL, 2H, in ^, and d, join abd for the reprefentation 
 of BAD. Note AD, is parallel to the ground line : fo is ad. 
 
 EXAMPLE V. 
 Plate VIII. Fig. 3. 
 
 Having an original Square given to find its VerfpeJlive Reprefentation, 
 
 Let LH be the horizontal line, GF, the ground line, O, the point of 
 fight, ABDE, the original fquare, the fides AD, BE, being perpendicular 
 to GF, and OC, parallel to BE, cutting the horizontal line at C, for 
 the vanifhing point of the fides AD, BE, and centre of the pidure, 
 continue AD, BE, till they interfedGF, at i, and 2, draw iC, 2C, then 
 produce the diagonal DB, to the ground line at 3, and make OH parallel 
 to DB, cutting HL, at H, H is the vanifhing point of DB, draw 3H, 
 cutting iC, 2C, at by and dy make ba^ de^ parallel to GF, (their original 
 being fo) cutting iC, 2C, in the points ^7, join the points abd e, for the 
 Perfpedive Reprefentation of ABDE. 
 
 EXAMPLE 
 
PRACTICAL PERSPECTIVE. 35 
 
 EXAMPLE VI. 
 Plate VIII. Fig. 4. 
 
 To give the reprefentation of a fquare, thai lies obliquely to the pi5fure. 
 
 Every thing being prepared as in the foregoing example, continue the 
 fides BA, and ED, of the original fquare, at the ground line GF ; to the 
 points 1, 2, 3,4, make OL parallel to A9, BE, and OH, parallel to BA, 
 ED, draw the lines 4H, 3H, iL, 2L, interfering each other at the 
 points, a, b, e, d, the PerfpeBive of A, B, D, E . 
 
 EXAMPLE VIL 
 Plate IX. Fig. i. 
 To give the reprefentation of a Tent agon, from an original given. 
 
 HL, the horizontal line, O, the point of fighty GF, the ground line, 
 A, B, C, D, E, the original^ N, P, Q^M, the interfering points on the 
 ground line of the fides AB, CB, DC, EA ; L, the vanifhing point of the 
 fide CB, H, of the fide AB, K, of the fide EA, and I of the fide CD. 
 Draw PH, QL, interfedting each other at b, and the lines KM, IN, 
 interfeds QL, PH, in c, and a ; draw the rays DO, EO, interfering NI, 
 MK, in d znde, join thefe points to finifh the projedion required. 
 
 Figures containing any number of fides are put in perfpedlive in the 
 fame manner. 
 
 EXAMPLE VIIL 
 Plate IX. Fig. 2. 
 To find the PerfpeSHvc of a circle or any given curved Ury;, 
 
 HL, the horizontal lincy EF, the ground line, O, the point of fight, 
 and A, B, C, an original circle given, D its center. Draw at pleafure 
 from the point B a line interfering the ground line at F, and as many 
 through the circle (parallel to BF) as you pleafe ; make OK parallel to 
 6F, then is K, the vanifhing point to them all. The point b of the 
 
 . F 2 original 
 
36 PRACTICAL PERSPECTIVE. 
 
 original B is found thus, draw FK, and from B draw the ray BO, inter- 
 fering FK at b the point fought, the point a is found by drawing EK, 
 and AO cutting EK. at the point a, the points i, 2, 3, and as many more 
 are found in the fame manner ; the ray DO gives on the line iK, the 
 point d for the center. Thefe points mud be joined artfully for the Per- 
 fpeftive reprefentation. 
 
 EXAMPLE IX. 
 Plate X. Fig. i. 
 To put an OBagon in Pcrfpc&ive. 
 
 HL, the horizontal line, O, the point of fight, and GD, the ground 
 line, AB, the original, C, the center, M, N, the interfedling points of 
 the lines MK, IN ; H L i 2, vanifliing points : viz. i for the fide B, 
 2 for the fide A, L, for MK, and H, of IN. Draw NH, ML, and, 
 through their interfeflion, draw a line 5 6 C , and 4 3 parallel to MN ^ 
 from L, through the point 3, draw a line cutting 6, 5, at 5, then draw 
 4L cutting 5, 6, at 6 J draw 3, i, gives 8 on ML, and 4, 2, gives 7, on 
 NH, then a line from 2, through 3, gives a, on NL, and a line from i, 
 through 4, gives b, on ML, and finiflies the odlagon. 
 
 The circle, (Fig. 3.) is put in Perfpeclive by the fame points and 
 manner. 
 
 EXAMPLE X. 
 Plate XI. Fig. 3. 
 To give a true Reprefentation of a Circle in any part of the Fi6iure^ by 
 having only a Line given as a Diameter. 
 
 Prepare the pidture in this manner. Let AB be the horizontal line, 
 OC, the point of difiance, for convenience to be transferred on the 
 line AB, in A, and B j C, the center. Set one foot of the compafles in 
 the point B, with the opening EO, defcribe the arch OL ; and from 
 AO, the arch OV ; let 3, 7, be a diameter given, through the mid- 
 dle 
 
PRACTICAL PERSPECTIVE. 37 
 
 die 9, draw C51, at pleafure, from A, and B, draw lines through the point 
 9, at pleafure. To find the points i, 2,4, 5, 6, and 8 j from B, through 7, 
 draw a line, interfedling the line 51, at i; from 3 to B, gives the point 5 ; 
 from V, through 7, aline drawn cuts the line A, 9, at the point 8, and 
 from 3, to V, gives the point 4, on the fame line ; from L, through 3, a 
 line cuts E9 at the point 2, and another from 7, to L, gives the point 6. 
 
 This method may ferve for an ocftagon, a fquare, and many other ufes, 
 and is the readieft way to reprefent any circles within each other. 
 
 CHAP. II. 
 
 Pradical Perfpedive of Solid Bodies. 
 
 EXAMPLE XI. 
 Plate XII. Fig. i. 
 To give the "Reprefent ation of a Cube from an Origi?jaJ given. 
 
 Let HL be the horizontal line, C, the center^ OC, the difiance, and H, 
 vani fling point of CB, and A BCD the plan of the original cube, find 
 the Perfpective fquare, a, b, c, d, of ABCD, as (by Ex. V. Plate VIII. 
 Fig, 3.) at point 2 on the ground line raife a perpendicular 2^ equal to any 
 height given, on every point abed raife perpendiculars at pleafure, from 
 the point of the line 2e, draw a line to the center C, cutting the per- 
 •pendiculars ^ d, at 6 and 7, from the points 6 and 7, draw lines parallel 
 to a b, cutting the perpendiculars a and c at 5, 4. 
 
 EXAMPLE XII. 
 Plate XII. Fig. 2. 
 T 9 fnd the Perfpeciive Reprefent ation of a Pyramid, the Ichncgraphy, or 
 Ground Plan CDEF being given, whofe Pofition is oblique to the PiSlure. 
 
 Let HL be the horizontal line, S the center, O point of fight, 
 
 L 
 
38 PRACTICAL PERSPECTIVE. 
 
 L vantJJnng point of the fides CD, EF, H vanifhing point of CE and 
 FD, the line AB on the ground line is a given height. Find the Perfpec^ 
 //wplan cdef (by Ex. VI. Plate VIII. Fig. 4.) draw ed, cf, to iind the 
 center a of the Perfpe(5live plan, on a raife a perpendicular at pleafure ; 
 from L through the center a draw a line to the ground line, at that in- 
 terfedlion fet the given line AB, draw BL cutting the perpendicular ah, 
 at the point 3, join bc^ bd^ be, bf, and compleat your pyramid. 
 
 EXAMPLE XIIL 
 Plate XII. Fig. 3. 
 T 0 put a Cube in FerfpeBi've Jlanding obliquely to the PiSlure. 
 
 Every thing ftanding as in the foregoing example, the Perfpedive 
 ground plan a b c d, the original ABCD, is found (by Ex. VI. Plate 
 VIII. Fig. 4.) at the point i fet a line iF for the given height, on every 
 point, a, h, c, d, raife perpendiculars at pleafure, draw FL, cutting the 
 perpendiculars a, d, at 5, 6, from H, through 5 and 6, draw lines, cutting 
 the perpendiculars, b, c, at the points 8 and 7, then is your folid figure 
 finifhed. 
 
 EXAMPLE XIV. 
 Plate XIII. Fig. i. 
 
 'To put a Chair in Perfpediive from a given original Plan, ivhofe Pojition 
 
 is parallel to the Pidiure. 
 
 Let ML be the horizontal line, N, the center and vanifliing point of 
 the fides AC, and BD, of the original plan X. M, the vanifliing point of 
 the diagonal DA, the fmall fquares on the original plan are the plans of 
 the legs ; find the Perfpedive plan by c, d, (Ex. V. Plate VIII. Fig. 
 3.) from the points 2, 3, on the ground line i J, draw 3N, 2N, cutting 
 KM; and there determine the little fquare, whofe parallel fide is ^a, 
 continue a line to 6 it v/ill give b6, and fo finifli the other fquares ; at d 
 
 and 
 
PRACTICAL PERSPECTIVE. 39 
 
 and and on every point ^5, 6, ^c. of the Perfpedlive plan, raife per- 
 pendiculars at pleafure. At the point 4 on the ground line i J, fet the 
 perpendicular RS, with the heights of every member of the chair, 
 marked 2;, jy, a;, 8, 7, and from every one of the points draw lines to 
 the point N, cutting the line / in I, t, r, fy hy from / draw 
 
 /U T, cutting the perpendicular a at T, draw TN cutting the line cky 
 at ky draw h parallel to U, T, then will three fides of the chair be com- 
 pleated ; a line from y to N finiflies the back. At the point U draw a 
 line to the point N to crofs the uprights, in thofe little lines U and k ; 
 To find the rail s, draw 7N, cutting lb at h, and from h, draw a fmall 
 line acrofs the perpendicular on the point 6, parallel to b a-y and from 
 that interfedlion draw a line through N, to the leg dy which will give the 
 upper fide of the rail s ; the fame muft be done for the other fide from 
 the line under z?" : it is eafy to imagine the rails Q, and P, only they 
 don't go acrofs the uprights but half way, as the line RS, on the point 
 4 is in the fame plane with the back of the chair. It finishes only that 
 fide, the other fides are found in the fame manner as the cube, (Fig. i. 
 Plate XII.) 
 
 EXAMPLE XV. 
 Plate XIII. Fig. 2. 
 
 To give the Keprejentation of a fqiiare Table. 
 
 Every thing ftanding as in the foregoing example, let ABCD, of 
 fig. 2. be the ground plan of the table, and the little fquares that of the 
 legs ; I, 7, 6, 2, their interfed:ing points on the ground line. The line 
 on the point 2, the beight of the table with the parts, markt 5, 4, 3 : 
 a, by c, dy the PerfpeBive plan of A, B, C, D, aq, pc, the thicknefs of 
 the feet, the rail f is carried parallel to FJ, cutting the perpendicular 
 ^H, through that interfedtion, and f draw lines to point N interfering 
 2 , the 
 
40 PRACTICAL PERSPECTIVE. 
 
 the back feet at h, and from thence a hne parallel to a, c, through the 
 leg b ; every line that ferves to fhew the thicknefs of the legs are contrived 
 in that manner, G, H, E, F, reprefents the top. 
 
 The fcheme will be fufficient without any farther explanation. 
 
 EXAMPLE XVI. 
 Plate XI. Fig. i. 
 To give the PerfpeSli'ue Reprefentation of a Cylinder^ i^Jianding perpen- 
 dlcular on the Groujid) by ofjlj a Line given in the PiSlure, as the Dia- 
 meter of a Circle. 
 
 Let OFI be the horizontal line, H, the point of dijlance laid on the 
 horizon, O, the center^ and V a poijit to determine the circle. Let aby 
 be the line, the circle ^, B, ^, is found, (by Ex. X. Plate XI. Fig. 3.) 
 On the points a^, B6, b, ij, c%, and on the center, raife perpendiculars at 
 pleafure, fuppofe BD, a height given, draw DO, cutting the lines, 
 Xy c, at e, and f and through e draw a line parallel to the horizontal 
 line cutting a, by at ^, hy from V through^ draw a line, which gives the 
 point I, on the perpendicular line 5, and a line from h to Y gives the 
 point 3 on the line 7, from 3 draw a line parallel to g by it gives the point 
 4 on the perpendicular 8, and through i a line parallel tog hy gives the 
 point 2 on the line 6, then join thofe points for the circle, on the point 
 /, raife a perpendicular to interfed: the circle above at r, and at point s, 
 eredl another to give the point w, the flep on which flands the cylinder 
 is to {how the perfpedive better. The line i r is farther from by towards 
 the center O, as the line w is from ^, nearer the ground line. 
 
 This reprefentation, which is not half, is as much as v^/e can fee of 
 a cyhnder, or any round body from that point of fight. 
 
 EXAMPLE 
 
PRACTICAL PERSPECTIVE. 41 
 
 EXAMPLE XVII. 
 Plate XI. Fig. 3. 
 Jo Jhew the PerfpcBtve of a Cylindrical Body lji?7g on the Ground at Right 
 
 Angle with the Figure. 
 
 The pidlure having every thing (landing as in the foregoing example, the hne 
 aC vanilhing at O, is the axis running through the cylinder. Every circle on 
 the body are parallel with the piSlure, and are flruck with compares j draw 
 I3i^, to the point O. On the ground line BG, fet the meafures for the dif- 
 tances between the circles, and length of the body. To find the circle c d e, 
 on the line BG, let the point A be the diftance for one circle : draw AH, 
 which gives the point e on B^^j at that point raife a perpendicular through 
 the line ^C, and a is the center for that circle. If the objedl fhould repre- 
 fent a cannon, as they diminifh at the muzle, let C be the finall end, draw 
 a perpendicular through C, cutting in h : let 3, 4, on the line eCy be equal 
 to the fmall end wanted j draw from 3 and 4 lines to O, cutting the line Qh, 
 in I and 2, fet one foot of the compafles in C as center, with the opening 
 Ci, or C2, defcribe a circle; divide your circles equally in as many parts a3 
 you pleafC; and join them together with ftraight lines, as C2y dR^&c. 
 
 EXAMPLE XVIIL 
 
 Plate XIV. 
 To put any kind of Wheels 'in Perfpe^ive. 
 
 In this example is the reprefentation of a water-wheel, with coggs. HL the 
 horizontal line, H the point of fight, E point of diftance laid on the horizon, 
 CM the ground line, VHV the nanijloing line of the plane BEGI, EF or 5D 
 the thicknefs ; from the point 5 draw a line 5G to the center H, and from 
 C draw CL, to terminate the line 5G, at G ; and from K, the center of the 
 circle to the point L, given e, on BG, from the line BG, finifh the perfpedllve 
 
 G fquare 
 
£,1 PRACTICAL PERSPECTIVE. 
 
 fquare EGIE, draw the diagonals GE, ^^I, inlerfef^ing at O, draw H0<^", 
 and e 0 f, then is the fquare prepared to draw the circles by the points VV, 
 as has been fliewn in (Plate XL Fig. 3.) then on^ the point D for the 
 thicknefs, draw DF, parallel to 5E, and with that line finidi another fquare 
 as is reprefented by the dotted line, and draw from each corner diagonals, 
 as in the firrt fquare : when jour fiifl circle e d f is finifhed, fet off any 
 given height for the coggSj on 5E as 6, draw 6Hj cutting ^'/^ in the point,, 
 to form the circle of the wheel, and fo on for the next. To divide the wheelj, 
 draw a quarter of a circle as K, and divide that in as many parts as you 
 pleaie, it is here in 4 ; from every one of thefe points draw perpendiculars to 
 the ground line, and from, their interfe.6lion with it draw lines to the point 
 of diftance L, cutting BG., and then draw perpendiculars through one half 
 of the perfpe<ftive circle, and from thofe points draw lines through the cenr 
 ter of the wheel acrofs to find the others. For example, to find the points 
 h andy^, from B draw a perpendicular, cutting the circle in. ^ and fy from / 
 thro' the point O, draw a line crofa the circle for the point/. For the thick- 
 nefs of the coggf, (fappofing them in the original to diminilh to the center) 
 from the point nof the cogg/, draw aline thro' the point O acrofs the circle, 
 gives the point/. Thofe lines are drawn to the lineVHV, and gives V,T, for 
 vaniihing points of the fides by this mean every circle is divided, from every 
 pointy^ n, draw lines parallel to the horizon, interfeding the outward circle 
 at p, and from the vanifhing point V of draw/>V, from ^ draw a line paral- 
 lel to n, cutting pY, at c, then is dc, a compleat fide of one of the coggs. 
 To find them without ufing vanifhing points, as it is often very troublefome 
 to get, make the circle c, then from d the parallel line cutting that circle 
 in c, join c p, the eight fpokes are the principal lines made ufe here for the 
 circle, and as it is fuppofed mufl: pafs through the middle of the thicknefs of 
 the wheel, proceed thus, divide 5, D, in P, and a line drawn from P, to the 
 point of diftance cn the line YUV, w.ill give the line through the middle of 
 
 the 
 
PRACTICAL PERSPECTIVE, 43 
 
 the fpoke ; but to do it without that point of diflance, draw the Hne m paral- 
 lel to the hcrizon^ divide that in two equal parts at m, draw m />, proceed for 
 every other, join the extremity of tiiofe crofs Une^ ia a circle; to fhew the in- 
 jGde thicknefs of xhe wheel. 
 
 C n A p. ill. 
 
 Relating to put in Perlpedlive any Objed or Objecfls in general. 
 
 EXAMPLE XIX. 
 Jiow ioput Doors in PerJpeSiive, and to defcribe Panneh thereon. 
 
 Let Plate XV. Fig. i. reprcfent the infide of a room, HL the horiz'jntaJ 
 line, C the center^ OC the diflance^ V a point to find the (emicircle on 
 the floor, (defcribed by the door opening) UR a line equal to the breadth 
 <of the door, continue U to the center C, draw a line from the point R to the 
 point of diftance (removed on the horizontal Hue, omitted on the plate for 
 want of room) interfeifting UCatX ; let Ull, be a given height, draw II C^ 
 and on Xraife a perpendicular cutting II C, at III, then isUX, il. III, the 
 aperture of the door; -draw RC, and froan X a line parallel to UR, cutting 
 RC at T, draw XR for a diagonal^ X the center, from V through T draw a 
 line cutting XR at Y, join UY F for a quarter of a circle. The other quarter, 
 if need be, is found, by finifhing the fquare XT/^Z, and drawing the diigo- 
 nal XZ, and from C and Y draw a line cutting XZ atjy ; join T)h for the 
 'Other quarter j chufs any point on the femicircle for the opening, as F, 
 draw a hne from that point through X, to the horizontal line at H,.for the 
 ■vanifhing point of lines parallel to that, and from H, through HI, draw a 
 -line at pleafure, on F, raife a perpendicular cutting X, III, at then III a.^ 
 XF reprefents a door opened. To form the pannels on the door, draw FE 
 parallel to the horizon j fct the foot of the compafTcs in H, and with the 
 
 G 2 opening 
 
44 PRACTICAL PERSPECTIVE. 
 
 opening HO draw the arch OL on t^e horizontal line, and L is the point 
 oi dijiance^ and H the ^oaning point of the door ; from the point L, through 
 X, dr^w a line, cutting FE at E, then EE is equal to the original of FXi, 
 fet on FE, the meafures i, 2, for the width of the pannels, and from them 
 points draw lines to the point L, cutting XF at 3, and 4, from any points 
 on Ya. For the height, draw lines to their vanifl-iing point H, cutting the 
 perpendicular raifed on 3, and 4, and terminating the pannels A and B. 
 
 The door, Fig. 2. at the end of the room is found in the fame manner ^ 
 the line KN the breadth, and KMPN a fquare made of that line, K is the 
 center, or hinge of the door, with the point V on the horizon j find the 
 point/*, and finifh the quarter of the circle, KW height of the door, the dot- 
 ted lines from K and W vanifhes to the fame point on the horizon. 
 
 Fig. 3. is the reprefentation of a trap-door on the floor. G9 the breadth, 
 and G D 7 9, a fquare in perfpedive of the aperture, the door, D Q^6 7, 
 is found the fame as Fig. i. or Fig. 2. KIDG is a dotted fquare, equal 
 the aperture, /O the vanlfhing line for it, C the center, OC the diflance, 
 and the points J. and L, to find the circle, then proceed as in the other 
 Example. 
 
 E X A M P L E XX. 
 Plate XVI. Fig. i. 
 How to give the PerfpeBive Reprefentation of RoomSy Doors, 5cc. I'he Point oj 
 Dijlance is fet on the Horizontal Line in this and the following Examples. 
 
 Let HL be the horizontal line, S the center, L point of dijlance, AU 
 breadth and depth of the room, AW the height, draw AS, US, interfed 
 US by the line AL in C, draw CB parallel to AU, then will ABCU re- 
 prefent the floor in perfpedlve. To finifh the fides, proceed as with a cube. 
 
 To fet a door on the fide of the room. On the ground line AU, let aU 
 be the width of the door, and U b the depth it muft be in the room, draw, 
 
 ^L cut- 
 
PRACTICAL PERSPECTIVE. 45 
 
 IL cutting US at F, from F draw FE at pleafure parallel to AU, draw aS 
 cutting FE at E, and the line EL will cut US at e, on e and F raife perpen- 
 diculars to make the door, the femicircle eEG and door is found as in the 
 foregoing example, K is the vanifliing point of the door I, H the point of 
 the door N, / of the door T, and I of the door M. The projedion of the 
 chimney-piece is found as the door, by fetting on the ground line the mea- 
 fures, and proceed as with the door, i, 2, reprefents the back part of an up- 
 per room, r another behind it, and T the door opening within it j E the under 
 room, R a room and pavement behind that, V vanifhing point of the ftairs. 
 
 EXAMPLE XXI. 
 Plate XVI. Fig. 2. 
 How to put Windows in a Room from any given Meafurement, 
 
 Let HL be the horizontal line. Aw the ground line, L point of dijiance^ H 
 the point of fight, the points m, k, /», q the breadth of the panes and frame, 
 and B, C, D, the height of them 3 lines from m, &c. to the point L, gives 
 on AH, the points i, 2, 3, 4; on thefe points raife perpendiculars j the per- 
 pendiculars I, 4, gives flW on the lineQH, for the fquare of the window- 
 frame. To give the thicknefs, and reprefent the fides, take OQ^for the 
 ihicknefs of the wall, draw OH from ^zdraw a line parallel to the horizontal 
 line, interfed:ing OH j then kk'ii the upper thicknefs, and where the line a 
 interfeds OFI, draw a line parallel to it ; a\ gives you the fide : where the 
 perpendiculars 1,2, 3, interfeft the under frame of the window, draw parallels 
 to HL, acrofs the window-feat, inrerfedling PH j at thofe points raife per- 
 pendiculars for the uprights of the fafhes, and from BCD, draw lines to the 
 points H to finifii the fquares, continue ^iWQ^to interfed: the line b 5, (at 
 the end of the room) at 5, draw 5/ parallel to HL, for the top of the window j 
 do the fame to get the bottom, as is (hewn by the dotted lines i/, draw a,7, 
 parallel to If^ and from /», n, p, draw lines to H, cutting h b, zie d c. To 
 
 find 
 
46 P Pv A C T 1 C A L :P R S p E C T I V 
 
 find the thicknefs continue Ok, to cut the line 5^ at 6, and from tlut 
 point draw a line parallel to / /, draw to the center H, interceding 6g at 
 5^, from that point draw a line parallel to f", for one fide of the window ; 
 and draw zH, interfering that line ; from that interfedlion draw a line 
 parallel to x z, for the thicknefs of the bottom. The crofs barrs are found by 
 drawing lines from the points 7, P, 9, to W, and from their interfedions on 
 4he line ^, draw lines parallel to x.z. 
 
 E X A Tvl P L E XXII. 
 Plate XVII. Fig. 2. 
 
 To -piit a Tufcan Fedejial in FerfpeBlve. 
 
 On the ground line AK, raife a perpendicular ac, for the height of the 
 members, as i, 2, 3, 4. c, for the projedion of thofe members take ^e, 2,d, 
 tfc. On any point of the ground line, raife the perpendicular AC j on it 
 jet the points 1,2, 3, 4 B C, equal to thofe in a c, as Ai the height of the 
 plinth, &c. At J draw GF parallel to AK, i G or i F is equal to half the width » 
 of the plinth, from G and F draw lines to the center H. On the ground line 
 fet the depth (equal to the line G F) from G to K, draw K to the poijit of 
 iijiance I, cutting G Hj at the point ,/, raife a perpendicular, and iinifii the 
 plinth ; from each corner draw the diagonals, Fj6, GL, interfeding each 
 ^ther in O, thetrenter of that fquare ; fet the different projedions from the 
 line a to d, and put them on from i towards D, draw iH, and from the firft 
 point on iD draw a line to I interfeding iH, at the interfeding .point draw 
 a line parallel to GF, cutting the diagonals GF, and from thofe interfedions 
 ■draw lines to the point H, interfeding the diagonal at on each point raife 
 pevpendlcularf, for the height of the fillet, then from the point 2, the height, 
 draw a line 2H, cutting the perpendicular on the line itl, fur its proper 
 ^icight ; from that interfedion draw a line parallel to GF, interfeding that 
 perppiidicular, and thpm on the diagonals G F, and fiaiHi the fillet in the 
 
 fame 
 
PRACTICAL PERSPECTIVE. 47 
 
 lame manner as the plinth ; the other members are laid on in the fame 
 manner. For the capital of the pedeflal, proceed as for the bafe. As the 
 fiat band BC projeds flufh with the phnth, from B to E lay the profile of the 
 members, draw BH, interfed: it by EL, and finifh every member before you 
 lay another on it. 
 
 EXAMPLE XXIIL 
 Plate XVIIL Fig. 2. 
 
 Ho'-j} to lay Menibers of Archhc^iire on cm anolher, 
 
 'Let ABCD be a fquare plan in Perfpe£live, where on it another is to be put 
 on equally every way, GP horizontal line, G center^ P diJIancCy r x is the 
 dijlance it is to be laid on ; draw the diagonals AC, BD, where rP interfcdis 
 xG, draw the line (3 parallel to AD, cutting the diagonals at a and^/, from 
 thofe points draw lines to G, cutting the diagonals B and C at the points l)Cy 
 then will the plan ab c dh^ placed equally on ABCD. 
 
 EXAMPLE XXIV. 
 Plate XVIL Fig. i. 
 In this Example is the PerfpcSlive of the Tufcan Eiitahlature. 
 
 Let X reprefent the original, on the line AD are fet the height of the 
 parts, liiz. i, 2, 3, 4, and from thofe as AB, \b, ic, &c. are the 
 diftances they are from that hne, (this figure is put on the plate upfide dov/n 
 for want of room) LH, the horizontal line, H the point of fight, and 
 L the point of dijlance, laid on the horizontal line. Let FE be equal 
 to the whole width of that member, divide it into two equal parts at a, let 
 fall the perpendicular ad j on that line fet themeafijres from AD, i, 2, 3, (Jc, 
 on oF fet their projediions from the line AD, as b, e, f, g, &c. with the line 
 FE make a perfpedtive fquare plan, draw diagonals acrofs, from rt' craw a 
 line to H, from b draw bL cutting aH at ri, through « draw a line parallel 
 .to FE cutting the diagonah at MM, draw MH till it cuts the diagonal; 
 
 cn 
 
48 PRACTICAL PERSPECTIVE. 
 
 on nMM raife perpendiculars. From the point i on ad draw a line to H, 
 cutting the perpendicular 71, and from that point draw a line parallel to FE, 
 cutting the perpendicular M M, and finifli that member as if it was a plinth ; 
 then on the corners at F and E draw the quarter round, and proceed the 
 fame for the reft, obferving that every member lies properly on the diagonal 
 of the preceding. . 
 
 EXAMPLE XXV. . ^ 
 In this Example is the VerfpeSlive of the Tufcan Bafe. 
 
 Let X in Plate XVIIL Fig. i. be an original fedlion made as in Plate 
 XVII. Fig. I. But to give the proper reprefentation of thefwell of the Torus, 
 I have divided the original in two equal parts, horizontally as marked 2, 
 and vertically marked x; and where the line x cuts the Torus, lines are 
 drawn parallel io x a againft a^^ in the points i and 3, to exprefs the fwell. 
 The bafe is fuppofed parallel with the pl(!3:ure, and the plinth BD, whole 
 height is AP, is put in Perfpedlive in the fame manner as a fquare; SL the 
 horizontal line, S the point of fight, and L for the dijlance ; on the furface of 
 the plinth, draw diagonals. Their interfedion 0 is the center of the bafe, 
 through that point 0 draw a line parallel to DB, for a diameter, and from the 
 S through 0 draw a line for the other, and given the point A on BD, with the 
 point V make the circles on the plinth, as has been fliewn in Plate XI. Fig. 3. 
 On A5 fet the height of every member, as i, 2, 3, ^c. on AJ are the mea- 
 fures for their projedions. The point ir, for one perfpedlive circle, is found 
 by the line JL, cutting AS in that point j the other points on that line are 
 found in the fame manner : thofe points are afterwards carried on one of the 
 femi-diameters, by continuing a line from thofe points to the point of dif- 
 tance interfering the femi-diameter, then with a pair of compafTes carry them 
 over to the other, 2.^ a be, then from V and thofe points finifli your circles, on 
 the center 0 raife the perpendicular Kc, and through every point i, 2, 3, 
 
 &;c. 
 
■I 
 
 , PRACTICAL PERSPECTIVE. 49 
 
 &c. on the line A5, draw lines to the point S, cutting in i, 2, 3, 6cc. On 
 that part of the circle you intend for a fedion, draw a line through the 
 center Ko, to the horizontal line j and the interfedlion will be the vanifliing 
 point, from that point draw lines through every point on Koy cutting the per- 
 pendiculars on the points for the circle of that fedion. For Example : Sup- 
 pofe for the fedion on the line AH, S is the uaniJJjing point for it, on every 
 point ab CyOn AS ralfe perpendiculars, and from S through every divifion on 
 K(? draw lines, cutting them. The firft point on A marked 2, for the mid- 
 dle or greateft fwell of the Torus j on the fecond, in two places toexprefs the 
 turning of the round j on the third, at ^ 4 and 5, the point b joining the 
 plinth, and 4 where the fillet begins, 4 and 5 the height of the fillet, on c at 
 d the fliaft begins, do the fame on as many parts of the circle as you think 
 necelTary to compleat your figure. Then join artfully the points 44, 33, 
 2 2, &c. in curves. R is part of a fedion behind that on the diagonal B, 
 a is part of another ; to finifh this figure to a great nicety, girt the fedions 
 with as many of thofe curves as will fill the fpaces. 
 
 When the learner has accuflomed himfelf in viewing a true reprefentation, 
 he may with only the circles compleat the Torus. 
 
 EXAMPLE XXVI. 
 Plate XVIII. Fig. 3. 
 
 In this Example is the PerfpeBive of the T ufcan Capital. 
 
 The line AJD is made ufe as a ground line, HL the horizontal line, 
 L the center of the pidure, H point of dijiance^ and V one point for 
 the circle, CD the width of the abacus, J the middle of it, a b d g the 
 projedion laid on from J towards D, Jr a line for the height of the parts, 
 as I, 2, 3 &c. KO the center of the column, ABDC an original fedion ; 
 proceed with this exadly as with the foregoing Example. 
 
 H EXAMPLE 
 
50 PRACTICAL PERSPECTIVE. 
 
 EXAMPLE XXVII. 
 Plate XIX. Fig. 3. 
 
 In this Example is the PerfpL Siiue of a double Crofs. 
 
 Let LH be the horizontal line, AD the ground h'ne, L the center y H the 
 point of dijiance^ ABCD the extent of the crofs bar, BC the thicknefs ; 
 draw AL, BL, CL, DL, and from A draw AH, gives the points b c'E 
 through thefe points draw lines parallel to AD for ag on AL, and d 
 on DL i on the point B fet the perpendicular BG, with the height of 
 the parts required j then on the points b c e fy plan of the upright poft, raife 
 perpendiculars at pleafure, and through the points on BG, draw GL, KL, 
 IL, cutting bo 2Li qry draw through the point r a line parallel to the ground 
 line, crofs the perpendicular lines cb as mrsty and another parallel to it 
 through q j IK, K«, is the end of one of the crofs bars, draw KL and IL; per- 
 pendiculars on ag dF will terminate the points ^Q£, &c. where the line 
 iQjnterfeds fp, draw a line parallel to w/, from tm draw lines to the point 
 L, a perpendicular on g will finifh the fide mh. 
 
 Plate XIX. Fig. 2. is the reprefentation of a bureau with the flap open. 
 b, i, k, is the circle defcribed by the flap opening, PL the horizontal line, L 
 the center, O, O, X, the vanij/jing line to BA/, O O two points to contrive 
 the circle, X the point of diflance on OLX, A;^M the flap, A the point it 
 turns on 5 proceed in this as in Plate XV. 
 
 EXAMPLE XXVin. 
 Plate XIX. Fig. i. 
 To find the Middle or Center of any Buildingy &c. 
 
 Let HV be the horizontal line, V the center, H the point of diflance, and 
 EC one fide of the building, through E draw HA, to the ground line AG, 
 divide AC in two equal parts at F, draw FH, cutting CE at D (the middle 
 of EC) raife the perpendicular DL, cutting RK at L, draw a line parallel to 
 
 2 AG, 
 
PRACTICAL PERSPECTIVE. 51 
 
 AG, interfering MV, divide that line in the middle, and defcribe the femi- 
 circle LQjfrom V, and through the center of that femicircle draw a line, it 
 will give the point Q.on it, which is the center of the arch MNR, as required. 
 EXAMPLE XXIX. 
 Plate XX. 
 
 In this example the eye is fo placed in the air as to look down in the building, 
 VUB and RUS are two va?n/hh/g lines at right angle, U is the center of the pic- 
 ture, R and S two points of dljlance j the line GI parallel to RS, for convenience, 
 muft be conceived as a ground line, ABCD an original arch given, PG is 
 equal the line BC, PA the thicknefs of the pilafter, PI is half the breadth, 
 and NO the extent of the capital and bafe ; interfedl PU by GR, to get the 
 point ^; LM drawn to R, gives on PU, hy /, the fame parts of the ori- 
 ginal, as by their capitals ; from them, lines are drawn parallel to GI, lU 
 terminating the fide NU gives nn for the projection of the bafe and capital. 
 To turn the perfpedive arch, W on the ground line being for its center, 
 draw WU cutting at a: j from R and S through x, draw lines at pleafure, 
 for diagonals x v, xy. To turn the arch, draw Z / 1;, Tj ty^ P 1 join 
 t,y,py Vt f. The fides of thefe pilaftersare parallel to VUB ; the window 
 12, 34 is found by drawing 2U, 5U ; for the thicknefs, draw lU, and 
 draw 4 3 parallel to RS, cutting lU at 3, and from 3 draw a line parallel to 
 VB ; finifliing the bottom of the window, if any depth for the window is 
 required, fet them on the front line, draw SU, and interfedl that line, by 
 another from a given poii.t, to a point of diftance, as sT draw s t parallel to 
 VB, then that is the fide of a fquare j and if from t you draw another to the 
 point of diftance, cutting SU, the window will be double its breadth : 
 In this fcheme the pavement being parallel to the piSlure^ every fquare re- 
 mains the fame. To find the point of difl:ance T, the line VB, and T^ 
 their interfedlion is the point required. Cielings arc reprcfented in the fame 
 manner. Thefe arches need only be inverted. 
 
 H 2 CHAP. 
 
PRACTICAL PERSPECTIVE. 
 
 CHAP. IV. 
 
 To reprefent inclined Planes and Objecls. 
 
 EXAMPLE XXX. 
 Plate XXL Fig. i. 
 To reprefent an inclined Plane, whofe inclination is given, having one fide 
 
 'parallel to it. 
 
 Let HL be the horizontal line, S the center, or point of fight, L the point 
 of diflance-y draw SO perpendicular to HL, draw LO parallel to Y the in- 
 clined furface, cutting SO at O, for the vanijlnng point of that plane j through 
 that point, draw VU parallel to HL ; fet one foot of the compaffes at O, 
 and the other in the point of diftance L, draw the arch LV ; then V is the 
 point of diftance to it. When the pidlure is thus prepared ; fet GB as one 
 fide, on the ground line GF, draw GO, BO, and from V, draw GV, cut- 
 ting BO at I, draw IK parallel to GB ; for the bottom that lies flat on thft 
 ground, draw BS j then finifh it by the perpendicular la. The fquare d 
 fS> ^^S* 2* foiJ^^d the fame i de being given. 
 
 EXAMPLE XXXI. 
 Plate XXI. Fig. 3. 
 In this Scheme is the PerfpeBive of a Prifm that has one Side perpendicular fa 
 the Ground, and rejleth on an inclined Plane. 
 
 Let HL be the horizontal line, S the center, and the point L the diftance, 
 Z the original prifm, X the inclined plane, draw SO perpendicular to HL; fet 
 the prifm Z at the point of diftanceX, and draw LO parallel to the upper fur- 
 face, and LV for the under, cutting VSO at O and V" j through thefe points 
 draw lines parallel to HS, and from V and O, with the diftance L defcribe 
 the arcs KL, LN j the furface of the wedge being parallel with the upper 
 
 furface 
 
PRACTICAL PERSPECTIVE. 53 
 
 furface of the prifm, the point O is the vanifhing point to both, and V 
 vanifliing point to the under lide of the prifm ; for the perfpedive of the 
 plane X, proceed as in the foregoing example, Fig. i. Let« / on theperfpec- 
 tive plane X be given (parallel to HL) through ^^L, from V draw lines at 
 pleafure, from N point of diftance through / draw a line cutting «fl at 
 ^, make parallel to nl, from ^ and b draw ^zO, either draw a per- 
 pendicular on / to find the point id^ or the line Ki?, make de parallel a 
 Fig. 4. is the ground plan of a fquare (whofe pofition is oblique to the pic- 
 ture) on an inclined plane, bd^ line given. From I and K the two points 
 of diftance, draw lines through b and interfering each other at a and c, 
 
 EXAMPLE XXXIL 
 Plate XXIL 
 
 In this Example are Stairs put in Perfpedfi've,Jianding parallel with the Perfpe&i've, 
 
 Make NL the horizontal line, S the center ^ N the dijlance, I the given 
 inclination, V the vaniQiing point of the inclination found, as by (Ex. XXX. 
 Plate XXI.) Let 3 4 on the ground line be the depth, and FE the height 
 of a ftep ; draw FS, ES and 3N, 4N cutting ES at 2 and i, there raife per- 
 pendiculars cutting FS at 5 W for the firft ftep of the flight K, then draw 5V, 
 WV, the point d is found by drawing a perpendicular from W, interfedting 
 5V at di then draw dS interfering WV, and fo on for as many fteps as you 
 pleafe. The rails of the flight C vanifhes at V, and the top of the landing 
 place at S ; and every fteps having their fides perpendicular to the pidure, 
 the points kqp,b^. are found as the point d, the flairs N that runs below, 
 are found by drawing a line from V through Y, and proceed as with the firfl: 
 flep K. X a ray of light, and parallel to the pidure, 6 6rr fhadow of the 
 rail D on the flairs. 
 
 EXAMPLE 
 
54 PRACTICAL PERSPECTIVE. 
 
 EXAMPLE XXXIIL 
 Plate XXIII. Fig. i. 
 
 In this Example is an inclined Chair in PerfpeSiivey and parallel with the PiSfure. 
 
 Let SQj3e an horizontal line, CD a ground line j make USN perpendi- 
 cular to SQ^, S the center, Q the dijlance, QU a vanifliing line for the 
 fide KA, or inclination above the horizon, NU the vanlfhing points, 
 QN the vanifliing line for the fide DA, CB, 6cc. or inclination below the 
 horizon; through the points N and U, draw lines perpendicular to NSU; 
 on them fet P and R as points of diftance, P equal to NO, and R equal UO, 
 on C2 j iD thicknefs of the legsj for the plane ABCD draw NCB, NDA, 
 then from P draw a line through D, cutting BC at B, r.iake BA parallel to 
 CD J for the fide BLKA, being a fquare, draw BU, AU and BR interfec- 
 ting AU at K, draw KL parallel to AB. For the fide ML, CB, draw CU 
 and LN cutting CU at M ; the rail is parallel to CD, and the fides va- 
 nishes at N. The diagonals FG, JP, interfeds at O, for one of the rails 
 FP, another &c. The length of this chair is known by the number of 
 diagonals, every two diagonals forming a fquare, whofe fide is equal to BA. 
 Fig. 2. is a chair finiflied. 
 
 EXAMPLE XXXIV. 
 Plate XXIV. Fig. i. 
 
 To put an inclined Cube in PerfpeBive, whofe Pofition is oblique in the Pi6iiire. 
 
 Let HL be the horizontal line, 8 the center, the perpendicular, SO the dif- 
 tance, and a c 2i given fide of a cube in the pidure: continue ac \.q the hori- 
 zontal line at H, draw HO and LO perpendicular to it; then H and L are 
 two vanifhing points. L forde, H for b d, acfg, through L draw CLA 
 perpendicular to HL j then with one foot of the compafies in the point L, 
 and with the opening LO, defcribe the arch OC ; make AL equal LO, C is 
 the vanifliing point of dfac^ bg, and A of the fides ady ef, c b ; draw 
 
 HA 
 
PRACTICAL PERSPECTIVE. 55 
 
 HA, vaniOiing line for the fide a b &c. draw SB perpendicular to AH. 
 To find the point of diftance E of that line, divide HA in two equal parts at 
 G ; fet one foot of the compafiTes in G, with the opening GH defcribe the fe- 
 micircle HE A ; through B draw a perpendicular line to HA, cutting the femi- 
 circle at E j join AE, HE, which is a right angle, as the original. Through 
 the given line a <r, draw Kad^ hcb^ at pleafure > at the point E fet ofF4 t^do^ 
 (that being the angle, the diagonal makes with the fide of a fquarCi) 
 at h and through h, draw E^D, from D through r, draw a line cutting 
 ad 2.t d, and draw dHy interfeding <: ^ at ^, draw dC^ bC, aC, and dL as a 
 diagonal, cutting aC at e, draw Ae at pleafure, cutting ^/C at f, and draw/^H, 
 interfeding 30 at^, and the cube is finilhed. If the diagonal dg is required, 
 find k on CH, as the point D on AH. 
 
 EXAMPLE XXXV. 
 Plate XXV. Fig. i. 
 In this Example in an inclined Chair in Perfpediive. 
 
 Every line points, &c. fianding as in the foregoing j the diagonals are got 
 by E, D and L; ah, cd, efgy are found as the cube j the back of the chair 
 is three times its width, as appears by the diagonals and which are 
 drawn to the point E. 
 
 C H A P. V. 
 
 Of finding the Shadows of given Figures, 
 
 « 
 
 Shadows are only the projedions of given figures on given furfaces, by 
 means of given luminous bodies j but to avoid the difficulties that would 
 attend the defcription of fhadows, if the maignitude of the luminous body 
 were taken in confideration, it being fufficient for pradice to regard only the 
 center of it ; therefore in thefe examples, a luminous body will be confidered 
 as a point, 
 
 EXAMPLE 
 
56 PRACTICAL PERSPECTIVE. 
 
 EXAMPLE XXXVI. 
 Plate XXV Fig. i. 
 
 Let HL be the horizontal line, C the center, and R ray of light, fup- 
 pofed to come from the fun, and parallel to the pidurej from every corner 
 of the Fig. A at top, draw lines parallel to R, as i 4, 2 3, and from their 
 feats <2 and b draw lines parallel to the ground line, interfeding them at 4 and 
 3, or elfe find only the point 4, as C is the vanifliing point of the fide abyi 2, 
 draw 4C, and at b draw a line parallel to the ground line, cutting 4C at 3. 
 
 The fliadow f e g oi the prifm B is found in this manner, L is the vanifh- 
 ing point of the fide c dcy make/' parallel to the ray R ; from the feat of c 
 at draw a line parallel HL, interfeding cfzif^ draw fh, and from g the 
 feat of dy draw a line parallel to^ f, cuttingy^L at^. 
 
 To find the points b c d of the fliadow of the inclined crofs D, from C 
 through 4, draw a line at pleafure ; at 2 draw a line parallel to 14, 
 cutting 4C at a, the feat of 2, from the point 2 make a line parallel to R, 
 and from the feat a a line parallel to HL, cutting it at 5 j the point c is found 
 by drawing ic parallel to 25 or R, and /\.c parallel to ab, the point d is found 
 the fame. 
 
 EXAMPLE XXXVIL 
 Plate XXV. Fig. 2. 
 In this Example the Light comes from behiiid the 'Picture. 
 
 HL the horizontal line, O the center, on HL at H raife the perpendi- 
 cular RH, R is the vanifhing point of the rays of the fun, and H is the vanifh- 
 ing point of the (hadow on the ground j to find the points A, B, C, (hadow 
 of the folid figure D, draw a line for R, through the point i, at pleafure, and 
 through a (the feat of ^) and H draw a line interfeding Ri at A, the point B 
 and C are found the fame. Obferve as O is the vanifhing point of the fide, 
 2 3, it is alfo of the fhadow BC, the fhadow of the crofs is found in the fame 
 manner j and the fliadow of it over the block S is found, by continuing that 
 
 of 
 
PRACTICAL PERSPECTIVE. 57 
 
 of the body "of the crofs on the ground, till it interfeds the fide r of the block, 
 and from that interfedion, draw a line parallel to the upright of the crofs, 
 till it comes to the furface at / ; and from that point and H, draw the fliadow 
 over it, (as it runs parallel with that on the ground) every point is found as 
 in Fig. D. The fliadow of the cylinder is found by fetting off as many 
 points on it at pleafure as a, find their feats on the ground as 5, 6, &c. 
 then proceed the fame as with the folid figure D, 5 is the feat of <r, and i is 
 the point of (hadow, 8cc. Note, when the fun is fuppofed behind the pic- 
 ture, the vanifliing point of light is above the horizon. 
 
 EXAMPLE XXXVin. 
 Plate XXV, Fig. 3. 
 In this Scheme the Light from the Sun is before the Pidfure. 
 
 HV the horizontal line, C the center, V the vaniOiing point of the fliadow 
 on the ground plane, L the vanifhing point of light coming before the pic- 
 ture J the point 3 the fhadow on the ground of point h of the ftairs, is found by 
 drawing a line through its feat a to V, and another from h to point L, in- 
 terfering aV at 3 the point fought. The points i, 2, are found the fame, 
 h, g vaniflies at C, as does the fliadow 3 4. The {hadow on the flep 3 
 is, by drawing 3 V, the fhadow of the ftick A on the flairs is projedled by 
 drawing AV, till it meets the bottom of the ftairs j make the fhadow e pa- 
 rallel to the flick where it meets the top, continue it towards V, the dotted 
 lines from the head of the flick to point L, terminates the fhadow of it on 
 the flat top of the fleps. The dotted lines of Fig. K, fufficiently explain 
 it, the fide e d is parallel to the horizon, fo is its fliadow. 
 
 N. B. When the light is before the pidure, the vanifhing point of light is 
 below the horizontal line. 
 
 I 
 
 EXAMPLE 
 
5$ PRACTICAL PERSPECTIVE, 
 
 EXAMPLE XXXIX. 
 Plate XXVL Fig. i. 
 In thii Example the Light is fuppofed before the PiBure, 
 
 V is the vanlfhing point of fliadow, W the vanifliing point of rays of 
 light, at any point on the quarter round where you want to projed a fhadow, 
 make the fedion as by Example XXV. The point 4, fhadow of b, is found 
 by drawing bY cutting the quarter round at 3, there make a fedlion, draw 
 interfering that fedtion at point 4, point 2, /hadow of a is found the 
 fame. 
 
 E X A M P L E XL. 
 Plate XXVI. Fig. 2. 
 
 X, S is the horizontal line, X the center, S the vanifhing point of the 
 fhadow on the ground, R that of light, come before the pidure, CAB is 
 a fquare board placed on the top of a cylinder, ^ ^ ^: the feat of A, B, C, on 
 the ground, and i, 2, 3, 4 its fhadow. The point J, the fhadow of A on 
 the cylinder is found by drawing A, S, cutting the top of the circumference, 
 and from that point let fall a perpendicular, draw AR, interfedling it at J, the 
 point fought J fet on as many points on ABC, as may ferve to compleat the 
 fhadow, and proceed with every one in the fame manner. E is the fhadow 
 of and d its feat on the ground, draw interfering the bottom of the 
 cylinder at D, there ralfe the perpendicular DF at pleafure, draw ^-R cutting 
 it at E, the line <^DE is the fhadow of the rail de, and the points G, F, &c,v 
 are found as point E. The fcheme is fufficlent to explain the refl, the lines 
 projedling the fhadows being on the plate. 
 
 EXAMPLE XLL 
 Plate XXVL Fig. 3. 
 
 In this fcheme C is the center of the pidlure, H the vanifhing point of 
 ihadow on the ground, and L the vanifhing point of rays of light. To find 
 
 the 
 
PRACTICAL PERSPECTIVE. 59 
 
 the points S, y fhadow of K on the cylinder, from any point 2 on the cir- 
 cumference of the bafe of the cylinder (which is parallel to the pidlure) draw 
 2C, and from its feat g draw ^C ; then from the feat of K draw lines to H 
 cutting in / and u, make Jy, uy perpendicular to the ground line, cutting 
 2C at ^, and y the points fought ; to find S take any point r, and proceed 
 as with point Qj^ any points of the fhadow of the circumference of the 
 inward cylinder on its furface, are found thus ; draw CL, and at^ hi, parallel 
 to it, cutting that circumference in a bl^ then drawing <7S, and ^C, meet- 
 ing in n, and BS IC meeting in m, n, m, are the points fought, h being the 
 ftat of the point s on the outward circumference, draw iL, /&H, and their in- 
 terfedion is the fhadow of point i. Fig. 4. is a hollow cylinder cut open, 
 every thing elfe {landing as in the foregoing example, ar, Bs, cf, are parallel 
 to CL, draw rC, sC, fCy and ah, bh, ch^ interfering each other at d^fy 
 fhadow ab c, 5 is the feat of point 3, by drawing 3L and 5H interfering 
 it at 2, give 2 for the {hadow of point 3 on the ground. 
 
 EXAMPLE XLIL 
 Plate XIX. Fig. i. 
 
 In this fcheme VH is the hortxmtal line, V the center make RS per- 
 pendicular to VH, S the vanifhing point of fliadow on the ground, R the 
 vanifliing point of rays of light. Then in the circumference of the arch 
 ONB (which is parallel to the pidlure and vanifliing at V) taking any point 
 O, and finding its feat A on the ground, draw OR, AS, interfeding at 0 the 
 point fought s the point n on the wall E, K, B, C, fhadow of N, is found by 
 drawing through P (the feat of N on the ground) PS cutting EC, (the feat 
 of the wall) in/', draw NR, and/;^ parallel to DL or EK, interfering NR 
 at n. In the fame manner are got all the points that form the fliadow of the 
 arch J Imkfor point R at the bottom of the plate. The points ^<:/in Fig 2. 
 are found in the fame manner as point n in the foregoing example j A is the 
 
 I 2 original 
 
6o PRACTICAL PERSPECTIVE. 
 
 original point, B its feat on the ground, W the vanifliing point of ihadows 
 on the ground, and A on the ground line ; BD the vanifhing point of rays 
 of light, c the interfedion, and d the fliadow of A. 
 
 EXAMPLE XLIIL 
 Plate XXVIL Fig. 3. 
 
 In this fcheme the light is from the candle L, ^d L is the point of light, 
 K its feat on the ground, and C the center of the pidlure ; from K draw a 
 line parallel to the ground line, cutting the feat of the fide of the room TPX 
 at T, draw TP parallel to KL, and LP parallel to KT, cutting TP at P, 
 feat of light on that fide the room, produce TP to X \ make LV parallel to 
 PX and XV, parallel to LP j point V is the feat of light on the cieling j 
 W is the feat of light on the farther end of the room, and is found by drawing 
 CK, where it cuts the interfedion of that fide of the room and the floor, raife 
 a perpendicular, and draw CL, interfering it at W. Having the different feats 
 of light, proceed in projedling the fhadows which are found in the fame manner 
 as from the fun. Point b fhadow of J, is found by drawing through K, and the 
 leg of the table, a line at pleafure, interfering it by at and every point of 
 fiiadow on the ground of the table is found the fame way ^ pointy fhadow of 
 
 is found by drawing L/> at pleafure, and through the feat of y and P, draw 
 a line cutting L^ at in this manner you may finifh the fhadow on the wall, 
 of the circle I y. To find point 8 on the fide of the room, thro' K 
 and 4, draw a line cutting the feat of that fide of the room at /, and draw a 
 line through L and the top of the chair at pleafure, then draw 7 8 perpendi- 
 cular to the ground line interfering at 8. The points i, i6, on the cieling, 
 ihadow of the crofs g,f, are found by drawing through the feat of c, /, 
 and lines cutting the interfedion of the cieling at n, (the crofs being per- 
 pendicular to the pidure, its vanifhing point is C) then through n draw Q,nh, 
 and through every interfedion for the points k. The fhadow of the ball on 
 
 the 
 
PRACTICAL PERSPECTIVE. 6i 
 
 the deling is better nnderftood by Fig. 3. / is the point of light, and S its 
 feat, C the center of the ball, draw C /, divide it in two equal parts in k ; fet 
 on the foot of the compafles in ^, with the opening -^C, cut the circumference 
 of the ball in E and F, draw the line EatF. On the diameter 3C4, make the 
 circle 1234, parallel to the ground plane, and make another paffing thro' 
 the points i, 2, and the line Qk j thro' the points x and V, draw a line cut- 
 ting the circle 2 6 5 i, at 5 and 6, and make the circle E6 5F. That upper 
 part of the ball is as much as can be enlightened by a light placed at /, or as 
 much as can be feen by a fpeilator at that diftance. Find the points e 6 5 
 feats of E, 6, F, 5 ; then from S the feat of light, and through e ^6 f, &c. 
 draw lines parallel to the horizon, interfec^ting them by lines thro' / and the 
 points E6, F5, at », r, w, s, join thofe points for the fhadow of the ball oa 
 the ground. 
 
 EXAMPLE XIV. 
 Fig. 2. In this fcheme R is a ray of light from the fun parallel to the pic- 
 ture,V the center of the pidlure, 1645a fedtion parallel to the ground plane, 
 123 4 another perpendicular to the line R (as the rays from the (nvt are to 
 be confidered as parallel, one half of the ball is enlightened.) The fha- 
 dow of pnlt on the ground is found by drawing thro' every point of the 
 circle 1234 lines parallel to Ri, as V3, P9, and lines parallel to the ho- 
 rizon, paffing thro' their feats, interfedling them at ^« / the little circle 
 on the ground is the feat of the great one 1234, and is marked by the fame 
 figures. 
 
 CHAP 
 
62 PRACTICAL PERSPECTIVE. 
 
 CHAP. VI. 
 
 Of finding the Refledlion of any Figure on poliflied Planes or 
 
 Standing Water. 
 
 EXAMPLE XLIV. 
 Plate XLV. 
 
 It is' well known, that the refledlion of figures on a polifhcd plane, or on 
 the furface of ftanding water, appear to be juft as much on one fide of the 
 plane, as the real objeds are on the other fide : fo that to find the refleded 
 reprefentation of any point, you muft draw a perpendicular to the refleding 
 plane from the real point, and in it take a point at the fame diflance on the 
 contrary fide of the plane. For infl:ance, to find the refledion of the per- 
 pendicular AB (Plate XXXVII.) kt AC be the furface of the refleding 
 plane, and A the feat of B, continue the perpendicular AB downwards, till 
 hb is equal to AB. The angle any original line makes with the refieding 
 plane, is the fame as the refledlon with that plane. 
 
 As in F (Fig. 2.) DC the furface of the water, the angle GDF is equal 
 to the angle CD/ of the refledion. Vanifliing points of any real objeds are 
 the vanifiiing points of their refledlons, asO, Fig. 3. is the vanifhing point 
 of the fquare board, and its refledions : the conftrudion of that figure is fuf- 
 ficiently clear by the fchcme only. 
 
 EXAMPLE XLV. 
 Plate XXVI. Fig. 3. 
 In this example the furface of the water is fuppofed even with the ground, 
 or the line h and the circles in the refledion are the fame fize as the 
 real ones, and the fame diftance from the line h g downwards, C is the va- 
 nilliing point, and every point of fhadow in the refledion may be found 
 
 in 
 
PRACTICAL PERSPECTIVE. 63 
 
 in the fame manner, as nm in the real figure, making /H equal to HL, and 
 ufing the point / inflead of L, 
 
 CHAP. VII. 
 
 Of the Perfpedive of Landfcapes, 
 
 EXAMPLE XLVI. 
 Plate XXVIII. Fig. i. 
 In this example is reprefented a landfcape that has a defcent marked Gah, 
 a rifing ground e dK n, and GCi a flat country, HL the horizontal line, C 
 the center, and L thediftance, BF the vanifhing line of a plane inclined be- 
 low the horizon, LB its inclination, Bthe center, andF the diftance on it, AS 
 the vanifhing line of a plane inclined above the horizon, LA its inclinationj 
 A the center, S the point of difliance on it j draw GB, IB, and OF, cutting 
 IB in &, make b a parallel to GI. For the fquare piece of water, draw ^C, BC', 
 then aL,, or bH, cutting aC at Cy make c d parallel to a b. For the afcent, draw 
 cA, dAy interfering i:A by ^S, at K, make K« parallel to cd. GYinl be- 
 ing perpendicular to the pidture, vanifhes at C, and D is the vanifliing point 
 of the houfe E. To find point k in the water {a 0 being the furface) draw 
 the perpendicular K-^ at pleafure, and a C cutting it at r [r is the feat of K on 
 the water) make rk equal to r K, the points are found the fame, ck is the re- 
 fledion of £-K, and j of K 72. 
 
 The fhadows which are fuppofed to be caft by the fun, arnl parallel to the 
 pidlure, the line R the inclination of the rays of light, make li parallel to it, 
 and bi parallel to the horizontal line, join I i b. For the (hadow of I / on 
 the flope, draw / m to the point C, cutting c d in m, join n w, then limn is 
 the fhadow of lln on the down-hill furfaee of the water and rifing ground. 
 The pofls T, U, and their fhadows, vanifh at B. Plate XXIX. will fuffi- 
 eiently explain the refl-. 
 
 3 EXAMPLE 
 
64 PRACTICAL PERSPECTIVE. 
 
 EXAMPLE XLVIL 
 Plate XXVIIL Fig. 2. 
 In this example HL is the horizontal line, C the center (or the point of 
 fight removed in the pidture) GD the ground-line, B a rifing ground, ND 
 the feat of it, NB the perpendicular height at B, equal to DA on the ground- 
 line, KNI being produced towards the horizontal line till it cuts it, that 
 point is the diftance ; and if KN, GD, be continued till they meet, from 
 their interfedion to D will be the real diftance between the point D and point 
 N. The level ground at a great diftance rifes to the horizontal line ; the 
 tops of the hills higher, but their feats are below it. The light which is 
 from the fun is parallel to the picture, and the line R the inclination of the 
 i^ys of light, every fliadow, and reflcdtlon in the water is found as in the 
 foregoing example. Plate XXX. reprefents the fame finifhed. 
 
 EXAMPLE XLVIIL 
 Plate XXXL 
 In this landfcape are reprefented feveral ftreets, whofe vaniftiing points are 
 H, C, L placed on the horizontal line HL, C is the center of the pidture ; 
 the houfes ftanding obliquely with the pidure, have one fide vaniftiing at 
 point L, the other at H3 the houfe F having one fide perpendicular to the 
 pidure vaniflies at C; tne ftreets are fuppofed to be at right angle with 
 each other j O an odagon bafon, the perfpedive of it is found by the points 
 H, L on the horizontal line, and/;»as diameter On VH, perpen- 
 dicular to HL, are fet the vanifliing points Q^. The fide of the roofs 
 vanifhes at S, the ftied C at and the roof v of the houfe ^' atV; 
 the oppofite fide forming the angle vanifhes at a point as far below the hori- 
 ZDntal line as the other is above it. Make ^C perpendicular to H L, conti- 
 nue one fide of the pediment g of the houfe F, cutting kC at k, the vanifli- 
 ing point for that fide. If lines are drawn from the top and bottom of the 
 houfe e to the point L, any line perpendicular to HL between thofe lines, as 
 
 If 2, 
 
P R A C T I C A L P E R S P E C T I V E, 65 : 
 
 I, 2y will be equal to the height of the houfc 0. Thro* i and 2 draw lines ! 
 
 to H, for the top and bottom of the houfes in that ftreet, the line RL is per- 1 
 
 pendicular on HL, and h is the vanifhing point of z the roof of the church B. j 
 
 The light is parallel to the pidure, and to the line RH j fo that the flia- ■ 
 
 dow a; of the point If is found by drawing 3^ through its feat parallel to j 
 
 HL, and g x parallel to HR cutting it in x. All the rays being parallel to j 
 
 RH, and H being the vanifhing point oi t a (the fide of the houfe before the I 
 
 church) HR is the vanifliing line of the plane made by the rays which pafs j 
 tlirough ihe line i'a, and LR being the vanifliing line of the church B j the 
 
 fhadow ui k 'caft on, R is the vanifliing point of the common interfedion J 
 of thofe two planes, that is of the fliadow / « of the line t a j therefore the 
 
 line / « is drawn through the ' point R, and not parallel to HR as ^a: for the j 
 fhadow 3X ; through Q^raw «j foi: the continuation of the fliadow /«, of 
 on the top of the flied, and j w perpendicular to HL finiflies the fliadow 
 
 {At a and the fide of the houfe. \ 
 
 EXAMPLE XLIX. 
 Plate XXXIL 
 
 i 
 
 In this landfcape the houfes A, B, D, E. have one fide perpendicular to j 
 
 the picture, therefore vaniflies at C, the center j the dotted lines drawn j 
 
 through the bottom of the houfes is their feat, (which is even with the fur- ; 
 
 face of the water) make every refledion as far below that line as the real ' 
 
 objedl is above it, as appears by the Fig. i, 2, 3, in the water anfwering the j 
 
 fame figures above it j point O is the vanifliing point for the boards F and :^ 
 its refiedion f the light is fuppofed coming from behind the pidlure, O is 
 the vanifliing point of the fliadow on the ground, but the vanifliing point of 
 the rays of light are out of the plate, to fliorten the fliadows. Plate XXXIIL 
 
 reprefents the fame finiflied. , ; 
 
 K EXAMPLE 
 
 I 
 
 4 
 
6^ PRACTICAL P E R S PjE C T I V E. 
 
 ' ^ X A ' M^ P^ L e • L.^^^^^'P'^^^''' 
 
 Plate XXXIV/ '^^^"^ "^^-^ "^^^ ,11 oJ 
 
 In this plate are feveral reprefentations of balls or fpheres ] mod of their 
 fedions being parallel to the pidurc are fimilar to their.originals j and tho' th^ 
 contour that (hould form the balls round are n,ot -fo, yet. their appearances 
 iire round to every fpedator. That reprefentations and their appearances may 
 be different, is very evident, by what has been demonftrated at the .eiid of 
 the theory (See Plate XXXIX. Fig» i.) . where CD i? the reprefenta* 
 tion, and Cd the appearance j for unlefs the axis of the fphere pafs through 
 the fped:ator's eye, and center of the pidure, the reprefentation of it cannot 
 be round, if according to the ftridl rules of perfpedtive. Fig. 6. in this 
 plate /hews de for the reprefentation of the line p^£>, and f of point C (the 
 center of Dw) on the pidure HK, but c on the line d e is not in the middle. 
 But h k the appearance of Dw, and / of C to a fpedator's eye, placed as ia 
 the fcheme, i appears to be the middle of h k. 
 
 EXAMPLE LII. 
 
 Plate XXXV. 
 
 . ,c, 
 
 Reprefents a landicape with a rifmg ground that afcends towards the hori- 
 zon, and anfwcrs to Plate XXX. That in Plate XXXVI . anfvi'ers to 
 Plate XXXIII. 
 
 EXAMPLE LIII. 
 Plat e XXXVU. Fig. 2. 
 
 Let AB be the Wizontal line, G the center of the pidure, and GD the 
 ground line. Take J the middle of the line AB as diameter, and defcribe a 
 femicircle AOB, at C raife CO perpendicular to AB, cutting the circumfe- 
 rence at O point of fight : Let A be the vanifliing point of Ug. To de- 
 termine 
 
P R A C T I C A L P E R S P E C T 1 V E. 67 
 
 termine the point ^, take AO for a radius, defcribe the arch OU, and AU 
 will be the diftance j fet any point G on the ground line, and draw GU 
 cutting Hg- in ^ the point fought: in this manner may be cut any line in 
 the pidlure : if B is the vanifliing point, F is the diftance,^ BO being radius, 
 and OF the arch ; make HD equal to HG, draw HB and FD, cutting 
 HB at </, then draw dK, ^B interfedting at hy and <6 d is the perfpeClive 
 of a fquare, whofe fides are equal to HG or DH, and the angle a right one» 
 as AOB. 
 
 EXAMPLE LIV. 
 Plate XXXVII. Fig. i. 
 
 In this example every thing (landing as in the laft, on the ground line fet 
 any meafure given, as HD for the front of a building, IL the center part of 
 it, K the middle of the pediment, MG its depth, and MH the diftance 
 the corner of that building is in the pidlure. Draw HF, MU interfedting 
 at hy draw /?B, (&A, and from the points I, K,L, D, draw lines to point F, 
 cutting hB at /, /, d, and from the points M, N, G to U cutting Z?A at ng. To 
 give the proper height of every parts of the building, produce hB to the 
 ' ground line at P, make PQ^ perpendicular to GD, fet on it the fcveral di- 
 menlions, and on /, k, I, d, n g raife lines perpendicular to the ground line, 
 and through every divifions on PQ^ draw lines to B cutting the perpendi- 
 culars <6, /, ky /, dy at b, r, x, and through the points b, R on draw lines to 
 A cutting the perpendicular n at T, and^ at V, join bTV for the roof. To 
 find its vanifhing point, make EA perpendicular to AB j produce bT, cutting 
 EA in E for the vanifhing point of the fide of the roof bT x u. If EA be 
 continued below the horizontal line, and TV" produced, their interfedlion 
 will be the vaniftiing point for TV. To finilh the pediment uvfu, draw 
 i^A, «B at their interfedlion j, raife a perpendicular to GD, draw TB, cut- 
 ling it at^, then draw A /f, Arw, make 0 w parallel to zr, draw cB, ivB, 
 
 draw 
 
68 PRACTICAL PERSPECTIVE. 
 
 draw a line through A and /, cutting oB at z j make z «, parallel to w Oy 
 cutting i£^B at w, through KF draw a line cutting oz zX. e, at that point 
 raife a perpendicular to GD, and through^ and A draw^ fy cutting ef2it f ; 
 tlien join w f «, draw atB (the top of the window) till it cuts ir at /, then 
 draw At c cutting oiv c, draw ^"3 for the top of the window on that part 
 of the building, proceed in the fame manner for the under part of them } 
 for the windows on the fide F of the houfe, it is only ufing the point A in- 
 ftead of point B. Thefe examples are fufficient for the young praflitioner to 
 draw landfcapes, views, &c. &c. in perfpedive, if he has already acquired 
 fome Ikill in drawing, as by thefe rules he will be enabled to adjuft and cor- 
 real his defigns, and be capable of performing them to the greateft nicety 
 and exadtnefs. 
 
 FINIS. 
 
 ERRATA. 
 
 Page 
 
 line 
 
 for 
 
 read 
 
 16 
 
 27 
 
 Fig. 7. 
 
 Fig. 4. 
 
 43 
 
 3 
 
 proceed for 
 
 proceed the fame for 
 
 52 
 
 7 
 
 parallel to it 
 
 parallel to the piflure 
 
 54 
 
 10 
 
 on 
 
 C2, iD. 
 
 55 
 
 6 
 
 
 45 deg- 
 
 56 
 
 25 
 
 for R 
 
 from R. 
 
 60 
 
 3 
 
 dele the femicolon after 
 
 
 
 
 the word line. 
 
 
 64 
 
 25 
 
 roofs 
 
 roofy; 
 
« 
 
• 
 
1 
 
I 
 
X/1 
 
{ 
 
XXXI 
 
{ 
 
yxxtii 
 
V, 
 
 I 
 
 i 
 
 i 
 
500(1/ 
 
ERR 
 
 ATA. 
 
 
 Lint. 
 
 4- 
 
 21. 
 
 6. 
 
 12. 
 
 • 16. 
 
 27. 
 
 26. 
 
 4- 
 
 28. 
 
 »7- 
 
 43- 
 
 3- 
 
 52. 
 
 7- 
 
 53- 
 
 14. 
 
 54- 
 
 10. 
 
 55- 
 
 6. 
 
 26. 
 
 25. 
 
 59. 
 
 7-' 
 
 59- 
 
 9- 
 
 59- 
 
 10. 
 
 60. 
 
 6. 
 
 62. 
 
 6. 
 
 6;. 
 
 14. 
 
 66. 
 
 23- 
 
 67. 
 
 Ji. 
 
 Fof 
 
 Fig. I. Plate Vir. 
 
 add 
 
 Fig. 2. 
 
 AD 
 
 BD 
 
 proceed for 
 Parallel to it 
 
 Parallel with the Perfpedlive 
 
 /:/e/e on iefere C i 
 
 45 do 
 
 for R 
 
 Point 
 
 aS 
 
 6S/C 
 Fig. 3. 
 Plate XLV. 
 ^/ek thro' Q_ 
 Fig. 2. 
 Fig. I. 
 
 Fig. 7. Plate I. 
 and 
 Fig. 4. 
 AB 
 
 BE 
 
 proceed from 
 Parallel to the Pifture 
 Parallel with thePifture 
 
 45 degrees, 
 from R 
 Point r 
 ah 
 ihlC 
 Fig. I. 
 
 Plate XXXVII, 
 
 Fig. I. 
 
 Fig. 
 Fig.