into nine equal squares, which are the outer upright HK parallel to AD, and you will have the second persurfaces of the wine cubes in the side AD of the square spective square GHKI of the parailelogram:-Go on in ABCD. this manner till you have drawn as many perspective From the points where the lines, which are parallel squares up toward S as you please. to AD in this square, meet the side CD thereof, draw Through the point e, where DG intersect g S, draw short lines to LN, all parallel to DL, and they will di bf parallel to D; and you will have formed the two vide that side into the outer upright surfaces of the nine perspective square bases A b c d and e f Dg of the two cubes which compose it: and then the outsides of all the pyramids at A and D. cubes that can be visible to an observer, placed at a pro From the point f (the upper outward corner of ef per distance from the corner D of the square, will be D g) draw fh toward the point of distance, till it meets finished. AS in h; then, from this point of meeting, draw h m As taught in Prob. 17. place the pyramid AE up- parallel to GI, and you will have formed the two perright on its square base A tv a, making it as high as spective squares Ghik and Im I n, for the square bases you please ; and the pyramid DH on its square base of the two pyramids at G and I. h u w D, of equal height with AE. Proceed in the same manuer to find the bases of all Draw EH from the top of one of these pyramids the other pyramids, at the corners of the rest of the to the top of the other; and EH will be parallel to perspective squares in the parallelogram ABCD, as AD. shown by the figure. Then, Draw ES and HS to the point of sight S, and HP to Having placed the first two pyramids at A and D the point of distance P, intersecting FS in F. upright on their square bases, as shown in Prob. 9. From the point F, draw FG parallel to EH; then and made them of any equal heights at pleasure, draw draw EG, and you will have a perspective square EFGH ES and FS from the tops of these pyramids to the point (parallel to ABCD) with its two diagonals EG and of sight S: place all the rest of the pyramids upright TH, intersecting one another in the centre of the square on their respective bases, making their tops touch the at I. The four corners of this square, E, F, G, H, straight lines ES and FS; and all the work, except the give the perspective heights of the four pyramids AE, shading part, will be finished. BF, CG, and DH; and the intersection I of the diagonals gives the height of the pyramid MI, the centre Prob. 24. To put a square pyramid of equal sized of whose base is the centre of the perspective square cubes in perspective. ABCD. Fig. 27. represents a pyramid of this kind; consist- Fig. 27, Lastly, place the three pyramids BF, CG, MI, up ing as it were of square tables of cubes, one table right on their respective bases at B, C, and M; and the above another; 81 in the lowest, 49 in the next, 25 required perspective representation will be finished as in in the third, 9 in the fourth, and 1 in the fifth or upthe figure. permost. These are the square numbers of 9, 7, 5, 3, and 1. Prob. 23. To put upright pyramids in perspective, on If the artist is already master of all the preceding the sides of an oblong square or parallelogram ; so operations, he will find less difficulty in this than in atthat their distances from one another shall be equal to tending to the following description of it: for it cannot the breadth of the parallelogram. be described in a few words, but may be executed in a In most of the foregoing operations we have consi In fig. 28. having drawn PS for the horizon, and ta- Fig. 28. dered the observer to be so placed, as to have an obj. ken S for the point of sight therein (the observer being lique view of the perspective objects : in this, we shall at O) draw AD parallel to PS for the side (next the suppose bim to have a direct view of fig. 26. that is, eye) of the first or lowermost table of cubes. Draw standing right against the middle of the end AD which AS and DS to the point of sight S, and DP to the is nearest to bis eye, and viewing AD under an angle point of distance P, intersecting AS in the point B. of 60 degrees. Then, from B, draw BC parallel to AD, and you will Having cut AD in the middle, by the perpendicular have the surface ABCD of the first table. line Ss, take S therein at pleasure for the point of sight, Divide AD into nine equal parts, as A a, a b, b and draw ES for the horizon, parallel to AD.-Here cd, &c. then make AK and DL equal to A a, and perSs must be supposed to be produced downward, below pendicular to AD. Draw KL parallel to AD, and the limits of the plate, to the place of the observer; and from the points of equal division at a, b, c, &c. draw SE to be produced towards the left hand beyond E, far lines to KL, all parallel to AK. Then draw h S to enough to take a proper point of distance therein, ac the point of sight S, and from the division points a, b, c, cording to the foregoing rules. &c. draw: lines with a black lead pencil, all tending toTake A d at pleasure, and Dg equal to A d, for the wards the point of sight, till they meet the diagonal BD breadths of the square bases of the two pyramids AE and DF next the eye : then draw AS and d S, and like From these points of meeting draw black lead lines wise DS and g S, to the point of sight S; and DG on to DC, all parallel to AD, then draw the parts of these to the point of distance, intersecting AS in G: then, lines with black ink which are marked 1, 2, 3, 4, &c. from G draw GI parallel to AD, you will have the between h E and DC. first perspective square AGID of the parallelogram Having drawn the first of these lines B q with black ABCD. ink, draw the parts a i, b k, cl, &c. (of the former From I draw. IH to (or toward) the point of dis- lines which met the diagonal BD) with black ink also ; . tance, intersecting AS in H: then, from H draw and rub out the rest of the black lead lines, which VOL. XVI. Part I. + Z would very short time. of the square. would otherwise confuse the following part of the work. rizon, and V P to the point of distance therein, intersectThen, draw LF towards the point of sight S; and, from ing SS in T; then draw TU parallel to SV, and you the points where the lines 1, 2, 3, 4, &c. meet the line have STUV, the surface of the fourth table, which beDC, draw lines down to LF, all parallel to DL; and ing reticulated or divided into 9 perspective small all the visible lines between the cubes in the first table squares, and the uppermost cube W placed on the midwill be finished. dlemost of the squares, all the outlines will be finished; Make i G equal and perpendicular to Bi, and q M and when the whole is properly shaded, as in fig. 27. equal and parallel to i G; then draw GM, which will the work will be done. be equal and parallel to i q. From the points k, l, m, n, &c. draw k, n, 10, m p, &c. all parallel to i G, and the PROB. 25. To represent a double cross in perspective. outsides of the seven cubes in the side Gq of tbe second table will be finished. In fig. 29. let ABCD and EFGH be the two per-Fig. 29. Draw GS and MS to the point of sight S, and MP spective squares, equal and parallel to one another, the to the point of distance P, intersecting GS in H; then, uppermost directly above the lowermost, drawn by the from the point of intersection H, draw HI parallel to rules already laid down, and as far asunder as is equal AD; and you will have the surface GHIM of the se to the given height of the upright part of the cross ; S being the point of sight, and P the point of distance, in cond table of cubes. From the points n, o, p, q, &c. draw black lead the horizon PS taken parallel to AD. lines toward the point of sight S, till they meet the Draw AE, DH, and CG; then AEHD and diagonal MH of the perspective square surface GHIM; DHGC shall be the two visible sides of the upright and draw SM, with black ink, toward the point of part of the cross ; of which, the length AE is bere sight. made equal to three times the breadth EH. From those points where the lines drawn from 1, 0, Through these points of division, at I and K, draw MO Divide DH into three equal parts, HI, IK, and KD. P, 9, &c. meet the diagonal MH, draw black lead lines to MI, all parallel to AD; only draw the whole first and PR parallel to AD; and make the parts MN, 10, Jine 1 with black ink, and the parts 2, 3, 4, &c. and PQ, KR, each equal to HI : then draw MP and OR unt, ou, p v, &c. of the other lines between y N and parallel to DH, From M and O, draw MS and OS to the point of MI, and GM and g 1, with the same; and rub out all the rest of the black lead lines, to avoid further confu- sight S; and from the point of distance P draw PN sion. Then, from the points where the short lines 1, cutting MS in 'T: from T draw TU parallel to MO, 2, 3, &c. meet the line MI, draw lines down to q E, and meeting OS in U; and you will have the upperall parallel to M q, and the outer surfaces of the seven most surface MTUO of one of the cross pieces of the cubes in the side ME will be finished ; and all these figure.-From R, draw RS to the point of sight S; last lines will meet the former parallels 2, 3, 4, &c. in and from U draw UV parallel to OR; and OUVR the line shall be the perspective square end next the eye of that E. 9 Make t O equal and perpendicular to y t, and cross part. P y equal and parallel to t 0; then draw OP, which will Draw PMX (as long as you please) from the point be equal and parallel to t y.--This done, draw OS and of distance P, through the corner M; lay a ruler to N PS to the point of sight S, and PP to the point of dis and S, and draw XN from the line PX :-then lay the tance P in the horizon. Lastly, from the point Q, ruler to 1 and S, and draw YZS.-Draw XY parallel where PP intersects Os, draw QR parallel to OP; to MO, and make XW and YB equal and perpendiand you will have the outlines OQRP of the surface of cular to XY: then draw WB parallel to XY, and the third perspective table of cubes. WXYB shall be the square visible end of the other From the points u, v, w, x, draw upright lines to cross part of the figure. OP, all parallel to i 0, and you will have the outer Draw BK toward the point of sight S; and from U surfaces of the five cubes in the side Oy of this third draw UP to the point of distance P, intersecting YS in Z: then, from the intersection Z, draw 2 a parallel table. From the points where these upright lines meet OP, to MO, and Z b parallel to HD, and the whole deli neation will be finished. draw lines toward the point of sight S, till they meet the diagonal PQ; and from these points of meeting will have a true perspective representation of a double This done, shade the whole, as in fig. 30. and you Fig. 3o. draw lines to PR, all parallel to OP, making the parts 2, 3, 4, 5, of these lines with black ink which lie between ZY and PR. Then, from the points where these Prob. 26. To put three rows of upright square objects lines meet PR, draw lines down to y N; which will bound the outer surfaces of the five cubes in the side in perspective, equal in size, and at equal distances PN of the third table. from each other, on an oblong square plane, the Draw the line d i with black ink; and, at a fourth breadth of which shall be of any assigned proportion part of its length between d and Z, draw au upright to the length thereof. line to S, equal in length to that fourth part, and ano. Fig. 31. is a perspective representation of an oblong Fig. 31. ther equal and parallel thereto from Z to V: then draw square plane, three times as long as it is broad, having SV parallel to d Z, and draw the two upright and equi a row of nine upright square objects on each side, and distant lines between & Z and SV, and you will have the one of the same number in the midde; all equally bigh, outer surfaces of the three cubes in the side SZ of the and at equal distances from one another, both long-wise fourth table. and cross-wise, on the same plane. Draw SS and VS to the point of sight S in the ho In fig. 32. PS is the horizon, S the point of sight, P Fig. 32. the cross. in fig. 31. the point of distance, and AD (parallel to PS) the Through the point x, where f c intersects 8 S, draw breadth of the plane. a line parallel to bf; and you will have three perspecDraw AS, NS, and DS, to the point of sight S; the tive squares, at b, l, and x, for the bases of the fourth point N being in the middle of the line AD: and draw cross row of objects. DP to the point of distance P, intersecting AS in the Go on in this manner, as you see in the figure, to point B : then, from B draw BC parallel to AD, and find the rest of the square bases, up to LM; and you you have the perspective square ABCD. will have 27 upon the whole oblong square plane, on Through the point i, where DB intersects NS, draw which you are to place the like number of objects, as a e parallel to AD; and you will have subdivided the perspective square ABCD into four lesser squares, as Having assumed AO for the perspective height of Aai N, Nie D, a B ki, and i k Ce. the three objects at A, N, and D (hg. 32.) next the From the point C (at the top of the perspective observer's eye, and drawn 0 18 parallel to AD, in square ABCD) draw CP to the point of distance P, order to make the objects at N and D of the same intersecting AS in E; then from the point E draw EF height as that at 0, and having drawn the upright parallel to AD; and you will have the second perspec lines 4 15, 7 W, 8 X, and D 22, for the heights at N tive square BEFC. and D; draw OS and RS, 15 S and WS, XS and Througb the point I, where CE intersects NS, draw 22 S, all to the point of sight S: and these lines will bf parallel to AD; and you will have subdivided the determine the perspectively equal heights of all the rest square BEFC into the four squares B blk, kif C, of the upright objects, as shown by the two placed at b Em I, and I m Ff. a and B. From the point F (at the top of the perspective To draw the square tops of these objects, equal and square BEFC) draw FP to the point of distance P, in- parallel to their bases, we only need give one example, tersecting AS in I; then from the point I draw IK pa which will serve for all. rallel to AD; and you will have the third perspective Draw 3 R and 2 Q parallel to A0, and up to the square EIKF. line RS; then draw PQ parallel to OR, and OPQR Through the point n, where FI intersects NS, draw shall be the top of the object at A, equal and parallel cg parallel to AD; and you will have subdivided the to its square base A 1 2 3:- In the same easy way the square EIKF into four lesser squares, Ecnm, mng F, tops of all the other objects are formed. clon, and no Kg. When all the rest of the objects are delineated, shade From the point K (at the top of the third perspective them properly, and the whole perspective scheme will square EIKF) draw KP to the point of distance P, in. have the appearance of fig. 31. tersecting As in L; then from the point L draw LM PROB. 27. To put a square box in perspective, contain, parallel to AD; and you will have the fourth perspec ing a given number of lesser square boxes of a depth tive square ILMK. Through the point p, where KL intersects NS, draw equal to their width. dh parallel to AD; and you will have subdivided the Let the given number of little square boxes or cells Fig. 33. square ILMK into the four lesser squares Id po,o phK, be 16, then 4 of them make the length of side of dLp, and p q Mh. the four outer sides a b, b c, cd, da, as in fig. 33. and Thus we have formed an oblong square ALMD, the depth a f is equal to the width a e. Whoever can whose perspective length is equal to four times its breadth, draw the reticulated square, by the rules laid down toand it contains 16 equal perspective squares.-If greaterwards the beginning of this article, will be at no loss length was still wanted, we might proceed further on about putting this perspective scheme in practice. toward S. PROB. 28. To put stairs with equal and parallel steps Take A 3, equal to the intended breadth of the side in perspective. of the upright square object AQ (all the other sides being of the same breadth), and AO for the intended In fig. 34. let a b he the given breadth of each step, Fig 34. height. Draw 0 18 parallel to AD, and make D 8 and a i the height thereof. Make b c, cd, de, &c. and 4 7 equal to A 3 ; then draw 3 S, 4 S, 7 S, and each equal to ab; and draw all the upright lines 8 S to the point of sight S; and among them we shall ai, bl, c n, d p, &c. perpendicular to a ħ (to which have the perspective square bases of all the 27 upright the horizon s S is parallel); and from the points i, l, objects on the plane. n, p, r, &c. draw the equidistant lines i B, 1 c, n D, Through the point 9, where DB intersects 8 S, draw &c. parallel to a h; these distances being equal to that I 10 parallel to AD, and you of i B from a h. v; and draw 2 16 parallel to z i, as far from it as you Through the point 21, where eb intersects 8 S, want the length of the steps to he. draw 14, 11 parallel to AD; and you will have the Toward the point of sight S draw the lines a 1, i 2, three perspective squares, a 14 15 16 17 18 19 20, and k 3, 14, &c. and draw 16 15,14 13,12 11, 10 9,87, 21 11 e 22, for the bases of the second cross row of 6 5,4 3, and 2 1, all parallel to A h, and meeting objects ; namely, the next beyond the first three at A, the lines w 15, u 13, s 11, &c. in the points 15, 13, 11, N, and D. 9, 7, S, 3, and 1 : then from these points draw 15 14, Through the point w, where CE intersects 8 S, draw 13.12, 11 10,9 8,7 6, 5 4, and 3 2, all parallel to ha; a line parallel to BC; and you will have three perspec and the outlines of the steps will be finished From tive squares, at B, k, and Ć, for the base of the third the point 16 draw 16 A parallel to h a, and A x 16 row of objects; one of which is set up at B. will be part of the flat at the top of the uppermost step. This ጊ 2 Fig. 35. Fig. 36. level of his eye. This done, shade the work as in fig. 35. and the whole PROB. 30. To put upright conical objects in perspective, as if standing on the sides of an oblong square, at diperspective, standing on a horizontal pavement of stances from one another cqual to the breadth of the oblong. squares. In fig. 36. having made S the point of sight, and spective circles inscribed in squares of the same diame In fig. 38. the bases of the upright cones are per-Fig. 3%. drawn a recticulated pavement AB with black lead Jines, which may be rubbed out again ; at any distance ter; and the cones are set upright on their bases by the from the side AB of the pavement which is nearest to same rules as are given for pyramids, which we need not repeat here. the eye, and at any point where you choose to begin the stair at that distance, as a, draw G a parallel to BA, ed the observer's eye to be above the level of the tops In most of the foregoing operations we bave considerand take a b at pleasure for the height of each step. of all the objects, as if he viewed them when standing Take a b in your compasses, and set that extent as many times upward from F to E as is equal to the first on high ground. In this figure, and in fig. 41. and required number of steps 0, N, M, L, K; and from fig. 42. we shall suppose him to be standing on low these points of division in EF draw 1 6, 2d, 39, 4h, ground, and the tops of the objects to be above the and E k, all equidistant from one another, and parallel to Fa: then draw the equidistant upright lines ub, In fig. 38. let AD be the perspective breadth of the Fig. 35. td, uf, vh, w k, and I m, all perpendicular to Fa: oblong square ABCD); and let A a and Dd (equal to then draw m b, touching the outer corners of these steps A a) be taken for the diameters of the circular bases of at m, k, h, f, d, and b; and draw n s parallel to mb, shall be AE and DF. the two cones next the eye, whose intended equal heights as far from it as you want the length of the steps K, L, Having made the point of sight in the horizon pa- rallel to AD, and found the proper point of distance cones on the left-hand side, and DS and d S for those on the right. 5, 14; 6, 15; 7, 16; 1, 17; and 2 s: which done, Having made the two first cones at A and D of draw n 5 and 06 parallel to Im, and the outlines of the equal beight at pleasure, draw ES and FS from their steps K, L, M, N, O will be finished. At equal distances with that between the lines mark- tops to the point of sight, for limiting the perspective ed 8 and 14, draw the parallel lines above marked 9 parallelogram ABCD into as many equal perspective heights of all the rest of the cones. Then divide the JO II 12 and 13; and draw perpendicular lines upwards from the points n, o, p, q, r, s, as in the figure. squares as you please ; find the bases of the cones at Make H m equal to the intended breadth of the flat the corners of these squares, and make the cones thereabove the square opening at the left hand, and draw on, as in the figure. If you would represent a ceiling equal and parallel HW toward the point of sight S, equal to the intended length of the flat: then draw WP parallel to Hm, and to ABCD, supported on the tops of these cones, draw EF, then EFGH shall be the ceiling; and by drawthe outlines of the flat will be finished. Take the width of the opening at pleasure, as from ing ef parallel to EF, you will have the thickness of the floor-boards and beams, wbich may be what you F to C, and draw CD equal and parallel to FE. Draw GH parallel to CD, and the short lines marked 33, 34, please. This shows how any number of equidistant pillars &c. just even with the parallel lines 1, 2, &c. From may be drawn of equal heights to support the ceiling the points where these short lines meet CD draw lines toward the point of sight Still they meet DE; then of a long room, and how the walls of such a room may be represented in perspective at the backs of these pil. from the points where the lines 38, 39, 40, &c. of lars. It also shows how a street of houses may be drawu the pavement meet C y, draw upright lines parallel to in perspective. depth of which shall bear any assigned proportion lo arch V, are done in the same manner with those in its width. the observer's eye is supposed to be placed, looking therein not nearer Sthan if AD was seen under an Fig. 37. in E; and you will have the first perspective square ABCD, both to right and left hand from 9, and divided all the equal to a third part of the intended depth. way in the same manner. Draw CV to the point of distance, intersecting AS From these points of equal dirision, 8, 9, 10, &c. then from the point E draw EF parallel to AD; draw lines to the point of sight S, and also to the two and you will liave the second perspective square BEFC, points of distance p and P, as in the figure. which, added to the former one, makes two thirds of the Now it is plain, that ach 9 is the perspective repreintended depth. sentation of A 9 BC, viewed by an observer even with Draw FWV to the point of distance, intersecting AS the corner C and diagonal C 9.-But if there are other in G; then from the point G draw GH parallel to such squares lying even with this, and having the same AD; and you will have the third perspective square position with respect to the line i 17, it is evident that EGHF, which, with the former two, makes the whole the observer, wbo stands directly even with the corner depth AGHD three times as great as the width AD, C of the first square, will not be even with the like. in a perspective view. corners G and K of the others; but will bave an obDivide AD into any number of equal parts, as sup- lique view of them, over the sides FG and IK, which pose 8; and from the division-points a, b, c, d, &c. are nearest his eye: and their perspective representadraw lines toward the point of sight S, and ending at tions will be eg f 6 and hki 3, drawn among the lines GH; then through the points where the diagonals BD, in the figure: of which the spaces taken up by each EC, GF, cut these lines, draw lines parallel to AD; side lie between three of the lines drawn toward the and you will have the parallelogram AGHD reticu- point of distance P, and three drawn to the otber point lated, or divided into 192 small and equal perspective of distance P. squares. PROB. Make AI and DM equal and perpendicular to AD; 34. To represent a common chair, in an oblique then draw IM, which will be equal and parallel to AD; perspective view. and draw IS and MS to the point of sight S. The original lines to the point of sight S, and points Fig. 43. Divide AI, IM, and MD, into the same number of of distance p and P, being drawn as in the preceding equal parts as AD is divided; and from these points of operation, choose any part of the plane; as l m n 13, on division draw lines toward the point of sight 8, ending which you would have the chair L to stand. There respectively at GK, KL, and LH. are just as many lines (namely two) between 1 and inFrom those points where the lines parallel to AD or 13 and n, drawn toward the point of distance p, at meet AG and DH draw upright lines parallel to AI the left hand, as between 1 and 13, or m and n, drawn and DM; and front the points where these lines meet to the point of distance P on the right: so that I m, IK and LM drawlines parallel to IM; then shade the mn, n 13, and 13 l, form a perspective square. work, as in the figure. From the four corners l, m, n, 13, of this square raise the four legs of the chair to the perspective perPROB. 32. To represent a semicircular arch in perspec; pendicular height you would have them : then make tive as if it were standing on two upright walls, equal the seat of the chair a sqnare equal and parallel to l m n in height to the height of the observer's eye. 13, as taught in Prob. 18. which will make the two After having gone through the preceding operation, sides of the seat in the direction of the lines drawn toFig. 42. this will be more easy by a bare view of fig. 42. than it ward the point of distancep, and the fore and back part could be made by any description; the method being of the seat in direction of the lines drawn to the other lique perspective view. In fig. 43. M is an oblong square table, as seen by Fig. 430 an observer standing directly even with C 9 (see Prob. PROB. 33. To represent a square in perspective, as view. 33.), the side next the eye being perspectively parallel ed by an observer standing directly even with one of to the side ac of the square abc 9.- The forementioned lines drawn from the line 1 17 to the two points of di- 9 BC be a true square, viewed by stance p and P, form equal perspective squares on the Choose any part of this plane of squares for the feet oblong square or parallelogramgrst for the part of the t, Fig. 43. |