OPT 
e,f, g, h, i, k, tending towards the point of fight S ; and 
draw BD for one of the diagonals of the fquare, the 
other diagonal AC being already drawn. Through the 
points r and s, where thefe diagonals cut ef and ik, draw 
Im parallel to AD. Through the centre-point x, where 
the diagonals cut gh, draw no parallel to AD. Laftly, 
through the points v and w, where the diagonals cut ef 
and ik, draw pq parallel to AD ; and the reticulated per- 
fpe&ive fquare will be finifhed. 
This fquare is truly reprefented, as if feen by an ob- 
ferver Handing at O, and having his eye above the hori¬ 
zontal plane ABCD on which it is drawn ; as if OS was 
the height of his eye above that plane : and the lines 
which form the fmall fquares within it have the fame 
letters of reference with thofe in fig. 12. which is drawn 
as it would appear to an eye placed perpendicularly 
above its centre x. 
Prob. IX. To put a circle in perfpeftive. 
If a circle be viewed by an eye placed direftly over 
its centre, it appears perfectly round; but, if it be ob¬ 
liquely viewed, it appears of an elliptical (liape. This 
is plain by looking at a common wine-glafs fet upright 
on a table. 
Make a true reticulated fquare, as fig. 12, of the fame 
diameter a3 you would have the circle ; and fetting one 
foot of your compafles in the centre x, defcribe as large 
a circle as the fides of the fquare will contain. Then 
having put this reticulated fquare into perfpeftive, as in 
fig. 13, obferve through what points of the crofs-lines 
and diagonals of fig. 12 the circle pafles ; and, through 
the like points in fig. 13, draw the ellipfis, which will be 
as true a perfpeftive reprefentation of the circle as the 
fquare in fig. 13 is of the fquare in fig-. 12. 
This is Mr. Fergufon’s rule for putting a circle in per- 
fpe&ive ; but the followinga-ules^ by Wolf are perhaps 
more univerfal. If the circle to be put in perfpe&ive be 
fmall, as fig. 14, Plate XIX. defcribe a fquare about it. 
Draw firft the diagonals of the fquare, and then the dia¬ 
meters ha and de, cutting .one another at right angles ; 
draw the ftraight lines^g- and be parallel to the diameter 
de. Through b and f, and likewife c andg, draw ftraight 
lines meeting DE, the ground-line of the picture in the 
points 3 and 4. To the principal point V, draw the 
ftraight lines rV, 3V, 4V, 2V, and to the points of diftance 
L and K, 2.L and iK. Laftly, join the points of inter- 
fe&ion, a, b, d,f, b, g, e, c, by the arcs ab, bd, df; and abdf 
hgeca will be the circle in perfpeftive. 
If the circle be large, fo as to make the foregoing prac¬ 
tice inconvenient, bifeet the ground-line AB, fig. 15. de- 
feribing from the point of bifedtion as a centre the fe- 
micircle AGB ; and from any number of points in the 
circumference C, F, G, H, I, &c. draw to the ground-line 
the perpendiculars Ci, F2, G3, I-I4, I5, &c. From the 
points A, 1, 2, 3, 4, 5, B, draw ftraight lines to the prin¬ 
cipal point, or point of fight V, likewife ftraight lines 
from B and A to the points of diftance L and K. 
Through the common interfe&ions draw ftraight lines 
as in the preceding cafe; and you will have the points 
a, c, f, g, h, i, b, reprefentatives of A, C, F, G, H, I, B. 
Then join the points a, c,f, &c. as formerly directed ; and 
you have the perfpeclive circle acfghibihgfca. 
Hence it is apparent how we may put not only a cir¬ 
cle, but alio a pavement laid with (tones of any form, in 
perfpeCtive. It is likewife apparent how ufeful the 
fquare is in perfpe&ive 5 for as, in the fecond cafe, a true 
fquare was deferibed round the circle to be put in 
perfpe&ive, and divided into feveral fmaller fquares, 
fo, in this third cafe, we make ufe of the femicircle 
only, for the fake of brevity, inltead of that fquare and 
circle. ■ 
Proe.X. To put a reticulated fquare in perfpe&ive, as 
feen by a perfon_not Handing right againft the middle 
I C S. 6G7 
of either of its fides, but rather nearly even with one 
of its corners. 
In fig. 16, let O be the place of an obferver, viewing 
the fquare ABCD almoft even with its corner D. Draw 
at pleafure SP for the horizon, parallel to AD, and 
make SO perpendicular to SP : then S (hall be the point 
of fight, and P the true point of diftance, if SP be made 
equal to SO. 
Draw AS and DS to the point of fight, and AP to the 
point of diftance, interfering DS in the point C ; then 
draw BC parallel to AD, and the outlines of the per- 
fpeftive fquare will be finilhed. This done, draw the 
lines which form the fmaller fquares, as taught in Prob. 
VIII. and the work will be completed. You may put a 
perfpe&ive circle in this fquare by the fame rule as it 
was done in fig. 13. 
Prob. XI. To put a cube in perfpe&ive, as if viewed by 
a perfon (landing almoft even with one of its edges, 
and feeing three of its fides. 
In fig. 17, let AB be the breadth of either of the fix 
equal fquare fides of the cube AG ; O the place of the 
obferver, almoft even with the edge CD of the cube ; S 
the point of fight, SP the horizon parallel to AD, and 
P the point of diftance taken as before. 
Make ABCD a true fquare ; draw BS and CS to the 
point of fight, and BP to the point of diftance, interfec- 
ting CS in G. Then draw FG parallel to BC ; and the 
uppermolt perfpe&ive fquare fide BFGC of the cube will 
be finifhed. 
Draw DS to the point of fight, and AP to the point 
of diftance, interfering DS in the point I: then draw 
GI parallel to CD ; and, if the cube be an opaque one, 
as of wood or metal, all the outlines of it will be finifhed; 
and then it may be fhaded as in the figure. But, if you 
want a perfpeftive view of a tranfparent glafs cube, all 
the fides of which will be feen, draw AH toward the 
point of fight, FH parallel to BA, and HI parallel to 
AD ; then AHID will be the fquare bafe of the cube, 
perfpe&ively parallel to the top BFGC ; ABFH will be 
the fquare fide of the cube parallel to CGID, and FGIH 
will be the fquare fide parallel to ABCD. 
As to the Jhading-part of the work, it is fuch mere 
children’s play, in comparifcn of drawing the lines 
which form the (liape of any object, that no rules need 
be given for it. Let a perfon fit with his left fide toward 
a window, and he knows full well, that, if any folid body 
be placed on a table before him, the light will fall on the 
left-hand fide of the body, and the right-hand fide will 
be in the (hade. 
Prob. XII. To put any folid in perfpe&ive. 
Put the bafe of the folid, whatever it be, in perfpeftive 
by the preceding rules. From each bounding-point of 
the bafe, raife lines reprefenting in perfpeCtive the alti¬ 
tude of the objeCt; by joining thefe lines and (hading 
the figure according to the directions in the preceding 
problem, you will have a fcenographic reprefentation of 
the objedl. 
This rule is general; but, as its application to particu¬ 
lar cafes may not be apparent, it will be proper to give 
the following example of it. 
Prob. XIII. To put a cube in perfpeClive, as feen from 
one of its angles. 
Since the bafe of a cube (landing of a geometrical 
plane, and feen from one of its angles, is a fquare feen 
from one of its angles, draw firft fuch a perfpeftive fquare, 
as at fig. 18. then raife from any point of the ground¬ 
line DE the perpendicular HI equal to the fide of the 
fquare, and draw to any point V in the horizontal line 
HR the ftraight lines VI and VH. From the angles d, b, 
and c, draw the dotted lines dz and ci parallel to the 
ground-line DE. Perpendicular to thofe dotted lines, 
t 1 and 
