606 
OPT 
of their interfe&ion with the ground-line. For example, 
to put the fquare ABCD (fig. 6.) into perfpe&ive. Draw 
from the projefting-point PV, jPW, parallel to AB, BC, 
and let AB, BC, CD, DA, meet the ground-line, in 
a, >c, f, / 3 ; and draw aV, fV, x W, ( 3 W, cutting each other 
in abed, the picture of the fquare ABCD. The demon- 
ftration is evident. 
This conftruCtion, however, runs the figure to great 
diftances on each fide of the middle line, when any of 
the lines of the original figure are nearly parallel to the 
ground-line. The following conftruCtion avoids this in¬ 
convenience : Let D, fig. 7, be the point of diftance. 
Draw the perpendiculars A a, B 13 , Cx, D$, and the lines 
Ac, B f, Cg, DA, parallel to PD. Draw Sa, S/ 3 , S*, Sf, and 
De, D/', Dg-, DA, cutting the former in a, b, c, d, the an¬ 
gles of the picture. It is not neceflary that D be the 
point of diftance, only the lines Ae, B f, &c. muft be pa¬ 
rallel to PD. 
In all the foregoing conftruCtions, the neceflary lines 
(and even the finiftied picture) are frequently confounded 
with the original figure. To avoid this great inconve¬ 
nience, the writers on perfpeCtive direct us to tranfpofe 
the figure ; that is, to transfer it to the other fide of the 
ground-line, by producing the perpendiculars Act., B| 3 , 
Cx, Df, till a A', /SB', &c. are refpeCtively equal to Ax, B/ 3 , 
&c. or, inftead of the original figure, to ufe only its tranf- 
pofed fubllitute A'B'C'D'. This is an extremely proper 
method. But in this cafe the point P muft alfo be tranf- 
pofed to P' above S, in order to retain the firft or moft na¬ 
tural and Ample conftruCtion, as in fig. 8. where it is evi¬ 
dent, that when BA=AB', and SPa^SP', and B'Pf is 
drawm cutting AS in h, we have bA : AS— B'A : P'S, 
z=BA : PS, and A is the picture of B ; whence follows 
the truth of all the fubfequent conftruCtions with the 
tranfpofed figure. 
Prob. IV. To put any curvilineal figure on the ground- 
plan into perfpective. 
Put a fufficient number of its points in perfpeCtive 
by the foregoing rules, and draw a curve-line through 
them. 
It is well known that the conic feCtions and fome other 
curves, when viewed obliquely, are conic feCtions or curves 
of the fame kinds with the originals, with different poll- 
tions and proportions of their principal lines, and rules 
may be given for deferibing their pictures founded on 
this property. But thefe rules are very various, uncon¬ 
nected with the general theory of perfpeCtive, and more 
tedious in the execution, without being more accurate, 
than the general rule now given. It would be a ufelefs 
affectation to infert them in this elementary treatife. 
We come in the next place to the delineation or figures 
not in a horizontal plane, and of folid figures. For this 
purpofe it is neceflary to demonftrate the following 
Theorem II. The length of any vertical line ftanding 
on the ground-plane is to that of its picture as the 
height of the eye to the diftance of the horizon-line 
from the picture of its foot. 
Let BC, fig. 2, be the vertical line ftanding on B, and 
let EF be a vertical line through the eye. Make BD 
equal to EF, and draw DE, CE, BE. It is evident that 
DE will cut the horizon-line in fome point d, CE will cut 
the piCture-plane in c, and BE, will cut it in A, and that Ac 
will be the picture of BC, and is vertical, and that BC is 
to be as BD to bd, or as EF to bd. 
Cor. The picture of a vertical line is divided in the 
fame ratio as the line itfelf. For BC : BM=Ac ; hn. 
Prob. V. To put a vertical line of a given length in per- 
fpeCtive, ftanding on a given point of the picture. 
Through the given point A, fig. 9, of the picture, draw 
SAAfrom the point of fight, and draw the vertical line AD, 
and make AE equal to the length or height of the given 
line. Join ES, and draw Ac parallel to AD, producing Ac, 
I c s. 
when neceflary, till it cut the horizontal line in d; and 
we have Ac : bd, =:AE : ADj that is, as the length of the 
given line to the height of the eye, and bd is the diftance 
of the horizon-line from the point A, which is the picture 
of the foot of the line. Therefore Ac is the required pic¬ 
ture of the vertical line. 
This Problem occurs frequently in views of architec¬ 
ture ; and a compendious method of folving it would be 
peculiarly convenient. For this purpofe, draw a vertical 
line ( XZ at the margin of the picture, or on a feparate paper, 
andjthrough any point V of the horizon-line draw VX. 
Set off XY, the height of the vertical-line, and draw 
VY. Then from any points A, r, on which it is required 
to have the pictures of iines equal to XY, draw A, s, r, t, 
parallel to the horizon-line, and draw the verticals su, tv : 
thefe have the lengths required, which may be transferred 
to A and r. This, with the third general conftruCtion for 
thebafe points, will fave all the confufion of lines which 
would arife from conftruCting each line apart. 
Prob. VI. To put any Hoping line in perfpeCtive. 
From the extremities of this line fuppofe perpendi¬ 
culars meeting the ground-plane in two points, which 
we (hall call the bafe points of the (loping line. Put 
thefe bafe points in perfpeCtive, and draw, by laft Pro¬ 
blem, the perpendiculars from the extremities. Join 
thefe by a ftraight line 5 and it will be the picture 
required. 
Prob. VII. To put a fquare in perfpeCtive, as feen by a 
perfon not ftanding right againft the middle of either 
of its fides, but rather nearly even with one of its cor¬ 
ners. 
V 1 
In fig. 10. let ABCD be a true fquare, viewed by an 
obferver, not ftanding at o, direCtly againft the middle 
of its fides AD, but at O, almoft even with its corner D, 
and viewing the fide AD under the angle AOD ; the an¬ 
gle AoD (under which he would have feen AD from o') 
being 60 degrees. 
Make AD, in fig. 11, equal to AD in fig. 10. and draw 
SP and Oo parallel to AD, Then, in fig. it, let O be 
the place of the obferver’s eye, and SO be perpendicular 
to SP; then S (hall be the point of fight in the horizon 
SP. Take SO in your compaffes, and fet that extent 
from S to P ; then P (hall be the true point of diftance, 
taken according to the foregoing rules. From A and D 
draw the ftraight lines AS and DS ; draw alio the ftraight 
line AP, interfering DS in C. Laftly, through the 
point of interfeCtion C draw BC parallel to AD ; and 
ABCD, in fig. 11, will be a true perfpeftive reprefenta- 
tion of the fquare ABCD in fig. 10. The point M is 
the centre of each fquare, and AMC and BMD are the 
diagonals. 
Prob. VIII. To put a reticulated fquare in perfpeCIive, 
as feen by a perfon ftanding oppofite to the middle of 
one of ics fides. 
A reticulated fquare is one that is divided into fe- 
veral little fquares, like net-work, as fig. 12. each fide 
of which is divided into four equal parts, and the whole 
furface into four times four (or iixteen) equal fquares. 
Having divided this fquare into the given number of 
fmaller fquares, draw the two diagonals AarC and B.rD. 
Make AD in fig. 13, equal to AD in fig. 12. and divide 
it into four equal parts, as Ae, eg, gi, and iD. Draw SP 
for the horizon, parallel to AD, and through the mid¬ 
dle point g of AD draw OS perpendicular to AD and 
SP. Make S the point of fight, and O the place of the 
obferver’s eye. 
Take SP equal to SO, and P (hall be the true point of 
diftance. Draw AS and DS to the point of fight, and AP 
to the point of diftance, interfeCling DS in C : then draw 
BC parallel to AD, and the outlines of the .reticulated 
fquare ABCD will be fini(hed. 
From the diviiion-points e, g, i, draw the ftraight lines 
e,f. 
