605 
OPT 
Cor. 4. All lines parallel to AB are reprefented by lines 
converging to V from the points where thefe lines inter- 
feCt the perfpeCtive-plane; and therefore V is the vaniffi- 
ing-point of all fuch parallel lines. 
Cor. 5. The pictures of all lines parallel to the perfpec- 
tive-plane are parallel to the lines tlremfelves. See fig. a. 
Cor. 6. If through V be drawn HVO parallel to KL, 
fig. 1, the angle EVH is equal to BAK. 
The propolition now demonftrated is not limited to 
any inclination of the piCture-plane to the ground-plane ; 
but it is ufual to confider them as perpendicular to each 
other, and the ground-plane as horizontal. Hence the 
line KL is called the ground-line , and OH the horizon- 
line / and OK, perpendicular to both, is called the height 
of the eye. If ES be drawn perpendicular to the piCture- 
plane, it will cut it in a point S of the horizon-line di¬ 
rectly oppofite to the eye. This is called the point of fight, 
or principal point. 
Cor. 7. The pictures of all vertical lines are vertical, 
and the pictures of horizontal lines are horizontal, be- 
caufe thefe lines are parallel to the perfpeCtive-plane. 
Cor. 8. The point of fight S is the vaniffiing-point of 
all lines perpendicular to the perfpeCtive-plane. 
The above propofition is a fufficient foundation for the 
whole practice of perfpeClive, whether on direCt or in¬ 
clined pictures, and ferves to fugged all the various prac¬ 
tical conductions, each of which has advantages which 
fuit particular purpofes. Writers on the fubjeCt have ei¬ 
ther confined themfelves to one condruCtion, from an af¬ 
fectation of fimplicity or fondnefs for fydem; or have 
multiplied precepts, by giving every condruCtion for 
every example, in order to give the fubjeCt an appearance 
of importance and difficulty. An ingenious practitioner 
will avoid both extremes, and avail himfelf of the advan¬ 
tage of each condruCtion as it happens to fuit his purpofe. 
We (hall now proceed to the practical rules, which re¬ 
quire no confideration of interfeCting planes, and are all 
performed on the perfpeCtive-plane by means of certain 
lubditutions for the place of the eye and the original 
figure. The general fubditution is as follows : 
Let the plane of the paper be fird fuppofed to be the 
ground-plan, and the fpeCtator to dand at F, fig. 3. Let 
it be propofed that the ground-plan is to be reprefented 
on a plane lurface, danding perpendicularly on the line 
GKL of the plan, and that the point K is immediately 
oppofite to the fpeCtator, or that FK is perpendicular to 
GL; then FK is equal to the didance of the fpeCtator’s 
eye from the picture. Now fuppofe a piece of paper laid 
on the plan with its draight edge lying on the line GL; 
draw on this paper KS perpendicular to GL, and make it 
equal to the height of the eye above the ground-plan. 
This may be much greater than the height of a man, be- 
caufe the fpeCtator may be danding on a place much railed 
above the ground-plan. Obferve alfo that KS mud be 
meafured on the fame fcale on which the ground-plan 
and the didance FK were meafured. Then draw HSO 
parallel to GL. This will be a horizontal line, and (when 
the picture is fet upright on GL) will be on a level with 
the fpeCtator’s eye, and the point S will be direCtly oppo¬ 
fite to his eye. It is therefore called the principal point , 
or point of Jight. The didance of his eye from this point 
will be equal to FK. Therefore make SP (in the line SK) 
equal to FK, and P is the projecting point, or fubditute 
for the place of the eye. It is fometimes convenient to 
place P above S, fometimes to one fide of it on the hori¬ 
zontal line, and in various other fituations; and writers, 
ignorant of, or inattentive to, the principles of the theory, 
have given it different denominations, fuch as point of 
dift ance, point of view , &c. It is merely a fubditute for the 
point E. in fig, 1. and its mod natural fituation is below, 
as in this figure. 
Fundamental Problem I. To put into perfpeClive 
any given point of the ground-plan. 
Firjl general ConJlruRion. —From B and P, fig. 3, draw 
Vol.XVII. No. izo6. 
I c s. 
any two parallel lines, BA, PV, cutting the ground-line 
and horizon-line in A and V; and draw BP, AV, cutting 
each other in h ; b is the picture of B. 
For it is evident that BA, PV, of this figure, are ana¬ 
logous to BA and EV of fig. 1. and that BA : PVr= 
bA : bV. 
If BA' be drawn perpendicular to GL, PV will fall on 
PS, and need not be drawn. A'V will be A'S. This is 
the mod eafy condruCtion, and nearly the fame with 
Fergufon’s. 
Second general ConftruSlion .—Draw two lines BA, BA", 
and two lines PV, PD, parallel to them; and draw AV, 
A"D, cutting each other in b; b is the picture of B by 
Cor. z. This condruCtion is the foundation of all the 
rules of perfpeClive that are to be found in the books on 
this fubjeCt. They appear in a variety of forms, owing to 
the ignorance or inattention of the authors to the princi¬ 
ples. The rule mod generally adhered to is as follows : 
Draw BA, fig. 4, perpendicular to the ground-line, and 
AS to the point of fight; and fet oft’ Aj 3 equal to BA. 
Set od' SD equal to the didance of the eye in the oppofite 
direction from S that |3 is from A, where B and E of fig. 1 
are on oppofite fides of the picture; otherwife fet them 
the fame way. D is called the point of didance. Draw 
/ 3 D, cutting AS in b. This is evidently equivalent to 
drawing BA' and PS perpendicular to the ground-line 
and horizon-line, and BA" and PD (fig. 3.) making an 
angle of 45 0 with thefe lines, with the additional puzzle 
about the way of fetting off A'A" and SD, which is 
avoided in the condruCtion here given. 
This ufual condruCtion, however, by a perpendicular 
and the point of didance, is extremely fimple and conve¬ 
nient; and two points of didance, one on each fide of S, 
ferve for all points of the ground-plan. But the fird ge¬ 
neral condruCtion requires dill fewer lines, if BA be 
drawn perpendicular to GL, becaufe PV will then coin¬ 
cide with PS. 
Third general Conflrudion .—Draw BA, fig. 4, from the 
given point B perpendicular to the ground-line, and AS 
to the point of fight. From the point of didance D fet 
off Dd equal to BA, on the fame or the contrary fide as S, 
according as B is on the fame or the contrary fide of the 
picture as the eye. Join d, A, and draw D b parallel to 
clA. b is the picture of B. For SD, Dr/, are equal to the 
didances of the eye and given point from the picture; and 
SD : Dd=.bS : bA. 
This condruCtion does not naturally arife from the 
original lines, but is a geometrical confequence from their 
pofition and magnitude ; and it is of all others the mod 
generally convenient, as the perpendicular didances of 
any number of points may be arranged along SD without 
confufion, and their direCt fituations transferred to the 
ground-line by perpendiculars fuch as BA ; and nothing 
is eafier than drawing parallels, either by a parallel ruler 
or a bevel-fquare, ufed by all who praCtife drawing. 
Prob. II. To put any draight line BC (fig. 5.) of the 
ground-plan in perfpeClive. 
Find the pictures b, c, of its extreme points by any of 
the foregoing condruCtions, and join them by the draight 
line be. 
Perhaps the following condruCtion will be found very 
generally convenient. Produce CB till it meet the ground¬ 
line in A, and draw PV parallel to it; join AV, and draw 
PB, PC, cutting AV in l>, c. V is its vaniffiing-point, by 
Cor. 3. of the Fundamental Theorem. 
Pros. III. To put any rectilineal figure of the ground- 
plan in perfpeClive. 
Put the bounding lines in perfpeCtive, and the problem 
is folved. 
The variety of condruCtions of this problem is very 
great, and it would fill a volume to give them all. The 
mod generally convenient, is to find the vanidiing-points 
of the bounding lines, and conned thefe with the points 
8 G of 
