242 
PERSPECTIVE MADE EASY. 
every line in tlie objects to be represented, 
that runs parallel to the line k 1. The 
vanishing point of any line commencing at 
the picture -sheet, or at this Sheet produced, 
and inclined to it in the up or down directions 
as well as sideways, is found in the point, 
where a line drawn through the eye parallel 
to the line whose vanishing point is wanted, 
meets the picture -sheet. The reasons given 
in paragraphs 3, 4, and 5, to prove that i, 
in the ground plan, and c, in the perspective 
view, mark the vanishing point of lines run- 
ning at right angles to the transparent plane, 
apply to lines running in the directions men- 
tioned in this remark. You will now be 
able to find the perspective of any line run- 
ning in any of the directions now mentioned, 
without the aid of the rule given in this paper, 
and you will also be able to make a variety 
of rules for finding the perspective of a point 
different from the rule that I have given. 
6. From what has been said in para- 
graphs 4 and 5, it may not be plain to 
every one, how that the points marked g 
in the ground plan, and shown by the 
points / d, and the corner of the cube 
under d in the elevation, should have their 
positions in the same line, h h p, perpen- 
dicular to the line a b in the perspective 
view ; or, in other words, it may not be 
evident, how in the case of every line in 
the objects to be represented, which has a 
perpendicular position, its perspective should 
stand perpendicular to the line a bin fig. 2. 
In order to understand this fully, let ab c, 
in the following figure, represent the end 
of a hollow cylinder, standing in a per- 
pendicular direction with a number of plane 
surfaces di,ei,f i, g i, and^c i, radiating from 
its centre i. 
Now, if this cylinder be cut parallel to its 
axis by any plane m n, the radiating 
planes will always be cut, so that their 
intersections with the cutting plane will be 
perpendicular ; this is so obvious, as to need no 
demonstration. But the lines whose posi- 
tions in the end-view of the cylinderare d e a 
f g, and which mark the places where the ra- 
diating planes meet the circumference of the 
cylinder, are perpendicular lines ; each of 
which may be considered a line in some ob- 
ject to be represented, and m n will represent 
the transparent plane. Let the eye have a ' 
position any where in the axis of the cy- 
linder — the rays of light reflected from the 1 ; 
whole line f, or from any part of it, to j 
the eye, will form a triangle in the plane j' 
fij and the intersection of m w with this |j 
triangle will be the perspective of the line, 
or part of the line, whose position is f ; but 
the intersection of the plane m n with the i 
plane fi^ is a perpendicular line ; so the part j 
of this intersection which forms the per- j 
sective of the line, or part of the line, whose | 
position is f, must be perpendicular. The ^ 
same reasoning applies if the lines in the | 
objects to be represented stand at any of j 
the other points, dea, or g, or even if the | 
line does not stand in a point in the circle j 
representing the circumference of the cylin- | 
der ; for in this case a new circle may be !! 
drawn, and every thing else can be shown as ij 
above. I may just mention it, for the jj 
thing can be demonstrated on the prin- j 
ciples now developed, that level lines in i 
the objects, running parallel to the picture- j 
sheet, are also level in the perspective view ; 
, and lines in the objects to be shown, that |! 
are inclined to the horizon at any angle, and j 
which keep parallel to the transparent plane, j 
run at the same angle to the line ab in the 
perspective view of these lines. The top and 
bottom lines of the front side of each cube, 
and the top and bottom lines of the front 
side of the six-sided prism C, also the out- 
side and inside lines that form the top angle j 
of the pyramid, and some other lines in the 
figures, illustrate this remark. The lines 1 
now noticed, though indefinitely produced, 
have no vanishing point. 
7. The eye should not be nearer to the 
picture -sheet than the greatest height or 
breadth of the picture ; and it should be 
placed in the ground plan, so that a line let 
fall from it perpendicular to the picture-sheet 
should bisect the angle xcb^ formed by lines 
drawn to it from the points which mark 
out the greatest width of the picture. The 
line din the ground plan does not bisect the 
angle a; cb ; but this was done to save room, 
and to show some parts of the objects that 
could not have been so well represented, if 
the position of the eye had been more nearly 
opposite to the centre of the picture. If 
the eye is very distant from the picture-sheet 
a perpendicular let fall from it to the pic- 
ture-sheet need not fall exactly on the 
centre of the picture. 
8. When the line drawn perpendicular to 
the line a b, in fig. 2, from the point in the 
ground plan whose perspective is wanted^ 
