0.5

1 cm

PERSPECTIVE.

107

Perspective of Solids.

Problem X.—The horizontal projections of two tetra-

hedrons being given, to draw the perspective of the solid.

Let AB ab {Fig. 221) be the projections of the picture,

C the point of sight, D D the points of distance, EFG h,

I K Lm, the horizontal projections of the two given tetra-

hedrons, one of which is placed on its base, and the other

on its summit.

(Fig. 221.)

Draw the perspectives of the horizontal projections,

then through h, the horizontal projection of the summit,

raise the perpendicular h H.

Problem XI.—The projections of two equal cubes being

Distance

given (Fig. 222), one placed on one of its angles, and the

other on one of its arriess, to draw them in perspective.

Here the operation is so simple that no explanation is

required.

Problem XII.—To draw three equal given cylinders in

perspective (Fig. 223).

It is not necessary to repeat the method of putting a

circle in perspective; but it may be well to observe, that

the upper surface of the cylinder may be so near the hori-

zon that it is physically impossible to inscribe an ellipse.

In such a case an approximate solution is all that can be

arrived at. A few principal points should be obtained,

and the ellipse traced by approximation.

{Fig. 223.)

This example presents a singularity, which at first sight

appears a paradox, and yet is nothing less than that. The

cylinder A, although evidently farther from the eye than

B, and seen, consequently, under a less angle, appears in

the picture to have a greater diameter. The

Distance reason is, that its optical cone is cut more obliquely

by the picture than that of B, and hence the in-

tersection of A is longer. This, which is held to

be a proof of the incorrectness of perspective, is,

on the contrary, a proof of its correctness; for, if

the subject is attentively considered, and we pro-

ceed to view the picture under the same condi-

tions as to distance and height of the eye, as

we have supposed to exist in viewing the object

itself, we shall perceive that the representations

of the objects must be seen under the same

angles as the objects themselves, and therefore,

although the diameter of the farther column

measures more in the representation than that

of the nearer one, yet, from the proper point

of view, the angle under which it is seen is less,

and therefore it will appear to be smaller. The

result, although geometrically correct, is yet a

distorted representation, when viewed in the way

we usually look at a picture. It is a correct

section of the cone of rays, but not made by a

plane so situated relatively to the object and spectator,

as we should place a picture or a transparent plane

through which to view the object. Before proceeding

farther, we shall take the opportunity of making some

remarks on the conditions under which objects may be

properly represented.

The greatest angle under which objects can be viewed

with distinctness is one of 90°. But when viewed under

this angle, it is with such an effort as to produce an un-

comfortable sensation. Let us, however, suppose that

107

Perspective of Solids.

Problem X.—The horizontal projections of two tetra-

hedrons being given, to draw the perspective of the solid.

Let AB ab {Fig. 221) be the projections of the picture,

C the point of sight, D D the points of distance, EFG h,

I K Lm, the horizontal projections of the two given tetra-

hedrons, one of which is placed on its base, and the other

on its summit.

(Fig. 221.)

Draw the perspectives of the horizontal projections,

then through h, the horizontal projection of the summit,

raise the perpendicular h H.

Problem XI.—The projections of two equal cubes being

Distance

given (Fig. 222), one placed on one of its angles, and the

other on one of its arriess, to draw them in perspective.

Here the operation is so simple that no explanation is

required.

Problem XII.—To draw three equal given cylinders in

perspective (Fig. 223).

It is not necessary to repeat the method of putting a

circle in perspective; but it may be well to observe, that

the upper surface of the cylinder may be so near the hori-

zon that it is physically impossible to inscribe an ellipse.

In such a case an approximate solution is all that can be

arrived at. A few principal points should be obtained,

and the ellipse traced by approximation.

{Fig. 223.)

This example presents a singularity, which at first sight

appears a paradox, and yet is nothing less than that. The

cylinder A, although evidently farther from the eye than

B, and seen, consequently, under a less angle, appears in

the picture to have a greater diameter. The

Distance reason is, that its optical cone is cut more obliquely

by the picture than that of B, and hence the in-

tersection of A is longer. This, which is held to

be a proof of the incorrectness of perspective, is,

on the contrary, a proof of its correctness; for, if

the subject is attentively considered, and we pro-

ceed to view the picture under the same condi-

tions as to distance and height of the eye, as

we have supposed to exist in viewing the object

itself, we shall perceive that the representations

of the objects must be seen under the same

angles as the objects themselves, and therefore,

although the diameter of the farther column

measures more in the representation than that

of the nearer one, yet, from the proper point

of view, the angle under which it is seen is less,

and therefore it will appear to be smaller. The

result, although geometrically correct, is yet a

distorted representation, when viewed in the way

we usually look at a picture. It is a correct

section of the cone of rays, but not made by a

plane so situated relatively to the object and spectator,

as we should place a picture or a transparent plane

through which to view the object. Before proceeding

farther, we shall take the opportunity of making some

remarks on the conditions under which objects may be

properly represented.

The greatest angle under which objects can be viewed

with distinctness is one of 90°. But when viewed under

this angle, it is with such an effort as to produce an un-

comfortable sensation. Let us, however, suppose that