0.5

1 cm

108

ENGINEER AND MACHINIST’S DRAWING-BOOK.

we view an object under an angle of 90°, and we may

consider the sum of the rays which can enter the eye

under that angle as forming a cone, having the eye at

its summit. Now, according to our proposition, the only

objects which can be distinctly seen are those contained

within the base of the cone. A picture is in this con-

dition. Let A B a b (Fig. 224) be the projections of the

base of a square picture. Now,

in order that this may be in-

scribed within the base of a right-

angled cone, it is necessary that

the axis of the cone be equal to

the radius of the circle of its base.

Thus from d, the centre of the

picture, with c a or c b as radius,

we describe a circle which con-

tains the whole of the picture. We

carry the radius in the horizontal

projection from c to d, and the

line cc will of course be the

shortest distance which we should

take in order to see distinctly the

picture A B. If we make a right

angle at d, we shall have an

isoceless triangle, of which the base o o will be equal to the

diameter of the base of the cone. Now this distance c c,

which is sufficient for the picture A B or a b, is too little

for the picture a G, which has the same base as A B, but a

greater height, or for a H, which has the same height, but

a greater width; for neither of these are contained entirely

within the optic cone o d o, but require the base of the

cone to be increased to p p. The distance, therefore, ought

to vary with the height or width of the picture. So much

for the principle; but we have said that a lesser angle

than 90" is better to be adopted, and it is not easy to

give any rule for this. In general, however, the distance

should never be less than the diagonal of the picture,

which would give oro as the angle under which the

square picture A B or ab should be viewed, and p s p for

the others.

The distance, too, should depend in some measure on

the height of horizon assumed, and in general, the higher

the horizon the greater should be the distance.

We now have to speak on a point to the misconception

of which we attribute much that has been said against

the art of perspective.

The central plane, it will be remembered, was defined

to be “ a plane passing vertically through the eye of the

spectator, and cutting the ground plane, the horizontal

plane, and the plane of the picture at right angles/' Now

let us take the horizontal projections of an object A A

(Fig. 225), the picture B B, the eye of the spectator c, and

the central plane cDa. The picture is parallel to the longest

side of the object, and the central plane bisects the angle

formed by the visual rays which proceed from the extre-

mities of the object to the eye. This, then, we assume to

be the correct conditions of a picture; but suppose it were

required to introduce into the same picture other objects

{Fig. 224.J

B B, as seen from the same point of view, we should no

longer place the picture parallel to the long sides of the

object, with the face of the spectator directed towards a,

{Fig. 225.)

treme visual rays by the central plane c E, and make the

line of the picture perpendicular to c E, as at F F.

Perspective has been divided into parallel and oblique

perspective, and this division has introduced the miscon-

ception animadverted upon. Parallel perspective, so called,

can only exist when the central plane, bisecting the angle

formed by the visual rays, bisects also the object to be

represented. Nevertheless, in views of interiors, of streets

and the like, it is unscrupulously used, although the in-

tersection of the central plane with the picture is at one-

third of the width of the latter.

{Fig. 226.)

In the figures CAB (Fig. 226), three different repre-

sentations of the same interior are given to illustrate these

remarks. In Fig. C the central plane bisects the visual

angle; and the lines of the further side of the apartment,

being parallel to the picture, are also parallel in the repre-

sentation. In A the central plane does not bisect the

visual angle, and is at one-third of the width of the pic-

ture ; and this is the condition animadverted on. If it be

desired to have the point of sight not in the centre of the

apartment, like C, but at one side, then the representa-

tion is only correct when it is like B, in which the central

plane bisects the picture, and the lines of the further side

of the apartment being no longer parallel to the picture,

have their proper vanishing points.

To revert to the example which has led to this digres-

sion, we observe by the figure 223, that the distortion

arises from assuming a position for the picture in which

the central plane does not bisect the visual angle. But

let us place the picture in the position E F, in which it is

perpendicular to the central plane O C, bisecting the visual

angle, and we have the cylinders in what is called oblique

perspective (a term which should not exist), and there is

no longer any distortion.

ENGINEER AND MACHINIST’S DRAWING-BOOK.

we view an object under an angle of 90°, and we may

consider the sum of the rays which can enter the eye

under that angle as forming a cone, having the eye at

its summit. Now, according to our proposition, the only

objects which can be distinctly seen are those contained

within the base of the cone. A picture is in this con-

dition. Let A B a b (Fig. 224) be the projections of the

base of a square picture. Now,

in order that this may be in-

scribed within the base of a right-

angled cone, it is necessary that

the axis of the cone be equal to

the radius of the circle of its base.

Thus from d, the centre of the

picture, with c a or c b as radius,

we describe a circle which con-

tains the whole of the picture. We

carry the radius in the horizontal

projection from c to d, and the

line cc will of course be the

shortest distance which we should

take in order to see distinctly the

picture A B. If we make a right

angle at d, we shall have an

isoceless triangle, of which the base o o will be equal to the

diameter of the base of the cone. Now this distance c c,

which is sufficient for the picture A B or a b, is too little

for the picture a G, which has the same base as A B, but a

greater height, or for a H, which has the same height, but

a greater width; for neither of these are contained entirely

within the optic cone o d o, but require the base of the

cone to be increased to p p. The distance, therefore, ought

to vary with the height or width of the picture. So much

for the principle; but we have said that a lesser angle

than 90" is better to be adopted, and it is not easy to

give any rule for this. In general, however, the distance

should never be less than the diagonal of the picture,

which would give oro as the angle under which the

square picture A B or ab should be viewed, and p s p for

the others.

The distance, too, should depend in some measure on

the height of horizon assumed, and in general, the higher

the horizon the greater should be the distance.

We now have to speak on a point to the misconception

of which we attribute much that has been said against

the art of perspective.

The central plane, it will be remembered, was defined

to be “ a plane passing vertically through the eye of the

spectator, and cutting the ground plane, the horizontal

plane, and the plane of the picture at right angles/' Now

let us take the horizontal projections of an object A A

(Fig. 225), the picture B B, the eye of the spectator c, and

the central plane cDa. The picture is parallel to the longest

side of the object, and the central plane bisects the angle

formed by the visual rays which proceed from the extre-

mities of the object to the eye. This, then, we assume to

be the correct conditions of a picture; but suppose it were

required to introduce into the same picture other objects

{Fig. 224.J

B B, as seen from the same point of view, we should no

longer place the picture parallel to the long sides of the

object, with the face of the spectator directed towards a,

{Fig. 225.)

treme visual rays by the central plane c E, and make the

line of the picture perpendicular to c E, as at F F.

Perspective has been divided into parallel and oblique

perspective, and this division has introduced the miscon-

ception animadverted upon. Parallel perspective, so called,

can only exist when the central plane, bisecting the angle

formed by the visual rays, bisects also the object to be

represented. Nevertheless, in views of interiors, of streets

and the like, it is unscrupulously used, although the in-

tersection of the central plane with the picture is at one-

third of the width of the latter.

{Fig. 226.)

In the figures CAB (Fig. 226), three different repre-

sentations of the same interior are given to illustrate these

remarks. In Fig. C the central plane bisects the visual

angle; and the lines of the further side of the apartment,

being parallel to the picture, are also parallel in the repre-

sentation. In A the central plane does not bisect the

visual angle, and is at one-third of the width of the pic-

ture ; and this is the condition animadverted on. If it be

desired to have the point of sight not in the centre of the

apartment, like C, but at one side, then the representa-

tion is only correct when it is like B, in which the central

plane bisects the picture, and the lines of the further side

of the apartment being no longer parallel to the picture,

have their proper vanishing points.

To revert to the example which has led to this digres-

sion, we observe by the figure 223, that the distortion

arises from assuming a position for the picture in which

the central plane does not bisect the visual angle. But

let us place the picture in the position E F, in which it is

perpendicular to the central plane O C, bisecting the visual

angle, and we have the cylinders in what is called oblique

perspective (a term which should not exist), and there is

no longer any distortion.