50 MATHEMATICS, 
shadow in the vertical view, as is clearly shown by drawing the rays d’d 
and 6°0'. 
C. Linear Perspective. 
We shall here confine ourselves to the mathematical principles of per- 
spective, as occasion will be had to speak of perspective in general and its 
application to the arts, in another part of the work. 
We have already remarked, in the introduction to projection, that in 
perspective the visual rays are all supposed to proceed from one point—the 
point of sight; while in projection they are supposed to be parallel to each 
other. We may here again employ the illustration of the plate of glass 
interposed between the spectator and the object, and upon which the per- 
spective image of the latter is represented. 
In fig. 57, pl. 4, let XY be the ground plan upon which a square, abcd, is 
supposed to be drawn, and whose perspective representation is to be 
obtained on the vertical glass plate RS. Let I be the station of the obser- 
ver, and A the point from which he sees the square abcd. This point is 
called the point of sight. The distance of the observer from the glass 
plate—the plane of projection or the plane of the picture—is determined 
by the line AA’; the point A’ is called the point of distance. If visual 
rays be drawn from the point of sight to all the corners of the square abed, 
these must necessarily intersect the glass plate or the plane of the picture. 
It is then necessary to find the points of intersection, so that by joining 
them by straight lines the perspective of the square may be obtained. For 
this purpose, in the horizontal plane draw perpendiculars through the points 
a, b, c, d, extending to the foot of the glass plate. In the position of the 
square here assumed, two such lines are all that is necessary, as two of the 
corners lie in one line. Drawing lines from f and g to A’, these must lie 
in the plane of the picture, and the same must be the case with their inter- 
sections with the visual rays. The points a’, b’, c', d’ will then be the per- 
spective representation of a, b,c, d, the corners of the square. By properly 
connecting the points a’, b’, c’, d’, by straight lines, the quadrilateral thus 
obtained will be the perspective of the square. 
The line DD’ supposed to be drawn at a height equal to that of the point 
of sight A, is called the horizon of the picture: it is the principal line in 
a picture, as by its height all the parts of the picture are regulated. All 
visual rays tend to the point of sight, and lines perpendicular to the plane 
of the picture will have their perspectives all tending to the point of sight. 
The point of sight is also called the vanishing point, because in it the lines 
appear to vanish. Other parallels not perpendicular to the plane of the 
picture meet in the horizon, but not in the point of sight. There may, 
therefore, be several vanishing points in the same picture, but only one - 
point of sight, since an object for one and the same picture can only 
be observed from a single point. 
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