GEOMETRIC CONSTRUCTION. (es 
the best way to obtain a complete mastery of the subject is to 
approach it from both points of view and then carefully com- 
pare the results. The broader outlook thus obtained more 
than compensates the reader for the extra time and labor ex- 
pended in learning two methods. 
87. Perspective Projection. Let two planes 2 and 7’ in- 
tersect ina line 1; from any point P, not in either plane, draw 
lines to all points of z. Each of the lines is cut by 2’ ina 
single point. Two points, A and A’ in a and x’ respectively, 
collinear with P are called corresponding points in the two 
planes. By means of the bundle of rays through P the points 
of x are projected into the points of a’. Every plane through 
P cuts x and x’ in a pair of corresponding lines that meet on I. 
It should be observed that to the line g, joining two points A 
and B in x, corresponds the line g’ joining A’ and B’, their cor- 
responding points in x’. Also to the point A, the intersec- 
tion of a pair of lines g and h in z, corresponds A’, the point 
of intersection of their corresponding lines g’ and h’ in z’. 
This method of constructing corresponding points and lines 
in a and z’ is called a perspective projection of xon za’. By 
means of this perspective projection whose vertex is at P, we 
establish a one-to-one correspondence of the points and lines 
of the two planes x and a’. This one-to-one correspondence 
is not without exceptions; but these exceptions may be re- 
moved by means of special assumptions. 
88. Line at Infinity. The plane through P parallel to 7’ 
cuts a in a line 7; the plane through P parallel to z cuts z’ in 
a line 7. If we assume with Euclid that through a given 
point P one and only one plane can be passed parallel to a 
given plane, and if we further assume that two parallel 
planes intersect in an infinitely distant line, then our one-to- 
one correspondence of points and lines is without exception. 
The line 7 in a corresponds to the line at infinity in 7’, and 
the line at infinity in a corresponds to the linez in za’. Thus 
by introducing the hypothesis that all infinitely distant points 
in a plane lie on a line at infinity the one-to-one correspond- 
