The axioms of descriptive geometry, by A.N. Whitehead.

26

A PROPERTY OF IDEAL LINES

[CH. III

form a triangle ABC, such that its sides AB, BC, CA belong to
L, M1, N respectively. Thus if N is a proper point, CA passes
through NA, the vertex of N: also since the lines BC and AB do not
intersect the line N1ML, the points A, B, C lie on the same side
of this line (cf. fig. 2): also since the lines BC and BA do not
intersect the line N2ML, either C lies on the same side of the line AB
as VN, or A lies on the same side of the line BC as N1. Assume that
C lies on the same side of AB as NV (cf. fig. 2). The rest of the proof
for figures 2 and 3 is now identical. In the plane 7r, let r be any line
Fig. 3.
belonging to L, on the side of AB remote from C. In the plane 7r,
take any point 0 on the side of r remote from C. Then the segments
OA and OB intersect r, say in A' and B'. Also the line A'N
intersects the segment OA, and does not intersect the segment A C;
hence it must intersect the segment OC, say in C'. Then, by projecting from any point O' in the plane wr' and by similar reasoning
to that in the second case, it is proved that the line B'C' belongs
to M. Then, as in the second case, by projecting from any point
0" in 7'", it follows that, if any two of the projective points L, M1, N
cohere with yr", so also does the third.
24. (a) It follows from ~ 23 that ally two planes, with which
both of two given projective points cohere, define the same projective