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

72

ANGLES

[CH. VIII

congruence group is elliptic, or when the congruence group is hyperbolic, and the line of intersection of p, and p2 passes through the
region enclosed by the absolute, then t, and t2 are necessarily not real.
Then the angle between the planes is defined to be 1 log tl1pt2p2}.
Thus if the two planes are given by
14x + mny + n2z + p =O, i.x + me2y + nz +p = 0,
and 0 is the angle between them, we have
Cos = - l 112+ m + 1 - n:12 + cpIp:......
cos:- -11- ~ ---....c(1).
l + 12 + + Cmp + n + c1'22 + + ~f2 + cp2f2
As before, there are two angles 0 and r - 0; but it can be proved that
the whole angle round a line is 27, owing to the existence of diametrically opposite regions in the neighbourhood of the line.
In the parabolic case, when c is indefinitely diminished, the angle
between the planes is given by
l+ m +viini —+ n 12(2).
cos............ (2
(12 + 1m12 +?22)2 (12 + n2. +.n2).
73. The same procedure can also be applied for the measurement
of the angle between two concurrent lines. Let 1I and 12 be any two
concurrent lines in a plane p. Let t, and t2 be the real or imaginary
tangents from the point (li. 1,) to the conic which is the section of the
absolute by the planep. When the congruence group is elliptic, or when
the congruence group is hyperbolic and the point (i. 1,) lies within the
region enclosed by the absolute, then t, and t2 are necessarily imaginary.
Then the angle between the lines is defined to be I1 log {tllt2l}. Thus
there are two angles 0 and 7 -0 between two intersecting lines, and
the whole angle round a point is 27r.
In the degenerate parabolic case the section of the absolute by the
plane p becomes two conjugate imaginary points in the plane at
infinity. These are known as the circular points at infinity. Then
t, and t2 are the imaginary lines from the point (1. 4) to these points
respectively*.
* This projective view of Euclidean Metrical Geometry was elaborated by
Laguerre in 1853, previously to the rise of the general theory which is explained
here.