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

CHAPTER VIII.

METRICAL GEOMETRY.
68. THE theory of distance follows immediately from that of congruence by noting two facts. In the first place let the anharmonic
ratio* of the range (PQRS) be denoted by {PQRS}; then if
A, A2, PI, P2, P3
are collinear points, we have
{A1PiA2P2} x {AP2A.P3} = {A1PA2P3},
or, in another form,
log {A1PIA2P2} + log {AP2A2P} = log {APAP}......(1).
In the second place, let Al and A2 be the two real or imaginary
points in which the line containing the points P1, P2, P3 meets the
real or imaginary absolute of some definite congruence group. Then
for any transformation of that group (a) the anharmonic ratios are
unaltered because the transformation is projective, and (/) the points
A, and A2 are transformed into the points in which the transformed
position of the line P1P2P3 cuts the absolute.
Thus if some multiplex of log {A1PA2P2} be defined as the
distance between the points Pi and P2, where A1 and A. are the
points where the line P1P2 cuts the absolute, then equation (1) secures
the characteristic addition property of distance in respect to collinear
points, and the second consideration secures the characteristic invariability of distances in a congruence transformation.
* Cf. Proj. Geom. ~ 38.
f This definition is due to Cayley, Sixth Memloir on Quantics, Phil. Trans.
1859 and Coll. Papers, vol. II., and to Klein, Ueber die sogenannte nicht-euklidische
Geometrie, Math. Ann. vol. iv. 1871.
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