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

28, 29]

ORDER OF IDEAL POINTS

31

Hence it is easily seen that the segment AB is divided into three
parts by reference to the harmonic conjugates of points in it with
respect to A and B. The part formed by the segment AK1 (cf. fig. 2),
Cs C1
B K2 K2 A
Fig. 2.
exclusive of A and K1, contains the points whose harmonic conjugates
lie on the side of A remote from B; the segment BK2, exclusive of
B and K2, contains the points whose harmonic conjugates lie on the
side of B remote from A; the segment K1K2, inclusive of K1 and K2,
contains the points for which the harmonic conjugates do not exist.
It is not necessary that the points K1 and K2 be distinct. If they
coincide, the segment K1K2, inclusive of K1 and K2, shrinks into a
single point K. Thus in Euclidean Geometry the middle point of any
segment AB is this degenerate portion of the segment.
It immediately follows that Fano's axiom* is satisfied for proper
projective lines. Hence, remembering that the harmonic relation is
projectivet, we have:
(XIV.) If A and B are distinct projective points, and C is a
projective point of the projective line AB, distinct from A and B,
then the harmonic conjugate of C, with respect to A and B, is
distinct from C.
Also the restriction to three dimensions follows at once from
Peano's Axiom XVI of ~ 6, giving the same restriction for Descriptive
Geometry. Hence we find:
(XV.) If a be any projective plane, and A be any projective point
not lying in a, any projective point P lies on some line joining A to
some projective point on a.
29. The order of the projective points on a projective line must
now be considered.
If the projective line is proper, the order of the proper projective
points on it will be defined to correspond to the order of the associated
points. Thus (cf. fig. 2 of ~ 28) if the points marked in the figure are
projective points, as C moves from A to K1, excluding K1, the projective point D, which is the harmonic conjugate to C with respect to A
and B, moves from C through all the proper projective points on the

* Cf. Proj. Geom. ~ 8.

t Cf. Proj. Geom. ~ 9 (s).