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

23-26]

AXIOMS OF PROJECTIVE GEOMETRY

27

line as any other pair of such planes. Hence two projective points
determine not more than one projective line.
(/f) Two projective points determine at least one projective line.
Fox if the points are proper, this is immediately evident. But in any
case let the projective points be A and B, and let 0 be any point.
There are at least two lines, a, and a2, which are members of A and
such that the plane a^a2 does not contain 0. Then the planes Oa1
and Oa2 intersect in a line which passes through 0 and is a member of
A. Hence through any point there passes a line which is a member of
a projective point. Hence through 0 there are lines belonging to A
and B respectively. But these lines determine a plane, with which A
and B both cohere. Similarly a second such plane can be determined.
Hence there is a projective line possessing both A and B.
25. The Axioms of Projective Geometry* can now be seen to be
true for the 'Projective Elements' as thus defined. Thus we have the
following theorems corresponding to those axioms of the previous tract,
of which the numbers are enumerated in the initial brackets.
(I, II, III.) There is a class of Projective Points, possessing at
least two members.
(IV, V, VI, VII, VIII.) If A and B are Projective Points, there
is a definite projective line AB, which (1) is a class of projective
points, and (2) is the same as the projective line BA, and (3) possesses
A and B, and (4) possesses at least one projective point distinct from
A and B.
Note that two improper projective points may possess no common
line.
(IX and X.) If A and B are projective points, and C is a projective point belonging to the projective line AB, and is not identical
with A, then (1) B belongs to the projective line AC, and (2) the
projective line A C is contained in the projective line AB.
(XI.) If A and B are distinct projective points, there exists
at least one projective point not belonging to the projective line AB.
26. Before considering the proof of the 'axioms' of the projective
plane i, some further propositions are required.
(a) Since a line exists through any given point and belonging
to any given projective point, it easily follows that the set of projective
* Cf. Proj. Geom. ~~ 4, 7, 8, 14.
t Cf. Proj. Geom., Axioms XII, XIII, XIV.