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

26, 27]

AXIOMS OF THE PROJECTIVE PLANE

29

projected into collinear groups. Let A, B,... M, N be projected
into A1, B1,... MI, N1.
Thus, in the plane 7r, two homological triangles A1BC1 and
A,'B/'C' are obtained, AiA1', BIB,', CICI' being concurrent in 0,;
also B1A1 and Bl'Al', BICi and B1'C,', AAC1 and Al'C/' are concurrent
respectively in L1, Ml, Ni. Hence, by (8) above, L1, M1, N1 belong
to the same projective line.
27. The next group of propositions correspond to the three axioms
concerning the projective plane.
(XII.) If A, B, C are three projective points, which do not
belong to the same projective line, and A' belongs to the projective
line BC, and B' to the projective line CA, then the projective lines
AA' and BB' possess a projective point in common.
If the projective plane ABC is proper, the theorem follows from
~ 26 (y). If the projective plane ABC is improper, consider any plane
with which all the projective points of the projective line BB' cohere.
A
B A' C
Such planes exist. Thus the associated projective plane of such a
plane is a proper projective plane containing the line BB'. But by
~ 26 (/,) the projective line AA' intersects this proper projective plane,
in the projective point D, say. Also by ~ 26 (E) the projections of
B, A', C from A on to this proper projective plane belong to the
same projective line. Hence D belongs to BB'. Thus A A' and BB'
intersect.
(XIII.) If A, B, C are three projective points, not belonging to
the same projective line, then there exists a projective point not
belonging to the projective plane ABC.
This follows immediately from Peano's Axiom XV given in ~ 6
above.