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

28

THEOREMS ON IDEAL ELEMENTS

[CH. III

points cohering with a plane form a proper projective plane; and that
conversely, any proper projective plane is the set of projective points
cohering with some plane.
(f) Any projective line intersects any given proper projective
plane. For through the vertex of any proper projective point on the
projective plane, a plane passes with which every point of the
projective line coheres (cf. ~ 23). This plane intersects the plane
associated with the projective plane in a line. Two such planes can
be found. The two lines in the plane associated with the projective
plane define a projective point which lies both in the projective line
and the projective plane.
(y) Two projective lines in a proper projective plane necessarily
intersect.
For let m and n be the projective lines and a be the proper
projective plane, and a, its associated plane. Take any point O
outside al. Then two planes Om and On exist, with which respectively all projective points of m and n cohere. These planes
intersect in a line through O, 1, say. Let A be any point in al.
The plane Al intersects al in a line, I', say. The two lines I and 1'
define a projective point which lies in both the projective lines m
and n.
(3) Desargues' Theorem holds for triangles formed by projective
lines and projective points in a proper projective plane.
By (y) immediately above, no exception arises from non-intersection. Then by taking a point external to the associated plane, two
trihedrons can be formed for which the theorem holds. Hence the
theorem holds for the proper projective plane.
(e) The projections upon a proper projective plane of three
projective points belonging to the same projective line also belong to
a projective line.
The theorem is immediately evident, if the centre of projection,
or if any one of the three projective points, is proper. Assume that
all the projective points are improper. Let L,, N be the three projective points, and S the projective point which is the centre of
projection. Let 7r be the proper projective plane on to which L, M, N
are to be projected. Let a be any plane with which L, M, N all
cohere. On a construct figure 3 of ~ 23. Project (remembering (/i)
above) the whole figure of associated projective points from S on to the
plane 7r. Then by the first case of the present theorem, all collinear
groups of projective points which possess a proper projective point are