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

21-23]

DEFINITIONS OF IDEAL ELEMENTS

23

intersect, the projective line is called 'proper'; and the line of intersection is the 'axis.' If the planes do not intersect, the projective
line is called 'improper.'
Since Projective Geometry has been developed* from the two
fundamental ideas of 'point' and 'straight line,' the other definitions
of projective elements must simply be those which have been given
in considering Projective Geometry. Thust a projective plane is the
class of those projective (ideal) points, which lie on any projective
line joining any given projective point A to any projective point on
any given projective line not possessing the given projective point A.
Definition. If a projective plane possesses any proper projective
points, it will be called a 'proper projective plane.' Otherwise it is an
'improper projective plane.'
The vertices of all the proper projective points on a proper projective plane will be seen to form a plane (cf. ~ 26 (a)).
Definition. A proper projective point and its vertex are said to be
'associated,' so likewise are a proper projective line and its axis, and
also a proper projective plane and the plane constituted by the vertices
of its proper projective points.
22. Since any two lines belonging to a projective point are
coplanar, it easily follows that any two lines of the projective point
can be used in place of the two special lines (a and b) used in the
definition (cf. ~ 21). Hence it can easily be proved that any plane,
containing one line of a projective point, contains an infinite number
of such lines. In other words, if a projective point is coherent with
a plane, an infinite number of the lines of the projective point lie
in the plane. In fact it follows that, through each point of the plane,
one line passes which belongs to the projective point.
23. If three projective points are incident in the same projective
line, then with any plane, with which two of the projective points
cohere, the third projective point also coheres.
First, if the three projective points are proper, the theorem is
immediately evident.
Secondly, let two of the projective points, M and N, say, be
proper, and let the third projective point, L, say, be improper. Let

* Cf. Proj. Geom.

t Cf. Proj. Geom. ~ 4.