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

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VARIOUS DEFINITIONS

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A sheaf of lines is a complete set of coplanar lines concurrent
at one point (the vertex). A sheaf of half-rays is a complete set of
coplanar half-rays emanating from one point (the vertex).
A bundle of lines is a complete set of lines concurrent at one point
(the vertex). A bundle of half-rays is a complete set of half-rays
emanating from one point (the vertex).
If p, q, r are three half-rays belonging to a sheaf of half-rays, then
r is said to 'lie between' p and q, if points A and B can be found on
p and q respectively, such that the segment AB intersects r.
It can be proved that if r lies between p and q, then p does not
lie between r and q.
The complete set of planes through a given line (the axis) is called
a sheaf of planes. The axis divides each plane into two half-planes.
These half-planes form a sheaf of half-planes.
If p, q, r are three half-planes belonging to a sheaf of half-planes,
then r is said to 'lie between' p and q, if points A and B can be found
on p and q respectively, such that the segment AB intersects r.
It can be proved that if r lies between p and q, then p does not lie
between r and q.
The theorems indicated in this and in the preceding sections, and
allied theorems, are not always very easy to prove. But their proofs
depend so largely upon the particular mode of formulation of the
axioms, that it would be outside the scope of this tract to enter into
a consideration of them. In the sequel we shall assume that the whole
class of theorems of the types, which have been thus generally indicated, can be proved from the axioms stated.
8. Formulations of the axioms of Descriptive Geometry have also
been given by Hilbert*, and by E. H. Mooret, and by B. Russell+,
and by O. Veblen~. Veblen's memoir represents the final outcome of
these successive labours, and his formulation will be given now. The
axioms are stated in terms of 'points' and of a relation among three
points called 'order.' Points and order are not defined.
I. There exist at least two distinct points.
* Grundlagen der Geometrie, Leipzig, 1899, English Translation by E. J.
Townsend, Chicago, 1902.
t On the Projective Axioms of Geometry, Trans. of the Amer. Math. Soc.,
vol. II., 1902.
+ The Principles of Mathematics, Cambridge, 1903, ch. XLVI.
~ A System of Axioms for Geometry, Trans. of the Amer. Math. Soc., vol. v.,
1904.