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

12-16]

ADVANTAGES OF IDEAL POINTS

15

excluding a particular plane; and further, as before, we confine our
consideration to the points whose coordinates are algebraic numbers.
15. It has been proved in ~~ 12 and 13 that a convex region of a
Projective Space is a Descriptive Space. The converse problem has
now to be considered in this and in the next chapter; namely, given
a Descriptive Space, to construct a Projective Space of which the
Descriptive Space is part. This effects a very considerable simplification in the investigation of the properties of Descriptive Space owing
to the superior generality of the analogous properties of Projective
Space. Thus a Projective Space affords a complete interpretation of
all the entities indicated in coordinate geometry. It is in order to
gain this simplification that the 'plane at infinity' is introduced into
ordinary Euclidean Geometry. We have in effect to seek the logical
justification for this procedure by indicating the exact nature of the
entities which are vaguely defined as the 'points at infinity'; and the
procedure is extended by shewing that it is not necessarily connected
with the assumption of the Euclidean axiom. This investigation is
the Theory of Ideal Points*, or of the generation of 'Proper and
Improper Projective Points' in Descriptive Geometry. The Euclidean
axiom will not be assumed except when it is explicitly introduced.
The remainder of this chapter will be occupied with the general
theorems which are required for the investigation.
16. If A be any point and I be any line not containing A, then
the plane Al divides the bundle of half-rays emanating from A into
three sets, (1) the half-rays in the plane Al, (2) the half-rays on one
side of the plane, (3) the half-rays on the other side of the plane.
The sets (2) and (3) are formed of half-rays supplementary one to
the other.
Lemma. With the above notation, it is possible to find a plane
through the line I and intersecting any finite number of the half-rays
either of set (2) or of set (3).
For let al,... an be n half-rays of one of the two sets. Let B1 be
any point on a,, and B, be any point on a2. Then either the plane
* Originally suggested by Klein (extending an earlier suggestion of von Staudt),
Math. Annal. vols. iv. and vi., 1871 and 1872; first worked out in detail by Pasch,
loc. cit., ~~ 6-9. In the text I have followed very closely a simplification of the
argument given by R. Bonola, Sulla Introduzione degli Enti Improprii in Geometria
Projectiva, Giornale di Matematiche, vol. xxxvIII., 1900.