ABSTRACT
All Geometries, as commonly developed, agree in starting with points as indefinables. That is, there is a certain class-concept point (which need not be the same in different Geometries), of which we assume that there are at least two, or three, or four instances, according to circumstances. Further instances, i.e. further points, result from special assumptions in the various cases. Where the three great types of Geometry begin to diverge is as regards the straight line. Projective Geometry begins with the whole straight line, i.e. it asserts that any two points determine a certain class of points which is also determined by any two other members of the class. If this class be regarded as determined in virtue of a relation between the two points, then this relation is symmetrical. What I shall call Descriptive Geometry, on the contrary, begins with an asymmetrical relation, or a line with sense, which may be called a ray; or again it may begin by regarding two points as determining the stretch of points between them. Metrical Geometry, finally, takes the straight line in either of the above senses, and adds either a second relation between any two points, namely distance, which is a magnitude, or else the consideration of stretches as magnitudes. Thus in regard to the relations of two points, the three kinds of Geometry take different indefinables, and have corresponding differences of axioms. Any one of the three, by a suitable choice of axioms, will lead to any required Euclidean or non-Euclidean space; but the first, as we shall see, is not capable of yielding as many propositions as result from the second or the third. In the present chapter, I am going to assume that set of axioms which gives the simplest form of projective Geometry; and I shall call any collection of entities satisfying these axioms a projective space. We shall see in the next chapter how to obtain a set of entities forming a projective space from a set forming a Euclidean or hyperbolic space; projective space itself is, so far as it goes, indistinguishable from the polar form of elliptic space. It is defined, like all mathematical entities, solely by the formal nature of the relations between its constituents, not by what those constituents are in themselves. Thus we shall see in the following chapter that the “points” of a projective space may each be an infinite class of straight lines in a non-projective space. So long as the “points” have the requisite type of mutual relations, the definition is satisfied.
