ABSTRACT
This is a formal definition of a projective space of three dimensions. Whatever class of entities fulfils this definition is a projective space. I have enclosed in brackets a passage in which no new properties of projective space are introduced, which serves only the purpose of convenience of language. There is a whole class of projective spaces, and this class has an infinite number of members. The existence-theorem may be proved to begin with, by constructing a projective space out of complex numbers in the purely arithmetical sense defined in § 360. We then know that the class of projective spaces has at least four members, since we know of four sub-classes contained under it, each of which has at least one member. In the first place, we have the above arithmetical space. In the second place, we have the projective space of descriptive Geometry, in which the terms of the projective space are sheaves of lines in the descriptive space. In the third place, we have the polar form of elliptic space, which is distinguished by the addition of certain metrical properties of stretches, consistent with, but not implied by, the definition of projective space; in the fourth place, we have the antipodal form of elliptic Geometry, in which the terms of the projective space are pairs of terms of the said elliptic space. And any number of varieties of projective space may be obtained by adding properties not inconsistent with the definition-for example, by insisting that all planes are to be red or blue. In fact, every class of 2α0 terms (i.e. of the number of terms in a continuous series) is a projective
space; for when two classes are similar, if one is the field of a certain relation, the other will be the field of a like relation. Hence by correlation with a projective space, any class of 2α0 terms becomes itself a projective space. The fact is, that the standpoint of line-Geometry is more fundamental where definition is concerned: a projective space would be best defined as a class K of relations whose fields are straight lines satisfying the above conditions. This point is strictly analogous to the substitution of serial relations for series which we found desirable in Part IV. When a set of terms are to be regarded as the field of a class of relations, it is convenient to drop the terms and mention only the class of relations, since the latter involve the former, but not the former the latter.
