ABSTRACT
The so-called axioms of Geometry, for example, when Geometry is considered as a branch of pure mathematics, are merely the protasis in the hypotheticals which constitute the science. This chapter executes the purely formal task imposed by these considerations, and sets forth the strict definitions of various spaces, from which, without indefinables and without primitive propositions, the various Geometries will follow. A projective space of three dimensions is any class of entities such that there are at least two members of the class; between any two distinct members there is one and only one symmetrical aliorelative. A Euclidean space of three dimensions is a class of terms containing at least two members, and such that any two of them have one and only one asymmetrical one-one relation of a class. The chapter briefly explains the nature of two-dimensional space, which is very different from Euclidean space.
