THE PROPOSITIONS OF INCIDENCE. INTRODUCTION OF A PLANE, AND OF A SPACE

Preliminary remarks. The point of view to which we have sought to guide the reader by the preceding discussion is that which we finally adopt as basis of the theory; it appears to possess a simplicity which justifies its being taken first. But it is general, and therefore abstract; as, for instance, in its use of the word line in such a sense that every two lines of a plane intersect one another. And it may appear to be artificial; as, for instance, in its adoption of Pappus' theorem. Moreover, it gives no recognition to at least two notions which are, probably, inseparable from any conception of space founded directly on experience. One of these notions is, that space consists of a limited part which is accessible, surrounded by an unlimited part which is inaccessible. The other notion, intimately connected with the former, is, that, when a point is given upon a line, there is thereby effected a distinction, between the points of the line on one side of the given point, and those on the other; or, when a line is given in a given plane, there is thereby effected a distinction between the points of the plane, according as they lie on one side, or the other, of the given line; or, when a plane is given in space, a similar separation is thereby made.