=102.= Projective Geometry proper, as we saw in Chapter I., does not employ the conception of magnitude, and does not, therefore, require those axioms which, in the systems of the second or metrical period, were required solely to render possible the application of magnitude to space. But we saw, also, that Cayley's reduction of metrical to projective properties was purely technical and philosophically irrelevant. Now it is in metrical properties alone--apart from the exception to the axiom of the straight line, which itself, however, presupposes metrical properties[116]--that non-Euclidean and Euclidean spaces differ. The properties dealt with by projective Geometry, therefore, in so far as these are obtained without the use of imaginaries, are properties common to all spaces. Finally, the differences which appear between the Geometries of different spaces of the same curvature--_e.g._ between the Geometries of the plane and the cylinder--are differences in projective properties[117]. Thus the necessity which arises, in metrical Geometry, for further qualifications besides those of constant curvature, disappears when our general space is defined by purely projective properties. =103.= We have good ground for expecting, therefore, that the axioms of projective Geometry will be the simplest and most complete expression of the indispensable requisites of any geometrical reasoning: and this expectation, I hope, will not be disappointed. Projective Geometry, in so far as it deals only with the properties common to all spaces, will be found, if I am not mistaken, to be wholly _à priori_, to take nothing from experience, and to have, like Arithmetic, a creature of the pure intellect for its object. If this be so, it is that branch of pure mathematics which Grassmann, in his _Ausdehnungslehre_ of 1844, felt to be possible, and endeavoured, in a brilliant failure, to construct without any appeal to the space of intuition.