The above statement of the three essential axioms of metrical Geometry is taken from Helmholtz as amended by Lie. Lie's own statement of the axioms, as quoted above, has been too much influenced by projective methods to give a historically correct rendering of the spirit of the second period; Helmholtz's statement, on the other hand, requires, as Lie has shewn, very considerable modifications. The above compromise may, therefore, I hope be taken as accepting Lie's corrections while retaining Helmholtz's spirit. =48.= But metrical Geometry, though it is historically prior, is logically subsequent to projective Geometry. For projective Geometry deals directly with that qualitative likeness, which the judgment of quantitative comparison requires as its basis. Now the above three axioms of metrical Geometry, as we shall see in Chapter III. Section B, do not presuppose measurement, but are, on the contrary, the conditions presupposed by measurement. Without these axioms, which are common to all three spaces, measurement would be impossible; with them, so I shall contend, measurement is able, though only empirically, to decide approximately which of the three spaces is valid of our actual world. But if these three axioms themselves express, not results, but conditions, of measurement, must they not be equivalent to the statement of that qualitative likeness on which quantitative comparison depends? And if so, must we not expect to find the same axioms, though perhaps under a different form, in projective Geometry? =49.= This expectation will not be disappointed. The above three axioms, as we shall see hereafter, are one and all philosophically equivalent to the homogeneity of space, and this in turn is equivalent to the axioms of projective Geometry. The axioms of projective Geometry, in fact, may be roughly stated thus: I. Space is continuous and infinitely divisible; the zero of extension, resulting from infinite division, is called a Point. All points are qualitatively similar, and distinguished by the mere fact that they lie outside one another. II. Any two points determine a unique figure, the straight line; two straight lines, like two points, are qualitatively similar, and distinguished by the mere fact that they are mutually external. III. Three points not in one straight line determine a unique figure, the plane, and four points not in one plane determine a figure of three dimensions. This process may, so far as can be seen _à priori_, be continued, without in any way interfering with the possibility of projective Geometry, to five or to _n_ points. But projective Geometry requires, as an axiom, that the process should stop with some positive integral number of points, after which, any fresh point is contained in the figure determined by those already given. If the process stops with (_n_ + 1) points, our space is said to have _n_ dimensions.