In the second chapter, we endeavoured, by a criticism of some geometrical philosophies, to prepare the ground for a constructive theory of Geometry. We saw that Kant, in applying the argument of the Transcendental Aesthetic to space, had gone too far, since its logical scope extended only to some form of externality in general. We saw that Riemann, Helmholtz and Erdmann, misled by the quantitative bias, overlooked the qualitative substratum required by all judgments of quantity, and thus mistook the direction in which the necessary axioms of Geometry are to be found. We rejected, also, Helmholtz's view that Geometry depends on Physics, because we found that Physics must assume a knowledge of Geometry before it can become possible. But we admitted, in Geometry, a reference to matter--not, however, to matter as empirically known in Physics, but to a more abstract matter, whose sole function is to appear in space, and supply the terms for spatial relations. We admitted, however, besides this, that all _actual_ measurement must be effected by means of _actual_ matter, and is only empirically possible, through the empirical knowledge of approximately rigid bodies. In criticizing Lotze, we saw that the most important sense, in which non-Euclidean spaces are possible, is a philosophical sense, namely, that they are not condemned by any _à priori_ argument as to the necessity of space for experience, and that consequently, if they are not affirmed, this must be on empirical grounds alone. Lotze's strictures on the mathematical procedure of Metageometry we found to be wholly due to ignorance of the subject. Proceeding, in the third chapter, to a constructive theory of Geometry, we saw that projective Geometry, which has no reference to quantity, is necessarily true of any form of externality. Its three axioms--homogeneity, dimensions, and the straight line--were all deduced from the conception of a form of externality, and, since some such form is necessary to experience, were all declared _à priori_. In metrical Geometry, on the contrary, we found an empirical element, arising out of the alternatives of Euclidean and non-Euclidean space. Three _à priori_ axioms, common to these spaces, and necessary conditions of the possibility of measurement, still remained; these were the axiom of Free Mobility, the axiom that space has a finite integral number of dimensions, and the axiom of distance. Except for the new idea of motion, these were found equivalent to the projective triad, and thus necessarily true of any form of externality. But the remaining axioms of Euclid--the axiom of three dimensions, the axiom that two straight lines can never enclose a space, and the axiom of parallels--were regarded as empirical laws, derived from the investigation and measurement of our actual space, and true only, as far as the last two are concerned, within the limits set by errors of observation.