It might be objected that this only proves the absence of quantitative difference between different spaces of positive space-constant, or between those of negative space-constant: the quantitative difference persists, it might be said, between those of positive curvature in general and those of negative curvature in general, or between both together and Euclidean space. This I entirely deny. There is no qualitatively similar unit, in the three kinds of space, by which quantitative comparison could be effected. The straight lines of one space cannot be put into the other: the two straight lines, in one space, whose product is the reciprocal of the measure of curvature, have no corresponding curves in the other space, and the measures of curvature cannot, therefore, be quantitatively compared with each other. That the one may be regarded as positive, the other negative, I admit, but their values are indeterminate, and the units in the two cases are qualitatively different. A debt of £300 may be represented as the asset of -£300, and the height of the Eiffel Tower is +300 metres; but it does not follow that the two are quantitatively comparable. So with space-constants: the space-constant is itself the unit for magnitudes in its own space, and differs qualitatively from the space-constant of another kind of space. Again, to proceed to a more philosophical argument, two different spaces cannot co-exist in the same world: we may be unable to decide between the alternatives of the disjunction, but they remain, none the less, absolutely incompatible alternatives. Hence we cannot get that coexistence of two spaces which is essential to comparison. The fact seems to be that Erdmann, in his admiration for Riemann and Helmholtz, has fallen in with their mathematical bias, and assumed, as mathematicians naturally tend to assume, that quantity is everywhere and always applicable and adequate, and can deal with more than the mere comparison of things whose qualities are already known as similar[95]. =80.= This suggests the fourth and last of the above points, that the _qualities_ of space, even if space could be successfully regarded as a magnitude, would have to be entirely omitted in such a manner of regarding it, and that, therefore, none of its important or essential properties would emerge from such treatment. For to regard space as a magnitude involves, as we saw, a comparison with something qualitatively similar, and an abstraction from the similar qualities. To some extent and by the help of certain doubtful arguments, such a comparison is instituted by Riemann and Erdmann; but when they have instituted it, they forget all about the common qualities on which its possibility depends. But these are precisely the fundamental properties of space, and those from which, as I shall endeavour to prove in Chapter III., the axioms common to Euclid and Metageometry follow _à priori_. Such are the dangers of the quantitative bias.