[3] Where the branch of experience in question is essential to all experience, the resulting apriority may be regarded as absolute; where it is necessary only to some special science, as relative to that science.
[4] I have given no account of these empirical proofs, as they seem to be constituted by the whole body of physical science. Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.
CHAPTER I.
A SHORT HISTORY OF METAGEOMETRY.
10. When a long established system is attacked, it usually happens that the attack begins only at a single point, where the weakness of the established doctrine is peculiarly evident. But criticism, when once invited, is apt to extend much further than the most daring, at first, would have wished.
"First cut the liquefaction, what comes last,
But Fichte's clever cut at God himself?"
So it has been with Geometry. The liquefaction of Euclidean orthodoxy is the axiom of parallels, and it was by the refusal to admit this axiom without proof that Metageometry began. The first effort in this direction, that of Legendre[5], was inspired by the hope of deducing this axiom from the others—a hope which, as we now know, was doomed to inevitable failure. Parallels are defined by Legendre as lines in the same plane, such that, if a third line cut them, it makes the sum of the interior and opposite angles equal to two right angles. He proves without difficulty that such lines would not meet, but is unable to prove that non-parallel lines in a plane must meet. Similarly he can prove that the sum of the angles of a triangle cannot exceed two right angles, and that if any one triangle has a sum equal to two right angles, all triangles have the same sum; but he is unable to prove the existence of this one triangle.
11. Thus Legendre's attempt broke down; but mere failure could prove nothing. A bolder method, suggested by Gauss, was carried out by Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically deducible from the others, we shall, by denying it and maintaining the rest, be led to contradictions. These three mathematicians, accordingly, attacked the problem indirectly: they denied the axiom of parallels, and yet obtained a logically consistent Geometry. They inferred that the axiom was logically independent of the others, and essential to the Euclidean system. Their works, being all inspired by this motive, may be distinguished as forming the first period in the development of Metageometry.
The second period, inaugurated by Riemann, had a much deeper import: it was largely philosophical in its aims and constructive in its methods. It aimed at no less than a logical analysis of all the essential axioms of Geometry, and regarded space as a particular case of the more general conception of a manifold. Taking its stand on the methods of analytical metrical Geometry, it established two non-Euclidean systems, the first that of Lobatchewsky, the second—in which the axiom of the straight line, in Euclid's form, was also denied—a new variety, by analogy called spherical. The leading conception in this period is the measure of curvature, a term invented by Gauss, but applied by him only to surfaces. Gauss had shown that free mobility on surfaces was only possible when the measure of curvature was constant; Riemann and Helmholtz extended this proposition to n dimensions, and made it the fundamental property of space.