Page:Philosophical Review Volume 3.djvu/249

L'ancienne et les nouvelles géométries.—I. L'espace réel est-il l'espace euclidien? J. Delbœuf. Rev. Phil., XVIII, II, pp. 449-484.

The article is intended to show that real space (A) has nothing in common with Euclidean space (B) but the name. Their differences are as follows: (1) B is homogeneous, i.e., admits of similar figures differing only in size. A is heterogeneous; the nature of figures in it depends on their absolute magnitudes, as well as on their proportions. In real space and time things differ according as they occupy different positions in them, and homogeneous space is abstracted from the heterogeneous real space. (2) B is everywhere identical with itself: A is not,—for every figure is changed as soon as it changes its place. (3) B is invariable; A is continually changing. (4) A exists, and is coherent; B does not exist, and is contradictory. If, e. g., there are two spheres, the one indefinitely large and the other indefinitely small, the latter may be contained in the former an indefinite number of times; and yet, in virtue of their similarity, a part of the smaller must correspond to each part of the larger. Hence also B does not admit of atoms, nor of sound and color theories based on wave-lengths, for the character of the sound and color depends on the absolute length of vibration. In short, if infinite divisibility were real, no science would be possible. (5) A is conceived of as unbounded; B always as finite, as finite and limited as one pleases. For its homogeneity involves its infinite divisibility, and serves every purpose of an infinity of extent. (6) A is continuous, while in B gaps and holes may be made. (7) B is penetrable, and admits of innumerable figures in the same place; A is impenetrable, once it is occupied. B, then, is an imaginary space, and exists only as universal conceptions do, in the human spirit. It is simple, because it is the result of a simplification; it is only as a theoretical abstraction, e. g., that a ray of light travels in a straight line,—its actual course depends on the medium through which it passes.—To clench his argument, and to show the impossibility of identifying the space we live in with the Euclidean space, in which relative proportions are all-important and absolute magnitudes do not matter, Professor Delboeuf, referring to his lecture on "Megamicros," supposes all the geometrical dimensions of our planet to have been reduced by one half, and points out, very ingeniously, how this would affect the physical characteristics of life. The conclusion of the whole is, that "homogeneity remains the exclusive and character-