Page:Philosophical Review Volume 1.djvu/666

650 genuine postulates, which combine a notion of shape with one of quantity. These postulates are: 1st, that the straight line is the distance between its termini; 2d, that, of lines having the same termini, those which envelop the others are the longer; 3d, that the perpendicular divides the space on either side of it into angles of equal content, and that a line making a complete revolution about a point describes four such angles; and 4th, that a point following the contour of a closed polygon and turning around each of its angles has, when it returns to its original direction, rotated through a sum of angles equal to four right angles. From this last postulate (which M. R. apparently adopts from M. A. J. H. Vincent) it follows almost immediately that the angles internal of a triangle are equal to two right angles, and from this theorem Euclid's postulate about parallels can be immediately deduced. The postulate of the parallels is thus neither more nor less demonstrable — or indemonstrable — than other propositions, and geometers have been wrong in treating it as an anomaly or scandal in their science.

Lobatschewsky's ignoring of it, and his assumption of triangles with their angles equal to less than two right angles, are constantly defended by saying that, since the consequences deduced lead to no conflict with the other postulates and axioms of Euclid, there may well be "something in" this imaginary geometry. M. Renouvier disposes of this talk very clearly. How could Lobatschewsky's assumption, however absurd in itself, when combined with Euclid's residual principles, lead to consequences which contradict the latter? This could only happen if the assumption itself contradicted the latter. But it only contradicts the postulate of parallels; and that, by universal consent, is a principle independent of all others. The "consequences," in the imaginary geometry, must then be internally consistent with each other, with the true premises, and with the absurdly assumed one, from which they are deduced; but they may be absurd nevertheless, absurd a priori, as one of their premises is absurd, and absurd a posteriori or in relation to the facts. Nothing in fact is more absurd than to treat this geometry as peculiarly empirical. Lack of space forbids an analysis of the rest of the paper, in which M. Renouvier discusses in a very luminous way the theory of measure and incommensurability, and the method of limits, and criticises the soi-disant empirical character of Riemann's and Helmholtz's n-dimensional speculations. The article may be cordially recommended to all who are interested in the philosophy of mathematics.

The second article, by M. Pillon, is on the Historical Evolution of Atomism, first as a cosmological, and second as a metaphysical, hypothesis. After Lasswitz'a great history, so recently completed, any