Page:Popular Science Monthly Volume 67.djvu/648

642 and first twenty-eight propositions. Hundreds of geometers tried at this. All failed. That eminent man, Legendre, was continually trying at this, and continually failing at it, throughout his very long life.

Naturally, some very respectable mathematicians were deceived.

The acute logician, De Morgan, accepted and reproduced a wholly fallacious proof of Euclid's parallel postulate, recently republished as sound by the Open Court Publishing Company, Chicago, 1898. A like pseudo-proof published in Crelle's Journal (1834) trapped even our well-known Professor W. W. Johnson, head mathematician of the U. S. Naval Academy, who translated and published it in the Analyst (Vol. III., 1876, p. 103), saying:

This demonstration seems to have been generally overlooked by writers of geometrical text-books, though apparently exactly what was needed to put the theory upon a perfectly sound basis.

But a more recent, a veritably shocking, example is at hand. On April 29, 1901, a Mr. Israel Euclid Rabinovitch submitted to the Board of University Studies of the Johns Hopkins University, in conformity with the requirements for the degree of doctor of philosophy, a dissertation in which, after an introduction full of the most palpable blunders, he proceeds to persuade himself that he proves Euclid's parallel postulate by using the worn-out device of attacking it from space of three dimensions, a device already squeezed dry and discarded by the very creator of non-Euclidean geometry, John Bolyai. And his dissertation was accepted by the referees. And since then Dr. (J. H. U.) Israel Euclid Rabinovitch has written, March 25, 1904:

As to Poincaré's assertion about the impossibility to [sic] prove the Euclidean postulate, it is no more than a belief—though an enthusiastic one [sic]—never proved mathematically, and in its very nature incapable of mathematical proof.

Poincaré is undoubtedly a great mathematician, perhaps the greatest now living; but his assertion of his inmost conviction, no matter how strongly put, can not pass for mathematical truth, unless mathematically proved.

His conclusion—shared also by many another noted mathematician as well as by the founders of the non-Euclidean geometries—can only be based on the fact of the existence of these last geometries, self-consistent and perfectly logical. But this is a poor proof of the impossibility to [sic] establish the Euclidean postulate.

If space is regarded as a point-manifold, it is Euclidean, and the postulate can be proved as soon as we are allowed to look for its establishment in three-dimensional geometry.

The two-dimensional elliptic geometry described by Klein, Lindemann and Killing, according to my opinion, is an absurdity for a point-space in the ordinary sense of the term.

Poincaré says that all depends upon convention. But still he deduces from this the perfectly gratuitous conclusion that therefore the parallel-postulate can not be proved.

Alongside this modern instance, too pathetic for comment, we may, however, be allowed to quote what one of the two greatest living mathematicians, Poincare, says in reviewing the work of the other, Hilbert's transcendently beautiful 'Grundlagen der Geometrie,' itself an outcome of non-Euclidean geometry: