642 POPULAR SCIENCE MONTHLY. 



and first twenty-eight propositions. Hundreds of geometers tried at 

 this. All failed. That eminent man, Legendre, was continually try- 

 ing 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 Eabinovitch submitted to the 

 Board of University Studies of the Johns Hopkins University, in con- 

 formity 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 Eabinovitch has written, March 25, 1904: 



As to Poincare^ 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 mathe- 

 matical proof. 



Poincare" 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 postu- 

 late 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. 



Poincare' 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 mathe- 

 maticians, Poincare, says in reviewing the work of the other, Hilbert's 

 transcendently beautiful ' Grundlagen der Geometrie,' itself an out- 

 come of non-Euclidean geometry: 



