NON-EUCLIDEAN GEOMETRY. 639 



THE VALUE OF NON-EUCLIDEAN GEOMETEY. 



By Professor GEORGE BRUCE HALSTED, 



KEN YON COLLEGE, GAMBIER, OHIO. 



Among conditions to a more profound understanding of even very elemen- 

 tary parts of the Euclidean geometry, the knowledge of the non-Euclidean 

 geometry can not be dispensed with. — E. Study. 



TT^LEMENTAKY geometry has been the most stable part of all 

 •*-** science. This was due to one book, of which Philip Kelland 

 says: 



It is certain, that from its completeness, uniformity and faultlessness, from 

 its arrangement and progressive character, and from the universal adoption 

 of the completest and best line of argument, Euclid's Elements stand pre- 

 eminently at the head of all human productions. In no science, in no depart- 

 ment of knowledge, has anything appeared like this work: for upwards of 

 2,000 years it has commanded the admiration of mankind, and that period has 

 suggested little towards its improvement. 



In all lands and languages, in all the world, there was but one 

 geometry. For the abstractest philosophy, for the most utilitarian 

 technology, geometry is of fundamental importance. For education it 

 is the before and after, the oldest medium and the newest ; older, more 

 classic than the classics, as new as the automobile. The first of the 

 sciences, it is ever the newest requisite for their ongo. Says H. J. S. 

 Smith : 



I often find the conviction forced upon me that the increase of mathe- 

 matical knowledge is a necessary condition for the advancement of science, 

 and, if so, a no less necessary condition for the improvement of mankind. I 

 could not augur well for the enduring intellectual strength of any nation of 

 men, whose education was not based on a solid foundation of mathematical 

 learning, and whose scientific conceptions, or, in other words, whose notions 

 of the world and of the things in it, were not braced and girt together with 

 a strong framework of mathematical reasoning. 



Of what startling interest then must it be that at length this cen- 

 tury-plant has flowered, a new epoch has unfolded. How did this 

 happen ? Euclid deduced his geometry from just five axioms and five 

 postulates. These were all very, very short and simple, except the last 

 postulate, which was in such striking contrast to the others that not its 

 truth, but the necessity of assuming or postulating it, was doubted from 

 remotest antiquity. The great astronomer Ptolemaeos (Ptolemy) 

 wrote a treatise purporting to prove it, and hundreds after him spent 

 their brains in like attempts. What vast effort has been wasted in this 

 chimeric hope, says Poincare, is truly unimaginable ! 



