< Page:Popular Science Monthly Volume 67.djvu
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NON-EUCLIDEAN GEOMETRY.

THE VALUE OF NON-EUCLIDEAN GEOMETRY.
By Professor GEORGE BRUCE HALSTED,

KENYON COLLEGE, GAMBIER, OHIO.

Among conditions to a more profound understanding of even very elementary parts of the Euclidean geometry, the knowledge of the non-Euclidean geometry can not be dispensed with.—E. Study.

ELEMENTARY 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 preeminently at the head of all human productions. In no science, in no department 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 mathematical 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 century-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 Ptolemæos (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 Poincaré, is truly unimaginable!

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