Page:Popular Science Monthly Volume 66.djvu/420

416 stops the march of ideas; the same discoveries, or discoveries almost equivalent, appear at nearly the same instant, and in places the most diverse. Without undertaking a discussion of this sort, which, besides, might appear useless or become irritating, it is, however, of importance to bring out a fundamental difference between the tendencies of the great geometers who, about 1830, gave to geometry a scope before unknown.

Some, like Chasles and Steiner, who consecrated their entire life to research in pure geometry, opposed what they called synthesis to analysis and, adopting in the ensemble if not in detail the tendencies of Poncelet, proposed to constitute an independent doctrine, rival of Descartes' analysis.

Poncelet could not content himself with the insufficient resources furnished by the method of projections; to attain imaginaries he created that famous principle of continuity which gave birth to such long discussions between him and Cauchy.

Suitably enunciated, this principle is excellent and can render great service. Poncelet was wrong in refusing to present it as a simple consequence of analysis; and Cauchy, on the other hand, was not willing to recognize that his own objections, applicable without doubt to certain transcendent figures, were without force in the applications made by the author of the 'Traité des propriétés projectives.'

Whatever be the opinion of such a discussion, it showed at least in the clearest manner that the* geometric system of Poncelet rested on an analytic foundation, and besides we know, by the untoward publication of the manuscripts of Saratoff, that by the aid of Descartes' analysis were established the principles which serve as foundation for the 'Traité des propriétés projectives.'

Younger than Poncelet, who besides abandoned geometry for mechanics where his works had a preponderant influence, Chasles, for whom was created in 1847 a chair of Géométric supérieure in the Faculty of Science of Paris, endeavored to constitute a geometric doctrine entirely independent and autonomous. He has expounded it in two works of high importance, the 'Traité de géométrie supérieure,' which dates from 1852, and the 'Traité des sections coniques,' unhappily unfinished and of which the first part alone appeared in 1865.

In the preface of the first of these works he indicates very clearly the three fundamental points which permit the new doctrine to share the advantages of analysis and which to him appear to mark an advance in the cultivation of the science. These are: (1) The introduction of the principle of signs, which simplifies at once the enunciations and the demonstrations, and gives to Carnot's analysis of transversals all the scope of which it is susceptible; (2) the introduction of