4 i2 POPULAR SCIENCE MONTHLY. 



A STUDY OF THE DEVELOPMENT OF GEOMETRIC 



METHODS.* 



BY M. GASTON DARBOUX, 

 SECRETAIRE PERPETUEL DE L'ACADEMIE DES SCIENCES. 



r PO appreciate the progress geometry has made during the century 

 -*- just ended, it is of advantage to cast a rapid glance over the 

 state of mathematical science at the beginning of the nineteenth 

 century. 



We know that, in the last period of his life, Lagrange, fatigued by 

 the researches in analysis and mechanics, which assured him, however, an 

 immortal glory, neglected mathematics for chemistry, which, according 

 to him, was easy as algebra, for physics, for philosophic specula- 

 tions. 



This mood of Lagrange we almost always find at certain mo- 

 ments of the life of the greatest savants. The new ideas which came 

 to them in the fecund period of youth and which they introduced 

 into the common domain have given them all they could have ex- 

 pected ; they have fulfilled their task and feel the need of turning their 

 mental activity towards wholly new subjects. This need, as we 

 recognize, manifested itself with particular force at the epoch of 

 Lagrange. At this moment, in fact, the program of researches opened 

 to geometers by the discovery of the infinitesimal calculus appeafed 

 very nearly finished up. Some differential equations more or less 

 complicated to integrate, some chapters to add to the integral calculus, 

 and one seemed about to touch the very outmost bounds of science. 



Laplace had achieved the explanation of the system of the world and 

 laid the foundations of molecular physics. New ways opened before 

 the experimental sciences and prepared the astonishing development 

 they received in the course of the century just ended. Ampere, Poisson, 

 Fourier and Cauchy himself, the creator of the theory of imaginaries, 

 were occupied above all in studying the application of the analytic 

 methods to mechanics, and seemed to believe that outside this new 

 domain, which they hastened to cover, the outlines of theory and science 

 were finally fixed. 



Modern geometry, a glory we must claim for it, came, after the 

 end of the eighteenth century, to contribute in large measure to the 

 renewing of all mathematical science, by offering to research a way 



* Read September 24, 1904, at the Congress of Arts and Science at St. Louis. 

 Translated by Professor George Bruce Halsted. 



