December 23, 1904.] 



SCIENCE. 



863 



the invariable relations of natural things.. 

 Considered under this point of view, it is 

 as extended as nature herself ; it defines all 

 sensible relations, measures times, spaces-, 

 forces, temperatures. This difficult science 

 forms slowly, but it retains all the prin- 

 ciples once acquired. It grows and 

 strengthens without cease in the midst of 

 so many errors of the human mind." 



The elegy is magnificent, but permeating 

 it we see the tendency which makes all 

 analysis uniquely an auxiliary, hoMrever in- 

 comparable, of the natural sciences, a tend- 

 ency, in conformity, as we have seen, with 

 the development of science during the pre- 

 ceding two centuries; but we reach just 

 here an epoch where new tendencies ap- 

 pear. 



Poisson having in a report on the Fun- 

 damenta recalled the reproach made by 

 Fourier to Abel and Jacobi of not hav- 

 ing occupied themselves preferably with 

 the movement of heat, Jacobi wrote to 

 Legendre : ' It is true that Monsieur Fourier 

 held the view that the principal aim of 

 mathematics was public utility, and the 

 explanation of natural phenomena; but a 

 philosopher such as he should have known 

 that the unique aim of science is the honor 

 of the human spirit, and . that from this 

 point of view, a question about numbers is 

 as important as a question about the sys- 

 tem of the world.' This was without 

 doubt also the opinion of the grand geom- 

 eter of Goettingen, who called mathematics 

 the queen of the sciences, and arithmetic 

 the queen of mathematics. 



It would be ridiculous to oppose one to 

 the other these two tendencies ;■ the har- 

 mony of' our science is in their synthesis.' 



The time was about to arrive .when one 

 would feel the need of inspecting the 

 foundations of the edifice, and of making 

 the inventory of accumulated wealth, using 

 more of the critical spirit. Mathematical 

 thought was about to gather more force by 



retiring into itself; the problems were ex- 

 hausted for a time, and it is not well for 

 all seekers to stay on the same road. More- 

 over, difficulties and paradoxes remaining 

 unexplained made necessary the progress 

 of pure theory. 



The path on which this should move was 

 traced in its large outlines, and there it 

 could move with independence without 

 necessarily losing contact with the problems 

 set by geometry, mechanics and physics. 



At the same time more interest was to 

 attach to the philosophic and artistic side 

 of mathematics, confiding in a sort of pre- 

 established harmony between our logical 

 and esthetic satisfactions and the necessi- 

 ties of future applications. 



Let us recall rapidly certain points in 

 the history of the revision of principles 

 where Gauss, Cauchy and Abel likewise 

 were laborers of the first hour. Precise 

 definitions of continuous functions, and 

 their most immediate properties, simple 

 rules on the convergence of series, were 

 formulated; and soon was established 

 under very general conditions, the possi- 

 bility of trigonometric developments, legiti- 

 matizing thus the boldness of Fourier. 



Certain geometric intuitions relative to 

 areas and to arcs give place to rigorous 

 demonstration. The geometers of the 

 eighteenth century had necessarily sought 

 to give accoiint of the degree of the gen- 

 erality of the solution of ordinary differ- 

 ential equations. Their likeness to equa- 

 tions of finite differences led easily to the 

 result ; but the demonstration so conducted 

 must not be pressed very close. 



Lagrange, in his lessons on the calculus 

 of functions had introduced greater pre- 

 cision, and .starting from Taylor's series, 

 he saw that the equation of order m leaves 

 indeterminate the function, and its m — 1 

 first derivatives for the initial , value of the 

 variable; we are not surprised that La- 



