DEVELOPMENT OF MATHEMATICAL ANALYSIS 505 



world." This was without doubt also the opinion of the grand geo- 

 meter 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 harmony of our science is in their synthesis. 



The time was about to arrive w r hen 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 exhausted for a time, and it is not well for all seekers 

 to stay on the same road. Moreover, 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 preestab- 

 lished harmony between our logical and esthetic satisfactions and the 

 necessities 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 possibility 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 account of the degree of the 

 generality of the solution of ordinary differential equations. Their 

 likeness to equations 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 intro- 

 duced greater precision, 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 Lagrange did not set himself the question 

 of convergence. 



In twenty or thirty years the exigencies in the rigor of proofs had 

 grown. One knew that the two preceding modes of demonstration 

 are susceptible of all the precision necessary. 



For the first, there was need of no new principle; for the second 

 it was necessary that the theory should develop in a new way. Up 

 to this point, the functions and the variables had remained real. 

 The consideration of complex variables comes to extend the field of 



