180 POPULAR SCIENCE MONTHLY 



hend their inner harmony;, which alone is beautiful and consequently 

 worthy of our efforts. 



The first examine I shall cite is so old we are tempted to forget 

 it; it is nevertheless the most important of all. 



The sole natural object of mathematical thought is the whole 

 number. It is the external world which has imposed the continuum 

 upon us, which we doubtless have invented, but which it has forced us 

 to invent. Without it there would be no infinitesimal analysis; all 

 mathematical science would reduce itself to arithmetic or to the theory 

 of substitutions. 



On the contrary, we have devoted to the study of the continuum 

 almost all our time and all our strength. Who will regret it; who will 

 think that this time and this strength have been wasted? Analysis 

 unfolds before us infinite perspectives that arithmetic never suspects; 

 it shows us at a glance a majestic assemblage whose array is simple 

 and symmetric ; on the contrary, in the theory of numbers, where reigns 

 the unforeseen, the view is, so to speak, arrested at every step. 



Doubtless it will be said that outside of the whole number there is 

 no rigor, and consequently no mathematical truth; that the whole 

 number hides everywhere, and that we must strive to render trans- 

 parent the screens which cloak it, even if to do so we must resign our- 

 selves to interminable repetitions. Let us not be such purists and 

 let us be grateful to the continuum, which, if all springs from the 

 whole number, was alone capable of making so much proceed therefrom. 



Need I also recall that M. Hermite obtained a surprising advantage 

 from the introduction of continuous variables into the theory of num- 

 bers ? Thus the whole number's own domain is itself invaded, and this 

 invasion has established order where disorder reigned. 



See what we owe to the continuum and consequently to physical 

 nature. 



Fourier's series is a precious instrument of which analysis makes 

 continual use, it is by this means that it has been able to represent 

 discontinuous functions; Fourier invented it to solve a problem of 

 physics relative to the propagation of heat. If this problem had not 

 come up naturally, we should never have dared to give discontinuity 

 its rights; we should still long have regarded continuous functions as 

 the only true functions. 



The notion of function has been thereby considerably extended and 

 has received from some logician-analysts an unforeseen development. 

 These analysts have thus adventured into regions where reigns the 

 purest abstraction and have gone as far away as possible from the real 

 world. Yet it is a problem of physics which has furnished them the 

 occasion. 



After Fourier's series, other analogous series have entered the do- 



