Jakuaey 23, 1914] 



SCIENCE 



115 



another. To represent these pieces, Euler 

 claimed that an equal number of different 

 analytic expressions, or arbitrary functions, 

 were necessary. Hence, as d'Alembert's 

 solution involved only one arbitrary func- 

 tion, it could not be the general solution of 

 the problem. 



In these considerations of Euler there is 

 a sharp antithesis between geometry and 

 analysis. In Euler 's thought the independ- 

 ent pieces of the above curve formed 

 "curvm discontinues sen mixtce seu irregu- 

 lar es.'^ There was a blind belief that the 

 definition of a curve in any interval by a 

 mathematical expression carried with it a 

 definite continuation of the curve beyond 

 the interval, the violation of which was a 

 violation of analysis. Thus the question 

 was raised as to the relative power of mathe- 

 matically eonstructible expressions and of 

 geometric representation, and it was de- 

 cided that geometric form transcends ana- 

 lytic expression rather than the converse. 



The dual character of this controversy 

 was changed into a triple one by Daniel 

 Bernoulli, who first introduced Fourier's 

 series into physics and obtained the solu- 

 tion of the equation of the vibrating string 

 with fixed end points under the form of a 

 trigonometric series, 



" . njri nirat 

 2/ = 2 anSin— ^ cos — ;— , 



n=l *' *' 



where I denotes the length of the string. 

 The separate terms of this series give the 

 tones and overtones of the vibrating string. 

 Inasmuch as this solution is compounded of 

 an infinite number of tones and overtones 

 of all possible intensities, Daniel Bernoulli 

 claimed that he had obtained the general 

 solution of the problem. 



For t = the above equation gives as the 

 initial form of the string, 



r 



The question then at once arose whether 



d'Alembert's arbitrary function was capa- 

 ble of expansion into such a sine series. 

 To Euler this seemed unthinkable. It was, 

 so to speak, against the laws of the game, it 

 was contrary to the rules of analysis that 

 arbitrary, non-periodic functions could be 

 represented in terms of periodic functions. 

 Hence to Euler, Bernoulli 's solution of the 

 problem appeared even more limited than 

 that of d 'Alembert. 



I have not the time to follow further this 

 controversy, nor to show how d 'Alembert 

 and Lagrange united with Euler in declar- 

 ing Daniel Bernoulli wrong in his claim. 

 Yet not withstanding this overwhelming 

 preponderance of authority Daniel Ber- 

 noulli was right. The controversy gradu- 

 ally languished without any clear conclusion 

 till 1807, twenty-five years after Bernoulli 's 

 death, when Fourier presented to the 

 French Academy one of the first of his com- 

 munications which were summed up in 

 1822 in his "Analytic Theory of Heat." 

 In this communication he startled La- 

 grange with the absolutely revolutionary 

 doctrine that an arbitrarily given curve or 

 function, irrespective of its nature, could 

 be represented in any interval by a trigo- 

 nometric series. Fourier sought no strict 

 proof of his assertion, but the concrete ex- 

 amples which he gave vindicated its force. 

 The precise limitations necessary to make 

 the assertion exactly true remained, and to 

 some extent still remain, for his successors 

 to ascertain. 



Fourier's result not merely vindicated 

 Daniel Bernoulli's claim for his series, but 

 showed that his claim fell far short of the 

 reality. At a single blow it shattered hope- 

 lessly the notion of Euler and his contem- 

 poraries that a mathematical function could 

 be carried continuously beyond the interval 

 of definition in only a single way. But 

 Fourier's examples went further than this. 

 The arbitrary curve which he represented 



