114 



SCIENCE 



[N. S. Vol. 2XXIX. No. 995 



nitely with n. Accordingly, the functions 

 defined by infinite trigonometric series are 

 obtained by compounding waves of varying 

 intensity and different wave-lengths and 

 may be almost infinitely complicated in 

 their behavior. This fact was fraught with 

 vital consequences for mathematical de- 

 velopment. 



A further distinction between the trigo- 

 nometric and power series appears in re- 

 spect to the values which their argument 

 may take. The convergent power series 

 P{x) has significance for at least a limited 

 domain of imaginary values of a;; on the 

 other hand, it is possible for trigonometric 

 series to define functions which have no 

 meaning except for real values of x. As, 

 therefore, the trigonometric series has a 

 functional content totally different from 

 that of the power series, its influence was 

 felt first, and primarily, in the development 

 of the notion of a function of a real 

 variable. 



The concept function was at first vague, 

 as vague and indefinite as our geometrical 

 intuitions. It had its root in the 17th cen- 

 tury in the analytic geometry of Descartes. 

 Here the variation of y with x along a curve 

 inevitably suggests the notion of a function. 

 The first published definition of the term 

 appeared in 1718 when John Bernoulli 

 defined a function of a variable as "an ex- 

 pression which is formed in any manner 

 from the variable and constants." Thirty 

 years later, in his ' ' Infinitesimal Analysis, ' ' 

 Euler defined it in like manner except that 

 the function is now an "analytic expres- 

 sion." "What is meant by "analytic ex- 

 pression" is not explained, but from his 

 definition of special classes of functions it 

 would appear that the term denoted an ex- 

 pression put together in terms of the vari- 

 able and constants by a finite or infinite 

 number of operations of addition, subtrac- 

 tion, mulitiplication, and division. Differ- 



entiation and integration were also un- 

 doubtedly permissible. 



About this time there began the famous 

 controversy over the mathematical repre- 

 sentation of a vibrating string. This satis- 

 fies the well-known differential equation 



d'w 



,dhjo 



where a is a certain constant, x the position 

 of a particle on the string when taut, and 

 w its transverse displacement at time t. A 

 solution of this problem for the case of 

 fixed end points was given by d'Alembert 

 in 1747 under the form 



w=f(x + at) — f{at — x), 



where f{x) denotes an arbitrary function 

 whose nature he apprehended too narrowly. 

 But he claimed to have the general solution 

 inasmuch as his solution involved an arbi- 

 trary function. 



This shot into mathematics the question : 

 What is an arbitrary function"! Even 

 to-day this question is a vexing one, owing 

 to disagreement in the point-set theory con- 

 cerning certain principles of logic which 

 cluster around the "Princip der Ausivahl" 

 as a center. But mathematicians had not 

 then arrived at the subtleties of the present 

 day. Their difficulties were really caused 

 more by imperfect notions concerning a 

 function than by the degree of arbitrariness. 

 On the basis of the above definition of a 

 function then current, Euler maintained 

 that d'Alembert 's solution was particular, 

 rather than the most general possible. He 

 rightly apprehended the nature of the phys- 

 ical problem and saw that the motion of the 

 string subsequent to the initial instant was 

 completely determined by the initial form 

 of the string and the initial velocities of 

 its points. Now the initial shape of the 

 string could be a continuous geometrical 

 curve composed of successive pieces whose 

 forms are absolutely independent of one 



