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SCIENCE 



[N. S. Vol. XXXIX. No. 995 



first of the two or three principal sources of 

 this theory is to be found inRiemann's ap- 

 plication of integration to discontinuous 

 functions in his memoir on ' ' the represent- 

 ability of a function by Fourier's series." 



The above example of Riemann is notable 

 for giving a mathematical expression for a 

 discontinuous function incapable of graph- 

 ical representation. I have already pointed 

 out how Euler conceived of graphs so 

 arbitrary as to be impossible of represen- 

 tation through an "analytic expression." 

 The scales were now turned decisively to 

 the other side, though it was not till later 

 that it was recognized that our geometric 

 figures have only an approximating char- 

 acter which our mathematical equations 

 refine. 



But the full power of mathematical ex- 

 pression was not realized until 1872-1875, 

 when Weierstrass startled the mathematical 

 world with an example (first published 

 by Du Bois Reymond in 1875) of a contin- 

 uous function having nowhere a derivative, 

 or, in other terms, of a continuous curve 

 without a tangent. The function given by 

 Weierstrass was a trigonometric series 



% b" cos a"irx, 



in which & is a positive constant less than 

 1 and a a fixed odd integer large enough 

 to make ab exceed a certain value. "Weier- 

 strass states also that Riemann is supposed 

 to have shown that the series 



represented a function of like property, but 

 the proof was not known. The failure of 

 the continuous function of Weierstrass to 

 be differentiable is due to the possession of 

 an infinite number of maxima and minima 

 in any interval, however small. 



This example completed the separation 

 between differentiable and continuous func- 



tions. It shows that the former are only a 

 subclass of the latter, a result not even sur- 

 mised by the boldest geometrical intuition. 

 This is the third influence of Fourier's 

 series which I wish to emphasize. So far 

 as I know, this is the only one of its results 

 which vitally affects geometric theory. It 

 reveals the transcendence of analysis over 

 geometrical perception. It signalizes the 

 fiight of human intellect beyond the bounds 

 of the senses. 



I return now to trace further the march 

 of the function theory of the real variable. 

 The second principal element in its forma- 

 tion seems to me to have been the concept 

 of uniform convergence. This also seems 

 to have been suggested chiefly by study of 

 trigonometric series. Originally it was sup- 

 posed that the sum of a convergent series 

 of continuous functions shared the common 

 properties of its terms and accordingly was 

 continuous. Even so great a mathemati- 

 cian as Cauchy fell for a time into this 

 error. The fallaciousness of this assump- 

 tion was first pointed out by Abel in 1826 

 in his well-known memoir on the binomial 

 series. Here he also discusses the series 



,,,, . , sin 20 , sin3<#> 



(2) 



every term of which is continuous. Clearly 

 the sum vanishes whenever ^ is a multiple 

 of TT. If <l> lies between mirand (m -(- l)7r, 

 the sum is <^/2 — vtt, where v denotes the 



half of m or m -f- 1, according as m is even 

 or odd. Consequently, when <f> passes 

 through an odd multiple of tt, the sum has 

 a discontinuity of amount v, as is indi- 

 cated in the adjoining graph. This re- 



