January 23, ]914] 



SCIENCE 



117 



a broader basis, Riemann perceived that 

 first of all it was necessary to sharpen and 

 widen the concept of an integral. Initially 

 Leibnitz had thought of integration as 

 a summation process, but this notion was 

 forced into the background by its definition 

 as the reverse of differentiation, until re- 

 vived by Cauchy in 1823. He then defined 

 the integral of a continuous function as 

 most of us were taught to define it. The 

 interval of integration was divided into n 

 parts Si, each 8j was multiplied by the value 

 of the function f{xi) at its beginning, and 

 the integral was defined as the limit of the 

 sum ^Sif{Xi) when the number of parts in- 

 creased indefinitely, their size diminishing 

 indefinitely. Because of the continuity of 

 the function this definition of the definite 

 integral was equivalent to that framed by 

 means of the reverse process of differenti- 

 ation. Riemann dismisses altogether the 

 requirement of continuity for the function, 

 and in forming the sum multiplies each sub- 

 interval 8i by the value of the function, not 

 necessarily at the beginning of the interval, 

 but at a point fj arbitrarily assumed in the 

 srubinterval. If, then, a limit exists for the 

 sum 28i/(^i), irrespective of the manner of 

 partitioning the interval and of the choice 

 of the points ii, this is called the integral. 

 Thus he redefined the fundamental concept 

 of the integral calculus, making it entirely 

 independent of the differential calculus. 

 This definition, often referred to as the Rie- 

 mann definition of an integral, has now be- 

 come the universally accepted one and is 

 the basis of scientific treatment of the in- 

 tegral calculus. Thus a second service of 

 Fourier's series has ieen in laying the foun- 

 dation of the modern integral calculus, and 

 in such ivise that it hid fair to completely 

 eclipse the differential calctdus in impor- 

 tance and reach. 



Riemann 's memoir may also be charac- 

 terized as the beginning of a theory of the 



mathematically discontinuous. The work 

 of Fourier had disclosed that mathematical 

 expressions could portray functions with 

 breaks, and the exacter but more limited 

 investigation of Diriehlet drew still further 

 attention to discontinuities. Riemann 's 

 definition of an integral did more; with 

 one leap it planted the discontinuous func- 

 tion firmly upon the mathematical arena. 

 In his integrable functions was comprised 

 a class of functions whose discontinuities 

 were infinitely dense in every interval, no 

 matter how small — though indeed, as we 

 now know, they are not totally discontin- 

 uous. One example which he gave was the 

 integrable function defined by the conver- 

 gent series, 



(x) , (2x) (3z) _ 



■I ~r p -?- 22 ' 32 ' ' 



in which {nx) denotes the positive or nega- 

 tive difference between nx and the nearest 

 integral value, unless nx falls half way be- 

 tween two consecutive integers, when the 

 value of (nx) is to be set equal to 0. The 

 sum of the series was shown to be discon- 

 tinuous for every rational value of x of the 

 form p/2n, where p is an odd integer rela- 

 tively prime to n. 



This example and others, such as that of 

 an integrable function with an infinite num- 

 ber of maxima and minima which was inca- 

 pable of representation by a Fourier's 

 series, were exceedingly stimulating. The 

 investigation so impressed the imagination 

 of Hermann Hankel as to call forth his no- 

 table memoir ' ' Uber die unendlich oft oszil- 

 lierenden und unstetigen Functionen" in 

 which he unfolds his principle of "conden- 

 sation of singularities, ' ' a memoir so impor- 

 tant that it has even been said to "entitle 

 him to be called the founder of the inde- 

 pendent theory of functions of a real vari- 

 able." It would appear to me that this 

 distinction could be assigned with equal 

 propriety to Riemann, for historically the 



