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SCIENCE 



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



must be in many ways perverted, if studied 

 without correlation to the other members. 

 But, after all, it is the Fourier series which 

 gave the initial push and chief impetus to 

 the construction of the function theory of 

 the real variable. 



This becomes still plainer if we take into 

 consideration the comparatively recent 

 point-set theory. Originally an off-shoot of 

 the real function theory and still often 

 treated by itself, it has been largely ab- 

 sorbed back into this theory, and its con- 

 cepts already permeate analysis. Its 

 founder was George Cantor, who was 

 trained in the exact yet fertile school of 

 Weierstrass. His earliest papers presaging 

 this theory relate to a trigonometric series. 



Two problems occupy his attention. The 

 first is to show that if the series 2 (<J„ sin nx 

 -\- bn cos nx) is convergent throughout an 

 entire interval, except possibly for a finite 

 number of points, the coefficients a„ and b„ 

 have for n = oo the limit 0. The second is 

 to establish the uniqueness of the develop- 

 ment of a function into a trigonometric 

 series; in other words, to prove that when 

 2 (On sin nx -j- 6„ cos nx) is identically 

 over an interval, then each and every a„ 

 and bn must be 0. The requirement of con- 

 vergence of the sum in the one case and of 

 its vanishing in the other, was originally 

 made for the entire interval, but Cantor 

 found that it could be remitted for certain 

 infiinite aggregates of points without affect- 

 ing the truth of the conclusions. He was 

 led consequently to introduce the notion of 

 the '' derivative of a point-set." Consider 

 with him the set of points for which the 

 requirement is omitted, and suppose that 

 they cluster in infinite number in the vicin- 

 ity of any point. This will be called a 

 limit-point of the set. The totality of these 

 limit-points is called the first derived set, 

 or first derivative. This derived set of 

 points may also have cluster points which 



form the second derivative ; and so on. 

 After introducing this concept. Cantor 

 proved that the requirement could be re- 

 mitted for any set of points whose nth deriv- 

 ative contains only a finite number of points 

 and whose (M + l)th derivative accord- 

 ingly vanishes. 



In these very early papers of Cantor we 

 have very clearly the beginning of his point- 

 set theory. His attention is here concen- 

 trated upon an infinite aggregate of points, 

 and the notion of the derived point-set was 

 the first of the concepts by means of which 

 he is able to distinguish between different 

 infinite aggregates of points. Prior to Can- 

 tor no effort was made to distinguish quali- 

 tatively between them. To be sure, mathe- 

 maticians were thoroughly conversant with 

 the distinction between a continuous curve 

 or set of points, on the one hand, and a 

 merely dense aggregate of points such as 

 the totality of points with rational coordi- 

 nates. The raw material lay at hand for a 

 beginning, especially in the work of Rie- 

 mann and others on integration. Cantor 

 alone saw the imperativeness of the need. 

 In comparing infinite sets of objects and 

 seeking a theory of the truly infinite he 

 blazed a new path for the human mind. 

 As a fifth and a mighty influence of Fou- 

 rier's series we have, therefore, to record 

 the historic origin of the theory of infinite 

 aggregates. 



Thus far in my sketch I have traced one 

 strong, single current of influence of the 

 Fourier's series. I have now to indicate 

 some other effects without close relation to 

 the foregoing. 



In Fourier's "Analytical Theory of 

 Heat" there are found what are said to be 

 the first instances of the solution of an in- 

 finite number of linear equations with an 

 infinite number of unknowns. He has, for 

 example, to determine the coefficients in the 

 equation : 



