March 28, 1902.] 



SCIENCE. 



501 



works. Each has the special interest that it 

 is devoted to that branch of the theory of 

 functions in which its author first attained 

 distinction. Hadamard aims to give a con- 

 cise, almost an encyclopedic, resume of the 

 present state of our knowledge concerning the 

 analytic continuation of Taylor's series. 

 Borel, on the other hand, gives a more detailed 

 exposition of a single chapter of this subject, 

 the divergent series. On this account his book 

 will have the greater interest for the mathe- 

 matical public and will be reviewed at some- 

 what greater length. 



Two other works of equal size and some- 

 what similar character have been previously 

 published by Borel, his 'Legons sur la theorie 

 des fonctions' (treating the 'Elements de la 

 theorie des ensembles et applications') and 

 his 'Legons sur les fonctions entieres.' 

 Together with the present work they form a 

 unique series, embodying the results of much 

 recent investigation in the theory of func- 

 tions. It is indeed a piece of rare good for- 

 tune in any province of mathematics to have 

 the important recent work thus promptly 

 picked out and thrown into accessible form 

 by such a mathematician as Borel. For this 

 reason the publication of these lectures can- 

 not be too warmly welcomed. 



It is safe to say that no previous book upon 

 divergent series has ever been written, Borel 

 opens up a field of research which is still very 

 new and promises rich reward to the in- 

 vestigator. In the process of evolution the 

 divergent series has passed through sev- 

 eral curious stages of development. At first 

 a divergent series was accepted on faith and 

 used with great naivete. Thus Leibnitz, for 

 example, when considering the expansion of 

 l/{l-\-x) into the series 1 — x-{-x^ — x'-\-.... 

 remarks that if x=l, the sum of n terms 

 takes alternately the values 1 and 0, and the 

 sum of the series must therefore be equal to 

 the mean value i. After the introduction of 

 exact analysis by Cauchy and Weierstrass 

 such a loose mode of treatment could no longer 

 be tolerated. The mass of inconsistencies to 

 which it would lead was clearly perceived, and 

 a divergent series was therefore considered by 

 the mathematician to be meaningless, good for 



nothing but to be thrown away. However, a 

 few of the great mathematicians were visibly 

 perturbed over the situation. Thus we find 

 Cauchy complaining in 1821 : 



"J'ai ete force d'amettre di verses proposi- 

 tions qui paraitront peut-etre un pen dures; 

 par exemple, qu'une serie divergente n'a pas 

 de somme." 



We know also that Abel was only prevented 

 by his premature death from attacking the 

 problem. But the view that a divergent series 

 had no place in mathematical analysis soon 

 became orthodox, and search after a legiti- 

 mate basis for its use was abandoned. Never- 

 theless the astronomers, in utter disregard of 

 this opinion, still continued to employ diver- 

 gent series and to obtain from them a suffi- 

 cient degree of approximation for practical 

 purposes. 



The impetus to a new mathematical treat- 

 ment of the subject may be said to have come 

 simultaneously from Stieltjes and Poincare, 

 although prior to this, in 1880, the legitimacy 

 of the conclusion of Leibnitz had been estab- 

 lished by Frobenius in a memoir which was 

 suggestive of the beautiful theory developed 

 later by Borel. According to the new view a 

 divergent power-series is considered as having 

 value in two distinct ways, either as enabling 

 one to find an approximate value of some cor- 

 responding function (Poincare and Stieltjes) 

 or as a source of another algorithm which is 

 convergent and therefore defines a proper 

 function (Stieltjes, 1894). 



The treatise of Borel begins with an inter- 

 esting historical introduction. The body of 

 the book can be divided roughly into four 

 parts, which take up successively the four chief 

 theories of divergent series; the asymptotic 

 theory of Poincare, the continued fraction 

 theory of Stieltjes, the theory of Borel — char- 

 acterized by the use of definite integrals con- 

 taining a parameter 2 — , and, finally, the 

 theory of Mittag-Leffler. The crowning 

 achievement is without doubt Borel's own 

 work, and his presentation of it is the most 

 interesting feature of the book. No adequate 

 idea, however, of the treatment of Stieltjes 

 can be obtained without direct reference to the 

 famous memoir of 1894, as Borel frankly 



