August 30, 19CX)] 



NATURE 



419 



nnembers were unable to be present at the first geheral 

 meeting, which was held on August 6 at 9.30 a.m. M. 

 Hermite was acclaimed president d'honneur ; M. Poincard, 

 president ; the vice-presidents (some in absentia) were 

 announced as MM. Czuber, Gordan, Greenhill, Lindelof, 

 Lindemann, Mittag-Leffler, Moore, Tikhomandritzky, 

 Volterra, Zeuthen, Geiser. The secretaries were MM. 

 Bendixson, Capelli, Minkowski, Ptaszycki, Whitehead ; 

 the general secretary, M. Duporcq. 



M. Poincare, on taking the chair, spoke a very few 

 words of greeting, and then called upon the speakers of 

 the day. M. Cantor, in his address, " Sur I'historio- 

 graphie des mathematiques," sketched the development 

 of this subject through Montucla (toujours un module 

 que tout historiog raphe des sciences doit suivre), 

 Kaestner, Cossali, Bossut, Chasles, Libri, Nesselmann, 

 Gerhardt, Arneth, Hankel, Boncompagni, up to authors 

 of the present day. He expressed the firm conviction 

 that the history of mathematics, from the beginning of 

 Lagrange's work, can only be written as a series of special 

 histories, with a final volume (Histoire des Idees) co- 

 ordinating the whole. M. Volterra, "Trois analystes 

 italiens, Betti, Brioschi, Casorati, et trois maniferes 

 d'envisager les questions d'analyse," compared and con- 

 trasted the work of these three mathematicians, and con- 

 sidered the influence their differing lines of thought and 

 expression have had on the development of Italian 

 analysis. 



Six sections had been arranged, with meetings extend- 

 ing over four days. While in general two sections were 

 sitting at the same hours, yet matters were so arranged 

 as to avoid, as far as possible, the conflict of interests 

 that had been felt at Zurich, where only one day was 

 devoted to the sectional meetings. These six sections, 

 with their presidents and secretaries, were as follows : — 



(i) Arithmetic and .Algebra: Hilbert, Cartan ; (2) 

 .Analysis : Painleve, Hadamard ; (3) Geometry : Darboux, 

 Niewenglowski ; (4) Mechanics and Mathematical 

 Physics : Larmor, Levi-Civita ; (5) Bibliography and 

 History: Prince Roland Bonaparte, d'Ocagne ; (6) 

 Teaching and Methods : Cantor, Laisant. 



Owing, however, to the unavoidable absence on some 

 days of the president of Section 5, and the small number 

 of papers in that section. Sections 5 and 6 sat together, 

 under the presidency, first of M. Cantor (Wednesday), 

 and then of M. Geiser (Friday) ; and at the Wednesday 

 morning sitting the two papers of most general interest 

 in the Congress were read. These were Hilbert's 

 address on the future problems of mathematics, valuable 

 as assisting the mathematician to orientate himself, and 

 Fujisawa's account of the mathematics of the old 

 Japanese school, of special interest as giving information, 

 not readily accessible otherwise, about a system of 

 mathematics that is now entirely obsolete. It appears 

 that the Japanese invented zero for themselves, and em- 

 ployed the circle as a symbol for zero ; that they used 

 imaginaries and complex numbers, and calculated the 

 value of IT correctly to forty-nine places of decimals. In 

 connection with this, M. Cantor remarked that the use of 

 zero is probably Baby.onian, and dates from about 

 1 700 15. c. 



M. Hilbert considered the origin and nature of the 

 problems of mathematics the study of which is most likely 

 to prove profitable ; the characteristics of a proper solu- 

 tion ; and the methods of attacking any problem that 

 offers special difficulties. If the problem is really in- 

 soluble, then for the advance of mathematics it is 

 essential that the impossibility be rigorously demon- 

 strated. He illustrated his argument by means of 

 selected problems that invite attack— problems regarding 

 the axioms of arithmetic and of physics, prime and 

 transcendental numbers, questions in the theory of 

 functions, and the determination of the arrangement of 

 the circuits that an algebraic curve can possess ; 



NO. 1609, VOL. 62] 



referring to a paper about to appear in the Nachrichten 

 der Kgl. Gesellschaft der Wissenschaften zu Gottingen, 

 1900, for a more complete list of definite problems that 

 demand investigation. 



Much interest was displayed in the papers read by M. 

 Mittag-Leffler at the Tuesday morning sitting of Section 

 2, " Sur fonction analytique et expression analytique," 

 " Sur une extension de la s^rie de Taylor." The domain 

 of an ordinary power-series is a circle that reaches to 

 the nearest singular point ; at all points inside this there 

 is convergence, at all points outside there is divergence ; 

 this the author generalised so as to obtain a certain 

 expression convergent within a particular region (an 

 etoile), and divergent without. He raised the question 

 whether an analytic expression can be found which shall 

 represent, throughout its domain of definition, an assigned 

 analytic function. A discussion followed between M M. 

 Borel, Hadamard and Painleve, as to the nature of the 

 connection between "analytic expression in a complex 

 variable jr" and " analytic function in jr." At the Thurs- 

 day sitting of Section i, M. Pade read a paper, "Apergu 

 sur les developpements recents de la th^orie des frac- 

 tions continues " ; in this he showed the dependence of 

 the expression of a function of .r as a continued fraction 

 on a certain diagram, in which each convergent is repre- 

 sented by a point whose co-ordinates are the degrees of 

 the numerator and denominator of the convergent ; and, 

 referring, to the discussion that followed Mittag-Lefiier's 

 paper, suggested that a continued fraction may be found 

 to be a suitable analytic form for any assigned analytic 

 function. 



The Friday morning combined sitting of Sections 5 

 and 6 was to a great extent occupied by the discussion 

 of a resolution offered by M. Leau, urging the Academy 

 to consider favourably the adoption of a universal lan- 

 guage, not with a view to displacing any of the existing 

 languages, but as a scientific medium auxiliary to these. 

 Some such resolution has been brought forward lately on 

 several similar occasions by the advocates of the latest 

 artificial language, Esperanto. The discussion showed, 

 on the part of mathematicians, very little sympathy with 

 the suggestion, and very little recognition of a need for 

 any such medium. As one speaker remarked, mathe- 

 matics already has a universal language, the language of 

 formulas ; and the general sense of the sections was evi- 

 dently that the existing diversity of languages need 

 cause no real difficulty, so long as writers are willing to 

 confine themselves to English, French, German and 

 possibly Italian, this view of the case being formulated 

 by a Russian, M. Vassilief The only result of the dis- 

 cussion was the rejection of M. Leau's motion, and the 

 recording of a wish that the Academy would discoun- 

 tenance any unnecessary diversity in the languages 

 employed for scientific purposes. The four languages 

 enumerated by M. Vassilief are those officially recognised 

 in the meetings of the Congress, though it was notice- 

 able that a great many of the speakers chose to speak in 

 French, possibly out of compliment to their hosts. 



Other communications of value, though of less general 

 interest, were the following : — In Section i, M. Ste- 

 phanos, Sur la separation des racines des equations 

 alg^briques ; in Section 2, M. Tikhomandritzky, Sur 

 I'evanouissement des fonctions e de plusieurs variables ; 

 M. Bendixson, Sur les courbes definies par les Equations 

 differentielles ; M. Jahnke, Zur Theorie der Thetafunc- 

 tionen von Zwei Argumenten ; in Section 3, M. Lovett, 

 On contact-transformations between the elements of 

 space ; M. d'Ocagne, Sur les divers modes d'application 

 de la methode graphique k I'art du calcul ; M. String- 

 ham, Orthogonal transformations in elliptic or in hyper- 

 bolic space ; M. Jamet, Sur le th^or^me de Salmon 

 concernant les cubiques planes ; in Section 4, M. Hada- 

 mard, Relations entre les caracteristiques reels et les 

 caract^ristiques imaginaires pour les Equations differ- 



