398 



NATURE 



[July 13, 1916 



approaches a line parallel to the C-axis " asymp- 

 totically " is certainly surprising, if Freundlich's 

 equation is accepted as correct. 



An ag-reeable feature of the book is the amount 

 of space devoted to presenting the historical 

 development of different branches of the subject, 

 many quotations from the original papers of 

 pioneer workers being given. In this connection 

 the author fixes 1750 as the earliest date at which 

 gold sols had been obtained by reduction. 

 "Aurum potabile," however — a red liquid pre- 

 pared by reducing gold chloride with oil of rose- 

 mary and undoubtedly a gold sol — had consider- 

 able vogue as a medicine much before that time, 

 being mentioned, e.g., by John Evelyn in his diary 

 under the date June 2'j, 1653. 



The references to literature — given at the end 

 of each chapter — are copious, and names and sub- 

 ject matter are well indexed. The book may be 

 thoroughly recommended to the large class of 

 students to whom a knowledge of colloidal science 

 is becoming increasingly necessary; to cover the 

 whole field it should be supplemented by a volume 

 dealing with emulsoid sols and gels, which latter 

 in particular are systems quite as fascinating, and 

 certainly as important, as sols. 



MATHEMATICAL PAPERS AND 

 ADDRESSES. 



(i) Proceedings of the London Mathematical 



Society. Second Series. Vol. xiv. Pp. 



xxxviii + 480. (London: F. Hodgson, 1915.) 



Price 255. 

 (2) Four Lectures on Mathematics. Delivered at 



Columbia University in 191 1 by Prof. J. 



Hadamard. Pp. v + 52. (New York: Columbia 



University Press, 191 5.) 



(i) A VOLUME of the L.M.S. Proceedings is 

 -^"^ not only a permanent record of achieve- 

 ment. At its first appearance it is a useful index of 

 the state of English mathematics at the time ; and 

 it also, from year to year, suggests the appear- 

 ance of new stars in the mathematical firmament. 

 It may be not without significance that, in the 

 present volume, there is a first contribution (we 

 believe) by. a Japanese gentleman, and another 

 by an Indian fellow-subject. Unless we are 

 greatly mistaken, or unkindly fate should inter- 

 vene, Mr. S. Ramanujan is likely to become an 

 arithmetician of the first rank. At any rate, his 

 paper on highly composite numbers is original, 

 profound, and ingenious, and shows complete 

 mastery of the new methods and notation in- 

 augurated by Landau. Mr. Tadahiko Kubota 

 provides one of the two papers in the volume 

 which have any claim to be called geometrical, 

 and of these it is the more truly such. Under 

 certain assumptions, most of which are explicit, 

 or nearly so, he proves the following theorem : 

 " If a closed convex surface be cut by every pencil 

 of parallel planes in homothetic curves, it is an 

 ellipsoid." The method of proof consists mainly 

 in showing that such a surface defines a polar 

 field precisely similar to that which is determined 

 NO. 24^7, VOL. Q71 



by an ellipsoid. The comparative simplicity of 

 the demonstration is very remarkable. 



The other geometrical paper, by Mr. E. H. 

 Neville, was suggested by the racecourse puzzle 

 of covering a circle by a set of five circular discs. 

 Unfortunately, the solution depends upon four 

 simultaneous trigonometrical equations, and as 

 these are treated analytically, the paper has only 

 a tinge of geometrical theory. Once more we 

 must express our regret that English mathe- 

 matics is so predominatingly analytical. Cannot 

 someone, for instance, give us a truly geo- 

 metrical theory of Poncelet's poristic polygons, of 

 of Staude's thread-constructions for conicoids? 



The other papers cover a wide range, from 

 group-theory at one end (Prof. Burnside) to tide- 

 theory at the other (Prof. Larmor). One of the 

 most important, in our opinion, is that of Mr. 

 and Mrs. W. H. Young on the reduction of sets 

 of intervals — one of the many notable extensions 

 of the famous Heine-Borel theorem. It would be 

 foolish to try to give a detailed estimate of all the 

 twenty-six papers. 



Prof. Love's address on mathematical research 

 is bright as well as stimulating, and many of his 

 crisp sayings deserve the most careful attention ; 

 for example, his remarks on exact solutions of 

 physical problems, on the difficulty of applying 

 the general theory of ordinary linear differential 

 equations, on "curiosity," on the danger of being 

 overwhelmed by the mass of literature, and so on. 

 We wish we could agree with his unqualified 

 assertion that "text-books and treatises include 

 always later additions to knowledge " ; perhaps 

 he regards productions that do not conform to 

 this statement as mere samples of those "books 

 that are no books " to which Lamb refers. Lastly,. 

 we may note that Prof. Love attaches due im- 

 portance to mathematical style in composition. 

 This is too often neglected ; simplicity, clearness,, 

 and appropriate notation ought, at any rate, to be 

 aimed at with all possible diligence. We rejoice, 

 too, that in this connection he boldly and truly 

 says that a mathematical book or paper is (or 

 should be) a work of art. 



(2) The United States have been pioneers in the 

 practice, now common, of inviting eminent 

 foreigners to give occasional lectures, or courses- 

 of lectures, on their chosen subject ; we do not 

 refer to lectures or addresses on ceremonial occa- 

 sions. Prof. J. Hadamard is renowned for his 

 original researches in function-theory ; in the 

 present short course of four lectures he deals with 

 the bearings on physics of various types of equa- 

 tions (differential, integral, integro-diflerential)^ 

 and, in a minor degree, of topology {analysis 

 situs). It is needless to say that they are highl] ' 

 suggestive and valuable ; their defect, such as it 

 is, is that in trying to cover a wide field the 

 author is obliged to be very concise, and in some 

 cases this leads to obscurity. As an example ot 

 what we mean, take p. 34. Substantially (unless 

 we mistake the author's intention). Prof. 

 Hadamard wishes to point out that physical 

 problems which have the same analytical solutioD 

 lead to different interpretations of the solution^ 



