NA TURE 



[December 7, 1905 



word. But we do not believe our author wishes to be 

 taken seriously here : — 



" Even if we postulate a Deity as the Originator 

 of all things, yet the whole history of science, which 

 is that of civilisation, proves that it is more profit- 

 able to seek the explanation of natural phenomena in 



natural laws (His laws) than in infractions of than — 

 in miracles." (The italics are mine.) 



We have one fault to find ; in a work on the prin- 

 ciples of heredity one would have expected a fuller 

 discussion than is actually given of biometric and 

 Mendelian methods of dealing with that phenomenon : 

 medical men reading the book will get a very meagre 

 idea of the nature of the investigation being carried 

 on and of the definite results already achieved by these 

 two sets of workers. 



IV. Reid does good service in dealing a blow 7 at 

 that teleology which is the curse of biological science 

 b) exposing the falsity of the old idea that the 

 " object " of bi-parental reproduction is to ensure a 

 sufficient degree of variability in each generation 

 for natural selection to operate upon. He cites as 

 evidence for this Dr. Warren's work on Daphnia 

 magna ; but docs not refer to a more recent and more 

 complete demonstration of the same truth bv the 

 same author in the case of Aphis, to be found in 

 Biometrika, vol. i., p. 129. 



I hese, however, are trifles, and do not detract from 

 the value of the book as a whole. A. D. D. 



MATHEMATICAL LECTURES FOR AMERICAN 



MATHEMATICIANS. 

 The Boston Colloquium. Lectures on Mathematics. 

 By Edward Burr Van Vleck, Henry Seelv White, 

 and Frederick Shenstone Woods. Pp. xii+188. 

 (New York : The Macmillan Company, 1905.) 

 Price 2 dollars net. 

 Lectures on the Calculus of Variations. Bv Dr. 

 Oskar Bolza. Pp. xvi + 272. (Chicago: The Uni- 

 versity Press, 1904.) Price 4 dollars net. 

 AMONG the many ways in which the American 

 Mathematical Society has endeavoured to 

 popularise and develop the study of higher mathe- 

 matics, not the least remarkable and useful is the 

 practice of holding " colloquia " in connection with 

 the summer meetings at intervals oi two or three 

 years. It had been felt that the mere reading of a 

 long string of disconnected papers does not produce 

 much lasting impression on the minds of the 

 audience. On the other hand, even a short course 

 of university lectures will often adequately cover a 

 wide range of mathematical stud)'. The societ) there- 

 fore decided in 1896 to arrange for courses of three 

 to six two-hour lectures, each dealing with a sub- 

 stantial part of mathematics. Four such colloquia 

 have been held, at Buffalo in 189(3, at Cambridge in 

 [898, at Ithaca in 1901, and at Boston in 19031 At 

 each of the first three two courses of lectures were 

 given, and Prof. Oskar Bolza 's course on " The 

 Simplest Type of Problems in the Calculus of Vari- 

 ations," given at the Ithaca colloquium of 1901, 

 NO. 1884, VOL. 73] 



forms the basis of one of the two volumes before us. 

 The chapters nearly follow the historic order laid 

 down in the introduction, which is also in close con- 

 formity with a logical sequence of treatment. The 

 study of the first and second variations of an integral 

 naturally leads to Weierstrass 's examination of the 

 conditions for a minimum and the distinction between 

 a " strong " and a " weak " minimum, a terminology 

 introduced by Kneser. The next steps are repre- 

 sented by Weierstrass's theory of parameter represent- 

 ation, Kneser's general theory based on the properties 

 of geodesies, and Hilbert's existence-theorem. For 

 Weierstrass's work (much of which is contained in 

 unpublished courses of lectures) the author has had 

 recourse to his own notes of a course (by Weierstrass) 

 which he attended in 1879, as well as to several other 

 sets of lecture notes, including one on Prof. Schwarz's 

 lectures at Berlin on the same subject. . 



At the next colloquium, held at Boston in 

 September, 1903, three courses of lectures were given. 

 The year marked the fiftieth anniversary of the 

 appointment of Prof. John Monroe Van Vleck to the 

 chair of mathematics at Wesleyan University, and it 

 was fitting to the occasion that all the lecturers were 

 Van Vleck's pupils, one of them being his son. Prof. 

 Henry S. White, of North-Western University, is re- 

 sponsible for the course of three- lectures on " Linear 

 Systems of Curves cm Algebraic Surfaces," Prof. 

 Frederick S. Woods, of the Massachusetts Institute 

 of Technology, for three lectures on " Forms of Non- 

 Euclidian Space," and Prof. Edward B. Van Vleck, 

 of Wesleyan University, lor six lectures on " Selected 

 Topics in the Theory of Divergent Series and Con- 

 tinued Fractions." A bibliography of literature on 

 continued fractions extending over twenty pages con- 

 cludes the last named discourse. 



Long- formula; involving x and v are like little 

 children — they ought to be "seen and not heard." 

 The success of these colloquia when originally de- 

 livered must have been in some considerable measure 

 due to the extent to which the authors have succeeded 

 in dealing with ideas and their symbolical represent- 

 ations without giving- tedious demonstrations in 

 ex I en so. 



INDUSTRIA I. REFRIGERATIOh . 

 Modern Refrigerating Machinery, its Construction, 

 Methods of Working, mid Industrial Applications. 

 By Prof. H. Lorenz. American Practice in Re- 

 frigeration. By 11. M. Haven and F. W. Dean. 

 Pp. x + 396. (New York: Wiley and Sons; 

 London : Chapman and Hall, Ltd., 1005.) Price 

 j 7s. net. 



IT is to be regretted that no treatise exists on this 

 subject which contains an exhaustive investi- 

 gation of the thermodynamical problems involved, and 

 of the physical properties ol the various gases used as 

 media, with special reference to their practical appli- 

 cation to refrigerating- machinery. In works on 

 thermodynamics, the matter is treated in general 

 terms. The physical constants are found in scattered 



