June 15, 1911] 



NATURE 



511 



go to prove the truth of evolution are skilfully and 

 attractively marshalled, but that is not the author's 

 chief object. What he aims at is to show that the 

 selection-hypothesis only applies to a limited field, that 

 use-inheritance has its restricted place, and so on ; 

 in short, that all o.r explanations are partial, and 

 that in none of them can we have complete con- 

 fidence. 



When we come to ask on what grounds this want 

 of confidence is asserted, we find the old objections 

 answered fifty years ago by Huxley put forward as 

 if they were discovered yesterday. "Darwinism 

 yields no information concerning the causes of vari- 

 ability " (p. 210); "the theory of natural selection is 

 the doctrine of chance " ; " the theory of mutation 

 plays with chance even more than selection." What 

 is Darwinism but a method? The author uses this 

 word as if it implied a corpus of knowledge which 

 could not be extended, and was to be judged by its 

 expression in the writings of casual essayists. In 

 only a single place does he refer either to the 

 "Origin" or to "Variation under Domestication," 

 though he quotes extensivelv from modern critics. 

 It is idle talk to say that Darwin did this, but not 

 that as though one man could do all, or as if Darwin 

 claimed a complete supremacy for his selection hypo- 

 thesis. Are we to have no confidence in the theorv 

 of natural selection because its discoverer was not 

 able to give a complete treatment of variation? One 

 would really conclude from this book that variation 

 was a subject to which Darwin gave no serious con- 

 sideration. The old goddess "chance" is once more 

 used as a stick to beat the dog "selection" with, 

 and we become rather tired at the iteration of argu- 

 ments urged and rebutted any time these thirtv 

 years. Even the old crude presentation of the Mil- 

 tonian "creation" is made to stand up in order to 

 receive fresh blows. 



The book is sadly in need of competent revision. 

 The number of misprints is really irritating. Amongst 

 the animals of the tropics we are told there is the 

 leopard and the gepard, a creature the nature of 

 which is wholly problematical. The choristers at 

 Rome have well developed " mammals " (p. 148) ; 

 "Omni vivum e vivo" (p. 97); "like all other infu- 

 sorians, no bell-animalcule is able to reproduce by 

 simple fission " (p. 230) ; the vermiform appendix 

 "has no function whatever, its object being rather 

 apparently to create suffering." On p. 129 Balano- 

 glossus is classed with Vermes, and the larvae of 

 Crinoids are tenned Bipinnaria, whilst on p. 130 the 

 larvae of starfish are correctly called by that name, 

 but the figure refers to an Auricularia or Holothurian 

 larva. In his condemnation of sexual selection the 

 author entirely overlooks the careful observations by 

 Mr. and Mrs. Pcckham on the spiders of the family 

 Attidae, and as these support the Darwinian position 

 the whole criticism falls to the ground. 



Apart from critical matters, the book is full of 

 interest, and its summary of recent work on heredity 

 is but one example of the wide reading and caretul 

 oxnosition which the author exhibits. Read with due 

 caution the book can only do good. 

 NO. 2172, VOL. 86] 



AN AMERICAN COLLOQUIUM. 

 The New Haven Mathematical Colloquium. By 

 E. H. Moore, E. J, Wilczynski, and Max Mason. 

 Pp. x + 222. (New Haven: Yale University Press; 

 Oxford : University Press, 1910.) Price 13s. 6d. 

 net. 



IN the autumn of 1906, at the meeting of the 

 American Mathematical Society, three short 

 courses of lectures were delivered to the assembled 

 experts, and are here published for the benefit of the 

 world at large, or at any rate for that of such persons 

 as are interested in the most recent aspects of pure 

 mathematics. The authors assume that their hearers 

 have a good knowledge of analysis, and the reviewer 

 must do the same, in order to keep within due limits. 



Prof. Moore's discourse is an introduction to general 

 analysis, which may be described as an essay on the 

 "functional theory" of Frechet and others. A free 

 use is made of the Peano stenography, and this is one 

 more sign of a fact which some of us will admit with 

 regret, namely, that students of the logical side of 

 mathematics must become proficient in Peanese. 



Two important ideas play a leading part in Prof. 

 Moore's discussion. The first is the dominance of one 

 function by another; m, is dominated by m, if, for 

 every argument p, the absolute value of Mo(/>) is not 

 less than the absolute value of M,(/>)- The other, 

 which appears to be both new and important, is that 

 of uniform convergence relatively to a function or. 

 Thus we have a notation 



Lt„/u„ = M (R; <r), 



meaning that when R is the range of the variable x, 

 and (m,, /*2» • • •) is a sequence of functions of x, this 

 sequence converges in such a way that, by taking n 

 lanre enough, and for all greater values of »j, 

 \fi-Hn\<e\<rlx)\, where e is any assigned positive 

 number; and if n can be determined by means of e 

 alone (without x), we have uniform convergence rela- 

 tive to 0-. When o- is constant, we come back to the 

 usual definition of uniform convergence. The second 

 part of the essay is on composition of classes, which 

 may be described as a generalisation of the theory of 

 the composition of moduli and ideals in the theory of 

 numbers. 



In connection with a certain notation, the author 

 remarks that "the intention is to discriminate sharply 

 between function and functional value." With this 

 we confess we are in sympathy, though, of course, wo 

 shall be told that a function is merely an enumeration 

 of values, either actually or potentially complete. In 

 the abstract, of course, this is undeniable; but con- 

 sider the function sin x, for example. Its property of 

 being periodic is intrinsic, and was actually realised 

 before there existed a table of its values, or rather it 

 was made part of a generalised definition of sin .r. 

 Again, the class of algebraic numbers, or that of 

 algebraic functions, surely has a significance apart and 

 beyond the aggregate of values associated with it. 



The next essay is upon projective differential geo- 

 metry, especially in connection with ruled surfaces. 

 Various results of great interest and generality arc 

 obtained hv the .luthdr; for instance, it nppenrs that 



i 



