410 



NATURE 



[January 25, 191 



The reader is not supposed to have any advanced 

 technical knowledge, and everything seems done to 

 help him to appreciate the important notions which 

 the last few years have developed. 



Among the subjects treated .n leaning of 



definitions, axioms, and postulates; tiic elementary 

 properties of classes, and of transfinite ordinal and 

 cardinal numbers; negative and complex numbers; the 

 three possible geometries in three-dimensional space; 

 spaces of more than three dimensions; variable, 

 function iikI limit. 



As ii Mil (,f ilio upper limit that is reached, 



it may i'< ni-i'*.! iliai we have Cantor's proofs that 

 the numerical continuum is not denumerable, and that 

 the class of algebraic numbers is; correspondence 

 between points of a line segment and those of a 

 square ; and a brief account of quaternions. 



In the more strictly geometrical part, Hilbert's 

 axioms are discussed in considerable detail, an illus- 

 tration is given (after Klein) of a system for which 

 Archimedes' axiom is not satisfied, and an account 

 is given of Fieri 's kinematical theory. This last is 

 comparatively novel, and will perhaps appeal to some 

 more vividly than Hilbert's. 



There are a few controversial points to which atten- 

 tion may be directed. On p. 43 we are told that 

 " the only test for the consistency of a body of proposi- 

 tions is that which connects with the abstract theory 

 a concrete representation of it." Even allowing the 

 widest sense to the term "concrete," this does not seem 

 justifiable. VVeierstrass proved that ordinary complex 

 algebra could not be consistently extended to a linear 

 algebra of three or more dimensions, and his proof 

 was as abstract as possible. The fact seems to be, 

 as Prof. Young practically admits elsewhere, that 

 there is no absolute test of consistency for any set of 

 assumptions ; all we can say is that, after applying 

 them in myriads of ways, we have not found any 

 inconsistency. 



Then there is the question of the term, " the class 

 of ordinary classes" (p. 219). Prof. Young takes the 

 view, which seems the right one, that this term is 

 intrinsically nonsensical, and involves a vicious circle. 



On p. 81 the symbol w is unfortunately used for 

 the cardinal number of all denumerable sets. It is 

 much better to keep it as the ordinal number of the 

 natural scale, and use o for the corresponding cardinal. 



Finally, with regard to the author's pedagogic atti- 

 tude. He emphasises rightly, more than once, that 

 with young pupils no attempt should be made to treat 

 the subject with logical rigour. On the contrary, a 

 large number of assumptions will be made, and everj'- 

 day experience constantly appealed to. This is a 

 sufficient answer to those who think that the 

 "logicians," as a body, wish to reduce mathematics 

 to dry bones. At the same time, a teacher ought to 

 know something about these logical methods, just in 

 order to avoid making dogmatic assertions which have 

 been shown to be false, and also that he may answer 

 inquiries without implanting erroneous ideas. For 

 this purpose Prof. Young's book may be heartily re- 

 commended; it is one more of the many good educa- 

 tional works on mathematics produced in the United 

 ^fates. G. B. M. 



NO. 2204, VOL. 88] 



OUR BOOK SHELF. 



Contemporary Chemistry: a Survey of the Present 

 State, Methods, and Tendencies of Chemical Science. 

 By E. E. Fournier d'Albe. Pp. xvi+180. (Lon- 

 don : Constable and Co., Ltd., 191 1.) Price 4*. net. 



If the author of this book had been content to give 

 it a more modest title, the task of the reviewer 

 would have been a pleasanter one. Mr. Fournier 

 d'Albe undoubtedly possesses a gift of expression, and 

 his breezy style is attractive and interesting. More- 

 over, due praise must be given to him for his effort to 

 popularise the fascinating phases of modern chemist' 

 Yet it is just here where the difficulty comes in. 

 the present book is intended for the general read 

 the author's semi-journalistic "lightning sketch' 

 assume too much, and are too lacking in coherence, 10 

 convey any definite impression. On the other hand, 

 the serious student of chemistry will find the auth< : " 

 fare, if appetising, rather scrappy and unsatisfying. 



The book indeed fails to give any rational and con- 

 nected account of the main lines of contemporary 

 chemical science. . Let us take, for example, the 

 chapter on "Affinity," where the author gives us little 

 more than a couple of hazardous calculations on t^-^f 

 attraction of electrical charges. Not a word is ^ 

 about the real work of to-day, namely, the numero _ 

 experimental methods of measuring chemical affinity, 

 and the collation and comparison of the vast amount 

 of data already obtained. We cannot have our N 

 ton before our Tycho Brahe and Kepler. 



A closer examination of the book reveals in many 

 cases the author's want of familiarity with chemistry, 

 whereof a few instances may be noted. On pp. 21-22 

 we hear about de Broglie, but no hint is given of the 

 work of Perrin, Svedberg, or Henri. On p. 36, it is 

 stated that "a single phase has therefore two inde- 

 pendent variabilities." On p. 38 there is a complete 

 confusion between passivity to change and Le Chate- 

 liers's theorem. The calculation on p. 55 appears to the 

 reviewer to be quite absurd. On p. 56 there occurs a 

 misleading confusion between ordinary and electrolytic 

 dissociation, whilst on p. 57 the use of the expression 

 "specific conductivity' " instead of molar conductivity 

 leads to dire results. P. 59 opens with the 

 sentence, " It might seem at first difficult to prove that 

 metallic sodium and uncombined chlorine exist in a 

 dilute solution of hydrochloric acid." Apart from the 

 obvious misprint, this is truly a case of " save us f ; 

 our friends." The cup is full when, 00 read 

 p. 92, the author says, "Yet to-day we believe 

 onlv that metallic sodium exists in sea water," 

 One wonders what the author's theory of the metallic 

 state would be like. Other slips of a similar nature 

 might be quoted, but it would be ungracious to 

 multiply instances. 



As a survey of the present state, methods, and 

 tendencies of contemporary chemistry, the _ book is 

 indeed very inadequate. But as a crisply written 

 re.idable sketch of many interesting things it t 

 stimulate interest where many a more ponderous 

 more accurate volume would fail. F. G. I ' 



Outlines of Biology. By Dr. P. Chalmers Mitch 1, 

 F.R.S. Revised and supplemented bv George P. 

 Mudge. Third edition, revised. Pp. xv + 348. 

 (London: Methuen and Co., Ltd., 1911.) Price 65. 

 net. 



This is a book which, as its author and reviser say 

 in their opening remarks, is intended to "cover the 

 ground of the student working for the First Examina- 

 tion of the Conjoint Board of Surgeons and Physicians, 

 London," and also as an elementan.- text-book for 



