NOVEMBEE 17, 1922] 



SCIENCE 



575 



norm by which the aggregate is defined. Or 

 again, page 239, "... an aggregate has a car- 

 dinal number only when it is one of a plurality 

 of equivalent aggregates, distinct from one an- 

 other." Again, what success can be expected 

 from an attempt to clarify the notion of ag- 

 gii-egate by the introduction of a new word, 

 "norm," itself of debatable meaning? Pro- 

 fessor Hobson intimates that something must 

 be "universally accepted" to be admitted as 

 mathematieal knowledge; buit he does not men- 

 tion the attack made by Brouwer and Weyl 

 upon some of the fundamental theorems in 

 Analysis. Still again, it may be asked, if 

 (page 6) "the justification [for a certain as- 

 sumption] is to be found in the fact that no 

 eontradictaon arises in the theory based on it," 

 why is not the multiplicative axiom justified? 

 Zermelo's Grundlagen have led to no contra- 

 diction. 



The present edition is "revised throughout" ; 

 "the parts of the subject dealt with in the first 

 five chapters of the fii-st edition have been ex- 

 panded into the eight chapters of the present 

 voliune." This expansion is due chiefly to the 

 recent developments in the Theory of Integra- 

 tion. The theories of Hellinger, Young and 

 Denjoy are also included. The mathematieal 

 world owes a debt of gratitude to Professor 

 Hobson for presenting in a smooth, connected 

 exposition a huge mass of research, a consid- 

 erable part of which is of recent origin. 



The style is, on the whole, very lucid, great 

 pains being taken to prepare the reader's mind 

 for the reception of new ideas. However, here 

 and there we find a lack of compactness — wit- 

 ness the proofs that cover pages 97 and 98, 

 and wliich may be compressed into one tenth of 

 the space by the use of points with rational co- 

 ordinates. The treatment retains, in p-ant, 

 something of the freshness of a memoir and 

 will thus prove more stimulating, in one way, 

 than treatises written with a oonstraining 

 finish. 



At this late date, strange to say, the defini- 

 tion of cardinal number as given by Professor 

 Hobson is not without an objectionable fea- 

 ture. Russel is right. Professor Hobson's 

 reference to the "degree of plurality" is like 



saying, "You know what I mean"; it does not 

 make his definition mathematically acceptable. 

 On page 259 occui-s the following statement: 

 "No elaborate theory is required for functions 

 which retain their complete generality, . . . 

 since few deductions of importance can be made 

 from that definition which will be valid for all 

 functions." It may be of interest to remark 

 that this view is rendered untenable in the 

 light of the results to be announced soon in the 

 Proceedings of the National Academy of Sci- 

 ences in a paper by the reviewer, entitled "New 

 Properties of All Real Functions." 



Henry Blumberg 

 Univeesitt of Illinois 



Analysis Situs. The Cambridge Colloquium 

 Lectures, Part II, Vol. V. By Oswald 

 Veblen. Published by the American Math- 

 ematical Society, 501 West 116th Street, 

 New York. 150 pp., octavo. 1922. 

 The Cambridge Colloquium lectures on 

 Analysis Situs were delivered in 1916, but the 

 publication having been postponed because of 

 the war, the lectures were completely rewritten 

 before publication, and the resulting book is a 

 treatise on the elements of Analysis Situs. It 

 is furthermore the only modern book on the 

 subject. By a study of this book it is possible 

 to acquire a knowledge of Analysis Situs with- 

 out going through the many widely scattered 

 memoirs as was formerly the case. Every one 

 interested in Analysis Situs will welcome Pro- 

 fessor Veblen's book as an important and use- 

 ful contribution to the subject. Part I by 

 G. C. Evans on Funetionals and their Applica- 

 tions was published ia 1918. 



H. L. Rietz 



SPECIAL ARTICLES 



ZOSTERA MARINA IN ITS RELATION TO 

 TEMPERATUREi 



In connection with some work on the tem- 

 perature control of the geogi-aphical distribu- 

 tion of the marine algas, it seems to be demon- 

 strated that the terms eui-ythermal and steno- 

 thermal apply only to the power of endurance 

 of a wider or narrower range of temperature 



1 Preliminary communication. 



