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SCIENCE 



[N. S. Vol. XLIV. No. 1127 



stance, on page 152, we find the following 

 statement : " Among English mathematicians 

 of the Peano School the Honorable Bertrand 

 Bussell stands preeminent. He is the author 

 of a ponderous and pretentious treatise en- 

 titled ' Principles of Mathematics.' " On page 

 192, we find the following sentence : " The 

 blunder of thinking that in a functional rela- 

 tion between two variables the one variable 

 necessarily alters its value when the value of 

 the other alters is, we hope, so far obsolescent 

 as to be peculiar at the present day to the 

 learned ordentliche Professor of the University 

 of Munich." 



On page 145, we find the following severe 

 stricture on authors of English text-books : 

 " Practically all the mathematical text -books 

 now in use in England and the United States, 

 either give no definition at all of variable and 

 constant, or reproduce almost verbatim the 

 definition of Newton. As, however, such text- 

 books are brought forth almost invariably by 

 mere compilers, rather than mathematicians of 

 authority, we turn to continental Europe, 

 where we find equally bad definitions from 

 more authoritative sources." On page 195 ap- 

 pears the statement that "inability to use 

 language with precision seems to be a failing 

 endemic among mathematicians, and Eiemann 

 was not immune " ; and on page 151 the reader 

 is enlightened by the comprehensive remark 

 that a mathematician " can seldom lay claim 

 to more than a narrow technical education." 



The fact that authors of a mathematical 

 work criticize rather harshly a considerable 

 number of eminent mathematicians and direct 

 attention to common failings of the tribe is 

 in itself no conclusive evidence against these 

 authors, but it naturally leads the mathemat- 

 ical reader to assume a somewhat critical atti- 

 tude with respect to such authors; especially 

 when, as in the present case, most of the 

 authors' criticisms relate to definitions or to 

 the choice of words. The critical reader of 

 the present volume will not need to look long 

 to find evidences tending to show that its 

 authors were not, at the time of writing, famil- 

 iar with some very well known mathematical 

 facts. 



For instance, on page 35, we find the follow- 

 ing statement : " The only mathematician that 

 we recall as making a specific distinction be- 

 tween quotient and ratio is Hamilton." As a 

 matter of fact this distinction is so common 

 that in the " Encyclopedie des Sciences Mathe- 

 matiques," tome I., volume I., page 44, it is 

 proposed to restrict the use of the symbol: as 

 an operational symbol to represent a ratio, in- 

 stead of continuing its use to represent both 

 a ratio and also the operation of division. 1 

 On page 177, and elsewhere, the common 

 erroneous assumption according to which the 

 word function was used by the older analysts 

 as synonymous with power is repeated not- 

 withstanding the fact that about seven years 

 ago there appeared in the " Encyclopedie des 

 Sciences Mathematiques," tome II., volume 1, 

 page 3, a clear exposition of the way in which 

 this error crept into the literature. 



The main question involved in a review of 

 the first volume of an extensive projected 

 series relating to fundamental questions in 

 mathematics is, however, not much affected by 

 occasional historical inaccuracies or by infelic- 

 itous statements relating to eminent mathe- 

 maticians and to mathematicians as a class, 

 even if these facts are not void of important 

 implications. To the reviewer the present 

 volume appears to be poorly adapted for the 

 mathematical reader, since the treatment is 

 often prolix and involves many considerations 

 of little mathematical import. According to 

 the preface, the key-note of the work " is the 

 distinction we find it necessary to make be- 

 tween quantities, values and variables on the 

 one hand, and between symbols and the quan- 

 tities or variables they denote or values they 

 represent, on the other." 



Probably most mathematicians will be more 

 interested in the definitions given by those 

 who have made important advances in the 

 fields to which these definitions are related 

 than in those given by men who appear to be 

 mainly interested in philosophical specula- 

 tions. This is especially true in case the 

 latter authors exhibit evidences of knowing 



i Cf. G. A. Miller, School Science and Mathe- 

 matics, Vol. 7 (1907), p. 407. 



