2S2 



Notices respecting JS 7 ew Book, 



We wish to emphasize, however, that the method of definition 

 given in detail by Professor Barton, and which apparently is pre- 

 ferred by himself, is perfectly logical. He has moreover put the 

 matter very clearly — at least considering that he only professes to 

 give an introduction to the subject. 



We recommend the volume very strongly to every teacher, and 

 we are confident that it will be well received. 



Fundamental Conceptions of Modern Mathematics. By Robert P. 

 Richardson and Edward H. Landis. Chicago and London. 

 The Open Court Publishing Company. 1916. 

 Numbers, Variables, and Mr. RusselVs Philosophy. By R. P. 

 Richardson and E. H. Landis. (Reprinted from the 'Monist' 

 of July 1915.) The Open Court Publishing Company. 1915. 

 The book and pamphlet naturally go together, part of the 

 criticism in the latter finding its place in the former. The 

 purpose of the book is best explained by some quotations from the 

 preface. It is — 



" to examine critically the fundamental conceptions of Mathe- 

 matics as embodied in the current definitions .... In ex- 

 pounding our own views we have often been obliged to find 

 fault with those of others ; but we have not gone out of our 

 way for the sake of mere criticism ; we have merely cleared 

 away false doctrine preparatory to replacing it with true. . . . 

 The keynote of our work is the distinction we find it necessary 

 to make between quantities, values, and variables, on the one 

 hand, and between symbols and the quantities or variables 

 they denote or values they represent, on the other." 

 The theme is attractive, and is to be worked out in thirteen 

 volumes, of which this is the first. It will be useful, meanwhile, 

 to see how the authors put their maxims into practice. 



Consider, for example, the meaning of Variable. We are prac- 

 tically told that all mankind since Newton have, in their endeavour 

 to define a variable, fallen into an estate of mathematical sin and 

 misery. After much criticism of these endeavours our Penn- 

 sylvanian authors say (pp. 154-5) that " while we have made 

 plain what a variable is not, and have described in what manner 

 it is constituted, care has been taken to avoid any statement as to 



what a variable is Any attempt to give a precise account 



•of the definition of the term 'variable ' w-ould require a somewhat 

 lengthy consideration of the philosophical theory of the categories, 

 which cannot be given in this place." We had hoped for bread, 

 and they give us a loaf-ticket ! After all, did it never strike them 

 that the mathematicians they criticize were in the same difficulty, 

 and that their so-called definitions were intended to be simply 

 ifirst notions which experience would amplify and even correct as 

 the mathematical student advanced in knowledge ? 



Consider, also, their discussion of quaternions and vectors. 

 Their view, we are told, is not precisely that of Hamilton. No 

 objection can be taken to snch a position. But what of their 

 treatment of Hamilton? They quote on p. 47 his definition that 



