432 



SCIENCE 



[N. S. Vol. XLV. No. 1166 



thors seem to have a mania for scattering liter- 

 ary fragments and may cultivate foreign jour- 

 nals merely for the sake of personal advertis- 

 ing. 



Entomological News, 28 : 141, after mention- 

 ing four journals which lasted an average of 

 five years each, says: 



In general it seems tliat the number of special- 

 ists in any one or two orders of insects is not sufii- 

 cient to support a special journal, and we know of 

 none such provided with an endowment fund guar- 

 anteeing its permanency. In this matter we must 

 stiU be entomologists, apparently, and yet the rec- 

 ord of general entomological journals contains 

 many a short-lived periodical. 



The significant point here is that, while we 

 are specialists as regards the literature we de- 

 sire, we are general entomologists as regards 

 the literature we have to pay for. 



As educational institutions the university is 

 local, while the printing press is cosmopolitan, 

 the only cosmopolitan university. The 'publi- 

 cation of scientific literature should not be 

 supported by requiring specialists to pay for 

 literature they do not need, any more than the 

 university should depend for its entire support 

 upon the tuition of its students. 



Chaeles Eobeetson 



Caelinville, Illinois 



fundamental conceptions of modern 

 mathematics 



To THE Editor of Science : In your issue of 

 August 4, there appeared a review of the first 

 part of our " Fundamental Conceptions of 

 Modern Mathematics," from the pen of Pro- 

 fessor G. A. Miller. Against a hostile criti- 

 cism, giving a portrayal of at least some of 

 the main theses of our book and attempting 

 to controvert them, we would have no inclina- 

 tion to protest. But all the important issues 

 raised by our treatise are ignored by Professor 

 Miller, who dwells upon features having no 

 bearing upon any of the arguments of our 

 work, or upon any of the doctrines which it is 

 the purpose of the arguments to uphold. 



Surely a reviewer can be justly expected to 

 take up at least one or two of the principal doc- 

 trines of a treatise of which he disapproves. 



and show that these doctrines are erroneous. 

 Our book contains an account of quantities 

 and their classification; an investigation into 

 what the symbols used by mathematicians 

 really stand for. We set forth the classifica- 

 tion of quantities into what we call sorts, 

 kinds and varieties, and show the importance 

 of this classification in the subdivision (orig- 

 inally conceived by De Morgan) of algebraic 

 science into single algebra, double algebra, etc. 

 A precise statement is given of what we appre- 

 hend to be the nature of the quantities dealt 

 with in quaternions and other systems of vector 

 analysis, and of their relation to the quantities 

 of ordinary algebra. We attempt to show that 

 any really scientific treatment of ordinary 

 imaginary quantities must be based on vector 

 analysis, all imaginary and complex abstract 

 quantities (save those of zero value) being, in 

 fact, relations between vectors. This is, we 

 hold, the only way to ascend, from a blind use 

 of imaginary and complex expressions without 

 any clear apprehension of what they denote, 

 to a rational comprehension of the matter; in 

 other words, from mere computation, and 

 manipulation of symbols, to true science. We 

 show further that the mathematicians who 

 look upon a variable as a quantity and those 

 who regard it as a symbol are equally in the 

 wrong; a variable being represented by a 

 symbol and being composed of quantities. We 

 consider the arrangement of the quantities of 

 a variable, and show the importance of this 

 commonly neglected attribute. We discuss 

 the peculiar arrangements which must be at 

 hand to justify the application of the theory 

 of monogenic functions, and show the rela- 

 tion of these multiplex arrangements (as we 

 call them) to the arrangements of the ele- 

 ments of the aggregates designated by Cantor 

 as melirfach geordnet. As the simplest of 

 variables we put forward the ordinary pro- 

 gressions of elementary mathematics which are 

 not usually recognized as variables at all. We 

 attempt to show clearly just what distinctions 

 should be drawn between a progression and a 

 series; and, including all progressions and all 

 series under the head of sequences, lay down 

 the conditions under which a variable is to be 



