Mat 4, 1917] 



SCIENCE 



433 



classified as a sequence. Passing to tlie ques- 

 tion of functional relation we take up inde- 

 pendent and dependent variables, and show 

 that these names correspond to at least three 

 separate distinctions, a fact not ordinarily 

 recognized by mathematicians. We give a 

 somewhat elaborate discussion of functional 

 relations, showing that what is ordinarily put 

 forward as the Dirichlet definition of function 

 does not adequately characterize a functional 

 relation, and moreover is not really the defi- 

 nition given by Dirichlet. We lay down what 

 we deem to be the conditions under which two 

 or more variables may be said to be in func- 

 tional relation with each other; and show that 

 previous authors, in their treatment of func- 

 tions, have not attained to a clear and precise 

 view of the essential characteristics of a func- 

 tional relation. 



To these salient features of our work Pro- 

 fessor Miller gives no heed whatsoever, though 

 assuredly they comprise topics of fundamental 

 importance in mathematics. He is content to 

 dismiss our inquiries by stigmatizing them as 

 relating chiefly "to definitions and the choice 

 of words." We plead that our work is con- 

 cerned chiefly with the unfolding of the con- 

 ceptions which words should awaken in our 

 minds, and not with the words themselves. To 

 purely verbal questions we give scant attention. 

 In our endeavors to attain to distinct and 

 exact conceptions of what is fundamental to 

 the inquiries of mathematics, we have found 

 that the portrayal of these conceptions, as set 

 forth by mathematicians of the highest emi- 

 nence, are not free from great imperfections. 

 We have spared no labor in obtaining and in 

 stating in full " the definitions given by those 

 who have made important advances in the 

 fields " into which we go ; and when unable to 

 assent to these deiinitions, we have carefully 

 set down our reasons for holding that they do 

 not truly depict the lineaments of the concep- 

 tions which they purport to unfold. A.nd Pro- 

 fessor Miller, though manifesting his disap- 

 proval of our criticisms, makes not the slight- 

 est attempt to show that our charges of error 

 are baseless, and that Baire, Pringsheim, 

 Riemann, Eussell, Weber, and the other 



authorities whom we controvert are not guilty 

 of the errors we ascribe to them. 



Three passages of our work are specifically 

 condemned by Professor Miller. All of these 

 are trivial and could be removed from the 

 work without affecting any of its doctrines or 

 any major or minor argument put forward in 

 defense of them. One fault that is imputed to 

 us is that "on page 1Y7 and elsewhere, the 

 common erroneous assumption according to 

 which the word function was used as synony- 

 mous with power is repeated." We merely say 

 that " the word function is said to have been 

 used by the older analysts as synonymous with 

 power." We took care to insert the qualifying 

 phrase is said, and so worded our remark is 

 neither an assumption nor an error. And no 

 reference to this usage occurs elsewhere. We 

 are also rebuked for saying that " The only 

 mathematician that we recall as making a 

 specific distinction between quotient and ratio 

 is Hamilton." We must acknowledge that 

 such a distinction has been made by others, 

 but we deny that the distinction we endeavor 

 to enforce is as common as Professor Miller 

 would have it appear. Finally we are chided 

 for applying to imaginary and complex quan- 

 tities the distinction between positives and 

 negatives. Yet, if precedent is to be a guide, 

 we can plead that both Gauss and Weierstrass 

 used the two adjectives with respect to imag- 

 inary quantities. 



Egbert P. Eichaedson, 

 Edward H. Landis 



QUOTATIONS 



SCIENTIFIC SNOBBERY 



One reason for the neglect of science is that 

 scientific men themselves frequently misrepre- 

 sent the objects for which they work. For ex- 

 ample, they often pretend that they perform 

 their labors merely for their own amusement. 

 We once heard it wittily said of such a man 

 that he takes out his watch before dinner and 

 exclaims, " Ha ! I have half an hour before I 

 must dress for dinner; I will just step over to 

 my laboratory and make a discovery." But 

 the public is not so easily deceived and there- 

 fore thinks in its dull way that the man of 



