May 18, 1917] 



SCIENCE 



467^ 



is then to deduce these invariants and to 

 give to them a physical interpretation. 



In the theory of invariants we have an 

 illustration of a fundamental fact concern- 

 ing the applications of any but the most 

 elementary mathematics, namely, that they 

 arise essentially as by-products of the lead- 

 ing discoveries. On the part of those who 

 make use of them they are often considered 

 as the essential and perhaps the only re- 

 sults of mathematical investigation. There 

 are some who have been impatient of those 

 studies which apparently have no connec- 

 tion with less abstract considerations. But 

 this is a short-sighted impatience. He who 

 wishes applications and applications alone 

 can not secure his ends better than by the 

 encouragement of theoretical investigations 

 even of an abstruse and remote character. 

 Within the separate natural sciences this 

 is so well understood that it is a matter 

 of surprise to observe that some individ- 

 uals persistently refuse to carry the con- 

 ception to its logical consequence as regards 

 the science of mathematics. But any one 

 who has meditated much on the character 

 of the progress of thought must certainly 

 have a profound realization of its truth. 



It seems that no body of thought has 

 been of more importance in human prog- 

 ress and at the same time been criticized 

 more freely than the science of mathe- 

 matics. Much of this criticism appears to 

 be good-natured and to amount to but 

 little more than a quasi-humorous way of 

 expressing the critic's own unashamed 

 ignorance. At first sight one might treat 

 this as harmless; but from the point of view 

 of the general interest it can hardly be 

 passed over in such a way. How this igno- 

 rance is to be overcome I can not say. Per- 

 haps one of the first requisites is to find 

 some means of overcoming the shameless- 

 ness with which individuals otherwise well 



trained contemplate their own ignorance 

 of mathematics. 



The mathematician himself is not dis- 

 turbed so far as the welfare of his own sci- 

 ence is concerned; but it is sometimes a 

 matter of pain to see the general loss 

 which arises from such ignorance and also 

 from the severer strictures of the more 

 pronounced adversaries of mathematics. 

 In no other ease, however, have the criti- 

 cisms been so severe as those meted out to 

 the infinitesimal calculus in the infancy of 

 its development; and never have the fond- 

 est hopes of the founders of a science been 

 so far surpassed by its actual achievements 

 as here where the subject has become cen- 

 tral in practically every field of pure and 

 applied mathematics. 



It will be instructive to examine briefiy 

 the criticisms thus met so early by the in- 

 finitesimal calculus. Some persons at- 

 tacked the certainty of its principles, 

 attempting to show that its conclusions 

 were at variance with those obtained by 

 methods previously known and accepted as 

 sound. Some who labored primarily with 

 matters of morality and religion attacked 

 the new departure of thought on general 

 grounds; they repulsed themselves by un- 

 wittingly displaying their ignorance of the 

 thing which they criticized. One man, 

 who entrenched himself in masses of calcu- 

 lation, pronounced the procedure of the 

 new calculus unsatisfactory because of the 

 indeterminacy of the form in which cer- 

 tain results appeared; but he afterwards 

 acknowledged his error and admitted that 

 he had been urged forward by malevolent 

 persons — a thing (let us believe) which 

 does not often happen among workers in 

 science. Christiaan Huygens, whose opin- 

 ion probably carried more weight than that 

 of any other scientific man in his day, be- 

 lieved that the employment of differentials 

 was unnecessary and declared that Leib- 



