514 POPULAR SCIENCE MONTHLY. 



be ultimately very closely connected. Their common nexus is, perhaps, 

 to be traced in the physiological ideas of which Helmholtz was the 

 most conspicuous exponent. To many minds such discussions are re- 

 pellant, in that they seem to venture on the uncertain ground of phi- 

 losophy. But, as a matter of fact, the current views on these subjects 

 have been arrived at by men who have gone to work in their own way, 

 often in entire ignorance of what philosophers have thought on such 

 subjects. It may be maintained chiefly, indeed, that the mathemati- 

 cian or the physicist, as such, has no special concern with philosophy, 

 any more than the engineer or the geographer. Nor, although this is 

 a matter for their own judgment, would it appear that philosophers 

 have very much to gain by a special study of the methods of mathe- 

 matical or physical reasoning, since the problems with which they are 

 concerned are presented to them in a much less artificial form in the 

 circumstances of ordinary life. As regards the present topic I would 

 put the matter in this way, that between mathematics and physics on 

 the one hand and philosophy on the other there lies an undefined bor- 

 derland, and that the mathematician has been engaged in setting things 

 in order, as he is entitled to do, on his own side of the boundary. 



Adopting tins point of view, it would be of interest to trace in 

 detail the relationships of the three currents of speculation which have 

 been referred to. At one time, indeed, I was tempted to take this as 

 the subject of my address; but, although I still think the enterprise a 

 possible one, I have been forced to recognize that it demands a better 

 equipment than I can pretend to. I can only venture to put before 

 you some of my tangled thoughts on the matter, trusting that some 

 future occupant of this chair may be induced to take up this question 

 and treat it in a more illuminating manner. 



If we look back for a moment to the views currently entertained not 

 so very long ago by mathematicians and physicists, we shall find, I 

 think, that the prevalent conception of the world was that it was con- 

 structed on some sort of absolute geometrical plan, and that the changes 

 in it proceeded according to precise laws; that, although the principles 

 of mechanics might be imperfectly stated in our text-books, at all 

 events such principles existed, and were ascertainable, and, when prop- 

 erly formulated, would possess the definiteness and precision which 

 were held to characterize, say, the postulates of Euclid. Some writers 

 have maintained, indeed, that the principles in question were finally 

 laid down by Newton, and have occasionally used language which sug- 

 gests that any fuller understanding of them was a mere matter of inter- 

 pretation of the text. But, as Hertz has remarked, most of the great 

 writers on dynamics betray, involuntarily, a certain malaise when ex- 

 plaining the principles, and hurry over this part of their task as quickly 

 as is consistent with dignity. They are not really at their ease until, 



