Si8 POPULAR SCIENCE MONTHLY. 



arisen in connection with the applications of geometry to purposes of 

 graphical representation. It is not necessary to dwell on the great 

 assistance which this method has rendered in such subjects as physics 

 and engineering. The pure mathematician, for his part, will freely 

 testify to the influence which it has exercised in the development of 

 most branches of analysis; for example, we owe to it all the leading 

 ideas of the calculus. Modern analysts have discovered, however, that 

 geometry may be a snare as well as a guide. In the mere act of draw- 

 ing a curve to represent an analytical function we make unconsciously 

 a host of assumptions which are difficult not merely to prove, but even 

 to formulate precisely. It is now sought to establish the whole fabric 

 of mathematical analysis on a strictly arithmetical basis. To those 

 who were trained in an earlier school, the results so far are in appear- 

 ance somewhat forbidding. If the shade of one of the great analysts 

 of a century ago could revisit the glimpses of the moon, his feelings 

 would, I think, be akin to those of the traveler to some medieval town, 

 who finds the buildings he came to see obscured by scaffolding, and is 

 told that the ancient monuments are all in process of repair. It is to 

 be hoped that a good deal of this obstruction is only temporary, that 

 most of the scaffolding will eventually be cleared away, and that the 

 edifices when they reappear will not be entirely transformed, but will 

 still retain something of their historic outlines. It would be contrary 

 to the spirit of this address to undervalue in any way the critical ex- 

 amination and revision of principles; we must acknowledge that it 

 tends ultimately to simplification, to the clearing up of issues, and the 

 reconciliation of apparent contradictions. But it would be a misfor- 

 tune if this process were to absorb too large a share of the attention of 

 mathematicians, or were allowed to set too high a standard of logical 

 completeness. In this particular matter of the { arithmetization of 

 mathematics ' there is, I think, a danger in these respects. As regards 

 the latter point, a traveler who refuses to pass over a bridge until he 

 has personally tested the soundness of every part of it is not likely to 

 go very far; something must be risked, even in mathematics. It is 

 notorious that even in this realm of ' exact ' thought discovery has often 

 been in advance of strict logic, as in the theory of imaginaries, for ex- 

 ample, and in the whole province of analysis of which Fourier's theo- 

 rem is the type. And it might even be claimed that the services which 

 geometry has rendered to other sciences have been almost as great in 

 virtue of the questions which it implicitly begs as of those which it 

 resolves. 



I would venture, with some trepidation, to go one step further. 

 Mathematicians love to build on as definite a foundation as possible, 

 and from this point of view the notion of the integral number, on 

 which (we are told) the mathematics of the future are to be based, is 



