September 12, 1913] 



SCIENCE 



353 



vast continent still largely undeveloped, 

 and there is the incidental problem of the 

 possibilities of geometry of position in any 

 number of dimensions, so important in so 

 many ways. But, in fact, a large propor- 

 tion of the more familiar general prin- 

 ciples, taught to-day as theory of functions, 

 have been elaborated under the stimulus of 

 the foregoing theory. Besides this, how- 

 ever, all that precision of logical statement 

 of which I spoke at the beginning is of par- 

 amount necessity here. What exactly is 

 meant by a curve of integration, what char- 

 acter can the limiting points of a region of 

 existence of a function possess, how even 

 best to define a function of a complex vari- 

 able, these are but some obvious cases of 

 difficulties which are very real and press- 

 ing to-day. And then there are the prob- 

 lems of the theory of differential equations. 

 About these I am at a loss what to say. 

 "We give a name to the subject, as if it were 

 one subject, and I deal with it in the fewest 

 words. But our whole physical outlook is 

 based on the belief that the problems of 

 nature are expressible by differential equa- 

 tions ; and our knowledge of even the possi- 

 bilities of the solutions of differential equa- 

 tions consists largely, save for some special 

 types, of that kind of ignorance which, in 

 the nature of the case, can form no idea of 

 its own extent. There are subjects whose 

 whole content is an excuse for a desired 

 solution of a differential equation; there 

 are infinitely laborious methods of arith- 

 metical computation held in high repute of 

 which the same must be said. And yet I 

 stand here to-day to plead with you for 

 tolerance of those who feel that the prose- 

 cution of the theoretic studies, which alone 

 can alter this, is a justifiable aim in life! 

 Our hope and belief is that over this vast 

 domain of differential equations the theory 

 of functions shall one day rule, as already 



it largely does, for example, over linear 

 differential equations. 



THEORY OF NUMBERS 



In concluding this table of contents, I 

 would also refer, with becoming brevity, to 

 the modern developments of theory of 

 numbers. Wonderful is the fascination 

 and the difficulty of these familiar objects 

 of thought — ordinary numbers. We know 

 how the great Gauss, whose lynx eye was 

 laboriously turned upon all the physical 

 science of his time, has left it on record 

 that in order to settle the law of a plus or 

 minus sign in one of the formulae of his 

 theory of numbers he took up the pen 

 every week for four years. In these islands 

 perhaps our imperial necessities forbid the 

 hope of much development of such a the- 

 oretical subject. But in the land of Kum- 

 mer and Gauss and Dirichlet the subject 

 to-day claims the allegiance of many eager 

 minds. And we can reflect that one of the 

 latest triumphs has been with a problem 

 known by the name of our English senior 

 wrangler. Waring — the problem of the 

 representation of a number by sums of 

 powers. 



Ladies and gentlemen, I have touched 

 only a few of the matters with which pure 

 mathematics is concerned. Each of those 

 I have named is large enough for one 

 man's thought; but they are interwoven 

 and interlaced in indissoluble fashion and 

 form one mighty whole, so that to be ig- 

 norant of one is to be weaker in all. I am 

 not concerned to depreciate other pursuits, 

 which seem at first sight more practical ; I 

 wish only, indeed, as we all do, it were pos- 

 sible for one man to cover the whole field 

 of scientific research; and I vigorously re- 

 sent the suggestion that those who follow 

 these studies are less careful than others of 

 the urgent needs of our national life. But 



