18 SECTIONAL ADDRESSES. 



times of Minkowski and Einstein, are all absolutely and equally real. 

 When a mathematician has constructed, or, to be more accurate, when 

 he has observed them, his ^professional interest ii:i the matter ends. 

 It may be the seven-point geometry that fits the facts the best, for 

 anything that mathematicians have to say. There may be three dimen- 

 sions in this room and five next door. As a professional mathematician, 

 I have no idea ; I can only ask the Secretary, or some other competent 

 physicist, to instruct me in the facts. 



The function of a mathematician, then, is simply to observe the 

 facts about his own hard and intricate system of reality, that astonish- 

 ingly beautiful complex of logical relations which forms the subject- 

 matter of his science, as if he wei'e an explorer looking at a distant range 

 of mountains, and to record the results of his observations in a series 

 of maps, each of which is a branch of pure mathematics. Many of 

 these maps have been completed, while in others, and these, naturally, 

 the most interesting, there are vast uncharted regions. Some, it seems, 

 have some relevance to the structure of the physical world, while others 

 have no such tangible application. Among them there is perhaps none 

 quite so fascinating, with quite the same astonishing contrasts of sharp 

 outline and mysterious shade, as that which constitutes the theory of 

 numbers. 



The number system of arithmetic is, as we know too well, not with- 

 out its applications to the sensible world. The currency systems of 

 Europe, for example, conform to it approximately; west of the Vistula, 

 two and two make something approaching four. The practical appli- 

 cations of arithmetic, however, are tedious beyond words. One must 

 probe a little deeper into the subject if one wishes to interest the ordinary 

 man, whose taste in such mxatters is astonishingly correct, and who 

 turns with joy from the routine of common life to anything strange 

 and odd, like the fourth dimension, or imaginary time, or the theory 

 of the representation of integers by sums of squares or cubes. 



It is impossible for me to give you, in the time at my command, any 

 general account of the problems of the theory of numbers, or of the 

 progress that has been made towards their solution even during the last 

 twenty years. I must adopt a much simpler method. I will merely 

 state to you, with a few words of comment, three or four isolated 

 questions, selected in a quite haphazard way. They are seemingly 

 simple questions, and it is not necessary to be anything of a mathe- 

 matician to understand them ; and I have chosen them for no better 

 reason than that; I happen to be interested in them myself. There is 

 no one of them to which I know the answer, nor, so far as I know, does 

 any mathematician in the world; and there is no one of them, with one 

 exception which I have included deliberately, the answer to^ which any 

 one of us would not make almost any sacrifice to know. 



1. When is a number the sum of tivo cubes, and whdt is the 

 number of ils representations? This is my first question, and first 

 of all I will elucidate it by some examples. The numbers 2 = 1^-|-1^ 

 and 9 = 2^-f 1^ are sums of two' cubes, while 3 and 4 are not: it is 

 exceptional for a number tO' be ol this particular form. The number 

 of cubes up to 1000000 is 100, and the number of numbers, up to this 

 limit and of the form required, cannot exceed 10000, one-hundredth of 



