TKAXSACTIONS OF SECTION A. 643 



obtained, then; are fuw tliat appeal to an unprofessional sympathy. Adams could 

 discover a new planet by subjecting observations made of the known planets to the 

 most profound calculations; and the world, not over curious about the process, 

 could appreciate the significant result. But such instances are rare ; for the 

 most part, our particular results must remain somewhat intangible, somewhat 

 incomprehensible, to those who dwell resolutely and completely outside the range 

 of mathematical Imowledge. 



What then am I to do ? It would be pleasant to me, though it might not 

 prove satisfying to you, to discourse of the present state of one branch or of several 

 branches of mathematics, and particularly to indicate what seem to be lines of possi- 

 ble and probable growth in the future. Instead of pursuing this course, I shall keep 

 my remarks of a general character as far as possible, and shall attempt, not merely 

 to describe briefly some of the relations of pure mathematics to other branches of 

 science, but also to make a bold claim that the unrestricted cultivation of pure 

 mathematics is desirable iu itself and for its own sake. Some — I should like to 

 believe many — who pre here will concede this claim to the fullest extent and 

 without reservation; but I doubt whether this is so in general. And yet the 

 claim is one which needs to be made before an English-speaking audience. For it 

 is a curious fact that, although the United Kingdom has possessed some of the very 

 greatest of pure mathematicians in the second half of this century, the subject has 

 there received but a scant share of attention as compared with that which it has 

 found in France, in Germany, in Italy, in Sweden and jNorway, or in the United 

 States. I am not oblivious of the magnificent contributions to other parts of our 

 science made alike by British leaders and British followers ; their fame is known to 

 the world. But apathy rather than attention has been the characteristic feature of 

 our attitude towards pure mathematics ; and it seems to me a misfortune, alike for the 

 intellectual activity of the nation and for the progress of the subject, that English 

 thought has had relatively so small an influence upon its vast modern developments. 



Now it is not enough for my purpose to be told that the British Association 

 includes all science in its scope, and consequently includes pure mathematics. A 

 statement thus made might be framed in a spirit of mere sufferance ; what I wish 

 to secure is a recognition of the subject as one which, being full of life and over- 

 flowing with a power of growth, is worthy of the most absorbing devotion. Tlie 

 most cursory examination of the opinions of scientific men leads at once to the 

 conclusion, that there are two views of the subject, both accurate so far as 

 they go, both inadequate whether alone or combined, which to some extent 

 explain if they do not justify what may be called the English attitude in the past. 

 Let me deal with these in succession. 



One of these estimates has been framed by what is called the practical man ; 

 he regards the subject as a machine which is to provide him with tables, as far 

 as tables can be calculated ; and with the simplest formulse and the most direct 

 rules, whenever tables cannot be calculated. Kesults, not methods, are his want ; 

 it is sufficient for him that an authoritative statement as to a result shall be made; 

 all else is ignored. And for what is beyond, in the shape of work that does 

 nothing to meet his special wants, or of the processes that have led to the results 

 he uses, he cares little or nothing. In fact, he would regard mathematics as a 

 collection of formulae and an aggregate of processes to grind out numerical results ; 

 whatever else there is in it, may be vain and is useless. In his view, it is to be 

 the drudge of the practical sciences. 



Now it is undoubtedly an advantage in any case that labour should be saved 

 and time economised ; and where this can be done, either bj' means of calculations 

 made once for all, or by processes that lead to results admitting simple formulation, 

 any mathematician will be glad, particularly if his own work should lead to some 

 such issue. But he should not be expected to consider that his science 

 has thus fulfilled its highest purpose; and perhaps he is not unreasonable if, when 

 he says that such results are but a very small part, and not the most interesting 

 part, of his science, he should claim a higher regard for the whole of it. Indeed, 

 I rather suspect that some change is coming ; the practical man himself is changing. 

 The developments in the training for a profession, for example, that of an 



