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NATURE 



[August 19, 1897 



Now t is undoubtedly an advantage in any case that labour 

 should be saved and time economised ; and where this can be 

 done, either by mears 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 develop- 

 ments in the training for a profession, for example, that of an 

 engineer, and the demands that arise in the practice of the 

 profession, are such as to force gradually a complete change of 

 view. When I look into the text-books that he uses, it seems 

 to me a necessity that an engineer should now possess a mathe- 

 matical skill and knowledge in some directions which, not so 

 very long since, could not freely be found among the profes- 

 sional mathematicians themselves. And as this change is 

 gradually effected, perhaps the practical man will gradually 

 change his estimate of the scope of mathematical science. 



I pass from the practical man to some of the natural philo- 

 sophers. Many of them, though certainly far from all of them, 

 expound what they consider proper and economical limits to 

 the development of pure mathematics. Their wisdom gives 

 varied reasons ; it speaks in tones of varied appreciation ; but 

 there can be no doubt as to its significance and its meaning. 

 Their aim is to make pure mathematics, not indeed the drudge, 

 but the handmaid of the sciences. The demand requires ex- 

 amination, and deserves respectful consideration. There is no 

 question of giving or withholding help in furthering, in every 

 possible fashion and with every possible facility, the progress of 

 natural philosophy ; there is no room for difference upon that 

 matter. The difference arises when the opinion is expressed 

 or the advice is tendered that the activity of mathematicians and 

 all their investigations should be consciously limited, and directed 

 solely and supremely, to the assistance and the furtherance of 

 natural philosophy. 



One group of physicists, adopting a distinctly aggressive 

 attitude in imposing limits so as to secure prudence in the pur- 

 suit of pure mathematics, regard the subject as useful solely for 

 arriving at results connected with one or other of the branches 

 of natural philosophy ; they entertain an honest dislike, not 

 merely to investigations that do not lead to such results, but to 

 the desirability of carrying out such investigations ; and some of 

 them have used highly flavoured rhetoric in expressing their 

 dislike. It would be easy — but unconvincing— to suggest that, 

 with due modifications in statement, they might find themselves 

 faced with the necessity of defending some of their own re- 

 searches against attacks as honestly delivered by men absorbed 

 in purely practical work. But such a suggestion is no reply, for 

 it does not in the least touch the question at issue ; and I prefer 

 to meet their contention with a direct negative. 



By way of illustration let me take a special instance : it is not 

 selected as being easier to confute than any other, but because 

 it was put in the forefront by one of the vigorous advocates of 

 the contention under discussion — a man of the highest scientific 

 distinction in his day. He wrote : "Measured [by the utility 

 of the power they give] partial differential equations are very 

 useful, and therefore stand very high [in the range of pure 

 mathematics] as far as the second order. They apply to that 

 point in the most important way to the great problems of nature, 

 and are worthy of the most careful study. Beyond that order 

 they apply to nothing." This last statement, it may be re- 

 marked, is inaccurate ; for partial differential equations, of an 

 order higher than the second, occur— to give merely a few ex- 

 amples — in investigations as to the action of magnetism on 

 polarised light, in researches on the vibrations of thick plates 

 or of curved bars, in the discussion of such hydrodynamical 

 questions as the motion of a cylinder in fluid or the damping of 

 air-waves owing to viscosity. 



Putting this aside, what is more important is the considera- 

 tion of the partial differential equations of the second order that 

 are found actually to occur in the investigations. Each case as 

 it arises is discussed solely in connection with its particular 

 problem ; one or two methods are given, more or less in the 

 form of rules ; if these methods fail, the attempt at solution 

 subsides. The result is a collection of isolated processes, about 

 as unsatisfactory a collection as is the chapter labelled Theory of 



NO. 145 1, VOL. 56] 



Numbers in many text-books on algebra, when it is supposed to 

 represent that great branch of knowledge. Moreover, this 

 method suffers from the additional disadvantage of suggesting 

 little or no information about equations of higher orders. 



But when the equations are considered, not each by itself but 

 as ranged under a whole system, then the investigation of the 

 full theory places these processes in their proper position, gives 

 them a meaning which superficially they do not exhibit, and 

 indicates the way in which each solution satisfies the general 

 conditions of existence of a solution. For the full theory of 

 partial differential equations of the second order in, say, two 

 independent variables establishes the conditions of existence of 

 a solution, the limitations upon the conditions which make that 

 solution unique, the range of variation within which that solu- 

 tion exists, the modes of obtaining expressions for it when it 

 can be expressed in a finite form, and an expression for the 

 solution when it cannot be expressed in a finite form. Of 

 course, the actual derivation of the solution of particular 

 equations is dependent upon analytical skill, as is always the 

 case in any piece of calculating work ; but the general theory 

 indicates the possibilities and the limitations which determine 

 the kind of solution to be expected. But not only does the 

 general theory effect much by way of coordinating isolated 

 processes — and, in doing so, lead to new results — but it gives 

 important indications for dealing with equations of higher 

 orders, and it establishes certain theorems about them merely 

 by simple generalisations. 



In fact, the special case quoted is one more instance, added 

 to the many instances that have occurred in the past, in which 

 the utilitarian bias in the progress of knowledge is neither the 

 best stimulus nor in the long run the most effective guide 

 towards securing results. It may be— it frequently is — at first 

 the only guide possible, and for a time it continues the best 

 guide, but it does not remain so for ever. It would be super- 

 fluous, after Cayley's address in 1883, to show how branches of 

 mathematical physics, thus begun and developed, have added 

 to knowledge in their own direction ; they have suggested, they 

 have even created, most fascinating branches of pure mathe- 

 matics, which, when developed, have sometimes proved of 

 reciprocal advantage to the source from which they sprang. 

 But for proper and useful development they must be free from 

 the restrictions which the sterner group of natural philosophers 

 would lay upon them. 



Now I come to another group of natural philosophers who 

 will unreservedly grant my contention thus far ; who will yield 

 a ready interest to our aims and our ideas, but who consider 

 that the possibility of applying our results in the domain of 

 physical science should regulate, or at least guide, advance in 

 our work. Some of these entertain this view because they 

 think that possibility of early application is, in the last resource, 

 the real test of useful development ; some, because they fear 

 that the profusion of papers annually published and the be- 

 wildering specialisation in each branch, are without purpose, 

 and may ultimately lead to isolation or separation of whole 

 sections of mathematics from the general progress of science. 



The danger arising from excess of activity seems to me un- 

 real ; at any rate there are not signs of it at home at the present 

 day, and I would gladly see more workers at pure mathematics, 

 though not of course at the expense of attention paid to any 

 other branch. But for results that are trivial, for investigations 

 that have no place in organic growth and development, or in 

 illustration and elucidation, surely the natural end is that they 

 soon subside into mere tricks of " curious pleasure or ingenious 

 pain." However numerous they may be, they do not possess 

 intrinsic influence sufficient to cause evil consequences, and any 

 attempt at repression will, if successful, inevitably and unwisely 

 repress much more. 



More attention must be paid to the suggestion that mathe- 

 maticians should be guided in their investigations by the 

 possibility of practical issues. That they are so guided to a 

 great extent is manifest from many of the papers written in that 

 spirit ; that they cannot accept practical issues as the sole guide 

 would seem sufficiently justified by the consideration that 

 practical issues widen from year to year and cannot be foreseen 

 in the absence of a divining spirit. Moreover, if such a principle 

 were adopted, many an investigation undertaken at the time for 

 its intrinsic interest would be cast on one side unconsidered, 

 because it does not satisfy an external test that really has 

 nothing to do with the case, and may change its form of 

 application from time to time. 



