August 19, 1897] 



NATURE 



2>77 



To emphasise this opinion that mathematicians would be 

 unwise to accept practical issues as the sole guide or the chief 

 guide in the current of their investigations, it may be sufficient 

 10 recall a few instances from history in which the purely 

 mathematical discovery preceded the practical application and 

 was not an elucidation or an explanation of observed phenomena. 

 The fundamental properties of conic sections were known to the 

 Greeks in the fourth and the third centuries before the Christian 

 era ; but they remained unused for a couple of thousand years 

 until Kepler and Newton found in thehi the solution of the 

 universe. Need I do more than mention the discovery of the 

 planet Neptune by Adams and Leverrier, in which the intricate 

 analysis used had not been elaborated for such particular 

 applications? Again, it was by the use of refined analytical 

 and geometrical reasoning upon the properties of the wave- 

 surface that Sir W. R. Hamilton inferred the existence of 

 conical refraction which, down to the time when he made his 

 inference, had been "unsupported by any facts observed, and 

 was even opposed to all the analogies derived from experience." 



It may be said that these are time-honoured illustrations, and 

 that objections are not entertained as regards the past, but 

 fears are entertained as regards the present and future. Very 

 well ; let me take one more instance, by choosing a subject in 

 which the purely mathematical interest is deemed supreme, the 

 theory of functions of a complex variable. That at least is a 

 theory in pure mathematics, initiated in that region and 

 developed in that region ; it is built up in scores of papers, and 

 its plan certainly has not been, and is not now, dominated or 

 guided by considerations of applicability to natural phenomena. 

 Vet what has turned out to be its relation to practical issues ? 

 The investigations of Lagrange and others upon the construction 

 of maps appear as a portion of the general property of conformal 

 representation ; which is merely the general geometrical method 

 of regarding functional relations in that theory. Again, the 

 interesting and important investigations upon discontinuous 

 two-dimensional fluid motion in hydrodynamics, made in the 

 last twenty years, can all be, and now are all, I believe, deduced 

 from similar considerations by interpreting functional relations 

 between complex variables. In the dynamics of a rotating 

 heavy body, the only substantial extension of our knowledge 

 made since the time of Lagrange has accrued from associating 

 the general properties of functions with the discussion of the 

 equations of motion. Further, under the title of conjugate 

 functions, the theory has been applied to various questions in 

 electrostatics, particularly in connection with condensers and 

 electrometers. And, lastly, in the domain of physical astronomy, 

 some of the most conspicuous advances made in the last few 

 years have been achieved by introducing into the discussion the 

 ideas, the principles, the methods, and the results of the theory 

 of functions. It is unnecessary to speak in detail of this last 

 matter, for I can refer you to Dr. G. W. Hill's interesting 

 " Presidential Address to the American Mathematical Society" 

 in 1895 > but without doubt the refined and extremely difficult 

 work of Poincare and others in physical astronomy has been 

 possible only by the use of the most elaborate developments of 

 some pure mathematical subjects, developments which were 

 made without a thought of such applications. 



Now it is true that much of the theory of functions is as yet 

 devoid of explicit application to definite physical subjects ; it 

 may be that these latest applications exhaust the possibilities in 

 that direction for any immediate future ; and it is also true that 

 whole regions of other theories remain similarly unapplied. 

 Opinion and divination as to the future would be as vain as 

 they are unnecessary ; but my contention does not need to be 

 supported by speculative hopes or uninformed prophecy. 



If in the range of human endeavour after sound knowledge 

 there is one subject that needs to be practical, it surely is 

 Medicine. Yet in the field of Medicine it has been found that 

 branches such as biology and pathology must be studied for 

 themselves and be developed by themselves with the single aim 

 of increasing knowledge ; and it is then that they can be best 

 applied to the conduct of living processes. So also in the 

 pursuit of mathematics, the path of practical utility is too 

 narro\y and irregular, not always leading far. The witness of 

 history shows that, in the field of natural philosophy, mathe- 

 matics will furnish more effective assistance if, in its systematic 

 development, its course can freely pass beyond the ever-shifting 

 domain of use and application. 



What I have said thus far has dealt with considerations 

 arising from the outside. I have tried to show that, in order 



NO. 145 I, VOL. 56] 



to secure tlie greatest benefit for those practical or pure sciences 

 which use mathematical results or methods, a deeper source 

 of possible advantage can be obtained by developing the subject 

 independently than by keeping the attention fixed chiefly upon 

 the applications that may be made. Even if no more were 

 said, it might be conceded that the unrestricted study of mathe- 

 matics would thereby be justified. But there is another side to 

 this discussion, and it is my wish now to speak very briefly 

 from the point of view of the subject itself, regarded as a branch 

 of knowledge worthy of attention in and for itself, steadily 

 growing and full of increasing vitality. Unless some account 

 be taken of this position, an adequate estimate of the subject 

 cannot be framed ; in fact, nearly the greater part of it will 

 thus be omitted from consideration. For it is not too much to 

 say that, while many of the most important developments have 

 not been brought into practical application, yet they are as 

 truly real contributions to human knowledge as are the dis- 

 interested developments of any other of the branches included 

 in the scope of pure science. 



It will readily be conceded for the present purpose that 

 knowledge is good in and by itself, and that the pursuit of pure 

 knowledge is aft occupation worthy of the greatest efforts which 

 the human intellect can make. A refusal to concede so much 

 would, in effect, be a condemnation of one of the cherished 

 ideals of our race. But the mere pursuit or the mere assiduous 

 accumulation of knowledge is not the chief object ; the chief 

 object is to possess it sifted and rationalised : in fact, organised 

 into truth. To .achieve this end, instruments are requisite that 

 may deal with the respective well-defined groups of knowledge, 

 and for one particular group, we use the various sciences. 

 There is no doubt that, in this sense, mathematics is a great 

 instrument ; there remains for consideration the decision as to 

 its range and function — are they such as- to constitute it an 

 independent science, or do they assign it a position in some 

 other science ? 



I do not know of any canonical aggregate of tests which a 

 subject should satisfy before it is entitled to a separate establish- 

 ment ; but, in the absence of a recognised aggregate, some 

 important tests can be assigned which are necessary, and may, 

 perhaps, be sufficient. A subject must be concerned with a 

 range of ideas forming a class distinct from all other classes ; it 

 must deal with them in such a way that new ideas of the same 

 kind can be associated and assimilated ; and it should derive a 

 growing vigour from a growing increase of its range. For its 

 progress, it must possess methods as varied as its range, 

 acquiring and constructing new processes in its growth ; and 

 new methods on any grand scale should supersede the older 

 ones, so that increase of ideas and introduction of new principles 

 should lead both to simplification and to increase of working 

 power within the subject. As a sign of its vitality, it must ever 

 be adding to knowledge and producing new results, even though 

 within its own range it propound some questions that have no 

 answer and other questions that for a time defy solution ; and 

 results already achieved should be an intrinsic stimulus to 

 further development in the extension of knowledge. Lastly, at 

 least among this list, let me quote Sylvester's words : " It must 

 unceasingly call forth the faculties of observation and com- 

 parison ; one of its principal methods must be induction ; it 

 must have frequent recourse to experimental trial and verifi- 

 cation, and it must afford a boundless scope for the highest 

 efforts of imagination and invention." I do not add as a test 

 that it must immediately be capable of practical application to 

 something outside its own range, though of course its processes 

 may be also transferable to other subjects, or, in part, derivable 

 from them. 



All these tests are satisfied by pure mathematics : it can be 

 claimed without hesitation or exaggeration that they are 

 satisfied with ample generosity. A complete proof of this 

 declaration would force me to trespass long upon your time, 

 and so I propose to illustrate it by references to only two or 

 three branches. 



First, I would refer to the general theory of invariants and 

 covariants. The fundamental object of that theory is the in- 

 vestigation and the classification of all dependent functions 

 which conserve their form unaltered in spite of certain general 

 transformations effected in the functions upon which they de- 

 pend. Originally it began as the observation of a mere 

 analytical property of a particular expression, interesting 

 enough in itself, but absolutely isolated. This then suggested 

 the inverse question : What is the general law of existence of 



