TRANSACTIONS OF SECTION A. 547 



outside. I have tried to show that, in order to secure the 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 mathematics 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 brouglit into practical application, yet they 

 are as truly real contributions to human knowledge as are the disinterested 

 •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 an occupation worthy of 

 the greatest efibrts 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 inde- 

 pendent 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 establishment as a science ; 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 form- 

 ing 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 restilts, 

 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 know- 

 ledge. Lastly, at least among this list, let me quote Sylvester's words: 'It must 

 unceasingly call forth the faculties of observation and comparison ; one of its 

 principal methods must be induction ; it must have frequent recourse to experi- 

 mental trial and verification, and it must aflbrd 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 co-variants. The 

 fundamental object of that theory is the investigation 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 depend. Originally it 

 began as the observation of a mere analytical property of a particular expression, 



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