512 TRANSACTIONS OF SECTION A. 



history of the actual movements of mathematical thought through the centuries 

 shows that these ideals are the very life-blood of the science, and warrants the 

 conclusion that a constant striving towards their attainment is an absolutely 

 essential condition of vigorous growth. These ideals have their roots in irresistible 

 impulses and deep-seated needs of the human mind, manifested in its efforts to 

 introduce intelligibility into certain great domains of the world of thought. 



There exists a widespread impression amongst physicists, engineers, and ether 

 men of science that the effect of recent developments of Pure Mathematics, by 

 making it more abstract than formerly, has been to remove it further from the 

 order of ideas of those who are primarily concerned with the physical world. 

 The prejudice that Pure Mathematics has its sole raison d'etre in its function of 

 providing useful tools for application in the physical sciences, a prejudice which 

 did much to retard the due development of Pure Mathematics in this country 

 during the nineteenth century, is by no means extinct. It is not infrequently 

 said that the present devotion of many mathematicians to the interminable dis- 

 cussion of purely abstract questions relating to modern developments of the 

 notions of number and function, and to theories of algebraic form, serves only 

 the purpose of deflecting them from their proper work into paths which lead 

 nowhere. It is considered that mathematicians are apt to occupy themselves too 

 exclusively with ideas too remote from the physical order in which Mathematics 

 had its origin and in which it should still find its proper applications. A direct 

 answer to the question cui bono? when it is raised in respect of a department of 

 study such as Pure Mathematics, seldom carries conviction, in default of a 

 standard of values common to those who ask and to those who answer the 

 question. To appreciate the importance of a sphere of mental activity different 

 from our own always requires some effort of the sympathetic imagination, some 

 recognition of the fact that the absolute value of interests and ideals of a par- 

 ticular class may be much greater than the value which our own mentality inclines 

 us to attach to them. If a defence is needed of the expenditure of time and 

 energy on the abstract problems of Pure Mathematics, that defence must be of 

 a cumulative character. The fact that abstract mathematical thinking is one 

 of the normal forms of activity of the human mind, a fact which the general 

 history of thought fully establishes, will appeal to some minds as a ground of 

 decisive weight. A great department of thought must have its own inner life, 

 however transcendent may be the importance of its relations to the outside. No 

 department of science, least of all one requiring so high a degree of mental con- 

 centration as Mathematics, can be developed entirely, or even mainly, with a 

 view to applications outside its own range. The increased complexity and 

 specialisation of all branches of knowledge makes it true in the present, however 

 it may have been in former times, that important advances in such a department 

 as Mathematics can be expected only from men who are interested in the subject 

 for its own sake, and who, whilst keeping an open mind for suggestions from 

 outside, allow their thought to range freely in those lines of advance which are 

 indicated by the present state of their subject, untrammelled by any preoccupa- 

 tion as to applications to other departments of science. Even with a view to 

 applications, if Mathematics is to be adequately equipped for the purpose of 

 coping with the intricate problems which will be presented to it in the future by 

 Physics, Chemistry, and other branches of physical science, many of these 

 problems probably of a character which we cannot at present forecast, it is 

 essential that Mathematics should be allowed to develop itself freely on its own 

 lines. Even if much of our present mathematical theorising turns out to be use- 

 less for external purposes, it is wiser, for a well-known reason, to allow the 

 wheat and the tares to grow together. It would be easy to establish in detail 

 that many of the applications which have been actually made of Mathematics 

 were wholly unforeseen by those who first developed the methods and ideas on 

 which they rest. Recently, the more refined mathematical methods which have 

 been applied to gravitational Astronomy by Delaunay, G. W. Hill, Poincare, 

 E. W. Brown, and others, have thrown much light on questions relating to the 

 solar system, and have much increased the accuracy of cur knowledge of the 

 motions of the moon and the planets. Who knows what weapons forged by the 

 theories of functions, of differential equations, or of groups, may be required 

 when the time comes for such an empirical law as Mendeleeff's periodic law of 



