INAUGUEAL ADDRESS BEFORE TEE BRITISH ASSOCIATION. 513 



Hence, the framework in which all the varied 

 manifestations of Nature have been set, that our 

 science has laid claim to be the arbiter of all 

 knowledge. She does not indeed contribute ele- 

 ments of fact, which must be sought elsewhere ; 

 but she sifts and regulates them ; she proclaims 

 the laws to which they must conform if those ele- 

 ments are to issue in precise results. From the 

 data of a problem she can infallibly extract all 

 possible consequences, whether they be those 

 first sought, or others not anticipated ; but she 

 can introduce nothing which was not latent in 

 the original statement. Mathematics cannot tell 

 us whether there be or be not limits to time or 

 space ; but to her they are both of indefinite ex- 

 tent, and this in a sense which neither affirms nor 

 denies that they are either infinite or finite. Mathe- 

 matics cannot tell us whether matter be continu- 

 ous or discrete in its structure ; but to her it is 

 indifferent whether it be one or the other, and 

 her conclusions are independent of either par- 

 ticular hypothesis. Mathematics can tell us 

 nothing of the origin of matter, of its creation 

 or its annihilation ; she deals only with it in a 

 state of existence ; but, within that state, its 

 modes of existence may vary from our most ele- 

 mentary conception to our most complex experi- 

 ence. Mathematics can tell us nothing beyond 

 the problems which she specifically undertakes ; 

 she will carry them to their limit, but there she 

 stops, and upon the great region beyond she is 

 imperturbably silent. 



Conterminous with space and coeval with time 

 is the kingdom of Mathematics ; within this range 

 her dominion is supreme ; otherwise than accord- 

 ing to her order nothing can exist ; in contradic- 

 tion to her laws nothing takes place. On her 

 mysterious scroll is to be found written for those 

 who can read it that which has been, that which 

 is, and that which is to come. Everything ma- 

 terial which is the subject of knowledge has 

 number, order, or position ; and these are her 

 first outlines for a sketch of the universe. If our 

 more feeble hands cannot follow out the details, 

 still her part has been drawn with an unerring 

 pen, and her work cannot be gainsaid. So wide 

 is the range of mathematical science, so indefi- 

 nitely may it extend beyond our actual powers 

 of manipulation, that at some moments we are in- 

 clined to fall down with even more than reverence 

 before her majestic presence. But so strictly lim- 

 ited are her promises and powers, about so much 

 that we might wish to know does she offer no in- 

 formation whatever, that at other moments we are 

 fain to call her results but a vain thing, and to 

 105 



reject them as a stone when we had asked for 

 bread. If one aspect of the subject encourages 

 our hopes, so does the other tend to chasten our 

 desires ; and he is perhaps the wisest, and in the 

 long-run the happiest among his fellows, who has 

 learned not only mathematics, but also the larger 

 lesson which they indirectly teach, namely, to 

 temper our aspirations to that which is possible, 

 to moderate our desires to that which is attain- 

 able, to restrict our hopes to that of which ac- 

 complishment, if not immediately practicable, is 

 at least distinctly within the range of conception. 

 That which is at present beyond our ken may, at 

 some period and in some manner as yet unknown 

 to us, fall within our grasp ; but our science 

 teaches us, while ever yearning with Goethe for 

 " Light, more light," to concentrate our attention 

 upon that of which our powers are capable, and 

 contentedly to leave for future experience the 

 solution of problems to which we can at present 

 say neither yea nor nay. 



It is within the region thus indicated that 

 knowledge in the true sense of the word is to be 

 sought. Other modes of influence there are in 

 society and in individual life, other forms of en- 

 ergy besides that of intellect. There is the po- 

 tential energy of sympathy, the actual energy 

 of work ; there are the vicissitudes of life, the di- 

 versity of circumstance, health, and disease, and 

 all the perplexing issues, whether for good or for 

 evil, of impulse and of passion. But although 

 the book of life cannot at present be read by 

 the light of science alone, nor the wayfarers be 

 satisfied by the few loaves of knowledge now in 

 our hands, yet it would be difficult to overstate 

 the almost miraculous increase which may be 

 produced by a liberal distribution of what we 

 already have, and by a restriction of our cravings 

 within the limits of possibility. 



In proportion as method is better than im- 

 pulse, deliberate purpose than erratic action, the 

 clear glow of sunshine than irregular reflection, 

 and definite utterances than an uncertain sound ; 

 in proportion as knowledge is better than sur- 

 mise, proof than opinion — in that proportion will 

 the mathematician value a discrimination between 

 the certain and the uncertain, and a just esti- 

 mate of the issues which depend upon one motive 

 power or the other. While on the one hand he 

 accords to his neighbors full liberty to regard 

 the unknown in whatever way they are led by 

 the noblest powers that they possess, so on the 

 other he claims an equal right to draw a clear 

 line of demarkation between that which is a mat- 

 ter of knowledge, and that which is at all events 



