8o 



NATURE 



{Nov. 23, 1876 



difficult this problem may prove is, perhaps, best attested 

 by the little advance that has been made towards its so- 

 lution." As an example, the researches of Cayley, 

 Bachmann, and Hermite on the algebraical problem of 

 the automorphies of a quadratic form, containing any 

 number of indeterminates, were alluded to. Omitting 

 many other points which were brought out, we can only 

 mention the second department of arithmetic, the theory 

 of congruences. In connection with this division, Prof. 

 Smith also dwelt in detail upon the subject of complex 

 numbers. "The last part of arithmetical theory to 

 which I would wish to direct the attention of some 

 of the younger mathematicians of this country is the 

 determination of the mean values, or the asymptotic 

 values of arithmetical functions. This is a field of 

 inquiry which presents enormous difficulties of its own ; 

 it is certainly one in which the investigator will not 

 find himself incommoded or crowded out by the number 

 of his fellow- workers. ' Nemo est fere mathematicorum,' 

 said Euler, in the last century ; ' qui non magnam 

 tempcris partem inutiliter consumpserit in investigatione 

 numerorum primorum ;' but I do not think that (as a 

 rule) the mathematicians of the present day have any 

 reason to reproach themselves on this score." The 

 speaker then pointed out what had been done in this 

 direction since the days of Euler. " I do not know that 

 the great achievements of such men as Tchebychef and 

 Riemann can fairly be cited to encourage other and less 

 highly gifted inquirers, but at least they may serve to 

 show two things — first, that nature has fixed no im- 

 penetrable barrier to the further advancement of mathe- 

 matical science in this direction ; and secondly, that the 

 boundary of our present knowledge lies so near us that at 

 any rate the inquirer has no very long journey to take 

 before he finds himself in the unknown land. It is this 

 peculiarity, perhaps, which gives such perpetual freshness 

 to the higher arithmetic. It is one of the oldest branches 

 — perhaps the very oldest branch — of human knowled.<;e, 

 but yet iis tritest truths lie close to some of its most 

 abstruse secrets. I do not know that any more striking 

 example of this could be furnished than by the theorem 

 of M. Tchebychef. To understand his demonstration 

 requires only such algebra and arithmetic as are at the 

 command of many a schoolboy ; and the method itself 

 might have been invented by a schoolboy with the genius 

 of Pascal or of M. Tchebychef." 



Passing on to other branches of analysis, Jacobi's me- 

 thod of approximation (" a natural extension of the 

 theory of continued fractions "), Lejeune Dirichlet's 

 researches on complex units and his great generalisation 

 of the theory of the Pellian equation, Liouville's treat- 

 ment of irrational quantities, Lambert's proofs that neither 

 TT nor TT^ nor e are rational with M. Hermite's extensions, 

 who, though he has proved that e is a transcendental 

 irrational, declines entering on a similar investigation for 

 the number ir, but leaves this to others, adding, " Nul ne 

 sera plus heureux que moi de leur succ^s, mai?, croyez 

 m'en, il ne laissera pas que de leur en couter quelques 

 efforts " — all came in for a notice. 



Another class of questions mentioned were those 

 which relate to the transcendental or algebraic cha- 

 racter of developments in the form of infinite series, 

 products, or continued fractions. The theorem of Eisen- 

 stein and M. Hermite's recent investigation of it, lately 

 communicated to the Society, "are amply sufficient to 

 awaken the expectation of great future discoveries in this 

 almost unexplored field of inquiry." 



Amongst important objects for mathematicians to set 

 before them were named the advancement of the integral 

 calculus (' confessedly all important in the applications of 

 mathematics to physics "). In this connection the theory 

 of differential equations and of singular solutions came in 

 for a detailed notice, as al^o did the subject of elliptic 

 functions. 



Towards the close of the address, Prof. Smith said : 

 " I am convinced that nothing so hinders the progress of 

 mathematical science in England as the want of advanced 

 treatises on mathematical subjects. We yield the palm 

 to no European nation for the number and excellence of 

 our text-books of the second grade ; I mean of such 

 text-books as are intended to guide the student as far as 

 the requirements of our University examinations in 

 honours are concerned. But we want works suitable for 

 the requirements of the student when his examinations 

 are over — works which will carry him to the frontiers of 

 knowledge in certain directions, which will direct him to 

 the problems which he ought to select as the objects of 

 his own researches, and which will free his mind from the 

 narrow views which he is apt to contract while getting up 

 work with a view to passing an examination, or, a little 

 later in his life, in preparing others for examination. Can 

 we doubt that much of the preference for geometrical and 

 algebraical speculation which we notice among our 

 younger mathematicians is due to the admirable works 

 of Dr. Salmon ; and can we also doubt that if other parts 

 of mathematical science had been equally fortunate in 

 finding an expositor, we should observe a wider interest 

 in, and a juster appreciation of, the progress which has 

 been achieved ? 



There are, of course, other works besides those of Prof. 

 Cayley and Dr. Salmon to which I might refer ; there is, 

 for example, the work of Boole, on Differential Equa- 

 tions ; and there are the great historical treatises of Mr. 

 Todhunter so suggestive of research, and so full of its 

 spirit ; we have also a recent work by the same author 

 on the functions of Laplace, Lamd. and Bessel. But the 

 field is not nearly covered . . . There are at least three 

 treatises which we sadly need, one on definite integrals, 

 one on the theory of functions in the sense in which that 

 phrase is understood by the school of Cauchy and of 

 Riemann, and one (though he should be a bold man 

 who would undertake the task) on the hyperelliptic and 

 Abelian integrals. 



Geometry, and some other subjects, were hardly more 

 than mentioned. 



" Varum haec ipse equidem spatiis exclusus iniquis 

 Prastereo, atque aliis post memoranda relinquo." 



" In these days, when so much is said of original re- 

 search, and of the advancement of scientific knowledge, I 

 feel that it is the business of our Society to see that, so far 

 as our own country is concerned, mathematical science 

 should still be in the vanguard of progress. I should not 

 wish to use words which may seem to reach too far, but I 

 often find the conviction forced upon me that the increase 

 of mathematical knowledge is a necessary condition for 

 the advancement of science, and, if so, a no less necessary 

 condition for the improvement of mankind. I should 

 tremble for the intellectual strength of any nation of men 

 whose education was not based on a solid foundation of 

 mathematical learning, and whose scientific conceptions, 

 or in other words, whose notions of the world and of the 

 things in it, were not braced and girt together with a 

 strong framework of mathematical reasoning. It is 

 something to know what proof is. and what it is not; and 

 where can this be better learned than in a bcic nee which 

 has never had to take one footstep backward, and which 

 is the same at all times and in all places. ... I shad be 

 more than satisfied if anything that may have fallen from 

 me may induce any one of us to thmk more highly than 

 he has hitherto done of the first and greatest of the 

 sciences, and more hopefully of the part which he himself ' 

 may bear in its advancement." 



The address, delivered in the author's effective style, 

 was frequently applauded by an appreciative group of 

 members. On the proposal of Prof. Cayley it was re- 

 solved (with the author's consent) that the address should 

 be printed in the Proceedings. 



