MATHEMATICAL AND PHYSICAL SCIENCE. 



[Diss. VI. 



lesson more and more loudly in the ears of mankind. 

 The era of Newton and Leibnitz was grandly distin- 

 guished by the continually increasing applications of 

 mathematics to physics, whereof Newton was the great 

 teacher; the century 1750-1850, whilst profiting 

 by the lessons of the past, has added almost a new 

 one in the eminently practical character of its science, 

 and in the no less scientific character of its practice. 

 The result of these gradual modifications of human 

 knowledge has not been in the slightest degree in- 

 jurious to the real progress of the more abstract in- 

 Notinju- gredient of the mixed sciences. Did mathematics 

 rious to ever flourish more vigorously than under Newton ] 

 science has P ure physical science had greater triumphs than 

 in the era of Volta, Watt, and Young ? It was 

 precisely because the new application of mathematics 

 stimulated their growth, because abstract relations 

 of quantity were vivified by concrete solutions of 

 physical problems, that a new geometry arose. 

 Dynamics could hardly be said to exist as a science 

 without the invention of Fluxions as a language by 

 which its conditions and results might be expressed ; 

 and from that time onwards, the necessities of the 

 natural philosopher have been the prime sources of 

 inspiration to the geometer, while the subjects have 

 become so blended that a mere discoverer in mathe- 

 matics has become a singularity. It would be hardly 

 possible to point out any mathematician of the 

 highest class since Newton, or but a few of the second 

 class, who have not contributed almost as much to 

 physical science as they have to analysis. Of purely 

 mathematical discoveries, the great majority have 

 been called forth by the immediate necessity arising 

 from some problem requiring solution in astronomy, 

 mechanics, optics, or heat. Lagrange's method of 

 Variations of arbitrary Constants in Integration, the 

 artifices for the computation of attractions by Laplace's 

 coefficients ; the introduction of the method of fac- 

 torials by Kramp in his solution of the problem of 

 refraction, and numberless improvements in the 

 Theory of Definite Integrals by Fourier and his suc- 

 cessors, sufficiently warrant the statement, and show 

 how richly the physical sciences have repaid to the 

 purely mathematical ones the debt which they origi- 

 nally owed. One other conclusion may be drawn 

 from these and parallel facts. It is that the com- 

 binations arising out of external phenomena are more 

 suggestive of the possible relations of number and 

 quantity than is the most unlimited stretch of fancy 

 and imagination ; and I believe it will be conceded 

 that, with few exceptions, theorems of the greatest 

 value and beauty have been more frequently dis- 

 covered during the attempt to solve some physical 

 or at least geometrical problem, than in compre- 

 hensive yet indefinite attempts to generalize the re- 

 lations of abstract magnitude. 

 (28.) These views are strikingly confirmed by the his- 



torical fact of the paucity of pure mathematicians, The pure 

 and of distinct mathematical treatises of a strictly M 

 original character in an age distinguished by the 

 diffusion of mathematical knowledge, and in countries 

 (like France) most celebrated for its triumphs. There 

 are not, perhaps, much more than half a dozen 

 really great mathematicians of the last seventy years, 

 who have not left treatises more numerous and more 

 distinguished on physical science, treated mathe- 

 matically, than on pure mathematics. Among the 

 exceptions which more immediately occur, are Monge, 

 Legendre, and Abel. And of distinct treatises, whilst 

 we have the MScanique Analytique, the MScanique 

 Celeste, the TMorie de la Chaleur, and numberless 

 others, containing precious mathematical develop- 

 ments, in connection with the applications which 

 suggested them, the purely mathematical memoirs of 

 the same period are to be sought chiefly in the form 

 of detached essays, in the ponderous volumes of 

 Academical Transactions. 



One point in the History of Mathematics has espe- (29.) 

 cial interest for the English reader, and as such may T 'eir pro- 

 be touched upon here with reference to the progress country. ' 

 of science for the last three quarters of a century. 

 The national pride of England in* the triumphs of 

 Newton impelled her ablest mathematicians to at- 

 tempt to carry forward the synthetic methods which 

 he had chiefly used, at least in his published works, 

 to the more arduous and intricate questions of Me- 

 chanics and Astronomy which presented themselves 

 for solution in the course of the 18th century. Mac- 

 laurin was almost the last Englishman of that period 

 whose mathematical writings came into direct com- 

 petition with the rising schools of Germany and 

 France. The labours of Matthew Stewart, and 

 Simpson were mostly geometrical ; those of Landen 

 and Waring, though profound, created little general 

 impression ; and, gradually, the extent and difficulty 

 of the foreign mathematics, increased by the use of the 

 Leibnitzian notation of differentials which was ab- 

 solutely unfamiliar in England, deterred almost every 

 one even from perusing the writings of Clairaut and 

 D'Alembert, Lagrange and Laplace. Of the conti- 

 nental mathematicians, Euler was probably the best 

 known, owing to the lucidity of his writings and 

 their eminently practical tendencies. Some idea may 

 be formed of the negation of mathematical talent in 

 Britain during the later portion of the last century, 

 when we find D'Alembert declaring, in 1769, that if 

 an Englishman is to be elected one of the eight 

 foreign associates of the Academy of Sciences, he 

 will vote for Earl Stanhope as the best mathemati- 

 cian there, as he believes, not having read any of 

 his works ! If the choice was to be free, he should 

 prefer M. de Lagrange ! I 1 A more cutting, though 

 unintentional satire on the state of Mathematics in 

 this country could not have been written. 



1 Letters of eminent persons, addressed to David Hume, edited by Mr Burton, p. 215. 



