5 i2 POPULAR SCIENCE MONTHLY. 



survey of the collected works of these writers will show how much, of 

 the very highest quality and import, would remain. However this may 

 be, the essential point, which can not, I think, be contested, is this, that 

 if these men had been condemned and restricted to a mere book knowl- 

 edge of the subjects which they have treated with such marvelous ana- 

 lytical ability, the very soul of their work would have been taken away. 

 I have ventured to dwell upon this point because, although I am myself 

 disposed to plead for the continued recognition of mathematical phys- 

 ics as a fairly separate field, I feel strongly that the traditional kind of 

 education given to our professed mathematical students does not tend 

 to its most effectual cultivation. This education is apt to be one-sided, 

 and too much divorced from the study of tangible things. Even the 

 student whose tastes lie mainly in the direction of pure mathematics 

 would profit, I think, by a wider scientific training. A long list of 

 instances might be given to show that the most fruitful ideas in pure 

 mathematics have been suggested by the study of physical problems. 

 In the words of Fourier, who did so much to fulfil his own saying, 

 " L' etude approfondie de la nature est la source la plus f eeonde des 

 decouvertes mathematiques. Non-seulement cette etude, en offrant 

 aux recherches un but determine, a l'avantage d'exclure les questions 

 vagues et les calculs sans issue; elle est encore un moyen assure de 

 former l'analyse elle-meme, et d'en decouvrir les elements qu'il nous 

 importe le plus de connaitre, et que cette science doit toujours conser- 

 ver: ces elements fondamentaux sont ceux qui se reproduisent dans 

 tous les effets naturels." 



Another characteristic of the applied mathematics of the past cen- 

 tury is that it was, on the whole, the age of linear equations. The 

 analytical armory fashioned by Lagrange, Poisson, Fourier and others, 

 though subject, of course, to continual improvement and development, 

 has served the turn of a long line of successors. The predominance of 

 linear equations, in most of the physical subjects referred to, rests on 

 the fact that the changes are treated as infinitely small. The electric 

 theory of light forms at present an exception ; but even here the linear 

 character of the fundamental electrical relations is itself remarkable, 

 and possibly significant. The theory of small oscillations, in partic- 

 ular, runs as a thread through a great part of the literature of the 

 period in question. It has suggested many important analytical re- 

 sults, and it still gives the best and simplest intuitive foundation for 

 a whole class of theorems which are otherwise hard to comprehend 

 in their various relations, such as Fourier's theorem, Laplace's expan- 

 sion, Bessel's functions, and the like. Moreover, the interest of the 

 subject, whether mathematical or physical, is not yet exhausted; many 

 important problems in optics and acoustics, for example, still await 

 solution. The general theory has in comparatively recent times re- 



