42 



SCIENCE. 



[N. S. Vol. XI. No. 263. 



necessary to impose restrictions, for the do- 

 main thus divided is still far too large to 

 be reviewed adequately in the brief interval 

 allotted to the present occasion. I have 

 therefore limited my considerations chiefly 

 to those branches of applied mathematics 

 which were already recognized as such at 

 the beginning of the present century. The 

 most important of these branches ap- 

 pear to be analytical mechanics, geodesy, 

 dynamical astronomy, spherical or obser- 

 vational astronomy, the theory of elastic- 

 ity, and hydromechanics. This rather 

 arbitrary subdivision may be made to 

 include several important branches not 

 enumerated, while it must exclude others 

 of equal or greater importance. Thus the 

 theory of heat diffusion which led Fourier 

 to the wonderful analysis which beai'S his 

 name may be alluded to under physical 

 geodesy ; the theories of sound and light 

 may be regarded as applications merely of 

 the theories of elasticity and hj'dromechan- 

 ics ; while the theories of electricity, mag- 

 netism, and thermodynamics, which are the 

 peculiar and perhaps most important de- 

 velopments of the present century, must be 

 excluded almost altogether. 



Another difficulty which besets one who 

 would speak of the progress in question is 

 that arising from the technicalities of the 

 subjects to be discussed. Beautiful and 

 important as these subjects are when ar- 

 raj'ed in their mathematical dress, and 

 thrilling as they truly are when rehearsed 

 with appropriate terminology in the quiet 

 of one's study, it must be confessed that 

 they are on the whole rather uninviting 

 for the purposes of semi- popular exposition. 

 In order to meet this difficult}' it seems 

 best to relegate technicalities which demand 

 expression in symbols to footnotes and, 

 while freely using technical terminolog}', to 

 translate it into the vernacular whenever 

 essential. Thus it is hoped to avoid the 

 dullness of undue condensation on the one 



hand, and the superficiality of mere literary 

 description on the other. 



The end of the last century marks one of 

 the most important epochs in the history of 

 mathematical science. This time, one hun- 

 dred years ago, the master work of La- 

 grange (1736-1813), the Mecanique Analy- 

 tique, had been published about eleven 

 years. The first two volumes of the 

 Mecanique Celeste of Laplace (1749-1827), 

 undoubtedly the greatest systematic treatise 

 ever published, had just been issued. 

 Fourier (1768-1830), whose mathematical 

 theory of heat was destined to play a 

 wonderful role in pure and applied mathe- 

 matics, was a soldier statesman in Egypt, 

 where with Napoleon he stood before the 

 pyramids while the centuries looked down 

 upon them.* Gauss (1777-1855), who with 

 Lagrange and Cauchy (1789-1857) must be 

 ranked among the founders of modern pure 

 mathematics, was a promising but little 

 known student whose Disquisitiones Arith- 

 meticae and other papers were soon to win 

 him the directorship of the observatory at 

 Gottingen. Poisson (1781-1840), to whom 

 we owe in large part the beginnings of 

 mathematical physics, had just started on 

 his brilliant career as a student and pro- 

 fessor in the Ecole Polytechnique. Bessel 

 (1784-1846), whose theories of observa- 

 tional astronomj' and geodesy were destined 

 soon to assume a prominence which they 

 still hold, was an accountant in a trading 

 house at Bremen. Dynamical astronomy, 

 the favorite science of the day was under 

 the dominating genius of Laplace, with no 

 one to dispute his preeminence, and with 

 only Lagrange and Poisson as friendly 

 competitors in the same field. Rational 

 mechanics as we now know it, was soon to 



* The bombastic words of Bonaparte, " Songez que 

 du baut de cea pyramides quarante siccles vous con" 

 templent," maj' be excused, perhaps, in view ot the 

 fact that Fourier, Monge, and Berthollet were present 

 on the occasion. 



