670 



SCIENTIFIC THOUGHT. 



these marvellous works of genius, science is probably 

 indebted for its greatest advances to those mathema- 

 ticians who, like Pllicker in Germany, Chasles in France, 

 and Cayley in England, employed the analytic and con- 

 structive methods alternately and with equal mastery. 



It is impossible — and it is not my object — to allot to 

 each of these original thinkers the special ideas intro- 

 duced by him into modern science ; but for the purpose 



like Johannes Miiller could not 

 understand how such simple things 

 could be brought before the 

 Academy of Sciences, whereas the 

 great mathematician Diiichlet was 

 full of praise of the ingenuity of 

 the method by which problems 

 were solved which the Calculus 

 of Variations attacked long after 

 Steiner, and then only in ways 

 which the synthetical method had 

 indicated (see Geiser, ' Zur Erin- 

 nerung an Jacob Steiner,' p. 28). 

 It must not be supposed, however, 

 that Steiner was an extreme purist 

 so far as geometrical methods were 

 concerned, for he says himself 

 " that of the two methods neither 

 is entitled to exclude the other ; 

 rather both of them will, for a long 

 time, have plenty to do in order to 

 master the subject to some extent, 

 and then only can an opinion as to 

 their respective merits be formed " 

 ('Ges. Werke,' vol. ii. p. 180). 

 An instance of a celebrated prob- 

 lem being treated alternately by 

 synthetic and analj'tic methods 

 is that of the Attraction of 

 Ellipsoids, in which the Theorem 

 of Maclaurin had created quite a 

 sensation. In spite of the ad- 

 miration which it evoked, both 

 Legendre and Poisson expressed 

 the opinion that the resources of 

 the synthetic method are easily 

 exhausted. The latter, whilst ad- 

 mitting "que la synthese ait 

 d'abord devance I'analyse," never- 



theless concludes that "la question 

 n'a ete enfin rdsolue completement 

 que par des transformations ana- 

 lytiques . . . auxquelles la syn- 

 thase n'aurait pu suppleer." This 

 expression of opinion was falsified 

 when Chasles presented to the 

 Academy, in the year 1837, a 

 memoir in which, through the 

 study of confocal surfaces, the 

 Theory of Maclaurin was synthet- 

 ically proved in its full generality. 

 Poinsot, who reported on this 

 memoir, attached the following re- 

 marks : " Ce m^moire remarquable 

 nous offre un nouvel exemple de 

 r^legauce et de la clarte que la 

 geometrie pent repandre sur les 

 questions les plus obscures et les 

 jjIus difficiles. ... II est certain 

 qu'on ne doit ndgliger ni I'une ni 

 I'autre ; elles sont au fond presque 

 toujours uuies dans nos ouvrages, 

 et forment ensemble comme I'in- 

 strument le plus complet de I'esprit 

 humain. Car notre esprit ne 

 marche guere qu'h, I'aide des signes 

 et des images; et quand il cherche 

 h, pendtrer pour la premiere fois 

 dans les questions difficiles, il n'a 

 pas trop de ces deux moyens et 

 de cette force particuliere qu'il ne 

 tire souveut que de leur concoura. 

 C'est ce que tout le monde peut 

 sentir, et ce qu'on peut recon- 

 naitre dans le Memoire meme." 

 (Chasles, ' Rapport sur les progr^ 

 de la geometric,' 1870, p. 105, &c.) 



