Mr. J. Cockle on the Method of Symmetric Products. 133 



values. Thus, for cubics, we employ two only in place of the 

 six which, when approached from first principles, Lagrange's 

 process involves (Berl. Mem. for 1770, p. 144; Gamier, Anal. 

 Alg. 2nd ed., Paris, 1814, p. 174; Young, p. 459). For bi- 

 quadi-atics we have three instead of the twenty-four which (Gar- 

 nier, pp. 184, 185 ; Young, pp. 464-6) are successively reduced to 

 twelve and to six, the latter having certain relations one with 

 another. In the theory of quiutics we do not take one hundred 

 and twenty values as our point of departure, but, starting from 

 four, we are conducted to twenty-four as involved in the dis- 

 cussion. 



53. Cardan's statement of Paciolo's views as to the impossi- 

 bility of solving cubics is dealt with by Cossali {Origine, &c., 

 vol. ii. pp.96, 97). Tschirnhausen* seems to have had no doubt 

 of the possibility of solving equations of any degree. Eulerf 

 did not despair of that of the fifth, notwithstanding obstacles 

 which Waring J deemed insuperable ; nor did Bezout § . Lagrange 

 (Berl. Mem. for 1770-71) showed the connexion of all\\ solutions 

 then known with the permutations of rational functions of the 

 roots, and pointed out difficulties in the theory of the high equa- 

 tions. Sir W. R. Hamilton has proved that the relations among 

 Lagrange's functions discovered by Badano do not lead to a 

 solution of equations of the fifth degree. The formulse of 

 Wronski do not, in the opinion of Gergonne and Peacock, differ 



* See the first and second pages (204, 205) of Tschirnhausen's paper» 

 printed in the Leipsic Acta Eruditorum for 1683, and entitled Metkodus 

 auferendi omnes terminos intermedios ex data equations. 



t See § 20 (pp. 230-1) of Euler's essay Deformis radicum cequationum 

 cuiusque ordinis coaiectatio (Pet. Com. [for 1732-33] vol. vi.) ; see also § 37 

 (pp. 92, 93) of his essay De resolutions eequationum cuiusvis gradus (Pet. 

 2<evv Com. [for 1762-63] vol. ix.). Gamier (p. 228) points out very clearly 

 the modification to which Euler subjected his original form of root. 



X See Wariug's Miscellanea Analytica, &c. (Camb. 1762), p. 47 ; see 

 also his Meditationss J/^/eiraicffi (Camb. 1770), pp. 120, 121, 122. The 

 researches of Bezout, alhuled to at p. v. of the Prcefatio of the latter work, 

 appear in the Paris Me'moires for 1762 and 1765 (not 1764). In explana- 

 tion of the remainder of the paragra))h which contains the allusion, Waring's 

 paper in the Philosophical Transactions for 1779 (pp. 86-104), not forget- 

 ting its introducttiiy part, should be referred to. (Et vid. Misc. An. p. 44.) 

 Lagrange appreciated Waring's abihties (Berl. Mem. for 1771 [published 

 in 1773], p. 202). 



§ Bezout, M6noire sur plusieurs classes d'^quations de tons les dsgr^s, 

 qui admettent une Solution alyibrique (Paris Memoires for 1762 [published 

 in 1764], pp. 17-52), arts. (3.), (6.), and (U.); see also his Memoire sur la 

 resolution yen&nle des equations de tons les degr4s (lb. for 1765 [published 

 in 176H], pp. 5:^3-552). Compare p. 549 with Art. (85.) of Lagrange's 

 discussion (Berl. Mem. for 1771, p. 187). 



II Lagrange's omission to notice T. Simpson's generalization of Ferrari's 

 solution of biquadratic is scarcely au exception to this remark. 



