356 Mr. J. Cockle's Analijsis of the Theorij of Equations, 



assumption as to the number of roots of the given equation. 

 The solution of a biquadratic by Descartes is also a direct 

 one ; and the same remark applies to the solutions of a cubic 

 given by Tschirnhausen at pp. 206, 207 of the Acta Ertiditoriim 

 (Leipsig) for 1683. 



Euler's solution of a biquadratic, just alluded to, will be 

 found in a paper by him, at pp. 216-231 of the sixth volume 

 of the Petersburgh Commentarii. His solution of a cubic 

 there given does not differ in any material respect from that 

 of Cardan ; but Euler has exhibited both these solutions as 

 consequences of one method. In fiict, if we omit the radical 

 signs which Euler attaches to the indeterminate quantities, 

 his method may in the case of cubics be said to be identical 

 with Cardan's, and in that of biquadratics to be a legitimate 

 extension of it. Euler's solution of a bicjuadratic is often ex- 

 hibited without the radical signs attached to the indeterminate 

 quantities. 



The earlier processes given by Bezout for the solution of a 

 cubic {Mevi. de VAcad. for 1762, pp. 23-25), and of a biqua- 

 dratic (Ibid. p. 52), are direct. But his subsequent ones 

 {Mem. de I' Acad, for 1765, pp. 533-552), and those given by 

 Euler in the ninth volume of the Novi Commentarii (pp. 70- 

 98), are iridirect. The fundamental assumption depends upon 

 our knowledge of the number of roots of a binomial equation 

 of the same degree as that under discussion. 



It must not be supposed that theie is any inferiority in the 

 indirect methods ; on the contrary, there is perhaps a greater 

 degree of uniformity and coherence in them than in the direct 

 processes. The method of Lagrange is an indirect one, and 

 depends essentially upon our knowledge of the number ofroots 

 of a given equation, and of the theory of their symmetric func- 

 tions. To the indirect class must be added the method of 

 Spence, and that given by Murphy in the Philosophical 

 Transactions for 1837 (pp. 161-178), and commented on by 

 Sir W. 11. Hamilton at pp. 256-259 of the eighteenth volume 

 of the Transactions of the Royal Irish Academy. 



I shall not stop to discuss particular cases in which equa- 

 tions admit of solution, — Demoivre's form of the equation of the 

 fifth degree for instance. Abel has however given very general 

 discussions of such cases, of which Dr. Peacock has given an 

 account at pp. 318-320 of the Sixth Report of the British 

 Association. 



Curious illustrations of the manner in which the different 

 departments of the theory of equations are, practically speaking, 

 blended one with the other, are afforded by the iacts, that 

 before we can apply Cardan's rule to a cubic, we must deprive 



