358 Mr. J. Cockle's Anahjsis of the Theory of Equations ^ 



shown in the Mechanics' Magazine (current and last volumes), 

 that the symbolical impossibility or contradiction expressed by 

 such an equation is convertible into an arithmetical one {d). 



(d). Previous to my first noticing an equation of this class (in 

 Hind's Algebra) I had looked upon all attempts to prove that every 

 equation had at least one root, as a superfluous labour ; and I know 

 that not a few algebraists still take the same view of it. The letter 

 of Mr. Homer's, referred to by Mr. Cockle, was in reply to one 

 which I addressed to Mr. Horner a few days before, when the sub- 

 ject was new to me, respecting the particular equation referred to ; 

 and I think the question oi fact, as to there being innumerable 

 equations which have no root, is set at rest by the discussion in that 

 letter. Not only the fact, however, is there established, but also 

 the true princtjjle which runs through such equations : and whilst I 

 have no reason to think that Mr. Horner was acquainted with 

 Garnier's solution, I need not urge upon any one who compares 

 what Gamier and Horner have done, the utter impossibility of the 

 solution of the former having furnished any suggestion for the dis- 

 cussion of the latter. Mr. Cockle's supplementary discussion, re- 

 ferred to above, appears to me to complete the inquiry. 



As regards the doubt thrown out by the editor of Wood's Algebra, 

 whether the fault be not in the method of searching for the root, it 

 may be remarked that such conjectures are scarcely allowable in 

 pure science except some method of bringing the question to a de- 

 cisive test be at least suggested. It may further be remarked, that 

 if we admit the proposed hypothesis, it would follow that, when the 

 equation and its congener were reduced to a single equation free 

 from radicals, this new equation would have a greater number of 

 roots than it had units of dimension ; viz. those alleged to belong 

 to the surd equation, and those which are admitted to belong to its 

 congener. This consequence is contradictory to one of the simplest 

 and best established of all our propositions respecting the composition 

 of equations ; and it never could have occurred to the able editor of 

 that valuable work when he proposed the hypothesis in question. 



2. The transformation by which the second term is taken 

 away from an equation has been long known. In the case of 

 a perfect cubic, it was absolutely necessary in order to the 

 application of the rule of Cardan. This transformation is 

 effected linearly, that is to say, the equation connecting 

 the roots of the original and of the transformed equation 

 only contains the first power of those roots, together with 

 an arbitrary quantity, which is determined so as to satisfy the 

 required condition. This it may be made to do by solving a 

 linear equation. In the same manner, by determiniug the 

 arbitrary quantity so as to satisfy a certain quadratic, cubic 

 or biquadratic, tiie third, fourth, or fifth terms of the trans- 

 formed equation might be made to vanish. But the linear 



