496 Mr. J. Cockle's Analysis of the Theory of Equations. 



apply it rigidly in the same sense. Great practical conveni- 

 ence will be found to result from this nomenclature ; but I 

 will observe that, as equations of the first four degrees are 

 designated by the respective names of linear, quadratic, cubic, 

 and biquadratic equations, so I have proposed to give to equa- 

 tions of the first four orders the respective titles of simple, 

 binary, tertiary, (or ternary,) and quaternary equations. This 

 will save much circumlocution, but it will be necessary to 

 banish the term simple as applied to equations of the first 

 degree. No inconvenience need attend the new application of 

 the term. The term * simple equation,' as hitherto used, is a 

 mere synonym for ' linear equation,' while, as applied to equa- 

 tions of the first degree, the latter term is far more appro- 

 priate, on the ground of analogy with 'quadratic,' &c. Ana- 

 logy would further justify us in using the abbreviations 'n-ic 

 equation ' to denote an equation of the n\X\ degree^ and « m-ary 

 equation ' to denote an equation of the wth order. The term 

 c m-ary w-ic' would completely define any algebraic equation. 

 While upon this subject I may take the liberty of referring 

 you to my remarks on it at pp. 509, 510, 582, &c. of vol. xlv. 

 of the Mechanics' Magazine. 



8. But, of these appellations e degree ' and ' order,' which 

 is to be Genus, and which Species? It is important to make 

 a correct choice, for an improper selection would vitiate our 

 arrangement, and introduce confusion instead of method. My 

 answer to the above inquiry is, that degree is to be regarded 

 as Genus, and order as Species. And this arrangement will 

 not only be found to be, philosophically speaking, more ac- 

 curate, but it is also more practically convenient, and, unques- 

 tionably, more in conformity with the historic development 

 of the Theory of Equations. It was only by reducing its so- 

 lution, to that of two simultaneous binary equations, that a 

 cubic was first solved. In other words — if we use the word 

 complex in contradistinction to simple — the solution of a simple 

 cubic involved the solution of complex equations of degrees 

 virtually, if not formally, lower than a cubic. Hence, in what 

 follows, I shall carry, as far as may seem desirable, the dis- 

 cussions of different orders of equations of a lower degree 

 before proceeding to those of equations of a higher degree. 

 The benefit of this, to a limited extent the ordinary course, 

 will be seen as we proceed. 



9. Of simple linear equations little more need be said than 

 that their solution is effected by algebraic addition, subtrac- 

 tion, multiplication and division, or one or more of those 

 operations, all of which we may include in the common name 

 reduction. Where several simultaneous complex linear equa- 



