352 Mr. J. Cockle's Analysis of the Theoj-y of Equations^ 



You will not be surprised to find that I now proceed to throw 

 that idea into form, and to execute my project of mapping out 

 the theory of equations, and pointing out what parts of the 

 field have been occupied by writers on the subject, but more 

 especially by the authors of recent English treatises. 



Algebra, arithmetical and symbolical, is capable of division 

 into three parts. The subject of identity might be made to 

 constitute tiie first of these divisions ; that of equality the 

 second ; and that of incongruity or absurdity the third. These 

 divisions are of a purely theoretical character; practically 

 speaking, the subjects of them are treated indiscriminately as 

 occasion requires (a). 



{a). Of these three cases (which are perfectly distinct) it may be 

 remarked, that the identity can only arise from the values of the 

 functions on both sides of the equation being csse7itially equivcdent 

 for all values of the terms that enter into the composition of the 

 equation. It can never, therefore, arise in the equations which 

 express the conditions of b. problem. On the other hand, the second 

 is the result of expressing a problem algebraically ; and it can never 

 appear as the result of an alleged theorem ; whilst tlie third may 

 make its appearance either from the expression of a theorem which 

 is not true, or from that of a problem, the conditions of which are 

 incompatible with each other. My views on this latter subject have 

 been briefly explained in an article (for temporary private reasons 

 given anonymously) which appeared awhile ago in the Phil. Mag. 

 vol. xxix. p. 171. 



It has been usual in treatises on algebra to speak of v — 1 (or of 

 a + /3 V^— 1) as though it were in some peculiar manner, the one 

 symbol of incongruity. It is, undoubtedly, the first form in which 

 contradictory conditions arc usually encountered in algebra ; and, 

 possiblj', with sufficient labour and ingenuity, all other forms or 

 indications of incongruity may be reduced to this type. Yet it is 

 altogether unnecessary to reduce, for instance, log ( — 1) as an 

 expression, or sin = 2 as an equation to such a form ; and it would 

 manifestlv, be a still greater waste of time to reduce more complex 

 functions to such a form when they already involve a perceptible 

 contradiction. As well might Euclid have imposed upon himself 

 the condition of reducing all his ex absurdo demonstrations to a con- 

 tradiction of the 10th axiom, because his first demonstration of a 

 proposition by the method (prop. iv. book i.) happened to be so 

 effecte'd. 



The assertion that one is equal to one, is an identity ; that x 

 is equal to one, an equation ; that t'wo is equal to one, an absur- 

 dity or contradiction. Whether they indicate identity, equality 

 or contradiction, these three species of proposition are all ex- 

 hibited in algebra under the form of equations : thus, 



i = l; x=l', 2 = 1; 



