ftEPORT ON CERTAIN BRANCHES OF ANALYSIS. 231 



considering the relations of formulae with a view to their equi- 

 valence, and also under other circumstances, which will be in- 

 dicated by such means as will destroy all traces of the equiva- 

 lence which would otherwise exist. 



The capacity, therefore, possessed by the signs of affection 



involving v' — 1 of admitting geometrical or other interpreta- 

 tions under certain circumstances, though it adds greatly to 

 our power of bringing geometry and other sciences under the 

 dominion of algebra, does not in any respect affect the general 

 theory of their introduction or of their relation to other signs : 

 for, in the first place, it is not an essential or necessary pro- 

 perty of such signs ; and in the second place, it in no respect 

 affects the form or equivalence of symbolical results, though it 

 does affect both the extent and mode of their application. It 

 would be a serious mistake, therefore, to suppose that such inci- 

 dental properties of quantities affected by such signs constituted 

 their real essence, though such a mistake has been generally 

 made by those who have proposed this theory of interpretation, 

 and has been made the foundation of a charge against them by 

 others, who have criticised and disputed its correctness*. 



* This charge is made by Mr. Davies Gilbert in a very ingenious paper in 

 the Philosophical Transactions for 1831, " On the Nature of Negative and Im- 

 possible Quantities." He says that those mathematicians take an incorrect 

 view of ideal quantities, — mistaking, in fact, incidental properties for those 

 "which constitute their real essence, — who suppose them to be principles of 

 perpendicularity, because they may in some cases indicate extension at right 

 angles to the directions indicated by the correlative signs + and — ; for with 

 an equal degree of propriety might the actually existing square root of a quan- 

 tity be taken as the principle of obliquity, in as much as in certain cases it 

 indicates the hypothenuse of a right-angled triangle. In reply to this last 

 observation, it may be observed, that I am not aware that in any case the 

 sign /)/— I has had such an interpretation given to it. 



It is quite impossible for me to give an abridged, and at the same time a fair 

 view of Mr. Davies Gilbert's theory, within a compass much smaller than the 

 contents of his memoir. But I might venture to say that his proof of the rule 

 of signs rests upon some properties of ratios or proportions which no arith- 

 metical or geometrical view of their theory vi'ould enable us to deduce. In con- 

 sidering, also, imaginar)' quantities as creations of an arbitrary definition, en- 

 dowed with properties at the pleasure of him who defines them, he ascribes to 

 them the same character as to all other symbols and operations of algebra ; 



but in saying "that quantities affected by the sign v' — 1 possess a. potential 

 existence only, but that they are ready to start into energy whenever that sign 

 is removed," he appears to me to assert nothing more than that symbols are 

 impossible or not, according as they are affected by the sign ^ — 1 or not. 

 Again, in examining the relation of the terms of the equation 



« (« — 1) __9 „ n (rt — 1) (M — 2) „ « 3 



