250 MR. WARREN ON THE SQUARE ROOTS 



as quantities to be subtracted, therefore their proofs are only applicable to the 

 diiFerence of two positive quantities, and not to negative quantities abstractedly 

 considered. These fundamental principles must therefore be looked upon as 

 hypotheses introduced into algebra in order to give to negative quantities a 

 representation and a real existence. And in like manner, in order to arrive at 

 the representation of the square roots of negative quantities, I have made the 

 following hypotheses : that all straight lines drawn in a given plane from a 

 given point in any direction whatever, may be algebraically represented both 

 in length and direction : that addition is performed in the same manner as 

 composition of motion in dynamics ; that four straight lines are proportionals, 

 both in length and direction, when they are proportionals in length, and the 

 fourth is inclined to the third at the same angle at which the second is inclined 

 to the first : and I have by means of these hypotheses as a foundation, esta- 

 blished all the common rules for performing algebraic operations, and thus 

 have proved, that the results arrived at by means of these hypotheses must be 

 correct : therefore I conclude, that these are true hypotheses, and true in the 

 same sense, that the hypotheses made by algebraists respecting the represen- 

 tation of negative quantities are true. In fact, if there be a question, whether 

 negative quantities can or cannot be represented geometrically ; the only way 

 in which such a question can be solved, is by making certain hypotheses with 

 respect to their geometric representation, and then showing that the results 

 arrived at from these hypotheses must be correct : and in like manner if there 

 be a question whether those quantities commonly called impossible can be 

 geometrically represented, the question must be solved in the same way ; viz. 

 by making certain hypotheses respecting them, and showing that the results 

 arrived at by means of these hypotheses must be correct. In this point of view, 

 the definitions and fundamental principles which I have laid down in my 

 treatise must be considered as mere hypotheses ; and mathematicians will be 

 satisfied of their correctness when they see that the results agree in every re- 

 spect with the results obtained by other independent processes. 



To the third objection, viz. that the geometric representation of the square 

 roots of negative quantities can be of no use to mathematicians, it will not be 

 necessary to say much in reply. 



In the works which have lately been written, either on pure or mixed mathe- 



