250 
MR. WARREN ON THE SQUARE ROOTS 
as quantities to be subtracted, therefore their proofs are only applicable to the 
difference 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 residts 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 rnathe- 
