244 MR. WARREN ON THE SQUARE ROOTS 



sibility ; and therefore it is absurd to suppose that the square roots of negative 

 quantities can have any real existence. A second objection is, that there is no 

 necessary connexion between algebra and geometry, and therefore that it is 

 improper to introduce geometric considerations into questions purely alge- 

 braic ; and that the geometric representation, if any exists, can only be analo- 

 gical, and not a true algebraic representation of the roots. A third objection 

 is, that this geometric representation, even if it be a correct representation of 

 the roots, is merely a matter of curiosity, and can be of no use to mathemati- 

 cians. — ^The object of this paper is to answer these objections. 



To the first objection, that impossible roots are merely signs of impossibility, 

 it may be i-eplied, that, though they are so in some questions they are not ne- 

 cessarily so in all, and that in this respect they resemble fractional and nega- 

 tive roots. It is obvious, that in all equations derived from suppositions, 

 which involve an impossibility or absurdity, the impossibility or absurdity will 

 show itself in the result of the operation : and this may appear as well by a 

 fractional root or by a negative root, as by a root commonly called impossible : 

 thus, if we have a question, which from its nature does not admit of a frac- 

 tional answer, and in resolving this question we arrive at an equation, which 

 only admits of fractional roots, these fractional roots are in this case a proof, 

 that the question involves an impossibility. Also if a question does not admit 

 of a negative answer, and in resolving it we arrive at an equation which only 

 admits of negative roots, in this case also we conclude that the question in- 

 volves an impossibility. So, in like manner, if in resolving a question which 

 does not admit of what are now commonly called impossible roots as answers, 

 we arrive at an equation, all of whose roots are impossible, we must conclude, 

 that the question involves an impossibility ; and we have no greater reason for 

 inferring from the last case, that what are called impossible roots have no real 

 existence, than we have for inferring from the two former cases, that fractional 

 or negative quantities have no real existence. This will be rendered clearer 

 by an example : Let a body revolve in a circle by the action of a centripetal 

 force, which varies inversely as the nth power of the distance ; and let it be 

 required to find the height from which a body must fall to the circle to acquire 

 the velocity of the body revolving in the circle. In this example, if n be greater 

 than 3, the velocity of the body revolving in the circle is greater than the velo- 



