372 



which period several objections have been made to this hypothesis. 

 The purpose of the present paper is to answer these objections. 



The first of these is, that impossible roots should be considered 

 merely as the indications of some impossible condition, which the pro- 

 position that has given rise to them involves ; and that they have 

 in fact no real or absolute existence. To this it is replied by the 

 author, that although such a statement may be true in some cases, 

 it is by no means necessarily so in all ; and that these quantities re- 

 semble in this respect fractional and negative roots, which, whenever 

 they are excluded by the nature of the question, are indeed signs of 

 impossibility, but yet in other cases are admitted to be real and 

 significant quantities. We have therefore no stronger reasons, a priori, 

 for denying the real existence of what are called impossible roots, 

 because they are in some cases the signs of impossibility, than we 

 should have for refusing that character to fractional or negative roots 

 on similar grounds. 



It has been objected, in the second place, that there is no necessary 

 connexion between algebra and geometry, but only one of analogy; 

 and that it is consequently improper to introduce geometric conside- 

 rations into questions purely of an algebraic nature. In answer to 

 this, the author contends that a necessary connexion may be shown 

 to exist between impossible roots, and the series expressive of the 

 ratio between the circumference of a circle and its diameter. This 

 he endeavours to prove by examining such values of the expansion of 

 1* as are functions of x ; whereby he is led to a series, the terms of 

 which involve both the square root of unity, and also the above-men- 

 tioned geometric ratio. In other cases he arrives, by methods which 

 are purely algebraic, to expressions containing sines and cosines, to- 

 gether with impossible roots. Hence the author infers that a neces- 

 sary connexion exists between algebra and geometry ; and that his 

 own hypothesis as to the geometric representation of the square 

 roots of negative quantities, is true in the same sense as the hypothe- 

 sis adopted by algebraists respecting the geometric representation of 

 negative quantities is true. 



To a third objection, derived from the alleged inutility of such a 

 geometric representation of the square roots of negative quantities, 

 the author replies, that from their frequent employment by mathe- 

 maticians, it is reasonable to expect that they will be of much greater 

 use when the true theory of their nature shall be established than 

 when it was unknown. 



If the hypothesis of the author is admitted, all questions in dyna- 

 mics where the motions of bodies are limited to one plane, will be 

 brought within the province of pure algebra. 



The author concludes by noticing a work- by M. Mourey, entitled 

 " La vraie Theorie des Quantites Negatives, et des Quantite's pretendues 

 Imaginaires," in which the same general views of the subject are pre- 

 sented as are entertained by the author. 



