PHILOSOPHICAL TRANSACTIONS. 



XIX. — Consideration of the objections raised against the geometrical representa- 

 tion of the square roots of negative quantities. By the Rev. John Warren^ M.A. 

 of Jesus College, Cambridge. Communicated by Thomas Young, M.D. Fo- 

 reign Secretary to the Royal Society. 



Read February 19, 1829. 



Some years ago my attention was drawn to those algebraic quantities, which 

 are commonly called impossible roots or imaginary quantities : it appeared 

 extraordinary, that mathematicians should be able by means of these quan- 

 tities to pursue their investigations, both in pure and mixed mathematics, and 

 to arrive at results which agree with the results obtained by other independent 

 processes ; and yet that the real nature of these quantities should be entirely 

 unknown, and even their real existence denied. One thing was evident re- 

 specting them ; that they were quantities capable of undergoing algebraic 

 operations analogous to the operations performed on what are called possible 

 quantities, and of producing correct results: thus it was manifest, that the 

 operations of algebra were more comprehensive than the definitions and funda- 

 mental principles ; that is, that they extended to a class of quantities, viz. 

 those commonly called impossible roots, to which the definitions and funda- 

 mental principles were inapplicable. It seemed probable, therefore, that there 

 was a deficiency in the definitions and fundamental principles of algebra ; and 

 that other definitions and fundamental principles might be discovered of a 

 more comprehensive nature, which would extend to every class of quantities 

 to which the operations of algebra were applicable ; that is, both to possible 

 and impossible quantities, as they are called. I was induced therefore to 



MDCCCXXIX. 2 I 



