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 
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MDCCCXXIX. 
