340 
MR. WARREN ON GEOMETRICAL REPRESENTATION 
From this observation it was evident that M. Mourey had arrived at the 
geometrical representation of all algebraic quantities whatever, and that in a 
larger work he entered fully into the subject ; but from his Preface it appeared 
also, that this larger work existed only in manuscript, and that circumstances 
would not permit the author to publish it at present. I was induced therefore 
to pursue my own investigations further ; and the result was, that I found (as 
M. Mourey had stated) that all algebraic quantities whatever are capable of 
geometrical representation, and are represented by lines all situated in the 
same plane : and my view in what I am now writing is to communicate this 
result to algebraists. 
This paper, therefore, is intended as a continuation of my “Treatise on the 
geometrical representation of the square roots of negative quantities;” and the 
object of it is to extend the geometrical representation to the powers of quan- 
tities, whose indices involve the square roots of negative quantities. 
Art. 1 . Def.) Mathematicians apply the words c possible’ and c impossible’ to 
algebraic quantities, the former signifying either positive or negative quantities; 
the latter, quantities involving the square roots of negative quantities. In this 
sense, as a matter of convenience, these words will be used in this paper ; it 
being understood at the same time that by the word ‘impossible’ no impossibility 
is necessarily implied, but on the contrary, that the quantities called impos- 
sible have a real existence, and are capable of geometric representation. 
2. Def.) Logarithms, according to the common definition given by mathe- 
maticians, must be possible quantities ; therefore as a general definition of 
logarithms will be given in this paper, it will be desirable for the sake of 
distinction to give a more limited name to the common definition of logarithms, 
and accordingly they will be called possible logarithms ; also for the like 
reason, the powers of quantities according to the common definition of powers, 
will be called possible powers. 
3. Def.) Let be any quantity whatever, and let £ be inclined to unity at 
an angle = 0, and let r be a positive quantity equal in length to f, and let the 
possible hyperbolic logarithm of r be v ; then v -f- 0 — 1 is called the general 
hyperbolic logarithm of g, and expressed thus 
