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. 



r 



Art. 1. Def.) Mathematicians apply the words ' possible' and ' 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- 

 linaticians, 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 = 6, and let r be a positive quantity equal in length to g, and let the 

 possible hyperbolic logarithm of r be w ; then v -{• 6 ^ — lis called the general 

 hyperbolic logarithm of g, and expressed thus g'. 



