FOUNDATION OF ALGEBRA. 291 



same time the power of adhering to the modes of derivation of the partial 

 view is too dearly paid for by a want of generality in the general one. 



In my last Paper I pointed out that the analogy of the definitions 

 of a + b and ab in arithmetic and algebra was perfect, insomuch that, 

 by an abstraction of the subject-matter of the former from those de- 

 finitions, the remaining words make definitions which will equally apply 

 to both views of the science. In fact, a + b is in both, a direction to 

 do with a what must be done with to make b ; while a b is a direction 

 to do with a what must be done* with 1 to make b. I now proceed 

 to disengage a h from its partial dependence on ab, and having established 

 an independent definition, to examine the analogies which exist between 

 «* in the ancient and modern view of the subject. 



Let R = {r, p), be a line of r units inclined to the unit-line at the 

 angle p ; and this being r cos p + r sin p J—l, let r cos p = R x , r sin p — R y . 

 It is in our power to suppose this line given by means of another, 

 R' = (r',p), by the conditions R x ' = <p(r,p), R v ' = ^(r,p), (p and ^ being 

 known functions, from which r and p can be determined in terms of 

 r' and p. The second line may be called the determinant of the first, 

 and the first line may be said to be determined from the second. Now 

 supposing only the operation + to have been defined in its most general 

 sense, we have from every form of <p and >//■ the means of instituting 

 a new process, as follows. Instead of adding two lines, add their de- 

 terminants, and let the sum of the determinants be the determinant of 

 a new line. If (r, p), (s, <r), be the given lines, and (t, t) the determined 

 line, we have then 



(t, t) = <p (r, p) + <p {s, a), x// (t, t) = ^ (r, p) + y}s (s, cr). 



* The most analogical view of a b is not a natural one, owing to the idea of a logarithm 

 being made subsequent to that of an exponent. But if the notion of a logarithm were ob- 

 tained, prior to the definitions of algebra, from two continuous linear motions, which severally 

 give equal increments in equal times, the one in difference and the other in ratio, the 

 exponent obtained from them would first enter as a logarithm, and would always retain 

 the character of a logarithm rendered into numbers, just as sin -1 A always retains that of 

 an angle rendered into numbers. Hence, a'° gt or o 1 " 8 ", which are the same things, would 

 be defined from e, introduced in the first process, as follows : a l0£ b is what arises from doing 

 to a that which must be done with e to form b. 



