FOUNDATION OF ALGEBRA. 293 



is (o, — 1 , which is the logometer of a line whose length has 0cos-, 



or 0, for its logarithm, inclined at the angle 0sin^, or 9. Hence « B ^~' 



is a unit of length, inclined at an angle ; or cos 9 + J - 1 sin 0. 



It will appear rather against the preceding definition, that it points 

 out e 6 ^ -1 to signify the same as cos + J— 1 sin 0, whatever e and -w 

 may be : for there is nothing in the preceding demonstration which 

 has reference to any particular value of these constants. And, in reality, 

 as far as this one definition is concerned, and its consequences, there 

 would be no limitation upon the meanings of w and «?. But when — 

 having invented this indefinite mode of constructing (1, 6), which leads 

 us to our result whatever may be e or n, and in fact contains a direct 

 and inverse use of e which prevents the value of that letter from af- 

 fecting the result — we equate this mode of producing (1, 0) to a definite 

 mode derived from another definition, we must expect to see the in- 

 definite character of the former changed, by the introduction of new 

 conditions, into the definite character of the latter. But this would 

 be no answer to the difficulty : for it would be admitting a new 

 fundamental rule among the laws of operation. Let the reasonableness 

 of the expectation just alluded to be ground for an assumption, and 

 we see that •n- (two right angles) and e are connected by the equation 

 6 V-i = cos 1 + J- 1 sin 1, or e depends upon the angle denoted by 1, 

 which is all that is necessary. But I say that this equation is a con- 

 sequence of the whole set of definitions only, without any new assump- 

 tion. In the first place, it is easily shewn that R s = JR*IlxIl...($ times) 

 when S=(s, 0) and s is integer. Next, from the definition of multipli- 

 cation it immediately follows that 



cos 80 + J — 1 sin s0 = (cos + J— 1 sin 0) (cos0+ J — I sin 0)...(s times); 



whence cos s0 + J — 1 sin s0 = (cos + J — 1 sin 0)'; 



whence it is cos 1 + J — 1 sin 1, and that only, which, raised to the in- 

 teger power of s, gives cos s + J — 1 sin * ; for it may readily be shewn 

 that no other line (r, p) can have the same * th power as (1, 1). It is 

 then the result of the definitions of addition and multiplication that 

 nothing but cos 1 + J — 1 sin 1, raised to the power of * (integer), gives 

 Vol. VII. Pakt III. Ii 



