294 PROFESSOR DE MORGAN, ON THE 



(1, *): it is the consequence of the definition of an exponential operation, 

 considered apart from the rest, that e^ _1 raised to the power of s, gives 

 (1, *) : the whole system therefore requires that W -1 = cos ] + J — 1 . sin 1; 

 which is thus proved previous to the equation of e"^ -1 , and cos 9 + J — l sin 9 

 from the definition of an exponent. Hence e depends only upon the an- 

 gular unit, which may be a degree, a minute, a right angle, or any 

 other, provided that e be taken accordingly. The proof that e = 1 + 1 

 -s- | + ... , when the angle 1 is that which has an arc equal to the 

 radius, must be a subsequent matter. 



If \ (r, p) represent the logometer, or complete algebraical logarithm, 

 of (r, p), the equation (r, p) — e x ( r, '' ) is an identical one; for the logometer 

 of e, or \(e, 0), being (1, 0), say that (t, r) is that of (r, p), whence, by 

 definition, K l, 0) x (t, r) or (/, t) is the logometer also of t^i r >i > ). Hence, 

 making 9 — r, we have e T ^ / " 1 = - 1, and taking the obvious truth that 

 lines equal (both in length and direction) have the same logometers, we 

 have 



M-i) 



a proposition which, not many years since, was one of the mysteries 

 of analysis. It is now a very simple geometrical proposition : the first 

 side means a line of -k units laid down positively on the unit-line ; the 

 second side means the logometer of a negative unit turned back through 

 a right angle. Now the logometer of a negative unit is a line of it units 

 erected positively perpendicular to the unit-line: whence the identity 

 of the two sides is manifest. 



On the analogy of the complete definition of 72 s with that of a h 

 in arithmetic, it can only be said that, so far as the latter is intelligible, 

 it is seen to coincide with the former : while the former itself intro- 

 duces an element which seems, up to this time, to defy analogy drawn 

 from arithmetic ; namely, the representation of a projection on the unit- 

 line by a logarithm, and of one on the perpendicular to it by an angle. 

 We see how this happens in the deduction of &J-\ = cos 9 + J— 1 sin 9, 

 and we also see that the general definition of an exponent may be derived 

 from the idea of exposition (to use an old phrase) of one symbol by an- 

 other, in such a manner as to reduce multiplication to the result of 



