1 12 
Mr. Woodhouse on the Truth of Conclusions 
Now the question concerning the logarithms of negative 
quantities, in a precise form, and freed from its verbal ambi- 
guities, is this ; is the symbol which, substituted for x in the 
developement of e x y makes y or e x = — 1, the sign of a real 
quantity or not ? 
In the expression e* y x is the logarithm of e x , and, by exten- 
sion, x \/ — ■ i is to be called the logarithm of e 
p* v / " 
1 is the symbol fori-f-^v/ — l — -~— 
xV- 
x* V" 
Now 
2 • 3 
- &c. 
1 I . 2 J . 
2.3.4 
or e x 
&c.) -f V — 1 [z — j _ 2 . 3 + &c -) 
may be said to be = cos. r-j-\/ — 1 sin. x. Hence, 
x s/ — 1 is the logarithm of cos. x + \/ • — 1 sin. x ; when the 
arc x is equal o, or 2 7r, or 4 tt, or 6 tt, or generally 2 m v, its 
cos. x = 1. 
Hence o is log. 1, 
or 2 it s/ — 1 is log. 1, 
4 %/ — 1, or generally 2 m^s/ 
or 
1, is log. 1. 
2 m w V — 1 
Hence, ify = 1, the equation y = e x becomes i = e 
[m being any number of the progression o, 1, 2, 3, 4, &c.) 
Again, if the arc x is equal tt, or 37?, or 5 7 r, or generally 
(2 w 4- 1 ) 7T, its cos. x ~ — 1. 
of— 1, either 7rv/ — 1, or 3 tvs / — i,or5 ?r\/ — i,or (2 w4-i) 7r V / — 1 5 
is to be called the logarithm; hence, if jy = — - 1, the equation 
y—. e * becomes — 1 = e^ z m + ; ~~ l . 
The meaning of the logarithms of 1 and — 1 are then thus 
to be understood. If in the series 
1 4- x 4- — 4- — 1 + &c. for x be substituted 
» ~ 1.2” 1 . 2 . 3 ~ 1. 2. 3. 4 1 
