DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 391 
making ^=1, we get cos(l)4-\/— 1 sin(l)=e'^"\ Hence while e' is the para- 
bolic base, e'^~‘ is the circular base. Or as [sece+tane] is the Napierian base, 
[cos(l)+\/ —1 sin(l)] is the circular or imaginary base. Thus 
[cos(l)+\/ —1 sin(l)]^=cosS^+\/~ 1 sinS-. 
Hence, speaking more precisely, imaginary numbers have real logarithms, but an 
imaginary base. We may always pass from the real logarithms of the parabola, to 
the imaginary logarithms of the circle, by changing tan0 into 1 sin^, sec0 into 
cosS^, and e' into e'^. 
As in the parabola the angle 9 is non-periodic, its limit being+g, while in the circle 
^ has no limit, it follows that while a number can have only one real or parabolic 
logarithm, it may have innumerable imaginary or circular logarithms. 
In the parabola we thus can show the geometrical origin of the magnitudes known 
as the base and the modulus. We might too form systems of circular trigonometry 
analogous to different systems of logarithms. We might refer the arc of a circle not 
to the radius, but to some other arbitrary fixed line, the diameter or any other sup- 
pose. Let the circumference be referred to the diameter, then t will signify a whole 
TT 
circumference instead of a semicircle, and ^ will represent a right angle. Having 
on this system, or any similar one, found the lengths of the arcs which correspond to 
certain functions, such as given sines or tangents, we should multiply the results by 
some fixed number, which we might call a modulus (2 in this example), to reduce 
them to the standard system ; but such systems would obviously be useless. 
If e be the angle which gives the difference between the parabolic arc and its pro- 
k 
tangent equal to g =2 ; is the angle which will give this difference equal to 2g, 
(e-i-e-Le) is the angle which will give this difference equal to 3g, and so on to any 
number of angles. Hence, in the circle, if be the angle which gives the circular 
arc equal to the radius, 2^ is the angle which will give an arc equal to twice the 
radius, and so on for any number of angles. This is of course self-evident in the 
case of the circle, but it is instructive to point out the complete analogy which holds 
in the trigonometries of the circle and of the parabola. 
LXIII. The geometrical origin of the exponential theorem may thus be shown. 
Assume two known logarithmic bases (seca+tana), and (secj3-f-tan|3), and let us 
investigate the ratio of the differences of the corresponding parabolic arcs and their 
protangents. 
Let sece-ftane be the Napierian base, and let one difference be xg and the other a/g. 
7 / 
The ratio of these differences is therefore -=z, if we make y—xz. Hence 
seca-l-tana=(sec€-f tane)*=e*, and (sec/3-l-tanf3)=e^ Therefore 
(seca-l-tana)^=e*'''^= (sec/3-1- tan(3)*. 
Or, as 3 /=j:z, (seca-l-tana)*=secj3-l-tan/3. 
MDCCCLII. 3 K 
