390 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
If we assume the theory of logarithms as known, we may at once arrive at this value, 
C (10 
for in general (sec 6+ tan 0) ; 
and as this is to be 1, we must have sec0+ tan0=e, as before. 
LXII. If we now extend this inquiry, and ask, what is the magnitude of the 
amplitude of the arc of the parabola which shall render the difference between the 
parabolic arc and its protangent equal to n times the distance between the focus and 
the vertex; we shall have, as before, by the terms of the question. 
But in general 
hence we must have 
{h.Q ) — g sec0 tanQ^wg. . 
{k.Q)-g sec9 tan0 =gJJ^ ; 
(348.) 
or sec0+ tan0=e’* (349.) 
Now we may solve this equation in two ways ; either by making w a given number, 
and then determine the value of sec^+t^iB^j which may be called the base. Or 
we may assign an arbitrary value to secQ-ftRR^j and then derive the value of ?i. 
Taking the latter course, let, for example, 
sec0+tan0= 10. Then w=logl0, 
or ^ is the modulus of the second system of logarithms. Hence, if we assume any 
number of systems of logarithms on the same parabola, and take their bases 
g(sec0+tan0), g(sec0'+tan0'), g(sec0"+tan6"), ...&c., 
the moduli of these successive systems will he the ratios of half the semiparameter to the 
successive differences between the base parabolic arcs and their protangents. 
In the Napierian system, g the distance from the focus to the vertex of the para- 
bola, is taken as 1. The difference between the parabolic arc and its protangent, when 
equal to g, gives g(sec0-|-tan0)=eg. In the decimal system g(sec0^-l-tan0^)=: lOg, 
and the difference between the corresponding parabolic arc and its protangent being 
wg, if we make this difference ng equal to the arithmetical unit, we shall have wg=:I, 
or g=^=modulus of the system. Hence in every .system of logarithms whatever, g 
the distance between the focus and the vertex of the parabola, is the modulus of the 
system. Every system of logarithms may be derived from the same parabola, but 
the Napierian system, in which the focal distance of the vertex is itself taken as the 
unit, may justly be taken as the natural system. In the same way w’e may consider 
that to be the natural system of circular trigonometry, in which the radius is taken 
as the unit. The modulus, in the trigonometry of the parabola, corresponds with the 
radius in the trigonometry of the circle. But while in the trigonometry of the parabola 
the base is real, in the circle it is imaginary. In the parabola, the angle of the base 
is given by the equation sec0-l-tan0=e. In the circle cos^+\/— 1 sinS^=e^'^"', and 
