DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 389 
The diameter of this circle is ^secip sec)(. 
The demonstration of these properties follows obviously from the figure. 
LXI. In the trigonometry of the circle, we find the formula 
tan^'^ . tan^d tan^3 
tan3-- 
■&c. 
3^5 7 
and if we develop, by common division, the expression 
1 COS^ 
j — sm^^ ~ cos0(l-|-sin^0-l~sin*0-|-sin°0-f- &c.) and integrate, 
(a.) 
'cos0 
= sin0-i 
sin^0 . sin'^0 . sin"^ 
&c. 
(b.) 
If we now inquire, what, in the circle, is the arc which differs from its protangent, 
by the distance between the vertex and its focus ; or, as the protangent is 0 in the 
circle, and the focus is the centre; the question may be changed into what is the 
trigonometrical tangent of the arc of a circle equal to the radius. This question is 
answered by putting 1 for ^ in (a.), and reverting the series 
tan^(l) tan^(l) tan^(I) 
1 = tan(l)- 
&c. ; 
vve should get, in functions of the numbers of Bernouilli, the value of tan(l), as 
is shown in most treatises on trigonometry. 
Let us now make a like inquiry in the case of the parabola, and ask what is the 
value of the amplitude which will give the difference, between the arc of the parabola 
and its protangent, equal to the distance between the focus and the vertex of the 
parabola. Now if 6 be this angle, we must have (^ . 0)— g sec0 tan0=g. But in 
A.0 
general, (A:.0)— g sec0 tan0=g 1^^. Hence we must have, in this case, If 
C dO 
we now revert the series (b.), putting 1 for we shall get from this particular 
iu' 
value of the series, 
1 = sin04 
sin^0 . sin®0 . sin^0 
7 
&c.. 
6^ — G” ^ 
an arithmetical value for sin0. This will be found to be, smO= e being the 
base of the Napierian logarithms. Hence sec0+ tan0=e, or if we write e for this 
particular value of 0 to distinguish it from every other, and call it the angle of the base, 
sece+ tane=e (345.) 
We are thus (for the first time it is believed) put in possession of the geometrical 
origin of that quantity, so familiarly known to mathematicians, the Napierian base. 
From the above equations we may derive 
sece=: 2 j tane = 2 — , (346.) 
or tane=lT75203015, whence e=-8657606, or e=49°. 36'. 15". 
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