82 REPORT — 1856. 



And if we develope by common division the expression 



_i cos0_ _ e( 1 + sin- + sin^ 6 + sin" + &c.), 



co80~l-sin'0 ^ ^ 



and integrate, 



C dd C a^n • fl , sin' 6 sin* sin' ^ , « „ ,u\ 



I =1 sec0a0=sin + -f 1 -__+&c. . . (b) 



JCOS0 J 3 5 7 



If we now inquire what, in the circle, is the magnitude of the trigonome- 

 trical tangent of the arc which differs from its subtangent, by the distance 

 between the vertex and its focus ; or, as the subtangent is in the circle, and 

 the focus is the centre, the question may be changed into this other, what is 

 the trigonometrical tangent of the arc of a circle which is equal in length 

 to the radius? This question would be answered by putting I for ;& in (a), 

 and reverting the series 



, , ... tan¥l) tan¥l) tanYl) , „ , . 



l = tan(l)-— A_Z+ -^ ^ + Scc. . . . (c) 



By this process we should get, in functions of the numbert of Bernoulli, the 

 value of tan (1), as is shown in most treatises on trigonon)etry. 



Let us now make a like inquiry in the case of the parabola, and ask what 

 is the value of the subtangent of the amplitude which will give the difference 

 between the arc of the parabola and tliis subtangent equal to the distance 

 between the focus and tlie vertex of the parabola. Now if be this angle, 

 we must have n(m . d)—m sec tan 0=m. But in general, as shown in IV., 



n(m . 0)— m sec tan d=m Isec dd. 



We must therefore have, in this case, Vsec Odd=l. If we now revert the 



series (b), putting 1 for f sec Odd, we shall get from this particular value of 



the series, namely , -a, sin^0 sin^ sin'O , ^,>. 



'' l = sm0H -— -L — — - J. — — - + &c., . . . (d; 



3 5/ 



an arithmetical value for sin 0*. This we shall find to be sm 0=^- -, e 



e' + e-' 



being the number called the base of the Napierian logarithms. Hence 

 sec 0+ tan 0=e ; or if we write e for this particular value of to distinguish 

 it from every other, 



sec e + tan € = 6 = 2-718281828, &c (25) 



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 



secf=e' + e-^ tane=^-^'' (26) 



2 ' 2 ^ ^ 



or tan 6=1-175201192, whence e=-8657606, 



or €=49° 36' 49". 



*= sin 6, then 



"{l-mxej \ ^3 + 6 + 7^9 / 



or 



/l+j«n0\ ^ /l+sine\»^^ HencMed+tane-.. 



J 



