of the Hyperbola. 123 
of the Hyperbola, and will more fully appear in the following 
pages. 
Mr. Landen’s methods of computing the aforesaid differ- 
ence next come under consideration : and, first, his method of 
computing it by means of the arches of two ellipses. 
16. We have seen above, that, when x becomes immensely 
great, u becomes = 1 ; and in this case the equation ( 6 ) in 
Art. 7 becomes H = 2 + ex -f- eE — • 2 (l-j- <?) V E; from which 
we have ex — H = 2 ( 1 -f e) V E — <?E — 2, another expression 
of the difference between the length of the asymptote and the 
infinite arch: which expression, however, is not so conve- 
nient for arithmetical calculation as the preceding. For here 
E denotes the quadrantal arch of an ellipsis of which the trans- 
verse semi-axis is 1, and the eccentricity e ; which arch is = 
N v • 1 _ — — __ 3-3-S' 6 __ 3-3-5-5-7* 8 
2.2 2. 2.4.4 2. 2. 4.4.6. 6 2.2.4.4.6.6.8.8 s 
E 
so that the computation of eE = - = 
N v • * — -L — _Jf!_ _ 3-3_‘5 a5 ... _i±i± ZlL_ 
2.2 2. 2. 4.4 2,2446.6 2.2.4.4.6.6.8.8 s 
will require as much labour as the computation of eG, which 
is the very difference sought. (See the preceding Art.) But, 
by this method, we have yet to compute the elliptic arch de- 
noted by 'E, of which the transverse semi-axis is 1 , the eccen- 
tricity ? = np (= ~’) and abscissa v = V -) . ( = ,/ (^) ) ; 
and then there must be a multiplication of this quantity by 2 
(1 + e), and, after that, a subtraction of eE + 2 from the pro- 
duct, to obtain the difference sought : all which labour is more 
than is required by the method described in the preceding 
article. Here, then, we have a striking instance (and a thou- 
sand more might be produced) of the inutility of rectifying 
the hyperbola by means of two ellipses. 
R 2 
