of the Hyperbola. 133 
The right-hand side of this equation, by multiplying both 
numerators and denominators by el, becomes el s/ 1 -f- e x : 1 
-4- -7 — 1 V - 3 ^L_ &c. which is = 
• 2e 2 .j.ee ' 2.4-6« 3 2.4 6.8e 4 ’ 11 
x v/ 1 + d- = V i e x V 1 e ~ l e. Hence we have 
2 Z, which is the very equation at the end of 
Art. 17, obtained by Landen’s method, the difference being 
only in the notation. 
It appears by this result, that the constant difference between 
the values of the ascending and descending series, here denoted 
by d, is equal to the difference between the lengths of the in- 
finite arch and its tangent, as was observed in Art. 14, and 
may be briefly proved thus : by the notation specified at the 
beginning of this Art. and the property of the curve, as 1 : e 
: : y : ey, the length of a line drawn parallel to the asymptote 
from the extremity of the ordinate to the transverse axis ; 
which line, when y becomes immensely great, will coincide 
with the tangent drawn from the same point, and will be equal 
to the corresponding portion of the asymptote. And it ap- 
pears by Art. 1 3 of my former Paper on the Rectification of 
the Hyperbola, (see Philos. Trans, for 1802, p. 461,) that the 
corresponding arch of the hyperbola, %, is = ey — d. We 
therefore have, in this case, d ■=. the asymptote — the infinite 
arch. 
2 6. Thus is Lan den's best theorem respecting the rectifi- 
cation of the hyperbola obtained by the common application 
of Sir Isaac Newton’s doctrine of infinite series. And I fur- 
ther observe, in transitu, that Fagnani’s theorem, respecting 
the rectification of the ellipsis, is attainable in the same easy 
manner. These devices are indeed very ingenious ; but their 
