144 Mr. Hellins on the Rectification 
from, or regard to, Landen’s theorem, we have obtained a 
pair of series for computing the value of d , which converge 
by the powers of and of which we can readily find as many 
terms as we please. And, by a similar process, (as was ob- 
served in Art. 26,) may Euler’s series for computing the 
quadrantal arch of an ellipsis be obtained, without any use of 
Fagnani’s theorem, or the “ tentative methods ,” and “ strange 
artifices ,” as Mr. Woodhouse * calls them, which appear in 
Euler’s paper. 
34. If we look back to Art. 18 and 20 of this Paper, we 
shall find that, when the transverse and conjugate semi-axes 
of an hyperbola are denoted by 1 and -L, respectively, (which 
hyperbola will be similar to that of which the semi-axes are 
a and i 5 ) the convergency of the first series, derived from 
Landen’s theorem, will be by the powers of the fraction 
— ■— r — ; — t, assisted by coefficients. 
1 + v (ad + 0 J 
And 
the same rate of convergency will obtain in the series given 
in Art. 29, by putting y = ; for then uu, by which the 
terms of that series are divided, will be = 1 e, e being = 
V aa -j- 1 ; so that, when aa is less than 1 -f- s/ a a -f- 1, 
the terms of the single series will decrease swifter than the 
terms of the pair of series ; and consequently half as many 
terms of the former as the latter will give a result equally 
near the truth. Fhe two quantities aa and 1 — j- aa + 1 are 
equal when aa = 3 ; and hence it appears, that the proper 
use of the pair of series above found, is, when a is considerably 
* See Philos. Trans, for 1804, p. 235. 
