i34< Mr. Hellins on the Rectification 
utility appears to me to be much less than has been imagined. 
It has been represented even in these Transactions, for the 
year 1804, p. 236, that Fagnani’s theorem is necessary to 
the investigation of Euler's series for computing the length 
of a quadrantal arch of an ellipsis ; yet, the fact is, that Fag- 
nani’s theorem is no more requisite on that occasion than 
Landen’s theorem is in the investigation of a similar series 
for computing the difference between the lengths of an infinite 
arch of an hyperbola and its asymptote, which will be given 
in this Paper. 
27. It appears by inspecting the values of z and z -f- d, ex- 
hibited in terms of e and /, in Art. 25, that wheny becomes 
equal to, or greater than, Theorem IV. will be more 
eligible for arithmetical calculation than Theorem II. It is 
obvious also, from the same article, that no more than one of 
those values need be computed in order to obtain the value 
of d. Putting, therefore, f for the value of % corresponding 
to y = 7? and s 
{ V e -f- ee 
L+ 
— A' — — : B 7 + 
ze 2 . 4 ^ 1 
2.4 6 e s 
C 7 , &c. 
( A 7 , B', C 7 , &c. being as there specified, ) we have f = S — d; 
and this value being written for £ in the equation d = 1 -J- e 
— 2 J, it becomes d— 1 -\-e + 2d — 2S; from which we have 
d = 2S — 1 — e, which is another convenient formula for com- 
puting the value of d. 
28. If the diagonal line of quantities in which l enters, either 
in the value of z or % d, before referred to, were written 
by itself, and the remaining perpendicular columns summed 
in the manner by me described in the Philos. Trans, for 1798, 
p. 548 et seq. the value of d might be obtained in a pair of 
