121 
of the Hyperbola . 
actions for 1802. And as those descending series differ from 
the ascending ones by a constant quantity, (as is there shown,) 
I will now add something to what was then said on the me- 
thods of computing the value of that constant quantity. 
14. It is very evident from what is done in the Philosophi- 
cal Transactions for 1802, from page 460 to page 465, that 
the constant quantity here spoken of is no other than the dif- 
ference between the length of the arch of the hyperbola and 
its tangent, “ when the point of contact” (to use Landen’s 
words,) “ is supposed to be carried to an infinite distance from 
“ the vertex of the curve :” * — which difference is undoubtedly 
the same as “ the difference between the length of the arch 
“ infinitely produced and its asymptote” — (as Simpson -f- ex- 
presses it.) And since each of these eminent mathematicians, 
and Mac Laurin also, (as I before observed,) has treated of 
this difference, it seems requisite that I should here give a 
brief statement of their methods of computing it, and compare 
them with such of my own as I offer to the public. 
1 5. If 1 be written instead of a, in Art. 435 of Simpson's 
Fluxions, the eccentricity will be ■*/ 1 -f- bb , which is denoted 
by e in this Paper ; and his dd = ~f b will be = ™ = es. Sub- 
stituting, therefore, 1 for a , and e for d , in the series by which 
he expresses the difference between the length of the asymp- 
tote and the infinite arch, it becomes 
k + 
+ 
3-3 s 
2.2.4 “ 2. 2. 4. 4.6 ' 2. z. 4.4. 6.6.8 s 
+ 
3-3-5 -5 £7 
&C. 
his A being = the quadrantal arch of a circle of which the 
radius is 1. 
In like manner, 1 being written instead of a, in the second 
* See Landen’s Memoirs, Vol. I. p. 23. 
MDCCCXI. R 
f See his Fluxions, Art. 435. 
