475 
of the conic Sections. 
discovering the same truth, without any knowledge of each 
other's works, I see no reason for disbelieving him. But I have 
seen no writings on this subject which contain any thing more 
than what is very common, besides those of the three gentle- 
men above mentioned, and Dr. Waring’s Meditationes Ana - 
lytic a ; and, while I have no inclination to detract from their 
merits, I may be allowed to say that I have borrowed nothing 
from their works. 
> 
29. With respect to Dr. Waring, (who was well known to 
be a profound mathematician, and I can testify that he was a 
good-natured man,) he has given, in page 470 of his Medita- 
tiones Analytic#, (published in 1776,) these two series, as ex- 
pressions of the length of an arch of an equilateral hyperbola ; 
viz. 
“ Arcus hyperbolicus exprimi possit per seriem — ~ + ~~x* 
« 1 r 7 _l - 3 x 11 — x ' 5 + -T- x' 9 , 
— 2*.2X7 d » 2 3 . 2.3XII 2*.2.34XI5 ‘ 2 5 . 2. 3. 4.5x19 
« &c. ubi x denotat abscissam ad asymptoton." 
“ Si vero requiratur descendens series, turn erit x — — x 
« _i_ _J r~ 7 -l! X~ l \ &c. quae, quoad coefficientes 
• 2 2 .zx 7 2 3 . 2.3x11 1 u 
tc attinet, prorsus eandem observat legem ac prsecedens." 
30. These series, as they now stand, are of little use. But, if 
proper corrections were applied to them, (which may easily be 
done from what has been shewn in this Paper, and in my Ma- 
thematical Essays,) and the first of them were transformed into 
another series converging by the powers of 7-^, they would 
become very useful for computing any arch of an equilateral 
* In the original, this term is erroneously printed, there being a 1 in the numerator, 
instead of a 3. 
