Mr. Hellins on the Rectification 
+ 
g \/(yy + i) == 11-2361,025, 
~““p B = °'° 0 00,297, 
sum of affirm, terms -{“ 1 1-2361,322; 
“ = 0 '° 44 & 4 < 7 2 > 
~ 4 ^ C= °' 0000 > 001 ’ 
— d — 0-1803,793, (Art. 15,) 
neg. terms and corr. O' 2 250,266; 
— 11 — * the sum is — 66. 
the difference is == 11*01 11,056, 
which is = z. 
28. Having now produced series, of good convergency, for 
computing the length of the arch from the vertex to the ordi- 
nate, (and consequently any portion of such an arch,) of any 
conical hyperbola, I shall conclude this Paper with a few 
remarks : reserving some other theorems which I have disco- 
vered for the purpose, till I shall have found an opportunity to 
describe nearly an equal number of theorems, which I have long 
had by me, for the Rectification of the Ellipsis. 
The utility of hyperbolic and elliptic arches, in the solution 
of various problems, and particularly in the business of com- 
puting fluents, has been shown by those eminent mathema- 
ticians, M c Laurin, Simpson, and Landen ; the last of whom 
hath written a very ingenious paper on hyperbolic and elliptic 
arches, which was published in the 1 st volume of his Mathema- 
tical Memoirs , in the year 1780. I have indeed heard, that some 
improvement in the rectification of the ellipsis and hyperbola 
had been produced, and some of the same theorems discovered, 
by a learned Italian, many years before Mr. Lan den’s Mathe- 
matical Memoirs were published; but, as Mr. Landen has 
declared that he had never seen nor heard any thing of that 
work, and as various instances are to be found of different men 
