i*3 
of the Hyperbola. 
taken in a series which will always converge ; and both he 
and Mr. Simpson have actually produced such a series, the 
one in the place before referred to, and the other in Art. 435 
of his Doctrine of Fluxions. And although this series, when 
the transverse axis of the hyperbola is much greater than the 
conjugate, will converge very slowly, yet ( as I have shown 
in the Philosophical Transactions for the year 1798,) the value 
of it to seven, or even to ten places of figures is, in all cases, 
attainable. 
As Mr. Landen had the character of a man of great pro- 
bity, and as he has, in various parts of his writings, shewn 
a regard for the memory of the eminent mathematicians 
above-mentioned, I cannot account for this misrepresenta- 
tion of the matter any other way than by supposing that, 
being old when he wrote this memoir, and incumbered with 
much other business, his memory failed him. His just praise 
on this occasion is, that his solution of the aforesaid problem 
is much better than those of his contemporaries. I shall have 
occasion to speak again of this problem in my observations 
towards the end of this paper ; but now proceed to the proof 
of the main point which I had in view in writing it. 
4. If the transverse axis of any conic hyperbola be called 2, 
and the conjugate axis 2 b; and if the abscissa counted from 
the center on the transverse axis be called x, the correspond- 
ing ordinate y, and the length of the arch from the vertex to 
the ordinate PI ; then, by Simpson's Fluxions , Art. 435, we 
have H == or, (putting ee=n + bb t ) H = 
x \/ (e exx—i) „ 
(xx—* -1) i-d d 
MDCCCXI. 
9 
