270 NEW SERIES for the 
cable to every poflible cafe of the problem to be refolved. Now 
this laft circumftance is the more remarkable, as it generally 
happens, that a feries which applies very well to the quadra- 
ture of a curve within certain limits, is quite inapplicable be- 
yond them. 
3. ALTHOUGH, in a general way, this Paper may be faid 
to treat of the quadrature of the Conic Sections, yet there is one 
of them, namely, the Parabola, which I fhall not at all notice ; 
becaufe, although its area may be found in a way analogous to 
that which is here employed in the cafe of the other two, yet 
the formula which would thence refult, muft, from its nature, 
be the fame as would be found by any other mode of proceed- 
ing. 
As the quadratures of the ellipfe, and any hyperbola may be 
deduced from thofe of the circle and equilateral hyperbola, I 
fhall, in the following Paper, treat only of the two laft; and 
as the quadrature of a fector of a circle, and the rectification 
of its bounding arch, are reducible the one to the other, it is a 
matter of indifference which of thefe we confider. I fhall, 
however, confine myfelf to the latter. 
4. IN treating of logarithms, I might, after the example of 
the earlier writers on this fubjeét, deduce the formule for their 
computation from thofe which we fhall find for the quadrature of 
the equilateral hyperbola. I prefer, however, treating this fub- 
ject in a manner purely analytical, without adverting at all to the 
hyperbola, being of opinion, that every branch of mathematics. 
ought, as much as poflible, to be deduced from its own pecu- 
liar principles; and therefore, that it would be contrary to 
good method, to haye recourfe to the properties of geometrical 
figure, when treating of a fubject entirely arithmetical. 
5. To 
