270 NEW SERIES for the 



cable to every poilible 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 (hall 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, mult, 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 hounding arch, are reducible the one to the other, it is a 

 matter of indifference which of thefe we conlider. I fhall, 

 however, confine myfelf to the latter. 



4. In treating of logarithms, I might, after the example of 

 the earlier writers on this fubjecl, deduce the formulae 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 have recourfe to the properties of geometrical 

 figure, when treating of a fubject entirely arithmetical. 



5- To 



