[ 3=t ] 



altitude, was known, viz. as i to 3 ; but all their diameters in 

 the triangle through the axis of the cone, to as many in the pa- 

 rallelogram through the axis of the cylinder, as i to 2. In like 

 manner it was known, that all the circles in the parabolic conoid 

 were to as many circles in a cylinder as i to 2 ; but all the dia- 

 meters of the former to thofe of the latter as 2 to 3. It was 

 alfo manifeft, that the right-lines of the triangle were arithmeti- 

 cally proportional, or as the numbers i, 2, 3, &c. therefore, the 

 circles of the cone (being in a duplicate ratio of their diameters) 

 as I, 4, 9, &c. Alfo the circles of the parabolic conoid (being 

 in the duplicate ratio of the ordinates, that is, in the ratio of 

 the diameters) v/ere as i, 2, 3, &c. therefore their diameters as 

 \/i, v'2, v^3, &c. He therefore hoped, that from knowing the 

 ratio of a feries of circles or fquares (which is the fame thing) 

 to as many equals, he fhould be able to difcover what was the 

 ratio of their diameters or fides to as many equals ; and that if 

 this were once proved univerfally, the quadrature of the circle 

 would follow of confequence. For as it was already known, 

 that all the parallel circles in a fphere were to as many in a 

 cylinder as 2 to 3 ; if it could be from thence difcovered what 

 was the ratio of the fum of all the diameters of the one to 

 ihe fum of all the diameters of the other, the quadrature of 

 the circle would be attained ; as the former fum conftitutes the 

 area of the circle, and the latter the area of the circumfcribed 

 fquare : the geometrical problem being tlius reduced to one 

 purely arithmetical. Obfcrving then the analogy between the 

 terms of certain infinite feries, and the ordinates of certain 

 curves, he difcovered rules for finding the fums of thefe feries, 

 and confeqtiently attained the quadrature of thofe curves whofe 

 ordinates were proportional to • the terms of thefe feries. In 



this 



