4Q0 PHILOSOPHICAL TRANSACTIONS. [aNNO I7OO. 



be equal to -Lr X 2 rp, or equal to the product of -iR, the half of the common 

 altitude of the wedges into the aggregate of the bases 2 kp. 



Which aggregate of the bases is the curved cylindric superficies itself, which 

 is equal to p X 2 r, or to the product of p, the periphery of the circular base, 

 drawn into the altitude 2 r, or equal to 4-rp x 4, four of the great circles 

 of the sphere. To which if we add tlie two opposite circular bases, there will 

 be made the whole superficies of the cylinder circumscribed to the sphere, 

 equal to six great circles, 4,- rp X 6 = 3rp. And the magnitude of the cylinder 

 = RRP = i RP X 2r, equal to the product of the circular base ^Ee drawn into 

 the altitude 2 R, as before. 



Now if the vertices of all these wedges that constitute the axis of the cylin- 

 der are conceived to be contracted into one point, so that these wedges or 

 prisms may now become so many pyramids, being on the same bases and the 

 same height, each will be of each, and therefore all of all, in a proportion sub- 

 sesqui-tertial, or as -^ to 4; and the superficies, which before was curved cylin- 

 drical, will now become spherical, because of all its points being equally remote 

 from the centre, the aggregate of the bases remaining as before, =: 2 rp, or 

 equal to four great circles, we shall then have the whole superficies of the 

 sphere = 2 rp =; ,' rp X 4 = four great circles; and equal to the whole curved 

 cylindric surface, and the parts respectively equal to the parts that belong to the 

 same parts of the axis; also the magnitude of the sphere |- rrp = -^ rp X 2 r, 

 equal to the product of 4- R? a third part of the common altitude of all the 

 prisms, drawn into 2 RP the aggregate of the bases, which is now become the 

 spherical surface. 



Therefore both the superficies and magnitude of the cylinder circumscribed 

 to the sphere, is sesqui-alter to the superficies and magnitude of the inscribed 

 sphere, or as 3 to 2 : there because the proportion is as six great circles ^ 3 pp 

 to four great circles = 2 rp ; here because the proportion is as rrp to -5- rrp : 

 which is the very invention of Archimedes so much celebrated. 



The same would be had a little shorter, if in the parallelopiped upon the 

 plane base 2 rp, of the altitude r, composed of minute parallelopipeds, all their 

 vertices were immediately supposed to be contracted into one point c : that the 

 aggregate of the bases continuing as before = 2 rp, those parallelopipeds may 

 be reduced to so many pyramids, having their vertices meeting at the centre of 

 the sphere, whose radius r is the common altitude of all the pyramids, and the 

 spherical superficies is the aggregate of all the bases. 



For i R, a third part of the common altitude, drawn into 2 rp, the aggregate 

 of the bases, exhibits as before the magnitude of the sphere ^ rrp, and the 

 surface of the sphere = 2 rp. 



