VOL. XXII. "J PHILOSOPHICAL TRANSACTIONS. 489 



By a like artifice may be shown how to compare the sphere and cylinder, which 

 Archimedes thought fit to chiise for his monument. 



If to the basis p, equal to the circumference of a circle, a height r be as- 

 sumed equal to radius, there will be made a rectangular parallelogram = rp, 

 fig. 12, pi. 12. This may be conceived as composed of an infinite number of 

 small parallelograms, of the same height, according to the received method of 

 indivisibles. 



Now if the vertices of all these are conceived to be contracted into one point, 

 fig. 13, so that of those minute parallelograms as many triangles may be made, 

 having the same basis and an equal height ; each will be half of each of the 

 others, and therefore all of all ; and the base being bent into the circumference 

 of a circle, a circle will be made whose radius is r, and centre c, which there- 

 fore is half the parallelogram or -^RP- 



This is Archimedes's dimension of a circle, which is equal therefore to a 

 right-angled triangle, one of whose sides about the right angle is equal to the 

 periphery, and the other to the radius of the proposed circle. For ^r, or half 

 the altitude of the triangle drawn into p the base, exhibits the magnitude of 

 that triangle 4-kp, which is equal to the circle. And the same may be accom- 

 modated to the circular sector, taking the arch a instead of the periphery p. 



Again, if to that parallelogram = rp as a base, be taken in like manner an 

 altitude r, fig. 14, in order to a hemisphere, there will be made a paralleiopiped 

 = RRP. This in the same manner may be conceived as composed of an infinite 

 number of small parallelopipeds of the same height, insisting upon the minute 

 areas of that plane, of all which the common altitude is r, and the aggregate of 

 bases = rp. Now if this parallelogram, the magnitude rp continuing, be sup- 

 posed to be bent into a cylindrical surface, (whose base p is now bent into the 

 periphery of a circle, and whose altitude is r) that those minute parallelopipeds 

 may be changed into so many wedges, or prisms, with triangular bases, each of 

 which are half their respective parallelopipeds, and therefore all are half of all, 

 having for their vertices so many points c, or minute lines, in the axis of the 

 cylinder, and thus filling it up; the cylinder will become half the paralleiopiped, 



or iRRP. 



Or in order to come at the entire sphere, if on each side be taken the altitude 

 R, so that the whole altitude may be d = 2 r, and if a convolution be made in 

 like manner, a cylinder will be produced as before, consisting of wedges or 

 prisms infinite in number, having their points or vertices in the axis of the 

 cylinder, which will be equal to rrp= 4 rp x 2 r, equal to the product of 4 rp, 

 or the circular base, into the altitude 2 r ; or, which is the same thing, it will 



VOL. IV. 3 R 



