BOOK 111. OP NATURAL PHILOSOPHY. 45? X MC'T wC^or 2XY X CY, is to 2AH X HC, the force of the same two particles situated in A, as CX2 to AC2. And therefore the difference of the parts, that is, the force of the two particles L and /, taken together, to wheel the earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX2 — CX2 to AC2. But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX2 will be to the like number of IX2 as 1 to 2 (by Lem. 1) ; and to the same number of AC2 as IX2 to 2 AC2 ; and the same number ol CX2 to as many AC2 as 2CX2 to 2AC2. Wherefore the united forcet of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX2 — 2CX2 to 2AC2 ; and therefore (by Lem. 1) to the united forces of as many particles in the cir cumference of the circle AE as IX2 — 2CX2 to AC2. Now if Pp. the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to insist, the matter in the circumference of every circle £K will be as IX2 ; and therefore the force of that matter to turn the earth about will be as IX2 into IX2 — 2CX2 : and the force of the same matter, if it was situated in the circumference of the circle AE, would be as IX2 into AC2. And therefore the force of all the particles of the whole matter situated without the sphere in the circumferences of all the circle? is to the force of the like number of particles situated in the circumfer ence of the greatest circle AE as all the IX2 into IX2 — 2CX2 to as many IX2 into AC2 ; that is, as all the AC2 — CX2 into AC2 — 3CX2 to as many AC2 — CX2 into AC2 : that is, as all the AC4 — 4AC2 x CX2 + 3CX4 to as many AC4 —AC2 X CX2 ; that is, as the whole fluent quantity, whose fluxion is AC4 — 4AC2 X CX3 + 3CX4, to the whole fluent quantity, whose fluxion is AC4 — AC2 X CX2 ; and, there fore, by the method of fluxions, as AC4 X CX — fAC2 X CX3 +