Page:Newton's Principia (1846).djvu/463

$\scriptstyle \times {\overline {MC+mC}}$, or 2XY $\scriptstyle \times$ CY, is to 2AH $\scriptstyle \times$ 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 l, 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 2AC²; and the same number of CX² to as many AC² as 2CX² to 2AC². Wherefore the united forces 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 circumference 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 IK 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 circles is to the force of the like number of particles situated in the circumference 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 $\scriptstyle \times$ CX² + 3CX⁴ to as many AC⁴ - AC² $\scriptstyle \times$ CX²; that is, as the whole fluent quantity, whose fluxion is AC⁴ - 4AC² $\scriptstyle \times$ CX² + 3CX⁴, to the whole fluent quantity, whose fluxion is AC⁴ - AC² $\scriptstyle \times$ CX²; and, therefore, by the method of fluxions, as AC⁴ $\scriptstyle \times$ CX - ⁴⁄₃AC² $\scriptstyle \times$ CX³ + ³⁄₅CX⁵ to AC⁴ $\scriptstyle \times$ CX - ⅓AC² $\scriptstyle \times$ CX³; that is, if for CX we write the whole Cp, or AC, as ⁴⁄₁₅ AC⁵ to ⅔AC⁵; that is, as 2 to 5.   Q.E.D.

LEMMA III.

The same things still supposed, I say, in the third place, that the motion of the whole earth about the axis above-named arising from the motions of all the particles, will be to the motion of the aforesaid ring about the same axis in a proportion compounded of the proportion of the matter in the earth to the matter in the ring; and the proportion of three squares of the quadrantal arc of any circle to two squares of its diameter, that is, in the proportion of the matter to the matter, and of the number 925275 to the number 1000000.

For the motion of a cylinder revolved about its quiescent axis is to the