Page:Newton's Principia (1846).djvu/398

 motion describe at the distance of 60 semi-diameters of the earth, is nearly 15$1/12$ Paris feet, or more accurately 15 feet, 1 inch, and 1 line $4/9$. Where fore, since that force, in approaching to the earth, increases in the reciprocal duplicate proportion of the distance, and, upon that account, at the surface of the earth, is 60 $$\times$$ 60 times greater than at the moon, a body in our regions, falling with that force, ought in the space of one minute of time, to describe 60 $$\times$$ 60 $$\times$$ 15$1/12$ Paris feet; and, in the space of one second of time, to describe 15$1/12$ of those feet; or more accurately 15 feet, 1 inch, and 1 line $4/9$. And with this very force we actually find that bodies here upon earth do really descend; for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris feet, and 8 lines ½ in length, as Mr. Huygens has observed. And the space which a heavy body describes by falling in one second of time is to half the length of this pendulum in the duplicate ratio of the circumference of a circle to its diameter (as Mr. Huygens has also shewn), and is therefore 15 Paris feet, 1 inch, 1 line $7/9$. And therefore the force by which the moon is retained in its orbit becomes, at the very surface of the earth, equal to the force of gravity which we observe in heavy bodies there. And therefore (by Rule I and II) the force by which the moon is retained in its orbit is that very same force which we commonly call gravity; for, were gravity another force different from that, then bodies descending to the earth with the joint impulse of both forces would fall with a double velocity, and in the space of one second of time would describe 30$1/6$ Paris feet; altogether against experience.

This calculus is founded on the hypothesis of the earth's standing still; for if both earth and moon move about the sun, and at the same time about their common centre of gravity, the distance of the centres of the moon and earth from one another will be 60½ semi-diameters of the earth; as may be found by a computation from Prop. LX, Book I.

The demonstration of this Proposition may be more diffusely explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn: the periodic times of these moons (by the argument of induction) would observe the same law which Kepler found to obtain among the planets; and therefore their centripetal forces would be reciprocally as the squares of the distances from the centre of the earth, by Prop. I, of this Book. Now if the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest mountains, the centripetal force thereof, retaining it in its orb, would be very nearly equal to the weights of any terrestrial bodies that should be found upon the tops of those mountains, as may be known by the foregoing computation. Therefore if the same little moon should be deserted by its centrifugal force that carries it through its orb; and so be