Page:A short history of astronomy(1898).djvu/279

§ 173] distance of the moon (which is 60 times the radius of the earth, or 20,000,000 feet); that is, it is $3,300 × 3,300⁄60 × 20,000,000$, which reduces to about $1⁄110$. Consequently, if the law of the inverse square holds, the acceleration of a falling body at the surface of the earth, which is 60 times nearer to the centre than the moon is, should be $60 × 60⁄110$, or between 32 and 33; but the actual acceleration of falling bodies is rather more than 32. The argument is therefore satisfactory, and Newton's hypothesis is so far verified.

The analogy thus indicated between the motion of the moon round the earth and the motion of a falling stone may be illustrated by a comparison, due to Newton, of the moon to a bullet shot horizontally out of a gun from a high place on the earth. Let the bullet start from in fig. 71, then moving at first horizontally it will describe a curved path and reach the ground at a point such as, at some distance from the point , vertically underneath its starting-point. If it were shot out with a greater velocity, its path at first would be flatter and it would reach the ground at a point beyond ; if the velocity were greater still, it would reach the ground at  or at ; and it requires only a slight effort of the imagination to conceive that, with a still greater velocity to begin with, it would miss the earth altogether and describe a circuit round it, such as. This is exactly what the moon does, the only difference being that the moon is at a much greater distance than we have supposed the bullet to be, and that her motion has not been produced by anything analogous to the gun; but the motion being once there it is immaterial how it was produced or whether it was ever produced in the past. We may in fact say of the moon "that she is a falling body, only she is going so fast and is so far off that she falls quite round to the other side of the earth, instead of hitting it; and so goes on for ever."

In the memorandum already quoted (§ 169) Newton speaks of the hypothesis as fitting the facts "pretty nearly"; but in a letter of earlier date (June 20th, 1686)