*The Law of the Inverse Square*

the sun; *i.e.* at twice the distance it is 14 as great, at three times the distance 19 as great, at ten times the distance 1100 as great, and so on.

The argument may perhaps be made clearer by a numerical example. In round numbers Jupiter's distance from the sun is five times as great as that of the earth, and Jupiter takes 12 years to perform a revolution round the sun, whereas the earth takes one. Hence Jupiter goes in 12 years five times as far as the earth goes in one, and Jupiter's velocity is therefore about 512 that of the earth's, or the two velocities are in the ratio of 5 to 12; the squares of the velocities are therefore as 5 × 5 to 12 × 12, or as 25 to 144. The accelerations of Jupiter and of the earth towards the sun are therefore as 25 ÷ 5 to 144, or as 5 to 144; hence Jupiter's acceleration towards the sun is about 128 that of the earth, and if we had taken more accurate figures this fraction would have come out more nearly 125. Hence at five times the distance the acceleration is 25 times less.

This **law of the inverse square,** as it may be called, is also the law according to which the light emitted from the sun or any other bright body varies, and would on this account also be not unlikely to suggest itself in connection with any kind of influence emitted from the sun.

173. The next step in Newton's investigation was to see whether the motion of the moon round the earth could be explained in some similar way. By the same argument as before, the moon could be shewn to have an acceleration towards the earth. Now a stone if let drop falls downwards, that is in the direction of the centre of the earth, and, as Galilei had shewn (chapter vi., § 133), this motion is one of uniform acceleration; if, in accordance with the opinion generally held at that time, the motion is regarded as being due to the earth, we may say that the earth has the power of giving an acceleration towards its own centre to bodies near its surface. Newton noticed that this power extended at any rate to the tops of mountains, and it occurred to him that it might possibly extend as far as the moon and so give rise to the required acceleration. Although, however, the acceleration of falling bodies, as far as was known at the time, was the same for