$ i 7 3] The Law of the Inverse Square 217 



the sun ; i.e. at twice the distance it is \ as great, at three 

 times the distance \ as great, at ten times the distance -j-J^ 

 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 ^ 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 x 5 to 12 x 12, 

 or as 25 to 144. The accelerations of Jupiter and of the 

 earth towards the sun are therefore as 25-4-5 to 144, 

 or as 5 to 144 ; hence Jupiter's acceleration towards the 

 sun is about -^ that of the earth, and if we had taken 

 more accurate figures this fraction would have come out 

 more nearly ^V 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 

 ^ith 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 down- 

 wards, 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 o\vn centre to bodies near its surface. Newton noticed 

 that this power extended at any rate to the tops of moun- 

 tains, 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 



