218 A Short History of Astronomy [CH. ix. 



terrestrial bodies wherever situated, it was probable that 

 at such a distance as that of the moon the acceleration 

 caused by the earth would be much less. Newton assumed 

 as a working hypothesis that the acceleration diminished 

 according to the same law which he had previously arrived 

 at in the case of the sun's action on the planets, that is 

 that the acceleration produced by the earth on any body 

 is inversely proportional to the square of the distance of 

 the body from the centre of the earth. 



It may be noticed that a difficulty arises here which did 

 not present itself in the corresponding case of the planets. 

 The distances of the planets from the sun being large 

 compared with the size of the sun, it makes little difference 

 whether the planetary distances are measured from the 

 centre of the sun or from any other point in it. The same 

 is true of the moon and earth ; but when we are comparing 

 the action of the earth on the moon with that on a stone 

 situated on or near the ground, it is clearly of the utmost 

 importance to decide whether the distance of the stone 

 is to be measured from the nearest point of the earth, 

 a few feet off, from the centre of the earth, 4000 miles 

 off, or from some other point. Provisionally at any rate 

 Newton decided on measuring from the centre of the 

 earth. 



It remained to verify his conjecture in the case of the 

 moon by a numerical calculation ; this could easily be done 

 if certain things were known, viz. the acceleration of a 

 falling body on the earth, the distance of the surface of 

 the earth from its centre, the distance of the moon, and 

 the time taken by the moon to perform a revolution round 

 the earth. The first of these was possibly known with fair 

 accuracy ; the last was well known ; and it was also known 

 that the moon's distance was about 60 times' the radius of 

 the earth. How accurately Newton at this time knew the 

 size of the earth is uncertain. Taking moderately accurate 

 figures, the calculation is easily performed. In a month of 

 about 27 days the moon moves about 60 times as far as 

 the distance round the earth; that is she moves about 

 60 x 24,000 miles in 27 days, which is equivalent to about 

 3,300 feet per second. The acceleration of the moon is 

 therefore measured by the square of this, divided by the 



