2i6 A Short History of Astronomy [CH. ix. 



the sun in a path which does not -differ much from a 

 circle. If we assume for the present that the path is 

 actually a circle, the planet must have an acceleration to- 

 wards the centre, and it is possible to attribute this to the 

 influence of the central body, the sun. In this way arises 

 the idea of attributing to the sun the power of influencing 

 in some way a planet which revolves round it, so as to 

 give it an acceleration towards the sun ; and the question 

 at once arises of how this "influence" differs at different 

 distances. To answer this question Newton made use of 

 Kepler's Third Law (chapter vn., 144). We have seen 

 that, according to this law, the squares of the times of 

 revolution of any two planets are proportional to the cubes 

 of their distances from the sun ; but the velocity of the 

 planet may be found by dividing the length of the path 

 it travels in its revolution round the sun by the time of 

 the revolution, and this length is again proportional to the 

 distance of the planet from the sun. Hence the velocities 

 of the two planets are proportional to their distances from 

 the sun, divided by the times of revolution, and conse- 

 quently the squares of the velocities are proportional to 

 the squares of the distances from the sun divided by the 

 squares of the times of revolution. Hence, by Kepler's 

 law, the squares of the velocities are proportional to the 

 squares of the distances divided by the cubes of the dis- 

 tances, that is the squares of the velocities are inversely 

 proportional to the distances, the more distant planet 

 having the less velocity and vice versa. Now by the 

 formula of Huygens the acceleration is measured by the 

 square of the velocity divided by the radius of the circle 

 (which in this case is the distance of the planet from the 

 sun). The accelerations of the two planets towards the 

 sun are therefore inversely proportional to the distances 

 each multiplied by itself, that is are inversely proportional 

 to the squares of the distances. Newton's first result 

 therefore is : that the motions of the planets regarded as 

 moving in circles, and in strict- accordance with Kepler's 

 Third Law can be explained as due to the action of the 

 sun, if the sun is supposed capable of producing on a 

 planet an acceleration towards the sun itself which is 

 proportional to the inverse square of its distance from 



