PHYSICAL ASTRONOMY. 231 



a. If, instead of descending, the body that has the 

 rectilineal motion should ascend from the centre, 

 while the revolving body retires from it, the same 

 thing holds of their velocities. 



b. When in these propositions a body is said to gra- 

 vitate to a centre, or to be urged toward it by a 

 centripetal force, no supposition is implied con- 

 cerning the nature of that force, or the cause from 

 which it proceeds. Nothing more is understood, 

 than the mere existence of a fact, like the tenden- 

 cy of bodies to fall at the surface of the Earth. 



238. If a body descending from rest at a gi- 

 ven point in a straight line, toward another gi- 

 ven point to which it gravitates, have its velo- 

 city always proportional to the square root of 

 its distance from the first of these points, divi- 

 ded by the square root of its distance from the 

 second, it is urged toward this last point, by a 

 force which is inversely as the square of the dis- 

 tance from it. 



a. Let A, fig. 22. be the point from which the body 

 falls, C the centre to which it tends ; let AD, per- 

 pendicular to AC, be equal to a given line ; and 

 as the centripetal force at A to the centripetal 

 force at any point B in the descent of the body, 

 so let AD be to the perpendicular BE, and let 

 DEF be the curve which passes through the ex- 

 tremities of all the perpendiculars so drawn ; then 



twice 



