NEWTON S PRINCIPIA. 79 



The moon in this way would fall to the earth in about four 

 hours less than five days.* 



The inquiry is closed with a solution of the general p 

 blem, of which the preceding solutions for the conic sec 

 tions, and for the force inversely as the squares of the 

 distances, are only particular cases ; and the times and 

 velocities are found from the places, or the places from the 

 times and velocities, where a body ascends from or de 

 scends to the centre, influenced by a centripetal force of 

 whatever kind. On the given straight line of ascent or 

 descent a curve is to be described whose co-ordinates are 

 the centripetal force at each point of the axis, or whose 

 equation isy = X, X being a function of x y the distance 

 from the beginning of the motion. The area of the curve 

 at each point is f y d x=f X d x-, and if that integral is 

 equal to Z 2 , Z is as the velocity at the distance a x, from 

 the centre. Another curve described on the same axis, 



... 1 . (*dx 



and whose equation is u = -=-, gives by its areas / -~- 



= , the time taken to move through the distance a x ; 

 it is equal to . This is easily demonstrated ; for, first, 

 if the velocity be v, and the time d t, the space being 



d x, we have the force?/ = -= ; and as dt= , there- 



d t v 



fore y = -3 , and y d x = v d v, and f y d x = ; but 



7P=.fydx\ therefore Z = - , and the velocity is as 

 the area Z. Again ; for the time in the other curve ; 



* It is comparing the greatest with the smallest things, to observe that the 

 time of the revolution of a planet round the sun, or the planetary year, 

 bears the same proportion to the time in which the planet would fall to the 

 sun, -which the square of the side of a bee s cell does to one of the six tri 

 angles, or to the sixth part of the rhomboidal plate. (See Appendix to vol. 

 i., Paley Illustrated.) 



