236 CENTRAL FORCES; 



and as that other force makes her move towards the earth 16 

 ,feet i inch, and gravity would make her move as much, her 

 motion would therefore be 32 feet | inch in a second at the 

 earth's surface, or as much in a minute in her orbit ; and her 

 velocity in her orbit would therefore be double of what it is, 

 or the lunar month would be less than 13 days and 16 hours. 

 It is, therefore, impossible that she can be drawn by any other 

 force, except her gravity, towards the earth.* 



Such is the important conclusion to which we are led from 

 this proposition, that the centripetal forces are as the squares 

 of the arcs described directly, and as the distances inversely. 

 This conclusion was the discovery of the great law of the 

 universe. The fruit of the consequences of this proposition 

 is the ascertaining the laws of curvilinear motion generally. 



The versed sine of the half of any evanescent arc (or sagitta 

 of the arc) of a curve in which a body revolves, was proved 

 to be as the centripetal force, and as the square of the times ; 

 or as F x T 2 . Therefore the force F is directly as the versed 

 sine, and inversely as the square of the time. From this it 

 follows that the central force may be measured in several 

 ways. The arc being Q C, we are to measure the central 

 force in its middle point P. Then the areas being as the 

 times ; twice the triangle S P Q, or Q L X S P is as T in the 

 last expression ; and, therefore, Q E being parallel to L P, the 



O T? 



central force at P is as ^-j^ . So if S Y be the perpen- 



O i P\ J-J v^ 



dicular upon the tangent P Y, because P E and the arc P Q, 

 evanescent, coincide, twice the triangle S P Q is equal to S Y 



O T? 



X Q P ; and the central force in P is as ^^ . Lastly, 



O JL X y - 



if the revolution be in a circle, or in a curve having at P the 

 same curvature with a circle whose chord passes from that 



* The proposition may be demonstrated by means of the Prop. XXXVI. 

 of Book I. of the Principia, as well as by means of the proposition of 

 which we have now been tracing the consequences (Prop. TV.). But in 

 truth the latter theorem gives a construction of the former problem (Prop. 

 XXXVI.), and from it may be deduced both that and Prop. XXXV. 



