41 



body falls about this space in one second upon the surface 

 of the earth. Therefore the force which deflects the moon 

 from the tangent of her orbit, is of the same amount, and 

 acts in the same direction, and follows the same proportions 

 to the time that gravity does. But if the moon is drawn 

 by any other force, she must also be drawn by gravity ; and 

 as that other force makes her move towards the earth 16 

 feet ^ 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. The great discovery of the law of 

 the universe, therefore, is unfolded in the very beginning 

 of the Principia. But the rest of the work is occupied 

 with tracing the various consequences of that law, and 

 first of all in treating generally of the laws of curvilinear 

 motion. The demonstration of the moon s deflection has 

 been now anticipated and expounded from the Third 

 Book, where it is treated with even more than the author s 

 accustomed conciseness. But there seemed good ground 

 for this anticipation, inasmuch as the Scholium to the 

 Fourth Proposition refers in general terms to the con- 



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

 of Book I., as well as by means of the proposition of which we have now 

 been tracing the consequences (Prop. IV). 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. 



