NEWTON S PKINCIPIA. 



19 



The other force acting once for all would make it, were 

 there no gravity acting, move in spaces proportioned to the 

 times simply. The latter or projecting force would make 

 it move through AB uniformly, or in spaces proportional 

 to the times ; the force of gravity would make it move 



through AP with a motion proportioned to the square 

 of the times ; therefore it will move in a curve passing 

 through M, if P M is equal, and parallel to AB; and 

 AP will be as the square of AB or PM, which is the 

 property of the conic parabola in . A P = P M 2 , m being 

 the parameter to the point A. 



The Scholium concludes by stating some consequences 

 of the equality of action and reaction, the third Law 

 of Motion, with respect to oscillation and impact, and 

 also with respect to mutual attractions ; of which conse 

 quences the most important is that the attraction or weight 

 of heavy bodies in respect of the earth, and of the earth 

 in respect of them, is equal. 



The great work itself, after these preliminary though 

 essential matters, proceeds to its proper subject. But 

 in order to show how the demonstrations are conducted, 

 a short treatise is prefixed upon the method of Prime and 

 Ultimate Ratios, in eleven Lemmas, with their corollaries. 



This method consists in considering all quantities as 

 c 2 



