18 



NEWTON S PRINCIPIA. 



The Scholium next, with equal brevity, states the 

 projectile motion of heavy bodies. If a body be impelled 

 in one direction by a force producing a uniform motion, 

 and in another direction at any angle with the former 

 by a force not uniform but accelerated, the diagonals 

 which it will move through will at every instant change 

 their direction towards the quarter to which the acce 

 lerating force tends. But a series of such diagonals is 

 a polygon of an infinite number of sides, infinitely 

 small : in other words, a curve line. Now in the case of 

 a projectile, this continued or accelerating force is such 

 as to make the body, if no other force acted on it, fall 

 through spaces proportional to the square of the times. 



velocity acquired at any moment P of the time A P be P M, and because 

 the velocity uniformly increases, or as the time, P M : A P : : B C : A B, 



and therefore the line A C is a straight line, and the triangles A P M, 

 ABC, are similar. But if q N is infinitely near P M, or P q represents the 

 smallest conceivable time, the motion during that time may be conceived 

 to be uniform and not accelerated. Now the space through which any 

 body moves is as the velocity multiplied by the time (s = v f), therefore the 

 space moved through in the time ~Pq is as P q x ^N. So the space moved 

 through in the time A B will be as the sum of all the small rectangles 

 P q x N &amp;lt;/, or as the triangle ABC. But the triangle A B C is to any other 

 of the triangles APM as AB 2 : AP 2 ; therefore the spaces are as the 

 squares of the times. The great general importance of this proposition 

 which Galileo first proved, makes it necessary to have the demonstration 

 clearly fixed in the reader s recollection. 



