54 OUTLINES OF NATURAL PHILOSOPHY. 



stance, it is at the beginning of the motion that the 

 saltus is made, the body is at the same instant both 

 at rest and in motion. 



100. Let AB (fig. 7.) be a straight line, along 

 which a body is accelerated by forces directed from 

 A to B ; let the perpendicular AD be equal to the 

 velocity which a body, accelerated by the force at 

 A, would acquire in one second ; and let DEF be 

 a curve so related to AB, that at any point what- 

 ever C, the ordinate CE may be to AD as the ac- 

 celerating force at C to the accelerating force at A; 

 then, if the body begin to descend from rest at A, 

 the square of its velocity at any point C, is double 

 of the curvilineal area ACED. 



For, by the construction, CE is proportional to the force 



at C, or is =-v, v being the velocity which the mo- 

 t 



ving body has acquired at C, and t the time of the 

 descent from A to C. Now C c is the momentary 

 increment of AC the space, and is therefore = v i ; 

 therefore, CE x C c= vv, and 2 CE x C c= 2 v v. 

 But CE x Cc is the momentary increment of the area 

 ACED, and 2 v v is the momentary increment of tr ; 

 therefore the square of the velocity of the moving 

 body, and twice the area of the curve ACED, in- 

 crease at the same rate, and they also begin to exist 

 &t the same time ; therefore they are always equal. 

 j's Prin. Lib. i. Prop. 39. 



a. If 



