82 PHILOSOPHICAL TRANSACTIONS. [aNNO 1776. 



2d minute, at the end of it he will have moved the sum of the journies of the 

 1st and 2d minute, that is, in the whole 4 yards from its original place. As he 

 has now generated a velocity in the ball of 4 yards per minute, in the 3d minute 

 he must travel 4 yards to keep up with the ball, and one more in generating the 

 equal increment of velocity ; so that in tlie 3d minute, he must travel 5 yards 

 to keep up the same impelling power on the ball that he aid in the 1st minute 

 in travelling 1, so that these 5 yards in the 3d minute, added to the 4 yards 

 that he had travelled in the 2 preceding minutes, sets him at the end of the 3cl 

 minute 9 yards from where he set out, having then given the ball a velocity 

 capable of carrying it uniformly forward at the rate of 6 yards per minute, as 

 before stated. We may now leave the further pursuit of these proportions, and 

 see how the account stands. He generated a velocity of 2 yards per minute in 

 the )st minute, the square of which is 4, when he had moved but 1 yard from 

 his place ; and he had generated a velocity of 6 yards per minute, the square of 

 which is 36, at the end of the 3d minute, when he had travelled Q yards from 

 his place. Now, since the square of the velocity, generated at the end of the 

 1st minute, is to that of the velocity generated at the end of the 3d minute, as 

 4 to 36, that is, as I to Q ; and since the spaces moved through by the man to 

 communicate these velocities, are also as 1 to g, it follows, that the spaces 

 through which the man must travel, in order to generate these velocities respec- 

 tively, preserving the impelling power perfectly equal, must be as the squares of 

 the velocities that are communicated to the ball ; for, if the man was to be 

 brought back again to his original place by a mechanical power, equally exerted 

 on the man equally resisting, this would be the measure of what the man has 

 done in order to give motion to the ball. It therefore directly follows, con- 

 formably to what has been deduced from the experiments, that the mechanic 

 power that must of necessity be employed in giving different degrees of velocity 

 to the same body, must be as the square of that velocity ; and if the converse 

 of this proposition did not hold, viz. that if a body in motion, in being stopped, 

 would not produce a mechanical effect equal or proportional to the square of its 

 velocity, or to the mechanical power employed in producing it, the effect would 

 not correspond with its producing cause. ■ ■■ 



Thus the consequences of generating motion on a level plane, exactly cor- 

 respond with the generating of motion by gravity ; viz. that though in 2 seconds 

 of time the equal impulsive power of gravity gives twice the velocity to a body 

 that it does in 1 second, yet this collateral circumstance attends it, that at the 

 end of the double time, in consequence of the velocity acquired in the first half, 

 the body has fallen from where it set forward 4 times the perpendicular ; and, 

 therefore, though the velocity is only doubled, yet 4 times the mechanical power 

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