ATTRACTION. 17 



of time is equal to the space described from the 

 beginning of the fall. But the sum of all lines 

 parallel to B C, taken indefinitely near to each 

 other, constitutes the area of the triangle. There- 

 fore the space described by a falling body, in the 

 time expressed by A B, with an uniformly accele- 

 rated velocity, of which the last degree is expressed 

 by B C, will be represented by the area of the 

 triangle ABC. 



Let us now suppose, that gravity ceased to act, 

 and that the body moved during another portion 

 of time, B F, equal A B, with the acquired velocity 

 represented by B C. As the space moved over is 

 found by multiplying the velocity by the time, the 

 rectangle BCFG will represent the space moved 

 over in this second portion of time, which is twice 

 the triangle ABC; and consequently twice the 

 space is moved over with the accelerating velocity 

 in the same time. 



But if we suppose gravity still to act, besides 

 the space BCFG, which it w r ould have moved 

 over by its acquired velocity, we must add the 

 triangle C G H for the effect of the constant 

 action of gravity; therefore, in this second portion 

 of time, the body moves over three times the space 

 as in the first. In like manner, it may be easily 

 seen by the figure, that in the next portion it 

 would move over five times the space; in the next, 

 seven times the space ; and so on in arithmetical 

 progression. And as the velocities of falling 

 bodies are in proportion to the spaces run over, it 

 follows that the velocities in each instant increase 

 as the numbers 1, 3, 5, 7» 9, &c. 



It follows from this, that the space run over is as 

 the square of the time; that is, in twice the time, 

 a body will fall with four times the velocity; in 



vol. i. c 



