82 OF ACCELERATING FORCES. 



234, Theorem. The forces are as the 

 spaces directly, and the squares of the times 

 inversely, beginning from the state of rest : 

 they are also as the squares of the velocities 

 directly, and as the spaces inversely. 



2iT w 



Since xi=.^ at ^, azz — : and since v^zza^t^, azz — 

 ^ tt att 



" Tx 



Scholium. Thus it ma)-^ be shown by experiment, that 

 if a body falls through one foot in a second by means of a 

 certain force, it will require a quadruple force to make it 

 fall through the same space in half a second ; and that, in 

 general, where the spaces are equal, the forces are as the 

 squares of the velocities. 



235. Theorem. The fluxions of the 

 squares of the velocities are as the fluxions of 

 the spaces, and as the forces conjointly, whe- 

 ther the forces be uniform or variable. 



In the evanescent time t\ the variation of the force 

 vanishes in comparison with the whole, so that it may be 

 considered as a uniform accelerating force, and v'n^at^ 

 (230); consequently dvzzad^: but d^i=?;df (231) ; there- 

 fore a^t^x^v^tdiV, and ad^rzzudvzzid (t;^) (49). 



Scholium. This proposition is one of the most im- 

 portant of the discoveries of Newton ; and it is of con- 

 sequence to bear in mind, that wherever the space and the 

 force remain the same, whether the force be uniform or 

 aot, the squares of any two velocities, with which a bodv 



