Motion  by  Impact  and  Gravity . 375 
rightly  affigned,  and  would  always  be  found  confonant 
to  experience,  yet  I was  extremely  furprized  to  find,  that 
when  the  quantity  a in  the  fecond  experiment  was  made 
exadtly  one-half  of  what  it  was  in  the  firft,  the  time  of 
defcending  through  the  fame  fpace  came  out  nearly  dou- 
ble of  what  it  was  before,  and  the  velocity  the  fame. 
Now  this  I knew  could  never  happen  unlefs  the  force  in 
the  firft  cafe  was  to  the  force  in  the  fecond  as  4 to  1 ; for 
when  the  fpaces  defcribed  are  the  fame,  the  accelerating 
forces  are  always  as  the  fquares  of  the  velocities,  or  in- 
verfely,  as  the  fquares  of  the  times.  This  confideration 
led  me  to  inquire  farther  into  the  ratio  of  thofe  forces  in 
the  cafe  defcribed,  in  order  to  difcover,  if  poffible,  whe- 
ther they  came  any  thing,  near  that  ratio,  which  of  ne- 
eeffity  they  ought  to  do; 
I confidered,  that  the  weight  of  the  axis  and  arms  of  the 
machine  was  inconfiderable,  compared  with  the  weight  of 
the  two  cylinders  of  lead,  and  alfo  that  the  quantity  a bore 
a very  fmall  proportion  to  the  length  of  the  cylindrical 
arms  of  fir.  And  fince  the  accelerating  force.is  always  as 
)°ras^^T7>  and  the  quantity  abv  or  e exprefles 
the  fum  of  all  the  particles  multiplied  by  the  fquares  of 
their  diftances  from  the  axis  of  motion,  it  is  plain  that 
E muft  far  exceed  alp\  and,  laftly,  fince  the  quantity  e ia 
6 the 
