Motion  by  Impaft  and  Gravity . 359 
Isaac  newton.  Let  us  attend  to  the  confequences  of 
thefe  two  different  principles  in  the  very  cafe  propofed 
by  j.  Bernoulli.  And  fird,  becaufe  ma-mxjr^~-'> 
by  tranfpoiition  we  have  mx  a-x~^~ - , which  is  faying 
no  more  than  that  the  motion  loft  by  c is  equal  to  the 
fum  of  the  motions  gained  by  a and  b,  eftimated  in  the 
fame  direction  CD.  By  a fimilar  procefs  from  the  fecond 
equation,  we  deduce  nixa-\-xxa-x-znqr\  and  there- 
fore the  comparifon  of  the  two  equations  gives 
The  quantity y therefore,  or  the  velocity  of  a or  b after 
the  droke,  mud  neceffarily  be  equal  to  the  fum  of  the 
two  quantities  y and  y . In  the  figure,  let  cd  reprefent 
the  velocity  of  c before  the  ftroke,  and  ch  the  velocity- 
after  it,  and  let  fall  the  perpendiculars  nn,  dl,  upon  the 
direction  ac.  It  eafily  appears,  that  c n is  equal  to 
— and  cl  equal  to  y,  becaufe  ch  r cn  ::  cd  : cl  ::  rad. : 
cof.  lcd  ::  q :p.  And  now  the  whole  controverfy  is  re- 
duced into  a narrow  compafs ; for  whether  the  two  prin- 
ciples alfumed  by  this  author  be  confident  with  experi- 
ence or  not;  it  is  impoflible  they  fhould  be  confident 
with  one  another,  unlefs  cn  + ch  diall  be  found  to  mea- 
fure  the  velocity  of  a in  the  direftion  cl.  Suppofe  cr  to> 
be  the  velocity  of  c after  impaft,  when  all  the  bodies  are: 
perfectly 
