38s Mr. Micuent onthe Means of difcovering the - 
whien it began to be urged by the aforefaid forces, its, velocity . 
would then be always proportional to the {quare Foot of the. 
fum or difference of the aforefaid area, and another area, whofe 
{quare root would ‘be proportional to the velocity which the 
body had before it began to be fo urged; that is, to the fquare 
root of the fum of thofe areas, if the motion acquired was in. 
the fame direction as the former motion, and the fquare root of 
the difference, if it was in a contrary direction. See cor. 2. to 
the abovefaid propofition. 
7. In order to find, by the foregoing propofition, the velo- 
city which a body would acquire. by falling towards any other 
central body, according to the common law of gravity, let Cin 
the figure (tab. III.) reprefent the centre of the central body, to- 
wards which the falling body is urged, and let CA bea line drawn 
from the point C, extending infinitely towards A. If then the 
line RD be fuppofed to reprefent the force, by which the fal- 
ling body would be urged at any point D, the velocity which it 
would have acquired by falling from an infinite height to the 
place D would be the fame as that which it would acquire by 
falling from D to C with the force RD, the area of the infi- 
nitely extended hyperbolic {pace ADRB, where RD is always 
inverfely proportional to the {quare of DC, being equal to the 
rectangle RC contained between the lines RD and CD. From 
hence we may draw the following corollaries. 
8. Cor. 1. The central body DEF remaining the fame, and 
confequently the forces at the fame diftances remaining the 
fame likewife, the areas of the retangles RC, rC will always 
be inverfely as the diftances of the points D, d from C, their 
fides RD, rd being inverfely in the duplicate ratio of the fides 
CD, Cd: and therefore, becaufe the velocity of a body falling 
from an mfinite height towards the pot C, is always in the 
fub-. 
