eat’. On be ORIGEN and 
one fide of it, is equal to the fum of the perpendiculars that 
fall on it from all the points on the other fide of it. | 
Or {till more generally, any number of points being given 
not in the fame plane, a point may be found, through which ° 
if any plane be fuppofed to pafs, the fum of all the perpendi- 
culars which fall on that plane from the points on one fide of 
it, is equal to the fum of all the perpendiculars that fall on the 
fame plane from the points on the other fide of it. 
Ir is unneceflary to obferve, that the point to be found in thefe 
propofitions, is no other than the centre of gravity of the given 
points’ and that therefore we have here an example of a Porifm 
very well known to the modern geometers, though not diftin- 
guifhed by them from other theorems. 
12. THe problem which follows appears to have led to the 
difcovery of more than one Porifm. 
PROP. IL PROB. Fie. 3. 
A circLE ABC, and two points D and E, in a diameter of 
it being given, to find a point F in the circumference of 
_the given circle, from which, if ftraight lines be drawn to 
the given points E and D, thefe ftraight lines fhall have to 
one another the given ratio of « to 6 *. 
Suppose the problem refolved, and that F is found, fo that 
FE has to FD the given ratio of « to @. Produce E F any how 
to B, bife@& the angle EFD bythe line FL, and the angle 
DFB by the line F M. 
Turn, becaufe the angle EF D is bifected by FL, EL is to 
LD as EF to FD, that is, ina given ratio; and as ED is 
given, each of the fegments EL, LD, is given, and alfo the 
point L. 
AGAIN, 
* The ratio of a tof is fuppofed that of a greater to a lefs. 
