1 66 On the ORIGIN 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 ftill 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. 



It is unnecefTary 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. III. P R O B. Fig. 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 nraight lines be drawn to 

 the given points E and D, thefe ftraight lines fhall have to 

 one another the given ratio of a, to (B *, 



Suppose the problem refolved, and that F is found, fo that 

 F E has to F D the given ratio of a to /3. Produce E F any how 

 to B, bifecl the angle EFD by the line FL, and the angle 

 D F B by the line F M. 



Then, becaufe the angle E F D is bifecled by F L, EL is to 

 LD as E F to FD, that is, in a 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 to j3 is fuppofed that of a greater to a lefs. 



