PROPERTIES of the CIRCLE. 63 



r 3 =r r* X GS -f- r X (3S . But BS is cut in G, in the fame 

 manner as GC is cut in S. Wherefore, if another circle be de- 

 fcribed, with BS as radius, and a line be drawn from one of the 

 angles of a fquare, defcribed on the diameter, through the centre, 

 to meet the circumference in a point, and if this point, and the 



other oppofite angle of the fquare be joined, 2 r + GS — r 3 



will in like manner be = 4 X r -f- GS , or 4 X BS , and 7/* -f 1 2 r . 



GS + 6r. GS +GS =4r 3 -f-i2r\GS-f 12 r. GS -f 4. GS . 

 Therefore 3 rs - 6 r. GS* + 3 GS 3 , and r 3 = 2 r. GS -f GS , 

 = r\ GS + r. GS*. Therefore 2 r. GS -f ^s' = S -f r. GS, and 



GS* + r. GS = r% and GS 3 = r\ GS — r. GS*. 



- If, therefore, from any point in the circumference of 

 the circle BDC, perpendiculars be drawn to the fides of 

 any regular figure circumfcribed about it, the fum of their 



cubes being = N X -. r 3 , (calling N the number of the 



2 



fides of the figure), is = N X 5 r. GS* + N X £„ GS ' j 

 and twice the fum of the cubes of thefe perpendiculars is N X 5. 



GS -f-NXior. GS ; that is, equal to five times a multiple by 



the number of the fides, of the figure of the cube on the fide of 

 an infcribed regular decagon, and ten times a multiple, by the 

 fame number, of the folid, which has the fquare of the fide of 

 the infcribed decagon for its bafe, and radius for its altitude ; 

 and if the perpendiculars be drawn from any point P, within the 

 circumfcribed figure, that is, not in the circumference of the 



circle-. 



