PROPERTIES of the CIRCLE, — 63 
y= xGS+rx GS. 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 {quare, 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, 27 -- GS — r° ’ 
will in like manner be = 4X7-+ GS ,or4XbS , and 773+ 127°. 
GS + 67. Gs +GS'=4 ri -+1277,GS + 127. es + 4. Gs. 
. Therefore 373 = 67. Gs ice 3GS*, and r? = 27.GS + GS’ 
=r. GS-++r. Gs. Therefore 27. GS hi =r+r.GS, and - 
eit as. GS = 7’, and Gs’ =r. GS—r. Gs. 
Ir, 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 xX 3. r3, (calling N the number of the 
fides of the figure), is = N X 57. Gs. +N x x GS. ; 
and twice the fum-of the cubes of thefe perpendiculars is N x oe 
Gs’ +N xX 107. 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, . 
