PROPERTIES of the CIRCLE. 6r 
AC’ + 3AC x AB+3ACx AB _ 
F: AC x AB’, and AC’ = 4. 
AC + AB’ — AB’ 
Df 8 
the cube of radius is equal to twice the difference between the 
cubes on the perpendicular to the fide of the infcribed pentagon, 
and half the fide of the infcribed decagon. 
x 2=2XCG—AF. Thus, in any circle, 
Proposition. Let any. regular figure of an odd number of 
fides, be circumf{cribed about a circle, and let (7) be any odd 
number, lefs than the number of the fides of the figure; and 
from any point within the figure let perpendiculars be drawn 
to the fides of the circumfcribing figure ; then the fum of the (2) 
powers of the parts by which thofe pérpendiculars, which are 
greater than radius, exceed it, is equal to the fum of the () pow- 
ers of thofe parts by which the perpendiculars, which are lefs 
than radius, fall fhort of it. 
_ Hence thefe problems, 
_Havine two equal given right lines, to cut one of them into 
two parts, and the other into three, fo that the cubes on the two 
parts, into which one of them is cut, fhall, together, be equal to 
the cubes on the three parts, into which the other is cut, taken ~ 
together. 
Anp having two equal right lines given, to cut one of them 
into feven parts, and the other into eight, fo that the cubes, the 
‘sth powers, the 7th, gth, 11th and 13th powers, of the feven 
parts, into which the one is cut, fhall, together, be refpectively_ 
equal to the cubes, the 5th, the 7th, the gth, the 11th, and the 
13th powers, of the eight parts, into which the other is cut. 
THE firft of thefe two problems is effected by a pentagon, 
infcribed in a circle; and the fecond, by a quindecagon in{cri- 
bed. 
Ir 
