PROPERTIES of the CIRCLE. 61 



3 AC X AF, and AC? = At? +,,AC x AB+,,ACxAB 



AC -f- AB 3 AB 3 * 3 



• , 8 X 2 = 2 x CG — AF . Thus, in any circle, 



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. 



Proposition. Let any. regular figure of an odd number of 

 fides, be circumfcribed about a circle, and let («) 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 (») 

 powers of the parts by which thofe perpendiculars, which are 

 greater than radius, exceed it, is equal to the fum of the (n) pow- 

 ers of thofe parts by which the perpendiculars, which are lefs 

 than radius, fall fhort of it. 



Hence thefe problems. 



Having 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. 



And having two equal right lines given, to cut one of them 

 into feven parts, and the other into eight, fo that the cubes, the 

 5th powers, the 7th, 9th, nth and 13th powers, of the feven 

 parts, into which the one is cut, fhall, together, be reflectively 

 equal to the cubes, the 5th, the 7th, the 9th, the nth, 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 j and the fecond, by a quindecagon infcri- 

 bed. 



If 



