28 INVESTIGATION of fome 



or, to fpeak algebraically, the fum of the fixth power of chords 

 drawn Trom either P or Q^ to the faid points, will be equal to 

 twenty times a multiple of the fixth power of radius, by the 

 number of the fides of the infcribed figure. 



In like manner, 



• 



4 2 ■ XT* 



Ac+Aa r + Oc +r — Oc , v - 2 ,2XOf 



-— = 2 r^ + 12 r X Oc + 



r 



B/ + B/ _ r + of + r — of -, — ? 2 x o/ 



— = — = 2r* + iir XOJ -f 



r 



4 



" 2 r3 -f 12 r X Oa + 



r r t 



&c. &c. &c. 



And third proportionals to radius, perpendiculars from P and Q^ 

 to right lines touching the circle in the points A, B, C, &c. and 

 the cubes of faid perpendiculars are, taken together, equal to 



2 n r 3 -J- 1 2 r X Oc + Of -f Od -f &c. +2X — 



r 



= (by Cor. ,.1 AP+BP+CP + &C. AQ>BQjfCCL + &c. 

 V 7 J J d*xr T d+xr 



But Oc 4* O/" + Od 4- &c. == — when the circumference 



2 



is equally divided in the points A, B, C, &c. or when a regular 



figure is circumfcribed about the circle, with its fides touching 



n r 1 

 the fame in faid points, and 12 r X " — ^~ — 6 n r$ m Alfo 



• oc t + of'+od , +ikc. is then _ azii'. Wherefore thefe 



»■ 4 



72 V 3 



third proportionals are taken together equal to 8 n r* + 3 — 



4 

 35 /; r 3 

 ~~~~7 ; and four times their aggregate is equal to $5 n *">• Or, 



to 



