PROPERTIES of the CIRCLE. 29 



to fpeak algebraically, eight times the fum of the fourth powers 

 of perpendiculars from either P or Q^ to the fides of a regular 

 figure of a greater number of fides than four circumfcribed 

 about the circle, and touching it in the points A, B, C, &c. are 

 equal to thirty-five times the multiple of the fourth power of 

 radius by the number of the fides of the figure. This is Dr 

 Stewart's 25th theorem. 



And if a regular figure of a greater number of fides than four 

 be infcribed in the circle, having its angles in the points A, B, C, 

 &o the eighth powers of the chords drawn from either P or Q^ 

 to the points A, B, G, &c. (to fpeak algebraically) is equal to 



70 n r 8 — n x — -♦ 2K r s = feventy times a multiple of the 



eighth power of radius by the number of the fides of the fi- 

 gure. 



In like manner, 



r + Of + r — Oc 5 , , e — 2 10 o? 



■ -\ =2r3 + 2orXOt 4.J., , 



r r 



J — — — = 2r3 + 2orxO/+ io x ^- r 



_ = 2 r3 4. 20 rxO^ +10X- 



r f 



&C &c 



Wherefore r +° c -frr-Or + r-fO/ 4-r-O/ , H^ 5 +7=gg 



&c. = 2»r3+iotfr 3 +.J 1 = .SZ — equal (by Cor. 3.) to 



AP + BP° 4- CP° + 8cc. + AQ° + BQ° + CO ° 4- &c. when 



d 5 /•' ! 



the circle is equally divided in the points A, B, C, &c. 



And 



