PROPERTIES of the CIRCLE. 27 



ber of the fides of the infcribed or circumfcribed figure ; or to 

 fpeak algebraically, the fum of the fourth powers of thS chords 

 is equal to fix times a multiple of the fourth power of the femi- 

 diameter of the circle, by the number of the fides of the figure. 

 This is Dr Stewart's 23d theorem. 



• In like manner, 



Ac -\-Aa =r + Oc +>— Oc -2^-{-6rXOc 



By + B«? =r + 0/+r-0/=2r 3 -f-6rXO/ 



Vd +Cb =r + Od + r — Od =2r"-\-6rXOd 

 &c. &c. Sec. 



And the cubes of perpendiculars from P and Qj:o right lines 

 touching the circle in the points A, B, C, &c are taken together 



= 2 n ri -f 6 r X O? -f 6? -f Oct -f &c. = (by Corollary 3.) 



AP 6 + BP 6 + Cp' + 8cc + AQ 6 + BQ+CQ+ &c . 



d s 



But O? + Of + Od + &c - — — > wnen the circumference 



*^ Jit 



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

 figure is circumfcribed about the circle, with its fides touching 

 the fame in faid points. Wherefore the cubes of perpendiculars 

 from P and Qj:o the fides of a regular figure of a greater num- 

 ber of fides than three circumfcribed about the circle, are taken 

 together r: 5 n r 3 . This is Dr Stewart's 19th theorem. 



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

 three be infcribed in the circle, having its angles in the points 

 A, B, C, &c. third proportionals to the cube of the diameter and 

 the cubes of chords drawn from P and Qjto the points A, B, C, 

 &c. will, taken together, be equal to 5 n n ; or third proportion- 

 als to the cube of the diameter and chords drawn from either P 





or Qj:o the faid angular points, will taken together, be := 



2 



