24 INVESTIGATION of fome 



chord CP, &c. is equal to the fum of the perpendiculars drawn 

 from the point P to right lines touching the circle in the points 

 A,B, C, &c. 



AQ* + BQ* + CQN-, &c. 



And — ^ ^*~2 =: fum of perpendiculars 



drawn from Qjto the fame lines. 



Again, fince by a well known property of the circle, 

 AP 2 +~Al£ = BF-f ^ = CT -fC^=&c. = 2r*-f 20^, 

 r denoting radius, the fum of the fquares of lines drawn from 

 the points A, B, C, &c. to any two points p, q, in the diameter 

 equally diftant from the centre, is = 2 n r 2 -f- 2 n X Op* zz a mul- 

 tiple of r 2 , by twice the number of the points A, B, C, &c. to- 

 gether with the fame multiple of the fquare of O p or O q. 



In like manner, aV -f AW" -f BV 2 -f BW'-f-CV'-f CW a +, &c. 

 s 2 n r % + 2 n X QV* — a multiple of r % by twice the number of 

 the points A, B, C, &c, together with the fame multiple of oV J 



orOW'- 



And fince the fquares of the chords AP, BP, CP, &c. are to* 

 gether equal to the fum of the fquares of the perpendiculars, 

 drawn from P to the right lines touching the circle in the points 



A, B, C, &c. together with the fum of the fquares of the per- 

 pendicular diftances of P from the diameters pairing through 

 thefe points, the fum of the fquares of Ap, Bp, Cp, &c. is in 

 like manner equal to the fum of the fquares of perpendiculars 

 from p to thefe lines, together with the fum of the fquares of 

 the perpendicular diftances from p to the faid diameters. 



In like manner, A? 2 + Bq* + Cq % +> &c - = fum of Squares 

 of perpendiculars from q to the lines touching the circle in A, 



B, C, &c together with the fum of the fquares of the perpendi- 

 cular diftances of q from the diameters paffing through A, B, 



C 5 &c. 



Wherefore 



