PROPERTIES cf the CIRCLE. i S 



Wherefore the fquares of the perpendicular diftances of ei- 

 ther P or Q^ from diameters pafling through the points of con- 

 tact A, B, C, &c, are, taken together, equal to the excefs of the 

 rectangle under half the diameter PQ^and the fum of perpendi- 

 culars from P and Q^to right lines touching the circle in the 

 points A, B, C, &c. above half the fum of the fquares of faid 

 perpendiculars = n r 2 — r s, (j being equal to the fum of per- 

 pendiculars from O, as, in what follows, to right lines touch- 

 ing the circle, of which OQ_ is the diameter, in the points 

 c, d,f, &c). And the fum of the fquares of thefe perpendicular 

 diftances from both P and Q^ is ~ 2 n r 1 — 2 r s. This is alfo 

 evident, from all angles in a femicircle being equal to right ones. 



For AP* -f ACT + BP 2 -f- BQ^-f CP *-f- CQ^+ &c - - n x ^OL 

 =: 4 71 r 2 ; and 4 n r — 2 n r % — 2 r s ~ 2 n r* — 2 r s. 



Consequently, when the whole circle is divided into equal 

 parts, in the points A, B, C, &c. Ap -f b/ + C~p + &c. = 

 Aq -f Bq -f Qq + &c. = w *;* + n X Op* ; an d AV* + BV* -f 



CV -f&c. =AW' + BW +CW+&Lc.= nr*-\-n x ov\ For 

 the fum of perpendiculars drawn from p to the fides of any 

 regular figure circumfcribing the circle, is then equal to the fum 

 of the perpendiculars drawn from q to the fides of the fame 

 figure. The fame obfervation holds with regard to perpendicu- 

 lars drawn from the points V, W. 



From the foregoing general inveftigation, when the circle is 

 fuppofed to be equally divided in the points A, B, C, &c Dr 

 Stewart's firft, fecond, third, and eleventh theorems can be im- 

 mediately derived. 



I shall, however, proceed regularly with the inveftigation; 

 and, in the firft place, take the fquares of the perpendiculars 

 from P and Qjo the right lines touching the circle in the points 

 A, B, C, & c . which perpendiculars are refpectively equal to A a, 

 Ac;B/,B^; C</,C^&c. 



Vol. VI.— P. I. D Now 



