26 INVESTIGATION of fome 



Now.Ac -f Aa —rJ^-cO +r — cO = 2 r* -f 2 X cO 



B/+B* rrr+O/ +r — O/ = 2 r* 4. 2 X O/ 



C? + C£ =r+0^ + r^—Od =2r*±2XOd 

 &c &c. & c . 



Wherefore the fum of the fquares of perpendiculars from 

 P, Q^to lines touching the circle in the points A, B, C, &o is 



= 2 n X r* + 2 XO/ + O/ + O/4-&C But the points c, d,f, 

 are in the circumference of a circle, of which the diameter is OQ^ 



or r, and by Cor. 3. the fum of Oc + Of +Od 4. &o == OQjx 

 into the fum of perpendiculars drawn from O to lines touch- 

 ing the circle, of which OQJs the diameter, in the points c, d,f 

 &o Call the fum of thefe perpendiculars s. Then we have 

 the fum of the fquares of perpendiculars drawn from P, Q^to 

 lines touching the circle APQJin the points A, B, C, &c. == 2 n -r z 



Jfirs - (Cor. 3. ) AP +BP 4-CP + &c + AQ^+BQ ^+CQ -f& c . 



~ " dT~ 



When the circumference is divided into equal parts by the" 



n 



points A, B, C, &c. or the angles at O are equal, s = - x OQ^ 



» 



or - x r and 2 n r -f- 2 r s — 3 n r*. 



If a regular figure be infcribed in the circle, having its 

 angles at the points A, B, C, &c. or a regular figure be cir- 

 cumfcribed about the circle, having its fides tangents to it 

 in the points A, B, C, &c. we get from the general expreffion 



AP 4 + BP + CP 4 + fee. or AQl + BQl+ CQ ? + &c. _ r3 

 r r ^ 



-\- 4r s — 4 n r 3 -f 2 n r'- zr 6 n r\ or third proportionals to ra- 

 dius, the chords drawn from either P or Q^to the points A, B, 

 C, fee. and the cubes of thefe chords equal, when taken toge- 

 ther, to fix times a multiple of the cube of radius by the num- 

 ber 



