PROPERTIES of the CIRCLE. 25 
Wuererore the fquares of the perpendicular diftances of ei- 
ther P or Q, from diameters paffing 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 = 72 —7s, (s 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 =247?—2rs. This is alfo 
evident, from all angles in a femicircle being equal to right ones. 
For AP +AQ + BP +BQ_+CP +cQ + & =27x PQ 
=4nr;and42r—anr—2rs=2nri—ars 
ConsEQUENTLY, when the whole circle is divided into equal 
parts, in the points A, B, C, &c. Ap + Bp +p +&a= 
Aq + Bq + Gq + cea SS nr nXxOp ; and AV + BV + 
CV + &. =AW +BW +CW +&c.=27°7+2%0V. For 
the fum of perpendiculars drawn from f to the fides of any 
regular figure circum{cribing the circle, is then equal to the fum 
of the perpendiculars drawn from g to the fides of the fame 
figure. The fame obfervation holds with regard to perpendicu- 
Jars 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 Seas theorems can be im- 
mers derived. F 
I sHatt, however, proceed regularly with the inveftigation ; 
and, in the firft place, take the fquares of the perpendiculars 
from P and Q to the right lines touching the circle in the points 
A, B, C, &c. which perpendiculars are refpeCtively equal to A 4, 
Mes BS Bis Cd, Ob; &é 
Vout. VI.—P. I. D Now 
