PROPERTIES of th CIRCLE. — 31 
nomial and refidual theorems, when m has to / the ratio of any 
two homogeneous magnitudes whatfoever, I muft refer the reader 
to my general demonftration of both in Baron MASEREs’s Scrip- 
tores Logarithmici, vol. 5. and to fome of the geometrical formu- 
le in my Univerfal Comparifon. , 
In like manner, if pg, p5, pi, &c. be perpendiculars refpec- 
tively to BO, CO, AO, &c. we have 7 +O i Preroys be gies OF 
A is 8 4.7405 we FSO 4 ec. 27 bP eR 
Oi + Or + Ob +&c. = 27r*-+-2. Op ,when the circle is equal- 
ly divided in the points A, B, C, &c. or when a regular figure 
is circumfcribed about it, with its fides touching it in thefe 
points. This is Dr Srewart’s third theorem, of which he gives. 
a demonftration of confiderable length, 
In like manner, 
F401 pr—Or +7408 +r—OF +7406 +7—O8 
+ &c. are equal to2"r? +67 ~ Oi 4 Og 4. Ob 4 &c. =2ar? 
+3 rxOP when the circle is equally divided.in the points, A,. 
B, C, &c. or when a regular figure circumfcribing it touches it 
in thefe points.. This is Dr Srewart’s 20th theorem. 
In like manner,. 
Sarees Nh See OS SS —— 17> 
r+0i aeaee madi aeas =0F" , 7 HOR +r—04 
x r 
+ &c. is equal’ to 2773+ 127x Oi + Og 4.05 4+ &e. + 2x: 
Pe ag) hea] Tt 
Ont Casati fe aay 
r 47° 
when the circle-is equally divided in the points: A, B, C,; &c. or 
whena regular figure, circum{cribing it, touches it in thefe points. 
Anda multiple of this by four, or eight times the aggregate of 
third: 
97 
= to22r? 1. 6rxOp x n+ 
