PROPERTIES of the CIRCLE. $i 



normal and refidual theorems, when m has to / the ratio of any 

 two homogeneous magnitudes whatfoever, I mud refer the reader 

 to my general demonftration of both in Baron M a seres' s Scrip- 

 tores Logarithmici, vol. 5. and to fome of the geometrical formu- 

 lae in my Univerfal Comparifon. 

 In like manner, if pg, pb t pi, &c be perpendiculars refpec- 



tively to BO, CO, AO, &c. we have r + Oi + r— O i + r + Og 



j^ r — 0/ + r + Qb 4. r — Ob 4. 8cc. = .2 » r* + 2 x 



O? + Og 2 -{-Ob -\- &c. = 2 n r % 4- #. 0/> , 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 Stewart's third theorem, of which he gives 

 a demonftration of confiderable length. 



In like manner, 



r + Qi +r — Oi i 4-r + o/ + r — Q g 3 + 7~+Ob 3 4" r — Ob* 



2 i 3 



4- Sec are equal to 2 n.r 3 4* 6 r x Oi 4- Qf 4. 0,6 4.&C. z2»/- 3 



4- $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 Stewart's 20th theorem. 



In like manner, 



r+0/* + r — Oi 4 j^ r + Q/ + r —Og , F+Q~b* + r — Ob" 

 r r r 



+ &c. is equal to 2 # r 3 + 12 rxOi + Qf 4.O/6 4 &c. 4- 2 x 



5/ + Q? 4. CM 4. &c. , £ ^ , ?>"£>? 



^ ^ -£ = to 2 n r % 4. 6 r x Op X « + — 7^> 



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. 

 And a multiple of this by four, or eight times the aggregate of 



third i 



