406 Demonstrations of Stexvarfs 



20rtR«. Hence P]VP+PN3+^c.=-iL?l. And2'FM~-fPN^ 

 +~&^ =i?ZL?! =5nR\ Q. E. D. 



Fropositioii IV, 



LET there be any regular figure, of a greater num- 

 ber of sides than four, circumscribed about a circle ; 

 and from any point in the circumference of the circle, 

 let there be drawn perpendiculars to the sides of the fig- 

 ure ; eight times the sum of the fourth powers of the 

 perpendiculars, will be equal to thirty-five times the mul- 

 tiple, by the number of the sides of the figure, of the 

 fourth power of the radius of the circle. 



Thus, if n denote the number of the sides of the fig- 

 ure ; and R, the radius of the circle ; eight times the 

 sum of the fourth powers of the perpendiculars, will be 

 equal to 35wR^ 



DEMONS TRA TIOJY, 



Let O (Plate III. Fig. 7) be any circle, and A, B, C, D, E, 

 &c. the'points of contact of the sides of any regular figure, of 

 more than four sides, circumscribed about it 5 and let the same- 

 construction be made, as in Proposition III. 



By a process of reasoning, similar to that in Prop. III., it may 



be shewn that PM'*=Z^-, and PN^=il_. And so of all 

 16K* 16R* 



the other perpendiculars. Therefore, PM* + PN* -f- ^^c. = 



?£!±Z^'-±i:2: But, (Cor. Prop. I.) PC » + PB » + ^c. == 

 16R^ ^ ^ 



70ftR«. Therefore, PM*-fPN*+^c;=_^_. Or, by mul- 



16R* ^ 



tiplication, 8.-PM*+PNM^=— --— =35nR*. Q,. E. D. 



Cor. In general, let there be any regular figure circumscri- 

 bed about a circle ; let n denote the number of its sides ; and 

 let m be any number less than n : let R denote the radius of 

 the circle ; and from any point in the circumference, let there 

 be drawn perpendiculars to the sides of the figure : the sum 

 of the mih powers of the perpendiculars will be equal tf» 



