Properties of the Circle, 409 



Case II. (Fig. 9.) Let the point V be taken without the cir- 

 €umference of the circle ; and let the same construction be 

 made as before. 



It may be shewn, as in Case 1st., that Vp — VH; and Va = 

 VS ; and so of all the other perpendiculars. 



Now Fp^Vp-VF=Vp-{VO-VO)=Vp-{d-'R.) Hence 



Vp'=(yp-d-R)'=Vp^-svp'-{d-n]+svp-(d-ny~{d-iiy. 



In like manner, Vq^ = Vq^-3yq'' ■ {d-K)-{-2>Y q • [d-RY-{d-i\Y, 

 And so of all the rest. 



Hence, Vp^ + Vq^ + ^c. = Vp^* + V^^ + 4'C. - 3(</-R) • 



But, (as in Case 1st.) Yp^+Yq^ + Uc — ^nd"^ ; and Vp* + 

 Yq^-\-hc.=z?_nd^', and Vp+Vj+^c. = nrZ; and(J-Il)3 + 

 {d-Uy Jr^c. = n{d-K)\ 



Therefore, Pp^ + Pg^ + &ic. = ^ nd^ - — nd^ (d-K) + 3nd 



2 2 



[d-'RY-n{d-'RY=n - — d^-— d'+l dHl + 3dr^-6dm + 



tu, ,^ ^ 



2 



Hence, 2 • P/4-P53 + 4^c.=2 • YH^+YS^^c. = 2nR3 + 

 3nd^K. Q.E. D. 



Proposition VI. 



LET there be any regular figure of a greater number 

 of sides than four, circumscribed about a circle ; and 

 from any pointy let there be drawn perpendiculars to the 

 sides of tlie figure ; and likewise a right line to the cen- 

 tre of the circle : eight times the sum of the fourth pow- 

 ers of the perpendiculars, will be equal to eight times the 

 multiple by the number of the sides of the figure, of the 

 fourth power of the radius of the circle ; together with 

 twenty-four times the multiple by the same number of 

 the fourth power of the line whose square is equal to the 

 rectangle contained by the radius and the line drawn to 

 the centre ; together with three times the multiple of 

 the fourth power of the line drawn to the centre of the 

 circlcj by the number of the sides of the figure. 



