Properties of the Circle. 407 



'^^'^--A————^^'"' where the numbers 1,3, 5, 7,&-c. 



are to be continued until the last be equal to 2m — 1 ; and the 

 numbers 1, 2, 3. 4. Sfc, until the last equal m. 



For, let the figure in the Proposition be supposed to hare 



n sides. Then, reasoning as before, PM'"4-PN'"-f &c. = 

 pr;2'» _(_ p K 2m I s,„ 



2^11^ But, (Cor. Prop. I.) PC^™+PB^+ &c. = 



« X -—J- .^__i X 2- R2- Whence PM'" + PN"" + &c. = 



1"2'3*4 wj 



l-2-3-4-'-"m ,l-2'3'4 m 



Froposition V. 



iLET there be any regular figure circumscribed about 

 ■acircle, of a greater number of sides than three ; and 

 from any point within the figure, let there be drawn per- 

 pendiculars to the sides of the figure ; and likewise let 

 there be drawn a right line to the centre of the circle : 

 twice the sum of the cubes of the perpendiculars will 

 be equal to twice the multiple of the cube of the radius 

 of the circle, by the number of the sides of the figure ; 

 together with thrice the multiple, by the same number, 

 of the solid, whose base is the square of the line drawn 

 to the centre, and altitude, the radius of the circle. 



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

 figure ; R, the radius of the circle ; and f/, the line drawn 

 from within the figure, to the centre of the circle ; twice 

 the sum of the cubes of the perpendiculars will be equal 

 to27zR3+3«(/2R. 



DEMONSTRATION* 



Let O (Plate III. Fig. 8) be any circle, and F, A, B, C, &c. 

 the points of contact of the sides of any regular figure, of more 

 tlian three sides, circumscribed about it. 



Case I. Let V be any point within the circle ; and draw 

 VH, VS, (^c. perpendicular to KG, GM, &c., the sides of the 



* JVote. The case where the point falls in the circumference of the 

 circle, was considered in Proposition III., and therefore is here omitted. 



i 



