FroperUes of ihe Circle. 401. 



And' wben m and n are not prime to each other, the sum of 

 the squares of these lines will be equal to 2oR2. But each one 

 of these lines is to be taken as many times as there are units in 

 t; tlierefore the sum of the squares of the lines belonging to all' 

 tlie arches, will be equal to 2ra-R2 —OnW,^. 



Proposition I, 



LET there be any regular figure inscribed in a circle ; 

 and from all the angles of the figure, let there be drawn 

 right lines to any point in the circumference of the cir- 

 cle : the sum of the fourth powers of the chords will be 

 equal to six times the multiple of the fourth power of 

 the radius of the circle, by the number of the sides of 

 the figure. 



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

 ure, and R, tlie radius of the circle ; the sum of the 

 fourth powers of the chords will be equal to 6hR\ 



DEMONSTRATION. 



Let O (Plate III. Fig. 5) be any circle, and A, B, C, SiC. the 

 angular points of any regular figure, incribed within it. Let P 

 be any point in the circumference; and let there be drawn to it, 

 the right lines PC, PB, he. from all the angular points of the 

 figure. 



It is evident, that the angular points of the figure, divide the 

 circumference into equal parts, the number of which is equal to 

 the number of the sides of the figure. Let now, the arches 

 PC, PCB, 4'C. be multiphed by the number 2. Then (Lemma 

 1st.) the arches 2PC, 2PCB, &c. will divide the circumference 

 into equal parts. And if right lines be drawn from P, to all the 

 points where the multiple arches terminate; then (Coroll. Lem- 

 ma 4.) the sum of the squares of the chords, belonging to all the 

 arches, will be equal to 2nR^. 



Draw through P, the diameter PQ,; and from the points C, 

 B, &£c. draw C(^, B(^, kc. Then (Cor. Lemma 2d.) the chord 



of 2PC will be equal to ^'I^. But 2qC=2VPQ^ - PC"^ ; 



and PQ,=2R j and let the chord of the arch 2PC be denoted. 



