Properties of the Circle. 403 



In general, let m be any number less than n ; then will the 

 sum of the iwith powers of the chords PC, PB, ^'C. be equal to 



nxii— --111^'— — -X2"'-R2'"; where the numbers 1,3,5, 

 1-2-3-4---V m 



7, <^c. are to be continued until the last equals 2ot — ] ; and the 



numbers 1, 2, 3, 4, &;c., until the last equals m. 



Proposition II. 



LET there be any regular figure inscribed in a circle ; 

 and from all the angles of the figure and the centre of the 

 circle, let there be drawn right lines to any point: the 

 sum of the fourth powers of the lines drawn from the 

 angles of the figure, will be equal to the multiple, by the 

 number of the sides of the figure, of the fourth power of 

 the radius of the circle ; together with 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 from the centre ; to- 

 gether with the multiple by the same number, of the 

 fourth power of the line drawn from the centre. 



DEMOJVS TRA TIOJV/' 



Let O (Plate III. Fig. 6) be any circle, and A, B, C, &c. the 

 angular points of any regular figure inscribed in it. Take any 

 point, as P; and through P, and the centre, draw the diametei- 

 POQ, produced if necessary. Draw also PC, PB, <^-c. from all 

 the angular points ; and from C, B, he. draw CG, BH, 4'C. per- 

 pendicular to the diameter. 



When P is taken without the circle, let it be denoted by P'; 

 and when within, by P". 



Now, because the angle CPP' is always obtuse, therefore 

 (^p'2_p p2_|.pc2^_2P'P-pG. And because the angle CPP'^ 

 is always acute, therefore CP"2=P"P2+PC2-2P"P-PG. But 



PG=^^-!. Therefore CP'»=PP2+PC' 4- ??-?-;?-. And 

 PGt PQ 



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

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



