PROPERTIES of the CIRCLE. 6 S 



that is, equal to five times a multiple, by the number of the fides 

 of the figure of the cube on the fide of the inferibed decagon, 

 together with fifteen times a multiple, by the fame number, of the 

 folid, which has the fquare on the fide of the inferibed decagon 

 as its bafe, and r for its altitude, together with five times a 

 multiple, by the fame number of the folid, which has r 2 for its 

 bafe, and the fide of the decagon for its altitude. 



Let the circumference of a circle be divided into any number 

 n of equal parts, and from any point in the circumference let 

 chords be drawn to the points of divifion, and let 3 m be any 

 number lefs than n, the fum of the 2 m powers of the lines 

 which have refpectively to 2 r the diameter, the ratios which 

 the cubes of the chords have refpeclively to 8 r 3 , the cube of 



, ,. . , 1. i.e. 7. . . 6 m — 1 -r rm 



the diameter, is equal to n X X — • 



1.2.3.4. ..3 m 2 "< 



Let the chords be denoted by A, B, C, D, &c. to n terms ; 



and let 8 r* : A 3 = 2 r : a, 8 r 3 : B 3 = 2 r : b, 8 r 3 : O = 2 r : c, 



A3 A 6m 



8 r3 : D J = 2 r : d, &c Then a = ~, and a™ = 2 Am b™~ 



4 ; 2 r 



-— -, &c ; and a"" + 3- + &c to terms, is = -£— 

 2 /" 2 / 



+ ' « 4w "+" & c * t0 n terms. If p = 3 /», we have a'* -f- 

 2 r 



A J/ B 4/> 



3"" + &c = p+m p+m -f +m p+m + &c But the fum of 

 2r 2 r 



the 2/> powers of the chords A, B, &c. is X '•*' :r/ * 2^ r 1 ^. 



1.2.3.4. . . p 



Vol. VI.— P.I. I Therefore 



