64 INVESTIGATION of Jome 



circle, twice the fum of their cubes will be equal to 2 N X 



j i Qp 



GS +2r.GS -f-2Nx6rX ; that is, equal to twice a 



multiple by the number of the fides of the figure of the cube on 

 the fide of the infcribed decagon, together with four times a mul- 

 tiple, by the fame number of the folid which has the fquare of 

 the fide of the decagon for its bafe, and r for its altitude, toge- 

 ther with thrice a multiple by the fame number of the folid, 

 which has the fquare of GP for its bafe, and r for its alti- 

 tude* 



In like manner, may the fixth powers of lines drawn from the 

 angles of any regular infcribed figure of a greater number of 

 fides than three, to any point either in, or not in the circumfer- 

 ence, be exprefTed in terms of the fide of an infcribed decagon, 

 fince their fum is a multiple of the fum of the cubes of the per- 

 pendiculars, to the fides of the circumfcribing figure, by 8 r 3 . 



Again, fince r -f- GS : r : : r : GS : : GS : r — GS, we have 



2 r + GS : r + GS : : r -f- GS : r : : r : GS : : GS : r — GS. 



Wherefore 3r-f-2GS — r + Gb sz 4 X 2 r -f* GS , or 

 26 r 3 + 51 r\ GS + 33 r. GS' -f 7 GS 3 = 32 H + 48 r\ GS + 

 24 r. GS' + 4 GS 3 , or 3 r\ GS -f 9 r. GS + 3 GS =6 r\ or 



r *-GS + 3/-.GS 2 + GS 3 = 2r3. 



Wherefore, fince four times the fum of the cubes of the 

 perpendiculars drawn from any point in the circumference of 

 the circle to the fides of any regular circumfcribing figure, is 

 N x 5 X 2 p ; four times the fum of thefe cubes is = N X 



S r\GS+i S r. GS +5GS = 5 N X r\ GS + 3 r. GS -f GS ; 



that 



