62 INVESTIGATION of feme 



If V be as much on the other fide of the centre G, towards L, 

 as it is towards C, the lines GN, GM, exchange their values or 

 magnitudes, as alfo do the lines GH, GF ; and the perpendicu- 

 lars to the fides of the circumfcribing figure then become 

 r — GM, r — GH, r -f GN, r + GF, r — GV ; and the fum of 



C D 

 their cubes N X n + N X 3 r. — — _|_ GN 3 + GF 3 — GM 5 — 



OH 3 — GV 3 ; which added to N X r* + N X 3 r. — -f- GM 3 



GH 3 -f GV 3 — GN 3 '--GF 3 , the fum of their cubes before 

 found, and the aggregate divided by 2, gives NXr 3 -f-Nx ^r. 



—— t the fum of their cubes, when D is in the line drawn from 

 2 



the centre G perpendicular to LG. 



Let a circle, (PL III. Fig. 7.), be defcribed on BC, with the 

 centre G, and let BF be a fquare on the diameter BC ; draw EGD 

 from E, through the centre G, to meet the circle in D, and join 

 D F. 



Then, fince BG X CS, or CG X CS = GS*, GC is cut in ex- 

 treme and mean proportion in the point S, and GS is the fide of 

 a regular decagon, infcribed in the circle. And fince the per- 

 pendicular from G to the fide of a regular infcribed pentagon, is 



BG + GS . BG + GS 



, BS is twice that perpendicular. But 



2 



-3 7=77T3 



^3 



or r+ 8 GS — ^ = f • Confequently BS 3 — GS 3 , or r"T~GS 3 



— GS 3 = 4 r\ Therefore 3 r J = 3 r X GS -\- 3 r X "GS*, and 



r 3 



