34 INVESTIGATION of fome 



Now, let a circle (Fig. 2.) be divided into an uneven number 

 of equal parts, by the points A, B, C, D, E, &c. and let PQ^be 

 any diameter ; from P let P^P^P^P^PI, &c. be drawn 

 perpendicular to the diameters parting through the points A, B, 

 C, &c. and from d let Qj, Q^/, Q^, QJ, Qj, &c. be perpen- 

 dicular to the fame diameters. 



Then it is evident, that A a, Ae are refpeclively equal to 

 perpendiculars drawn from P, Q> to a tangent to the circle in 

 the point A ; and fince O a = O e, their fum A a + A e = 



r — O a + r + O a. In like manner, the fum of the perpendicu- 

 lars from P, Q^to the tangent at B is = r — Oc + r + O c, to the 

 tangent at C is = r — O k + r + Ok, to the tangent at D is 

 = r + O b + r — O b, and to the tangent at E is rz r + Q d + 

 r—6 d. But r— O a + r — Oc + r — O k + r + Ob + r + Od = 



r + Oa+r + Qc + r + Ok + r—O b + r — O d- 2 xOb + O d =: 



2XOa + Oc + Ok and Ob + Od — Oa + Oc + Ok, and fince 



r—O a + r—O c + r—O k + r + Qb + r + d =r + a + 



r — Gc + r + Qk + r — Ob + r — Od , we have this equation 

 4Xr xO b + rxOd zz 4.X r xO a + rxOc + rxO k, or 0^ + 

 Odz=.Oa + Oc + Ok. 



But if from a point in the circumference of a circle, perpen- 

 diculars be drawn to the alternate fides of a regular figure of an 

 even number of fides circumfcribing the circle, or, which comes 

 to the fame thing, beginning with any one fide, perpendiculars 

 be drawn to the ift, 3d, 5th, 7th, &c. fides, the fum of thefe 

 perpendiculars, the fum of their fquares, the fum of their cubes, 



~i "> th n 2 th 



&c. to the fum of their - — 1 or powers, is refpeftively 



equal to the fum of the perpendiculars drawn from the fame 



point 



