4 INVESTIGATION of fome 
w 
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 Pa, P4, Pc, Pd, Pk, &c. be drawn 
perpendicular to the diameters pafling through the points A, B, 
C, &c. and from Q let Qe,Q f, Qs, Q4, Qi, Kc. be perpen- 
dicular to the fame diameters. 
THEN it is evident, that Aa, Ae are refpectively equal. to 
perpendiculars drawn from P, Q; to a tangent to the circle in 
the point A; and fince Oa = Og, their fum Aa + Ae = 
7—Oa+r+Oa, In like manner, the fum of the perpendicu- 
lars from P, Q to the tangent at Bis = r—Oc +7+ Oc, to the 
tangent at C is = r—Ok + 7 + Of, to the tangent at D is 
=r+04+7—O4, and to the tangent at E is = r+Od + 
r—Od. But r—Oa+7r—Oc 4r—Ohk +r +O0b4+r+Od= 
r+Oat+rt+Oc+rt+tOk+r—O0b4+r—Od; 2xO0b+0d= 
2xOat+Oc+Ok and OF8+O0d =0a+0c+O8, and fince 
r—Oa + r—Oc +7—-OF +7408 +7tOd =rtO0a 4 
x—Oe +7+0k +7—Ob + 7—Od. , we have this equation 
4xrxOb+rxOd = 4xrxOatrxOct+rxOQk, or OFF 
Od=Oa+O0Oc+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 comés 
to the fame thing, beginning with any one fide, perpendiculars 
be drawn to the rft, 3d, 5th, 7th, &c. fides, the fum of thefe 
perpendiculars, the fum of their fquares, the fum of their cubes, 
n va n—2'h 
&c. to the fum of their te or 
powers, is refpectively 
equal to the fum of the perpendiculars drawn from the fame 
point 
