66 INVESTIGATION of fome 



Therefore a™ + b«* + &c. = n X '*'>''' ' X — = 



WX 1.2.3.4.-3'" 2- 



If m=i, « 3 + ^ + &c.=«X^x' l = ^ ; and the 



1.2.3 2 4 



diameter, (or 2 r) X *r + Z> 2 -j- c 1 ■+• &c. = fum of the cubes of 

 perpendiculars drawn from any point in the circumference, to 

 the fides of a regular circumfcribing polygon of n number of 

 fides, and a -f- b 2 -f- &c. is to the fum of the fquares of thefe 

 perpendiculars as 5 to 6 ; and if the perpendiculars to the fides 

 of the polygon correfponding to the chords A, B, C, D, &c and 

 drawn from the fame point in the circumference that thefe 

 chords are drawn from, be denoted by P, Q^_R, S, &c. a -f- b -f- 



, AxP , BxO , CxR i DxS , 



c -f &c. = — ^— + — —M — " + —fz~ + &c. 2'" r m x 



2 r 2 r 2r 2 r 



I. ■3. C«7- • • 6" — x 

 of the 3 m powers of thefe perpendiculars, r= n X '* — - — 



.1.2.3.4. . . 3 



X r,*h 



Theorem O. From any point C, (PI. III. Fig. 8.), let the chord 

 GA be drawn ; let GAF be a tangent to the circle at A ; and let AD 

 be perpendicular to the diameter BC, and CF, BG to GF. The 



right line which has to BC (2 r), the ratio of AG 3 to BC , or the 



triplicate ratio of the chord of the arc AC, to the diameter, is 



ACXCF ACXCD r , . j _ " 



— rrp — , or — |.p — zz a fourth proportional to the diameter, 



the chord and the perpendicular drawn from one extremity of 



the 



