40 INVESTIGATION of ft 



ome 



variable. But their 4th, 6th, 8th, &c. are not invariable, when 

 AB, BC, &c. are unequal among themfelves. 



It is manifeft, that when AB, BC, &c. are equal among them- 

 felves, whatever be the number of the points A, B, G, &c. or 

 whatever be the part of the circumference they take in or extend 

 to, the fum of the fquares, cubes, &c. of perpendiculars drawn 

 from P and Q_to lines touching the circle in thefe points, is the 

 fame with the fum of the fquares, cubes, &c. of perpendiculars 

 drawn from P and Q^to the fides of a regular circumfcribing 

 figure having the fame number of fides as there are points, A, B, 

 C, &c. Thus, if the number of the points be five, and thefe be 

 comprehended in a femicircle, a quadrant, or any fector, the fum 

 of the fquares, cubes, and fourth powers of perpendiculars 

 drawn from P and Qj:o lines touching the circle in thefe points, 

 is the fame with the fum of the fquares, cubes, and fourth 

 powers of perpendiculars drawn from P and Qj:o the fides of a 

 regular pentagon circumfcribing the circle. And fo on. 



Perpendiculars drawn from P, (fig. 2.), one extremity of 

 the diameter PQ^ to the fides of the figure of an uneven or 

 odd number of fides circumfcribing the circle, and touching it 

 in the points A, B, C, D, E, &c. are refpeclively equal to per- 

 pendiculars drawn from Q^the other extremity of the diameter, 

 to the fides of a circumfcribing figure of double the number of 

 fides, which touch the circle in the points H, I, K, L, G, &c. or 

 \\g— Bc,Ge = Aa,Lb = Ed,Kf = D b, li — Ck, &c. ; 

 and perpendiculars drawn from Q_to the fides of the figure of 

 an odd number of fides, are refpectively equal to perpendiculars 

 drawn from P to the fides of a figure of double the number of 

 fides, which touch the circle in the points H, I, K, L, G, &c. or 

 Df- Kb, Ci - Ik, Bg - He, A e ~ G a, E b ~ Ld, &o 

 Wherefore, the fum of the m powers of perpendiculars, drawn 

 from P and Q_to the fides of any circumfcribing figure of an 

 odd number of fides, is equal to half the fum of the m powers 



of 



