PROPERTIES of the CIRCLE. 35 



point to the other fides, viz. the 2d, 4th, 6th, 8th, &c the fum 

 of their fquares, the fum of their cubes, &c. to the fum of their 



powers, but not in powers above (n being the num- 

 ber of the fides). 



Thus for inflance, if a regular hexagon circumfcribe a circle, 

 and from any point in the circumference perpendiculars be 

 drawn to the alternate fides, that is, to the fides of an equilate- 

 ral triangle circumfcribing it, the fum of thefe perpendiculars, 

 and the fum of their fquares, are refpectively equal to the fum 

 of the perpendiculars drawn to the other three fides, and the 

 fum of their fquares. For the fum of the perpendiculars to the 

 three fides of an equilateral triangle, is equal to half the fum of 

 the perpendiculars to the fides of the hexagon, and the fum of 

 their fquares in the one, equal to half the fum of their fquares 

 in the other. But this does not hold in regard to the fum of 

 their cubes, as the fum of the cubes of perpendiculars to the fides 

 of the triangle is not invariable. 



In like manner, if perpendiculars be drawn from a point in 

 the circumference to any four fides of a regular circumfcribing 

 octagon, taking them alternately, that is, to the fides of a cir- 

 cumfcribing fquare, their fum, the fum of their fquares, and the 

 fum of their cubes, are reflectively equal to the fum of perpen- 

 diculars to the other four fides, the fum of their fquares and the 

 fum of their cubes. But this does not hold in regard of the 

 fum of their fourth powers, which to the fides of a fquare are 

 not invariable. 



In like manner, the fum of perpendiculars to the alternate 

 fides of a regular circumfcribing decagon, that is, to the fides of 

 a pentagon, the fum of their fquares, the fum of their cubes, 

 and the fum of their fourth powers, are refpectively equal to the 



E 2 fum, 



