PROPERTIES of the CIRCLE. 35 
point to the other fides, viz. the 2d, 4th, 6th, 8th, &c. the fum 
of their {quares, the fum of their cubes, &c. to the fum of their 
<i 2 ( being the num- 
+~ 
2 : oo 
powers, but not in powers above 
ber of the fides). 
Tuus for inftance, if a regular hexagon circum{cribe 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 {quares, 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 dra 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 refpectively equal to the fum of perpen- 
diculars to the other four fides, the fum of their {quares and the 
fam of their cubes. But this does not hold in regard of the 
fum of their fourth powers, which to the fides of a {quare are 
not invariable. 
In like manner, the fum ne perpendiculars. to the erases 
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, 
