40 INVESTIGATION of fome 
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, C, &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 circum{fcribing 
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 Q to 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 Q to 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 circumferibing the circle, and touching it 
in the points A, B, C, D, E, &c. are refpectively 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 
Het Ra OP= Nal hea E dep Spe, TECK eer; 
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=Ké, Ci=lh, Bg = He, Ae = Ga Es ald, &es 
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 
