36 INVESTIGATION of fome 
fum, the fum of the fquares, the fum of the cubes, and the fum 
of the fourth powers of perpendiculars to the other five fides. 
But this equality does not hold in the fifth powers, which to the 
10—2 
fides of a pentagon are not invariable. For = 4. And 
fo on. 
N. B. Tue fame holds true if the perpendiculars be drawn 
from any point within the figure for odd powers, and either 
within or without, in even ones. 
Bur as it was obferved in the preceding page, that the equa- 
lity between the fum of the powers of perpendiculars, drawn | 
from any point in the circumference of a circle, to the alternate 
fides of any regular figure of an even number of fides, and the 
fum of the powers of perpendiculars drawn from the fame point 
n—2" 
to the other fides, exifted only to the ZT. Power; fo the equa- 
lity between the fum of the powers of perpendiculars drawn 
from the extremities P and Q of any diameter to the fides of a 
regular figure of an odd number of fides circumfcribing the 
circle, and the fum of perpendiculars from either of thefe, or any 
point in the circumference, to the fides of a regular circumfcri- 
bing figure of double the number of fides, exifts only to the 
n—2™ power. 
A wipe field is here opened for the geometrical folution of 
both determinate and indeterminate problems. 
For inftance, having two equal right lines given, to cut one 
into two parts, and the other into three, fo that the fum of the 
fquares on the two parts, into which the one is cut, fhall be 
equal 
