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 



fides of a pentagon are not invariable. For l^ZZ^ = * ^nd 

 fo on. 



N. B. The 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. 



But 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 



to the other fides, exifted only to the "-— power j 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 

 » — 2 th power. 



A wide 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 



