38 INVESTIGATION of fi 



ome 



Solution. 



With a fourth part of either of the equal given right lines 

 as radius defcribe a circle. If a regular decagon circumfcribe 

 the circle, and from any point in the circumference, that is nei- 

 ther one of the points, where the fides of the figure touch the 

 circle, nor at an equal diftance between the points of contact, 

 perpendiculars be drawn to the fides of the octagon, thefe taken 

 alternately are the parts into which the given right lines are re- 

 quired to be divided. 



If the point coincide with one of the points of contact, one of 

 the given lines is cut into three parts, and the other into four. 



If the point be equally diftant from two points of contact, the 



i ft perpendicular is zz the 8th, the 2d = 7th, the 3d =: 6th, and 



q 2 6" 



the 4th = 5th. * ■ rr - zz 3 the higheft power. 



With fuch problems one might proceed without end. 



Since (fig. 1.) AP + BP + CP + &c. AQ_ + BQ^ + CQ^ + &c. 

 are equal to the fquares of lines drawn to P and Q^ from the 

 angles of a regular infcribed figure of the fame number of fides 

 with the irregular circumfcribing figure, or from the points 

 where the fides of a regular circumfcribing figure touch the 

 circle, it is evident, that the fum of the fquares of perpendicu- 

 lars drawn from P and Q^to the fides of any circumfcribing fi- 

 gure, regular or irregular, of a given number n of fides, together 

 with the fquares of the perpendicular di (lances of P and Q^from 

 the diameters pafling through the points of contact A, B, C, &c. 



viz. P a + P b + P e + &c. + Qj + Q^d + Q/* + &c = 



2XP« i +BZ>+P<? + &c is an invariable quantity. For 



Pa + aO zz Fb + bO := P <? + cQ =PO whether the 



angles 



