38 INVESTIGATION of fome 
SOLUTION. 
Wit# a fourth part of either of the equal given right lines 
as radius defcribe a circle. If a regular decagon circum{cribe 
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 o€tagon, 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 mgt of contact, one of 
the given lines is cut into three parts, and the other into four. 
Ir the point be equally diftant from two points of contact, the 
1ft perpendicular is = the 8th, the 2d = 7th, the 3d = 6th, and 
a 
the 4th = sth. = = = 3 the higheft power. 
Witu 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 2 of fides, together 
with the fquares of the perpendicular diftances of Pand Q from 
the diameters pafling through the points of conta@ A, B, C, &c. 
2 —_—— oer} pe Pea FETS = 2 . 
viz Pa + Pb + Pe +&.+ Qe +04 +Of 4. Ser 
2x Pa+Bo+Pe + &c. is an invariable quantity. For 
2 nl — —? 
Pa+a0 =Pb +40 = Pe +20 = PO whether the 
angles 
