PROPERTIES of the CIRCLE. . 57 
WHEREFORE, thé fum of the fourth powers of the perpendicu- 
lars drawn from thé point D to the fides of the figure, is = N 
Xrt+Nx ay x GD ae NG . x GD: ; and eight times this 
fum = N X 87+ + 247°. GD + 3°GD.. This is Prop. 29. 
of Dr SrewarT’s Theorems. 
Wuen GD = 7, the fum of thefe fourth powers is NX 
4ré+ aya) ara Nx Cr a which is Prop. 28. 
oF Dr SreWARt’ s Theorems 
AND Gace BC. AB 4 AT aT &c. =Nx aS , we get 
2+ 28 2% r* 
AG + AB 4 AT 4, ec." NX 33, ae = Nx BEEZ 
27’. | 
Awd by proceeding in this-way, (the law of continuation be- 
ing evident), we get Propofitions 39, 40, 41, &c. of Dr Stewart's 
Theorems, ‘fince the powers of DM, DV, DH, DN, &c. however 
high, may always be exprefled by thofe of DG and 7. The 
fame reafoning ‘holds in all even ‘powers, when the point D 
is without the figure,” by taking the powers of DH, +7, &c. 
when 'DH, ‘Secs 18 esha — than 7; inftead of the poeeks of 
- DH, &c.' >” 
‘Let any regular figure of an odd number of fides, (PI. ‘TL. 
Fig. 5.), circumfcribe the circle, and touch it in the points B, E,. 
C, QL; and from any point D, let perpendiculars DP, DR, 
DS, DO, DT, ’be drawn ‘to the ‘fides of the figure; and DF, 
DM, DN, DH, ‘DV, perpendiculars to the diameters paffing 
through the points of contact: 
‘Vou. VI.—P. I. H _ THEN 
