PROPERTIES of the CIRCLE. 27 
ber of the fides of the infcribed or circumfcribed figure; or to 
{peak algebraically, the fum of the fourth powers of thé chords 
is equal to fix times a multiple of the fourth power of the femi- 
diameter of the circle, by the number of the fides of the figure. 
This is Dr SrEwart’s 23d theorem. 
s In like manner, 
Any Ae SP IPOR P= Or Sat 67 xOe 
By +Be —7+O0f +7—Of =27+6rxOf 
G2 460 =r Od +7—08 = 2+ 6rxOd 
&c. &e. &c. 
And the cubes of perpendiculars from P and Q to right lines 
touching the circle in the points A, B, C, &c. are taken together 
—a2nr+6rxOc + Of +0d + &. = (by Corollary 3.) 
—6- —6 —‘ at ae .C) 
AP + BP +P + 8c. + 8Q +O +6Q + Be 
a ; 
aot of + Od + ke = — when the circumference 
is equally divided in the points A, B, C, &c. or when a regular 
figure is circumfcribed about the circle, with its fides touching 
the fame in faid points. Wherefore the cubes of perpendiculars 
from P and Q to the fides of a regular figure of a greater num- 
ber of fides than three circumfcribed about the circle, are taken 
together = 5293. This is Dr Srewart’s 19th theorem. 
Anp if a regular figure of a greater number of fides than 
three be infcribed in the circle, having its angles in the points 
A, B, C, &c. third proportionals to the cube ef the diameter and 
the cubes of chords ‘drawn from P and Q to the points A, B, G, 
&c. will, taken together, be equal to 5 775; or third proportion- 
als to the cube of the diameter and chords drawn from either P 
_ 52 
— 
or Q to the faid angular points, will taken together, be 
? 
D2 ‘or, 
