PROPERTIES of the CIRCLE. 57 



Wherefore, the fum of the fourth powers of the perpendicu- 

 lars drawn from the point D to the fides of the figure, is r^ N 



X r* -f N X 3 r z X GD + N X g X GD ; and eight times this 



fum = N X 8 r4 -f 24 r\ GD + 3 • GD . This is Prop. 29. 

 of Dr Stewart's Theorems. 



When GD == r, the fum of thefe fourth powers is N X 



4 r* + |r4-NX^^ = Nx l&JfcS. x r\ which is Prop. 28. 

 8 8 i' 2 '3«4 



of Dr Stewart's Theorems. 



A r AC , AB 8 , AT' . XT v 3 c H- 



And fince _ + _ + _+, &c. = N X ^-, we get 



AC 8 + AB ? + AT 8 +, &c. =NX 4r- & r 8 = N X ^^^ x 



8 1.2.3.4 



24 r 8 . 



And by proceeding in this way, (the law of continuation be- 

 ing evident), we get Proportions 39, 40, 41, &c. of Dr Stewart's 

 Theorems, fince the powers of DM, DV, DH, DN, &c. however 

 high, may always be exprefTed by thofe of DG and r. The 

 fame reafoning holds in all even powers, when the point D 

 is without the figure, by taking the powers of DH, + r, &c. 

 when DH, &c. is greater than than r, inflead of the powers of 

 r ± DH, &c. 



Let any regular figure of an odd number of fides, (PI. III. 

 Fig- 5-)> circumfcribe the circle, and touch it in the points B, E, 

 C, Q^ L ; 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 pafTmg 

 through the points of contact : 



Vol. VI.— P.I. H Then 



