396 Demonstrations of Stewarfs 



2 



chord of 3A = ??-^™^~^?-E-^. In like manner, also, the 



.3 



, , -,. 2FPEH-4rPEHEN 

 chord of 4A = g- 



In general, the chord of wA, will be equal to 

 2FP ' EH- rX^i^^P • EHEN + — X Yx2FP • EHEN- 



tx?X_?x5Fp'^EH-EN'4S.c. . „„^,^ ;, pp ,^ ,^_ 

 noted by a; EH, by 6; and EN, by d, will become, hx 



— '"-I m-2 ^-^ , m-.3 7n-4 '"-* ,, »«-4 »n-S w-6 '""'^ ,, 



2a — y- X 2a : d^+ -7- X T" X 2a • «^— T X ^ X T" X 2a • f/° 



21-^^' ; and this series will terminate when the numerator of 

 of one of the factors in the co-efficient of the power of 2a has 

 become m—m. 



Lemma HI. 



IF there be any regular figure circumscribed about a 

 circle ; and from any point within the figure, perpendi- 

 culars be drawn to all the sides of the figure ; the sum 

 of all the perpendiculars will be equal to the multiple of 

 the radius of the circle, by the number of the sides of 

 the figure. 



DEMOJYSTRATIOJV. 



Let O (Plate III. Fig. 3) be any circle, and A, B, C, he. any 

 regular figure circumscribed about it. Let H be any point 

 within the figure, and draw HI, HK, HP, &c. perpendicular to 

 all the sides of the figure ; and also draw the right lines HE, 

 HF, HC, HB, &c to all the angular points of the figure. From 

 O, the centre of the circle, draw OG, ON, &c. to all the points 

 of contact, between the sides of the figure, and the circumfer- 

 ence ; and also, OE, OD, OA, he. to all the angular points of 

 the figure. 



HI'EF 



Now, the area of the triangle EHF ; and the area 



"of FHC= ; and so of all the other triangles, whose 



vertices are at H. But because the figure is regular, the bases 



