22 INVESTIGATION of ft 



omc 



1 hat truly elegant and inventive geometer the late Dr 

 Matthew Stewart, publifhed at Edinburgh, in 1746, without 

 demonflrations, a number of general theorems, of great ufe in 

 the higher parts of mathematics, and much calculated for impro- 

 ving and extending geometry. Such of them as refer to the 

 circle, and to regular figures infcribed in, and circumfcribed 

 about it, have not, as far as I can understand, been yet demon- 

 ftrated. Thefe, with an endlefs variety of other theorems, are 

 derivable, as corollaries, from the following general though 

 limple geometrical inveftigation, that occurred to me fifteen 

 years ago, and which, I fuppofe, has remained fo long unknown 

 and unattended to chiefly on account of its Simplicity. 



LetA,B,C,&c. (PI. II. Fig. 1.) be any number of points in the 

 circumference of a circle, and let that number be denoted by n. 

 Let RA, RS, ST, &c. be tangents to the circle, in the points A, 



B, C, &c. ; and let POQ_be any diameter. Let Qc , Qd, Qf, 

 &c. be perpendiculars from the point Q^ to the diameters paf- 

 fing through the points A, B, C, &c, and Fa, P b, P c t &c. per- 

 pendiculars from the point P to the fame diameters. 



Then it is evident, that PQ^= XF + AQ^ = BP* + BQ^n 

 CP 2 + CC£ == &c. Wherefore PQ^X n = AP + B? + CP* -f , 

 &c. +AQl+BQ^+CQ^-f, &c. But AP* = AG X A^rzPQ^ 

 xA^BP'r PQjx B e, CP 2 = PQj< C b, &c. j and A~Q^+ BQ^-f 

 C(£ -f, &c. - ?QX At + B/+C^+, &c. Now ha, B e % 

 C b, &c are refpectively equal to perpendiculars drawn from P 

 to the tangents RA, RS, ST, &c, as are A c, B/, C d, &.c. equal 

 to perpendiculars drawn from Q^to the fame tangents. Con- 

 fequently the fum of all the perpendiculars drawn from the 

 points P and Q^ to lines touching the circle in the points, A, B, 



C, &c. is equal to PQX », or a multiple of the diameter by n. 

 The fame may be proved othewife ; for fince O a — O c, A a 



~Gc, Atf-f-Ac— the diameter. In like manner, B e -f Bf— 

 the diameter, and C b -f C d — diameter, &.c. 



In 



