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2 INVESTIGATION of fom 
Tuar truly elegant and inventive geometer the late Dr 
Marttuew Srewarm, publifhed at Edinburgh, in 1746, without 
demonftrations, 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 underftand, been yet demon- 
(trated. Thefe, with an endlefs variety of other theorems, are 
derivable, as corollaries, from the following general though 
fimple geometrical imyeftigation, that otcurred. to me fifteen 
years ago, and which, I fuppofe, has remained fo long unknown 
and unattended to chiefly on account of its fimplicity. 
Let A,B,C, &c. (Pl. IL. Fig. 1.) be any number of points in the 
circumference of a circle, and let that number be denoted by 2. 
Let RA, RS, ST, &c. be tangents to the circle, in the points A, 
B, C, &c.; and let POQ be any diameter. Let Qe, Qd, QS, 
&c. be perpendiculars from the point Q to the diameters paf- 
fing through the points A, B, C, &c., and Pa, P b, Pe, &c. per- 
pendiculars from the point P to the fame diameters. 
Tuen it is evident, that PQ. = AP +AQ =BP +BQ = 
CP. + cQ, = &c. Wherefore PO. x C= AP + BP + ce +, 
&e. + AQ’ + BQ + cQ'+, &c. But AP?=AGXAa= PQ. 
x Aa, BP =PQx Be, cP =PQ*« Cd, &c. ; and AQ’ + BQ+ 
oe &e. = PQX Ac+Bf+Cd4, &. Now Aa, Be, 
Cb, &c. are refpeCtively equal to perpendiculars drawn from P 
to the tangents RA, RS, ST, &c., as are Ac, Bf, Cd, &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 PQ. X%, or a multiple of the diameter by 2. 
Tue fame may be proved othewife ; for fince Oa = Oc, Aa 
— Gc, Aa+Ac=the diameter. In like manner, Be+Bf= 
the diameter, and C4-+Cd= diameter, &c. 
In 
