TOL. LXVIII.] PHILOSOPHICAL TRANSACTIONS. 413 



his wishes, having at length met with a method by which the business has been 

 effected in perhaps -f or -f of the time that would have been required in the 

 other way. 



Of all the methods of dividing the plan into a great number of small parts, 

 he found that to be the most convenient for the computation, in which it is first 

 divided into a number of rings by concentric circles, and these again divided into 

 a sufficient number of parts by radii drawn from the common centre, that centre 

 being the observatory where the plummet is placed on which the effect of attrac- 

 tion is to be computed. By this means the plan is divided into a great number 

 of small quadrilateral spaces, 1 of the opposite sides of which are small portions 

 of adjacent circles, and the other 2 are the intercepted small parts of 2 adjacent 

 radii, as appears by fig. ] 1, pi. 4; in which, for the present, let the circles and 

 their radii be supposed to be drawn at any distances whatever from each other, 

 till it shall appear from the theorem to be investigated what may be the properest 

 distances and positions of those lines. In this figure a is the observatory, an 

 the meridian, wae an east-and-west line, bcde one of the little spaces, and f 

 its centre or the foot of the axis of the pillar whose base is bcde; the figure 

 AWNEA being a horizontal or level section through the point a. Join a, f, and 

 with the centre a describe the mid circle gfh. Let a denote the length of the 

 axis on the point p, or the mean height of the pillar on the base bd; and s the 

 sine of the angle of elevation of that pillar as observed at a, to the radius i, or 

 s = ^ — jr. Then will the magnitude of that column, or its quantity of 

 matter, be expressed by (bc + ed) X ^be X a, which is supposed to be all col- 

 lected into the axis; consequently, if the attraction of each particle of matter be 

 in the reciprocal duplicate ratio of its distance, the attraction of the matter in 

 the pillar, so placed, on the plummet at a, in the direction of the meridian an, 



■11 1 BC + ED , , « . BC + ED GH 



Will be —-^ — XBEXaX-Xc= —■ — X be X «c = — X be X .?c 



ZA F a * A F A F 



nearly, supposing f to be equally distant from bc and ed, and c the cosine of 

 the angle pan to the radius i. 



But — ; X c is nearly equal to d the difference of the sines of the angles ban, 

 CAN, as is thus demonstrated. Draw gk, fl, km, perpendicular, and gp parallel 

 to AW; and draw the chord gh. Then ak, am are the sines of the angles gan, 

 HAN, to the radius af, their difference being km = gp; also fl is the cosine of 

 fan to the same radius: consequently gp : fl = (/ : c. But the triangles lfa, 

 pgh are equiangular, and therefore gp : fl = gh : af. Consequently gh : af 

 ^ d:c; or — X c = d. This equation is accurately true when gh is the chord 

 of the arc; and as the small arc differs insensibly from its chord, the same equa- 

 tion is sufficiently near the truth when gh is the arc itself. Substituting now d 



