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CALCULATIONS TO ASCERTAIN THE TRACT 26.

be assigned, by which the effect of its attraction may be com-puted. But, to retain the proper degree of accuracy in thiscomputation, it is evident that the plan must he divided intoa great number of parts, perhaps not less than a thousand foreach observatory, in order that they may be sufficiently small,and by this means forming about 2000 of such pillars of mat-ter, whose attractions must be separately computed, as abovementioned. The labour and time necessary for such compu-tation, it is evident, would be very great, perhaps not lessthan those employed in all the preceding computations ofthe sections, and all the other points and lines concerned inthis business. For this reason, I was desirous of obtaining atheorem, or method, by which the attractions of the smalland numerous pillars might be computed, with the same de-gree of accuracy, but with less expence of labour and time,than when computed separately as above mentioned. Andin this inquiry the success has been equal to my wishes, hav-ing at length devised a method, bj' which the business hasbeen effected in perhaps one-fourth or one-fifth of the time,that would have been required in the other way.

Of all the methods of dividing the plan into a great num-ber of small parts, I have found that to be the most convenientfor the computation, in which it is first divided into a numberof rings by concentric circles, and these again divided intoa sufficient number of parts by radii drawn from the commoncentre, that centre being the observatory where the plummetis placed, on which the effect of attraction is to be computed.By this means, the plan is divided into a great number ofsmall quadrilateral spaces, two of the opposite sides of whichare small portions of adjacent circles, and the other two arethe intercepted small parts of two adjacent radii, asappears byfig. 1, pi. 3, in which, for the present, let the circles and theirradii be supposed to be drawn at any distances whatever fromeach other, till it shall appear, from the theorem to be in-vestigated, what maybe the most convenient distances andpositions of those lines. In this figure, a is the observatory, anthe meridian, wae an east-and-west line, bcde one of the