413 PHILOSOPHICAL TRANSACTIONS. [aNNO 1778. 



points in the sections, whether vertical, horizontal, or irregular. These places or 

 points were determined by drawing lines from each extremity of the base so as to 

 form with it angles equal to those which were observed on the ground for each 

 corresponding pole; the intersections of these lines are the places of the poles, 

 which having marked with a fine dot or point of ink, and written close to each 

 point the proper number expressing its relative altitude, and cleaned the paper by 

 rubbing out the lines forming the angles by which the points were determined, 

 there remained only the points with the figures expressing their altitudes dis- 

 tinctly exhibited in the plan. 



It remains now to apply all the foregoing calculations and constructions to the 

 determination of the effect of the attraction in the direction of the meridian. 

 And here it soon occurred, that the best method was to divide the plan into a 

 great number of small parts, which may be considered as the bases ot as many 

 vertical columns or pillars of matter, into which the hill and the adjacent ground 

 may be supposed to be divided by vertical planes, forming an imaginary group of 

 vertical columns, something like a set of basaltine pillars, or like the cells in a 

 piece of honey-comb; then to compute the attraction of each pillar separately in 

 the direction of the meridian; and lastly, to take the sum of all these computed 

 effects for the whole attraction of the matter in the hill, &c. Now the attraction 

 of any one of these pillars on a body in a given place may be easily determined, 

 and that in any direction, to a sufficient degree of accuracy, because of the 

 smallness and given position of the base; for, on account of its smallness, all 

 the matter in the pillar may be supposed to be collected into its axis or vertical 

 line erected on the middle of the base, the length of which axis, as the mean 

 altitude of the pillar, is to be estimated from the altitudes of the points in the 

 plan which fall within and near the base of the pillar: then, having given the 

 altitude of this axis, with the position of its base, and the matter supposed to 

 be collected into it, a theorem can easily be given by which the effect of its at- 

 traction may be computed. But to retain the proper degree of accuracy in this 

 computation, it is evident that the plan must be divided into a great number of 

 parts, perhaps not less than 1000 for each observatory, in order that they may 

 be sufficiently small, and by this means forming about 2000 of such pillars of 

 matter, whose attractions must be separately computed, as mentioned above. 

 The labour and time necessary for such computation, it is evident, would be very 

 great, perhaps not less than those employed in all the preceding computations of 

 the sections, and all the other points and lines concerned in this business. For 

 this reason it was desirable to obtain a theorem or method by which the attractions 

 of the small and numerous pillars might be computed with the same degree of 

 accuracy, but with less expence of labour and time than when computed sepa- 

 rately as above-mentioned. And in this inquiry the success has been equal to 



