VOL. LXVIII.J PHILOSOPHICAL TRANSACTIONS. 411 



proportion, as radius is to the tangent of the angle of elevation or depression, so 

 is the given base to the altitudinal difference between the two given points, ex- 

 clusive of the height of the theodolite or other instrument, which was afterwards 

 allowed for. From the resolution of all these triangles, and taking the means 

 of the many corresponding results, were obtained the numbers, which show how 

 many feet the points denoted by he letters standing against them are below the 

 level of the point n or the western cairn on the ridge of the hill. They are all 

 referred to this point n at the western extremity of the ridge of the hill, because 

 it is the most elevated point in the whole survey. 



These depressions, and those of several other principal points, were first care- 

 fully computed by means of various different bases, as so many places whence 

 the sections were to commence. 



These sections are very numerous, made in all directions from the primitive 

 points before- mentioned, and many of them extended to great distances, indeed 

 far beyond the bounds of the plan, so as to include the nearest hills and valleys 

 of the surrounding country. They are mostly made in vertical planes in the 

 manner described in the article of the Philos. Trans, before referred to, except- 

 ing some few, which are level sections in planes parallel to the horizon, and some 

 indeed irregular, being neither vertical nor horizontal. To compute the relative 

 altitude of each point in these sections, it is evident, requires the resolution of 

 2 different triangles, viz. a horizontal triangle by which its place in the plan is 

 ascertained, and a vertical triangle of which one side is the elevation or depression 

 of the point. Of these sections there are above 70, containing near 1000 points, 

 whose places in the plan and relative altitudes have been computed; so that the 

 number of triangles, whose numeral resolutions have been performed in the 

 course of this business, amounts to several thousands. Before the abstract of 

 the computation of the sections, is set down at large the calculation of one of 

 them, to show the manner in which they have been computed in the readiest and 

 easiest way. After which is given a large table of the results of all these calcu- 

 lations, comprising 72 sections, and some thousand points in the sections, whose 

 altitudes or depressions are arranged in order, and illustrated by drawings of ap- 

 propriate figures. 



Having finished the computation of the relative altitudes of all the points in 

 the sections, the next consideration is how they are to be applied in determining 

 the attraction of the hill. In whatever manner this last mentioned operation 

 may be performed, it is evident, that all the points and sections with their alti- 

 tudes must be entered in the plan. Therefore, having accurately constructed a 

 large plan of the ground, as before mentioned, containing all the principal lines 

 or bases, at the extremities of which either vertical or horizontal augles were 

 taken, from them are then determined in this plan the places of all the other 



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