VOL. LXVIIl.] PHILOSOPHICAL TRANSACTIONS. 415 



come contrary to those of the parts above the horizontal plane ; that is, the 

 effect of the southern vacuity is affirmative, and that of the northern one nega- 

 tive. But for the northern observatory, when the attraction towards the south 

 is to be found, the contrary names take place ; that is, in the elevations the 

 southern parts are affirmative, and the northern parts negative ; but in the 

 vacuities or depressions, the northern parts are affirmative, and the southern 

 ones negative. 



According to the foregoing method the plan of the ground was divided into 

 20 rings, by equidistant concentric circles, described about each observatory as 

 a centre ; and each quadrant was divided into 12 parts or sectors, by lines form- 

 ing, with the meridian, angles whose sines are in arithmetical progression ; by 

 which means the space in each quadrant was divided into 240 small parts, making 

 almost 1000 of such parts in the whole round for each observatory, or near 

 2000 for the two observatories. This was judged to be a sufficiently great 

 number of parts to afford a very considerable degree of accuracy ; or at least 

 that number was as great, and the parts as small, as was well consistent with 

 the degree of accuracy afforded by the number of points whose relative altitudes 

 had been determined. 



In this division the common breadth of the rings, or the common difference 

 of the radii, is 666^ feet ; and the common difference of the sines of the 

 angles formed by the radii and the meridian, is J^ of the radius ; and conse- 

 quently those angles are expressed in degrees and minutes as here follows, viz. 

 4'' 47', 9° 36', 14" 29', 19° 28', 24'' 37', 30° O', 33° 4l', 41" 48'i, 48° 35', 



56° 26%, 66° 26'4-, 90° o'. 



Of this kind were made two large plans, one divided for each observatory, 

 from v/hich were estimated the mean altitudes of the pillars erected on the 

 spaces into which they are divided. These altitudes are easily estimated when 

 several of the points fall near and in the small spaces or bases, especially when 

 they are near the middle of them ; but, numerous as the points are, there were 

 many bases in which none at all were contained, nor even near them. This 

 circumstance at first gave much trouble and dissatisfaction, till he fell upon the 

 following method by which the defect was in a great measure supplied, and by 

 which he was enabled to proceed in the estimation of the altitudes both with 

 much expedition and a considerable degree of accuracy. This method was the 

 connecting together by a faint line all the points which were of the same relative 

 altitude : by so doing, he obtained a great number of irregular polygons lying 

 within, and at some distance from each other, and bearing a considerable degree 

 of resemblance to each other : these polygons were the figures of so many level 

 or horizontal sections of the hills, the relative altitudes of all the parts of them 

 being known ; and as every base or little space had several of them passing through 



