41 6 PHU.OSOPHICAL TRANSACTIONS. [aNNO 1778. 



it, he was thus enabled to determine the altitude belonging to each space with 

 much ease and accuracy. In this estimation he could generally be pretty sure of 

 the altitude to within 10 feet, and often within 5, which on an average might be 

 about the 100th part of the whole altitude ; and when we consider that the 

 number of such estimated altitudes is very great, and that it is probable the 

 small errors among them would nearly balance each other, the defect of those 

 that might be reckoned too little being compensated by the excess in those 

 which might be taken too great, we need not hesitate to pronounce, that the 

 error arising from the estimation of the altitudes is probably still much less than 

 that part. 



It was necessary to determine these altitudes of the pillars, in order to com- 

 pute the sines of the angles of elevation subtended by them, as the theorem re- 

 quires the use of these sines ; and the very easy method used in deducing the 

 latter from the former is explained after registering the altitudes of all the pillars 

 as they were computed. This register consists of l6 tables, viz. 4 quadrants of 

 spaces in the altitudes, and 4 in the depressions, for each observatory, as speci- 

 fied in the titles of them. The numbers are feet, like all the other dimensions. 

 The numbers on the same horizontal line from left to right are such as are all in 

 the same ring; and those in one and the same vertical column are in the same 

 sector, or between the same two radii ; the number of the ring, counted from 

 the common centre, is written in the left-hand margin ; and the number of the 

 vertical column, or distance of the space or sector from the meridian, at the 

 top ; also the radius of each ring, that is, the line from the common centre to 

 the middle of the ring, is written on the same line with it, in the right-hand 

 margin. It may be further remarked, that in such little spaces as were cut 

 through by the boundary line between elevations and depressions, thus making 

 but a part of such spaces in each of those denominations, each space was ac- 

 counted as a whole one ; but then the mean altitude or depression in each part 

 was diminished in the proportion of the whole space to the part of it so included 

 in the boundary. The altitudes and depressions are set down first with respect 

 to the southern observatory o, and then for the northern observatory ? ; and in 

 each, the altitudes are placed first. 



After these l6 long tables. Dr. H. adds, it remains now to find the sines of 

 the vertical angles subtended by all the foregoing altitudes and depressions, since 

 the sum of these sines is what we are in quest of. Each altitude or depression 

 is the perpendicular of a right-angled triangle, of which the given radius stand- 

 ing on the same line with it in the right-hand margin is the base, or other side 

 about the right angle ; and by the resolution of the right angled triangle, for 

 each perpendicular, the same number of corresponding sines will be found. 

 But with such data the tangent of the angle is nuicli easier to be found than the 



