710 PHILOSOPHICAL TRANSACTIONS. [aNNO \775. 



them by angles with a base, to be measured in a level spot in the vale below the 

 hill, and then to make use of the said distance as a secondary base for determining 

 the sides of the polygon, and the distance of the 1 stations of the observatory. 

 Had the 2 cairns been visible from the 2 stations of the observation, 1 triangles 

 would have sufficed for connecting the 1 stations together. But notwithstand- 

 ing that this was not the case, and that only the 1 cairns were visible from 

 each other, yet all the angles of these 2 triangles were measured by Mr. Burrow 

 in the following method, suggested by himself. He went with the theodolite 

 to the neighbouring hill on the south side of Schehallien, which runs parallel 

 to it ; and, by varying his situation, found a point whence the western cairn and 

 southern observatory appeared by the theodolite to be in one vertical plane; and, 

 removing the theodolite, he planted a pole there. In like manner he planted 

 another pole on the same hill, in a vertical plane with the southern observatory 

 and eastern cairn. Then returning to the observatory, he took the horizontal 

 angle contained between the 2 poles, which it is evident is equal to its opposite 

 angle, or that contained between the cairns. And going to the west cairn, he 

 took the angle contained between the east cairn and the pole planted on the 

 opposite hill, in a line with the southern observatory and west cairn, which is the 

 same with the angle between the east cairn and southern observatory. An<l 

 lastly, going to the east cairn, he took the angle contained between the western 

 cairn and the pole placed on the opposite hill in a line with the east cairn and 

 southern observatory. Thus were the 3 angles found of the triangle made by 

 the southern observatory and 2 cairns. In the like manner were the angles of 

 the triangle made by the northern observatory and 2 cairns found afterwards. 

 And, as a proof that the angles of the 2 triangles were rightly determined, their 

 sum in the first case differed from 180° by little more than 2 minutes ; and in the 

 second case by only half a minute. 



Notwithstanding the advantages which attended this method of finding the 

 distance of the 2 stations of the observatory, I thought it proper to make use 

 also of the other method of doing the same thing by a small number of trian- 

 gles, carried directly across the hill, thinking it expedient, in a matter of such 

 consequence, to rely on no single operation ; but, as far as possible, to confirm 

 every deduction by another found in an independent manner. I had caused 2 

 poles to be set as far up the hill of Schehallien as they could be placed ; one as 

 near the western, and the other the eastern cairn, as they could be, so as to be 

 visible from the southern station of the observatory : also 2 others in like 

 manner visible from the north observatory ; one of which was very near the 

 east cairn, and the other only 269 feet distant from the westernmost of the 

 2 poles visible from the south observatory ; so narrow was the ridge of the hill 

 in that part, though it grew wider both to the west and east, but much more 



