VOL. LXVIII.J PHILOSOPHICAL TKANSACTIONS. 409 



base in Glenmore, the valley on the south of Schihallien ; in which, after adding 

 together all the repetitions of the 20 foot measuring rods, with the numerous 

 small intervals between them, and reducing the whole on account of the difference 

 of temperature and the gradual wearing of the brass standard, there results 

 301 1.4 17 feet for the corrected measure of this base, or the true length of the 

 line called rb in the survey. 



The 2d base, in the vale of Rannoch, on the north side of mount Schihallien, 

 was likewise measured in the same manner, and when reduced on all the above 

 mentioned accounts, comes out finally 5895.399 feet, for the correct measure of 

 that base, called ay. Besides these two principal bases, several other shorter 

 lines were also measured, the better to connect together those bases and the 

 several parts of the mountain. 



Having now obtained, to a great degree of accuracy, the measured lengths of 

 two lines which were to serve as bases for all the future calculations, the next 

 consideration was how to make the properest use of them. Every other line or 

 distance, drawn or conceived to be drawn, must be calculated from them by the 

 help of the angles observed, either at their extremities, or at all the other points 

 and stations in the survey and plan. As these two bases are situated in the low 

 parts of the country, from which but a very few of the other principal stations 

 are visible, one method evidently is to compute immediately from these bases such 

 of the great lines in the survey whose extremities are visible from them ; and 

 then from these calculated lines to compute others next to them, and so on quite 

 around and within the whole figure. In this manner several values of each line 

 will arise, both from the double computations by the two measured bases, and 

 from the various sets of triangles which can be formed from the numerous hori- 

 zontal angles which were observed at the several stations. But in this mode of 

 computation, after great labour and pains, Dr. H. had frequently the mortifica- 

 tion to find that the several values of the same lines would differ so greatly one 

 from another, that it was often very doubtful whether he could rely on any of 

 them, or even on the mean among them all. These differences arose from the 

 small errors in the observed angles, which in some degree are unavoidable; and 

 indeed they were so small, that the sum of the angles of the several triangles 

 which were used in the calculation seldom differed by more than a minute or 2 

 from 180°. But in a long connected chain of triangles, dependent on each other, 

 the effects of such small errors at length become too great to be tolerated in a 

 computation requiring much acci>racy. Another method is, first to compute 

 from both bases the length of the line kn extended along the ridge of the hill 

 from east to west, and from it, as a secondary base, compute all the other lines 

 in the plan. This method admits of much more accuracy than the former, sup- 

 posing this secondary base to be truly assigned; because that, from the elevated 



VOL. XIV. 3 G 



