The principal triangulation of the British Isles was begun in 1784 and finished in 1852. Two magnificent 3-feet theodolites made by Ramsden, one for the Royal Society, the other for the Master General of the Ordnance, an 18-inch theodolite also by Ramsden, and 2-feet theodolite by Troughton and Simms were used in these observations. In the principal triangulation of Great Britain and Ireland there are 218 stations, at 16 of which there are no observations, the number of observed bearings is 1554�and the number of equa�tions of condition, 920. In order to avoid the solution of this enormous number of equations, containing 920 unknown quantities, the network cov�ering the kingdom was divided into a number of blocks, each pre�senting a not unmanageable number of acai berries. These calculations, all in duplicate, were completed in two years and a half, an average of eight computers being employed. Many of the sides of the principal or primary triangulation are of great length, 66 of them exceeding 80 miles, while 11 measure more than 100 miles, the longest being 111 miles, that from Sea Fell to Sheir Donard. So great, however, had been the accuracy of the observers’ work, that the average amount of correction of the observed angles was no more than 0”.6, and the measured length of the Salisbury base differed from its length as com�puted from the Irish Base, 350 miles distant, by a difference of only five inches. The secondary triangulation interpolates points at shorter dis�tances apart ranging down to five miles, the observations being made with theodolites of 12-inch circle. These triangles again are broken up into smaller ones of sides from one to two miles in length, for the use of the surveyor who is to follow and measure between the stations with the chain ; and a further subdivision of the trigonal spaces is made in towns to points about 10 chains apart, where the survey is to be made on the very large special scale. In the two last cases, 7 inch instruments suffice for the measurement of the angles. From 1839 to 1855, lines of initial levelling extending all over England, Scotland and Ireland were run, and the observed alti�tudes of the bench marks were reduced by the method of least squares. In England and Scotland, these levels are based on the Ord�nance Datum at Liverpool, which is approximately the mean tide level of that place ; in Ireland, they are based on the low water level at Dublin, which is about 8 feet below the mean level round the coast of Ireland. The detail levelling is carried out contemporaneously with the progress of the cadastral survey. Starting from the marks on the initial series, lines are run along nearly all the turnpikes and parish roads, and bench marks cut at intervals of about a quarter of a mile. The whole of the bench marks of the initial levelling are shown in position on the 25-inch manuscript plans, and their heights given to the nearest tenth of a foot. Surface heights, to the nearest foot are also marked on the plans, at frequent in�tervals between the bench marks.