GEODESY meridian. The first star on passing the central meridional wire is bisected by the micrometer; then the telescope is rotated very carefully through 180° round the vertical axis, and the second star on passing through the field is bisected FIG. 4.—Zenith Telescope. by the micrometer on the centre wire. thus measured the difl"erence of the zenith distances, and The micrometer has the calculation to get the latitude is most simple. Of course it is necessary to read the level, and the observa- tions are not necessarily confined to the centre wire. In fact if 72, s be the north and south readings of the level for the south star, 9;’, s’ the same for the north star, l the value of one division of the level, m the value of one division of the micrometer, 7', r’ the refraction corrections, ,u, p.’ the micrometer readings of the south and north star, the micrometer being supposed to read from the zenith, then, supposing the observation made on the centre wire,— tp=§(3+3') +§(p. — ,u’)m+ §~’n+7z,’— s — s’)l+ §(r— 7"). It is of course of the highest importance that the value m of the screw be well determined. This is done most effectually by observing the vertical movement of a close cireumpolar star when at its greatest azimuth. In a single night with this instrument a very accurate result, say with a probable error of about O"'3 or O"'4, could be obtained for latitude from, say, twenty pair of stars ;_ but when the latitude is required to be obtained with the highest possible precision, four or five fine nights are necessary. The weak point of the zenith telescope lies in 167 the circumstance that its requirements prevent the selection of stai's whose positions are well fixed; very frequently it is necessary to have the declinatinns of the stars selected for this instrument specially observed at fixed observatories. The zenith telescope is made in various sizes from 30 to 54 inches in focal length 3 a 30-inch telescope is sufficient for the highest purposes, and is very portable. The zenith telescope is a particularly pleasant instrument to Work with, and an observer has been known (a sergeant of Royal Engineers, on one occasion) to take every star in his list during eleven hours on a stretch, namely, from 6 o'clock P.M. until 5 A..I., and this on a very cold November night on one of the highest points of the Grampians. Observers accustomed to geodetic operations attain considerable powers of endurance. Shortly after the commencement of the observations on one of the hills in the Isle of Skye a storm carried away the wooden houses of the men and left the observatory roofless. Three observatory roofs were sub- sequently deinolislied, and for some time the observatory was used without a roof, being filled with snow every night and emptied every morning. Quite difl'erent, however, was the experience of the same party when on the top of Ben Nevis, 4406 feet high. For about a fortnight the state of the atmosphere was unusually calm, so much so, that a lighted candle could often be carried between the tents of the men and the observatory, whilst at the foot of the hill the weather was wild and stormy Calculation of Tria2zgulatiou. The surface of Great Britain and Ireland is uniformly covered by triangulation, of which the sides are of various lengths from 10 to 111 miles. The largest triangle has one angle at Siiowdon in ‘Vales, another on Slieve Donard in Ireland, and a third at Scaw Fell in Cumberland ,' each side is over a hundred miles, and the spherical excess is 64". The more ordinary method of triangulation is, however, that of chains of triangles, in the direction of the meridian and perpendicular thereto. The principal triangulations of France, Spain, Austria, and India are so arranged. Oblique chains of triangles are formed in Italy, Sweden, and Nor- way, also in Germany and Russia, and in the United States. Chains are composed sometimes merely of con- secutive plain triangles ,' sometimes, and more frequently in India, of combinations of triangles forming consecutive polygonal figures. I11 this method of triaiigulating, the sides of the triangles are generally from 20 to 30 miles in length—seldom exceeding 40. The inevitable errors of observation, which are insepar- able from all angular as well as other measurements, in- troduce a great difficulty into the calculation of the sides of a triangulation. Starting from a given base in order to get a required distance, it may generally be obtained in several different ways—that is, by using different sets of triangles. The results will certainly differ one from another, and probably no two will agree. The experience of the computer will then come to his aid, and enable him to say which is the most trustworthy result; but no experi- ence or ability will carry him through a large network of triangles with anything like assurance. The only way to obtain trustworthy results is to employ the method of least squares, an explanation of which will be found in FIGURE OF THE EARTH (vol. vii. p. 605). We cannot here give any illustration of this method as applied to general triangula- tion, for it is most laborious, even for the simplest cases. Ve may, however, take the case of a simple chain—com- mencing with the consideration of a single triangle in which all three angles have been observed. Siippose that the sum of the observed angles exceeds the proper amount by a small quantity 6 : it is required to assign proper cor-
rections to the angles, so as to cause this error to disappear. To