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ly wearing into disuse, and already the statute acre seems to
be generally adopted in the counties south of the Forth.
21. LevelLING is a delicate and important branch of general surveying. It may be performed very expeditiously by help of a large theodolite, capable of measuring with precision the vertical angle subtended by a remote object, the distance being calculated, and allowance made for the effect of the earth’s convexity and the influence of refraction. But the more usual and preferable method is to employ an instrument designed for the purpose, and termed a spirit-level, which is accompanied by a pair of square staves, each composed of two parts that slide out into a rod of ten feet in length, every foot being divided centesimally. Levelling is distinguished into two kinds, the simple and the compound; the former, which rarely admits of application, assigns the difference of altitude by a single observation; but the latter discovers it from a combined series of observations carried along an irregular surface, the aggregate of the several descents being deducted from that of the ascents. The staves are therefore placed successively along the line of survey, at suitable intervals according to the nature of the ground and not exceeding 400 yards, the levelling instrument being always planted nearly in the middle between them, and directed backwards to the first staff, and then forwards to the second. The difference between the heights intercepted by the back and the fore observation, must evidently give at each station the quantity of ascent or descent, and the error occasioned by the curvature of the globe may be safely overlooked, as on such short distances it will not amount at each station to the hundredth part of a foot. To discover the final result of a series of operations, or the difference of altitude between the extreme stations, the measures of the back and fore observations are all collected severally, and the excess of the latter above the former indicates the entire quantity of descent.
As an example of levelling, I shall take the concluding part of a survey, which my friend Mr Jardine, civil engineer, has recently made for the Town-Council of Edinburgh, with a degree of accuracy seldom attempted, in tracing the descent from the Black and Crawley springs, near the summits of the
Pentland chain, to the Reservoir on the Castlehill, with a view to the conducting of a fresh supply of water from those heights. To avoid unnecessary complication, however, I shall only notice the principal stations. The figure annexed represents a profile or vertical section of the ground, LV is the level of the Black spring, and the several perpendiculars from it denote the varying depth of the surface, referred to the base assumed 700 feet below. The stations marked are as follow :
L Lowest point in the Meadow.
aco'355 L–I. H odoo
M 370 4.59 2.04 2.55
Black spring, being 620.05 feet above the level of the Meadow, is therefore 520.1 feet higher than the belt of the reservoir. The numbers exhibited in the last column, are obtained by taking the differences of the aggregates of the two preceding columns. Where the ground either sinks or rises suddenly, some intermediate observations are here grouped together into a single amount. Thus, three observations were made between O and P, two between P and Q, three between Q and R, five between R and S, three between T and U, and no fewer than nine between U and V. The slight sketch between the perpendiculars from Q and R, shows the mode of planting and directing the instrument.
The mode of levelling on a grand scale, or determining the heights of distant mountains, will receive illustration from the third volume of the Trigonometrical Survey, which Colonel Mudge has been kindly pleased to communicate to me before its publication. I shall select the largest triangle in the series, being one that connects the North of England with the Borders of Scotland. The distance of the station on Cross Fell to that on Wisp Hill, is computed at 235018.6 feet, or 44.51.1 miles, which, reckoning 6094; feet for the length of a minute near that parallel, corresponds, on the surface of the globe, to an arc of 38° 33'.7. Wisp Hill was seen depressed 30, 48m from Cross Fell, which again had a depression of 2' 31" when viewed from Wisp Hill. The sum of these depressions is 33' 19", which, taken from 38' 33".7, the measure of the intercepted arc, or the angle at the centre, leaves 5' 14".7, for the joint effect of refraction at both stations. The deflection of the visual ray produced by that cause, which the French philosophers estimate in general at 079, had therefore amounted only to .06805, or a very little more than the fifteenth part of the intercepted arc. Hence, the true depression of Wisp Hill was 30' 48"–16' 39".5=14'8".5; and consequently, estimating from the given distance, it is 967 feet lower than Cross Fell.
From Wisp Hill, the top of Cheviot appeared exactly on the same level, at the distance of 185023.9 feet, or 35,0424 miles. Wherefore, two-thirds of the square of this last number, or 819, would, from the scholium at page 276, express in feet the approximate height of Cheviot above Wisp Hill. But refraction gave the mountain a more towering elevation than
it really had ; and the measure being reduced in the former ratio of 38'33.7 to 33' 19, is hence brought down to 708 feet. Again, the distance 292012.7 feet, or 55.3054 miles, of Cross Fell from Cheviot, corresponds to an arc of 47' 54".8, which, reduced by the effect of refraction, would leave 41'23".8 for the sum of the depressions at both stations. Consequently, Cheviot had, from Cross Fell, a true depression of only 23' 44–20, 41.9 or 3, 2.1, and is therefore lower than that mountain by 258 feet. These results agree very nearly with each other. The height of Cross Fell above the level of the sea being 2901, that of Wisp Hill is 1934, and that of Cheviot 2642 or 2643. In the Trigonometrical Survey, the latter heights are stated at 1940 and 2658; a difference of small moment, owing to a balance of errors, or perhaps to the adoption of some other data with respect to horizontal refraction, and which do not appear on record. From the same valuable work, I am tempted to borrow another example, which has more local interest. From Lumsdane Hill, the north top of Largo Law, at the distance of 189240.1 feet, or 35.84 miles, appeared sunk 9 32” below the horizon. Here the intercepted arc is 31' 3", and the effect of the earth’s curvature, modified by refraction, is 13, 24”.8; whence the true elevation of Largo Law was 13/24".8–9'32", or 3' 52.8, which makes it 213 feet higher than Lumsdane Hill, or 938 feet above the level of the sea. In the Trigonometrical Survey, this height is stated at 952; but I am inclined to prefer the former number, having once found it by a barometrical measurement, in weather not indeed the most favourable, to be only 935 feet. Through the kindness of Captain Colby of the Royal Engineers, who has for several years so ably conducted the survey under the direction of Colonel Mudge, I am enabled to subjoin some more examples, from the observations made last season. From Dunrich Hill the station on Cross Fell appeared depressed 19'21", at the distance of 349,343 feet or 66.1634 miles. This corresponds on the same parallel to an intercepted arc of 57'19"; the half of which, diminished by one-twelfth of the whole, gives 23’53, for the effect of curvature modified by
refraction. Cross Fell had therefore an elevation of 4'32", the excess of 23'53" above 19'21", which, at the given distance, makes it to be 461 feet higher than Dunrich Hill. Consequently, the altitude of Dunrich Hill above the level of the sea is 2901–461, or 2440 feet. This altitude, determined from nearer bases, was only 2421 feet. Again, from Cairnsmuir upon Deugh, at the height of 2597 feet above the sea, the top of Ben-Lomond appeared with a depression of 18' 24", the distance being nearly 352,004 feet, or 66,6673 miles. The intercepted, arc on the earth's surface was hence 57.45%", and the effect of curvature, as modified by refraction, 24'4". Wherefore, R: tan 6’40”, the real elevation : : 352,004: 580, which, added to 2597, gives 3177 for the altitude of Ben-Lomond. We shall select another example, which affords an approximation to the diameter of our globe. From the station at the observatory on the Calton-hill, at the altitude of 350 feet, the horizon of the sea was found depressed 18' 12". But refraction being supposed to have diminished the effect by onetwelfth part, if the eleventh part be added of this remaining quantity, there will result 19' 3" for the true measure of depression. The angle at the centre is consequently the half of 1943" or 9' 51"; wherefore, tan 9'51;": R :: 350: 122,048 feet, or 23.1152 miles, the distance at which the extreme visual ray-grazes the sea. Again, tan 9 51}" : R :: 23.1152 : 4030 miles, the radius of the earth, a near approximation to the real measure, or 3956. It should be noticed, that the state of the tide would have some effect in modifying the angle of depression. Thus, on the 12th May 1816, at 7+ p.m. the depression towards the mouth of the Firth of Forth, between the Isle of May and the Bass Rock, was found to be 18' 14"; but it was 18' 16" in a direction more to the north and near the Fife coast, because the sea had ebbed nearly five hours, the current outwards running first along the northern shore. On the following day, at three quarters after twelve o’clock, and therefore two hours and a half before high water, the depression about the middle of the Firth was 18'9", and only 18, 6" on the northern shore, the tide then flowing up principally in the middle of the channel.