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 44.511 miles, which, reckoning 60944 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′ 48′′ 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′ 43′′ for the true measure of depression. The angle at the centre is consequently the half of 19′ 43′′ or 9′ 51′′; wherefore, tan 9′ 511⁄2": 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 74 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.

22. MARITIME SURVEYING is of a mixed nature: It not only determines the positions of the remarkable headlands, and other conspicuous objects that present themselves along the vicinity of a coast, but likewise ascertains the situation of the various inlets, rocks, shallows and soundings which occur in approaching the shore. To survey a new or inaccessible coast, two boats are moored at a proper interval, which is carefully measured on the surface of the water; and from each boat the bearings of all the prominent points of land are taken by means of an azimuth compass, or the angles subtended by these points and the other boat are measured by a Hadley's sextant. Having now on paper drawn the base to any scale, straight lines radiating from each end at the observed angles, as in Prop. 21. of the Trigonometry, will by their intersections give the positions of the several points from which the coast may be sketched. But a chart is more accurately constructed, by combining a survey made on land, with observations taken on the water. A smooth level piece of ground is chosen, on which a base of considerable length is measured out, and station staves are fixed at its extremities. If no such place can be found, the mutual distance and position of two points conveniently situate for planting the staves, though divided by a broken surface, are determined from one or more triangles, which connect with a shorter and temporary base assumed near the beach. A boat then explores the offing, and at every rock, shallow, or remarkable sounding, the bearings of the station staves are noticed. These observations furnish so many triangles, from which the situation of the several points are easily ascertained.-When a correct map of the coast can be procured, the labour of executing a maritime survey is materially shortened. From each notable point of the surface of the water, the bearings of two known objects on the land are taken, or the intermediate angles subtended by three such objects are observed. In the first case, those various points have their situations ascertained by Prop. 21. and the second case by Prop. 25. of the Trigonometry. To facilitate the last construction, an instrument called the Station-Pointer has been invented, consisting of three brass rulers, which open and set at the given angles.

23. THE nice art of observing has in its progress kept pace with the improved skill displayed in the construction of instruments. Surveys on a vast scale have lately been performed in Europe, with that refined accuracy which seems to mark the perfection of science. After the conclusion of the American war, a memoir of Count Cassini de Thury was transmitted by the French Government to our Court, stating the important advantages which would accrue to astronomy and navigation, if the difference between the meridians of the observations of Greenwich and Paris were ascertained by actual measurement. A spirit of accommodation and concert fortunately then prevailed. Orders were speedily given for carrying the plan into execution; and General Roy, who was charged with the conduct of the business on this side of the Channel, proceeded with activity and zeal. In the summer of 1784, a fundamental base, rather more than five miles in length, was traced on Hounslow Heath, about 54 feet above the level of the sea, and measured with every precaution, by means of deal rods, glass tubes, and a steel chain, allowance being made for the effects of the variable heat of the atmosphere in expanding those materials. The same line was, seven years afterwards, remeasured with an improved chain, which yet gave a difference on the whole of only three inches. The mean result, or 27404.2 feet, at the temperature of 62° by Fahrenheit's scale, is therefore assumed as the true length of the base. Connected with this line, and commencing from Windsor Castle, a series of thirty-two primary triangles was, in 1787 and 1788, extended to Dover and Hastings, on the coast of Kent and Sussex. Two triangles more stretched across the Channel. The hori zontal and vertical angles at each station were taken with singular accuracy by a theodolite, which the celebrated artist Ramsden had, after much delay, constructed, of the largest dimensions and the most exquisite workmanship. At the same period, a new base of verification was measured on Romney Marsh, 15 feet above the sea, and found, after various reductions, to be 28535.6773 feet in length. This base, computed from the nearest chain of triangles dependent on that of Hounslow Heath, ought to have been 28533.3; differing scarcely more than two feet on a distance of eighty miles. The mean, or 28534.5, is adopted for calculating the adjacent