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berland and Westmoreland, into Scotland, and uniting with the collateral chain of Misterton Carr from Yorkshire and Northumberland, is prolonged to the heights immediately beyond the Firth of Forth. We look forward with anxiety to the conclusions of this arduous undertaking. The mountains and islands near the western coast of Scotland will furnish triangles of vast extent. Colonel Mudge will not omit, we are confident, the opportunities that such stations may afford to determine the quantity of horizontal refraction, noting at the same time the variable state of the atmosphere. The indications of the hygrometer would then require attention. We have perfect reliance in the accuracy of his observations; yet it would be desirable in all cases, as in the French operations, that the third angle of each triangle were actually measured. It would likewise be satisfactory, in surveying the more mountainous tracts, that the barometer should always accompany the theodolite, that both modes of determining the altitudes of the stations might be compared. The triangulation has been extended along the east coast of Scotland as far as the county of Banff and the borders of Ross-shire. It has also been carried towards the same points from Cumberland, through the heights of Galloway and Dumfries-shire, to the summit of Ben-Lomond; and from Dumbartonshire and the vicinity of Glasgow in a north-easterly direction, connecting all the remarkable mountains of Perthshire. The sands of Belhelvie, a few miles westward of Aberdeen, the spot formerly pointed out by General Roy, is now selected for a base of verification, which Colonel Mudge intends to measure in person this summer. It would no doubt be very desirable to have another intermediate base determined nearer the west side of the island. For this purpose, the plain between Kinniel and Carron, in the Carse of Falkirk, might seem eligible. Besides the principal triangles thus determined, a multitude of subordinate ones were ascertained in the progress of the survey, which serve to connect all the remarkable objects that occurred over the face of the country. The capital points were hence established for constructing the most accurate charts and provincial maps. A number of royal military sur... veyors, of approved skill, have since been constantly employed in filling up the secondary triangles, and embodying the skeleton plans. The various materials are collected at the drawing-room of the Tower, and there adjusted, reduced and combined. Under the same able direction, an extensive establishment has been formed in those spacious apartments, where a voluminous series of maps, on the largest scale, are not only delineated but engraved. This truly national work advances with great activity, and has already proved highly advantageous to the public service. The Ordnance Maps, in elaborate accuracy, and even beauty of execution, surpass every thing hitherto designed. The publication of these valuable geographical details, after having been suspended for some years, is again free. Five parts have already appeared, including Devonshire, Essex, Sussex, Dorsetshire, Kent, the Isle of Wight, Hampshire and Cornwall. Other maps are in a state of great forwardness, as far northward as the parallel from Caernarvon through Shrewsbury and Warwick to twenty miles beyond Boston in Lincolnshire. The completion of a work of such vast magnitude must require proportional time and perseverance. The maritime counties will probably be first given to the public, and the districts of the interior afterwards delivered. For a concise and perspicuous exemplification of all the refinements adopted in the practice of trigonometrical surveying, I have much satisfaction in referring to the late work of Baron Zach sur l’Attraction des Montagnes; nor can I omit this opportunity of testifying my respect and regard for that able and very learned astronomer, in whose interesting society I made a delightful excursion, in the month of August 1814, from Lyons by Orange to Vaucluse, and thence by Avignon to Marseilles, where he was then residing, as chamberlain to her Highness the Dowager Duchess of Saxe-Gotha.
22. To determine geometrically the altitude of a mountain requires, it hence appears, a nice operation performed with some large instrument. The barometrical mensuration of heights is therefore, in most cases, preferred, as much easier and often more exact. This curious application was early suggested, by the objections themselves which ignorance opposed to Torrio" celli's immortal discovery of the weight of our atmosphere. But more than a century elapsed before the improvements in mechanics had completely adapted the machine to that purpose, and experiment combined with observation had ascertained the proper corrections. Barometers of various constructions are now made quite portable, and which indicate with the utmost precision the height of the mercurial column supported by the pressure of the atmosphere. The air which invests our globe, being a fluid extremely compressible, must have its lower portions always rendered denser by the weight of the incumbent mass. To discover the law that connects the densities with the heights in the atmosphere, it is only requisite, therefore, to apply the fact which experiment has established,—that the elasticity counterbalancing the pressure is exactly proportioned to the density. The elasticity of the air at any point of elevation, is hence measured by a column possessing the same uniform density, with a certain constant altitude. Let AB denote the height of this equiponderant column, and the perpendicular BI its density; and suppose the mass of air below to be distinguished into numerous strata, having each the same thickness BC. It is evident that the weight of the minute stratum at B will be expressed by BC; whence AB is to AC, or BI to CK, as the pressure at B to the augmented pressure at C, and therefore the density at C is denoted by CK. Again, having joined IC,
and drawn KD parallel, BI: CK :: BC : CD ; and consequently CD will, on the same scale of density, express the weight of the stratum at C. Hence, AC is to AD, as CK to DL, or as the density at C is to that at D. It thus appears, that, repeating this process, the densities BI, CK, DL, &c. of the successive strata form a continued geometrical progression. But the same relation will evidently obtain at equal though sensible intervals. Thus, the density of the atmosphere is re
duced nearly to one half, for every 3% miles of perpendicular ascent. At 7 miles in height, the corresponding density is one-fourth ; at 10% miles, one-eighth; and at 14 miles, onesixteenth. The difference of altitude between two points in the atmosphere, is hence proportional to the difference of the logarithms of the corresponding densities or vertical pressures. But the heights of mountains may be computed from barometrical measurement to any degree of exactness, by a simple numerical approximation. Since AB, AC, AD, &c. are continued proportionals, it follows that AB: BC : : AB+ AC+AD, &c.; BC+CD+DE, &c. or BH. Let n denote the number of sec
tions or strata contained in the mass of air, and ; (AB+AH)
will nearly express the sum of the progression AB, AC, AD, &c.; wherefore, AB+AH : BH :: 2AB: nBC, or the absolute difference of altitude. The height AB of the equiponderant column, reduced to the temperature of freezing water, is nearly 26,000 feet; and hence this general rule,_As the sum of the mercurial columns is to their difference, so is the constant number 52,000 to the approximate height. This number is the more easily remembered, from the division of the year into weeks. Two corrections depending on the variation of temperature are besides required. 1. Mercury expands about the 5,000th part of its bulk, for each degree of the centigrade scale; and hence the small addition to the upper column will be found, by oremoving the decimal point four places to the left, and multiplying by twice the difference between the degrees of the attached thermometers. 2. But the correction afterwards applied to the prin‘cipal computation is of more consequence. Air has its volume increased by one 250th part, for each degree of heat on the same scale. If, therefore, the approximate height, having its decimal point shifted back three places, be multiplied by twice the sum of the degrees on the detached thermometers, the £roduct will give the addition to be made. If it were worth while to allow for the effect of centrifugal force in diminishing the pressure of the aërial column, this will be easily done before the last multiplication takes place, by adding to twice the degrees on the detached thermometers the fifth part of the nean temperature corresponding to the latitude.
An example will elucidate the whole process. In August 1775, General Roy observed the barometer on Caernarvon Quay at 30.091 inches, the attached thermometer being 15°.7, and the detached 15°.6 centigrade, while on the Peak of Snowdon the barometer stood at 26.409, the attached thermometer marking 10°.0, and the detached 8°.8. Here, twice the difference of the attached thermometers is 11°.4, which multiplied into .00264 gives .030, for the correction of the upper barometer. Next, 30.091 + 26.439 : 30.091 + 26.439, or 56.530: 3.652 : : 52000 : 3359. Again, twice the sum of the degrees marked on the detached thermometers is 48.8, which multiplied into 3.359 gives 164; wherefore, the true height of Snowdon above the Quay of Caernarvon is 3359-H 164, or 3533 feet. The correction for centrifugal force is only 7 feet more.
This mode of approximation may be deemed sufficiently near, for any heights which occur in this island; but greater accuracy is attained by assuming intermediate measures. To illustrate this, I shall select another example. At the very period when General Roy was making his barometrical observations at home, Sir George Shuckburgh Evelyn found the barometer to stand at 24.167 on the summit of the Mole, an insulated mountain near Geneva, the attached and detached thermometers indicating 14°.4 and 13°4, while they marked 16°.3 and 17°.4 at a cabin below and only 672 feet above the lake, the altitude of the barometer at this station being 28.132. Now, 3.8x.0024=.009, and 24.167+009–24.176; the arithmetical mean between which and 28.132 is 26.154; and hence, separately, 50.330: 1.978:: 52000: 2044, and 54.286 : 1.978:: 52000: 1895. Wherefore, joining these two parts, 2044+ 1895, or 3939 expresses the approximate height. The final correction is 61.6× 3.939-243, or 254 feet, if allowance be made for the effect of centrifugal force, and consequently the Mole has its summit elevated 4865 feet above the lake of Geneva, and 6063 above the level of the sea.
In general, let A and A + nb denote the correct lengths of, the columns of mercury at the upper and the lower stations; the approximate height of the mountain will be expressed by