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GEODESICAL OPERATIONS CONNECTED WITH A TRIGONOMETRICAL SURVEY.
In the words of Sir J. Herschel," Astronomical Geography has for its objects the exact knowledge of the form and dimensions of the earth, the parts of its surface occupied by sea and land, and the configuration of the surface of the latter regarded as protuberant above the ocean, and broken into the various forms of mountain, table land, and valley."
The form of the earth is popularly considered as a sphere, but extensive geodesical operations prove its true figure to be that of an oblate spheroid, flattened at the poles, or protuberant at the equator; the polar axis being about part shorter than the equatorial diameter *. This result is arrived at by the measurement of arcs of the meridian in different latitudes, by which it is ascertained beyond the possibility of doubt, that the length of a degree at the equator is the least that can be measured, and that this length increases as we advance towards the pole; whence the
* The exact determination of arcs of the meridian measured in France, and also the comparison of the three portions into which the arc of the meridian between Clifton and Dunnose was divided, presenting the same anomaly of the degrees appearing to diminish as they approach the pole, are opposed to the figure of the earth being exactly a homogeneous or oblate ellipsoid; but its approximation to that figure is so close, that calculations based upon it are not affected by the supposed slight difference. The proximity of the extreme stations to mountainous districts was supposed to have been partly the cause of this discrepancy, as the attraction of high land, by affecting the plummet of the Zenith Sector, might have vitiated the observations for the difference of latitude between two stations. A survey was undertaken by Dr. Maskeylene solely to establish the truth of this supposition, the account of which is published in the "Philosophical Transactions" for 1775. A distance of upwards of 4000 feet was accurately measured between two stations, one on the north and the other on the south side of a mountain in Perthshire. The difference of latitude between these extremities of the measured distance was, from a number of most careful observations, determined to be 54" 6. Geodesically this arc ought to have been only 42"-9, showing an error of 11"-7, due to the deflection of the plummet.
greater degree of curvature at the former, and the flattening at the latter, is directly inferred.
Our "diminutive measures" can only be applied to comparatively small portions of the surface of the earth in succession; but from thence we are enabled, by geometrical reasoning, to conclude the form and dimensions of the whole mass.
There are two difficulties attending the measurement of any definite portion of the earth's circumference, (such as one degree, for instance*,) in the direction of the meridian, independent of those caused by the distance along which it is to be carried: the first is, the necessity of an undeviating measurement in the true direction of a great circle; and the second, the determination of the exact spot, where the degree ends.
The earth having on its surface no landmarks to guide us in such an undertaking, we must have recourse to the heavens; and though by the aid of the stars † we can ascertain when we have accomplished exactly a degree, it is far more convenient to fix upon two stations as the termini of the arc to be measured, having as nearly as possible, the same longitude, and to calculate the length of the arc of the meridian contained between their parallels from a series of triangles connected with a measured base, and extending along the direction of the arc. From the value thus obtained, compared with the difference between the latitudes of the two termini determined by a number of accurate astronomical observations, can be ascertained of course the length of one degree in the required latitude.
The measurement of an arc of the meridian, or of a parallel, is perhaps the most difficult and the most important of geodesical operations, and nothing beyond a brief popular description of the
* More than an entire degree (about 100 miles) was actually measured on the ground in Pennsylvania, by Messrs. Mason and Dixon, with wooden rectangular frames, 20 feet long each, laid perfectly level, without any triangulation. Page 10, "Discours Préliminaire, Base du Système Métrique," and "Philosophical Transactions" for 1768.
†The stars whose meridional altitudes are observed for the determination of the latitude should be selected among those passing through, or near, the zenith of the place of observation, that the results may be as free as possible from any uncertainty as to the amount of refraction. With proper care and a good instrument, the latitude for so important a purpose ought to be determined within one second of space, unless local causes interfere to affect the result.
modes of proceeding which have been adopted in this country, and elsewhere, can here be attempted. For the details of the absolute measurement of the bases from which the elements of the triangles were deduced, as well as the various minute but necessary preliminary corrections, and the laborious analysis of the calculations by which the length of the arcs were determined from these data, reference must be made to the standard works descriptive of these operations.
At the end of the second volume of the "Account of the Operations on the Trigonometrical Survey of England and Wales," will be found all the details connected with the measurement of an arc of the meridian, extending from Dunnose in the Isle of Wight, to Clifton, in Yorkshire. The calculations are resumed at page 354 of the third volume; the length of one degree of the arc resulting from which, in latitude 52° 30′, (about the centre of England,) being equal to 364,938 feet.
An arc of a parallel was also measured in the course of the trigonometrical survey between Beachy Head and Dunnose, in 1794, but fault has been since found with the triangulation, and corrections have been applied to the longitudes deduced therefrom, which are alluded to in "The Chronometer Observations for the difference of the longitudes of Dover and Falmouth," by Dr. Tiarks, published in "The Phil. Trans. for 1824," and in Mr. Airy's paper "On the Figure of the Earth."
The arc measured by Messrs. Mechain and Delambre between the parallels of Dunkirk and Barcelona, described in detail in the "Base du Système Métrique Décimal," had for its object, as the title of the work implies, not only the determination of the figure of the earth, but also that of some certain standard, which, being an aliquot part of a degree of the meridian in the mean latitude of 45°, might be for ever recognised by all nations as the unit of measurement. To have any idea of the labour and science devoted to this purpose, it is necessary to refer to the work itself, in which will be found the reasons for preferring a portion of the measurement of the surface of the globe involving only the consideration of space, to the length of a pendulum vibrating seconds having reference both to time and space. In addition to the determination of this standard of linear measurement, which was
denominated the "metre," and defined to be the ten-millionth part of the quarter of a great circle passing through the poles *, the committee, consisting of all the most distinguished scientific men on the Continent, agreed also upon a standard of weight derived from the same source. A cube, each side part of the metre, or a " decimetre," (chosen on account of its convenient size,) was supposed to be filled with distilled water of the temperature of ice just melting; and the weight of the fluid constituted the "killogramme." This temperature was selected as being pointed out by nature, and independent of any artificial gradations; and also, as being the point at which the density of water is nearly a maximum, as it expands immediately on solidifying; although down to about 40° it continues gradually to condense. No other substance, either liquid or solid, combines so many recommendations; but the difficulty that arose was to construct a solid mass representing this weight of water, which might be kept as a standard; their method of overcoming this is explained at pp. 563, 626, and the following pages of the third volume. "Bodies of unequal specific gravities may weigh equally in one state of the atmosphere, but not so in one of either greater or less density, and a vacuum was therefore of necessity resorted to." In the words of the report, (vol. iii. p. 565,) "C'est au poids du decimètre cube d'eau distillée, à sa plus grande densité, qu'on doit faire égal le poids d'une masse solide donnée, tous les deux étant supposés dans le vide; voilà a quoi se reduisoit la question de la fixation de l'unitè de poids." In the end, cylinders of platinum and of brass were constructed, of precisely the same weight as the killogramme of water, both weighed in a vacuum. These two, from the difference of their masses, evidently would not
* The French Commissioners, however, having in their calculations employed as their value of the earth's compression, now known to be incorrect, the metre, strictly speaking, can no longer be so defined. The determination of the value of the English standard,—the yard, has been recommended by the commissioners appointed in 1841 for the restoration of the standards of weight and measures after the injury done to the original standard by the burning of the House of Commons in which it was deposited, to be effected by joint reference to the three standards extant upon which most reliance can be placed; viz., those belonging to the Royal Society; the Royal Astronomical Society; and the Board of Ordnance; instead of having recourse to the standard previously established by act of Parliament, of the length of a pendulum vibrating seconds at a fixed temperature in the latitude of London. Mr. Baily states this length at the level of the sea, in vacuo, at the temperature of 62° Fahr., by Sir G. Shuckburgh's scale, to be 39.1393 inches.
weigh alike in the air. A brass cylinder, (of which several were made,) was kept as the standard for public use; the platinum presented to the "Institut," to be deposited there as "le représentatif d'une masse d'eau prise à son maximum de condensation, contenue dans le cube du decimètre, et pesée dans le vide."
During the progress of these operations, observations were made by Borda, (whose repeating circles of 16 and 16 inches diameter were used in triangulation,) on the length of a pendulum vibrating seconds at the level of the sea, in the latitude of 45°, at one determinate temperature. The length of this pendulum (of platina) was ascertained in millimetres, and was declared by the Committee to be so accurate, as to serve, in case of any accident happening to the standard, to construct again the unit of measurement without another reference to an arc of the meridian.
The prolongation of the measurement of this arc from Barcelona to Formentera, the most southerly of the Balearic Isles, and its connection with England and Scotland, was published in 1821 by Messrs. Biot and Arago (under whom the operations were conducted), in a work entitled "Recueil des Observations Géodesiques, Astronomiques, et Physiques." The whole arc measured amounted nearly to 124°, and was crossed at about half its length by the mean parallel of 45°.
The following table, taken from Mr. Airy's Figure of the Earth," published in the "Encyclopædia Metropolitana," shows the length of the principal arcs of meridian and parallel that have been measured in different latitudes :