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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 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 :
The detailed accounts of the measurements of these arcs are to be found in the works of Puissant, Cassini, Biot, Arago, Borda, in Colonel Lambton's papers in the "Philosophical Transactions" (1818 and 1823), and in the works of Captain Everest, published in 1839; and a popular description of the different methods adopted for the measurement of the bases, in each of these operations, is given in the paper "On the Figure of the Earth," in the "Encyclopædia Metropolitana," from which the foregoing table was extracted.
The conclusion drawn by Professor Airy from the above measures, is that "the measured arcs may be represented nearly enough on the whole, by supposing the earth's surface at the level of the sea, or at the level at which water communicating freely with the sea would stand, to be an ellipsoid of revolution whose polar semiaxis is 20853810 English feet, or 3949.583 miles; and whose equatorial radius is 20923713 feet, or 3962-824 miles. The ratio of the axis is 298.33 to 299-33: and the ellipticity (measured by the quotient of the difference of the axis by the smaller) is 79353, or '003352. The meridional quadrant is 32811980 feet, and one minute 6076-2777 feet."
Mr. Baily assumes the proportion between the polar axis and the equatorial diameter to be as 304 to 305, whence the compression amounts to 35.
The most general valuation of the compression is 3, and in the numerous tables of compression, given by Dr. Pearson in his invaluable work on Practical Astronomy, it varies from 3 to 325.
Instructions for conducting the measurement of arcs of the meridian will be found in Francoeur, page 148, and also in Puissant's "Géodesie," vol. i. p. 242, and in the 12th chapter of
"Woodhouse's Trigonometry." Below is given a popular account of the methods of procedure.
The line AX in the figure annexed (fig. 1) represents a portion of an arc of the meridian, on which it is required to measure the length of one degree. A and L are the two stations selected as the extreme points to be connected by a series of triangles ABC, BCD, DCE, &c., running along the direction of the meridian
which passes through A. The vertices of these triangles, particularly the station L, are purposely chosen as near as possible to this meridian line; and the distance from A to X, the intersection of a perpendicular to the meridian drawn through L, (the distance
LX being short,) or more correctly to X', the point of intersection with this meridian of the parallel drawn through L, becomes the distance to be attained by calculation. The length of A B, or of any other side, is first accurately determined with reference to some measured base, and the angles at the vertices of all the triangles observed with the most rigid accuracy; and after the necessary corrections for spherical excess have been made, with the reductions to the centre and to the horizon if required*, the sides of the triangles are calculated from these data, as if projected on the surface of the globe, at the mean level of the sea. The azimuths of all these sides also require to be known, that is, the angles they respectively make with the meridian, which can be calculated from CAX, or any other azimuth which has been observed; and the latitudes of the two extreme stations must be ascertained with all the minuteness of which the best instruments are capable †, for comparison with the distance obtained by calcution between them. The first method that was adopted of ascertaining from these data the required length of AX, is termed that of oblique-angled triangles, described in Francœur's "Géodesie," page 151; in "Puissant," vol. i. page 243; in the "Base du Système Métrique;" and in p. 277 of Woodhouse's "Trigonometry." It consists in calculating the distances A M, M M', &c., on the meridian line between the intersections of the sides of these triangles, or their prolongations, as at N ; their sum evidently gives the total length A X.
The preliminary steps of the second method are the same; but instead of finding the distances A M, M M', &c., the perpendiculars to the meridiant Bb, Cc, Dd, are calculated (page 246, Puissant's "Géodesie," vol. i.), the azimuths of all the sides being known; and from thence are obtained the distances on the meridian Ab, Ac, cN, &c., and of course the total length AX. This method was introduced by Mr. Legendre, and has been partly adopted in the calculation of the arc measured between Dunkirk
* Francœur's "Géodesie,” p. 132; Airy's "Figure of the Earth," p. 199.
No less than 3900 observations were made for the, determination of the latitude of Formentera.
Perpendiculars to the meridian in a sphere cut the equator in two points diametrically opposite, but not in an ellipsoid of revolution, or in an irregular spheroid.