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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 793′33, 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 meridian ‡ 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.
and Barcelona described in the "Base du Système Métrique,” as also on that between Dunnose and Clifton, it being considered not only more expeditious, but also more correct. Another advantage of this method is (if all the triangles are intersected by the meridian), that by calculating the various portions of which the arc is composed from the right-angled triangle formed on each side of the meridian separately, one result serves as a check upon the other.
A modification of this method is described in Puissant's Géodesie," page 248, which consists in constructing through the vertices of the triangles parallels both to the meridian AX and the perpendicular A Y, without taking any account of the spherical The intersections of these lines form, with the sides of the triangles, right-angled triangles, of which those sides are the hypothenuses; and the azimuth of each being known, all the elements can be ascertained, as is evident by reference to fig. 2. In this manner, the distances of several places from the perpendicular, and the meridian passing through the observatory of Paris, were calculated by Cassini.
The third method ("Puissant," vol. i. page 316) of ascertaining the length of the arc AX is by determining the geographical positions of the vertices of the triangles extending along the meridian, and calculating the difference of their parallels of latitude projected on the meridian, the sum of these being the measure of the arc.
The measure of an arc of a parallel is calculated by a similar process, which is described at page 319 of the same work.
The methods of calculating, geodesically, the latitudes, longitudes, and azimuths of the different stations from one meridian, with the rigid accuracy required in such operations as the measurement of an arc of the meridian or parallel, will be found fully explained in the 12th chapter of Woodhouse's "Trigonometry;" in the 18th chapter of Puissant's "Géodesie;" and in "Francœur." Their determination by astronomical observations will be treated of hereafter.
On the supposition that the earth is a sphere, the calculations are resolved into the solution of spherical triangles.
The accurate length of the arc on the surface of the earth, between two very distant places whose latitude and longitude have
been determined, is, on account of the spheroidal figure of the globe, a problem of great difficulty, and of no real practical utility;—it is fully investigated in Puissant's "Géodesie," vol. i., page 296*. Between stations, however, within the limits of triangulation, it is often useful to calculate the distance as a check upon the geodesical operations; and in the length of an extended line of coast, or in a wild country, where triangulation may be, from local obstacles or want of means, quite impossible, the solution of this problem is of great importance for the purpose of laying down upon paper the positions of a certain number of fixed stations, between which the interior survey has to be carried on; and it is, within such bounds, one of easy application, particularly in the latter case, where the observations themselves are generally taken with portable instruments, and not with minute accuracy.
In the accompanying figure, P is the pole of the earth (considered as a sphere), and S and S' the two stations, whose latitude and longitude are determined; the angle SPS' is evidently measured by the difference of their longitude, and PS and PS' are their respective latitudes; the solution of the spherical triangle PSS' then gives the length of the arc SS'.
If it is possible, when observing at S and S', to determine the azimuths of these stations from each other, that is, the angles PSS' and PS'S, a more accurate result will be obtained, as these angles can be determined with precision, whereas the angle P depends upon the correctness of the observations for longitude at each station, which with portable instruments is always, at best, but a close approximation; and the errors in the determination of each may lie in the same, or in different directions. In geodesical operations, if it be possible, the reciprocal azimuths of stations should
* See also Francœur's "Géodesie," p. 208.
+ In cases where the difference of longitude between the two stations can be ascertained by means of signals, or by the interchange of chronometers, as explained in the next chapter, the measure of the angle P may be obtained with great accuracy.