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Detailed descriptions of the various methods that have been at different times adopted to insure the correct measurement of base lines on the Continent, may be found in all standard works on geodesical operations*. A popular account of the mode of conducting these measurements, and of the nature of the rods, &c., used, is also given in Mr. Airy's "Figure of the Earth," in the "Encyclopædia Metropolitana," commencing at page 206.
A base measured on any elevated plain is thus reduced to its proper measure at the level of the sea.
Call A B the measured base at any elevation
A a above the level of the sea
a b its value at this level
Cb the radius of the earth
b R h,
And the altitude above the sea A a
as ascertained by levelling, or by the barometer. Then R+h: R:: B: b. &b=Rh
And B-b the difference of the measured and re-
21008000 feet; if,
The radius of the earth may be considered then, the log of the base, in feet, be added to the log of the altitude, and the log of the sum of the radius and altitude be subtracted therefrom, the remainder will be the log of a number to be
* "Recueil des Observations Géodesiques, par Biot et Arago" "Puissant, Traité de Géodesie "-" Base du Système Métrique decimal;" and the works of Cassini, Francœur, Colonel Lampton, &c.
The bases of the original arc of Mechain and Delambre, described in the "Base du Système Métrique," were measured with rods of platinum two toises long; to each bar was attached at one end a rod of brass. The proportion of the expansion of brass and platinum being known, the expansion of the platinum rod was inferred from the observed difference of expansion of the two rods. The rods were laid in boxes, and placed on trestles; and their ends not brought into contact, but measured with a slider. The temperature was reduced to thirteen degrees of Reaumur. The length of the base of Perpignan was 6006.28 toises; and that of Melun 6075.9 toises. The calculation of the Perpignan base of verification from that of Melun differed only eleven inches from its actual measurement on the ground.
These platinum bars are described in page 203, vol. i. Puissant's "Géodesie." Few bases have ever been measured solely for the determination of the value of an arc of the meridian, or of a parallel, but have formed at the same time the foundations of the survey of a country.
deducted from the measured base, to reduce it to its value at the level of the sea. This correction, though generally trifling, is not to be neglected when the base is measured on ground of any considerable elevation.
Mr. Airy, in page 198 of the "Figure of the Earth,” in the Encyclopædia Metropolitana," gives this formula :-" If r be the earth's radius, or the radius of the surface of the sea (which is known nearly enough), h the elevation, the measured lengths must be multiplied by the fraction + or 12 or they must be diminished by the part of the whole. If the surface slopes uniformly, the mean height may be taken; if it is very irregular it may be divided into several parts."
The reduced length a b of the base
AB is thus found, and if the length of the chord is required, it is found by subtracting 247
Beside the marks at the extremities of a base line-which, if it is to form the groundwork of a survey of considerable extent, should be constructed so as to be permanent, as well as minute— intermediate points should be carefully determined and marked during the progress of the measurement by driving strong pickets, or sinking stones into the ground, with dots upon a plate of metal, or some other indication of the exact termination of the chain, clearly defined upon them. These marks serve for testing the accuracy of the different portions, and reciprocally comparing them with each other. It has been already remarked, that the length of the base on the Ordnance Survey of Ireland was not obtained entirely by measurement, an addition of two miles having been made
to its measured length by calculation. This calculation was also contrived to answer the purpose of verifying the measurement of intermediate portions of the base between marks left for the purpose, as alluded to in the last paragraph; and which will be explained by reference to the figure given below, in which AB represents the portion of the base actually measured, and BC, that to be added by calculation, for the purpose of extending the base to C, to obtain a more eligible termination.
The points E and D have been marked during the measurement, and are thus made use of:—
The stations F and G are selected, so that the angles at E may be nearly right angles, and the points themselves nearly equidistant from the line, and about equal to AE. Similar conditions determine the positions of H, I, K, and L. At A the whole of the objects visible are most accurately observed with a large theodolite, which is then taken to the other points on the line, as well as
those selected on either side of it, where all the angles are measured. From A E, then, and the three observed angles, GE and EF are determined, from each of which in the triangles GED and DEF the side ED is obtained, the distances thus found forming two checks on its measured length; ID and DH are in like manner calculated from A D and also from ED as bases, and each of these again furnish data for the determination of DB. Lastly, BL and BK are found from AB, and also from EB; from the mean results of which BC, the required addition to the measured base, is obtained.
Even if the entire base had been measured, the above is an excellent method of verifying the accuracy of the intermediate component parts; and is also a test of the instrument used for measuring the angles. The stations H, K, L, &c., will also answer for minor trigonometrical points, and will be found useful in the course of the work.
The next process, as has been stated, is the Triangulation, which, combined with the measurement of a base line, just described, forms the preliminary step, not only in a correct trigonometrical survey, but in the more delicate operations of the determination of the difference of longitudes between two meridians, such as those of the observatories of Greenwich and Paris, and the measurement of an arc of the meridian to obtain the length of a degree in different latitudes, from whence to deduce the figure and magnitude of the earth.
THE most conspicuous stations are selected as trigonometrical points, and are chosen with reference to their relative positions; as the nearer these triangles approach to being equilateral, the less will be the error in the calculation of the sides resulting from any slight inaccuracy in the observed angles.
The base being generally of trifling length, compared with the distances between the points of the principal triangles to be ultimately deduced from it, the sides of these triangles must be from the first gradually increased as rapidly as is consistent with the remark in the previous paragraph, till they arrive at their greatest limit *, determined in an extensive survey by the distance at which these points can be rendered clearly visible. As early as 1822, the reflection of the sun from a plane mirror was employed in Hanover for the purpose of rendering distant stations visible; and this method was adopted by General Colby and Captain Kater in verifying General Roy's triangulation for connecting the meridians of Paris and Greenwich. The station on Hanger Hill tower could not be seen from Shooter's Hill (only 10 miles distant), owing to the dense smoke of London, but was rendered clearly visible by tin plates attached to the signal post so as to reflect the sun towards the station at stated times on a certain day. The same plan was tried the
'Laplace a demontré par le calcul des probabilités qu'il ne faut employer que le moins grand nombre possible de triangles du premier ordre couvrant l'étendue entiere du pays, en leur donnant les plus grandes dimensions permises par les localités, et par la puissance des lunettes des instruments." Francœur, "Geodesie," page 110.
The distances between some of the trigonometrical points on the Ordnance Survey of Ireland exceed 100 miles, and have been deduced from the original base of about 10 miles. Observations may be made on a station which would be hid by intervening high ground were it not elevated above its real place by refraction, but periods should always be chosen for observing angles when extraordinary refraction is not remarkable, on account of its very irregular action.