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MEASUREMENT OF A BASE LINE.
In fixing upon an appropriate site for the measurement of a base line, a level plain should obviously be selected, where both ends of the base would be visible from the nearest trigonometrical points. Steel chains, glass, deal, and platinum rods, have at different times been used for the purpose of determining its length; but each of these units of measurement, whichever is preferred, must be supported in such manner on trestles that it may be laid perfectly level. The whole thus forms a portion of a great circle, which has ultimately to be reduced to its proper measure at the level of the sea, and at one mean temperature.
The exact measurement of a base, though apparently easy, is perhaps the most difficult, and the most important part of a trigonometrical survey, as upon its accuracy that of every triangle depends. In the account of the operation on the Trigonometrical Survey of England and Wales, published in 1801, will be found detailed accounts of the first base measured on Hounslow Heath, in 1784, with Ramsden's steel chain, at first intended solely for the purpose of connecting by triangulation the Observatories of Paris and Greenwich, but afterwards made the first step in the trigonometrical survey of England. This base was measured a second time with prepared deal rods,* and again by a combination of these two methods, the mean of the three valuations being 27404.0137 feet at the level of the sea. The details of the base of verification (i.e. the actual measurement of the side of a remote triangle, whose length had been previously obtained by calculation) in Romney Marsh, in 1787, are also given in the same work, as well as
* The deal rods were first laid, as it is termed, “in coincidence;” that is, lines drawn across them, near their extremities, were made to coincide most accurately by fine
screws, as in the sketch,
but this method occupying a con
siderable time, their spherical ends were afterwards brought in contact and the measurement was continued in this manner, so that no decision was arrived at as to the comparative accuracy of the two modes; that by coincidence would, however, appear likely to be more minutely correct than the one adopted.
the remeasurement of the original base on Hounslow Heath, in 1791, and of another base of verification on Salisbury Plain, in 1794, which is stated to have corresponded exactly with its mean length, as obtained by calculation in three different triangles.
No account has as yet been published of the mode adopted by Colonel Colby to obtain the accurate value of the base measured on the Ordnance Survey of Ireland, but as the beautiful contrivance resorted to for preventing any expansion or contraction between the points marked at the extremities of the metal rods which were used for the purpose, instead of allowing for the alteration produced in their length by every change of temperature, is original, as well as the more correct mode of obtaining a visual instead of an actual contact of their extremities; and as the method itself is perhaps susceptible of a greater degree of accuracy than any that has been practised either in England or on the Continent, a short description is given below of the means employed on the measurement of this base, on the east side of Loch Foyle, in the county of Londonderry.
Two bars, one of iron and the other of brass, ten feet long, placed parallel to each other, were riveted together at their centres, it having been previously ascertained, by numerous experiments, that they expanded and contracted in their transitions from cold to heat, and the reverse, in the proportion of three to five. The latter was coated with some non-conducting substance to equalise the susceptibility of the two metals to change of temperature; and across each extremity of these combined bars was fixed a tongue of iron, with a minute dot of platinum, almost invisible to the naked eye, and so situated on this tongue, that, under every degree of expansion or contraction of the rods, the dots at each end always remained at the constant distance of ten feet.* This will be better understood by reference to the sketch below.
A is the iron bar (about five-eighths of an inch wide and one and a
* In most methods of measuring the length of a base line the temperature requires to be noted as every chain or rod is laid ; and the expansion or contraction consequent upon every change in the thermometer calculated and allowed for.
half deep), the expansion of which is represented by three ; B the brass bar (of the same size), the expansion of which is five, the two being riveted together at the centre C; D E and d e are the iron tongues pinned on to the bars, so as to admit of their expansion, with the platina dots at D and d. The tongues are by construction made perpendicular to the rods at a mean temperature of 60. Fahrenheit, and the expansion taking place from their common centre, when A expands any quantity which may be expressed by three, B expands at the same time a quantity equal to five, and the position of the tongues is changed to D F, d f, the dots D and d remaining unalterably fixed at the exact distance of ten feet. It is evident from this construction, that the dots at the extremities of these bars could not, if desired, be brought either into actual contact or coincidence, but a more correct plan was adopted, which consisted of laying each rod so that the dot at its extremity should always be at a fixed distance from that at the end of the next rod. This was effected by means of powerful microscopes, attached to the end of similar short compound bars,* six inches long, mounted on a stand, by which means they could be laid perfectly horizontal by a spirit level, the microscopes in these bars occupying the position of the dots on the longer rods. These dots, after the rods had all been carefully levelled, were brought exactly under the microscopes by means of three micronometer screws attached to the box in which each rod was laid, so that it could be moved to either side, backwards or forwards, elevated or depressed, as required, the rods being laid on supports equidistant from the centre of the box, that they might always have the same bearing. The point of starting was a stone pillar, with a platina dot let into its centre, with a transit instrument placed over it, by which the line was laid out with the greatest precision, with the assistance of sights at each end of the bars, an average of about 250 feet being completed in one day, and five boxes, giving a length of 52 feet, being levelled and laid together.
* This was the usual distance between the foci of the microscopes; but to meet cases where the uneven surface rendered it difficult to bring the short bars to a level at this distance, it was sometimes diminished to one half. Microscopes of different lengths were used where the inclination of the ground rendered it necessary to lay the boxes on different levels, so that the platina dots might be brought in the focus of each microscope.
About 400 feet of this measured base was across the river Roe, and clumps of pickets were driven at intervals of about 5 feet 3 inches apart from centre to centre, by a small pile engine, on the heads of which the boxes containing the compound rods rested. At the end of each day's work a triangular stone was sunk at the end of the last bar laid, with a cast-iron block fitting over it, having a brass plate with a silver disk let into the middle of the brass, which was adjustable by means of screws. This disk was brought exactly under the focus of the extreme microscope, and served as a starting point the following day, a sentinel being always left in charge of this stone, which was further secured by a wooden cover screwed over it.
The total length of the measurement of this base amounted to about eight miles; two miles were subsequently added by a method described in the next page, making the entire distance between the two extremities rather more than 10 miles.
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 to be found in Mr. Airy's "Figure of the Earth,” in the “Encyclopædia Metropolitana," commencing at page 206.
* “Recueil des Observations Geodesiques, par Biot et Arago". Puissant, Traité de Geodesie”—“Base du Système Metrique decimal ;” and the works of Cassini, Franceur, Colonel Lampton, &c.
The bases of the original arc of Mechain and Delambre, described in the “ Base du Système Metrique,” 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, or 38.408 feet; and that of Melun 6075.9 toises, or 12,950 feet. The calculation of the Perpignan base of verification from that at Melun differed only eleven inches from its actual measurement on the ground.
These platinum bars are described in page 203, vol. i. Puissants “ Geodesie.” 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.
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
B ab its value at this level
as ascertained by levelling, or by the barometer. Then R+h:R::B:6.& b = R. B
B.R B.h duced base = B
Rth Rth The radius of the earth may be considered = 21008000 feet; if, 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 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 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
A B is thus found, and if the
length of the chord is required, it
is found by subtracting