The Adventures of Three Englishmen and Three Russians in Southern Africa/Chapter VII
CHAPTER VII.
THE BASE OF THE TRIANGLE.
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The work undertaken by the Commission was a triangulation for the purpose of measuring an arc of meridian. Now the direct measurement of one or more degrees by means of metal rods would be impracticable. In no part of the world is there a region so vast and unbroken as to admit of so delicate an operation. Happily, there is an easier way of proceeding by dividing the region through which the meridian passes into a number of imaginary triangles, whose solution is comparatively easy.
These triangles are obtained by observing signals, either natural or artificial, such as church-towers, posts, or reverberatory lamps, by means of the theodolite or repeating-circle. Every signal is the vertex of a triangle, whose angles are exactly determined by the instruments, so that a good observer with a proper telescope can take the bearings of any object whatever, a tower by day, or a lamp by night. Sometimes the sides of the triangles are many miles in length, and when Arago connected the coast of Valencia in Spain with the Balearic Islands, one of the sides measured 422,555 toises.
When one side and two angles of any triangle are known, the other sides and angle maybe found; by taking, therefore, a side of one of the known triangles for a new base, and by measuring the angles adjacent to the base, new triangles can be successively formed along the whole length of the arc; and since every straight line in the network of triangles is known, the length of the arc can be easily determined. The values of the sides and angles may be obtained by the theodolite and repeating circle, but the ”first” side, the base of the whole system, must be actually measured on the ground, and this operation requires the utmost care.
When Delambre and Méchain measured the meridian of France from Dunkirk to Barcelona, they took for their base a straight line, 12,150 metres in length, in the road from Melun to Lieusaint, and they were no less than 42 days in measuring it. Colonel Everest and Matthew Strux designed proceeding in the same way, and it will be seen how much precision was necessary.
The work was begun on the 5th of March, much to the astonishment of the Bochjesmen, who could not at all understand it. Mokoum thought it strange for these learned men to measure the earth with rods six feet long; but any way, he had done his duty; they had asked him for a level plain, and he had found it for them.
The place was certainly well chosen. Covered with dry, short grass, the plain was perfectly level as far as the horizon. Behind lay a line of hills forming the southern boundary of the Kalahari desert; towards the north the plain seemed boundless. To the east, the sides of the table-land of Lattakoo disappeared in gentle slopes; and in the west, where the ground was lower, the soil became marshy, as it imbibed the stagnant water which fed the affluents of the Kuruman.
“I think, Colonel Everest,” said Strux, after he had surveyed the grassy level, “that when our base is established, we shall be able here also to fix the extremity of our meridian.”
“Likely enough,” replied the Colonel. “We must find out too, whether the arc meets with any obstacles that may impede the survey. Let us measure the base, and we will decide afterwards whether it will be better to join it by a series of auxiliary triangles to those which the arc must cross.”
They thus resolved to proceed to the measurement of the base. It would be a long operation, for they wanted to obtain even more correct results than those obtained by the French philosophers at Melun. This would be a matter of some difficulty: since when a new base was measured afterwards near Perpignan to verify the calculations, there was only an error of 11 inches in a distance of 330,000 toises.
Orders were given for encamping, and a Bochjesman village, a kind of kraal, was formed on the plain. The waggons were arranged in a circle like the houses, the English and Russian flags floating over their respective quarters. The centre was common ground. The horses and buffaloes, which by day grazed outside, were driven in by night to the interior, to save them from attacks of the wild beasts around.
Mokoum took upon himself to arrange the hunting expedition for revictualling; and Sir John Murray, whose presence was not indispensable in the measurement of the base, looked after the provisions, and served out the rations of preserved meat and fresh venison. Thanks to the skill and experience of Mokoum and his companions, game was never wanting. They scoured the district for miles round, and the report of their guns resounded at all hours.
The survey began on the next day, Zorn and Emery being charged with the preliminaries.
“Come along,” said Zorn, “and good luck be with us.”
The primary operation consisted in tracing a line on the ground where it was especially level. This chanced to be from S.E. to N.W., and pickets being placed at short intervals to mark the direction, Zorn carefully verified the correctness of their position by means of the thread-wires of his telescope.
For more than eight miles (the proposed length of the base) was the measurement continued, and the young men performed their work with scrupulous fidelity.
The next step was to adjust the rods for the actual measurement, apparently a very simple operation, but which, in fact, demands the most continuous caution, as the success of a triangulation in a great measure is contingent on its preciseness.
On the morning of the 10th, twelve wooden pedestals were planted along the line, securely fastened in their position, and prepared to support the rods. Colonel Everest and Matthew Strux, assisted by their young coadjutors, placed the rods in position, and Nicholas Palander stood ready, pencil in hand, to write down in a double register the figures transmitted to him.
The rods employed were six in number, and exactly two toises in length. They were made of platinum, as being (under ordinary circumstances) unaffected by any condition of the atmosphere. In order, however, to provide against any change of temperature, each was covered with a rod of copper somewhat shorter than itself, and a microscopic vernier was attached, to indicate any contraction or expansion that might occur. The rods were next placed lengthwise, with a small interval between each, in order to avoid the slight shock which might result from immediate contact. Colonel Everest and Matthew Strux with their own hands placed the first rod. About a hundred toises farther on, they had marked a point of sight, and as the rods were each provided with iron projections, it was not difficult to place them exactly in the proper direction. Emery and Zorn, lying on the ground, saw that the projections stood exactly in the middle of the sight.
“Now,” said Colonel Everest, “we must define our exact starting-point. We will drop a line from the end of our first rod, and that will definitely mark the extremity of our base.”
“Yes,” answered Strux, “but we must take into account the radius of the line.”
“Of course,” said the Colonel.
The starting-point determined, the work went on. The next proceeding was to determine the inclination of the base with the horizon.
“We do not, I believe, pretend,” said Colonel Everest, “to place the rod in a position which is perfectly horizontal.”
“No,” answered Strux, “it is enough to find the angle which each rod makes with the horizon, and we can then deduce the true inclination.”
Thus agreed, they proceeded with their observations, employing their spirit-level, and testing every result by the vernier. As Palander was about to inscribe the record, Strux requested that the level should be reversed, in order that by the division of the two registers a closer approximation to truth might be attained. This mode of double observation was continued throughout the operations.
Two important points were now obtained: the direction of the rod with regard to the base, and the angle which it made with the horizon. The results were inscribed in two registers, and signed by the members of the Commission.
There were still two further observations, no less important, to be made: the variation of the rod caused by differences of temperature, and the exact distance measured by it. The former was easily determined by comparing the difference in length between the platinum and copper rods. The microscope gave the variation of the platinum, and this was entered in the double register, to be afterwards reduced to 16° Centigrade.
They had now to observe the distance actually measured. To obtain this result, it was necessary to place the second rod at the end of the first, leaving a small space between them. When the second rod was adjusted with the same care as the former, it only remained to measure the interval between the two. A small tongue of platinum, known as a slider, was attached to the end of the platinum bar that was not covered by the copper, and this Colonel Everest slipped gently along until it touched the next rod. The slider was marked off into 10,000ths of a toise, and as a vernier with its microscope gave the 100,000ths, the space could be very accurately determined. The result was immediately registered.
Michael Zorn, considering that the covered platinum might be sooner affected by heat than the uncovered copper, suggested another precaution: accordingly they erected a small awning to protect the rod from the sun's rays.
For more than a month were these minutiæ patiently carried on. As soon as four bars were adjusted, and the requisite observations complete, the last of the rods was carried to the front. It was impossible to measure more than 220 to 230 toises a day, and sometimes, when the wind was violent, operations were altogether suspended.
Every evening, about three quarters of an hour before it became too dark to read the verniers, they left off work, after taking various anxious precautions. They brought forward temporarily the rod “No. 1,” and marked the point of its termination. Here they made a hole, and drove in a stake with a leaden plate attached. They then replaced “No. 1” in its original position, after observing the inclination, the thermometric variation, and the direction. They noted the prolongation measured by rod “No. 4,” and then with a plumb-line touching the foremost end of rod “No. 1,” they made a mark on the leaden plate. They carefully traced through this point two lines at right angles, one signifying the base, the other the perpendicular. The plate was then covered with a wooden lid, the hole filled in, and the stake left buried till the morning. Thus, if any accident had happened to their apparatus during the night they would not be obliged to begin afresh. The next day, the plate was uncovered, and rod “No. 1” replaced in the same position as on the evening before, by means of the plumb-line, whose point ought to fall exactly on the point intersected by the two straight lines.
These operations were carried on for thirty-eight days along the plain, and every figure was registered doubly, and verified, compared, and approved, by each member of the Commission.
Few discussions arose between Colonel Everest and his Russian colleague; and if sometimes the smallest fraction of a toise gave occasion for some polite cavillings, they always yielded to the opinion of the majority. One question alone called for the intervention of Sir John Murray. This was about the length of the base. It was certain that the longer the base, the easier would be the measurement of the opposite angle. Colonel Everest proposed 6000 toises, nearly the same as the base measured at Melun; but Matthew Strux wished that it should be 10,000 toises, since the ground permitted. Colonel Everest, however, remained firm, and Strux seemed equally determined not to yield. After a few plausible arguments, personalities began: they were no longer two astronomers, but an Englishman and a Russian. Happily the debate was interrupted by some days of bad weather, which allowed their tempers to cool. It was subsequently decided by the majority that they should “split the difference,” and assign 8000 toises as the measurement of the base. The work was at length completed. Any error which occurred, in spite of their extreme precision, might be afterwards corrected by measuring a new base from the northern extremity of the meridian.
The base measured exactly 8037.75 toises, and upon this they were now to place their series of triangles.