Chapter 7 Chapter 7 Bases of Triangles

We know that the geodetic survey to be carried out by the Commission is a triangulation exercise for the purpose of obtaining the length of a meridian arc.From a standpoint of mathematical precision, however, it is simply not feasible to connect lengths of a metal ruler end to end to measure the length of an arc of meridian spanning a latitude or a few latitudes.And there is no single piece of land that is completely tiled within a few hundred kilometers at any point on Earth to allow this tricky experiment to be efficiently conducted.Thankfully, we can use a more precise method: Divide the land passing through the meridian into triangles in the sky, so the measurement is relatively easy.

These triangles were obtained with the aid of sophisticated instruments: theodolites or double-measured theodolites, natural or artificial signs such as clock towers, street lamps, poles, etc.Each sign can form a triangle in the air, the angle of which can be measured with the above-mentioned instrument.In fact, any object, a clock tower by day or a street lamp by night, can be accurately determined by a good observer with the help of the cross mark.Triangles thus obtained often have bases several miles long.It is in this way that Arago connects the Spanish coast of Valencia and the Balearic Islands with a huge triangle, one of which is 82,555 tuises long.

Now, according to a geometric principle, any triangle can be fully known as long as the length of one side and the measure of two angles are known, because the lengths of the other two sides and the third angle can be obtained according to known conditions. The degree of the angle.Therefore, take one side of a known triangle as the base of a new triangle, and then measure the degree of the angle adjacent to this base, so that new triangles can be continuously established until the longitude to be measured The end point of the arc.In this way, the lengths of all the straight lines in the triangular series can be known, and then through a series of trigonometric calculations, the length of the meridian arc passing between the two endpoints of the triangular series can be easily determined.

I just said that as long as one side and two angles of a triangle are known, the whole triangle can be known.The degree of the angle can be accurately measured by theodolite or re-measurement theodolite, but the basis of the whole scheme of the first side must be measured directly on the ground with exceptional accuracy, which is the most difficult work in the whole triangulation. When Delambre and Mechan measured the longitude between Dunkirk and Barcelona, ​​they took Seine.The base of the triangulation is the distance of a straight line from Morin to Lear St. in the Marne.This base is 12,150 meters long, and it took more than forty-five days to measure it.How did the two scientists arrive at a precision?This is Colonel Everett and Mathieu.What Strux's experiments will tell, they will do in the same way as the first two French scientists.We'll see how accurately they should operate.

The preliminary work of the geodesy was begun on the 5th of March, much to the amazement of the Bushmen, who knew nothing of it.It was a scientist's joke to Morcumb to connect lengths of large six-foot rulers to measure land.In any case, he had done his duty, and he was asked to find a level plain, which he found. For a direct measurement of the hem, this area is well chosen.The plain is covered with sparse dry turf, extending flat and clearly to the end of the horizon.The surveyors of Moran Road must have not been so lucky.At the southern end of the plain, a line of rolling hills constitutes the southern end of the Kalahari Desert, but there is no end to the north, and the eastern part is the gradually disappearing low slopes, which belong to the Lataku Plateau.

In the western part of the plain, the land continued to decline and became a watery marsh. This stagnant backwater was the source of water for the tributaries of the Juluman River. Colonel Everett, Mathieu.From observation, said Strux, I think that once the bottom edge of the foundation is established, we can establish the end of the meridian on this flat grass. I think as you do, Mr. Strux.Colonel Everett replied that once we have established the exact longitude here, we shall confirm it again on the map, provided that the arc of meridian passes through no insurmountable obstacles preventing the geodetic survey.

I don't think so.Russian astronomer said. We will know.Here, the British astronomer said, we measure the base base, since it is suitable for this operation, and then decide whether it can be connected to the series of triangles that the meridian arc will cross. Having thus decided, they intend to begin immediately with base hem measurements.The work goes on for a long time because the members of the committee want to do it with exacting precision.They were to accurately beat the geodetic survey of Morin, France, which was done so perfectly that when a new foundation base was later measured to test those trigonometric calculations at the southern end of the triangulation near Perpignan, it was only within three An error of eleven inches between the direct measurement and the calculated value was found for a distance of thirteen tuises long.

As soon as the order was given to set up camp, a small Bushman-like village, surrounded by a protective fence, began to appear provisionally on the plain.Four-wheelers are lined up to look like real houses, and the town is divided into British and Russian quarters, each with its own flag flying over it.In the middle of the town is a public square.Outside a circle of four-wheeled carts, drivers herd horses and buffalo, which they drive into the circle at night to escape the ravenous beasts that haunt the southern African interior. Morcum is in charge of organizing hunts to feed the town.John.Your Excellency Ma Rui does not need to participate in the basic bottom edge measurement, mostly because he is busy with food.Therefore, it is important to cherish the stored meat and supply fresh game meat to the expedition team every day.Fortunately, Morcum was industrious, and his companions were quick-witted, so there was never a shortage of game.The plains and mountains for miles around the camp were their hunting grounds, filled with the sound of European weapons.

On March 6, the geodetic experiment began.One or two of the youngest scientists on the committee are responsible for the initial work. Get on the road, man, Michelle.Zorn happily greeted William.said Amory, may the gods of precision help us! The first job is to draw a straight line on the flattest part of the ground.The terrain establishes a southeast|northwest direction for this right-angled edge.The straightness of the line is sampled by means of small wooden stakes placed in the ground at small intervals to form as many benchmarks.Michelle.Zorn used the cross marker mirror to check and confirm whether the placement of the post is accurate.

This straight line is to be measured in about nine miles, which is the assumed length for which astronomers intend to prescribe.Each small stake has a level ruler attached to the top to make metal ruler placement easy.This job takes several days to complete successfully.The two young men did it meticulously. The next step is to connect the straightedges that directly measure the base of the first triangle end to end. This task may seem simple, but on the contrary, it requires great care, and to a large extent determines the success of the triangulation. no. This is the preparation for placing the ruler, which will be described later.

On the morning of March 10, some wooden bases were made on the ground along the direction of the lifted straight line.There are twelve of these bases, built up through the bottom on three iron screws, with only a few French inches of play, which prevent them from loosening and hold them in a constant position. On these plinths are very cleverly placed small pieces of wood to support and hold the ruler in the small frame.These small frames fix the direction of the straightedge, but do not hinder their thermal expansion and contraction, which must be considered in the experiment. When the twelve bases were fixed and the tops were covered with small wooden blocks, Colonel Everett and Mathieu.Strux was in charge of placing the ruler on it, and the two young men also participated in the most difficult work.Nicola.Ballandre held a pencil, ready to record the value told him in a double note. A total of six straightedges were used, the lengths of which were determined with absolute accuracy in advance and compared with the units of length commonly used in the geodesy of Toise, France. Each of these rulers is two tuises long and six French minutes wide. ], one ligah thick, made of platinum, a metal that does not deteriorate under normal circumstances, and does not oxidize under any hot or cold conditions.However, it should be considered that these rulers will expand with heat and contract with cold at different temperatures.So they thought of equipping each ruler with a metal thermometer, which is based on the principle that various metals expand differently at high temperatures.That's why a slightly shorter copper ruler is placed on top of each ruler.There is a vernier at the end of the copper ruler, which can accurately indicate the relative length of the ruler, from which the absolute length of the platinum can be deduced.In addition, the vernier changes can be calculated so accurately that no matter how small the expansion of the platinum ruler can be calculated.From this we can see how high the accuracy of this experiment is.The vernier is also equipped with a microscope, capable of accuracy to 0.025 Tuise. The straightedges were placed end to end on the small wooden blocks, but none of them touched each other, because any brief contact would cause collisions to be avoided.Colonel Everett and Mathieu.Strux himself placed the first ruler on the block.On the first small stake, about a hundred tuises away, a horizontal scale was built, and since both ends of the straightedge were equipped with iron points just perpendicular to the central axis of the straightedge, it was easy to put The ruler is placed exactly in the required direction.Amory and Zorn, who had been following behind, were now prostrate on the ground again, checking that the two points of the ruler were at the midpoint of the level.This ensures that the ruler is positioned in the correct orientation. Now, said Colonel Everett, the point of departure for the experiment should be determined exactly by a line perpendicular to the end of the first straightedge.None of the hills will act appreciably on this line, so the apex of the bottom edge of the foundation can be accurately marked on the ground. Yes, says Strux, but only if we take into account one-half the thickness of the wire at the point of contact. I think so.Colonel Everett said. The starting point is precisely fixed and the work continues.But it's not enough just to place the ruler exactly in the straight line of the foundation's bottom edge, you should also take into account its inclination relative to the horizon. I don't think we can ever hope to have a straightedge in a perfectly horizontal position.Colonel Everett said. I don't think so either, said Strux, just measure the angle each ruler makes with the horizon with a level, and deduce the true length from the measured lengths. The two scientists came to an agreement.So I started to use a special level to measure the angle between the ruler and the horizon.This level is made of a movable alidade, a hinge resting on a wooden square.A vernier indicates the angle of inclination by the coincidence of its scale with that of a ruler marking an arc of ten degrees in fifth-minute increments. The angle of the ruler is measured and the result is confirmed.Just as Ballendre was about to record the results, Strux suggested turning the level over and reading the difference between two radians. This difference was the angle of inclination required to be measured, and the work was checked.The advice of Russian scientists was adopted in experiments like this. So far, two important points have been observed: the orientation of the ruler relative to the base of the foundation and the angle it makes with the horizontal.The two numerical results were recorded on two separate notes, each signed in the margins by the names of the members of the committee. Two equally important observations are required to conclude work on the first straightedge: its variation with temperature and the accuracy of its measurements. As for its change with temperature, it can easily be marked by comparing it with the difference in length of the copper ruler.The microscope, observed once in succession by Colonel Strux and by Colonel Everett, was able to mark the absolute value of the change of the platinum ruler, which was recorded in two notebooks to be deduced at sixteen degrees Celsius.When Barendre got the measured value, everyone checked it again immediately. Now the actual measured length needs to be marked.To get this result it is necessary after the first straightedge; place the second straightedge on the small wooden block with a small space between the two straightedges.The second ruler was placed in the same manner after they had carefully checked that the four iron points of the two rulers were in a straight line at the midpoint of the level. All that remains is to measure the distance between the two rulers.At the part of the end of the first ruler not covered by the brass ruler, a small platinum slide slides unusually slightly in a slide.Colonel Everett moved the slide into contact with the second ruler.Since the scale of the slide is accurate to 0.010 tons, and it is located on the side of the chute, the vernier equipped with a microscope can be accurate to 0.100, so it can be accurately calculated. interval.The values ​​were quickly recorded in two notebooks and immediately rechecked. On Zorn's suggestion, they took another cautious step in favor of more precise measurements.Under sunlight, the part of the platinum ruler covered by the copper ruler heats up relatively slowly.To prevent this difference in temperature, they built a small canopy a few French inches above the ruler, so as not to obstruct the observation.Only in the morning and afternoon, when the sun slanted into the roof and fell on the ruler, they opened the roof from the side to block the main sunlight. This operation was carried out patiently and meticulously for more than a month.When the four platinum rulers were placed one after another, and their directions, angles, expansions, and true lengths had been checked, the scientists removed the first ruler with its base and support, and followed the fourth ruler with the same The rules start working again.Although the scientists are very skilled, these operations still take a long time.They could only measure 220 to 230 tuises per day, especially when the weather was bad and strong winds would destabilize the instrument, so they had to interrupt the experiment. Every afternoon, three quarters of an hour before the vernier calipers could not be read due to insufficient light, the scientists cautiously stopped their work to resume the next morning.A straight edge marked with a number 1 was used temporarily to mark a point in the ground where it would reach, at which point the scientists dug a hole and inserted a stake topped with a lead plate.After the angle, temperature change and direction of the first ruler have been measured, put it back to a fixed place, and record the extension distance measured with the fourth ruler, and then use a vertical tangent to the top of the first ruler to line, make a mark on the board at the top of the stake.At this point, two straight lines (one of which is in the same direction as the vertical base) intersecting at right angles are carefully drawn.The lead plates were covered with wooden round caps, the holes were re-plugged, and the stakes were buried in the ground until the next day.That way, nothing messes up the instrument during the night, and you don't have to start the experiment from scratch. The next day, remove the cover, and place the first ruler in the same place as yesterday, by means of a vertical line whose end point falls on the intersection of two vertical lines. This was the series of experiments they conducted on this level plain for thirty-eight days.All figures are recorded in duplicate and checked, checked and confirmed by all members. There was little dispute between Colonel Everett and his Russian colleagues.The figure of 0.400 tuise displayed on the vernier sometimes caused the two people to exchange a few polite and blunt words.But all members have the right to express their opinions, and everyone must obey the opinions of the majority. Only one issue clearly divided the two opponents, John.His Excellency Marui had to mediate.That's the question about the length of the base of the first triangle.Obviously, the longer the base, the more open the apex angles of the triangle and the easier it is to measure.However, the length cannot be extended indefinitely, and Colonel Everett suggested a base length of six thousand tuises, almost the length of the base of the Moran Road.Strux wanted to extend it to ten thousand tuises, as the ground allowed. Colonel Everett seemed uncompromising on this issue, and Strux seemed equally unwilling to budge.After exposing each other's reasonable arguments, the members began to take their stand.The ethnic issue is on the verge of breaking out.The two scientists were no longer scientists, but a British representative and a Russian representative.Fortunately, a bad weather that lasted for several days interrupted the debate, and the scientists calmed down. Finally, the majority decided that the length of the base was about 8000 tons, which happened to be impartial. All in all, the experiment went well and with precision.As for the accuracy of its altitude, another experiment on the same meridian in the northern hemisphere will allow it to be tested. In the end, the directly measured length of the base was 8037.65 tuises, above which a series of triangles would be constructed spanning several latitudes across southern Africa.