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, a more precise method is available: dividing the land across the meridian into "air" triangles, which makes the measurement 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 forms an "air" triangle, the angle of which can be measured with the aforementioned instrument.In fact, any object—a clock tower by day, a street lamp by night—can be accurately determined by a good observer with the help of a crosshair.Triangles thus obtained often have bases several miles long.It is in this way that Arago connects the coast of Valencia, Spain, 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 length of the other two sides and the third angle can be obtained according to known conditions. degrees of an 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 meridian 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.Angles can be accurately measured in degrees with a theodolite or re-measurement theodolite, but the first side, the basis of the whole scheme, must be measured directly on the ground with extraordinary precision, and this is the trickiest job in all triangulation.
When Delambre and Méchan measured the longitude between Dunkirk and Barcelona, they took as the base of their triangulation a straight line in the Seine-et-Marne from the road from Morins to Les Saintes.This base is 12,150 meters long, and it took more than 45 days to measure it.How did the two scientists arrive at a precision?This is what the experiments of Colonel Everett and Mathieu Strux will tell, in the same way as the two previous 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 a large six-foot ruler 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 a low and gentle hillside that gradually disappears, belonging 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," said Mathieu Strux, "I think from observation that once the bottom edge of the foundation is established, we can establish the end of the warp on this level grass."
"I think as you do, Mr. Strux," replied Colonel Everett. "Once we have established the exact longitude here, we shall confirm it on a map. If this arc of meridian passes through the The region will not encounter insurmountable obstacles preventing geodesy."
"I don't think so," said the Russian astronomer.
"We'll find out," said the English astronomer. "We'll measure the base base here, since it's suitable for the operation, and then decide whether we can connect it 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 had to beat the geodesy of Morin, France, so perfectly that they later measured a new foundation base near Perpignan, the southern tip of the triangulation, to test those trigonometric calculations. An error of 11 French inches between the direct measurement and the calculated value was found over a distance of 33 Tuises.
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.Mr. John Murray does not need to participate in the basic hem measurement, but is mostly 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.
"On the road, man," said Michel Zorn cheerfully to William 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 topography establishes a southeast-northwest orientation for this leg.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.Michel Zorn checks and confirms the correct placement of the poles with a crosshair microscope.
This straight line is to be measured in about nine miles, the assumed length for which astronomers intend to prescribe.A leveling rod is attached to the top of each small stake 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 rulers 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 on three iron screws through the bottom, with only a few French inches of clearance to keep them from loosening and to keep them in their constant position.
On these plinths are very cleverly placed small pieces of wood to support the ruler and secure it in the small frame.These small frames fix the orientation of the rulers, but do not hinder their thermal expansion and contraction, which must be considered in the experiment.
When the 12 bases were fixed and the tops were covered with small wooden blocks, Colonel Everett and Mathieu Strux were in charge of placing rulers on them, and the two young people also participated in this difficulty Biggest job.Nicolas Ballandre, pencil in hand, was always ready to record the values told to him in a double note.
A total of six rulers were used, the lengths of which were determined with absolute accuracy in advance and compared with the French Toise, the commonly used unit of length in geodesy.
These rulers were each two tuvoises long, six wide, and one ligne thick, and were made of platinum, a metal which does not deteriorate under ordinary conditions, nor does it oxidize under any conditions of heat or cold.But 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 shorter brass 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 elongation of the ruler, from which the absolute elongation of 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 and can be accurate to 0.025 Tuwaz.
The rulers were placed end to end on the small wooden blocks, but none of them touched each other, because any brief contact caused collisions.Colonel Everett and Mathieu Strux themselves laid out the first ruler on the small block.There is a leveling staff built on the first small wooden post about 100 tuwaz away. Since both ends of the ruler are equipped with iron points that are just perpendicular to the central axis of the ruler, it is easy to adjust the ruler accurately. placed in the desired 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 leveling staff.This ensures that the ruler is placed 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 ruler. No hill will have any appreciable effect on this line, so Ability to accurately mark the top of the base bottom edge on the ground."
"Yes," said Strux, "but only if we take into account 1/2 the thickness of the wire at the point of contact."
"I thought so too," said Colonel Everett.
The starting point is precisely fixed and the work continues.But it is not enough to place the ruler exactly in the straight line of the base of the foundation, it is also necessary to take into account its inclination relative to the horizon.
"I don't think we can hope to have the ruler perfectly level," said Colonel Everett.
"I don't think so either," said Strux, "just measure the angle each ruler makes with the horizon with a spirit 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 marked with ten-degree arcs in 5-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 ruler: its change 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 was observed once by Colonel Strux and Colonel Everett, and was able to mark the absolute value of the change of the platinum ruler. This value was recorded in two notebooks for calculation at 16°C.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 ruler; place the second ruler on the small wooden block with a small space between the two rulers.The second ruler was placed in the same manner—after they had carefully checked that the four iron points of both rulers were in a straight line at the midpoint of the leveling staff.
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 copper ruler, there is a small piece of platinum slide that slides very slightly in the chute.Colonel Everett moved the slider into contact with the second ruler.Since the scale of the slide is accurate to 0.010 tons and is located on the side of the chute, the vernier equipped with a microscope can be accurate to 0.100, so the interval deliberately left between the two rulers can be accurately calculated.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.In order to prevent this difference in temperature variation, they built a small canopy a few French inches above the ruler, so that it would not hinder the observation.Only in the morning and afternoon, when the sunlight 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 put in place one after another, and their directions, angles, expansions, and true lengths had been checked, the scientists removed the first ruler and its base and support, and recreated it with the same rule behind the fourth ruler. start working.Although the scientists are very skilled, these operations still take a long time.They can only measure 220-230 Tuise per day, especially when the weather is bad and the strong wind will destroy the stability of the instrument, they have 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.The ruler marked "No. 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 No. 1 ruler have been measured, put it back to a fixed place, and record the extension distance measured with the No. 4 ruler, and then use a vertical tangent to the top of the No. 1 ruler. 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.
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 is a series of experiments they conducted on this flat plain for 38 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 scale 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.
There was only one issue which created a marked difference between the two opponents, and the Lord John Murray 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 10,000 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 8 kilotons, 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.
Finally, the directly measured length of the base is 8037.65 Tuises, above which a series of triangles will be built spanning several latitudes in southern Africa.
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