Reports of how Eratosthenes calculated the size of the earth briefly. Time travel: Eratosthenes – the father of geography

Eratosthenes' contributions to the development of geography, the great Greek mathematician, astronomer, geographer and poet, are outlined in this article.

Eratosthenes' contribution to geography. What did Eratosthenes discover?

The scientist was a contemporary of Aristarchus of Samos and Archimedes, who lived in the 3rd century BC. e. He was an encyclopedist, keeper of the library in Alexandria, philosopher, correspondent and friend of Archimedes. He also became famous as a surveyor and geographer. It is logical that he should summarize his knowledge in one work. And what book did Eratosthenes write? They would not have known about it if not for Strabo’s “Geography,” who mentioned it and its author, who measured the circumference of the Earth’s globe. And this is the book “Geography” in 3 volumes. In it he outlined the foundations of systematic geography. In addition, the following treatises belong to his hand: “Chronography”, “Platonist”, “On Average Values”, “On Ancient Comedy” in 12 books, “Revenge, or Hesiod”, “On Sublimity”. Unfortunately, they reached us in small snatches.

What did Eratosthenes discover in geography?

The Greek scientist is rightfully considered the father of geography. So what did Eratosthenes do to deserve this honorary title? First of all, it is worth noting that he is the one in scientific circulation introduced the term “geography” in its modern sense.

He is responsible for the creation of mathematical and physical geography. The scientist made the following assumption: if you sail west from Gibraltar, you can reach India. In addition, he tried to calculate the sizes of the Sun and Moon, studied eclipses and showed how the length of daylight hours depends on geographic latitude.

How did Eratosthenes measure the radius of the Earth?

In order to measure the radius, Eratosthenes used calculations made at two points - Alexandria and Syena. He knew that June 22, the summer solstice, celestial body illuminates the bottom of the wells exactly at noon. When the Sun is at its zenith in Siena, it is 7.2° behind in Alexandria. To obtain the result, he needed to change the zenith distance of the Sun. What instrument did Eratosthenes + use to determine the size? It was a skafis - a vertical pole fixed at the bottom of a hemisphere. By placing it in a vertical position, the scientist was able to measure the distance from Syene to Alexandria. It is equal to 800 km. Comparing the difference in zenith between the two cities with the generally accepted circle of 360°, and the zenith distance with the circumference of the earth, Erastosthenes made a proportion and calculated the radius - 39,690 km. He was only slightly wrong; modern scientists have calculated that it is 40,120 km.

The ancient Egyptians noticed that during the summer solstice the sun illuminated the bottom of deep wells in Siene (now Aswan), but not in Alexandria. Eratosthenes of Cyrene (276 BC -194 BC)

) a brilliant idea appeared - to use this fact to measure the circumference and radius of the earth. On the day of the summer solstice in Alexandria, he used a scaphis - a bowl with a long needle, with which it was possible to determine at what angle the sun was in the sky.

So, after measuring the angle turned out to be 7 degrees 12 minutes, that is, 1/50 of a circle. Therefore, Siena is 1/50 of the circumference of the earth from Alexandria. The distance between cities was considered equal to 5,000 stadia, therefore the circumference of the earth was 250,000 stadia, and the radius was then 39,790 stadia.

It is unknown which stage Eratosthenes used. Only if it is Greek (178 meters), then its radius of the earth was 7,082 km, if Egyptian, then 6,287 km. Modern measurements give a value of 6.371 km for the average radius of the earth. In any case, the accuracy for those times is amazing.

People have long guessed that the Earth they live on is like a ball. One of the first to express the idea that the Earth was spherical was the ancient Greek mathematician and philosopher Pythagoras (c. 570-500 BC). Greatest Thinker antiquity Aristotle watching lunar eclipses, noticed that the edge of the earth's shadow falling on the Moon always has round shape. This allowed him to confidently judge that our Earth is spherical. Now, thanks to the achievements space technology, we all (and more than once) had the opportunity to admire the beauty globe from photographs taken from space.

A reduced likeness of the Earth, its miniature model is a globe. To find out the circumference of a globe, just wrap it in drink and then determine the length of this thread. You cannot walk around the vast Earth with a measured contribution along the meridian or equator. And no matter in what direction we begin to measure it, insurmountable obstacles will certainly appear along the way - high mountains, impassable swamps, deep seas and oceans...

Is it possible to find out the size of the Earth without measuring its entire circumference? Of course you can.

It is known that there are 360 degrees in a circle. Therefore, to find out the circumference, in principle, it is enough to measure exactly the length of one degree and multiply the measurement result by 360.

The first measurement of the Earth in this way was made by the ancient Greek scientist Eratosthenes(c. 276-194 BC), who lived in the Egyptian city of Alexandria, on the shores of the Mediterranean Sea.

Camel caravans came to Alexandria from the south. From the people accompanying them, Eratosthenes learned that in the city of Syene (present-day Aswan) on the day of the summer solstice, the Sun was overhead on the same day. Objects at this time do not provide any shadow, but Sun rays penetrate even the deepest wells. Therefore, the Sun reaches its zenith.

By astronomical observations Eratosthenes found that on the same day in Alexandria the Sun is 7.2 degrees from the zenith, which is exactly 1/50 of the circumference. (In fact: 360: 7.2 = 50.) Now, in order to find out what the circumference of the Earth is, all that remained was to measure the distance between the cities and multiply it by 50. But Eratosthenes was not able to measure this distance running through the desert. The guides of the trade caravans could not measure it either. They only knew how much time their camels spent on one journey, and believed that from Siena to Alexandria there were 5,000 Egyptian stadia. This means the entire circumference of the Earth: 5000 x 50 = 250,000 stadia.

Unfortunately, we do not know the exact length of the Egyptian stage. According to some data, it is equal to 174.5 m, which gives the earth’s circumference 43,625 km. It is known that the radius is 6.28 times less than the circumference. It turned out that the radius of the Earth, but Eratosthenes, was 6943 km. This is how the size of the globe was first determined more than twenty-two centuries ago.

According to modern data, the average radius of the Earth is 6371 km. Why average? After all, if the Earth is a sphere, then in theory the Earth’s radii should be the same. We will talk about this further.

Way precise measurement long distances was first proposed by the Dutch geographer and mathematician Wildebrord Siellius (1580-1626).

Let's imagine that it is necessary to measure the distance between points A and B, hundreds of kilometers away from each other. The solution to this problem should begin with the construction of a so-called reference geodetic network on the ground. In its simplest form, it is created in the form of a chain of triangles. Their tops are chosen in elevated places, where so-called geodetic signs are built in the form of special pyramids, and always so that from each point the directions to all neighboring points are visible. And these pyramids should also be convenient for work: for installing a goniometer instrument - a theodolite - and measuring all the angles in the triangles of this network. In addition, one side of one of the triangles is measured, which lies on a flat and open area, convenient for linear measurements. The result is a network of triangles with known angles and the original side - the basis. Then comes the calculations.

The solution begins with a triangle containing the basis. Using the side and angles, the other two sides of the first triangle are calculated. But one of its sides is also a side of the triangle adjacent to it. It serves as the starting point for calculating the sides of the second triangle, and so on. In the end, the sides of the last triangle are found and the required distance is calculated - the arc of the meridian AB.

The geodetic network necessarily relies on astronomical points A and B. Using the method of astronomical observations of stars, their geographical coordinates (latitudes and longitudes) and azimuths (directions to local objects) are determined.

Now that the length of the arc of the AB meridian is known, as well as its expression in degree measure(as the difference between the latitudes of astropoints A and B), it will not be difficult to calculate the arc length of 1 degree of the meridian by simple division first magnitude to second.

This method of measuring long distances earth's surface received the name triangulation - from Latin word“triapgulum”, which means “triangle”. It turned out to be convenient for determining the size of the Earth.

The study of the size of our planet and the shape of its surface is the science of geodesy, which translated from Greek means “earth measurement.” Its origins should be attributed to Eratosthesnus. But scientific geodesy itself began with triangulation, first proposed by Siellius.

The most grandiose degree measurement 19th century founder Pulkovo Observatory V. Ya. Struve.

Under the leadership of Struve, Russian surveyors, together with Norwegian ones, measured the arc “extending from the Danube along western regions Russia to Finland and Norway to the coast of the Arctic Ocean. Total length this arc exceeded 2800 km! It contained more than 25 degrees, which is almost 1/14 of the earth's circumference. It entered the history of science under the name “Struve arc”. The author of this book in post-war years I had a chance to work on observations (angle measurements) at state triangulation points adjacent directly to the famous “arc”.

Degree measurements showed that our Earth is not exactly a sphere, but is similar to an ellipsoid, that is, it is compressed at the poles. In an ellipsoid, all meridians are ellipses, and the equator and parallels are circles.

The longer the measured arcs of meridians and parallels, the more accurately the radius of the Earth can be calculated and its compression determined.

Domestic surveyors measured the state triangulation network over almost half of the territory of the USSR. This allowed the Soviet scientist F.N. Krasovsky (1878-1948) to more accurately determine the size and shape of the Earth. Krasovsky ellipsoid: equatorial radius - 6378.245 km, polar radius - 6356.863 km. The compression of the planet is 1/298.3, that is, by this part the polar radius of the Earth is shorter than the equatorial radius (in linear measure - 21.382 km).

Let's imagine that on a globe with a diameter of 30 cm we decided to depict the compression of the globe. Then the polar axis of the globe would have to be shortened by 1 mm. It is so small that it is completely invisible to the eye. This is how the Earth appears completely round from a great distance. This is how the astronauts observe it.

Studying the shape of the Earth, scientists come to the conclusion that it is compressed not only along the axis of rotation. The equatorial section of the globe in projection onto a plane gives a curve that also differs from regular circle, though quite a bit - hundreds of meters. All this indicates that the figure of our planet is more complex than it seemed before.

Now it is absolutely clear that the Earth is not a regular geometric body, that is, an ellipsoid. In addition, the surface of our planet is far from smooth. It has hills and high mountain ranges. True, there is almost three times less land than water. What, then, should we mean by the underground surface?

As is known, oceans and seas, communicating with each other, form a vast expanse of water on Earth. Therefore, scientists agreed to take the surface of the World Ocean, which is in a calm state, as the surface of the planet.

What to do in continental areas? What is considered the surface of the Earth? Also the surface of the World Ocean, mentally continued under all the continents and islands.

This figure, limited by the surface of the average level of the World Ocean, was called the geoid. All known “heights above sea level” are measured from the surface of the geoid. The word "geoid", or "Earth-like", was specifically coined to name the shape of the Earth. In geometry, such a figure does not exist. A geometrically regular ellipsoid is close in shape to the geoid.

On October 4, 1957, with the launch of the first artificial satellite Earth, humanity has entered space age. Active exploration of near-Earth space began. At the same time, it turned out that satellites are very useful for understanding the Earth itself. Even in the field of geodesy, they said their “weighty word.”

As is known, classical method The study of the geometric characteristics of the Earth is triangulation. But previously, geodetic networks were developed only within continents, and they were not connected to each other. After all, you cannot build triangulation on the seas and oceans. Therefore, the distances between the continents were determined less accurately. Due to this, the accuracy of determining the size of the Earth itself was reduced.

With the launch of the satellites, surveyors immediately realized: “sighting targets” appeared on high altitude. Now it will be possible to measure large distances.

The idea of the space triangulation method is simple. Synchronous (simultaneous) satellite observations from several distant points on the earth's surface make it possible to bring their geodetic coordinates to unified system. This is how triangulations built on different continents were linked together, and at the same time the dimensions of the Earth were clarified: equatorial radius - 6378.160 km, polar radius - 6356.777 km. The compression value is 1/298.25, that is, almost the same as that of the Krasovsky ellipsoid. The difference between the equatorial and polar diameters of the Earth reaches 42 km 766 m.

If our planet were a regular sphere, and the masses inside it were distributed evenly, then the satellite could move around the Earth in a circular orbit. But the deviation of the Earth’s shape from spherical and the heterogeneity of its interior lead to the fact that above various points The force of gravity on the earth's surface is not equal. The force of gravity of the Earth changes - the orbit of the satellite changes. And everything, even the slightest change in the movement of a low-orbit satellite, is the result of the gravitational influence on it of one or another earthly bulge or depression over which it flies.

It turned out that our planet also has a slightly pear-shaped. Her North Pole is raised above the plane of the equator by 16 m, and the Southern one is lowered by approximately the same amount (as if depressed). So it turns out that in a section along the meridian, the figure of the Earth resembles a pear. It is slightly elongated to the north and flattened at South Pole. There is polar asymmetry: This hemisphere is not identical to the Southern one. Thus, based on satellite data, the most accurate idea of the true shape of the Earth was obtained. As we see, the figure of our planet noticeably deviates from the geometric correct form ball, as well as from the figure of an ellipsoid of revolution.

The sphericity of the Earth makes it possible to determine its size in a way that was first used by the Greek scientist Eratosthenes. Eratosthenes' idea is as follows. On the same geographic meridian of the globe, we select two points \(O_(1)\) and \(O_(2)\). Let us denote the length of the meridian arc \(O_(1)O_(2)\) by \(l\), and its angular value by \(n\) (in degrees). Then the length of the 1° arc of the meridian \(l_(0)\) will be equal to: \ and the length of the entire circumference of the meridian: \ where \(R\) is the radius of the globe. Hence \(R = \frac(180° l)(πn)\).

The length of the meridian arc between the points \(O_(1)\) and \(O_(2)\) selected on the earth's surface in degrees is equal to the difference geographical latitudes these points, i.e. \(n = Δφ = φ_(1) - φ_(2)\).

To determine the value of \(n\), Eratosthenes used the fact that the cities of Siena and Alexandria are located on the same meridian and the distance between them is known. Using a simple device, which the scientist called “scaphis,” it was established that if in Siena at noon on the summer solstice the Sun illuminates the bottom of deep wells (is at the zenith), then at the same time in Alexandria the Sun is \(\ frac(1)(50)\) fraction of a circle (7.2°). Thus, having determined the arc length \(l\) and the angle \(n\), Eratosthenes calculated that the length of the earth's circumference is 252 thousand stadia (a stadia is approximately equal to 180 m). Considering the rudeness measuring instruments of that time and the unreliability of the initial data, the measurement result was very satisfactory (real average length The meridian of the Earth is 40,008 km).

Accurate measurement of the distance \(l\) between points \(O_(1)\) and \(O_(2)\) is difficult due to natural obstacles (mountains, rivers, forests, etc.).

Therefore, the arc length \(l\) is determined by calculations that require measuring only a relatively small distance - basis and a number of corners. This method was developed in geodesy and is called triangulation(Latin triangulum - triangle).

Its essence is as follows. On both sides of the arc \(O_(1)O_(2)\), the length of which must be determined, several points \(A\), \(B\), \(C\), ... are selected at mutual distances of up to 50 km , in such a way that from each point at least two other points are visible.

At all points, geodetic signals are installed in the form of pyramidal towers with a height of 6 to 55 m, depending on terrain conditions. At the top of each tower there is a platform for placing an observer and installing a goniometric instrument - a theodolite. The distance between any two neighboring points, for example \(O_(1)\) and \(A\), is selected on a completely flat surface and taken as the basis of the triangulation network. The length of the base is very carefully measured with special measuring tapes.

The measured angles in triangles and the length of the basis allow trigonometric formulas calculate the sides of the triangles, and from them the length of the arc \(O_(1)O_(2)\) taking into account its curvature.

In Russia, from 1816 to 1855, under the leadership of V. Ya. Struve, a meridian arc with a length of 2800 km was measured. In the 30s In the 20th century, high-precision degree measurements were carried out in the USSR under the leadership of Professor F.N. Krasovsky. The length of the base at that time was chosen to be small, from 6 to 10 km. Later, thanks to the use of light and radar, the length of the base was increased to 30 km. The accuracy of meridian arc measurements has increased to +2 mm for every 10 km of length.

Triangulation measurements showed that the arc length of the 1° meridian is not the same under different latitudes: near the equator it is 110.6 km, and near the poles it is 111.7 km, i.e. it increases towards the poles.

The true shape of the Earth cannot be represented by any known geometric solid. Therefore, in geodesy and gravimetry, the shape of the Earth is considered geoid, i.e., a body with a surface close to the surface of a calm ocean and extended under the continents.

Currently, triangulation networks have been created with complex radar equipment installed on ground points, and with reflectors on geodetic artificial Earth satellites, which allows you to accurately calculate distances between points. A significant contribution to the development of space geodesy was made by a native of Belarus, the famous geodesist, hydrographer and astronomer I. D. Zhongolovich. Based on the study of the dynamics of the movement of artificial Earth satellites, I. D. Zhongolovich clarified the compression of our planet and the asymmetry of the Northern and Southern Hemispheres.

Traveling from Alexandria to the south, to the city of Siena (now Aswan), people noticed that there in the summer on the day when the sun is highest in the sky (summer solstice - June 21 or 22), at noon it illuminates the bottom of deep wells, that is, it happens just above your head, at the zenith. Vertical pillars do not provide shade at this moment. In Alexandria, even on this day the sun does not reach the zenith at noon, does not illuminate the bottom of the wells, objects give shadow.

Eratosthenes measured how much the midday sun in Alexandria is deflected from the zenith, and obtained a value equal to 7 ° 12′, which is 1/50 of a circle. He managed to do this using a device called a scaphis. Skafis was a bowl in the shape of a hemisphere. In its center there was a vertical fortification

On the left is the determination of the height of the sun using a scaphis. In the center is a diagram of the direction of the sun's rays: in Siena they fall vertically, in Alexandria - at an angle of 7°12′. On the right is the direction of the sun's ray in Siena at the moment of the summer solstice.

Skafis is an ancient device for determining the height of the sun above the horizon (in cross-section).

needle. The shadow of the needle fell on the inner surface of the scaphis. To measure the deviation of the sun from the zenith (in degrees) by inner surface scaphis drew circles marked with numbers. If, for example, the shadow reached the circle marked with the number 50, the sun was 50° below the zenith. Having constructed a drawing, Eratosthenes quite correctly concluded that Alexandria is 1/50 of the circumference of the Earth from Syene. To find out the circumference of the Earth, all that remained was to measure the distance between Alexandria and Siena and multiply it by 50. This distance was determined by the number of days that camel caravans spent traveling between cities. In units of that time it was equal to 5 thousand stadia. If 1/50 of the circumference of the Earth is equal to 5000 stadia, then the entire circumference of the Earth is 5000x50 = 250,000 stadia. Translated into our measures, this distance is approximately 39,500 km. Knowing the circumference, you can calculate the radius of the Earth. The radius of any circle is 6.283 times less than its length. Therefore, the average radius of the Earth, according to Eratosthenes, turned out to be equal to the round number - 6290 km, and diameter - 12,580 km. So Eratosthenes found approximately the dimensions of the Earth, close to those determined by precision instruments in our time.

How information about the shape and size of the earth was checked

After Eratosthenes of Cyrene, for many centuries, no scientist tried to measure the earth’s circumference again. In the 17th century a reliable way to measure large distances on the Earth's surface was invented - the triangulation method (so named from the Latin word "triangulum" - triangle). This method is convenient because obstacles encountered along the way - forests, rivers, swamps, etc. - do not interfere with the accurate measurement of large distances. The measurement is carried out as follows: directly on the surface of the Earth, the distance between two closely located points is very accurately measured A And IN, from which the remote ones are visible tall objects- hills, towers, bell towers, etc. If from A And IN through a telescope you can see an object located at a point WITH, then it is not difficult to measure at the point A angle between directions AB And AC, and at the point IN- angle between VA And Sun.

After that, along the measured side AB and two angles at the vertices A And IN you can build a triangle ABC and therefore find the lengths of the sides AC And sun, i.e. distances from A before WITH and from IN before WITH. This construction can be done on paper, reducing all dimensions several times, or using calculations according to the rules of trigonometry. Knowing the distance from IN before WITH and pointing the telescope of a measuring instrument (theodolite) from these points at an object in any new point D, in the same way measure distances from IN before D and from WITH before D. Continuing the measurements, they seem to cover part of the Earth’s surface with a network of triangles: ABC, BCD etc. In each of them, all sides and angles can be determined sequentially (see figure).

After the side is measured AB first triangle (basis), the whole thing comes down to measuring the angles between two directions. By constructing a network of triangles, you can calculate, using the rules of trigonometry, the distance from the vertex of one triangle to the vertex of any other, no matter how far apart they are. This is how the issue of measuring large distances on the Earth's surface is resolved. Practical use The method of triangulation is far from simple. This work can only be carried out by experienced observers armed with very precise goniometric instruments. Usually, special towers have to be built for observations. Work of this kind is entrusted to special expeditions that last for several months and even years.

The triangulation method has helped scientists clarify their knowledge of the shape and size of the Earth. This happened under the following circumstances.

Famous English scientist Newton(1643-1727) expressed the opinion that the Earth cannot have the shape of an exact sphere because it rotates on its axis. All particles of the Earth are under the influence of centrifugal force (force of inertia), which is especially strong

If we need to measure the distance from A to D (and point B is not visible from point A), then we measure the basis AB and in triangle ABC we measure the angles adjacent to the basis (a and b). Using one side and two adjacent corners, we determine the distance AC and BC. Next, from point C, using the telescope of the measuring instrument, we find point D, visible from point C and point B. In the triangle CUB, we know the side NE. It remains to measure the angles adjacent to it, and then determine the distance DB. Knowing the distances DB u AB and the angle between these lines, you can determine the distance from A to D.

Triangulation scheme: AB - basis; BE - measured distance.

at the equator and absent at the poles. Centrifugal force at the equator acts against gravity and weakens it. The balance between gravity and centrifugal force was achieved when the globe “inflated” at the equator, and “flattened” at the poles and gradually took on the shape of a tangerine, or, in other words, scientific language, spheroid. An interesting discovery made at the same time confirmed Newton's assumption.

In 1672, a French astronomer found that if accurate watch transport from Paris to Cayenne (in South America, near the equator), then they begin to lag by 2.5 minutes per day. This lag occurs because the clock pendulum swings slower near the equator. It became obvious that the force of gravity, which makes the pendulum swing, is less in Cayenne than in Paris. Newton explained this by the fact that at the equator the surface of the Earth is further from its center than in Paris.

French Academy science decided to check the correctness of Newton's reasoning. If the Earth is shaped like a tangerine, then a 1° meridian arc should lengthen as it approaches the poles. It remained to use triangulation to measure the length of an arc of 1° at different distances from the equator. The director of the Paris Observatory, Giovanni Cassini, was assigned to measure the arc in the north and south of France. However southern arc he turned out to be longer than the northern one. It seemed that Newton was wrong: the Earth is not flattened like a tangerine, but elongated like a lemon.

But Newton did not give up his conclusions and insisted that Cassini had made a mistake in his measurements. A scientific dispute broke out between supporters of the “tangerine” and “lemon” theories, which lasted 50 years. After the death of Giovanni Cassini, his son Jacques, also director of the Paris Observatory, in order to defend his father’s opinion, wrote a book in which he argued that, according to the laws of mechanics, the Earth should be elongated like a lemon. To finally resolve this dispute, the French Academy of Sciences equipped in 1735 one expedition to the equator, another to the Arctic Circle.

The southern expedition carried out measurements in Peru. A meridian arc with a length of about 3° (330 km). It crossed the equator and passed through a series of mountain valleys and the highest mountain ranges of America.

The work of the expedition lasted eight years and was fraught with great difficulties and dangers. However, the scientists completed their task: the degree of the meridian at the equator was measured with very great accuracy.

The Northern Expedition worked in Lapland (as the northern part of the Scandinavian and West Side Kola Peninsula).

After comparing the results of the expeditions, it turned out that the polar degree is longer than the equatorial degree. Therefore, Cassini was indeed wrong and Newton was right in claiming that the Earth is shaped like a tangerine. Thus ended this protracted dispute, and scientists recognized the correctness of Newton's statements.

Nowadays, there is a special science - geodesy, which deals with determining the size of the Earth using precise measurements of its surface. The data from these measurements made it possible to quite accurately determine the actual figure of the Earth.

Geodetic work to measure the Earth was and is carried out in various countries. Similar work has been carried out in our country. Back in the last century, Russian surveyors carried out very precise work on measuring the “Russian-Scandinavian arc of the meridian” with an extension of more than 25°, i.e., a length of almost 3 thousand. km. It was called the “Struve arc” in honor of the founder of the Pulkovo Observatory (near Leningrad) Vasily Yakovlevich Struve, who conceived this enormous work and supervised it.

Degree measurements are of great practical importance, primarily for drawing up accurate maps. Both on the map and on the globe you see a network of meridians - circles going through the poles, and parallels - circles parallel to the plane earth's equator. A map of the Earth could not be compiled without a long and painstaking work surveyors who determined step by step over many years the position of different places on the earth's surface and then plotted the results on a network of meridians and parallels. To have accurate maps, it was necessary to know the actual shape of the Earth.

The measurement results of Struve and his collaborators turned out to be very important contribution into this work.

Subsequently, other surveyors measured with great accuracy the lengths of the arcs of meridians and parallels in different places earth's surface. From these arcs, with the help of calculations, it was possible to determine the length of the diameters of the Earth in the equatorial plane (equatorial diameter) and in the direction earth's axis(polar diameter). It turned out that the equatorial diameter is longer than the polar one by about 42.8 km. This once again confirmed that the Earth is compressed from the poles. According to the latest data from Soviet scientists, the polar axis is 1/298.3 shorter than the equatorial one.

Let's say we would like to depict the deviation of the Earth's shape from a sphere on a globe with a diameter of 1 m. If the ball at the equator has a diameter of exactly 1 m, then its polar axis should be only 3.35 mm Briefly speaking! This is such a small value that it cannot be detected by eye. The shape of the Earth, therefore, differs very little from a sphere.

One might think that the unevenness of the earth's surface, and especially Mountain peaks, the highest of which Chomolungma (Everest) reaches almost 9 km, must greatly distort the shape of the Earth. However, it is not. On the scale of a globe with a diameter of 1 m a nine-kilometer mountain will be depicted as a grain of sand with a diameter of about 3/4 stuck to it mm. Is it possible to detect this protrusion only by touch, and even then with difficulty? And from the height at which our satellite ships fly, it can only be distinguished by the black speck of shadow cast by it when the Sun is low.

In our time, the size and shape of the Earth are very accurately determined by scientists F.N. Krasovsky, A.A. Izotov and others. Here are the numbers showing the size of the globe according to the measurements of these scientists: the length of the equatorial diameter is 12,756.5 km, polar diameter length - 12,713.7 km.

Studying the path taken by artificial Earth satellites will make it possible to determine the magnitude of the force of gravity in different places above the surface of the globe with such accuracy that could not be achieved in any other way. This in turn will make it possible to further refine our knowledge of the size and shape of the Earth.

Gradual change in the shape of the earth

However, as we managed to find out with the help of the same space observations and special calculations made on their basis, the geoid has complex look due to the rotation of the Earth and the uneven distribution of masses in earth's crust, but quite well (with an accuracy of several hundred meters) is represented by an ellipsoid of rotation, having a polar compression of 1:293.3 (Krasovsky ellipsoid).

Nevertheless, until very recently it was considered a well-established fact that this small defect was slowly but surely leveled out due to the so-called process of restoration of gravitational (isostatic) equilibrium, which began approximately eighteen thousand years ago. But just recently the Earth began to flatten again.

Geomagnetic measurements, which since the late 70s have become an integral attribute of scientific research programs of satellite observation, have consistently recorded the alignment gravitational field planets. In general, from the point of view of mainstream geophysical theories, the gravitational dynamics of the Earth seemed quite predictable, although, of course, both within the mainstream and outside it there were numerous hypotheses that differently interpreted the average and long term prospects this process, as well as what happened in past life of our planet. Quite popular today is, say, the so-called pulsation hypothesis, according to which the Earth periodically contracts and expands; There are also supporters of the “contraction” hypothesis, which postulates that in the long term the size of the Earth will decrease. There is also no unity among geophysicists regarding what phase the process of post-glacial restoration of gravitational equilibrium is in today: most experts believe that it is quite close to completion, but there are also theories that claim that its end is still far away or that it has already stopped.

Nevertheless, despite the abundance of discrepancies, until the end of the 90s of the last century, scientists still did not have any compelling reasons to doubt that the process of post-glacial gravitational alignment is alive and well. The end of scientific complacency came rather abruptly: after spending several years checking and rechecking the results obtained from nine different satellites, two American scientists, Christopher Cox of Raytheon and Benjamin Chao, a geophysicist at the Goddard Control Center space flights NASA came to an amazing conclusion: since 1998, the “equatorial coverage” of the Earth (or, as many have dubbed this dimension Western media, its “thickness”) began to increase again.
The sinister role of ocean currents.

Cox and Chao's paper, which claims "the discovery of a large-scale redistribution of Earth's mass," was published in the journal Science in early August 2002. As the study authors note, " long-term observations behavior of the Earth's gravitational field showed that the post-glacial effect that leveled it in the last few years suddenly appeared more powerful opponent, approximately twice as strong as its gravitational influence."

Thanks to this “mysterious enemy,” the Earth again, as in the last “era of the Great Glaciation,” began to flatten, that is, since 1998, in the region of the equator there has been an increase in the mass of matter, while it has been outflowing from the polar zones.

Terrestrial geophysicists do not yet have direct measurement techniques to detect this phenomenon, so in their work they have to use indirect data, primarily the results of ultra-precise laser measurements of changes in the trajectories of satellite orbits that occur under the influence of fluctuations in the Earth’s gravitational field. Accordingly, speaking about “observed mass movements earthly matter", scientists proceed from the assumption that they are responsible for these local gravitational fluctuations. The first attempts to explain this strange phenomenon and undertaken by Cox and Chao.

The version about some underground phenomena, for example, the flow of matter in the earth's magma or core, looks, according to the authors of the article, quite doubtful: in order for such processes to have any significant effect gravitational effect, supposedly much more is required long time than a ridiculous four years by scientific standards. As possible reasons that caused the thickening of the Earth along the equator, they name three main ones: oceanic influence, melting of polar and high mountain ice and certain “processes in the atmosphere.” However, last group factors are also immediately dismissed by them - regular measurements of the weight of the atmospheric column do not give any grounds for suspicion of the involvement of certain air phenomena in the occurrence of the discovered gravitational phenomenon.

Cox and Chao's hypothesis about the possible influence of ice melting in the Arctic and Antarctic zones on the equatorial bulge seems far from clear. This process is like essential element the notorious global warming global climate, of course, to one degree or another may be responsible for the transfer of significant masses of matter (primarily water) from the poles to the equator, but theoretical calculations made by American researchers show: in order for it to turn out to be a determining factor (in particular, it “blocked "consequences of a thousand-year "growth of positive relief"), the size of the "virtual block of ice" melted annually since 1997 should have been 10x10x5 kilometers! Geophysicists and meteorologists do not have any empirical evidence that the process of ice melting in the Arctic and Antarctic in recent years could have taken on such proportions. According to the most optimistic estimates, the total volume of melted ice floes is at least an order of magnitude smaller than this “super iceberg”; therefore, even if it had some influence on the increase in the equatorial mass of the Earth, this influence could hardly be so significant.

As the most likely reason for the sudden change in the Earth's gravitational field, Cox and Chao today consider oceanic influence, that is, the same transfer of large volumes of water mass in the World Ocean from the poles to the equator, which, however, is associated not so much with the rapid melting of ice, how many with some not entirely explainable sharp fluctuations ocean currents, occurring in recent years. Moreover, as experts believe, the main candidate for the role of a disturber of gravitational peace is Pacific Ocean, more precisely, the cyclical movements of huge water masses from its northern regions to the southern ones.

If this hypothesis turns out to be correct, humanity in the very near future may face very serious changes in the global climate: the ominous role of ocean currents is well known to everyone who is more or less familiar with the basics of modern meteorology (what is El Niño worth). True, the assumption that the sudden swelling of the Earth along the equator is a consequence of the climate revolution already in full swing seems quite logical. But, by and large, it is still hardly possible to really understand this tangle of cause-and-effect relationships based on fresh traces.

The obvious lack of understanding of the ongoing “gravitational outrages” is perfectly illustrated by a short excerpt from Christopher Cox’s own interview with Nature news correspondent Tom Clark: “In my opinion, it is now possible to high degree certainty (hereinafter it is emphasized by us. - ‘Expert’) to speak only about one thing: the ‘weight problems’ of our planet are likely to be temporary and not a direct result human activity"However, continuing this verbal balancing act, the American scientist immediately once again prudently stipulates: “Apparently, sooner or later everything will return ‘to normal’, but perhaps we are mistaken about this."

Home → Legal consultation→ Terminology → Area units

Units of land area measurement

The system for measuring land areas adopted in Russia

1 weave = 10 meters x 10 meters = 100 sq.m
1 hectare = 1 ha = 100 meters x 100 meters = 10,000 sq.m = 100 acres
1 square kilometer= 1 sq. km = 1000 meters x 1000 meters = 1 million sq. m = 100 hectares = 10,000 acres

Reciprocal units

1 sq.m = 0.01 acres = 0.0001 hectares = 0.000001 sq.km
1 hundred square meters = 0.01 hectares = 0.0001 sq. km

Conversion table for area units

Area units	1 sq. km.	1 Hectare	1 Acre	1 Sotka	1 sq.m.
1 sq. km.	1	100	247.1	10.000	1.000.000
1 hectare	0.01	1	2.47	100	10.000
1 acre	0.004	0.405	1	40.47	4046.9
1 weave	0.0001	0.01	0.025	1	100
1 sq.m.	0.000001	0.0001	0.00025	0.01	1

unit of area in metric system measures used for measurements land plots.

Abbreviated designation: Russian ha, international ha.

1 ha equal to area square with side 100 m.

The name "hectares" is formed by adding the prefix "hecto..." to the name of the area unit "ar":

1 ha = 100 are = 100 m x 100 m = 10,000 m2

a unit of area in the metric system of measures is equal to the area of a square with a side of 10 m, that is:

1 ar = 10 m x 10 m = 100 m2.
1 tithe = 1.09254 hectares.

land measure, used in a number of countries using English system measures (UK, USA, Canada, Australia, etc.).

1 acre = 4840 sq. yards = 4046.86 m2

The most commonly used land measure in practice is the hectare, an abbreviation for ha:

1 ha = 100 are = 10,000 m2

In Russia, a hectare is the basic unit of measurement of land area, especially agricultural land.

On the territory of Russia, the unit “hectare” was introduced into practice after October revolution, instead of tithe.

Ancient Russian units of area measurement

1 sq. verst = 250,000 sq.
fathoms = 1.1381 km²
1 tithe = 2400 sq. fathoms = 10,925.4 m² = 1.0925 ha
1 tithe = 1/2 tithe = 1200 sq. fathoms = 5462.7 m² = 0.54627 ha
1 octopus = 1/8 tithe = 300 square fathoms = 1365.675 m² ≈ 0.137 hectares.

The area of land plots for individual housing construction and private plots is usually indicated in acres

One hundred is the area of a plot measuring 10 x 10 meters, which is 100 square meters, and therefore called sotka.

Here are a few typical examples dimensions that a land plot of 15 acres can have:

In the future, if you suddenly forget how to find the area of a rectangular plot of land, then remember a very old joke when a grandfather asks a fifth-grader how to find Lenin’s area, and he answers: “You need to multiply the width of Lenin by the length of Lenin”)))

It is useful to familiarize yourself with this

For those who are interested in the possibility of increasing the area of land plots for individual housing construction, private household plots, gardening, vegetable farming, owned, it is useful to familiarize yourself with the procedure for registering additions.

From January 1, 2018, the exact boundaries of the plot must be recorded in the cadastral passport, since buying, selling, mortgaging or donating land without accurate description borders will simply be impossible. This is regulated by amendments to the Land Code. A total revision of borders at the initiative of municipalities began on June 1, 2015.

On March 1, 2015, a new the federal law"About amendments to Land Code RF and individual legislative acts RF" (N 171-FZ "dated 06/23/2014, in accordance with which, in particular, the procedure for purchasing land plots from municipalities has been simplified& You can familiarize yourself with the main provisions of the law here.

With regard to the registration of houses, bathhouses, garages and other buildings on land plots owned by citizens, the new dacha amnesty will improve the situation.

I will try not only to answer the question, but also to describe the measurement method, which, in my opinion, is very original. In general, I hope it turns out interesting, and most importantly, informative.

How Eratosthenes measured the circumference of the Earth

Today, perhaps, any schoolchild can cope with this, but then, more than 2000 years ago, it was almost impossible to do. Moreover, in those days, most people believed that the world was a flat disk, from the edge of which one could fall into the abyss. However, the scientist who lived in Alexandria forever went down in history as the first who managed to calculate the size of our planet. But how did he do it, because in his arsenal there were practically no special devices? He used the data that the Egyptians had, namely the fact that on the day of the summer solstice the rays of the luminary reach the bottom of the deepest wells in the city of Siena. However, this phenomenon is not observed in Alexandria. So, in 240 BC, a scientist used an ordinary bowl with a needle to understand the angle of the star in the sky. Next, the following calculations were made:

in Siena it is noon - there is absolutely no shadow, that is, the angle is 0°;
in Alexandria, which is located almost 5000 stadia (about 800 km), the angle was 7° 12′ - therefore, 1/50 of a circle;
After calculations, it was found that the circumference was at least 250 thousand stadia or almost 40 thousand km.

As you can see, taking into account a small error, the result corresponds to reality. In general, it is obvious that Eratosthenes turned out to be an excellent scientist for his time.

How the Earth is measured today

Nowadays, there is a special science - geodesy, which deals with solving such problems. Experts use many instruments to calculate angular distances. For example, to determine the exact shape of the planet, fluctuations in gravity in different areas are compared, and satellites are used to determine the angles.

The device is like the vertex of a triangle, naturally imaginary, and the remaining angles rest on different areas surface of the Earth.

Test tasks

1. According to the ideas of the ancient Indians, the Earth was considered

a) flat

b) convex

c) spherical

d) geoid

2. First to determine the size of the globe

a) Pythagoras

b) Aristotle

c) Eratosthenes

d) Ptolemy

3. The length of the earth's equator line is about

c) 40,000 km

4. What instrument do you think Eratosthenes used to determine the size of the globe?

a) ruler

b) compass

c) telescope

d) metronome

5. One of the first evidence of the sphericity of the Earth was obtained as a result of observations of

a) ships sailing out to sea

b) sunrise

c) northern lights

d) the flight of a spaceship

6. Fill in the gaps in the text.

The ancient Greek scientist Aristotle collected a lot of evidence of the sphericity of the Earth. The most serious of them was based on observations made during lunar eclipses. Later, another scientist calculated the size of the globe. His name was Erastophenes.

Thematic workshop.

Read the text and answer the question.

The ancient Babylonians' ideas about the Earth were based on observations of natural phenomena. However, limited knowledge did not allow them to correctly explain these phenomena.
In ancient times in western Asia there existed Babylonian kingdom. The Babylonians imagined the Earth as a mountain, on the western slope of which the Babylonian kingdom was located. They noticed that to the south of Babylon there was a sea, and to the east there were mountains, which they did not dare cross. Therefore, it seemed to them that the Babylonian kingdom was located on the western slope of the “world” mountain. This mountain is round, and it is surrounded by the sea, and on the sea, like an overturned bowl, rests the solid sky - heavenly world. In the sky, as on Earth, there is land, water and air. The celestial land is the belt of the constellations of the zodiac, like a dam stretching among the celestial sea. The Sun, Moon and five planets move along this belt of land.

Under the Earth there is an abyss - hell, where the souls of the dead descend. At night the Sun passes through this dungeon from western edge Earth to the east, so that in the morning they will again begin their daily journey across the sky. Watching the Sun set over the sea horizon, people thought that it went into the sea and should also rise from the sea.

How were the Babylonians’ ideas about the structure of the world influenced by the real features of the location of their country? Give some examples.

The Babylonians noticed that to the south of Babylon there was a sea, and to the east there were mountains, which they did not dare cross. Therefore, it seemed to them that the Babylonian kingdom was located on the western slope of the “world” mountain.

The Sun and Moon, as well as five planets, move along the land belt.

Cartographic workshop.

Apply to contour map digital designations of the listed geographical objects.

1 - North America

2 - Atlantic Ocean

3 - Eurasia

4 - Madagascar island

5 - Pacific Ocean

6 - Arabian Sea

People have long guessed that the Earth they live on is like a ball. One of the first to express the idea that the Earth was spherical was the ancient Greek mathematician and philosopher Pythagoras (c. 570-500 BC). The greatest thinker of antiquity, Aristotle, observing lunar eclipses, noticed that the edge of the earth's shadow falling on the Moon always has a round shape. This allowed him to confidently judge that our Earth is spherical. Now, thanks to the achievements of space technology, we all (more than once) had the opportunity to admire the beauty of the globe from photographs taken from space.

Is it possible to find out the size of the Earth without measuring its entire circumference? Of course you can.

The first measurement of the Earth in this way was made by the ancient Greek scientist Eratosthenes (c. 276-194 BC), who lived in the Egyptian city of Alexandria, on the shores of the Mediterranean Sea.

Camel caravans came to Alexandria from the south. From the people accompanying them, Eratosthenes learned that in the city of Syene (present-day Aswan) on the day of the summer solstice, the Sun was overhead on the same day. Objects at this time do not provide any shadow, and the sun's rays penetrate even the deepest wells. Therefore, the Sun reaches its zenith.

Through astronomical observations, Eratosthenes established that on the same day in Alexandria the Sun is 7.2 degrees from the zenith, which is exactly 1/50 of the circumference. (In fact: 360: 7.2 = 50.) Now, in order to find out what the circumference of the Earth is, all that remained was to measure the distance between the cities and multiply it by 50. But Eratosthenes was not able to measure this distance running through the desert. The guides of the trade caravans could not measure it either. They only knew how much time their camels spent on one journey, and believed that from Siena to Alexandria there were 5,000 Egyptian stadia. This means the entire circumference of the Earth: 5000 x 50 = 250,000 stadia.

A method for accurately measuring large distances was first proposed by the Dutch geographer and mathematician Wildebrord Siellius (1580-1626).

Now that the length of the arc of the AB meridian is known, as well as its expression in degrees (as the difference in the latitudes of astropoints A and B), it will not be difficult to calculate the length of the arc of 1 degree of the meridian by simply dividing the first value by the second.

This method of measuring large distances on the earth's surface is called triangulation - from the Latin word "triapgulum", which means "triangle". It turned out to be convenient for determining the size of the Earth.

The most ambitious degree measurement of the 19th century was headed by the founder of the Pulkovo Observatory, V. Ya. Struve. Under the leadership of Struve, Russian surveyors, together with Norwegian ones, measured an arc stretching from the Danube through the western regions of Russia to Finland and Norway to the coast of the Arctic Ocean. The total length of this arc exceeded 2800 km! It contained more than 25 degrees, which is almost 1/14 of the earth's circumference. It entered the history of science under the name “Struve arc”. In the post-war years, the author of this book had the opportunity to work on observations (measurements of angles) at state triangulation points adjacent directly to the famous “arc”.

The longer the measured arcs of meridians and parallels, the more accurately the radius of the Earth can be calculated and its compression determined.

Studying the shape of the Earth, scientists come to the conclusion that it is compressed not only along the axis of rotation. The equatorial section of the globe in projection onto a plane gives a curve that also differs from a regular circle, although quite a bit - by hundreds of meters. All this indicates that the figure of our planet is more complex than it seemed before.

What to do in continental areas? What is considered the surface of the Earth? Also the surface of the World Ocean, mentally continued under all the continents and islands.

On October 4, 1957, with the launch of the first artificial Earth satellite in our country, humanity entered the space age. Active exploration of near-Earth space began. At the same time, it turned out that satellites are very useful for understanding the Earth itself. Even in the field of geodesy, they said their “weighty word.”

As you know, the classic method for studying the geometric characteristics of the Earth is triangulation. But previously, geodetic networks were developed only within continents, and they were not connected to each other. After all, you cannot build triangulation on the seas and oceans. Therefore, the distances between the continents were determined less accurately. Due to this, the accuracy of determining the size of the Earth itself was reduced.

With the launch of the satellites, surveyors immediately realized that “sighting targets” had appeared at high altitudes. Now it will be possible to measure large distances.

The idea of the space triangulation method is simple. Synchronous (simultaneous) satellite observations from several distant points on the earth's surface make it possible to bring their geodetic coordinates to a single system. This is how triangulations built on different continents were linked together, and at the same time the dimensions of the Earth were clarified: equatorial radius - 6378.160 km, polar radius - 6356.777 km. The compression value is 1/298.25, that is, almost the same as that of the Krasovsky ellipsoid. The difference between the equatorial and polar diameters of the Earth reaches 42 km 766 m.

If our planet were a regular sphere, and the masses inside it were distributed evenly, then the satellite could move around the Earth in a circular orbit. But the deviation of the Earth’s shape from spherical and the heterogeneity of its interior lead to the fact that the force of attraction over different points of the earth’s surface is not the same. The force of gravity of the Earth changes - the orbit of the satellite changes. And everything, even the slightest change in the movement of a low-orbit satellite, is the result of the gravitational influence on it of one or another earthly bulge or depression over which it flies.

It turned out that our planet also has a slightly pear-shaped shape. Its North Pole is raised above the plane of the equator by 16 m, and the South Pole is lowered by approximately the same amount (as if pressed in). So it turns out that in a section along the meridian, the figure of the Earth resembles a pear. It is slightly elongated to the north and flattened at the South Pole. There is polar asymmetry: This hemisphere is not identical to the Southern one. Thus, based on satellite data, the most accurate idea of the true shape of the Earth was obtained. As we can see, the figure of our planet noticeably deviates from the geometrically correct shape of a ball, as well as from the figure of an ellipsoid of revolution.

Reports of how Eratosthenes calculated the size of the earth briefly. Time travel: Eratosthenes – the father of geography

Eratosthenes' contribution to geography. What did Eratosthenes discover?

What did Eratosthenes discover in geography?

How did Eratosthenes measure the radius of the Earth?

Units of land area measurement

The system for measuring land areas adopted in Russia

Conversion table for area units

The area of ​​land plots for individual housing construction and private plots is usually indicated in acres

It is useful to familiarize yourself with this

How Eratosthenes measured the circumference of the Earth

How the Earth is measured today

The area of land plots for individual housing construction and private plots is usually indicated in acres