Now you know that in the fabulous Universe of our distant ancestors, the Earth did not even resemble a ball. The inhabitants of Ancient Babylon imagined it as an island in the ocean. The Egyptians saw it as a valley stretched from north to south, with Egypt in the center. And the ancient Chinese at one time depicted the Earth as a rectangle... You smile, imagining such an Earth, but have you often thought about how people guessed that the Earth is not a limitless plane or a disk floating in the ocean? When I asked the guys about this, some said that people learned about the sphericity of the Earth after their first trips around the world, while others recalled that when a ship appears over the horizon, we first see the masts, and then the deck. Do these and some similar examples prove that the Earth is a sphere? Hardly. After all, you can drive around... a suitcase, and the upper parts of the ship would appear even if the Earth had the shape of a hemisphere or looked like, say,... a log. Think about this and try to depict what was said in your drawings. Then you will understand: the examples given only indicate that The earth is isolated in space and possibly spherical.

How did you know that the Earth is a ball? What helped, as I already told you, was the Moon, or rather, lunar eclipses, during which the round shadow of the Earth is always visible on the Moon. Set up a small “shadow theater”: illuminate objects of different shapes (triangle, plate, potato, ball, etc.) in a dark room and notice what shadow they create on the screen or just on the wall. Make sure that only the ball always forms a circle shadow on the screen. So, the Moon helped people learn that the Earth is a ball. To this conclusion, scientists in Ancient Greece(for example, the great Aristotle) ​​came back in the 4th century BC. But for a very long time, the “common sense” of man could not come to terms with the fact that people live on the ball. They couldn’t even imagine how it was possible to live on the “other side” of the ball, because the “antipodes” located there would have to walk upside down all the time... But wherever a person is on the globe, everywhere a stone thrown up will be under the influence of force gravity of the Earth to fall down, that is, to the earth's surface, and if it were possible, then to the center of the Earth. In fact, people, of course, nowhere except circuses and gyms have to walk upside down and upside down. They walk normally anywhere on Earth: the earth’s surface is under their feet, and the sky is above their heads.

Around 250 BC, Greek scientist Eratosthenes for the first time measured the globe quite accurately. Eratosthenes lived in Egypt in the city of Alexandria. He guessed to compare the height of the Sun (or its angular distance from a point above his head, zenith, which is called - zenith distance) at the same point in time in two cities - Alexandria (in northern Egypt) and Siena (now Aswan, in southern Egypt). Eratosthenes knew that on the day summer solstice(June 22) Sun in noon illuminates the bottom of deep wells. Therefore, at this time the Sun is at its zenith. But in Alexandria at this moment the Sun is not at its zenith, but is 7.2° away from it. Eratosthenes obtained this result by changing the zenith distance of the Sun using his simple goniometric instrument - the scaphis. This is simply a vertical pole - a gnomon, fixed at the bottom of a bowl (hemisphere). The skafis is installed so that the gnomon takes a strictly vertical position (directed to the zenith). A pole illuminated by the sun casts a shadow divided into degrees inner surface skafisa. So at noon on June 22 in Siena the gnomon does not cast a shadow (the Sun is at its zenith, its zenith distance is 0°), and in Alexandria the shadow from the gnomon, as can be seen on the scaphis scale, marked a division of 7.2°. In the time of Eratosthenes, the distance from Alexandria to Syene was considered to be 5,000 Greek stadia (approximately 800 km). Knowing all this, Eratosthenes compared an arc of 7.2° with the entire circle of 360° degrees, and a distance of 5000 stadia with the entire circle globe(let's denote it by the letter X) in kilometers. Then from the proportion

it turned out that X = 250,000 stadia, or approximately 40,000 km (imagine, this is true!).

If you know that the circumference of a circle is 2πR, where R is the radius of the circle (and π ~ 3.14), knowing the circumference of the globe, it is easy to find its radius (R):

It is remarkable that Eratosthenes was able to measure the Earth very accurately (after all, today it is believed that the average radius of the Earth 6371 km!).

But why is it mentioned here? average radius of the Earth, Aren't all the radii of a ball the same? The fact is that the figure of the Earth is different from the ball. Scientists began to guess about this back in the 18th century, but it was difficult to find out what the Earth really was like - whether it was compressed at the poles or at the equator. To understand this, the French Academy of Sciences had to equip two expeditions. In 1735, one of them went to carry out astronomical and geodetic work in Peru and did this in the equatorial region of the Earth for about 10 years, and the other, Lapland, worked in 1736-1737 near the Arctic Circle. As a result, it turned out that the arc length of one degree of the meridian is not the same at the Earth's poles and at its equator. The meridian degree turned out to be longer at the equator than at high latitudes (111.9 km and 110.6 km). This can only happen if the Earth is compressed at the poles and is not a ball, but a body similar in shape to spheroid. At the spheroid polar radius is smaller equatorial(the polar radius of the earth's spheroid is almost shorter than the equatorial radius 21 km).

It is useful to know that the great Isaac Newton (1643-1727) anticipated the results of the expeditions: he correctly concluded that the Earth is compressed, which is why our planet rotates around its axis. In general, the faster a planet rotates, the greater its compression should be. Therefore, for example, the compression of Jupiter is greater than that of the Earth (Jupiter manages to rotate around its axis in relation to the stars in 9 hours 50 minutes, and the Earth only in 23 hours 56 minutes).

And further. The true figure of the Earth is very complex and differs not only from a sphere, but also from a spheroid rotation. True, in this case we're talking about about the difference not in kilometers, but... meters! Scientists are still engaged in such a thorough clarification of the figure of the Earth, using for this purpose specially conducted observations with artificial satellites Earth. So it is quite possible that someday you will have to take part in solving the problem that Eratosthenes took on a long time ago. This is very what people need case.

What is the best figure for you to remember on our planet? I think that for now it is enough if you imagine the Earth in the form of a ball with an “additional belt” put on it, a kind of “splash” on the equator region. Such a distortion of the Earth’s figure, turning it from a sphere into a spheroid, has considerable consequences. In particular, due to the attraction of the “additional belt” by the Moon, the earth’s axis describes a cone in space in about 26,000 years. This movement of the earth's axis is called precessional. As a result, the role of the North Star, which now belongs to α Ursa Minor, is alternately played by some other stars (in the future it will become, for example, α Lyrae - Vega). Moreover, due to this ( precessional) movement of the earth's axis Zodiac signs more and more do not coincide with the corresponding constellations. In other words, 2000 years after the Ptolemaic era, the “sign of Cancer,” for example, no longer coincides with the “constellation Cancer,” etc. However, modern astrologers try not to pay attention to this...

The earth is round - this is common knowledge. What else do we know about its shape and size? Which of us can remember from memory how many kilometers the circumference of the Earth is at the equator? What about the meridian? Who knows when and how the circumference of the earth was first measured? Meanwhile, these facts are extremely interesting.

The circumference of the Earth was first measured by Eratosthenes, who lived in the city of Siena. At that time, scientists already knew that the Earth is spherical in shape. Observing the celestial body at different times of the day, Eratosthenes noticed that at the same time the sun, being observed from Syene, is located exactly at the zenith, while in Alexandria on the same day and hour it deviates by a certain angle.

Observations were carried out annually. Having measured this angle using astronomical instruments, the scientist found that it was 1/50 of the full circle.

As you know, a complete circle is equal to 360 degrees. Thus, it is enough to know the chord of an angle of 1 degree (i.e., the distance between points on the Earth’s surface lying on rays with an angular distance between them of 1 degree). Then the resulting value should be multiplied by 360.

Taking the distance between the cities of Alexandria and Syene (5 thousand Egyptian stadia) as the length of the chord and assuming that these cities lie on the same meridian, Eratosthenes made the necessary calculations and named the figure that equaled the circumference of the Earth - 252 thousand Egyptian stadia.

For that time, this measurement was quite accurate, because there were no reliable methods for measuring the distance between cities, and the path from Siena to Alexandria was measured by the speed of the camel caravan.

Subsequently, scientists different countries They repeatedly measured and clarified the value that is the circumference of the Earth. In the 17th century, a Dutch scientist named Sibelius came up with a way to measure distances using the first theodolites - special geodetic instruments. This method was called triangulation and is based on the construction large quantity triangles with the measurement of the basis of each of them.

The triangulation method is still used today; the entire earth's surface is virtually divided and lined into large triangles.

Russian scientists also contributed to these studies. In the 19th century, the circumference of the Earth was measured by V. Ya. Struve, who led the research.

Until the mid-17th century, the Earth was considered a sphere of regular shape. But later, some facts were accumulated indicating a decrease in the force of gravity from the equator to the pole. Scientists fiercely debated the reasons for this; the most plausible theory was considered to be the compression of the Earth from the poles.

To test this hypothesis French Academy Two independent expeditions were organized (in 1735 and 1736), which measured the length of the equatorial and polar degrees, respectively, in Peru and Lapland. At the equator, the degree, as it turns out, is shorter!

Subsequently, other, more accurate measurements confirmed that the polar circle of the Earth is 21.4 km shorter than the equatorial one.

Currently, high-precision measurements have been made using the latest research methods and modern instruments. In our country, the data obtained by Soviet scientists A. A. Izotov and F. N. Krasovsky have been officially approved. According to these studies, the circumference of our planet along the equator is 40075.7 kilometers, along the meridian - 40008.55 km. The equatorial radius of the globe (the so-called semi-major axis) is equal to 6378245 meters, the polar (semi-minor axis) is 6356863 meters.

510 million sq. kilometers, of which only 29% belongs to land. The volume of the earth's "ball" is 1083 billion cubic meters. kilometers. The mass of our planet is characterized by the figure 6X10^21 tons. Of this, about 7% comes from water resources.

Source: apxiv

I periodically have the feeling that many simple things are specifically presented in such a way that the reader does not understand anything and stupidly memorizes it, or feels his insignificance in front of the sophistication of science. This entirely refers to Eratosthenes’ enchanting method of measuring the circumference of the globe, known from school textbooks. Maybe he actually calculated in such a perverted way, but why replicate this nonsense from school?

Let's look at how you can confuse your brain with a simple question using the example of calculating the circumference of the Earth in nautical miles, which is a special case of measuring the latitude of an area and the length of the path traveled along the meridian.

If to modern man given the task of calculating the circumference of the Earth in nautical miles, in the vast majority of cases he will look at the Internet/reference books and solve something like this: the circumference of the Earth, for example along the Parisian meridian, 40,000 km, using a calculator, will divide by a modern nautical mile 1,852 km and get 21,598.3 nautical miles miles, which will be close to reality.

Now I’ll show you how to calculate the circumference of the Earth in your head and absolutely accurately. To do this, you only need to know one thing: “The nautical mile is a unit of measurement of distance used in navigation and aviation. Initially, the nautical mile was defined as the length of the arc of a great circle on the surface of the globe measuring one minute of arc.”

There are 60 minutes in one angular degree, 360 degrees in a circle, that is, in a circle 360x60 = 21,600 angular minutes, which in this case corresponds to the circumference of the globe at 21,600 nautical miles. And this is absolutely accurate, since the circumference of the globe along the meridian is the standard, and the arc minute-mile is a derived unit. Since the Earth is not a perfect spheroid, but slightly curved, the miles on different meridians will differ slightly from each other, but this is completely unimportant for navigation, because an arc minute is also an arc minute in Africa.

The latitude of the area can be measured accurately to the nearest degree even with primitive devices like a protractor with a plumb line, which is not very different from the quadrant actually used by sailors and is essentially the same as an astrolabe:

For more accurate measurements of angles, a sextant was subsequently invented (Marine argo - sextant):

Modern people have little idea of ​​what analog computers are and how to use them. In order to calculate the distance between two points in the meridional direction, you just need to measure the latitudes of the points, and the difference in latitude, expressed in minutes of arc, will be the distance between them in nautical miles. Everything is simple, convenient and practically applicable.

If you really want to find out how many stages, fathoms, arshins, or Egyptian cubits there are in a nautical mile, you need to carefully use your knees to measure the distance between points with a known distance in nautical miles-arcminutes. But why? How is this practically applicable?

Eratosthenes allegedly measured angles with an accuracy of arc seconds and the difference in latitudes of Alexandria was 7° 6.7", that is, 7x60 = 420 + 6.7 = 426.7 nautical miles (arc minutes). It seems that what else is needed? But for some reason it takes days of camel travel and stages.There is a feeling of something far-fetched - a fake or a hoax.

Eratosthenes' method according to V. A. Bronstein, Claudius Ptolemy, Chapter 12. Ptolemy's works in the field of geography:

“As is known, Eratosthenes’ method was to determine the arc of the meridian between Alexandria and Syene on the day of the summer solstice. On this day, according to the stories of people who visited Syene, the Sun at noon illuminated the bottom of the deepest wells and, therefore, passed through the zenith. Consequently, the latitude Syene was equal to the angle of inclination of the ecliptic to the equator, which Eratosthenes determined at 23 ° 51 "20". On the same day and hour in Alexandria, the shadow from the vertical column of the gnomon covered 1/50 of the circle, the center of which was the tip of the gnomon. This means that the Sun at noon was 1/50th of a circle from the zenith, or 7° 12". Taking the distance between Alexandria and Syene as 5000 stadia, Eratosthenes found that the circumference of the globe was 250,000 stadia. The question of the exact length of the stage adopted by Eratosthenes, for a long time was the subject of debate, since there were stages ranging from 148 to 210 m in length<60>. Most researchers accepted the length of the stage as 157.5 m ("Egyptian" stage). Then the circumference of the Earth is equal, according to Eratosthenes, 250,000-0.1575 = 39,375 km, which is very close to actual value 40,008 km. If Eratosthenes used the Greek (“Olympic”) stage with a length of 185.2 m, then the circumference of the Earth was already 46,300 km.

According to modern measurements<97>the latitude of the Museum in Alexandria is 31° 11.7", the latitude of Aswan (Siena) is 24° 5.0", the difference in latitude is 7° 6.7", which corresponds to the distance between these cities of 788 km. Dividing this distance by 5000, we obtain the length of the stage, used by Eratosthenes, 157.6 m. Does this mean that he used the Egyptian stadion?

This question is more complicated than it might seem. The mere fact that Eratosthenes gave a clearly rounded number - 5000 stades (and, say, not 5150 or 4890) does not inspire confidence in him. And if Eratosthenes’ estimate was overestimated by at least 15%, we find that he used the Egyptian stade of 185 m. This issue cannot yet be resolved.”

Let us now pay attention to the following circumstances:

Aswan (Siena) and Alexandria are not located on the same meridian; the difference in longitude is 3°, that is, about 300 kilometers.

Eratosthenes did not measure the distance, but took it based on the days of travel of the camels, which clearly did not walk in a straight line.

It is completely unclear what instrument Eratosthenes used to measure angles with an accuracy of seconds

It is not clear what stage Eratosthenes used to measure distances, etc.

But at the same time he seemed to get a fairly accurate result! Or have historians made adjustments to the result?

From Wikipedia: “Eratosthenes says that Syene and Alexandria lie on the same meridian. And since the meridians in space are large circles, the meridians on Earth will certainly be the same large circles. And since this is the solar circle between Siena and Alexandria, then the path between them on Earth necessarily goes in a large circle. Now he says that Siena lies on the circle of the summer tropic. And if the summer solstice in the constellation Cancer occurred exactly at noon, then the sundial at this moment in time would necessarily not cast a shadow, since the Sun would be exactly at its zenith; things are indeed this way in [a strip of width] 300 stadia. And in Alexandria at the same hour the sundial casts a shadow, since this city lies south of Siena. These cities lie on the same meridian and on a great circle. On the sundial in Alexandria we draw an arc passing through the end of the shadow of the gnomon and the base of the gnomon, and this segment of the arc will produce a large circle on the bowl, since the sundial bowl is located on the large circle. Next, imagine two straight lines descending underground from each gnomon and meeting in the center of the Earth. The sundial at Siena is perpendicular to the Sun, and an imaginary straight line runs from the Sun through the top of the sundial gnomon, producing one straight line from the Sun to the center of the Earth. Let us imagine another straight line drawn from the end of the shadow of the gnomon through the top of the gnomon to the Sun on the bowl in Alexandria; and it will be parallel to the already named straight line, since it has already been said that straight lines from different parts of the Sun to different parts The earths are parallel (how does he know this?). A straight line drawn from the center of the Earth to the gnomon in Alexandria forms equal opposite angles with these parallel ones. One of them - with the vertex in the center of the Earth, when meeting the straight lines drawn from the sundial to the center of the Earth, and the other - with the vertex at the end of the gnomon in Alexandria, when meeting with the straight line going from this end to the end of its shadow from the Sun, where these lines meet at the top. The first angle rests on an arc from the end of the shadow of the gnomon to its base, and the second on an arc with its center at the center of the Earth, drawn from Siena to Alexandria. These arcs are similar to each other because they are supported equal angles. And what relation does the arc on the cup have to its circle, the same relation does the arc from Siena to Alexandria have [to its circle]. But it was found that on the cup it makes up a fiftieth of its circle. Therefore, the distance from Syene to Alexandria will necessarily be a fiftieth of the great circle of the Earth. But it is equal to 5,000 stadia. Therefore the whole circle will be equal to 250,000 stadia. This is the method of Eratosthenes."

The number obtained by Eratosthenes was later increased to 252,000 stades. It is difficult to determine how close these estimates are to reality, since it is not known exactly what stage Eratosthenes used. But if we assume that we are talking about Greek (178 meters), then its radius of the earth was 7,082 km, if Egyptian (157.5), then 6,287 km. Modern measurements give the average radius of the Earth a value of 6,371 km, which makes the above calculation an outstanding achievement and the first sufficiently accurate calculation of the size of our planet." via

I would like to draw your attention to the fact that in Wikipedia, in addition to adjusting the results, it also first talks about Eratosthenes’ measurement of the circumference of the Earth, and ultimately draws a conclusion about the accuracy of calculating the radius of the Earth. In general, there is an elderberry in the garden, and a man in Kyiv, although they are interconnected.

The diagnosis is very simple: in textbooks they will continue to replicate the method of Eratosthenes, which does not provide anything for understanding the essence and practical applicability, but they will not say a word about the connection “nautical mile - arc minute” as an example of the proportional thinking of the ancients, because the modern trend is tailored to discrete computers, and we have to talk about the analogue computers of antiquity anew.

Thor Heyerdahl not only put forward some theories, he personally conducted many investigative experiments to test his statements, unlike keyboard warriors and many armchair scientists. So his works, IMHO, should be in the “must read” mode.

Chapter "Possible ocean routes to America and from America to Columbus" :

“When one gets acquainted with Heyerdahl's theory, as it was in 1961, it becomes clear that he approaches the question of migration with certain reservations. Heyerdahl takes into account the enormous difficulties that man of the past had to face.

Such restraint is necessary because the view of migration across the vast expanses of the oceans has now changed everywhere. For a very long time it was believed (especially in the USA) that the settlement of the New World occurred only through the Bering Strait and at a certain period of time in the distant past. And coincidences with certain features of highly developed cultures of the Old World were entirely explained by parallel development.

Now this cultural-historical doctrine of Munro has been revised. More and more people are inclined to admit that the Asian peoples made a number of distant voyages and discoveries. If we talk about the Atlantic Ocean, it is believed that the Normans were not the first to cross it. At a time of rapid flowering of migration theories, it is very useful to read Heyerdahl’s analysis, which, in addition to the sometimes disorienting geographical map, also takes into account winds and currents.

This report is short review possible ocean routes practically accessible to man in ancient times when sailing to and from America. I do not at all claim that Columbus’s predecessors actually sailed along all the routes discussed below, although it is obvious that on these routes ancient man no insurmountable obstacles lay in wait. And the purpose of the review is not to delve into the problems of ancient interpenetration of cultures - I analyze only purely practical issues that arise for those who assume the possibility of transoceanic communications between individual regions of the Old and New Worlds.

There is no doubt that the ocean hindered the geographical spread of primitive man much more seriously than the desert, swamp, jungle or tundra. But in the ocean, unlike other geographical obstacles, there are “paths” that can be compared to rivers. This is why the assertion that man had very little hope of surviving a long transoceanic voyage seems precocious. Significant adjustments are required for certain areas.

Modern ethnologists, as a rule, ignore two important circumstances. They do not take into account, firstly, that the distance between two polar points lying at opposite ends of the globe (like the North and South Poles) along the equator is no shorter than the distance between them along the great circle in any hemisphere and, secondly, that the travel distance covered by the ship from one geographical point to another is practically not equal to the distance measured on the map, Furthermore– the path in one direction is not equal to the path in the opposite direction.

The first circumstance can be illustrated by the following typical example. Taking apart interesting discovery(some common features in the ceramics of Japan and Ecuador), (ii) Newsweek (February 19, 1962, p. 49) states that the Equatorial Countercurrent is “heading straight toward Ecuador,” while “the Japan Current is taking a detour across the North Pacific.” . The usual, widely used turn of phrase is only misleading. After all, in fact, the Kuroshio (Japanese Current), which supposedly makes a detour, is the shortest and most direct of the two named paths. This can be verified if, instead of the deceptive Mercator projection (it is often used for world maps; in this projection, the surface of the globe is reduced to the surface of a cylinder, so the circumpolar regions are greatly distorted), we turn to the globe, which conveys the real picture incomparably more accurately.

It seems that few ethnologists realize that if you sail from the Malacca Peninsula to Ecuador through the Aleutian Islands, you will get a straight line between these two points (you couldn't imagine a straighter route). It makes no sense to look for the shortest path along the equator: after all, it follows the curvature of the globe in the same way as any other great circle arc, only this is not visible on a flat map of the Pacific Ocean.

China and Peru are also polar. The straight line distance between the Pacific coast of South China and Peru through the equator is no shorter than through the North or South Pole. Between these two opposite shores of the Pacific Ocean it is impossible to draw a line straighter or shorter than that which on the Mercator projection describes an imaginary arc through the extreme north of the Pacific Ocean. Connect the coast of Southern China with Peru along the equator on a globe with a wire and move the wire upward, securing both ends, it will settle down even on a route passing through the Bering Sea.

Calling the equator the shortest route between Southeast Asia and South America is as incorrect as saying that the shortest route from the North Pole to the South Pole is along the Greenwich meridian.

It should be remembered that the vast Pacific Ocean is not a smooth plain, but a regular hemisphere, equally sloping in all directions. Then the prerequisites for the travel of indigenous ships in the unknown ocean look in a completely different light. The primitive navigator, no matter which direction he went, saw himself in the center of a flat circle; he had no map that could confuse him.

The second circumstance, which absolutely requires great caution when studying ancient ocean voyages, is associated with the incorrect determination of the travel distance between fixed points in the sea. The absolute distance between two points can be expressed in miles, usually it differs from the actual distance that needs to be covered. We simply do not know anything about the distance traveled by the ancient navigator, since we do not know the relationship between the speed of the current in this area and the technically possible own speed of the ship. The lower the ship's own speed, the greater the discrepancy between the measured and actually traveled distance.

This is why the travel distance for a modern ocean liner can be completely different from that of a primitive vessel, even if they are traveling in the same straight line, over the same area of ​​stationary ocean floor. How great this difference is can be shown by the example of a transoceanic voyage on an aboriginal ship in which the author participated.

The absolute distance from Peru to the Tuamotu Islands is approximately 4000 miles. But in fact, the Kon-Tiki raft, having passed from Peru to the Tuamotu, crossed only about 1000 miles of ocean surface. If we imagine a primitive ship capable of traveling at the same speed and also in a straight line, but in the opposite direction, it would have to travel 7,000 miles along the ocean surface to get from the Tuamotu to Peru. The fact is that during the voyage the surface of the ocean itself shifted by about 3000 miles (about 50 degrees of the circumference of the globe). So, in terms of travel distance, the Tuamotu Islands are only 1,000 miles from Peru, while the distance from Tuamotu to Peru for one crossing the ocean at the speed of the Kon-Tiki raft is 7,000 miles.

Likewise, the absolute distance between Peru and the Marquesas Islands is approximately 4,000 miles. But average speed the current in this area is approximately 40 miles a day, which means that if a native vessel goes west at its own speed of 60 miles a day, it actually covers 60 plus 40, that is, 100 miles, in a day, and covers the entire distance in 40 days. In the opposite direction, at the same natural speed, it will make 60 minus 40 miles, that is, 20 miles per day, and the journey from the Marquesas Islands to Peru will take 200 days.

K – Columbus’s route from Africa to the Gulf of Mexico; E - Leiv Eiriksson's route from Northwestern Europe to the northeastern part North America; U – Urdaneta route from Indonesia to North-West America and Mexico; C – Saavedra route from Mexico to Micronesia and Indonesia; M – Mendanc route from the Andean coast to Polynesia and Papua Melanesia.

Even if the ship's own speed is only 40 miles per day, it will still go west at a speed of 40 plus 40, or 80 miles, and in 50 days it will reach the Marquesas Islands. And in the opposite direction, at a speed of 40 minus 40, that is, zero miles per day, it will not break away from the archipelago at all.

These examples are applicable not only to the area about which we have spoken, they, to one degree or another, extend to any transoceanic voyages of primitive ships. Along with the curvature of the surface of the great oceans, such a calculation of travel distance plays a decisive role in the author's subsequent reasoning. Calculations of both curvature and travel distance now form the basis of modern maritime navigation, and even before, when there were no maps, these factors were taken into account by everyone who paved the way to and from America. And they were probably no less important for those who went out into the unknown ocean when there were no descriptions yet, if we, of course, accept the idea that prehistoric man dared to cross a huge water desert, this ever-moving hemisphere.

There are three main ocean routes to the New World (two via Atlantic Ocean and one via the Pacific) and two main routes from the New World (both via the Pacific). These routes are so well defined that they can be named after their historically famous discoverers."

The chapter "Cultivated Plants - Evidence of Pre-Columbian Contact with the Americas" discusses: coconut, bottle gourd, banana, cotton (including tetraploid 26-chromosome), pineapple, Peruvian cherry (Physalis peruviana) and Argemone, yam beans, yam itself ( Dioscorea sp.), hibiscus (Hibiscus tiliaceus), common bean (Phaseolus vulgaris), lima bean (Phaseolus lunatus), bean-related plant Canavalia sp.

Some quotes from the chapter “The balsa raft and the role of guar in the indigenous navigation of South America”:

“A rough sketch of a balsa raft under sail was made by the Dutch admiral Spielbergen (16) during his circumnavigation of the world in 1614-1617. Spielbergen reports that on this raft a crew of five natives went out to fish for two months. The catch delivered to Paita was which lies 120 miles south of the Peruvian port of Tumbes, was enough to supply all the Dutch ships in the bay with provisions. Spielbergen's drawing is interesting in that the crew is shown in action. Two Indians are busy with the sail, the other three are maneuvering guar - wide boards stuck into the cracks between the logs; neither oars nor any rudder are visible. Such retractable centerboards were mastered by European shipbuilders only in 1870, that is, two hundred and fifty years later.

In the text, Spielbergen says nothing about the guar; he only concludes that the raft turned out to be an excellent vessel.

One hundred and thirty years passed before two Spanish naval officers, Juan and Ulloa, were so interested in the navigational techniques of the Indians that they decided to penetrate the secret of the indigenous guara. They published an excellent drawing of a balsa raft at sea, conveying such details as the structure of a two-legged mast with sails and rigging, the location of the deckhouse in the middle part of the ship, the “galley” with an open fireplace and a supply of water in jugs at the stern, the placement of retractable centerboards in the bow and stern parts. Juan and Ulloa strongly argued that the Indian crew, having mastered the art of maneuvering retractable centerboards, could sail the balsa raft like an ordinary ship in any wind.

They wrote: “Until now we have spoken only about the design and use of rafts, but main feature The advantage of these ships is that they sail, tack and are driven to the wind no worse than keel ships and are almost not subject to demolition. This is achieved using not a rudder, but another device, namely boards three to four meters long and about half a meter wide, which are installed vertically between the logs of the base both at the bow and at the stern.

Immersing some boards deep into the water and lifting others, they backstay, are brought to the wind, change tack, go into a drift - in short, they perform all the maneuvers available to ordinary ships. An invention still unknown to the most enlightened nations of Europe... If you immerse in water guara on the bow, the ship is brought to the wind, if you lift it, it will go to the backstay or go down to the wind. And if you immerse the guara at the stern in the water, the raft will go into the backstay, and if you lift it, it will be driven and go steeper to the wind.

This is the way in which the Indians steer balsa rafts; sometimes they place five or six guaras to prevent drift, and it is clear that the deeper the guara are immersed in the water, the greater the resistance of the vessel on that side, since the guars act as retractable keels (like later centerboards) used on small sailing ships. The method of controlling the guar is so easy and simple that when the raft is on the desired course, they then use only one of them, submerging or raising it as needed" (17).

These ancient Peruvian ship control techniques made such a strong impression on both authors that they strongly suggested adopting them in Europe."

... "Then stories about Peruvian navigation techniques were published by the famous scientist and traveler Alexander von Humboldt (1810) and his English colleague Stevenson (1825). (20) Stevenson left an excellent description of the balsa rafts that were still used along the coast of the former state Chimu as far as Huanchaco, south of Chicama. The largest rafts had bamboo huts with four or five rooms; such rafts sailed against the wind and current for hundreds of miles with a load of 25-30 tons, not counting the crew and their provisions."

... "The French maritime researcher Pari went to the northwestern part of South America to study a balsa raft there. He described this raft in his major work on non-European ships, published in 1841-1843. A little over a hundred years ago, Pari wrote: “In Peru they still use the same rafts that the aborigines built in ancient times; they are so adapted to local conditions that they are preferred to all other vessels...”

Volume 38. Measuring the world. Calendars, measures of length and mathematics Guevara Yolanda

Chapter 4 Measuring the Earth

Dimension of the Earth

The study of the movements of celestial bodies helped determine the units of time, but man was also interested in the shape and size of the world in which he lived, and he wanted to measure the Earth. Ptolemy not only contributed to the measurement of the heavens, but also became the indisputable authority on everything related to the measurement of the Earth, describing in his Geography the entire known world of his time. In the 15th–16th centuries, with the discovery of new territories, Europeans expanded the boundaries of the familiar world and made amendments to the work of Ptolemy. At the end of the 17th century, more careful measurements of the size of the Earth were made using triangulation. Thus the foundations of geodesy were laid. There were two points of view regarding the shape of the Earth: according to the first, the Earth was flattened at the poles, according to the second, at the equator. The differences between the supporters of these two points of view resulted in heated debate, and it was decided to find the truth by measuring the length of the meridian arc of one degree. The measurements were to be made by two expeditions at two points that were as far apart in latitude as possible from each other.

The first ideas about the shape and size of the Earth

In ancient times, most people believed that the inhabited Earth was flat - at least, it looked that way, if you do not take into account the unevenness of the terrain. However ancient greek philosophers began to consider other hypotheses. Anaximander is credited with the concept that the Earth had a cylindrical shape, was elongated in length and was located in the center of the celestial sphere. According to this concept, only the upper disk of the cylindrical Earth was inhabited. It is believed that Anaximander compiled a map of the Earth, which he later corrected and improved Hecataeus of Miletus(c. 550 BC - c. 476 BC). This map depicted the then-known regions of Europe, Asia and Africa, located on a disk surrounded by a river-ocean. Greece was located in the central part of the disk.

Although it is always difficult to accurately estimate the size of ancient units of measurement, it is believed that the diameter of the disk depicted on Hecataeus's map was approximately 8,000 kilometers.

Map Hecatea I century BC e.

If the Earth was flat, did it have an end? Hecataeus apparently believed so. But why then did the ocean surrounding the land not overflow? Perhaps it rested against some kind of wall where the sky connected with the sea? How was the Earth held in place? As you can see, the hypothesis about the flat shape of the Earth raised many difficult questions. The ancient Greeks theorized that the Earth was spherical and made compelling arguments to support this hypothesis, as we covered in Chapter 2. But how did Greek thinkers determine the size of the Earth?

ARGUMENTS ARISTOTLE IN FAVOR OF THE SPHERICAL SHAPE OF THE EARTH

Aristotle gave a number of arguments against the idea that the Earth is flat. For example, he pointed out that the height of stars above the horizon varies depending on the point of observation. So, a traveler going south saw that the constellations were rising higher and higher above the horizon. This meant that the horizon in the south formed a certain angle with the horizon seen by an observer in the north. Therefore, the Earth could not be flat. Likewise, the shadow cast by the Earth on the Moon during partial lunar eclipses always had a circular border, regardless of the Moon's altitude above the horizon. What body, other than a sphere, could cast a circular shadow in all directions?

Measuring the dimensions of the spherical Earth. Eratosthenes

During the Hellenistic period, Alexandria became the scientific center of Greek civilization thanks to two important institutions - a museum and a library. It was there that the circumference of the Earth was first calculated. This was done by a Greek sage, mathematician and geographer. Eratosthenes of Cyrene(276 BC - 194 BC).

As head of the Library of Alexandria, he had access to a variety of different data recorded on papyri. Eratosthenes knew that in the city of Syene (now Aswan), located south of Alexandria, at noon local time on the summer solstice, the sun's rays reached the bottom of deep wells, and vertical poles did not cast shadows. At the same time, in Alexandria the gnomon cast a shadow.

Engraving depicting the ancient Library of Alexandria.

Eratosthenes suggested: since the Sun is at a great distance, its rays fall on the Earth in parallel. If the Earth was flat, as many people still believed in those days, then the same objects on the same day and hour should cast the same shadow, regardless of where they are. But the shadows of the objects were different, therefore the Earth was not flat. At noon on the day of the summer solstice in Alexandria, Eratosthenes, using a gnomon, measured the angle at which the sun's rays are separated from the vertical. This angle was 1/50 of a circle (7°12?). Assuming that the Earth is spherical (360°), and Alexandria is located north of Siena on the same meridian, by simple reasoning (see figure) he determined that the central angle between the two radii of the Earth corresponding to Siena and Alexandria is also 1/50 of a circle (7°12?).

Scheme of reasoning Eratosthenes.

Eratosthenes knew that the distance between these cities was 5,000 stadia (about 800 kilometers), and he determined the circumference of the Earth using a simple proportion. The circumference of the Earth was supposed to be 50 times greater than the distance between Alexandria and Siena, that is, 250 thousand stadia. He rounded the result of the calculations and took one degree equal to 70 stadia, thus the total length of the earth's circumference was 252 thousand stadia.

Unfortunately, we do not know the exact length of the stage used by Eratosthenes in his calculations. The Greek stage is approximately equal to 185 m - in this case, the circumference of the earth is 46,620 km (16.3% more than it actually is). But if we assume that the scientist used the Egyptian stage, which was equal to 157.5 m, then his result is 39690 km (in this case, the error is less than 2%).

Eratosthenes' reasoning was unmistakable, but a small remark should be made regarding the accuracy of his measurements: Syene is not located on the same meridian as Alexandria, and the Sun is seen from Earth as a disk located at a finite distance, so it cannot be considered an infinitely distant point source of light. In addition, in ancient times, measuring distances over land was unreliable and became a source of error. If we take into account the errors in all the data that Eratosthenes used in his calculations, it becomes obvious that the result he obtained was surprisingly accurate.

Earth maps: latitude and longitude, geographical position and map projections

Ptolemy worked in Alexandria several centuries later than Eratosthenes. In his “Geography”, using strict scientific methods, he described the entire world known to the ancient Greeks. Ptolemy outlined mathematical methods for drawing up accurate maps using various projections, and also indicated geographical coordinates almost 10 thousand points of the world known at that time. In plotting these points on a map, he constructed a grid of parallels and meridians and applied concepts such as latitude and longitude. The prime meridian on Ptolemy's map was located near Canary Islands, the zero parallel is near the equator. He located the northern tip of the inhabited world on the parallel of Thule Island.

Apparently, the dimensions of the Earth used by Ptolemy were smaller than the actual ones: he assumed that the length of a one-degree arc of the equator was approximately 80 kilometers, thus the length of the Earth's circumference was slightly less than 30 thousand kilometers. Ptolemy enjoyed enormous authority during the Renaissance, and only thanks to this sailors dared to cross the ocean in search of new lands.

The problem of representing a curved surface on a plane is solved mathematical methods. In this sense, Ptolemy also made significant contributions to cartography. It is believed that even before him, Hipparchus divided the earth's circumference into 360° and built a grid of parallels and meridians. Hipparchus studied ways to depict a spherical surface on a flat map and, according to some scientists, used stereographic projection to solve this problem. The geographer and cartographer had a great influence on Ptolemy Marin of Tire(approx. 60 - approx. 130), who was the first to take the meridian of the Canary Islands as the zero meridian, and the parallel of Rhodes as the origin of latitude. Apparently, he also proposed using a cylindrical projection for making maps.

To depict the surface of the Earth on a plane, Ptolemy developed conical and pseudoconical projections. With their help, he was able to depict different areas on the same plane. earth's surface on different scales. In his conical projection, he represented parallels in the form of concentric arcs of circles, meridians in the form of straight lines converging at a focus that coincided with the North Pole. In the second, pseudo-conical projection of Ptolemy, the meridians were also depicted as curved lines converging at the pole, due to which he was able to depict a larger area of ​​the earth's surface with less distortion.

Conical projection Ptolemy, given in his “Geography” (“Geographicae enarrationis libri octo”), published in Lyon and Vienna in 1541.

Ptolemy's conic projection was used until the 15th century, until the borders known world have not expanded significantly. With new discoveries, this projection turned out to be insufficient for drawing maps of the world, and it began to be used only in maps of individual regions.

No map projection of the globe can preserve both areas and angles at the same time, but it is possible to preserve areas and angles to varying degrees of accuracy depending on the type of projection—particularly the projections believed to have been created by Hipparchus, Marinus, and Ptolemy.

In a stereographic projection at an arbitrary point on the sphere A, different from the pole R(projection focus), a point on the plane is assigned, defined as the point of intersection of the line RA and planes. And vice versa, to each point of the plane IN corresponds to a single point A, different from R, which is defined as the point of intersection of the sphere with the line RV. Ptolemy explains this projection in his Planisphere and uses it to depict the celestial sphere on a plane. Later, this projection was used by the Arabs in the manufacture of astrolabes - instruments for determining the position of stars in the sky.

Stereographic projection.

In cylindrical projection, the surface of the globe is projected onto a cylinder touching it at a point lying on the equator. The resulting map is distinguished by small distortions near the equator and huge distortions in the polar regions. This projection preserves angles but not areas - they increase as you move away from the equator and approach either of the two poles.

In conic projection, points on the globe are projected onto a cone, with one of the poles chosen as the focus. Subpolar regions are distorted in this projection, but the hemisphere in which the pole chosen as focus is located will be depicted with high accuracy. On a map constructed in a conic projection, distortions along the parallel of tangency are small and increase with distance from it.

The Arabs adopted much of the cultural heritage from the Greeks, but were more practical than the Greeks when it came to cartography and location tasks: they revised and corrected cartographic data as they explored new lands. At the end of the 13th century, large centers of cartography were located in the Mediterranean - in Genoa, Venice and Palma de Mallorca, where nautical maps were produced, and research was of a clearly applied nature. With the advent of the compass in Europe, when creating nautical charts, calculations began to be used that linked the coordinates of a ship with distances to various ports.

These maps, which focused on sea routes, are called portolans. They reflect the shape of coasts, coastal topography, river mouths, wind directions, and so on. A significant number of such maps were produced in the 14th and 15th centuries.

The best of the portolans made in Mallorca is the “Catalan Atlas” Abraham Cresques 1375 The illustration shows a copy of this map made in the 19th century.

The 16th century was the pinnacle of navigation: in less than 100 years, so many new lands were discovered that the area of ​​the known world doubled. Maps of the Earth improved, and for the first time it was possible to obtain direct evidence of the spherical shape of the Earth: Ferdinand Magellan (1480–1521) And Juan Sebastian Elcano (1476–1526) committed trip around the world. And soon the question of measuring the globe arose again.

FIRST DIRECT EVIDENCE OF THE SPHERICAL SHAPE OF THE EARTH

The first trip around the world (1519–1522), which became direct evidence of the spherical shape of the Earth, was started by Ferdinand Magellan and completed by Juan Sebastian Elcano. Magellan led an expedition of five ships that set sail from the city of Sanlucarde Barrameda in the Spanish province of Cadiz on September 20, 1519. The navigator crossed the Atlantic and reached the coast of Brazil near Rio de Janeiro. He then proceeded towards the La Plata River and further south to Patagonia. There Magellan discovered the strait, which now bears his name, and sailed his ships through it. His team had to endure many hardships, but the expedition crossed the Pacific Ocean, discovered the island of Guam in the Mariana Islands archipelago and reached the Philippines in March 1521. There, in the Philippines, on April 27, 1521, Ferdinand Magellan died. After his death, the expedition was led by Juan Sebastian Elcano. Setting off from the Moluccas, he crossed Indian Ocean, circumnavigated Africa and arrived in Sanlúcar de Barrameda on September 6, 1522 on the ship Victoria. Thus ended the first trip around the world.

Measuring meridian arcs through triangulation

In 1669–1670, the French astronomer Abbé Jean Piccard became the first to calculate the size of the Earth with sufficient accuracy. To do this, he applied the principles of triangulation and used the method of the Leiden astronomer, mathematician and professor Willebrord Snell (1580–1626) . Snell planned and carried out measurements in 1615, and in 1617 he described his methods in the book Eratosthenes Batavus ("Dutch Eratosthenes"), thereby laying the foundations of geodesy. His method of measuring the circumference of the Earth was to determine the length of the meridian arc through triangulation.

From a geometric point of view, triangulation is the use of triangles and their trigonometric properties to calculate unknown parameters (sides and angles) based on known ones. In geodesy, triangulation is a method that allows one to determine the size of the Earth by covering its surface with a network of adjacent triangles. Triangulation measurements begin with a competent selection of the vertices of the triangle and determining the exact length of one of the sides of the triangle.

Brilliant writer Jules Verne (1828–1905) in his novel “The Adventures of Three Russians and Three Englishmen in South Africa” clearly describes the sequence of actions during triangulation:

“To better understand what the geodetic operation called triangulation is, let us borrow the following geometric constructions from the textbook “New Lessons in Cosmography” by Mr. A. Garce, mathematics teacher at the Lyceum Henry IV. With the aid of the figure here appended this curious procedure will be easily understood:

"Let AB- meridian whose length is required to be found. Carefully measure the base (basis) AC, coming from the tip A meridian to the first position WITH. Then on both sides of this meridian we select additional positions D, E, F, G, H, I and so on, each of which allows us to see the neighboring position, and using a theodolite we measure the angles of each of the triangles ACD, CDE, EDF and so on, which they form among themselves. This first operation makes it possible to determine the parameters of various triangles, since in the first the length is known AC and angles and you can calculate the side CD; in the second - side CD and angles, and sides are easily calculated DE; in the third - the side is known DE and corners and you can get the side E.F. and so on. Then we determine the inclination of the meridian relative to the base AC why we measure the angle MAC ACM known side AC and the angles adjacent to it and you can calculate the first segment A.M. meridian. The angle is calculated in the same way M and side CM; thus in a triangle MDN turns out to be a known side DM = CD - SM and adjacent angles, and you can calculate the second segment MN meridian, angle N and side DN. Thus, in a triangle NEP side becomes known EN = DE - DN and adjacent angles and the third segment can be determined NP meridian, and so on. It is clear that in this way the total length of the axle is obtained in parts AB».

Thus, to carry out triangulation, it is necessary to determine as accurately as possible the length of the side of the triangle, which we will call the base, since all other calculations depend on the result of this measurement (in practice it turns out to be the most complex and time-consuming). The base should be as long as possible to minimize possible errors. From both ends of the base, measurements are taken of the angles that the base makes with the other two sides of the triangle. These two sides converge at a well-chosen third vertex. This defines the first triangle of the network.

Knowing two angles and a side (base) of a triangle, we can easily calculate the third angle and the two remaining sides using trigonometric methods. This way we will completely define the triangle and can choose any of its three sides as the base of the second, adjacent triangle. If we sequentially add more and more adjacent triangles to the network, then ultimately the triangulation network will cover two extreme points the meridian arc we want to measure, and we will determine the astronomical latitude and longitude of these points.

Next, using the known length of the base, it is necessary to find the length of its horizontal projection. In general, the vertices of a triangle are not necessarily at the same height, so they should be projected onto a horizontal plane or reference surface. Snell found a way to make corrections to the triangulation formulas to take into account the curvature of the Earth.

The basis for the systematic use of modern triangulation networks was the results of the first measurements made by Snell, as well as his calculated distance between the cities of Alkmaar and Bergen op Zoom in the Netherlands. These cities were located approximately on the same meridian and were separated from each other by one degree of longitude. Snell chose the distance from his home to the local church tower as the length of the base. He constructed a network of 33 triangles and measured their angles using a 2x2 meter quadrant. After taking measurements, he determined that the distance between the cities was 117,449 yards (107.393 km). The actual distance between these cities is approximately 111 km.

Using Snell's methods, Picard measured the distance corresponding to one degree of longitude of the Parisian meridian. He built a network of thirteen triangles, starting from the city of Malvoisin near Paris to the clock tower of the town of Sour Don near Amiens. The base of the network of triangles was measured along the surface of the Earth, and the angles of the triangles were measured from points located on towers, bell towers or other elevations from which the vertices of neighboring triangles could be seen.

Picard was the first to use a quadrant in measurements, supplemented with a telescope, and also designed his own measuring instruments. He used movable quadrants, complemented by spotting scopes, as well as a micrometer by the French astronomer Adrien Ozu, which ensured measurement accuracy of several arcseconds. The operating principle of a micrometer is based on the movement of a screw, in which small distances, too small for direct measurements, are marked on a measuring scale. When triangulating, it was necessary to determine the difference in height between observation points, as well as their height relative to the reference plane. Picard managed to level with an accuracy of about 1 centimeter per kilometer.

JEAN PICARD (1620–1682)

French astronomer Jean Piccard, educated at the Jesuit school of La Flèche, worked with Pierre Gassendi, a mathematics teacher at the Collège Royale in Paris (now the Collège de France). In 1655, after the death of Gassendi, Picquart became a teacher of astronomy in this educational institution, and in 1666 - a member of the newly created French Academy of Sciences. He designed a micrometer - a device for measuring the diameters of celestial bodies (Sun, Moon and planets). In 1667, Piccard added a telescope to the quadrant, making it much more convenient for observations. The researcher significantly improved the accuracy of Earth measurements by using Snell's triangulation method, and also used scientific methods in drawing up maps. In 1671, together with the Danish astronomer Ole Roemer at the Uraniborg Observatory, he observed about 140 eclipses of Jupiter's moon Io. Based on the data obtained, Roemer obtained the first quantitative estimate of the speed of light.

Piccard's goal was to determine how many toises (the so-called unit of length he used) was the length of the straight line between Malvoisin and Sourdon, as well as their difference in latitude, measured along the circumference of the meridian. Thus, it was necessary to make two measurements: geodetic (in toises) and astronomical (in degrees, minutes and seconds).

He carefully measured the length of the straight road between Villejuif and Juvisisur-Orge (it amounted to 5663 toises), and obtained the rest of the results by triangulation. He used the Chatelet toise, or Parisian toise, as a unit of measurement (later, at the end of the 18th century, it was adopted as 1.949 m). According to the measurement results, the length of the meridian arc of one degree was 57,060 toises.

Thanks to the high precision of his measuring instruments and the improvements made by Picquart, he is believed to have been the first to give a fairly accurate estimate of the radius of the Earth. He found that one degree of latitude is equal to 110.46 km, which corresponds to the radius of the Earth at 6328.9 km (today the equatorial radius of the Earth is estimated at 6378.1 km, the polar radius at 6356.8 km, the average radius at 6371 km) . Picard's data was used by Isaac Newton to create his theory of gravity.

Five triangles from a triangulation network Picara.

After Picard, length measurements along the Parisian meridian were carried out by triangulation Giovanni Domenico Cassini (1625–1712) , head of the Paris Observatory, and his son Jacques Cassini (1677–1756) , who succeeded his father in his post. Jacques Cassini measured the length of the meridian arc between Dunkirk and Perpignan and published the results in 1720. Later, in 1733–1740, together with his son, Caesar François Cassini, he first built a triangulation network that covered the entire country. In 1745, thanks to his work, the first accurate map of France appeared.

Later, triangulation networks were also built in other countries. For example, the UK triangulation project called Principal Triangulation of Great Britain was started in 1783, and was completely completed only in the middle of the 19th century.

The first project to compile an accurate map of Spain was proposed by Jorge Juan in 1751, but the first sheets of the National topographic map Spain saw the light of day only in 1875.

Location and orientation.

Navigation and the longitude problem

To determine the position of a point on a plane, you can use a Cartesian coordinate system with perpendicular axes: the x-axis ( X) and ordinate axis ( at). Value pair ( x, y) uniquely determines a single point on the plane. Similarly, in order to accurately determine the position of any point on the surface of the Earth (we will consider it spherical), it is enough to know two numbers - latitude and longitude (geographic coordinates of the point). In this case, the role of the coordinate axes will be played by the equator and the great circle passing through the poles, that is, the meridian chosen as the base one (0° meridian).

The latitude of a point on the Earth's surface is the angular distance between the equator and that point, measured from the center of our planet along the meridian passing through that point. Latitude is measured in degrees, minutes and seconds and ranges from 0° to 90°. In addition, it is indicated in which hemisphere, Northern or Southern, the point is located, for example 41°24?14? northern latitude (N). Consequently, all points located on the same parallel of the Earth (the circumference of a circle parallel to the equator) have the same latitude.

Latitude can be calculated using astronomical methods. The simplest method for the Northern Hemisphere was to find the North Star in the sky ( North Pole world) and measure the angle between the line of sight and the horizontal plane on which the observer is located. The resulting angle will be the desired latitude. In the Southern Hemisphere, you should act in a similar way, choosing the Southern Cross for observations. There are other methods for determining latitude during the day - for example, you can measure the height of the Sun above the horizon at noon and use tables that indicate the position of the Sun relative to the ecliptic on the day of observation.

Latitude and longitude of the point R on the sphere.

Longitude is the value of the angle between the prime meridian (more precisely, the half-meridian), chosen as the origin (0°), and the meridian passing through this point. This angle is measured from the center of the Earth along the equator. Longitude values ​​range from 0° to 180°. In addition, it is indicated in which direction from the prime meridian the longitude was measured - to the east or to the west, for example, 2°14?50? West longitude (W). Consequently, all points located on the same semi-meridian between the two poles of the Earth have the same longitude.

Latitude and longitude are measured from the equator and the meridian chosen as the origin (this meridian is called the zero meridian, its longitude is 0°).

Today the prime meridian is usually considered to be Greenwich, but before it many other meridians were used as prime meridians.

As we have already said, determining the latitude of a ship at sea is not difficult. It is also relatively easy to find out the longitude of a ship if it has land visible from it. But if it is on the open sea, then determining longitude is associated with serious difficulties.

This task gained enormous importance after the discovery of America by Christopher Columbus. At that time, longitude was calculated approximately, based on the distance traveled by a ship from west to east or vice versa. To determine the speed of the ship, sailors used a log, which was a freely rotating reel with a rope wound around it. Knots were tied on the rope at regular intervals, and a weight was attached to its end. The sailor threw the log behind the stern, and when the first knot hit his hand, he gave the command, and another sailor began counting the time using an hourglass. When all the sand was poured from the upper vessel of the clock into the lower one, the second sailor reported this to the first, and he indicated the number of knots that went overboard, for example, “three and a half knots” or “six knots and a quarter.” The speed of ships is still measured in knots.

Of course, such a primitive method of determining longitude was accompanied by significant errors that led to catastrophic consequences. Therefore, in the 17th - early 18th centuries, the task of determining longitude became a strategic priority for all powers that had interests overseas.

Theoretically, calculating longitude can be reduced to determining the time difference between the reference point (the port of departure or the prime meridian) and the point at which the ship is located. When the sun passes through the meridian of the observer (that is, the meridian of the ship), then, knowing the exact time at the reference point, it is possible to determine the longitude of the ship, that is, the angular distance to the reference point, and therefore to the prime meridian. This method works because the time difference between two meridians can be converted into degrees of longitude. Since the Earth makes a full rotation of 360° in 24 hours, in 1 hour it rotates 1/24 of a revolution, that is, 13°. If in an hour, that is, in 60 minutes, the Earth rotates 13°, then a difference of 4 minutes corresponds to one degree of longitude.

Therefore, longitude can be calculated by determining the time difference between two points using observations and astronomical measurements. The idea was put forward of determining longitude from observations of eclipses, but this method is not very suitable on the open sea, and eclipses were rarely observed.

OBSERVING eclipses to calculate longitude

Let's assume that we know at what time the eclipse will be observed in a certain place (on land, in an observatory, etc.), while we are in the open sea. If we determine when the eclipse was observed in local time, we can calculate the longitude of the place where we are. To use this method, we will need tables that indicate at what time an eclipse will occur at a certain point (of course, we cannot do without mathematical calculations). In the 16th century, determining longitude from observations of eclipses was convenient on land, but not on the open sea - it was very difficult to fix measuring instruments due to the motion, and most importantly, eclipses were observed rarely: from two to five solar eclipses occur per year. If we also take into account lunar ones, then there are at least two and no more than seven eclipses per year, with an average of four. Over the entire 20th century, 375 eclipses were observed: 228 solar and 147 lunar. Already rare eclipses are not always visible: observations can be hampered by unfavorable weather conditions.

The insufficient frequency of eclipses was overcome thanks to Galileo's discovery of the moons of Jupiter in 1610. Jupiter's moons disappear from view and reappear as they rotate around it. These eclipses are observed several thousand times a year, and their timing can be accurately predicted. This method could indeed be used to determine longitude, but in the open sea the rolling motion interfered, and observations could only be made at night, in clear weather, and only at certain times of the year.

The problem of determining longitude on the open sea remained unsolved for quite some time. The local time on the ship could be determined by the Sun. But how can you find out the time at the starting point without having a sufficiently accurate clock? The accuracy of pendulum clocks was reduced, among other factors, by the motion of the ship; in addition, the period of oscillation of the pendulum differed at different latitudes, and as a result, the clocks were in a hurry or late. The ship's clock could not keep time at the port of departure; this caused significant errors in determining longitude.

In 1714, the British Parliament offered a huge prize of 20 thousand pounds sterling to anyone who could present a method or instrument for determining the longitude of a ship on the high seas. The prize went to the English watchmaker John Harrison (1693–1776), who, after several decades of work, was able to produce a very accurate chronometer. In 1761, the chronometer was loaded onto a ship bound for Jamaica for testing. The chronometer lasted 147 days, and upon return to England the deviation was only 1 minute 34 seconds. The problem of determining longitude was solved. Today, the exact position of the ship can be determined thanks to the GPS system, which we will talk about in Chapter 6.

Non-spherical Earth. Scientific expeditions to the Viceroyalty of Peru and Lapland

When measuring the Earth, including Picard's measurements, it was believed that it had the shape of a perfect sphere. A few years after Picard's experiment, in 1671–1673, the French astronomer Jean Richet (1630–1696) , assistant to Giovanni Domenico Cassini, traveled to Cayenne in French Guiana, where he made important discovery: he noticed that in Cayenne the oscillations of the pendulum were slower than in Paris, and he was the first to understand that the gravitational force of the Earth differs in different parts of it. He made the right conclusion: the change in gravity was explained by the fact that Cayenne was further from the center of the Earth than Paris. When news of the discovery reached Europe, it caused great excitement among members of the French Academy of Sciences. Upon returning to his homeland, Richet began making a pendulum that would count seconds - in other words, the period of oscillation of the pendulum in Paris should have been exactly one second. The same pendulums were made in other parts of the earth, and it turned out that the length of the pendulum varied depending on the latitude. According to the theories known at that time, everything pointed to the fact that if the force with which the Earth attracts a pendulum to itself is different at different points, then the Earth cannot have the shape of a perfect sphere.

Newton took into account Richet's results in his famous "Mathematical principles of natural philosophy", published in 1687, which laid out the foundations of mechanics. He proposed a mathematical description of the shape of the Earth, linking it with his ingenious theory of gravity. Newton considered our planet as a homogeneous liquid body of rotation and concluded: The Earth must be flattened at the poles. In his opinion, the Earth was flattened by 1/230. In other words, if we assume that the cross section of the Earth is an ellipse, then its major axis will be 1/230th longer than the minor axis.

In 1720, Jacques Cassini’s work “On the Size and Shape of the Earth” was published in France, where Newton’s hypothesis was refuted. Cassini supported his point of view with the results of his own astronomical observations and geodetic measurements of the Collioure - Paris - Dunkirk meridian (however, some members of the French Academy of Sciences considered these measurements not to be entirely accurate).

Cassini called Newton's arguments speculative and pointed out that the Earth is an ellipsoid, oblate at the equator. What does the Earth look more like - a watermelon or a melon? A controversy ensued, involving scientists from the Royal Society of London and the French Academy of Sciences. As a result, the discussion began to be seen as a confrontation between French and English science.

To put an end to the controversy, the French Academy of Sciences decided to measure the length of the meridian arc corresponding to the central angle of one degree, at points as far apart as possible. For this purpose, two scientific expeditions of astronomers, mathematicians, naturalists and other scientists were organized. The first expedition led Pierre Louis Moreau de Maupertuis (1698–1739) , went to Lapland. Its members were Pierre Charles Le Monnier, Alexis Claude Clairaut, Charles Etienne Louis Camus, the Swede Anders Celsius and the Abbe Houtier. The second expedition, which went to the Viceroyalty of Peru, on the territory of modern Ecuador, was led by an astronomer Louis Gaudin (1704–1760) .

The participants of the expedition were the geographer Charles Marie de la Condamine, the astronomer and hydrographer Pierre Bouguer, the botanist Antoine Laurent de Jussieux and the Spaniards Jorge Juan and Antonio de Ulloa. Creole scientist Pedro Vicente Maldonado joined the expedition in Guayaquil. Also included in the expedition were the watchmaker Hugo, the engineer and draftsman Morinville, the captain of the frigate Couplet, the surgeon and botanist Seignerg, the instrument maker Gaudin de Odonnet, the nephew of Louis Gaudin, the cartographer and military engineer Vergen.

At that time, the Viceroyalty of Peru, located in the equatorial Andes, was Spanish territory, so the expedition members had to ask permission from the Spanish crown. Permission was given with the condition that two young gifted officers of the Cadiz Academy of Midshipmen, Jorge Juan and Antonio de Ulloa, would join the expedition.

Participants in the expedition to Lapland (1736–1737), thanks to the abilities and insight of the mathematician Clairaut, received desired results relatively quickly.

The Swedish military helped them in setting up observation posts. Scientists carried out triangulation during long summer days and covered a distance of 100 kilometers between the cities of Kittis and Torneo. Astronomical measurements were made in spring and autumn, when the nights were already quite long and at the same time not too cold. The base of the triangulation was measured along the frozen river bed. The final result of measurements carried out by members of the Maupertuis expedition was as follows: at an average latitude of 66°20? the length of a meridian arc of one degree was equal to 37,438 toises. If we compare this result with the result of Piccard's measurements, carried out near Paris at a latitude of about 48 ° (57060 toises), it becomes obvious that the Earth is a spheroid, oblate at the poles.

Goniometric measurements during triangulation. Illustration for the novel Jules Verne"The Adventures of Three Russians and Three Englishmen in South Africa."

The expedition to America, in turn, lasted for ten years and turned into a real epic. The participants set off from La Rochelle in the spring of 1735 and arrived in Quito a year later. They had to face a variety of problems: in addition to constant scientific disputes, the members of the expedition were hampered by the harsh climate, difficult terrain, numerous financial troubles, and in 1741 they had to split into two groups. Measurements and triangulation were particularly difficult due to the terrain of the Andes and high altitude, exceeding 4 thousand meters. Scientists decided to build a large-scale triangulation of 43 triangles to cover a segment of 354 kilometers and measure the arc of the meridian not at 1°, but at 3°. Bouguer (1749) determined that the length of a meridian arc of one degree is equal to 56,763 toises, and Juan and Ulloa (1748), as well as La Condamine (1751), obtained a result of 56,768 toises. If we recall the analogy with a watermelon or melon that Voltaire proposed, we can say that the Earth is more like a watermelon. The results of measurements and mathematical calculations seemed to confirm that Newton was right.

JORGE JUAN AND ROYAL OBSERVATORY IN SAN FERNANDO (CADIZ)

Spanish navigator Jorge Juan and Santasilla (1713–1773) , who participated in an expedition to measure the meridian arc at the equator, made a significant contribution to the development of Spanish science in the 18th century. Traces of his work have survived to this day - he, among other things, founded the Royal Observatory in San Fernando (Cádiz) in 1757. Modern Royal Institution and Observatory naval forces- not only the heart of astronomical and geodetic research, but also a scientific research and cultural center run by the Spanish army. The center's staff calculates ephemeris, determines exact time, publishes marine astronomical yearbooks and the results of meteorological, seismic and magnetic observations. The institute is responsible for determining the official Spanish time (Coordinated Universal Time, or UTC) and for maintaining the standards of Spain's official units of measurement.

Chapter 1 Who is John? In order to find out which of the two twin brothers is named John, you need to ask one of them: “Is John telling the truth?” If the answer to this question is “yes,” then regardless of whether the asked twin is lying or always tells the truth, he must