Presentation on the topic "Archimedes". Archimedes presentation for a mathematics lesson on the topic He was a talented mathematician

Archimedes Presentation by Albert Aghababyan 9 “B” class

Archimedes was born in 287 BC. in Syracuse on the island of Sicily. Archimedes' father, the astronomer and mathematician Phidias, was closely related to Hiero, the tyrant of Syracuse. The father instilled in his son a love of mathematics, mechanics and astronomy from childhood. In Alexandria of Egypt - the scientific and cultural center of that time - Archimedes met famous Alexandrian scientists. He corresponded with Eratosthenes until the end of his life. It was here that Archimedes became acquainted with the works of Democritus, Eudoxus and other outstanding Greek geometers. After leaving Alexandria, Archimedes returned to Sicily. In Syracuse he was surrounded by attention and did not need funds. Because of its long history, Archimedes' life is closely intertwined with legends.

Archimedes was a remarkable practical mechanic and theorist, but his main work in life was mathematics. According to Plutarch, Archimedes was simply obsessed with her. He forgot about food and did not take care of himself at all. His works related to almost all areas of mathematics of that time: he did remarkable research in geometry, arithmetic, and algebra. He found all the semiregular polyhedra that now bear his name, significantly developed the doctrine of conic sections, and gave a geometric method for solving cubic equations, the roots of which he found using the intersection of a parabola and a hyperbola. Archimedes also carried out a complete study of these equations, that is, he found under what conditions they would have real positive different roots and under what conditions the roots would coincide. Snub cube Cuboctahedron Truncated tetrahedron

He considered his best achievement to be determining the surface and volume of a sphere - a problem that no one had been able to solve before him. Archimedes asked to knock out a ball inscribed in a cylinder on his grave. In his essay Quadrature of a Parabola, Archimedes proved that the area of a segment of a parabola cut off from it by a straight line is 4/3 of the area of the triangle inscribed in this segment. To prove this, Archimedes calculated the sum of an infinite series:

The ratio of the circumference of a circle to its diameter, calculated by Archimedes, was of great importance for the development of mathematics. In his work “On the Measurement of a Circle,” Archimedes gave his famous approximation for a number: the “Archimedean number.” Moreover, he was able to estimate the accuracy of this approximation: . To prove this, he constructed inscribed and circumscribed 96-gons for the circle and calculated the lengths of their sides. In mathematics, physics and astronomy, it is very important to be able to find the largest and smallest values of changing quantities - their extrema. For example, how can you find the cylinder with the largest volume among the cylinders inscribed in a sphere? All such problems can currently be solved using differential calculus. Archimedes was the first to see the connection between these problems and the problems of determining tangents and showed how to solve problems involving extrema. Archimedes' ideas were almost two millennia ahead of their time. Only in the 17th century were scientists able to continue and develop the works of the great Greek mathematician.

Archimedes became famous for many mechanical designs. The endless screw he invented for scooping water moves water through the pipe to a height of up to 4 m. It is still used in Egypt.

Archimedes built a planetarium or “celestial sphere”, during the movement of which one could observe the movement of the five planets, the rising of the Sun and the Moon, the phases and eclipses of the Moon, the disappearance of both bodies beyond the horizon. After the death of Archimedes, the planetarium was taken by Marcellus to Rome, where it aroused admiration for several centuries

In his treatise “On Leverage,” Archimedes established the RULE OF LEVER EQUILIBRIUM

Archimedes was the first to come up with a block design, study its mechanical properties and apply it in practice.

There is a legend about how King Hiero instructed Archimedes to check whether the jeweler had mixed silver in his golden crown. The integrity of the product could not be compromised. Archimedes could not complete this task for a long time. The solution came by chance when he lay down in the bathtub and noticed the displacement of liquid. Archimedes shouted: “Eureka!” - “Found it!”, and ran naked into the street. He realized that the volume of a body immersed in water is equal to the volume of displaced water. Thus, Archimedes found out that silver was mixed into gold, exposed the deceiver and discovered the fundamental law of hydrostatics!

The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse by the Romans in 212 BC. e. But at that time he was already 75 years old! Powerful throwing machines built by Archimedes threw heavy stones at the Roman troops. Thinking that they would be safe at the very walls of the city, the Romans rushed there, but at that time light, short-range throwing machines pelted them with a hail of cannonballs. Powerful cranes grabbed ships with iron hooks, lifted them up, and then threw them down, so that the ships turned over and sank. According to legend, during the siege, the Roman fleet was burned by the defenders of the city, who, using mirrors and shields polished to a shine, focused the sun's rays on them on the orders of Archimedes.

According to the first, in the midst of the battle, he sat on the threshold of his house, thinking deeply about the drawings he made right on the sand of the road. At this time, a Roman soldier running past stepped on the drawing, and the indignant scientist rushed at the Roman shouting: “Don’t touch my drawings!” This phrase cost Archimedes his life. The soldier stopped and coldly killed the old man with his sword.

The second version says that the Roman commander Marcellus specifically sent a warrior to search for Archimedes. The warrior found the scientist and said: “Come with me, Marcellus is calling you.” - What kind of Marcellus?! I have to solve the problem! The angry Roman drew his sword and killed Archimedes.

Thank you for your attention

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Archimedes This is an amazing person whose name people have remembered for more than 2,000 years. He was a talented mathematician, mechanic and engineer. Every schoolchild is familiar with the number π, the rule of balance of a lever, the “golden” rule of mechanics, the law of floating of bodies, etc. The name of Archimedes lives in legends.

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Biography Archimedes was born in 287 BC. in Syracuse on the island of Sicily. Archimedes' father, the astronomer and mathematician Phidias, was closely related to Hiero, the tyrant of Syracuse. The father instilled in his son a love of mathematics, mechanics and astronomy from childhood. In Alexandria of Egypt - the scientific and cultural center of that time - Archimedes met famous Alexandrian scientists. He corresponded with Eratosthenes until the end of his life. It was here that Archimedes became acquainted with the works of Democritus, Eudoxus and other outstanding Greek geometers. After leaving Alexandria, Archimedes returned to Sicily. In Syracuse he was surrounded by attention and did not need funds. Because of its long history, Archimedes' life is closely intertwined with legends.

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Legend of the Crown There is a legend about how King Hiero ordered Archimedes to check whether the jeweler had mixed silver in his golden crown. The integrity of the product could not be compromised. Archimedes could not complete this task for a long time. The solution came by chance when he lay down in the bathtub and noticed the displacement of liquid. Archimedes shouted: “Eureka!” - “Found it!”, and ran naked into the street. He realized that the volume of a body immersed in water is equal to the volume of displaced water. Thus, Archimedes found out that silver was mixed into gold, exposed the deceiver and discovered the fundamental law of hydrostatics!

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Legends Legend tells that the luxurious ship Syrokosia, built by Hieron as a gift to the Egyptian king Ptolemy, could not be launched. Archimedes built a system of blocks (pulley hoist), with which he was able to do this work with the help of a few people.

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The second group consists of works on geometric analysis of statistical hydrostatic problems: “On the equilibrium of plane figures.” The famous law of hydrostatics, which entered science as Archimedes’ law, was formulated in the treatise “On Floating Bodies.” Any body immersed in a liquid is subject to a buoyant force directed upward and equal to the weight of the liquid displaced by it. Archimedes' law is also true for gases. FA = ρzh·g∙VT = Rzh

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Siege of Syracuse The engineering genius of Archimedes manifested itself with particular force during the siege of Syracuse by the Romans in 212 BC. e. But at that time he was already 75 years old! Powerful throwing machines built by Archimedes threw heavy stones at the Roman troops. Thinking that they would be safe at the very walls of the city, the Romans rushed there, but at that time light, short-range throwing machines pelted them with a hail of cannonballs. Powerful cranes grabbed ships with iron hooks, lifted them up, and then threw them down, so that the ships turned over and sank. According to legend, during the siege, the Roman fleet was burned by the defenders of the city, who, using mirrors and shields polished to a shine, focused the sun's rays on them on the orders of Archimedes.

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Legends of Death According to the first, in the midst of the battle, he sat on the threshold of his house, pondering in depth over the drawings he made right on the road sand. At this time, a Roman soldier running past stepped on the drawing, and the indignant scientist rushed at the Roman shouting: “Don’t touch my drawings!” This phrase cost Archimedes his life. The soldier stopped and coldly killed the old man with his sword.

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Mathematical works Archimedes was a remarkable practical mechanic and theorist, but the main work of his life was mathematics. According to Plutarch, Archimedes was simply obsessed with her. He forgot about food and did not take care of himself at all. His works related to almost all areas of mathematics of that time: he did remarkable research in geometry, arithmetic, and algebra. He found all the semiregular polyhedra that now bear his name, significantly developed the doctrine of conic sections, and gave a geometric method for solving cubic equations, the roots of which he found using the intersection of a parabola and a hyperbola. Archimedes also carried out a complete study of these equations, that is, he found under what conditions they would have real positive different roots and under what conditions the roots would coincide. Snub cube

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The surviving works of Archimedes can be divided into three groups: The first group is the determination of the areas of curvilinear figures or, accordingly, the volumes of bodies. Archimedes found a general method that allows one to find any area or volume. Using his method, he determined the areas and volumes of almost all bodies that were considered in ancient mathematics. He considered his best achievement to be the determination of the surface area and volume of a sphere. Archimedes' ideas formed the basis of integral calculus.

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The third group includes various mathematical works: For example, how can you find the cylinder with the largest volume among the cylinders inscribed in a sphere? In his work “On the Measurement of the Circle,” Archimedes gave his famous approximation of the number π: “Archimedean number.” He was able to assess the accuracy of this approximation: To prove it, he constructed inscribed and circumscribed 96-gons for the circle and calculated the lengths of their sides.

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Archimedes' screw Archimedes became famous for many mechanical designs. The endless screw he invented for scooping water moves water through the pipe to a height of up to 4 m. It is still used in Egypt.

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An Archimedean spiral is a flat curve, the trajectory of point M moving from point O at a constant speed along a beam rotating around the pole O at a constant angular speed. Equation in polar coordinates: r = a∙f, where a is a constant.

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“Celestial Sphere” of Archimedes Archimedes built a planetarium or “celestial sphere”, during the movement of which one could observe the movement of the five planets, the rising of the Sun and the Moon, the phases and eclipses of the Moon, the disappearance of both bodies beyond the horizon. After the death of Archimedes, the planetarium was taken by Marcellus to Rome, where it aroused admiration for several centuries

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He discovered the “golden” rule of mechanics: the number of times a mechanism gives a gain in strength, the same number of times it gives a loss in distance “Give me a fulcrum, and I will turn the whole world upside down.”

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The scientific work of Archimedes was so ingenious, and his amazing inventions were so unique, that even during his lifetime, incredible legends and stories were formed around the figure of the scientist. Archimedes united in himself, on the one hand, a brilliant mathematician who outlined fundamentally new ways for the development of this science, and on the other, a remarkable engineer who surpassed all his predecessors and contemporaries in terms of technical skill. Archimedes is a pioneer of mathematical physics. Mathematics in his works is systematically applied to the study of problems in natural science and technology. Archimedes is one of the founders of mechanics as a science.

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THE GREAT ARCHIMEDES, HIS DISCOVERIES AND INVENTIONS The material was prepared by E.A. Alikueva, Physics Teacher, School No. 1981, Moscow. 2017

Birth Archimedes (Ἀρχιμήδης; 287 - 212 BC) Ancient Greek mathematician, physicist and engineer from Syracuse Made many discoveries in geometry. He laid the foundations of mechanics and hydrostatics, and was the author of a number of important inventions.

Childhood Archimedes' father may have been the mathematician and astronomer Phidias. According to Plutarch, Archimedes was closely related to Hiero II, the tyrant of Syracuse.

Training To study, Archimedes went to Alexandria of Egypt - the scientific and cultural center of that time. To study, Archimedes went to Alexandria of Egypt - the scientific and cultural center of that time.

Discoveries Archimedes became famous for his numerous mechanical designs. The Archimedes screw (auger) he invented for scooping up water is still used in Egypt. “This invention,” Galileo wrote about the Archimedean screw, “is not only magnificent, but simply wonderful, since we see that the water rises in the screw, continuously descending.”

Archimedes' discoveries test and create a theory of five mechanisms, called "simple mechanisms." This is a lever (“Give me a fulcrum,” said Archimedes, “and I will move the Earth”), a wedge, a block, an endless screw and a winch..

Inventions The legend tells that Archimedes managed to move the heavy multi-deck ship Syracuse with one movement of his hand thanks to the system of blocks he developed, so

Inventions Polyspast - a system (of N pieces) of movable and fixed blocks connected in common holders, providing a 2N multiple gain in strength

Short range throwing machine

Inventions "Archimedes's Paw", a unique lifting machine and the prototype of a modern crane. Outwardly, it looked like a lever protruding beyond the city wall and equipped with a counterweight. If a Roman ship tried to land near Syracuse, this “manipulator” would grab its bow and turn it over. (the weight of Roman triremes exceeded 200 tons, and the penteras could reach all of 500), flooding the attackers.

Inventions The Roman fleet anchored near the city. According to legend, Archimedes constructed a large mirror, with the help of which he “concentrated” sunlight on the enemy fleet and burned it to the ground.

Discoveries In physics, Archimedes introduced the concept of the center of gravity, established the scientific principles of statics and hydrostatics, and gave examples of the use of mathematical methods in physical research.

Archimedes' Law There is a legend that the idea of this law came to Archimedes while he was taking a bath; with an exclamation of “Eureka!” he jumped out of the bath and ran naked to write down the scientific truth that had come to him.

Archimedes' Law

Crown of King Hieron King Hieron, who lived 250 BC. instructed him to check the honesty of the craftsman who made the golden crown. Archimedes calculated the buoyant force equal to the weight of water in the volume of the crown. Having then determined the volume of the crown, he was able to calculate its density. The density of the corona material turned out to be less than the density of pure gold. The master was exposed in deception.

In astronomy Construction of a “planetarium” to observe the movement of the five planets of the solar system, the rising of the Sun and the Moon Also built an instrument to determine the apparent diameter of the sun

In mathematics, he substantiated the method of calculating the area of a parabolic segment, and did this two thousand years before the discovery of integral calculus. In his work “On the Measurement of a Circle,” Archimedes first calculated the number “pi” - the ratio of the circumference to the diameter - and proved that it is the same for any circle.

The Death of Archimedes Archimedes died during the siege of Syracuse: he was killed by a Roman soldier at a time when the scientist was absorbed in searching for a solution to the problem he had set himself.

The Palimpsest of Archimedes The Palimpsest of Archimedes is a Christian book compiled in the 12th century from “pagan” parchments of the 10th century. To do this, the previous writings were washed off them, and a church text was written on the resulting material. Fortunately, the palimpsest was made poorly, so the old letters were visible in the light. In 1906, it turned out that these were three previously unknown works of Archimedes.

In memory One of the large lunar craters (82 kilometers wide) was named after Archimedes.

Sources of information http://elementy.ru/trefil/21067/Zakon_Arkhimeda http://class-fizika.narod.ru/7_archim.htm http://900igr.net/kartinki/fizika/Zakon-Arkhimeda/Zakon-Arkhimeda. html https://ru.wikipedia.org/wiki/ http://alternathistory.com/voennye-mashiny-arkhimeda https://www.google.ru/search?

around 287–212 BC

The fact that he addressed messages about his discoveries to mathematicians associated with Alexandria, such as Eratosthenes, confirms the opinion that Archimedes was one of the active successors of Euclid who developed the mathematical traditions of the Alexandrian school. Returning to Syracuse, Archimedes remained there until his death during the capture of Syracuse by the Romans in 212 BC.

A native of the Greek city of Syracuse on the island of Sicily, Archimedes was a close associate of King Hiero, who ruled the city (and probably his relative). Perhaps Archimedes lived for some time in Alexandria, the famous scientific center of that time.

The date of birth of Archimedes (287 BC) is determined based on the testimony of a Byzantine historian of the 12th century. John Tzetz, according to which he “lived seventy-five years.” The vivid pictures of his death, described by Livy, Plutarch and Valerius Maximus, differ only in details, but agree that Archimedes, who was engaged in geometric constructions in deep thought, was hacked to death by a Roman soldier. In addition, Plutarch reports that Archimedes “is said to have bequeathed to his relatives and friends to install on his grave a cylinder described around a ball, indicating the ratio of the volume of the described body to the inscribed one,” which was one of his most famous discoveries. Cicero, who in 75 BC. I was in Sicily, I found a tombstone peeking out from the thorny bushes and on it - a ball and a cylinder

In our time, the name of Archimedes is associated mainly with his remarkable mathematical works, but in antiquity he also became famous as the inventor of various kinds of mechanical devices and tools, as reported by authors who lived in a later era. True, the authorship of Archimedes is in doubt in many cases. So, it is believed that Archimedes was the inventor of the so-called. the Archimedean screw, which served to lift water to the fields and was the prototype of ship and air propellers, although, apparently, this kind of device was used before. What Plutarch says in the Life of Marcellus does not inspire much confidence either.

It says that in response to King Hiero's request to demonstrate how a heavy load could be moved with little force, Archimedes “took a three-masted cargo ship, which had previously been pulled ashore with great difficulty by many people, seated many people on it and loaded it with ordinary cargo. After that, Archimedes sat down at a distance and began to effortlessly pull the rope thrown over the pulley, causing the ship to “float” towards him easily and smoothly, as if on water.”

Archimedes screw: 1 - engine, 2 - screw, 3 - casing

It is in connection with this story that Plutarch cites the remark of Archimedes that “if there was another Earth, he would move ours by moving to that one” (a more famous version of this statement is reported by Pappus of Alexandria: “Give me where to stand, and I will move the Earth "). The authenticity of the story told by Vitruvius is also questionable, that King Hiero allegedly instructed Archimedes to check whether his crown was made of pure gold or whether the jeweler appropriated part of the gold by alloying it with silver.

More reliable is the testimony of Pappus that Archimedes wrote an essay on the manufacture of the [celestial] sphere, which probably dealt with the construction of a planetarium model that reproduced the visible movements of the Sun, Moon and planets, as well as, possibly, a star globe depicting the constellations. In any case, Cicero reports that both instruments were captured by Marcellus as trophies in Syracuse. Finally, Polybius, Livy, Plutarch and Tzetz report on the grandiose ballistic and other machines built by Archimedes to repel the Romans.

He was a friend of King Hieron,
What is the most important subject for a king?
You guessed it - crown!
Hieron wanted
Make a new crown.
He measured out the gold strictly.
I took not a little and not a lot -
Just how much is needed.
Jeweler to order.
A month later Hieron
The jeweler brought the crown
Hiero took the crown,
Looked at it from all sides
Sparkles with pure gold...
But anything can happen,
And even worse - copper
(if you don’t have a conscience)..."

Hiero: “Here is the crown, Archimedes,
Gold or not?
Archimedes: It sparkles like pure gold.

Hieron: But you know, anything happens!
And you can add silver to gold cunningly,
And even worse - copper,
If you don't have a conscience.
I began to doubt something:
Was the work done honestly?
Is it possible, tell me, to determine this?
But don’t scratch the crown, don’t saw it..."

What is known? WEIGHT of the crown,
Well, how to find VOLUME?
I thought at night, I thought during the day.
And one day while washing in the bath
He sank to his waist.
Water spilled onto the floor -
He guessed then
How to find the VOLUME of the crown
And rushed to Hieron
Not shod and not dressed...
And the people shout after:
What happened, Archimedes?
Maybe an earthquake?
Or is there a fire in the city?
The whole market was alarmed!
They even closed shops.
Noise, and screams, and confusion!
He rushed past the guards."

Archimedes: “Eureka! I found a solution!”

Archimedes: “I came up with an idea, Hiero!”

In a palace

Archimedes: “Eureka! Revealed the secret!

Hiero: “Get dressed, Archimedes!
Here are the sandals, chiton,
And you’ll tell me everything later!”

Archimedes: “Let the scales be carried here
And a large vessel with water...
Deliver everything to Hieron!
(Servants bring everything)
We put the crown on the scales
And now it’s exactly the same
We are looking for a gold bar...
(Finds a piece of gold equal in weight to the crown)"

Hieron: “Everything is clear!”

Archimedes: “No, wait!
We now lower our crown into this bowl.
Hieron! Look here -
The water in the bowl has risen!
I’ll put a line around the edge.”

Hieron: “And the crown?”

Archimedes: “I’m taking it out.
Let’s put the gold in the water...!”

Hieron: “Gold in the water? Let's say...

Archimedes: The water has risen again
I'm putting a mark..."

Hieron: “Where?”

Archimedes: “Well, of course, along the edge.”

Hieron: “I don’t understand anything.
I only see two lines:
This one is higher, this one is lower.
But what is the main conclusion?

Archimedes: “Equal weight.
The volume is not equal!
You see, Hieron.
I have now discovered the law.
The law is very simple:
The body will displace..."

Hieron: “Wait!
Are you saying the volume is not equal?
My master is an obvious fraudster!
For a false crown
He will answer according to the law!”

A lot of water has flown under the bridge since then,
But they remember Archimedes' law"

Teacher: Now we understand the words of D.M. Balashov, which are written on the board, and why the buoyant force was called the Ar Force

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Prepared by mathematics teacher of Municipal Educational Institution Secondary School No. 36 of Kaliningrad Kovalchuk Larisa Leonidovna

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Archimedes is the pinnacle of scientific thought of the ancient world. Subsequent scientists - Heron of Alexandria (1st-11th centuries BC), Pappus of Alexandria (III century AD) - added little to the legacy of Archimedes.

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Archimedes was born in 287 BC in the Greek city of Syracuse, where he lived almost his entire life. His father was Phidias, the court astronomer of the ruler of the city of Hiero. Archimedes studied in Alexandria, where the rulers of Egypt, the Ptolemies, gathered the best Greek scientists and thinkers, and also founded the world's largest library.

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Archimedes' main works concerned various practical applications of mathematics (geometry), physics, hydrostatics and mechanics. In his work “Parabolas of Quadrature,” Archimedes substantiated the method for calculating the area of a parabolic segment, and he did this two thousand years before the discovery of integral calculus. In his work “On the Measurement of a Circle,” Archimedes first calculated the number “pi” - the ratio of the circumference to the diameter - and proved that it is the same for any circle.

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The mathematical method of Archimedes, associated with the mathematical works of the Pythagoreans and with the work of Euclid that completed them, as well as with the discoveries of Archimedes’ contemporaries, led to the knowledge of material space, to the knowledge of the theoretical form of objects located in this space, the form of a perfect, geometric form, to which objects are more or less approaching and the laws of which must be known in order to influence the material world.

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Archimedes studied the forces that move objects or bring them into equilibrium, inventing a new branch of mathematics in which material bodies, reduced to their geometric form, at the same time retain their weight. This geometry of weight is rational mechanics, it is statics, as well as hydrostatics, the first law of which was discovered by Archimedes (the law that bears his name), according to which a force equal to the weight of the fluid displaced by it acts on a body immersed in a liquid.

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The famous "Eureka!" was pronounced not in connection with the discovery of Archimedes' law, but in connection with the law of the specific gravity of metals - a discovery that also belongs to the Syracusan scientist. According to legend, one day the ruler of Syracuse turned to Archimedes. He ordered to check whether the weight of the golden crown corresponded to the weight of the gold allocated for it. To do this, Archimedes made two ingots: one of gold, the other of silver, each of the same weight as the crown. Then he put them one by one in a vessel with water and noted how much its level had risen. Having lowered the crown into the vessel, Archimedes established that its volume exceeded the volume of the ingot.

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Archimedes tests and creates a theory of five mechanisms known in his time and called "simple mechanisms." These are a lever (“Give me a fulcrum,” said Archimedes, “and I will move the Earth”), a wedge, a block, an endless screw and a winch. The invention of the endless screw led him to the invention of a bolt, constructed from a screw and a nut.

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In 212 BC, during the defense of Syracuse against the Romans during the Second Punic War, Archimedes designed several war machines that allowed the townspeople to repel the attacks of the superior Romans for almost three years. One of them was a system of mirrors, with the help of which the Egyptians were able to burn the Roman fleet. Archimedes died during the siege of Syracuse: he was killed by a Roman soldier at a time when the scientist was absorbed in searching for a solution to the problem he had set himself. Archimedes screw - water pumping machine.