What was the first task the king gave to Archimedes? The famous problem of Hieron

There is a legend about how Archimedes came to the discovery that the buoyant force is equal to the weight of the liquid in the volume of the body. He reflected on the task given to him by the Syracusan king Hieron (250 BC).

King Hiero instructed him to check the honesty of the master who made the golden crown. Although the crown weighed as much as the gold that went into it, the king suspected that it was made of an alloy of gold with other, more cheap metals. Archimedes was instructed to find out, without breaking the crown, whether there was an impurity in it or not.

It is not known for certain what method Archimedes used, but we can assume the following: First, he found that a piece of pure gold was 19.3 times heavier than the same volume of water. In other words, the density of gold is 19.3 times greater than the density of water.

Archimedes had to find the density of the corona matter. If this density were the density of water is not 19.3 times, but a smaller number of times, which means the crown was not made of pure gold.

Weighing the crown was easy, but how to find its volume? This is what made it difficult for Archimedes, because the crown was of a very complex shape. This problem tormented Archimedes for many days. And then one day, when he, while in the bathhouse, plunged into a bathtub filled with water, he suddenly A thought struck me that provided a solution to the problem. Jubilant and excited by his discovery, Archimedes exclaimed; "Eureka! Eureka!”, which means; "Found! Found!".

Archimedes weighed the crown first in the air, then in the water. From the difference in weight, he 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. And knowing the density, answer the king’s question: are there any impurities of cheap metals in the golden crown?

Legend says that the density of the corona substance turned out to be less than the density of pure gold. Thus, the master was exposed as a deceiver, and science was enriched with a remarkable discovery. Historians say that the problem of the golden crown prompted Archimedes to study the question of the floating of bodies. The result of this was the appearance of the wonderful essay “On Floating Bodies,” which has come down to us.

The seventh sentence (theorem) of this work was formulated by Archimedes as follows:

Bodies that are heavier than a liquid, being lowered into it, are all immersed deeper until they reach the bottom, and, staying in the liquid, lose so much weight, how much the liquid weighs, taken in the volume of the bodies.

Ex. Assuming that the golden crown of King Hiero weighs 20 N in air and 18.75 N in water, calculate the density of the crown’s substance. Believing that there was gold only silver was mixed in, determine how much gold was in the crown and how much silver. When solving the problem, consider the density of gold to be rounded equal to 20,000 kg/m3, the density of silver - 10,000 kg/m3.

It is not for nothing that Greece is considered the cradle of Western culture, because it was on this blessed land, washed by the warm waves of the Mediterranean Sea, that brilliant scientists lived and worked. The list of names of people who laid the foundations of modern science could take more than one page. We will focus on one of them - mathematics, physics, engineer. A lot of information about his truly great mind has been preserved, and the legend of Archimedes is known to every schoolchild. We will tell you what kind of person he is and what all generations of people owe to him.

A little about genius

The legend of Archimedes is undoubtedly interesting. But first we want to tell you a little about the scientist himself. The biography of the famous Greek has come to us in the accounts of such ancient authors as Titus Livius, Vitruvius, Cicero, Polybius, Plutarch. Each of them lived much later than Archimedes, so it cannot be said that the events described by them are reliable.

The future genius was born in Syracuse, in Sicily. Perhaps Archimedes was a relative of the city's ruler, Hieron II. His father, Phidias, a famous astronomer and mathematician, instilled in him a passion for science. And he studied in Alexandria, the largest cultural and scientific center of that time.

Long before the legend of Archimedes appeared, the genius met outstanding people, Conon and Eratosthenes, with whom he corresponded throughout his life. He spent hours in the famous library, which contained over seven hundred thousand manuscripts. It was there that Archimedes had the opportunity to familiarize himself with the works of Democritus and Eudoxus, which he often mentioned later in his works.

Biographers claim that, after completing his studies, Archimedes returned to his hometown, where he was held in high esteem and did not need funds at all.

Scientist and crown

There is more than one legend about Archimedes, there are many of them, because the scientist constantly invented, researched, and created something. The most popular of them is familiar to us from school. This is the legend of Archimedes about the crown. Let's briefly tell its essence.

One day, the cruel king Hiero wanted to check whether the jeweler had deceived him when he made a golden crown for him. He ordered the scientist to determine whether his jewelry was really made of the purest precious metal. The difficulty was in determining the volume of the crown, since it had an irregular shape. Reflecting on the problem, Archimedes found a way to cope with it: immerse the product in water and measure the volume of liquid displaced by it. Then, as the legend about Archimedes tells, the genius exclaimed “Eureka!”, which translated means “found.” And this discovery entered the science of hydrostatics as

How to turn the Earth upside down?

But we also know another legend about Archimedes (photo below). Biographers say that the ruler of Syracuse ordered the construction of a heavy multi-deck ship, which was intended as a gift to Ptolemy, the Egyptian king. But there was no way to launch it into the water, and that’s where Archimedes came to the rescue. He built a whole system of blocks around the ship and, using the power of leverage, easily completed the task. It was then that the inventor’s aphorism was born: “Give me a point of support, and I will change the world.”

Saved by Syracuse

The scientist's amazing inventions saved his hometown from destruction. This is another legend about Archimedes (you probably studied it in physics). So, according to the biographers of the engineering genius, in 212 BC. e. Syracuse was besieged. At the time of the second, our hero was approximately 75 years old. But his mind was still quick and inquisitive.

So, Archimedes developed drawings of powerful throwing machines that threw stones at the troops of the commander Marcellus. Fleeing from such shelling, she rushed to the walls of Syracuse. But an unpleasant surprise also awaited them there - light throwing machines. In addition, the townspeople (probably not without the help of a scientist) built cranes that captured ships, lifted them up, and then threw them down and sank them. The invaders retreated.

Another version says that during the siege the fleet of the Eternal City was burned by fire that arose when mirrors or incendiary mixtures were used. However, if the previous legends have been verified by modern scientists and confirmed, then the fire from Syracuse is still considered a beautiful fairy tale.

End of life

As a result of treason, Syracuse was nevertheless captured by the Romans in the same year. Archimedes, who saved the city earlier, was killed. There are four versions of the scientist’s death, but they all boil down to the fact that the old man was hacked to death by soldiers. The military leader Marcellus was very upset when he learned of the death of a famous man, and gave him a decent funeral. The murderers were executed. Today in Syracuse you can see the stone tomb of Archimedes, built two centuries after his death. But the scientist continues to live in the hearts of people as the very past, as the savior of his hometown and a devoted servant of science.

The first who penetrated to the essence of thought, “into the dialectic of concept(s),” was the genius of Hegel.

The genius of Pythagoras is that he grasped the universal (the square ICOR, unity, the fusion of opposites, where ““contained together both immediacy and mediation”),” “TRANSITION from one to another, and That's the most important thing."

In order to more boldly enter the “realm of pure thought”, in order to more clearly feel the dramatic nature of the search for a solution, we will consider another specific Hamlet, borderline situation; the essence of the solution to the famous problem of Archimedes.

"The Legend of Archimedes"

There is a legend that Archimedes came to the discovery of the magnitude of the force that pushes a body out of liquid and gas, reflecting on the problem given to him by the king of Syracuse (250 BC).

King Hiero instructed him to check the honesty of the master who made the golden crown. Although the crown weighed as much as the gold that was put on it, the king suspected that it was made from an alloy of gold with other, cheaper metals. Archimedes was instructed to find out, without breaking the crown, whether there was an impurity in it or not.

It is not known for certain what method Archimedes used (dialectical!! Author), but we can assume the following. First, he found that a piece of pure gold was 19.3 times heavier than the same volume of water. In other words, the density of gold is 19.3 times greater than the density of water.

Archimedes had to find the density of the corona matter. If this density turned out to be greater than the density of water not by 19.3 times, but by a smaller number of times, it means that the crown was not made of pure gold.

Weighing the crown was easy, but how to find its volume? This is what made it difficult for Archimedes, because the crown was of a very complex shape.

This problem tormented Archimedes for many days. And then one day, while in the bathhouse, he immersed himself in a bathtub filled with water, a thought suddenly struck him that provided a solution to the problem.

Jubilant and excited by his discovery, Archimedes exclaimed: “Eureka! Eureka!”, which means “Found! Found!”

Archimedes weighed the crown first in the air, then in the water. From the difference in weight, he determined 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 determine its density. And knowing the density, answer the king’s question: are there any impurities of cheap metals in the golden crown?

Legend says that the density of the corona substance turned out to be less than the density of pure gold. Thus, the master was exposed as a deceiver, and science was enriched with a remarkable discovery.

Historians say that the problem of the golden crown prompted Archimedes to study the question of the floating of bodies. The result of this was the appearance of the wonderful essay “On Floating Bodies,” which has come down to us.

The seventh sentence (theorem) of this work was formulated by Archimedes as follows:

“Bodies that are heavier than a liquid, being lowered into it, sink deeper and deeper until they reach the bottom, and, while in the liquid, lose as much weight as the liquid weighs taken in the volume of the bodies.”

“At first he (Archimedes. Auth.) found that a piece of pure gold is 19.3 times heavier than the same volume of water.”

Where did the physicist get this water?

From there, where does the mathematician get the equality of the squares M"K"O"R" and MCOR in the proof of the Pythagorean theorem.

Archimedes needed to “find out, without breaking the crown, whether there is an admixture in it or not.”

Nothing more is given to him.

“To find out whether there is an impurity in it (the crown) or not” is an easy task. Take the crown directly and melt it, and then compare the weight of the volume of the molten crown with an equal volume of pure gold.

"Without breaking the crown"!!

But “there is a contradiction”!!

So it is categorically “impossible” (!!) to allow contradictions. A condition that carries a contradiction is unsolvable. It is impossible to solve such a problem, “it is illegal already because it excludes any possibility of moving (“and this is the most important thing.” Author) from the first to the second. An abyss is formed between them , which cannot be filled with anything."

"Aristotle answers: (Archimedes will allow. Author) if he is allowed to "cross the border."

And Hegel: “This answer is correct, contains everything.”

And who will allow it?

So, Archimedes faced opposites: to melt and at the same time not to melt. “In this case, a contradiction is revealed that requires resolution.” “Knowledge is the eternal, infinite approach of thinking to an object. The reflection of nature in human thought must be understood not “dead”, not “abstract”, not motionless, not without contradictions, but in the eternal process of movement, the emergence of contradictions and their resolution."

How to melt the crown at the same time without melting it, i.e. preserving it!!?

This is what “Archimedes tormented for many days”!

".So that the same thing at the same time should be and not be" !!

"If there is a contradiction, it is obvious that one and the same person cannot at the same time consider the same thing to exist and not to exist."

"The ordinary idea grasps difference and contradiction, but does not transition from one to another, and this is the most important thing."

First of all, Archimedes dives into the question. He drowns in it, is absorbed by it. The question tortures him, tears him apart.

“The connecting thread broke for days.

How can I connect their fragments!”

("Hamlet". W. Shakespeare.)

“Wit grasps a contradiction, expresses it, brings things into relation to each other, makes “the concept shine through the contradiction,” but does not express the concept of things and their relationships.”

While immersing his body in the bath, Archimedes suddenly saw more water appearing in the bath out of nothing.

His body was melting, dissolving before our eyes, turning into liquid, water!!

“A thought suddenly struck him that provided a solution to the problem.”

"The thinking mind (mind) sharpens the dull distinction of the different, the simple diversity of ideas, to a significant difference, to the opposite. Only contradictions and diversity raised to the top become mobile (regsam) and living in relation to one another - acquire that negativity that is in INTERNAL PULSATION OF MOVEMENT AND VITALITY."

Reason is death at the same time immortality; the essence of sacrifice is at the same time salvation; the essence of salvation is somersault through death (to be saved is to come out of (from) the mouth); the essence of the idea.

The architect Vitruvius spoke about a problem solved two hundred years earlier by the physicist Archimedes. Since then, this story has been retold countless times, and the problem itself, solved by Archimedes, has become one of the most famous historical problems.

Scientific research, says Vitruvius, absorbed Archimedes to such an extent that he had to be reminded about sleep and food. Even in the bath, while rubbing, he continued to draw geometric shapes in the sand. One day while bathing, Vitruvius continues, Archimedes was thinking about the difficult task set before him by King Hieron.

As you know, this king wished to bring a golden crown as a gift to the temple. He entrusted the work to one jeweler, giving him the appropriate amount of gold. Soon the work was completed, but there were rumors that the master had replaced some of the gold with silver. Archimedes, whom the king instructed to investigate this matter, thought for a long time about resolving the issue. It appeared suddenly while he was sitting in the bath. Overjoyed, Archimedes jumped out of the bath and ran through the streets of Syracuse, repeating: “Eureka!” (Found!).

This is exactly how, according to Vitruvius, Archimedes discovered the most important law of hydrostatics. We invite readers to understand exactly how Archimedes applied this law to the solution of the problem set before him by Hiero. It should be taken into account that for this purpose Archimedes' law can be used in two ways. While you are looking for them, we will continue the story about the famous historical problem.

LUCKY FIND

Two thousand years after Vitruvius spoke about the discovery of Archimedes, the Greek scientist Kerameus discovered in the monastery of St. Sava, near Jerusalem, a palimpsest - a parchment from which the original text had been removed in order to make a new entry on it. Parchment was very expensive in the Middle Ages, and monastic chroniclers and scribes mercilessly washed away and erased ancient writings. But this time, extraordinary luck awaited the scientists.

The ancient text, which turned out to be a set of works by Archimedes, was not erased, but only washed away. In 1906, Professor Heiberg managed to read it, and several of the works of Archimedes, which we had previously known only from references and passages in the works of ancient scientists, were read from beginning to end. Among the newly discovered texts of Archimedes was his essay “On Floating Bodies,” which sets out the conclusion of “Archimedes’ law.” There were no references to Hiero’s problem and the incident in the public baths in this work.

"STUPID FABLE" ABOUT ARCHIMEDES

Academician A. N. Krylov in his “Essay on the Development of the Theory of the Ship” examined in detail the content of the newly discovered work of Archimedes.

“This work of Archimedes,” he wrote, “consists of two books or chapters, the first of which contains two main provisions, or postulates, and nine provisions, of which seven establish the general doctrine of floating bodies...” Having outlined the main provisions of Archimedes and shown However, how complicated the path of his reasoning was, Academician Krylov notes: “We must remember that all geometric concepts, starting from the area of ​​a circle, the area of ​​a parabola, the volume of a cylinder, a sphere, a spherical segment, the doctrine of the center of gravity of bodies, about their equilibrium - all this created by Archimedes himself; then only a small idea will appear of the extraordinary power of his genius and the absurdity of the fable repeated by historians that Archimedes, sitting in a bathtub in public baths, found his law ... "

Thus, the study of the accidentally found work of Archimedes dispelled the legend that the discovery of an important law of nature was made as a result of sudden insight. But this does not mean that everything in the legend of the Crown of Hiero is fictitious. It is likely that 2200 years ago it was for this reason that the theoretically derived Archimedes law was first applied in practice. It is interesting that the next case of deliberate application of this law dates back to 1666.

This year, an extraordinary event occurred in one of the English coastal towns. When the king became aware of him, he hurried with his retinue to the shipyards of this city, where warships were being built. And this is what he saw here.

On the shore stood a frigate, ready to be launched, with “ports” gaping in its sides - holes for gun barrels. The command to begin lowering the vessel was expected any minute.
- What kind of wild innovation? - one of those present exclaimed. - Now there will be a disaster! Who knows how deep the ship will sink into the water? What if water rushes into all the holes in the sides?

In fact, since time immemorial, shipbuilders have been making holes for gun barrels after the finished and equipped ship was on the water. But shipbuilder Anton Dean, based on Archimedes’ law, calculated in advance to what level the ship would sink and where a “port” for cannons should be made on its sides.

Having become inspector of shipbuilding of the English fleet in 1684, Dean ordered in all cases to weigh in advance the parts of the hull of ships, as well as all cargo included in their equipment, supplies, military weapons, etc. Since then, the law discovered by Archimedes has been more than two thousand years back, underlies the theory of the buoyancy of ships.

P.S. Ancient chronicles tell: in general, Archimedes owned many different ingenious inventions. Even modern testing using a lie detector is rooted in Archimedean observations that a person’s pulse quickens when excited. By the way, all modern lie detectors work on this principle: when a person tells a lie, he gets excited, this excitement gives rise to an increased pulse, which the detector actually records.

"Archimedes 1" - "Celestial Sphere" by Archimedes. Archimedes screw. Equation in polar coordinates: r = a?f, where a is a constant. Biography. Legends about death. In his treatise “On Leverage,” Archimedes established the RULE OF LEVER EQUILIBRIUM. The Legend of the Crown. Truncated tetrahedron. The angry Roman drew his sword and killed Archimedes. I have to solve the problem!

“Scientists - mathematicians” - Mathematical names. Shawl Michel (1793–1880), French mathematician. Euler Leonhard (1707-1783), Swedish mathematician. Riemann Bernhard (1826-1866), German mathematician. Jacobi polynomials, Jacobi determinant - Jacobian. Geometry of Lobachevsky. A Möbius strip is a surface that has only one side. Cartesian coordinates.

"Mathematics and natural sciences" - Thermal phenomena. Man complements nature. Chemical phenomena. The structure of the atom. Electromagnetic field. Aristotle. Arithmetic. Mechanical vibrations. Diversity of living organisms. Sound. The structure of a living organism. Work, power, energy. The principle of interpenetration and mutual assistance. The Book of Nature is written in the language of mathematics.

“Great mathematicians” - The coordinate system proposed by Descartes received his name. Euclid. Archimedean spiral. Leibniz Gottfried Wilhelm. Carl Friedrich Gauss. Pythagoras of Samos. Keldysh Mstislav Vsevolodovich. Kovalevskaya Sofya Vasilievna. Great mathematicians. Gauss was the only son of poor parents. Archimedes. "Method" (or "Ephod") and "Regular Heptagon".

"Archimedes' Law" - Archimedes' Screw. Submarines. Hydrostatic weighing. Ships. Divers. Archimedes' law. Swimming tel. ARCHIMEDES (287 BC – 212 BC). “Here is the crown, Archimedes, is it gold or not?” Planes, helicopters. Archimedes' bestseller in modern scientific research. The sage Archimedes lived in Syracuse...

“Mathematics as a science” - Sobolev was born on October 22, 1793 in the Nizhny Novgorod province. Sobolev Sergey Lvovich. Numerator. On the history of mathematics. Lyubachevsky is a professor at Moscow University and the Imperial Technical School. Competition "Counting machine". Triangle. Alexandrov's parents were school teachers. Zhukovsky Nikolai Egorovich.

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