What was the first task the king gave to Archimedes? Archimedes' inventions

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, when lowered into it, all sink 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.

Nick. Gorkavy

Other scientific tales Nick. Gorkavoy were published in the journal “Science and Life” in 2010-2013.

Domenico Fetti. Archimedes is thinking. 1620 Painting from the Old Masters Gallery, Dresden.

Edward Vimon. Death of Archimedes. 1820s.

Tomb of Archimedes in Syracuse. Photo: Codas2.

Ortigia Island, historical Center Syracuse, hometown of Archimedes. On these shores, Archimedes burned and sank Roman galleys. Photo: Marcos90.

Greek Theater in Syracuse. Photo: Victoria|photographer_location_London, UK.

Archimedes turns the Earth over with a lever. Ancient engraving. 1824

The image of Archimedes on the Fields Gold Medal, the highest honor given to mathematicians. The inscription in Latin: “Transire suum pectus mundoque potiri” - “To transcend your human limitations and conquer the Universe.” Photo by Stefan Zachov.

Each new tale of the writer and astrophysicist, doctor of physical and mathematical sciences Nikolai Nikolaevich Gorkavy (Nick. Gorkavy) is a story about how important discoveries were made in one or another field of science. And it is no coincidence that the heroes of his popular science novels and fairy tales were Princess Dzintara and her children - Galatea and Andrei, because they are from the breed of those who strive to “know everything”. The stories told by Dzintara to children were included in the collection “Star Vitamin”. It turned out to be so interesting that readers demanded a continuation. We invite you to familiarize yourself with some fairy tales from the future collection “The Makers of Times.” Here is the first publication.

Greatest Scientist ancient world the ancient Greek mathematician, physicist and engineer Archimedes (287-212 BC) was from Syracuse, a Greek colony on the largest island of the Mediterranean - Sicily. Ancient Greeks, creators European culture, settled there almost three thousand years ago - in the 8th century BC, and by the time of the birth of Archimedes, Syracuse was prosperous cultural city, where their philosophers and scientists, poets and speakers lived.

The stone houses of the townspeople surrounded the palace of King Hieron II of Syracuse, high walls defended the city from enemies. Residents loved to gather in stadiums, where runners and discus throwers competed, and in bathhouses, where they not only washed, but relaxed and exchanged news.

That day, the baths on the main square of the city were noisy - laughter, screams, splashing water. Young people swam in a large pool, and older people, holding silver goblets of wine in their hands, had a leisurely conversation on comfortable couches. The sun peeked into the courtyard of the baths, illuminating the doorway leading to a separate room. In it, in a small pool that looked like a bathtub, sat alone a man who behaved completely differently from the others. Archimedes - and it was he - closed his eyes, but by some elusive signs it was clear that this man was not sleeping, but was thinking intensely. In recent weeks, the scientist became so deep in his thoughts that he often forgot even about food and his family had to make sure that he did not go hungry.

It began with the fact that King Hieron II invited Archimedes to his palace, poured him the best wine, asked about his health, and then showed him a golden crown made for the ruler by the court jeweler.

“I don’t know much about jewelry, but I do know about people,” Hieron said. - And I think that the jeweler is deceiving me.

The king took a gold bar from the table.

I gave him the exact same ingot and he made a crown out of it. The weight of the crown and the ingot are the same, my servant checked this. But I still have doubts: is there silver mixed into the crown? You, Archimedes, are the greatest scientist of Syracuse, and I ask you to check this, because if the king puts on a false crown, even the street boys will laugh at him...

The ruler handed the crown and ingot to Archimedes with the words:

If you answer my question, you will keep the gold for yourself, but I will still be your debtor.

Archimedes took the crown and the gold bar and left royal palace and since then I have lost peace and sleep. If he can’t solve this problem, then no one can either. Indeed, Archimedes was the most famous scientist of Syracuse, studied in Alexandria, was friends with the head of the Library of Alexandria, mathematician, astronomer and geographer Eratosthenes and other great thinkers of Greece. Archimedes became famous for his many discoveries in mathematics and geometry, laid the foundations of mechanics, and was responsible for several outstanding inventions.

The puzzled scientist came home, put the crown and the ingot on the scales, lifted them by the middle and made sure that the weight of both objects was the same: the bowls swayed at the same level. Archimedes knew the density of pure gold; he had to find out the density of the crown (weight divided by volume). If there is silver in the crown, its density should be less than that of gold. And since the weights of the crown and the ingot are the same, then the volume of the false crown should be greater than the volume of the gold ingot. The volume of the ingot can be measured, but how can one determine the volume of the crown, which has so many complexly shaped teeth and petals? This problem tormented the scientist. He was an excellent geometer, for example, he decided difficult task- determination of the area and volume of a sphere and a cylinder circumscribed around it, but how to find the volume of a body of complex shape? A fundamentally new solution is needed.

Archimedes came to the bathhouse to wash off the dust of a hot day and refresh his head, tired from thinking. Ordinary people while bathing in the bathhouse, they could chat and chew figs, and Archimedes’ thoughts about the unsolved problem did not leave him either day or night. His brain searched for a solution, clinging to any clue.

Archimedes took off his chiton, put it on the bench and walked up to the small pool. Water splashed in it three fingers below the edge. When the scientist plunged into the water, its level rose noticeably, and the first wave even splashed onto the marble floor. The scientist closed his eyes, enjoying the pleasant coolness. Thoughts about the volume of the crown habitually swirled in my head.

Suddenly Archimedes felt that something important had happened, but could not understand what. He opened his eyes in annoyance. From the outside large swimming pool Voices and someone's heated argument could be heard - it seemed about the last law of the ruler of Syracuse. Archimedes froze, trying to understand what had happened? He looked around: the water in the pool did not reach the edge by only one finger, and yet when he entered the water, its level was lower.

Archimedes stood up and left the pool. When the water calmed down, she was again three fingers below the edge. The scientist climbed into the pool again - the water obediently rose. Archimedes quickly estimated the size of the pool, calculated its area, then multiplied it by the change in water level. It turned out that the volume of water displaced by its body is equal to the volume of the body, if we assume that the densities of water and human body almost identical and each cubic decimeter, or a cube of water with a side of ten centimeters, can be equated to a kilogram of the weight of the scientist himself. But during the dive, Archimedes’ body lost weight and floated in the water. In some mysterious way, the water displaced by the body took away his weight...

Archimedes realized that he was on the right path, and inspiration carried him on its mighty wings. Is it possible to apply the found law on the volume of displaced fluid to the crown? Certainly! You need to lower the crown into water, measure the increase in the volume of liquid, and then compare it with the volume of water displaced by the gold bar. Problem solved!

According to legend, Archimedes, with a victorious cry of “Eureka!”, which means “Found!” in Greek, jumped out of the pool and, forgetting to put on his chiton, rushed home. I urgently needed to check my decision! He ran through the city, and the residents of Syracuse waved their hands at him in greeting. Still, it’s not every day that the most important law of hydrostatics is discovered, and it’s not every day that you can see a naked man running through the central square of Syracuse.

The next day the king was informed about the arrival of Archimedes.

“I solved the problem,” said the scientist. - There really is a lot of silver in the crown.

How did you know this? - the ruler asked.

Yesterday, in the baths, I guessed that a body that is immersed in a pool of water displaces a volume of liquid equal to the volume of the body itself, and at the same time loses weight. Returning home, I conducted many experiments with scales immersed in water, and proved that a body in water loses exactly as much weight as the liquid it displaces weighs. Therefore, a person can swim, but a gold bar cannot, but it still weighs less in water.

And how does this prove the presence of silver in my crown? - asked the king.

“Tell me to bring a vat of water,” Archimedes asked and took out the scales. While the servants were dragging the vat to the royal chambers, Archimedes put the crown and the ingot on the scales. They balanced each other.

If there is silver in the crown, then the volume of the crown is greater than the volume of the ingot. This means that when immersed in water, the crown will lose more weight and the scales will change their position,” said Archimedes and carefully immersed both scales in the water. The bowl with the crown immediately rose up.

You are truly a great scientist! - exclaimed the king. - Now I can order a new crown for myself and check whether it is real or not.

Archimedes hid a grin in his beard: he understood that the law he had discovered the day before was much more valuable than a thousand golden crowns.

Archimedes' law has remained in history forever; it is used when designing any ships. Hundreds of thousands of ships ply the oceans, seas and rivers, and each of them floats on the surface of the water thanks to the force discovered by Archimedes.

When Archimedes grew old, his measured studies in science suddenly ended, as did quiet life townspeople, the rapidly growing Roman Empire decided to conquer the fertile island of Sicily.

In 212 BC. a huge fleet of galleys filled with Roman soldiers approached the island. The advantage in strength of the Romans was obvious, and the commander of the fleet had no doubt that Syracuse would be captured very quickly. But that was not the case: as soon as the galleys approached the city, powerful catapults struck from the walls. They threw heavy stones so accurately that the invaders' galleys were shattered into splinters.

The Roman commander was not at a loss and commanded the captains of his fleet:

Come to the very walls of the city! At close range, catapults will not be afraid of us, and archers will be able to shoot accurately.

When the fleet, with losses, broke through to the city walls and prepared to storm it, a new surprise awaited the Romans: now light throwing vehicles pelted them with a hail of cannonballs. The lowering hooks of powerful cranes grabbed the Roman galleys by the bows and lifted them into the air. The galleys overturned, fell down and sank.

The famous ancient historian Polybius wrote about the assault on Syracuse: “The Romans could quickly take possession of the city if someone had removed one old man from among the Syracusans.” This old man was Archimedes, who designed throwing machines and powerful cranes to protect the city.

The quick capture of Syracuse failed, and the Roman commander gave the command to retreat. The greatly reduced fleet retreated to a safe distance. The city held firm thanks to the engineering genius of Archimedes and the courage of the townspeople. The scouts reported to the Roman commander the name of the scientist who created such an impregnable defense. The commander decided that after the victory he needed to get Archimedes as the most valuable military trophy, because he alone was worth an entire army!

Day after day, month after month, men stood guard on the walls, shot with bows and loaded catapults with heavy stones, which, alas, did not reach their target. The boys brought water and food to the soldiers, but they were not allowed to fight - they were still too young!

Archimedes was old, he, like children, could not shoot from a bow as far as young and strong men, but he had a powerful brain. Archimedes gathered the boys and asked them, pointing to the enemy galleys:

Want to destroy the Roman fleet?

We are ready, tell us what to do!

The wise old man explained that he would have to work hard. He ordered each boy to take a large copper sheet from the already prepared pile and place it on smooth stone slabs.

Each of you must polish the sheet so that it shines in the sun like gold. And then tomorrow I will show you how to sink Roman galleys. Work, friends! The better you polish the copper today, the easier it will be for us to fight tomorrow.

Will we fight ourselves? - asked the little curly boy.

Yes,” Archimedes said firmly, “tomorrow you will all be on the battlefield along with the soldiers.” Each of you will be able to accomplish a feat, and then legends and songs will be written about you.

It is difficult to describe the enthusiasm that gripped the boys after Archimedes' speech, and they energetically began polishing their copper sheets.

The next day, at noon, the sun burned scorchingly in the sky, and the Roman fleet stood motionless at anchor in the outer roadstead. The wooden sides of the enemy galleys heated up in the sun and oozed resin, which was used to protect the ships from leaks.

Dozens of teenagers gathered on the fortress walls of Syracuse, where enemy arrows could not reach. In front of each of them stood a wooden shield with a polished copper sheet. The shield supports were made so that the copper sheet could be easily turned and tilted.

“Now we’ll check how well you polished the copper,” Archimedes addressed them. - I hope everyone knows how to make sunbeams?

Archimedes approached the little curly-haired boy and said:

Catch the sun with your mirror and direct the sunbeam into the middle of the side of the large black galley, just under the mast.

The boy rushed to carry out the instructions, and the warriors crowded on the walls looked at each other in surprise: what else was the cunning Archimedes up to?

The scientist was pleased with the result - a spot of light appeared on the side of the black galley. Then he turned to the other teenagers:

Point your mirrors at the same place!

Wooden supports creaked, copper sheets rattled - a flock of sunbeams ran towards the black galley, and its side began to fill with bright light. The Romans poured onto the decks of the galleys - what was happening? The commander-in-chief came out and also stared at the sparkling mirrors on the walls of the besieged city. Gods of Olympus, what else did these stubborn Syracusans come up with?

Archimedes instructed his army:

Keep your eyes on the sunbeams - let them always be directed to one place.

Not even a minute had passed before smoke began to billow from a shining spot on board the black galley.

Water, water! - the Romans shouted. Someone rushed to draw sea water, but the smoke quickly gave way to flames. The dry, tarred wood burned beautifully!

Move the mirrors to the adjacent galley on the right! - Archimedes commanded.

In a matter of minutes, the neighboring galley also began to fire. The Roman naval commander came out of his stupor and ordered to weigh anchor in order to move away from the walls of the cursed city with its main defender Archimedes.

Unfastening the anchors, putting the rowers on the oars, turning the huge ships around and taking them out to sea at a safe distance is not a quick task. While the Romans were running hecticly along the decks, choking from the choking smoke, the young Syracusans were transferring mirrors to new ships. In the confusion, the galleys came so close to each other that the fire spread from one ship to another. In their haste to set sail, some ships unfurled their sails, which, as it turned out, burned no worse than the tar sides.

Soon the battle was over. Many Roman ships burned out in the roadstead, and the remnants of the fleet retreated from the city walls. There were no losses among the young army of Archimedes.

Glory to the great Archimedes! - the delighted residents of Syracuse shouted and thanked and hugged their children. A mighty warrior in shining armor firmly shook the curly-haired boy's hand. His small palm was covered with bloody calluses and abrasions from polishing the copper sheet, but he did not even wince when shaking hands.

Well done! - the warrior said respectfully. “The people of Syracuse will remember this day for a long time.”

Two millennia passed, but this day remained in history, and not only the Syracusans remembered it. Residents different countries They know the amazing story of Archimedes burning the Roman galleys, but he alone would not have done anything without his young assistants. By the way, quite recently, already in the twentieth century AD, scientists conducted experiments that confirmed the full functionality of the ancient “superweapon” invented by Archimedes to protect Syracuse from invaders. Although there are historians who consider this a legend...

Oh, it's a pity I wasn't there! - exclaimed Galatea, who was listening attentively with her brother to the evening fairy tale that their mother, Princess Dzintara, was telling them. She continued reading the book:

Having lost hope of capturing the city by force of arms, the Roman commander resorted to the old tried and tested method - bribery. He found traitors in the city, and Syracuse fell. The Romans stormed into the city.

Find me Archimedes! - ordered the commander. But the soldiers, intoxicated by victory, did not understand well what he wanted from them. They broke into houses, robbed and killed. One of the warriors ran out to the square where Archimedes was working, drawing a complex geometric figure in the sand. Soldiers' boots trampled the fragile drawing.

Don't touch my drawings! - Archimedes said menacingly.

The Roman did not recognize the scientist and struck him with a sword in anger. This is how this great man died.

Archimedes' fame was so great that his books were often rewritten, thanks to which a number of works have survived to this day, despite the fires and wars of two millennia. The history of the books of Archimedes that have come down to us was often dramatic. It is known that in the 13th century, some ignorant monk took the book of Archimedes, written on durable parchment, and washed away the formulas of the great scientist in order to get blank pages to write down prayers. Centuries passed, and this prayer book fell into the hands of other scientists. Using a strong magnifying glass, they examined its pages and discerned traces of the erased precious text of Archimedes. The book of the brilliant scientist was restored and printed in large quantities. Now it will never disappear.

Archimedes was a real genius who made many discoveries and inventions. He was ahead of his contemporaries not even by centuries - by millennia.

In the book “Psammitus, or Calculus of Grains of Sand,” Archimedes retold the bold theory of Aristarchus of Samos, according to which the great Sun is located in the center of the world. Archimedes wrote: “Aristarchus of Samos... believes that the fixed stars and the Sun do not change their place in space, that the Earth moves in a circle around the Sun, located at its center...” Archimedes considered the heliocentric theory of Samos convincing and used it to estimate the size spheres of fixed stars. The scientist even built a planetarium, or “celestial sphere,” where one could observe the movement of the five planets, the rising of the sun and moon, its phases and eclipses.

The rule of leverage, which Archimedes discovered, became the basis of all mechanics. And although the lever was known before Archimedes, he outlined its complete theory and successfully applied it in practice. In Syracuse, he single-handedly launched the new multi-deck ship of the king of Syracuse, using an ingenious system of blocks and levers. It was then, appreciating the full power of his invention, that Archimedes exclaimed: “Give me a fulcrum, and I will turn the world around.”

Archimedes' achievements in the field of mathematics, which, according to Plutarch, he was simply obsessed with, are invaluable. His main mathematical discoveries relate to mathematical analysis, where the scientist’s ideas formed the basis of integral and differential calculus. The ratio of the circumference of a circle to its diameter, calculated by Archimedes, was of great importance for the development of mathematics. Archimedes gave an approximation for the number π (Archimedean number):

The scientist considered his highest achievement to be his work in the field of geometry and, above all, the calculation of a ball inscribed in a cylinder.

What kind of cylinder and ball? - asked Galatea. - Why was he so proud of them?

Archimedes was able to show that the area and volume of a sphere are related to the area and volume of the described cylinder as 2:3.

Dzintara rose and removed from the shelf a model of the globe, which was soldered inside a transparent cylinder so that it was in contact with it at the poles and at the equator.

I have loved this geometric toy since childhood. Look, the area of ​​the ball is equal to the area of ​​four circles of the same radius or the area of ​​the side of a transparent cylinder. If you add the areas of the base and top of the cylinder, it turns out that the area of ​​the cylinder is one and a half times the area of ​​the ball inside it. The same relationship holds for the volumes of a cylinder and a sphere.

Archimedes was delighted with the result. He knew how to appreciate the beauty of geometric shapes and mathematical formulas- that is why it is not a catapult or a burning galley that adorns his grave, but an image of a ball inscribed in a cylinder. Such was the desire of the great scientist.

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 fact that he addressed messages about his discoveries to mathematicians associated with Alexandria, for example 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.

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 circumscribed 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 looking out from the thorny bushes and on it - a ball and a cylinder.

Legends about Archimedes.

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 Lives of Marcellus. 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 easily and smoothly, as if on water, “float” towards him.” 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. “Thinking about this problem, Archimedes once went into a bathhouse and there, plunging into the bath, he noticed that the amount of water overflowing was equal to the amount of water displaced by his body. This observation prompted Archimedes to solve the problem of the crown, and he, without hesitating a second, jumped out of the bath and, as if he were naked, rushed home, shouting at the top of his voice about his discovery: “Eureka! Eureka!" (Greek: “Found! Found!”).”

More reliable is the testimony of Pappus that Archimedes owned the work About production[heavenly]spheres, which was probably about building 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.

Mathematical works.

The surviving mathematical works of Archimedes can be divided into three groups. The works of the first group are devoted mainly to the proof of theorems on the areas and volumes of curvilinear figures or bodies. This includes treatises About the ball and cylinder, About measuring a circle, About conoids and spheroids, About spirals And About squaring a parabola. The second group consists of works on geometric analysis of static and hydrostatic problems: On the equilibrium of plane figures, About floating bodies. The third group includes various mathematical works: On the method of mechanical proof of theorems, Calculus of grains of sand, Bull problem and preserved only in fragments Stomachion. There is one more job - A book about assumptions(or Book of Lemmas), preserved only in Arabic translation. Although it is attributed to Archimedes, in its current form it is clearly by another author (since there are references to Archimedes in the text), but it may provide evidence that goes back to Archimedes. Several other works attributed to Archimedes by ancient Greek and Arab mathematicians have been lost.

The works that have reached us have not retained their original form. So, apparently, Book I of the treatise On the equilibrium of plane figures is an excerpt from a larger work Mechanical elements; in addition, it differs markedly from Book II, which was clearly written later. Proof mentioned by Archimedes in his essay About the ball and cylinder, was lost by the 2nd century. AD Job About measuring a circle is very different from the original version, and sentence II in it is most likely borrowed from another work. Title About squaring a parabola It could hardly have belonged to Archimedes himself, since in his time the word “parabola” was not yet used as the name of one of the conic sections. Texts of such works as About the ball and cylinder And About measuring a circle, most likely, were subject to changes in the process of translation from Dorian-Sicilian to the Attic dialect.

In proving theorems about the areas of figures and the volumes of bodies bounded by curved lines or surfaces, Archimedes constantly uses a method known as the “method of exhaustion.” It was probably invented by Eudoxus (heyday of activity around 370 BC) - at least, that’s what Archimedes himself thought. Euclid also resorts to this method from time to time in Book XII. Began. Proof by exhaustion is essentially an indirect proof by contradiction. In other words, the statement “A is equal to B” is considered true when accepting the opposite statement, “A is not equal to B,” leads to a contradiction. The main idea of ​​the exhaustion method is that the correct figures are inscribed (or described around it, or inscribed and described at the same time) into the figure whose area or volume needs to be found. The area or volume of inscribed or described figures is increased or decreased until the difference between the area or volume to be found and the area or volume of the inscribed figure becomes less than a given value. Using various variants of the exhaustion method, Archimedes was able to prove various theorems that are equivalent in modern notation to the relations S = 4p r 2 for the surface area of ​​the ball, V = 4/3p r 3 for its volume, the theorem that the area of ​​a parabola segment is equal to 4/3 the area of ​​a triangle having the same base and height as the segment, as well as many other interesting theorems.

It is clear that, using the method of exhaustion (which is more a method of proof rather than discovery of new relations), Archimedes must have had some other method that would allow him to find the formulas that constitute the content of the theorems he proved. One of the methods for finding formulas is revealed in his treatise On the mechanical method of proving theorems. The treatise sets out mechanical method, in which Archimedes mentally balanced geometric figures, as if lying on the scales. Having balanced a figure with an unknown area or volume with a figure with a known area or volume, Archimedes noted the relative distances from the centers of gravity of these two figures to the point of suspension of the balance beam and, according to the law of the lever, found the required area or volume, expressing them respectively through the area or volume of the known figure. One of the basic assumptions used in the exhaustion method is that the area is considered as the sum of an extremely large set of “material” lines that are closely adjacent to each other, and the volume is considered as the sum of plane sections that are also closely adjacent to each other. Archimedes believed that his mechanical method did not have demonstrative value, but allowed one to obtain a preliminary result, which could subsequently be proven by more rigorous geometric methods.

Although Archimedes was primarily a geometer, he made a number of interesting excursions into the field of numerical calculations, even if the methods he used were not entirely clear. In sentence III of the essay About measuring a circle he established that the number p is less than and greater than . It is clear from the proof that he had an algorithm for obtaining approximate values square roots from large numbers. It is interesting to note that he also gives an approximate estimate of the number , namely: . In the work known as Calculus of grains of sand, Archimedes sets out an original system for representing large numbers, which allowed him to write down the number , where itself R equals . He needed this system to calculate how many grains of sand it would take to fill the Universe.

At work About the spiral Archimedes investigated the properties of the so-called. Archimedean spiral, wrote down the characteristic property of the points of the spiral in polar coordinates, gave the construction of a tangent to this spiral, and also determined its area.

In the history of physics, Archimedes is known as one of the founders of the successful application of geometry to statics and hydrostatics. In Book I of the essay On the equilibrium of plane figures he gives a purely geometric derivation of the law of the lever. In essence, his proof is based on reducing the general case of a lever with arms inversely proportional to the forces applied to them, to the special case of an equal-arm lever and equal forces. The entire proof from beginning to end is permeated with the idea of ​​geometric symmetry.

In his essay About floating bodies Archimedes applied a similar method to solving problems of hydrostatics. Based on two assumptions formulated in geometric language, Archimedes proves theorems (propositions) regarding the size of the immersed part of bodies and the weight of bodies in a liquid with both higher and lower density than the body itself. In sentence VII, which talks about bodies denser than liquid, the so-called Archimedes' law, according to which “any body immersed in a liquid loses, in comparison with its weight in the air, as much as the weight of the liquid displaced by it.” Book II contains subtle considerations regarding the stability of the floating segments of a paraboloid.

The influence of Archimedes.

Unlike Euclid, Archimedes was remembered in antiquity only occasionally. If we know anything about his works, it is only thanks to the interest that they had in Constantinople in the 6th–9th centuries. Euthokyus, a mathematician born in the late 5th century, commented on at least three of Archimedes' works, apparently the most famous at the time: About the ball and cylinder, About measuring a circle And On the equilibrium of plane figures. The works of Archimedes and the commentaries of Eutocias were studied and taught by the mathematicians Anthimius of Trallus and Isidore of Miletus, the architects of the Cathedral of St. Sophia, erected in Constantinople during the reign of Emperor Justinian. The reform of teaching mathematics, which was carried out in Constantinople in the 9th century. Leo of Thessaloniki appears to have contributed to the collection of Archimedes' works. Then he became known to Muslim mathematicians. We now see that the Arab authors lacked some of the most important works Archimedes, such as About squaring a parabola, About spirals, About conoids and spheroids, Calculus of grains of sand And About the method. But in general, the Arabs mastered the methods outlined in other works of Archimedes, and often used them brilliantly.

Medieval Latin-speaking scholars first heard about Archimedes in the 12th century, when two translations of his work appeared from Arabic into Latin About measuring a circle. The best translation was that of the famous translator Gerard of Cremona, and it served as the basis for many expositions and expanded versions over the next three centuries. Gerard also owned a translation of the treatise Words of the sons of Moses Arabic mathematician of the 9th century. Banu Musa, which cited theorems from the work of Archimedes About the ball and cylinder with a proof similar to that given by Archimedes. At the beginning of the 13th century. John de Tinemuet translated the work About curved surfaces, which shows that the author was familiar with another work of Archimedes - About the ball and cylinder. In 1269, the Dominican William of Moerbeke translated from ancient Greek the entire corpus of Archimedes' works, except Sand grain calculus, Method and short essays Bull problem And Stomachion. For the translation, William of Moerbeke used two of the three Byzantine manuscripts known to us (manuscripts A and B). We can trace the history of all three. The first of these (Manuscript A), the source of all copies made during the Renaissance, appears to have been lost around 1544. The second manuscript (Manuscript B), containing Archimedes' works on mechanics, including the About floating bodies, disappeared in the 14th century. No copies were made of it. The third manuscript (manuscript C) was not known until 1899, and began to be studied only in 1906. It was manuscript C that became a precious find, as it contained a magnificent essay About the method, previously known only from fragmentary fragments, and the ancient Greek text About floating bodies, disappeared after being lost in the 14th century. manuscript B, which was used in the translation into Latin by William of Moerbeke. This translation was in circulation in the 14th century. in Paris. It was also used by Jacob of Cremona when in the mid-15th century. he undertook a new translation of the corpus of works of Archimedes included in manuscript A (i.e., with the exception of the work About floating bodies). It was this translation, slightly corrected by Regiomontanus, that was published in 1644 in the first Greek edition of the works of Archimedes, although some translations by William of Moerbeke were published in 1501 and 1543. After 1544, Archimedes's fame began to increase, and his methods had a significant influence on such scientists as Simon Stevin and Galileo, and thereby, albeit indirectly, influenced the formation of modern mechanics.

"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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Were devoted to mechanics, it would be natural to begin our conversation with a consideration of how the basic ideas of Greek mechanics arose and how they developed. The word “mechanics” itself comes from the Greek merhane- mehane, which originally meant a lifting machine used in Greek theaters to lift and lower Greek gods onto the stage, which were supposed to resolve the intricate course of the drama being presented; This is where the often used saying comes from: deus ex machina - god from the machine. Later, the word mechane began to be used to refer to military vehicles, and then to machines in general.

As the historian Diodorus Siculus says, Archimedes invents the cochlea, or Archimedes screw, which serves to raise water. The Archimedes screw (Fig. 1) is an invention with the help of which in the distant past rivers were pumped or even completely drained.

Rice. 1 Archimedes screw

The Archimedes catapult, or ballista (Fig. 2, Fig. 3) is an invention of Archimedes, which supposedly appeared around 399 BC. The catapult was used as a weapon in different wars; antique two-arm torsion-action machine for throwing stones. Later, in the first centuries of our era, ballistae began to mean arrow throwers.

Archimedes also proved that it was possible to pull heavy loads with less force than usual; the inventor ordered the heavy ship to be pulled ashore and filled with cargo. Standing near the pulley (reel side), Archimedes began to pull the rope tied to the ship without any significant effort.

Fig.4. Archimedes' paw

Archimedes' paw (Fig. 4) is the prototype of a modern crane. Outwardly, it looked like a lever protruding beyond the city wall and equipped with a counterweight. Polybius wrote in “World History” that if a Roman ship tried to land on the shore near Syracuse, this “manipulator,” under the control of a specially trained driver, would grab its bow and turn it over (the weight of Roman triremes exceeded 200 tons, and penteras could reach all of 500) , swamping the attackers.

Rice. 5. Planetarium

Cicero wrote that after Syracuse was sacked, Marcellus took from there two instruments - “spheres”, the creation of which is attributed to Archimedes. The first was a kind of planetarium, and the second simulated the movement of luminaries across the sky, which suggested the presence of a complex gear mechanism in it.

The Romans were shocked to see Archimedes' machines in action. Plutarch writes that sometimes things reached the point of absurdity: when they saw some kind of rope or log on the wall of Syracuse, the invincible Roman legionnaires fled in panic, thinking that now another hellish mechanism would be used against them.

Until recently, this evidence was considered dubious, but in 1900, near the Greek island of Antikythera, at a depth of 43 meters, the remains of a ship were found, from which the remains of a certain device were raised - an “advanced” system of bronze gears dating back to 87 BC. This proves that Archimedes could well have created a complex mechanism - a kind of “computer” of ancient times.

Archimedes took the lead in many discoveries in the field of exact sciences. Thirteen treatises of Archimedes have reached us. In the most famous of them, “On the Sphere and the Cylinder” (in two books), Archimedes establishes that the surface area of ​​a sphere is 4 times the area of ​​its largest cross-section; formulates the ratio of the volumes of the ball and the cylinder described near it as 2:3 - a discovery that he valued so much that in his will he asked to erect a monument on his grave with the image of a cylinder with a ball inscribed in it and the inscription of the calculation.

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. The basic principles of statics are formulated in the essay “On the Equilibrium of Plane Figures.” Archimedes considers the addition of parallel forces, defines the concept of the center of gravity for various figures, and gives a derivation of the law of leverage.

Using the principle of integration, Archimedes discovered the number pi. Subsequently, its meaning was constantly clarified. In 1882, German mathematician Ferdinand von Lindemann proved that pi is infinite. In the 20th century, computers were able to calculate approximately a billion decimal places. The computer made it possible to discover a comprehensive solution to the famous “bull problem.” The smallest answer to it was found in 1880 and was expressed in a number consisting of 206,545 digits. One hundred years later, in 1981, scientists used a computer to calculate more than a billion decimal places. Modern Syracuse has preserved almost no traces of its former greatness. Tourists are often taken to the so-called “Tomb of Archimedes” in the Grotticelli necropolis. In fact, this Roman burial does not contain the remains of the famous scientist.

The Palimpsest of Archimedes is a Christian book compiled in the 12th century from “pagan” parchments from 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 (from the Greek palin - again and psatio - erase) was made poorly, so the old letters were visible in the light (or even better - under ultraviolet light). In 1906, it turned out that these were three previously unknown works of Archimedes.

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 bathroom and suddenly noticed the effect of fluid displacement (he 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, and this helped him expose the deceiver.