Archimedes

Archimedes of Syracuse (in ancient Greek, Ἀρχιμήδης Arkhimḗdēs from αρχι archi (pre-eminence, dominion) and Ημαδομαι emadomai (to worry), would mean: "he who cares"; Syracuse (Sicily), ca. 287 BC.-ibidem, ca. 212 BC) was a Greek physicist, engineer, inventor, astronomer, philosopher, and mathematician. Although few details of his life are known, he is considered one of the most important scientists of antiquity. Among his advances in physics are his fundamentals in hydrostatics, statics and the explanation of the principle of the lever. He is credited for having designed innovative machines, including siege weapons and the Archimedean screw, which is named after him. Modern experiments have proven claims that Archimedes went on to design machines capable of pulling enemy ships out of the water or setting them on fire using a series of mirrors.

Archimedes is considered to be one of the greatest mathematicians of antiquity and, in general, of all history. He used the exhaustive method to calculate the area under the arc of a parabola with the sum of an infinite series, and gave an extremely accurate approximation of the number pi. He also defined the spiral that bears his name, formulas for the volumes of surfaces of revolution, and an ingenious system for expressing very large numbers.

Archimedes died during the siege of Syracuse (214-212 BC), when he was killed by a Roman soldier, despite orders that no harm be done to him.

Unlike his inventions, Archimedes's mathematical writings were not widely known in antiquity. Alexandrian mathematicians read and cited him, but the first comprehensive compilation of his work was not made until c. 530 d. C. by Isidoro de Mileto. Commentaries on the works of Archimedes written by Eutocios in the sixth century first opened them up to a wider audience. The relatively few copies of Archimedes' written works that survived through the Middle Ages were an important source of ideas during the Renaissance, while the discovery in 1906 of Archimedes' unknown works in the Archimedean Palimpsest has helped to understand how he obtained their mathematical results.

Biography

Archimedes bronze statuette located in the Archenhold observatory in Berlin. It was sculpted by Gerhard Thieme and opened in 1972.

There is little reliable data on the life of Archimedes. However, all sources agree that he was a native of Syracuse and that he died during the outcome of the siege of Syracuse. Archimedes was born c. 287 BC C. in the seaport of Syracuse (Sicily, Italy), a city that at that time was a colony of Magna Graecia. Knowing the date of his death, the approximate date of birth is based on a statement by the Byzantine historian John Tzetzes, who stated that Archimedes lived to the age of 75. According to a reading hypothesis based on a corrupt passage from The sand counter—whose title in Greek is ψαμμίτης (Psammites)—Archimedes mentions the name of his father, Phidias, an astronomer.

Plutarch wrote in his work Parallel Lives (Life of Marcellus, 14, 7.) that Archimedes was related to the tyrant Hiero II of Syracuse. a friend of Archimedes, Heraclides, wrote a biography of him but this book has not survived, thus missing the details of his life. It is unknown, for example, if he ever married or had children.

Among the few certain facts about his life, Diodorus Siculus provides one according to which it is possible that Archimedes, during his youth, studied in Alexandria, in Egypt. The fact that Archimedes refers in his works to scientists whose activity took place in that city supports the hypothesis: in fact, Archimedes refers to Conon of Samos as his friend in On the sphere and cylinder, and two of his works (The Method of Mechanical Theorems and the Cattle Problem) are dedicated to Eratosthenes of Cyrene.

Archimedes died c. 212 BC C. during the Second Punic War, when Roman forces under General Marco Claudio Marcelo captured the city of Syracuse after a two-year siege. Archimedes especially distinguished himself during the siege of Syracuse, in which he developed weapons for the city's defense. Polybius, Plutarco, and Tito Livio precisely describe his work in defending the city as an engineer, developing artillery pieces and other devices capable of keeping the enemy at bay. Plutarch, in his stories, goes so far as to say that the Romans were so nervous about Archimedes' inventions that the appearance of any beam or pulley on the city walls was enough to cause panic among the besiegers.

Cicero and the magistrates discovering the tomb of Archimedes in Syracuseof Benjamin West (1797). Private collection.

Archimedes was assassinated at the end of the siege by a Roman soldier, contravening the orders of the Roman general, Marcellus, to spare the life of the great Greek mathematician. There are various versions of Archimedes' death: Plutarch, in his account, gives up to three different versions. According to the most popular account of him, Archimedes was contemplating a mathematical diagram when the city was taken. A Roman soldier ordered him to go meet the general, but Archimedes ignored this, saying that he had to solve the problem first. The soldier, enraged at the response, killed Archimedes with his sword. However, Plutarch also gives two other, lesser-known accounts of Archimedes' death, the first of which suggests that he might have been killed while trying to surrender to a Roman soldier, and while asking for more time to solve a problem at the time. that he was working. According to the third story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought they were valuable items. Tito Livio, for his part, limits himself to saying that Archimedes was bent over some drawings that he had drawn on the ground when a soldier who did not know who he was killed him. In any case, by all accounts, General Marcelo was furious at Archimedes' death, because he considered him a valuable scientific asset, and had previously ordered that he not be injured.

A dial has 2/3 exact volume and cylinder surface that surrounds it. A sphere and a cylinder were placed above the tomb of Archimedes, fulfilling their will.

The last words attributed to Archimedes were "Don't disturb my circles," referring to the circles in the mathematical drawing he was supposedly studying when he was interrupted by the Roman soldier. The phrase is often quoted in Latin as Noli turbare circulos meos, but there is no evidence that Archimedes uttered the words and they do not appear in Plutarch's accounts.

Cicero describes the tomb of Archimedes, which he would have visited, and indicates that a sphere inscribed within a cylinder had been placed on it. Archimedes had proved that the volume and area of the sphere are two-thirds that of the cylinder that inscribes it, including its bases, which was considered the greatest of his mathematical discoveries. In the year 75 B.C. 137 years after his death, the Roman orator Cicero was serving as quaestor in Sicily and heard stories about Archimedes' tomb, but none of the locals were able to tell him exactly where it was. Finally, he found the tomb near the Agrigento gate in Syracuse, in a neglected and overgrown condition. Cicero cleaned the tomb, and was thus able to see the carving and read some of the verses that had been written on it.

Stories about Archimedes were written by historians of ancient Rome long after his death. Polybius's account of the siege of Syracuse in his Histories (Book VIII) was written about seventy years after Archimedes' death, and was used as a source of information by Plutarch and Titus. light. This account offers little information about Archimedes as a person, focusing instead on the war machines that he was said to have built to defend the city.

Discoveries and inventions

The Golden Crown

Archimedes may use its principle of floating to determine whether the golden crown was less dense than pure gold.

One of the best known anecdotes about Archimedes tells how he invented a method to determine the volume of an object with an irregular shape. According to Vitruvius, Hiero II ordered the manufacture of a new crown in the shape of a triumphal crown, and asked Archimedes to determine if the crown was made only of gold or if, on the contrary, a dishonest goldsmith had added pyrite to it. realization. Archimedes had to solve the problem without damaging the crown, so he could not melt it down and turn it into a regular body to calculate its mass and volume, from there, its density. While he was taking a bath, he noticed that the water level in the tub rose when he got in, and thus realized that this effect could be used to determine the volume of the crown. Because water cannot be compressed, the crown, when submerged, would displace an amount of water equal to its own volume. By dividing the weight of the crown by the volume of water displaced, the density of the crown could be obtained. The density of the crown would be less than the density of gold if other less dense metals had been added to it. When Archimedes, during his bath, became aware of the discovery, it is said that he ran naked through the streets, and that he was so excited by his discovery that he forgot to get dressed. According to the story, in the street he shouted "Eureka!" (Ancient Greek: "εὕρηκα" meaning "I have found it!").

However, the story of the golden crown does not appear in the known works of Archimedes. Furthermore, it has been doubted that the method the story describes was feasible, because it would have required an extreme level of accuracy to measure the volume of water displaced.

Instead, Archimedes could have sought a solution in which he applied the principle of hydrostatics known as Archimedes' principle, described in his treatise On Floating Bodies. This principle states that any body immersed in a fluid experiences a push from bottom to top equal to the weight of the displaced fluid. Using this principle, it would have been possible to compare the density of the golden crown with that of pure gold using a balance. Placing the crown under investigation on one side of the scale and a sample of pure gold of the same weight on the other, the scale would be submerged in water; if the crown had less density than the gold, it would displace more water due to its larger volume and would experience a greater thrust than the gold sample. This buoyancy difference would tip the balance accordingly. Galileo believed that this method was "probably the same as that used by Archimedes, since, in addition to being highly accurate, it is based on demonstrations discovered by Archimedes himself." Around 1586, Galileo Galilei invented a hydrostatic balance for weighing metals. in air and water that apparently would be inspired by the work of Archimedes.

Syracusia and the Archimedean screw

The Archimedes screw can efficiently raise water

Much of Archimedes' work in engineering arose to meet the needs of his hometown of Syracuse. The Greek writer Athenaeus of Naucratis recounts that Hiero II commissioned Archimedes to design a huge ship, the Syracusia, which Archias of Corinth built under his supervision. The ship could be used for luxurious voyages, cargo supplies and as a warship. Finally its name was changed to Alexandria, when it was sent as a gift, along with a cargo of grain, to King Ptolemy III of Egypt.

The Syracusia is said to have been the largest ship in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and included among its facilities decorative gardens, a gymnasium, and a temple dedicated to the goddess Aphrodite. Because a ship of this size would let large amounts of water pass through the hull, the Archimedean screw was supposedly invented in order to remove water from the bilge. Archimedes' machine was a mechanism with a screw-shaped blade inside a cylinder. It was spun by hand, and could also be used to transfer water from low-lying bodies of water to higher irrigation canals. In fact, the Archimedean screw is still used today to pump liquids and semi-fluid solids, such as coal, ice, and grain. Archimedes' screw, as described by Marcus Vitruvius in Roman times, may have been an improvement on the pumping screw that was used to irrigate the Hanging Gardens of Babylon.

The Claw of Archimedes

Polybius narrates that the intervention of Archimedes in the Roman attack on Syracuse was decisive, to the point that he thwarted the Roman hope of taking the city by assault, having to modify his strategy and go to a long-lasting siege, a situation that it lasted eight months, until the final fall of the city. Among the devices that he used for such a feat (catapults, scorpions and cranes) is one that is his own invention: the so-called manus ferrea . The Romans brought the ships as close as they could to the wall to hook their ladders to the fortifications and be able to access the battlements with their troops. Then the claw came into action, consisting of a crane-like arm from which dangled a huge metal hook. When the claw was dropped on an enemy ship the arm would swing upward, lifting the bow of the ship out of the water and causing an ingress of water at the stern. This disabled enemy engines and caused confusion, but it was not the only thing it did: using a system of pulleys and chains, it suddenly dropped the ship causing a list that could lead to capsizing and sinking. There have been modern experiments with in order to test the feasibility of the claw, and in a 2005 documentary titled Superweapons of the Ancient World (Superweapons of the Ancient World) a version of the claw was built and it was concluded that it was a feasible device.

Archimedes' heat ray

Print that reproduces the use of mirrors in the defense of the city of Siracusa during the Roman siege

According to tradition, as part of his work in the defense of Syracuse, Archimedes could have created a system of burning mirrors that reflected sunlight, concentrating it on enemy ships and setting them on fire. However, the sources that collect these facts are late, the first of them being Galen, already in the 2nd century. Luciano de Samosata, also a historian of the II, wrote that, during the siege of Syracuse (213-211 BC), Archimedes repelled an attack by Roman soldiers with fire. Centuries later, Anthemius of Tralles mentions burning mirrors as a weapon used by Archimedes. The artifact, sometimes referred to as the "Archimedean heat ray", would have served to focus sunlight on approaching ships, causing them to burn.

The credibility of this story has been the subject of debate since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect considering only the technical abilities available to Archimedes. It has been suggested that a large number of highly polished bronze or copper shields could have been used. like mirrors, in order to focus the sunlight towards a single ship. In this way the principle of the parabolic reflector could have been used, in a similar way to a solar oven.

Unlike Descartes, Georges-Louis Leclerc de Buffon did believe that Archimedes' feat was perfectly possible. To prove it, he carried out a series of experiments and demonstrations, among which a spectacular exhibition in the royal gardens in 1747 stands out. Using a device with 168 flat mirrors of about 40 centimeters, he managed to burn a pile of firewood at a distance of 60 meters. He concluded that Archimedes probably worked at a distance of 30-45 meters when he set fire to the Roman ships.

In 1973 the Greek scientist Ioannis Sakkas carried out a test of the Archimedean heat ray. The experiment took place at the Skaramangas naval base on the outskirts of Athens, and this time 70 mirrors were used, each covered with a copper cover and around 1.5m high and 1m wide. The mirrors were directed at a plywood model of a Roman warship at a distance of around 50m. When the mirrors were precisely focused, the ship went up in flames in a matter of a few seconds. The model was painted with a layer of bitumen, which could have aided combustion.

In October 2005, a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 30 cm square mirrors focused on a wooden model of a ship at a distance of 30 m. Flames sprang up in part of the ship, but only after the skies had cleared and the ship had been stationary for about ten minutes. It was concluded that the weapon was a viable mechanism under these conditions. The institute group repeated the experiment for the television show MythBusters, using a wooden fishing boat as a target, in San Francisco. Again there was charring, as well as a small amount of flame. To catch fire, wood needs to reach its flash point, which is around 300°C.

When the Mythbusters aired the experiment conducted in San Francisco in January 2006, the claim was categorized as false, due to the length of time and climate required for combustion. They also pointed out that, because Syracuse faces the sea to the East, the Roman fleet should have attacked during the morning for optimal reflection of light by the mirrors. Additionally, conventional weapons such as flaming arrows or catapults would have been a much easier way to set a ship on fire at close ranges.

Other discoveries and inventions

Although Archimedes did not invent the lever, he did write the first known rigorous explanation of the principle involved in operating it. According to Pappus of Alexandria, due to his work on levers he remarked: "Give me a foothold and I will move the world" (Greek, δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω). Plutarch describes how Archimedes designed the hoist system, allowing sailors to use the lever principle to lift objects that would otherwise have been too heavy to move.

Archimedes has also been credited with increasing the power and accuracy of the catapult, as well as inventing the odometer during the First Punic War. The odometer was described as a car with a gear mechanism that dropped a ball into a container after each mile driven. Also, in attempting to measure the apparent dimension of the sun, using a ruler, Archimedes, to try to reduce the imprecision of the measurement, he tried to measure the diameter of the pupil of the human eye. Using that data in his calculations he came up with a better estimate of the solar diameter.

Cicero (106 BC-43 BC) mentions Archimedes briefly in his dialogue De re publica, in which he describes a fictional conversation in 129 BC. C. It is said that, after the capture of Syracuse c. 212 B.C. C., General Marco Claudio Marcelo brought back to Rome two mechanisms that were used as tools for astronomical studies, which showed the movements of the Sun, the Moon and five planets. Cicero mentions similar mechanisms designed by Thales of Miletus and Eudoxus of Cnidos. The dialogue says that Marcelo kept one of the mechanisms as his personal loot from Syracuse and donated the other to the Temple of Virtue in Rome. According to Cicero, Gaius Sulpicius Gaul demonstrated the Marcellus mechanism, and described it thus:

Hanc sphaeram Gallus cum moviet, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fert eadem illa defectio, et incideret luna tum in eam metame esset umbra terrae, cum sol e regione.

When Galo moved the globe, it happened that the Moon followed the Sun so many turns in that invention of bronze as in the sky itself, so also in the sky the solar balloon came to have that same distance, and the Moon came to that position in which its shadow was on the Earth, when the Sun was on the line.

This description corresponds to that of a planetarium. Pappus of Alexandria said that Archimedes had written a (now lost) manuscript about the construction of these mechanisms entitled On Making Spheres. Modern research in this area has focused on the Antikythera mechanism, another mechanism from classical antiquity probably designed for the same purpose. Building mechanisms of this type should have required a sophisticated knowledge of differential gears and this was thought to be beyond the scope of available technology at the time, but the discovery of the Antikythera mechanism in 1902 confirmed that this class of artifacts were known to the ancient Greeks.

Math

While Archimedes' inventive side is perhaps the most popular, he also made important contributions to the field of mathematics. In this regard, Plutarch said of him that "he regarded as ignoble and ministerial all occupation in mechanics and all art applied to our uses, and he put only his desire to excel in those things that carry with them the beautiful and excellent, without mixing anything." servile, diverse and separate from the rest".

Archimedes used the exhaustive method to achieve the approximate value of the π number.

Archimedes was able to use the infinitesimals in a similar way to modern integral calculus. Through reduction to absurdity ( reductio ad absurdum ), he was able to solve problems by approximations with a certain degree of precision, specifying the limits between which the correct answer was found. This technique is called the exhaustive method, and it was the system he used to approximate the value of the number π. To do this, he drew a regular inscribed polygon and another circumscribed to the same circumference, so that the length of the circumference and the area of the circle are bounded by those same values of the lengths and areas of the two polygons. As the number of sides of the polygon increases, the difference shortens, and a more accurate approximation is obtained. Starting from polygons with 96 sides each, Archimedes calculated that the value of π should lie between 3¹⁰⁄₇₁ (approximately 3.1408) and 3¹⁄₇ (approximately 3.1429), which is consistent with the true value of π. He also proved that the area of the circle was equal to π multiplied by the square of the radius of the circle. In his work On the Sphere and the Cylinder, Archimedes postulates that any magnitude, added to itself a sufficient number of times, can exceed any other given magnitude, a postulate that is known as the Archimedean property of numbers. real.

In his work on the Measurement of the Circle, Archimedes gives an interval for the value of the square root of 3 out of ²⁶⁵⁄₁₅₃ (approximately 1.7320261) and ¹³⁵¹⁄₇₈₀ (approximately 1.7320512). The true value is approximately 1.7320508, so Archimedes' estimate turned out to be very accurate. However, he introduced this result into his work without explaining what method he had used to obtain it.

Archimedes showed that the area of the parabolic segment of the upper figure is equal to 4/3 of the inscribed triangle of the lower figure.

In his work on Squaring the Parabola, Archimedes proved that the area defined by a parabola and a straight line was exactly equal to ⁴⁄₃ the area of the corresponding inscribed triangle, as can be seen in the figure on the right. To get that result, he developed an infinite geometric series with a common ratio of ¹ ⁄ ₄ :

␡ ␡ n=0∞ ∞ 4− − n=1+4− − 1+4− − 2+4− − 3+ =43.{displaystyle sum _{n=0}^{infty }4^{-n}=1+4^{}-1+4^{-2}+4^{-3}+cdots ={4 over 3}.;}

The first term of this sum is equivalent to the area of the triangle, the second would be the sum of the areas of the two triangles inscribed in the two areas delimited by the triangle and the parabola, and so on. This proof uses a variation of the infinite series ¹⁄₄ + ¹⁄₁₆ + ¹⁄₆₄ + ¹⁄₂₅₆ +..., whose sum is shown to be equal to ¹⁄₃.

In another of his works, Archimedes faced the challenge of trying to calculate the number of grains of sand that the universe could contain. To do so, he challenged the idea that the number of grains was large enough to be counted. He wrote:

There are some, King Gelon, who believe that the number of grains of sand is infinite in multitude; and when I refer to the sand I refer not only to that which exists in Syracuse and the rest of Sicily but also to that which can be found in any region, whether inhabited or uninhabited.

Archimedes

In order to deal with the problem, Archimedes designed a calculation system based on the myriad. It is a word that comes from the Greek μυριάς (murias) and that was used to refer to the number 10,000. He proposed a system in which a power of a myriad of myriads (100 million) was used. and concluded that the number of grains of sand needed to fill the universe would be 8×10⁶³.

Writings

The works of Archimedes were originally written in Doric Greek, the dialect spoken in ancient Syracuse.

The written work of Archimedes has not been as well preserved as that of Euclid, and seven of his treatises are known only through references made by other authors. Pappus of Alexandria, for example, mentions On Making Spheres and other work on polyhedra, while Theon of Alexandria cites a commentary on refraction from a lost work entitled Catoptrica. During his lifetime, Archimedes disseminated the results of his work through his correspondence with the mathematicians of Alexandria. Archimedes' writings were collected by the Byzantine architect Isidore of Miletus (c. AD 530), while commentaries on Archimedes' works written by Eutocios in the sixth century helped spread the his work to a wider audience. Archimedes' work was translated into Arabic by Thábit ibn Qurra (836-901 AD), and into Latin by Gerard of Cremona (c. 1114-1187 AD). During the Renaissance, in 1544, the Editio Princeps (First Edition) was published by Johann Herwagen in Basel, with the works of Archimedes in Greek and Latin.

Retained works

It is said that Archimedes said about the lever: "Give me a point of support and I will move the world"

On the balance of planes

In two volumes

The first book consists of fifteen propositions with seven axioms, while the second consists of ten propositions. In this work, Archimedes explains the law of the lever, stating the following:

The quantities are in balance at distances mutually proportional to their weights.

Archimedes uses derivative principles to calculate the areas and centers of gravity of several geometric figures, including triangles, parallelograms and parables.

On the measure of a circle

It is a short work, consisting of three propositions. It is written in the form of a letter to Dositeo de Pelusio, a student of Conón de Samos. In Proposition II, Archimedes shows that the value of the number π (Pi) is greater than 223/71 and less than 22/7. This figure was used as an approximation of π throughout the Middle Ages and even today it is used when an approximate figure is required.

On the spirals

This work, composed of 28 propositions, is also directed to Dositeo. The treaty defines what is now known as the spiral of Archimedes. This spiral represents the geometric place in which the points corresponding to the positions of a point are located that is moved out from a fixed point with a constant speed and along a line that rotates with a constant angular velocity. In polar coordinates, (r, θ) ellipse can be defined through the equation

:r=a+bθ θ {displaystyle ,r=a+btheta }

being a and b Real numbers. This is one of the first examples in which a Greek mathematician defines a mechanical curve (a curve drawn by a turning point).

On dial and cylinder

Two volumes

In this treatise, also addressed to Dositeo, Archimedes reaches the mathematical conclusion of which he would be most proud, that is, the relationship between a sphere and a cylinder circumscribed with the same height and diameter. The volume is 43π π r3{displaystyle {tfrac {4}{3}}pi r^{3}} for the field, and 2π π r3{displaystyle 2pi r^{3} for the cylinder. The surface area is 4π π r2{displaystyle 4pi r^{2}} for the field, and 6π π r2{displaystyle 6pi r^{2}} for the cylinder (including its two bases), where r{displaystyle r} is the radius of the sphere and the cylinder. The dial has an area and volume equivalent to two thirds of the cylinder. At the request of the Archimedes himself, the sculptures of these two geometric bodies were placed on his grave.

About conoids and spheroids

This is a work in 32 propositions and also aimed at Dositeo in which Archimedes calculates the areas and volumes of the sections of cones, spheres and paraboloids.

On floating bodies

In two volumes

In the first part of this treaty, Archimedes explains the law of liquid balance, and proves that water adopts a spherical form around a center of gravity. This may have been an attempt to explain the theories of contemporary Greek astronomers, like Eratosthenes, who claimed that the earth is spherical. The liquids described by Archimedes are not self-gravitational, because he assumes the existence of a point to which all things fall, from which the spherical form derives.

In the second part, Archimedes calculates the balance positions of paraboloid sections. This was probably an idealization of the shapes of the hulls of the ships. Some of its sections float with the base under the water and the top over the water, in a way similar to how the icebergs float. Archimedes defines in his work the principle of floatability as follows:

Every body submerged in a liquid experiences a vertical thrust and upwards equal to the liquid weight evicted.

The square of the parable

In this work of 24 propositions, directed to Dositeo, Archimedes test through two different methods that the area fenced by a parable and a straight line is 4/3 multiplied by the area of a triangle of equal base and height. Obtains this result by calculating the value of a geometric series that adds to the infinite with the radio 1/4.

Ostomachion

In this work, whose most complete treatise that describes it was found within the Palimpsest of Archimedes, Archimedes presents a dissection puzzle similar to a tangram. Archimedes calculates the areas of 14 pieces that can be assembled to form a square. An investigation published in 2003 by Reviel Netz of Stanford University argued that Archimedes was trying to determine how many forms the pieces could be assembled to form a square. According to Netz, the pieces can form a square of 17 152 different ways. The number of provisions is reduced to 536 when solutions that are equivalent to rotation and reflection are excluded. This puzzle represents an early example of a combination problem.

The origin of the name of the puzzle is uncertain; it has been suggested that it may have arisen from the Greek word for throat, stómakhos (στόμαχος). Ausonio refers to puzle as Ostomachion, a Greek word composed of the roots bridesmaidosteon, ‘bone’) and μχγmachē, ‘light’). The puzzle is also known as Loculus Archimedes or like the Archimedes Box.

The problem of the cattle of Archimedes

This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a 44-line poem, in the Herzog August Library in Wolfenbüttel, Germany, in 1773. It is aimed at Eratosthenes and the mathematicians of Alexandria and, in it, Archimedes challenges them to count the number of reses in the Sun's Handle, solving a number of simultaneous diophantic equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by A. Amthor in 1880, and the answer is a very large number, approximately 7,760271×10²⁰⁶⁵⁴⁴.

The sand counter

In this treatise, Archimedes counts the number of grains of sand that would enter the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos, and contemporary ideas about the size of the Earth and the distances of several celestial bodies. Using a system of numbers based on the capacity of the myriad, Archimedes concludes that the number of grains of sand that would be required to fill the universe would be 8×10⁶³In modern notation. The introductory letter states that the father of Archimedes was an astronomer called Fidias. The sand counter or Psammites is the only surviving work of Archimedes in which his vision of astronomy is concerned.

The method of mechanical theorems

This treaty, which was considered lost, was re-founded thanks to the discovery of the Palimpsest of Archimedes in 1906. In this work, Archimedes employs infinitesimal calculation, and shows how the method of fractioning a figure in an infinite number of infinitely small parts can be used to calculate its area or volume. Archimedes may have considered that this method lacked sufficient formal rigor, so he also used the exhaustive method to reach the results. Like The problem of cattle, The method of mechanical theorems was written in the form of a letter addressed to Eratosthenes of Alexandria.

Apocryphal works

The Book of Lemmas or Liber Assumptorum is a treatise of fifteen propositions on the nature of circles. The oldest copy of the text is written in Arabic. The scholars Thomas Heath and Marshall Clagett argued that it could not have been written by Archimedes in that version, since he himself is quoted in the text, which suggests that it was modified by another author. The Lemmas may be based on an older work, now lost, written by Archimedes.

It has also been said that Archimedes already knew Heron's formula to calculate the area of a triangle knowing the measure of its sides. However, the first reliable reference to the formula is given by Heron of Alexandria in the I d. C.

The Archimedean Palimpsest

«Stomachion» is a puzzle of dissection in the Palimpsest of Archimedes

The Archimedes palimpsest is one of the main sources from which the work of Archimedes is known. In 1906, Professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin scroll of prayers written in the 13th century. He discovered that it was a palimpsest, a document with text that has been overwritten on top of an earlier erased work. Palimpsests were created by scraping ink off existing works and then reusing the material they were printed on, which was a common practice in the Middle Ages because vellum was expensive. The oldest works that could be found on the palimpsest were identified by scholars as 10th-century copies of previously unknown treatises by Archimedes. The scroll spent hundreds of years in the library of a Constantinople monastery, before being sold. to a private collector in the 1920s. On October 29, 1998, it was sold at auction to an anonymous buyer for $2 million at Christie's, New York. The palimpsest contains seven treatises, including the only hitherto known copy of the work On Floating Bodies in the original in Greek. It is also the only source of The Method of Mechanical Theorems, referred to by Suidas and believed to be lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than could be found in earlier texts.

The palimpsest is housed at the Walters Art Museum in Baltimore, Maryland, where it has undergone various modern tests, including the use of ultraviolet light and X-rays to read the overwritten text.

The treatises contained in the Archimedes Palimpsest are: On the balance of planes, On spirals, Measurement of a circle, On the sphere and cylinder, On floating bodies, The method of mechanical theorems and Stomachion.

Acknowledgments

The Fields Medal represents a portrait of Archimedes

In 1935 it was decided to name a lunar crater (29.7° N, 4.0° W) located in the eastern part of the Mare Imbrium in his honor "Archimedes". The lunar mountain range "Montes de Arquímedes" also bears his name (25.3° N, 4.6° W) and the asteroid (3600) Archimedes (3600 Archimedes).

The Fields Medal, an award given for outstanding mathematical achievement, bears a portrait of Archimedes, along with his proof of the mathematical relationship between the areas and volumes of the sphere and cylinder. The inscription around Archimedes' head is a quote attributed to him, which reads in Latin Transire suum pectus mundoque potiri> ("To overcome oneself and rule the world").

Arquímedes has appeared on stamp issues in East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).

Attributed to Archimedes, the exclamation "Eureka!" is the motto of the state of California. In this case, however, the word refers to the time of the discovery of gold near Sutter's Mill in 1848, which sparked the California Gold Rush.

Further reading

In English