ARCHIMEDES OF SYRACUSE

Greek mathematician who flourished in Sicily. He is generally considered to be the greatest mathematician of ancient times. Most of the facts about his life come from a biography about the Roman soldier Marcellus written by the Roman biographer Plutarch.

Archimedes performed numerous geometric proofs using the rigid geometric formalism outlined by Euclid, excelling especially at computing areas and volumes using the method of exhaustion. He was especially proud of his discovery for finding the volume of a sphere, showing that it is two thirds the volume of the smallest cylinder that can contain it. At his request, the figure of a sphere and cylinder was engraved on his tombstone. In fact, it is often said that Archimedes would have invented calculus if the Greeks had only possessed a more tractable mathematical notation. By inscribing and circumscribing polygons on a circle, for instance, he was able to constrain the value of (pi) between 3 10/71 and 3+1/7.

Archimedes was also an outstanding engineer, formulating Archimedes' principle of buoyancy and the law of the lever. Legend has it that Archimedes discovered his principle of buoyancy, which states that the buoyancy force is equal to the weight of the liquid displaced, while taking a bath, upon which he is supposed to have run naked through the streets of Syracuse shouting "Eureka!" (I have found it). Archimedes is also purported to have invented the Archimedean screw. Some of Archimedes's geometric proofs were actually motivated by mechanical arguments which led him to the correct answer. During the Roman siege of Syracuse, he is said to have single-handedly defended the city by constructing lenses to focus the Sun's light on Roman ships and huge cranes to turn them upside down. When the Romans finally broke the siege, Archimedes was killed by a Roman soldier after snapping at him "Don't disturb my circles," a reference to a geometric figure he had outlined on the sand.

CURRICULUM VITAE

Born	About 287 BC in Syracuse, Sicily. At the time Syracuse was an independent Greek city-state with a 500-year history.
Died	212 or 211 BC in Syracuse when it was being sacked by a Roman army. He was killed by a Roman soldier who did not know who he was.
Education	Probably studied in Alexandria, Egypt, under the followers of Euclid.
Family	His father was an astronomer named Phidias and he was probably related to Hieron II, the king of Syracuse. It is not known whether he was married or had any children.
Inventions	Many war machines used in the defense of Syracuse, compound pulley systems, planetarium, water screw (possibly), water organ (possibly), burning mirrors (very unlikely).
Fields of Science Initiated	On plane equilibriums, Quadrature of the parabola, On the sphere and cylinder, On spirals, On conoids and spheroids, On floating bodies, Measurement of a circle, The Sandreckoner, On the method of mechanical problems.
Major Writings	On plane equilibriums, Quadrature of the parabola, On the sphere and cylinder, On spirals, On conoids and spheroids, On floating bodies, Measurement of a circle, The Sandreckoner, On the method of mechanical problems.
Place in History	Generally regarded as the greatest mathematician and scientist of antiquity and one of the three greatest mathematicians of all time (together with Isaac Newton (English 1643-1727) and Carl Friedrich Gauss (German 1777-1855)).

HIS LIFETIME

Archimedes is considered one of the three greatest mathematicians of all time along with Newton and Gauss. In his own time, he was known as "the wise one," "the master" and "the great geometer" and his works and inventions brought him fame that lasts to this very day. He was one of the last great Greek mathematicians.

Born in 287 B.C., in Syracuse, a Greek seaport colony in Sicily, Archimedes was the son of Phidias, an astronomer. Except for his studies at Euclid's school in Alexandria, he spent his entire life in his birthplace. Archimedes proved to be a master at mathematics and spent most of his time contemplating new problems to solve, becoming at times so involved in his work that he forgot to eat. Lacking the blackboards and paper of modern times, he used any available surface, from the dust on the ground to ashes from an extinguished fire, to draw his geometric figures. Never giving up an opportunity to ponder his work, after bathing and anointing himself with olive oil, he would trace figures in the oil on his own skin.

Much of Archimedes fame comes from his relationship with Hiero, the king of Syracuse, and Gelon, Hiero's son. The great geometer had a close friendship with and may have been related to the monarch. In any case, he seemed to make a hobby out of solving the king's most complicated problems to the utter amazement of the sovereign. At one time, the king ordered a gold crown and gave the goldsmith the exact amount of metal to make it. When Hiero received it, the crown had the correct weight but the monarch suspected that some silver had been used instead of the gold. Since he could not prove it, he brought the problem to Archimedes. One day while considering the question, "the wise one" entered his bathtub and recognized that the amount of water that overflowed the tub was proportional the amount of his body that was submerged. This observation is now known as Archimedes' Principle and gave him the means to solve the problem. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka! eureka!" (I have found it!). The fraudulent goldsmith was brought to justice. Another time, Archimedes stated "Give me a place to stand on and I will move the earth." King Hiero, who was absolutely astonished by the statement, asked him to prove it. In the harbor was a ship that had proved impossible to launch even by the combined efforts of all the men of Syracuse. Archimedes, who had been examining the properties of levers and pulleys, built a machine that allowed him the single-handedly move the ship from a distance away. He also had many other inventions including the Archimedes' watering screw and a miniature planetarium.

Though he had many great inventions, Archimedes considered his purely theoretical work to be his true calling. His accomplishments are numerous. His approximation of between 3-1/2 and 3-10/71 was the most accurate of his time and he devised a new way to approximate square roots. Unhappy with the unwieldy Greek number system, he devised his own that could accommodate larger numbers more easily. He invented the entire field of hydrostatics with the discovery of the Archimedes' Principle. However, his greatest invention was integral calculus. To determine the area of sections bounded by geometric figures such as parabolas and ellipses, Archimedes broke the sections into an infinite number of rectangles and added the areas together. This is known as integration. He also anticipated the invention of differential calculus as he devised ways to approximate the slope of the tangent lines to his figures. In addition, he also made many other discoveries in geometry, mechanics and other fields.

The end of Archimedes life was anything but uneventful. King Hiero had been so impressed with his friend's inventions that he persuaded him to develop weapons to defend the city. These inventions would prove quite useful. In 212 B.C., Marcellus, a Roman general, decided to conquer Syracuse with a full frontal assault on both land and sea. The Roman legions were routed. Huge catapults hurled 500 pound boulders at the soldiers; large cranes with claws on the end lowered down on the enemy ships, lifted them in the air, and then threw them against the rocks; and systems of mirrors focused the sun rays to light enemy ships on fire. The Roman soldiers refused to continue the attack and fled at the mere sight of anything projecting from the walls of the city. Marcellus was forced to lay siege to the city, which fell after eight months. Archimedes was killed by a Roman soldier when the city was taken. The traditional story is that the mathematician was unaware of the taking of the city. While he was drawing figures in the dust, a Roman soldier stepped on them and demanded he come with him. Archimedes responded, "Don't disturb my circles!" The soldier was so enraged that he pulled out his sword and slew the great geometer. When Archimedes was buried, they placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

Writings by Archimedes

This scroll explans the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.

In this scroll, Archimedes defines what is now called Archimedes' spiral. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician. Using this curve, he was able to square the circle.

In this scroll Archimedes obtains the result he was most proud of: that the area and volume of a sphere are in the same relationship to the area and volume of the circumscribed straight cylinder.

In this scroll Archimedes calculates the areas and volumes of sectios of cones, spheres and paraboloids.

In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water around a center of gravity will adopt a spherical form. This is probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal gravitation.

In the second part, a veritable tour-de-force, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, which is reminiscent of the way icebergs float, although Archimedes probably wasn't thinking of this application.

In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by triangulating the area and summing the geometric series with ratio 1/4.

This is a Greek puzzle similar to Tangram. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of combinatorics to solve a problem.

Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers cattle in the Herd of the Sun by solving a number of simultaneous diophantine equations, some of them quadratic. This problem is one of the famous problems solved with the aid of a computer.

In this scroll, Archimedes counts the number of grains of sand fitting inside the universe. This book mentions Aristarchus' theory of the solar system, contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the inroductory letter we also learn that Archimedes' father was an astronomer.

In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneered the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. Archimedes did probably consider these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional method of exhaustion to prove them. This particular work is found in what is called the Archimedes Palimpsest. Some details can be found at how Archimedes used infinitesimals.

MECHANICS

The Golden Crown Mystery -- Archimedes Principle

Legend has it that Archimedes discovered his famous theory of buoyancy - Archimedes Principle - while taking a bath. He was so excited that he ran naked through the streets of Syracuse shouting "Eureka, eureka (I have found it)!". Another legend describes how Archimedes uncovered a fraud against King Hieron II of Syracuse using his principle of buoyancy. The king suspected that a solid gold crown he ordered was partly made of silver. Archimedes first took two equal weights of gold and silver and compared their weights when immersed in water. Next he compared the weights of the crown and a pure silver crown of identical dimensions when each was immersed in water. The difference between these two comparisons revealed that the crown was not solid gold.

The Lever

Magnitudes are in equilibrium at distances reciprocally proportional to their weights.

This is the statement of the Law of the Lever that Archimedes gives in Propositions 6 and 7 of Book I of his work entitled On the Equilibrium of Planes. While it is commonly stated that Archimedes proves this law in these two propositions, there has been considerable debate as to what Archimedes really proved, what his stated postulates mean, what hidden assumptions he used, and what he may have thought he proved

Geometry

Archimedes discovered pi.

He performed numerous geometric proofs using the rigid geometric formalism outlined by Euclid, excelling especially at computing areas and volumes using the method of exhaustion.

Archimedes, although he achieved fame by his mechanical inventions, believed that pure mathematics was the only worthy pursuit. He was a brilliant mathematician who helped develop the science of geometry. His methods anticipated the integral calculus 2,000 years before Newton and Leibniz.

Although many solid figures having all kinds of surfaces can be conceived, those which appear to be regularly formed are most deserving of attention. Those include not only the five figures found in the godlike Plato, that is, the tetrahedron and the cube, the octahedron and the dodecahedron, and fifthly the icosahedron, but also the solids, thirteen in number, which were discovered by Archimedes and are contained by equilateral and equiangular, but not similar, polygons.

The thirteen "Archimedean solids

Truncated Tetrahedron - The first is a figure of eight bases, being contained by four triangles and four hexagons.
Cuboctahedron - After this come three figures of fourteen bases, the first contained by eight triangles and six squares,
Truncated Octahedron - the second by six squares and eight hexagons,
Truncated Cube - and the third by eight triangles and six octagons
Rhombicuboctahedron - After these come two figures of twenty-six bases, the first contained by eight triangles and eighteen squares,
Truncated Cuboctahedron -the second by twelve squares, eight hexagons and six octagons.
Icosidodecahedron - After these come three figures of thirty-two bases, the first contained by twenty triangles and twelve pentagons,
Truncated Dodecahedron - and the third by twenty triangles and twelve decagons.
Snub Cube - After these comes one figure of thirty-eight bases, being contained by thirty-two triangles and six squares
Rhombicosidodecahedron - After this come two figures of sixty-two bases, the first contained by twenty triangles, thirty squares and twelve pentagons,
Truncated Icosidodecahedron - the second by thirty squares, twenty hexagons and twelve decagon
Snub Dodecahedron - After these there comes lastly a figure of ninety-two bases, which is contained by eighty triangles and twelve pentagons.

INVENTIONS

Archimedes Screw

This machine for raising water, allegedly invented by the ancient Greek scientist Archimedes for removing water from the hold of a large ship. One form consists of a circular pipe enclosing a helix and inclined at an angle of about 45 degrees to the horizontal with its lower end dipped in the water; rotation of the device causes the water to rise in the pipe. Other forms consist of a helix revolving in a fixed cylinder or a helical tube wound around a shaft.

Modern screw pumps, consisting of helices rotating in open inclined troughs, are effective for pumping sewage in wastewater treatment plants. The open troughs and the design of the screws permit the passage of debris without clogging.

The Steam Engine of Archimedes

Martial weapon where launched balls of weight one talant (roughly 23 kilos) in distance of 6 stadia (roughly 1.100 m.).. It is the first worldwide arm that functioned with steam compresion. Archimedes invented it during the siege of Syracuse from the Romans (213-211 B.C.). With this weapon dealt also Leonarnto da Vinci, witch named Arhitronito (from the words Archimedes and tronnymi), and made the first constructional drawings of arm.

The Burning Mirror

Archimedes invented many machines, which were used as engines of war. These were particularly effective in the defense of Syracuse when the Romans under the command of Marcellus attacked it.

During the Roman siege of Syracuse, he is said to have single-handedly defended the city by constructing lenses to focus the Sun's light on Roman ships and huge cranes to turn them upside down. When the Romans finally broke the siege, Archimedes was killed by a Roman soldier after snapping at him ``Don't disturb my circles,'' a reference to a geometric figure he had outlined on the sand.

The Compound Pulley

Other inventions of Archimedes such as the compound pulley also brought him great fame among his contemporaries.

Archimedes had stated in a letter to King Hieron that given the force, any given weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king's arsenal, which could not be drawn out of the dock without great labor and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavor, but only holding the head of the pulley in his hand and drawing the cords by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea.

Heron’s Odometer

The Heron’s odometer was most probably an invention made by Archimedes. Vitruvius was the first writer who mentioned this machine (1^st cent BC).It’s full description is found in Heron’s Work Dioptra or On Dioptra.The odometer was used in order to calculate road distances, while a later variation known as nautical odometer was used measure distances at sea. The function of the mechanism relies on a system of cogwheels (gears) which, assembled in attermones Kochlies ,transferred the movement of the wheels onto three circular disks at the upper part of the instrument, where the travelled distance was measured.

The Spheres and Planetaria

Archimedes is supposed to have made two "spheres" that Marcellus took back to Rome - one a star globe and the other a device (the details of which are uncertain) for mechanically representing the motions of the Sun, Moon, and planets.

One was a solid sphere on which were engraved or painted the stars and constellations, which Marcellus placed in the Temple of Virtue. Such celestial globes predate Archimedes by several hundred years and Cicero credits the famed geometers Thales and Eudoxos with first constructing them.

The second sphere, which Marcellus kept for himself, was much more ingenious and original. It was a planetarium: a mechanical model that shows the motions of the sun, moon, and planets as viewed from the earth.

Cicero writes that Archimedes must have been "endowed with greater genius that one would imagine it possible for a human being to possess" to be able to build such an unprecedented device.

Many other ancient writers also refer to Archimedes' planetarium in prose and in verse. Several viewed it as proof that the cosmos must have had a divine creator: for just as Archimedes' planetarium required a creator, so then must the cosmos itself have required a creator.

Cicero reverses the argument to contend that since the cosmos had a divine creator, so then must Archimedes be divine to be able to imitate its motions.

The Greek mathematician Pappus of Alexandria, who lived in the fourth century AD, writes that Archimedes wrote a now-lost manuscript entitled On Sphere-making. Pappus also states that it was the only manuscript that Archimedes wrote on "practical" matters. No physical trace of Archimedes' planetarium survives. Cicero refers to it as a "bronze contrivance" while Claudian describes it as "a sphere of glass."

The 1752 engraving of Rowley's orrery suggests how Archimedes' planetarium might have looked. On this orrery the sun, moon and planets revolve along a flat surface driven underneath by a hidden gear works.

Spherical bands surrounding the flat surface represent the celestial equator, the arctic circle, a movable horizon, and the ecliptic marked with the zodiacal signs.

In 1900 a shipwreck discovered off the shore of the Greek island of Antikythera uncovered an unexpected treasure. The ship dated from the first century BC and was sailing from the Greek island of Rhodes. Amidst its cargo was a complicated gear works in a deteriorated state about the size of a cigar box.

The device, now called the Antikythera mechanism, was analyzed by Derek De Solla Price of Yale University, who concluded that it was an ancient planetarium in which the positions of the heavenly bodies were indicated by dials on the face of the device.

The gear works are about as complicated as those in a modern mechanical clock and represent the earliest physical evidence of an advanced metallic mechanism. Price gives evidence that this mechanism was in the Archimedean tradition and strongly suggests that Archimedes' planetarium was its forerunner. A complete presentation of Price's research can be found in Gears from the Greeks.

THE DEATH OF ARCHIMEDES

The Romans in the Second Punic War killed Archimedes in 212 BC during the capture of Syracuse after all his efforts to keep the Romans at bay with his machines of war had failed. Plutarch recounts three versions of the story of his killing which had come down to him.

"Archimedes ... was ..., as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through."

"A Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him."

"As Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him."

Archimedes was buried Syracuse, where he was born, were he grew up, where he worked, and where he died.

On his grave their is an inscription of pi, his most famous discovery. They also placed on his tombstone the figure of a sphere inscribed inside a cylinder and the 2:3 ratio of the volumes between them, the solution to the problem he considered his greatest achievement.

Given the magnitude and originality of Archimedes' achievement, the influence of his mathematics in antiquity was rather small. Those of his results that could be simply expressed--such as the formulas for the surface area and volume of a sphere--became mathematical commonplaces, and one of the bounds he established for

, 22/7, was adopted as the usual approximation to it in antiquity and the Middle Ages. But his mathematical work was not continued or developed, as far as is known, in any important way in ancient times, despite his hope expressed in Method that its publication would enable others to make new discoveries. It was not until some of his mathematical treatises were translated into Arabic in the late 8th or 9th century that attempts were made to extend his results, particularly in the determination of the volumes of solids of revolution. Several meritorious works by Arabic mathematicians of the early medieval period were inspired by their study of Archimedes.

But the greatest effect of his work on that of later mathematicians came in the 16th and 17th centuries with the printing of texts derived from the Greek, and eventually of the Greek text itself, the editio princeps, in Basel in 1544. The Latin translation of many of Archimedes' works by Federico Commandino in 1558 contributed greatly to the spread of knowledge of them, which was reflected in the work of the foremost mathematicians and physicists of the time, including Johannes Kepler and Galileo. David Rivault's edition and Latin translation (1615) of the complete works, including the ancient commentaries, was enormously influential in the work of some of the best mathematicians of the 17th century, notably René Descartes and Pierre de Fermat. Without the background of the rediscovered ancient mathematicians, amongst whom Archimedes was paramount, the development of mathematics in Europe in the century between 1550 and 1650 is inconceivable. It is unfortunate that Method remained unknown to both Arabic and Renaissance mathematicians (it was only rediscovered in the late 19th century), for they might have fulfilled Archimedes' hope that it would prove of use to his successors in discovering theorems.

BIBLIOGRAPHIC SOURSES

Archimedes: The Ingenious Engineer by Christos D. Lazos, Aiolos Publishers, Athens, 1995

Mechanics and Technology in Ancient Greece by Christos D. Lazos, Aiolos Publishers,

(in Greek). Μηχανική και Τεχνολογία στην Αρχαία Ελλάδα Χρήστος Δ. Λάζος, ΑΙΟΛΟΣ

Οι Μαθηματικοί της Αρχαίας Ελλάδας:Βαγγέλη Σπανδάγου, Ρούλα Σπανδάγου, Δέσποινα Τραυλού. The Mathematicians of Ancient Greece by Vagellis Spandagos,

ARCHIMEDES OF SYRACUSE (287 – 212 DC)

CURRICULUM VITAE

HIS LIFETIME

Writings by Archimedes

MECHANICS

The Golden Crown Mystery -- Archimedes Principle

The Lever

Geometry

Archimedes discovered pi.

The thirteen "Archimedean solids

Truncated Tetrahedron

Cuboctahedron

Truncated Octahedron

Truncated Cube

Rhombicuboctahedron

Truncated Cuboctahedron

Icosidodecahedron

Truncated Dodecahedron

Snub Cube

Rhombicosidodecahedron

Truncated Icosidodecahedron

Snub Dodecahedron