Archimedes

For other uses, see Archimedes (disambiguation).

Archimedes of Syracuse (Greek: Άρχιμήδης)
Era	Ancient philosophy
Region	Classical Greek philosophy
School	Euclid of Alexandria Natural philosophy
Main interests	mathematics, physics, engineering, astronomy
Notable ideas	Hydrostatics, Levers, Infinitesimals

Archimedes of Syracuse (Greek: Άρχιμήδης c. 287 BC – c. 212 BC) was an ancient Greek mathematician, physicist and engineer. Although little is known about his life, he is regarded as one of the most important scientists in classical antiquity. In addition to making important discoveries in the field of mathematics and geometry, he is credited with producing machines that were well ahead of their time.

The Ancient Roman historians showed a strong interest in Archimedes and wrote several biographies relating to his life and works, while the few copies of his treatises that survived through the Middle Ages were a major influence on scientists during the Renaissance.

Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today. Carl Friedrich Gauss, himself frequently called the most influential mathematician of all time, is said to have remarked that Archimedes was one of the three epoch-making mathematicians, with the others being Sir Isaac Newton and Ferdinand Eisenstein.

Biography

Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, which was then a colony of Magna Graecia. The date of his birth is based on an assertion by the Byzantine Greek historian John Tzetzes that he lived for seventy-five years. In The Sand Reckoner Archimedes gives his father's name as Phidias, an astronomer about whom nothing is known. Plutarch wrote that Archimedes was related to King Hieron II, the ruler of Syracuse. A biography of Archimedes was written by his friend Heracleides but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children. Archimedes is believed to have spent part of his youth being educated in Alexandria, Egypt where he was a contemporary of Conon of Samos and Eratosthenes. Some of Archimedes' mathematical works were written in the form of letters to Eratosthenes, who was the chief librarian in Alexandria.

Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two year long siege. According to the popular account given by Plutarch in his Parallel Lives, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. Plutarch also gives a lesser-known account of the death of Archimedes which suggests that he may have been killed while attempting to surrender to a Roman soldier. According to this story, Archimedes was carrying mathematical instruments, and was killed because the soldier thought that they were valuable items. General Marcellus was reportedly angered by the death of Archimedes, as he had ordered him not to be harmed.

The last words attributed to Archimedes are "Do not disturb my circles" (Greek: μή μου τούς κύκλους τάραττε), a reference to the circles in the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. This quote is often given in Latin as "Noli turbare circulos meos", but there is no reliable evidence that Archimedes uttered these words and they do not appear in the account given by Plutarch.

Archimedes' tomb had a carving of his favorite mathematical diagram, which was a sphere inside a cylinder of the same height and diameter. Archimedes had proved that the volume and surface area of the sphere would be two thirds that of the cylinder. In 75 BC, 137 years after his death, the Roman orator Cicero visited the tomb in Syracuse which had become overgrown with scrub. Cicero had the tomb cleaned up, and was able to see the carving and read some of the verses that had been added as an inscription.

The account of the siege of Syracuse given by Polybius in his Universal History was written some seventy years after the death of Archimedes, and was used as a source by Plutarch and Livy. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city.

Discoveries and inventions

The most famous anecdote told about Archimedes is how he discovered the principle of buoyancy. According to Vitruvius, a new crown in the shape of a laurel wreath had been made for King Hieron, and Archimedes was asked to determine whether it was of solid gold, or whether silver had been added by a dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down in order to measure its density as a cube, which would have been the simplest solution. While taking a bath, he noticed that the level of the water rose as he got in. He realized that this effect could be used to determine the volume of the crown, and therefore its density after weighing it. The density of the crown would be lower if cheaper and less dense metals had been added. He then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" "I have found it!" (Greek: "έυρηκα!")

The story about the golden crown does not appear in the known works of Archimedes, but in his treatise On Floating Bodies he gives the principle known in hydrostatics as Archimedes' Principle. This states that a body immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid.

The Archimedes' screw was operated by hand and could raise water efficiently

Another invention bearing his name is the Archimedes' screw, a machine with a revolving screw shaped blade inside a cylinder. It was turned by hand, and used to drain ships or transfer water from a low-lying body of water into irrigation canals. Versions of the Archimedes' screw are still in use today in developing countries. The Archimedes' screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.

While Archimedes did not invent the lever, he wrote the earliest known rigorous explanation of the principles involved, which are the transmission of force through a fulcrum and moving the effort applied through a greater distance than the object to be moved. His Law of the Lever states: Magnitudes are in equilibrium at distances reciprocally proportional to their weights. According to Pappus of Alexandria, his work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth." (Greek: "δος μοι που στω και κινω την γην") Plutarch describes how Archimedes designed block and tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move.

A large part of Archimedes' work in engineering arose from fulfilling the needs of his home city of Syracuse. The Greek writer Athenaeus of Naucratis describes how King Hieron II commissioned Archimedes to design a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a naval warship. The Syracusia is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and contained garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes' Screw was said to have been developed in order to remove the bilge water.

Archimedes may have used mirrors acting as a parabolic reflector to burn ships attacking Syracuse

During the Second Punic War, Archimedes is said to have repelled an attack by Roman forces by using a "burning glass" to focus sunlight on the approaching ships, causing them to catch fire. This claim, sometimes called the "Archimedes death ray", has been the subject of ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes.

It has been suggested that a large array of highly polished bronze shields acting as mirrors could have been employed to focus sunlight on to a ship. This would have used the principle of the parabolic reflector in a manner similar to a solar furnace. In October 2005 a group of students from the Massachusetts Institute of Technology carried out an experiment with 127 one foot square mirror tiles, focused on a mocked-up wooden ship at a range of around 100 feet. Flames broke out on a patch of the ship, but only after the sky had been cloudless and the ship had remained stationary for around ten minutes. It was concluded that the weapon was a feasible device under these conditions. The MIT group repeated the experiment for the television show MythBusters, using a wooden fishing boat in San Francisco as the target. Again some charring occurred, along with a small amount of flame. When Mythbusters broadcast the result of the San Francisco experiment in January 2006, the claim was placed in the category of "myth" due to the length of time and ideal weather conditions required for combustion to occur.

Critics of the MIT experiments have argued that the moisture content of the wood needs to be taken into consideration. However, the flash point of wood is around 300 degrees Celsius (572 degrees Fahrenheit), and this is hotter than the maximum temperature produced by domestic ovens. In real life, a ballista (catapult) armed with flaming bolts would have been a more dangerous weapon, while the effects of the "Archimedes death ray" might have been limited to confusing or temporarily blinding people on board a ship.

The Claw of Archimedes is another weapon that he is said to have designed in order to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped on to an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it. As with the "Archimedes death ray" there have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitled Superweapons of the Ancient World built a version of the claw and concluded that it was a workable device. No contemporary drawings of the Claw of Archimedes exist, although the weapon may have been similar in design to a trebuchet.

Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer is said to have been a cart with a gear mechanism that dropped a ball into a container after each mile traveled.

Cicero wrote that after the capture of Syracuse, General Marcellus took two mechanical devices back to Rome that were used as aids in astronomy. He credits Thales and Eudoxus with constructing these devices. The motions of the Sun, Moon and five planets were shown by one device, and it was demonstrated to Cicero some 150 years later by a man named Gallus. Cicero described the event as follows:

Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in [caelo] sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione. — When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze [contrivance] as in the sky itself, from which also in the sky the Sun's globe became [to have] that same eclipse, and the Moon came then to that position which was [its] shadow [on] the Earth, when the Sun was in line.

The device described by Cicero is a planetarium or orrery. Pappus of Alexandria stated that Archimedes had written a manuscript (now lost) on the construction of these devices entitled On Sphere-Making. Recent research in this area has been focused on the Antikythera mechanism, another device from classical antiquity that was probably designed for the same purpose. Constructing devices of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.

Mathematics

Although he is often regarded as a designer of mechanical devices, Archimedes also made important contributions to the field of mathematics. Plutarch wrote: “He placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life.”

Archimedes used the method of exhaustion to approximate the value of π

Some of his mathematical proofs involve the use of infinitesimals in a way that is similar to modern integral calculus. By assuming a proposition to be true and showing that this would lead to a contradiction, Archimedes was able to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the value of π (Pi). He did this by drawing a larger polygon outside a circle, and a smaller polygon inside the circle. As the number of sides of the polygon increases, it becomes a more accurate approximation of a circle. When the polygons had 96 sides each, he calculated the lengths of their sides and showed that the value of π lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). This was a remarkable achievement, since the ancient Greek number system was awkward and used letters rather than the positional notation system used today. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle.

He used the method of exhaustion to show that the value of the square root of 3 lay between 265/153 (approximately 1.732) and 1351/780 (approximately 1.7320512). The modern value is around 1.7320508076, making this a very accurate estimate.

Another noted mathematical work by Archimedes is The Sand Reckoner. In this work he set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote: "There are some, King Gelon (Gelon II, son of Hieron II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited." To solve the problem, Archimedes devised a system of counting based around the myriad. This was a word used to mean infinity, based on the Greek word for uncountable, murious. The word myriad was also used to denote the number 10,000. He proposed a number system using powers of myriad myriads (100 million) and concluded that the number of grains of sand required to fill the universe would be 8×10⁶³ in modern notation.

In the field of geometry, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height (see illustration on right).

He expressed the solution to the problem as a geometric progression that summed to infinity with the ratio 1/4:

$\sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}\;.$

If the first term in this series is the area of the triangle in the illustration then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines in the illustration, and so on. This proof is a variation of the infinite series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

It has been suggested that Heron's formula for calculating the area of a triangle from the length of its sides was known to Archimedes. However, the first reliable reference to this formula is given by Heron of Alexandria in the 1st century AD.

The Archimedes Palimpsest

Main article: Archimedes Palimpsest

Archimedes' work was not as widely recognized in classical antiquity as that of Euclid, and several of his treatises are believed to have been lost when the Library of Alexandria was damaged at various periods in its history. Some writings survived through copies collected by the Byzantine architect Isidore of Miletus (c. 530 AD) and translations into Latin and Arabic made during the Middle Ages . During the Renaissance, the Editio Princeps (First Edition) was published in Basel in 1544 by Johann Herwagen and contained the works of Archimedes in Greek and Latin. Around the year 1586 Galileo Galilei invented a hydrostatic balance for weighing metals in air and water after apparently being inspired by the work of Archimedes.

The most important document containing Archimedes' work is the Archimedes Palimpsest. A palimpsest is a document written on vellum that has been re-used by scraping off the ink of an older text and writing new text in its place. This was often done in the Middle Ages since animal skin parchments were expensive. In 1906, the Danish professor Johan Ludvig Heiberg realized that a goatskin parchment containing prayers written in the 13th century AD also carried an older work written in the 10th century AD, which he identified as previously unknown copies of works by Archimedes. The parchment spent many years in a monastery library in Constantinople before being sold to a private collector, and reappeared at an auction at Christie's in London in October 1998 where it was sold to an anonymous buyer for $2 million. The palimpsest contains seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. Most importantly, it contains the only known source of the Method of Mechanical Theorems and Stomachion, which had previously been thought lost. The palimpsest is now stored at the Walters Art Museum in Baltimore, Maryland, where it has been subjected to a range of modern tests including the use of ultraviolet and x-ray light to read the overwritten text.

The treatises contained in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, The Measurement of the Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.

Writings by Archimedes

Archimedes is said to have remarked about the lever: "Give me a place to stand on, and I will move the Earth."

• On the Equilibrium of Planes (2 volumes)

This treatise explains the Law of the Lever, and uses it to calculate the areas and centers of gravity of various geometric figures including triangles, paraboloids, and hemispheres.

• On Spirals

In this treatise he defines what is now called the Archimedes spiral. This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.

• On the Sphere and the Cylinder

In this treatise, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. He proves that the sphere will have exactly two thirds of the volume and area of the cylinder. A carving of this proof was used on his tomb.

• On Conoids and Spheroids

In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

• On Floating Bodies (2 volumes)

In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This was probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. His fluids are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.

In the second part, 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, similar to the way that icebergs float.

• The Quadrature of the Parabola

In this treatise he proves that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric progression that sums to infinity with the ratio 1/4. (See the diagram in Mathematics).

File:Archimedes greece 1983.jpg

Archimedes appears on a Greek postage stamp from 1983. He is demonstrating the quadrature of the parabola, while the weighing scale demonstrates the principle of buoyancy.

• Stomachion

This is a Greek puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the various pieces. 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 an early use of combinatorics to solve a problem.

• Archimedes' cattle problem

Archimedes wrote a letter to the scholars in the Library of Alexandria, who had apparently questioned the importance of his work. He challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine 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 computer in 1965, and the answer is a very large number, approximately 7.760271×10²⁰⁶⁵⁴⁴.

• The Sand Reckoner

In this treatise, Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the heliocentric theory of the solar system proposed by Aristarchus of Samos (concluding that "this is impossible") , contemporary ideas about the size of the Earth and the distance between various celestial bodies. By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×10⁶³ in modern notation. The introductory letter contains the information that Archimedes' father was an astronomer named Phidias. The Sand Reckoner or Psammites is the only surviving work in which Archimedes discusses his views on astronomy.

• The Method of Mechanical Theorems

This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses infinitesimals, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. Archimedes may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results.

Posthumous recognition

The Fields Medal carries a portrait of Archimedes

There is a crater on the Moon named Archimedes in his honor, and a lunar mountain range, the Montes Archimedes.

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with his proof concerning the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to him which reads in Latin: Transire suum pectus mundoque potiri (Rise above oneself and grasp the world).

The exclamation of Eureka! attributed to Archimedes is the state motto of California.

See also

• Archimedes' Axiom
• Archimedes' cattle problem
• Archimedes number
• Archimedes Paradox
• Archimedes' use of infinitesimals
• 1/4 + 1/16 + 1/64 + 1/256 + · · · This infinite series was used by Archimedes.
• Heron of Alexandria
• Zhang Heng

Books

• E. J. Dijksterhuis, Archimedes, 1987, Princeton University Press, Princeton, ISBN 0-691-08421-1. Republished translation of the 1938 study of Archimedes and his works by an historian of science.
• T L Heath, Works of Archimedes, Dover Publications, ISBN 0-486-42084-1. Complete works of Archimedes in English.
• Dennis L Simms, Archimedes the Engineer, Continuum International Publishing Group Ltd , ISBN 0-720-12284-8
• Mary Gow, Archimedes: Mathematical Genius of the Ancient World, Enslow Publishers, Inc, ISBN 0-7660-2502-0
• Martin Gardner, Archimedes, Mathematician and Inventor, The Macmillan Company

External links

• Archimedes—The Greek mathematician and his Eureka moments—BBC Radio 4 discussion from In Our Time, broadcast 25 January 2007 (requires RealPlayer).
• Archimedes' Book of Lemmas at cut-the-knot
• Archimedes and the Rhombicuboctahedron by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Archimedes Home Page
• O'Connor, John J.; Robertson, Edmund F., "Archimedes", MacTutor History of Mathematics Archive, University of St Andrews
• The Archimedes Palimpsest web pages at the Walters Art Museum.
• NOVA program on Archimedes Palimpsest
• The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
• The Archimedes Water Clock and other examples of ancient technology
• Archimedes—The Golden Crown points out that in reality Archimedes may well have used a more subtle method than the one in the classic version of the story.
• Archimedes' Quadrature Of The Parabola Translated by Thomas Heath.
• Archimedes' On The Measurement Of The Circle Translated by Thomas Heath.
• The Works Of Archimedes by Thomas Heath.
• Works by Archimedes at Project Gutenberg
• Archimedes and his Burning Mirrors, Reality or Fantasy?
• Biography of Archimedes
• Squaring the circle History Topic at MacTutor
• Archimedes—The Greatest Scientist Ever
• New commentary on Aristotle found by researchers in Archimedes Palimpsest—27 April 2007