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Surviving works

Se cuenta que Arquímedes dijo sobre la palanca: "Deme un punto de apoyo y moveré el mundo"

• Sobre el equilibrio de los planos (dos volúmenes)

El primer libro tiene quince proposiciones con siete axiomas, mientras el segundo tiene diez proposiciones. En esta obra, Arquímedes explica la ley de la palanca, la cual afirma:

Las magnitudes están en equilibrio a distancias recíprocamente proporcionales a sus pesos.

Arquímedes usa los principios derivados para calcular las áreas y los centros de gravedad de varias figuras geométricas, incluyendo triángulos, paralelogramos y parábolas.​

• Sobre la medida de un círculo

Esta es una obra corta consistente en tres proposiciones. Esté escrito en forma de una carta a Dositeo de Pelusio, quien fue un estudiante de Conon de Samos. En la proposición II, Arquímedes muestra que el valor de π (Pi) es mayor que 223/71 y menor que 22/7. Esta cifra fue usada como una aproximación de π a través de la Edad Media y aún hoy es usada cuando se requiere de una cifra aproximada.

• Sobre las espirales

Esta obra de 28 proposiciones también está dirigida a Dositeo. El tratado define lo que hoy se conoce coom espiral de Arquímedes. Es el lugar geométrico de los puntos correspondientes a las posiciones de un punto, a través del tiempo, que es movido hacia afuera desde un punto fijo con una velocidad constante junto con una línea que rota con una velocidad angular constante. Equivalentemente, en coordenadas polares (r, θ) puede ser descrito por la ecuación

$\,r=a+b\theta$

con a y b como números reales. Este es un ejemplo temprano de la curva mecánica (una curva trazada por un punto) considerado por un matemático griego.

• Sobre la esfera y el cilindro (dos volúmenes)

En este tratado dirigido a Dositeo, Arquímedes obtiene el resultado del que estaría más orgulloso, a saber, la relación entre una esfera y un cilindro cirscunscrito con la misma altura y diámetro. El volumen es ${\tfrac {4}{3}}\pi r^{3}$ para la esfera, y $2\pi r^{3}$ para el cilindro. El área de la superficie es $4\pi r^{2}$ para la esfera, y $6\pi r^{2}$ para el cilindro (incluyendo sus dos bases), donde $r$ es el radio de la esfera y el cilindro. La esfera tiene un área y un volumen equivalentes a dos tercios de los del cilindro. A pedido del propio Arquímedes, se colocaron sobre su tumba las esculturas de un cilindro y de una esfera.

• Sobre los conoides y los esferoides

Este es un trabajo en 32 proposiciones dirigido a Dositeo. En este tratado, Arquímedes calcula las áreas y los volúmenes de las secciones de cono (geometría)s, esferas y paraboloides.

• On Floating Bodies (two 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 may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating, since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.

Archivo:Archimedes greece 1983.png

Archimedes is commemorated on a Greek postage stamp from 1983.

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. Archimedes' principle of buoyancy is given in the work, stated as follows:

Any body wholly or partially immersed in a fluid experiences an upthrust equal to, but opposite in sense to, the weight of the fluid displaced.

• The Quadrature of the Parabola

In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods 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 series that sums to infinity with the ratio 1/4.

• (O)stomachion

This is a dissection 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 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. The figure given by Dr. Netz is that the pieces can be made into a square in 17,152 ways.​ The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded.​ The puzzle represents an example of an early problem in combinatorics.

The origin of the puzzle's name is unclear, and it has been suggested it is taken from the Ancient Greek word for throat or gullet, stomachos (στόμαχος).​ Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of ὀστέον (osteon - bone) and μάχη (machē - fight). The puzzle is also known as the Loculus of Archimedes or Archimedes' Box.​

• Archimedes' cattle problem

This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes 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. Amthor​ in 1880, 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, 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 states 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. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

Apocryphal works

Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with fifteen propositions on the nature of circles. The earliest known copy of the text is in Arabic. The scholars T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.​

It has also been claimed 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 the formula is given by Heron of Alexandria in the 1st century  AD.​

Archimedes Palimpsest

Stomachion is a dissection puzzle in the Archimedes Palimpsest

The foremost document containing the work of Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople and examined a 174-page goatskin parchment of prayers written in the 13th century AD. He discovered that it was a palimpsest, a document with text that has been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, which was a common practice in the Middle Ages as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th century AD copies of previously unknown treatises by Archimedes.​ The parchment spent hundreds of years in a monastery library in Constantinople 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 in New York.​ The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of the Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. 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 in the Archimedes Palimpsest are: On the Equilibrium of Planes, On Spirals, Measurement of a Circle, On the Sphere and the Cylinder, On Floating Bodies, The Method of Mechanical Theorems and Stomachion.

Legacy

The Fields Medal carries a portrait of Archimedes.

There is a crater on the Moon named Archimedes (29.7° N, 4.0° W) in his honor, and a lunar mountain range, the Montes Archimedes (25.3° N, 4.6° W).​

The asteroid 3600 Archimedes is named after him.​

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).​

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).​

The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance the word refers to the discovery of gold near Sutter's Mill in 1848 which sparked the California Gold Rush.​

A movement for civic engagement targeting universal access to health care in the US state of Oregon has been named the "Archimedes Movement," headed by former Oregon Governor John Kitzhaber.​

See also

• Archimedes' axiom
• Archimedes number
• Archimedes paradox
• Archimedean property
• Archimedes' screw
• Archimedean solid
• Archimedes' twin circles
• Archimedes' use of infinitesimals
• Diocles
• Methods of computing square roots
• Pseudo-Archimedes
• Salinon
• Steam cannon
• Vitruvius
• Zhang Heng

Notes and references

Notes

a. ↑La plantilla {{note label}} está obsoleta, véase el nuevo sistema de referencias.In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.

b. ↑La plantilla {{note label}} está obsoleta, véase el nuevo sistema de referencias.The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar. Of the surviving works by Archimedes, T. L. Heath offers the following suggestion as to the order in which they were written: On the Equilibrium of Planes I, The Quadrature of the Parabola, On the Equilibrium of Planes II, On the Sphere and the Cylinder I, II, On Spirals, On Conoids and Spheroids, On Floating Bodies I, II, On the Measurement of a Circle, The Sand Reckoner.

c. ↑La plantilla {{note label}} está obsoleta, véase el nuevo sistema de referencias.Boyer, Carl Benjamin A History of Mathematics (1991) ISBN 0471543977 "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula — k = √(s(s − a)(s − b)(s − c)), where s is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken chord' … Archimedes is reported by the Arabs to have given several proofs of the theorem."

References

• Wikisource en inglés contiene el artículo de la Encyclopædia Britannica de 1911 sobre Archimedes.

Further reading

• Boyer, Carl Benjamin (1991). A History of Mathematics. New York: Wiley. ISBN 0-471-54397-7. 
• Dijksterhuis, E.J. (1987). Archimedes. 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.
• Gow, Mary (2005). Archimedes: Mathematical Genius of the Ancient World. Enslow Publishers, Inc. ISBN 0-7660-2502-0. 
• Hasan, Heather (2005). Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1404207745. 
• Heath, T.L. (1897). Works of Archimedes. Dover Publications. ISBN 0-486-42084-1.  Complete works of Archimedes in English.
• Netz, Reviel and Noel, William (2007). The Archimedes Codex. Orion Publishing Group. ISBN 0-297-64547-1. 
• Pickover, Clifford A. (2008). Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0195336115. 
• Simms, Dennis L. (1995). Archimedes the Engineer. Continuum International Publishing Group Ltd. ISBN 0-720-12284-8. 
• Stein, Sherman (1999). Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 0-88385-718-9. 

The Works of Archimedes online

• Text in Classical Greek: PDF scans of Heiberg's edition of the Works of Archimedes, now in the public domain
• In English translation: The Works of Archimedes, trans. T.L. Heath; supplemented by The Method of Mechanical Theorems, trans. L.G. Robinson