# Archimedes

Archimedes of Syracuse ( Greek Ἀρχιμήδης ὁ Συρακούσιος Archimḗdēs ho Syrakoúsios ; * around 287 BC, probably in Syracuse ; † 212 BC there ) was a Greek mathematician , physicist and engineer . He is considered one of the most important mathematicians of the ancient world . His works were still important in the development of higher analysis in the 16th and 17th centuries .

## Life

Archimedes in his circles: Sculpture on the square in front of the Freiherr-vom-Stein-Gymnasium (Fulda)

Little is known about the life of Archimedes and much is considered legend .

Archimedes, born around 287 BC. BC probably in the port city of Syracuse in Sicily, was the son of Pheidias, an astronomer at the court of Hieron II of Syracuse. With this and his son and co-regent Gelon II he was friends and possibly related.

During a longer stay in Alexandria , Archimedes met the mathematicians Konon , Dositheos and Eratosthenes there, with whom he later corresponded.

When he returned to Syracuse, he did math and practical physics (mechanics). His throwing machines were used in the defense of Syracuse against the Roman siege in the Second Punic War . During the conquest of Syracuse in 212 BC After three years of siege by the Roman general M. Claudius Marcellus , he was killed by a Roman soldier, much to the regret of Marcellus, who wanted to arrest him alive. Plutarch reports several versions of the circumstances in his biography of Marcellus, after one he was busy with a mathematical proof and asked a soldier who was raiding the city not to disturb him, whereupon he killed him. The words Noli turbare circulos meos ( Latin for: “Do not disturb my circles”), which Archimedes is said to have spoken, became proverbial .

According to Plutarch , Archimedes wanted in his will a tomb with a representation of a sphere and cylinder , as he was obviously particularly proud of his treatise perì sphaíras kaì kylíndrou ("On sphere and cylinder"). In the Tusculan Talks , Cicero reports that during his time as quaestor in Sicily (75 BC) he looked for the grave and found it overgrown with undergrowth.

A biography written by his friend Heracleides has not survived.

## Fonts

The main fonts preserved are:

• About the balance of flat surfaces , Greek Περὶ ἐπιπέδων ἰσορροπιῶν, Peri epipédōn isorrhopíai , Latin De planorum aequilibriis , transcribed in two books.
• Quadrature of the parabola , Latin De quadratura parabolae . Content: area of ​​a parabolic segment.
• About the method , Latin De methodo . Preserved as a fragment in the Archimedes palimpsest found by Heiberg .
• About sphere and cylinder , Greek Περὶ σφαίρας καὶ κυλίνδρου, transcribed Peri sphaíras kai kylíndrou , Latin De sphaera et cylindro , 2 volumes. Contents: volume of sphere and cylinder.
• About spirals , Latin De lineis spiralibus . Content: Area of ​​an object he invented, the spiral line. The Archimedes' spiral was probably invented by his friend Konon.
• About conoids and spheroids , Latin De conoidibus et sphaeroidibus . Contents: volumes of hyperbolas and ellipses.
• About floating bodies , 2 books, peri ochoumenon transcribed in Greek , De corporibus fluitantibus in Latin . Contents: volume and specific weight of bodies, hydrostatics.
• Loop measurement , Greek Κύκλου μέτρησις transcribed Kýklou métrēsis , Latin Dimensio circuli .
• The bill of sand , transcribed in Greek Psammites , Latin Arenarius . Content: Representation of arbitrarily large numbers, heliocentric worldview of Aristarchos of Samos .

The order of the main scriptures up to the sand calculation given here corresponds to the chronological order as it was given by Thomas Heath , whereby the quadrature of the parabola between books 1 and 2 was classified by Equilibrium of Flat Areas and About the Method between Equilibrium of Flat Areas , Book 2 , and About sphere and cylinder . But there was also criticism of the chronology.

In the square of the parable , the recent death of his friend Konon is mentioned, so that this writing dates back to 240 BC. Can be dated BC. According to the above-mentioned relative dating, most of Archimedes' works were created afterwards. According to Archimedes, the book about spirals was written many years after Konon's death. According to Ivo Schneider, it was written around 230 BC. Is to be dated. Schneider classified the methodology at the end of the 220s and the swimming bodies as the last work in the last eight years of life, but probably before 216 BC. Because of the subsequent war events.

There are references to some writings that have been lost today, for example about polyhedra and levers (mentioned by Pappos ), about the representation of numbers (mentioned by Archimedes in his sand calculator ) and about mirrors ( Catoptrica , mentioned by Theon of Alexandria ). From the incompleteness of Archimedes' mechanical writings ( equilibrium of flat surfaces , quadrature of the parabola ) and several references from Archimedes (and, for example, from Heron of Alexandria ), it was concluded that parts of his mechanics had been lost, which the Drachmann group tried to reconstruct. These partially reconstructed mechanical writings are chronologically at the beginning of the works of Archimedes.

There are some references to lost Archimedes' writings in Arabic translation, such as a book about the postulate of parallels , which is listed in Ibn al-Nadim's catalog of books and which may have influenced the treatment of the subject in Thabit Ibn Qurra.

## plant

Medieval fantasy portrait of Archimedes

Archimedes was equally creative in both mathematics and today's physics.

### physics

He was also credited with inventing and combining various machine elements (such as screws, cable pulls with corrugated gears, pulley blocks and gear wheels), which he also demonstrated in practice. After Plutarch, he preferred abstract thinking and looked down on practical applications and the work of an engineer, although he did them on behalf of his King Hieron, with contempt. For this reason he did not leave a treatise on practical inventions. His writings on mechanics and hydrostatics are strictly axiomatic based on the example of geometry .

#### Law of leverage

Archimedes formulated the laws of levers (in his work On the Equilibrium of Flat Surfaces ) and thereby created the theoretical basis for the later development of mechanics . He himself developed the scientific basis of statics for statically determined systems from the lever law . The description of the lever itself can be found in older Greek writings from the school of Aristotle.

He is said to have said (as Pappos and others narrated): " Δός μοι ποῦ στῶ, καὶ τὴν γῆν κινήσω " ("Give me a fixed point and I will turn the world off its hinges"). The term Archimedean point is based on this . When he once said this to Hieron, the latter demanded practical proof from Plutarch, and Archimedes managed, among other things, with pulleys (Plutarch) and rope winches, the movement of a large fully laden ship by a single man.

#### Archimedean principle

According to Vitruvius , Archimedes should check the gold content of a crown consecrated to the gods by ruler Hieron II , but without damaging it. The king suspected the goldsmith of having betrayed him. To solve the given problem, he once dipped the crown and then a gold bar (and a silver bar) that weighed just as much as the crown into a full water container and measured the amount of overflowing water. The crown displaced more water than the gold bar. This proved that the crown had a lower specific weight and was therefore not entirely made of gold. According to legend, Archimedes discovered the Archimedes' principle while bathing. The amount of water that ran out of the brimful water tank that he displaced with his body volume when he got into the bathroom. Happy about his discovery he is with the exclamation " Eureka (ancient Greek:" ηὕρηκα / hɛːǔ̯rɛːka / , "I found it!") Have run naked into the street. The anecdote about checking the gold content of Hieron's crown by water displacement has been criticized - it would have been difficult to carry out with the resources of the time and is probably a legend. As early as 1586, Galileo Galilei suspected that Archimedes would have used scales instead to measure weights under buoyancy.

The Archimedes' principle can be applied to any floating body. It is a fact that must be taken into account in shipbuilding. In his hydrostatic experiments, he also discovered the principle of communicating vessels .

### mathematics

Bronze sculpture intended to represent Archimedes at the Archenhold Observatory in Treptower Park , Berlin ( Gerhard Thieme 1972)

#### Area calculations

Archimedes proved that the circumference of a circle is related to its diameter in the same way as the area of the circle is related to the square of the radius . He did not call this ratio (now known as pi or circle number ) π (pi), but gave instructions on how to approach the ratio to an arbitrarily high level of accuracy, probably the oldest numerical method in history. Archimedes anticipated the ideas of integral calculus much later with his considerations on area and volume calculation (including with an exact quadrature of the parabola ) . He went beyond the exhaustion method attributed to Eudoxus by Knidos (exhaustion method ); for example, he was already using a form of Cavalieri's principle .

In 1906, Johan Ludvig Heiberg (1854–1928), a Danish philologist and professor at the University of Copenhagen , found a manuscript dated to the 10th century in Istanbul , which contained, among other things, a copy of Archimedes' work The Method .

In it he reveals a mechanical method with which he had achieved many of his results before he proved them in a geometrically strict manner. The method corresponds to weighing the volumes or areas to be compared, but in geometric form. In his description, Archimedes also mentions an older method used by Democritus , which may be the weighing of models.

#### Heptagon after Archimedes

Thabit Ibn Qurra translated a treatise by Archimedes on the construction of a regular heptagon , known as the heptagon according to Archimedes . The construction was incomplete, but it was completed by Abu Sahl al-Quhi .

This construction of the heptagon according to Archimedes after Abu Sahl al-Quhi - more precisely an approximation - uses, it has been handed down, the construction method of insertion (Neusis) . In this case, a corner of the ruler serves as a pivot point to find the end point of a route to be determined with the help of the ruler edge and by corresponding "wobbling". The way Archimedes himself found the length of this route - e.g. B. using a conic section or a special curve, as shown in heptagon after Archimedes - has not survived.

#### Place value-based number system

Archimedes also developed a digit-based number system based on 10 8 .

He used it in order to be able to mathematically grasp astronomically large numbers (up to the size of 10 64 ) - this at a time when his surroundings already equated a myriad (lit. 10,000) with "infinite". The reason for this was the treatise on floating bodies and the number of sand , also known as the sand calculator for short , which he dedicated to the son of Hieron II, Gelon. It says: “There are people, King Gelon, who are of the opinion that the number of sand is infinitely great […] Others do not believe that the number is infinite, but that no number has yet been named could exceed its quantity. ” Since Gelon is addressed as king, the script was written after 240 BC. When he became co-regent (and before Gelon's death in 216 BC).

He refuted these notions by estimating and naming the number of grains of sand that covered all the beaches on earth in the treatise . He went even further and calculated the number of grains of sand it would take to fill the whole universe with sand. At that time, however, the universe was imagined to be much smaller - namely as a sphere about the size of our solar system . Archimedes' calculation therefore says that an imaginary sphere the size of our solar system would fit about 10 64 grains of sand.

#### Archimedes' axiom

Although named after him, the Archimedean axiom does not come from Archimedes, but goes back to Eudoxus of Knidos , who introduced this principle as part of his theory of magnitude .

#### Archimedean solids

The original work by Archimedes has not been preserved. However, there is still a work by the mathematician Pappos (approx. 290-350 AD), in which it is mentioned that Archimedes described the 13 Archimedean solids .

### technology

Archimedes significantly influenced the technology of his time and the later development of technology, especially mechanics. He himself constructed all kinds of mechanical devices, not least machines of war.

#### Archimedean screw

Archimedes is credited with inventing the so-called Archimedes' screw , to which he was inspired after seeing the simple devices for field irrigation there while studying in Egypt. The principle of the Archimedean screw is used today in modern conveyor systems, so-called screw conveyors .

A painting of Archimedes' Claw

It may have been developed by Archimedes as a bilge pump for ships, because after Atheneus of Naukratis , King Hieron commissioned Archimedes to build the largest ship of the time, the Syracusia .

#### Machines of war during the siege of Syracuse

According to Plutarch, Archimedes is said to have stopped the Romans during their protracted siege with the war machines he had developed: For example, he developed throwing machines and catapults or cable winches , which moved an entire ship, fully loaded and with its entire crew, by pulling on a single rope. They also included powerful grab arms that grabbed enemy boats and supposedly tore them to pieces.

The claw of Archimedes is said to have been a weapon against attacking fleets, which was built into the city wall of Syracuse and used during the siege against the Roman fleet. The exact function of this weapon is unclear, however. In ancient writings, the weapon is represented as a lever with a large iron hook. As early as 425 BC The city of Syracuse had a naval weapon described as an "iron hand" with which one could board ships ( Thucydides , Pel. Kr. IV, 25), possibly a grappling hook .

Copper engraving on the title page of the Latin edition of the Thesaurus opticus , a work by the Arab scholar Alhazen . The illustration shows how Archimedes is said to have set Roman ships on fire with the help of parabolic mirrors.

#### Burning mirror

In addition, Archimedes is said to have set the ships of the Romans on fire, even over great distances, with the help of mirrors that deflected and focused the sunlight. This is reported by Lukian of Samosata and later by Anthemios of Tralleis . There is a fierce controversy about this that has lasted for over 300 years. Historically speaking, the sources, translation issues ( pyreia was often translated as burning mirror , although it only means "inflammation" and also includes arrows ) and the legend that only appeared centuries later. Physical counter-arguments are the necessary minimum size and focal length of such a mirror, the minimum temperature to be reached for igniting wood (approx. 300 degrees Celsius) and the time that the piece of wood to be ignited must remain constantly lit. Technical counter-arguments discuss the manufacturability of such mirrors at the time, the assembly of a mirror or mirror system and the operability. A modern critic of the legend was the pyrotechnician Dennis L. Simms. Several experiments were carried out to ensure feasibility. In 2005, students at the Massachusetts Institute of Technology and the University of Arizona successfully ignited a model of a ship's wall 30 meters away using 127 small mirrors after an attempt had previously failed with two mirrors. However, the sky had to be cloudless and the ship had to be irradiated for around 10 minutes. A repeated attempt on a fishing boat on the television program MythBusters with 500 volunteers (broadcast in January 2006) with the participation of MIT students in the port of San Francisco , which came to similar results, was therefore classified as a failure. It was also noted that the sea in Syracuse is to the east, so the Roman fleet should have attacked in the morning, and that projectiles and incendiary arrows would have been more effective. Possibly the story originated as a conclusion from the lost script of Archimedes Katóptrika ( optics ).

#### More inventions

After Cicero ( De re publica ) Marcellus brought two mechanical planetariums developed by Archimedes back to Rome. Similar devices, according to Cicero, were built by Eudoxus of Knidos and Thales of Miletus - archaeological evidence of such instruments was later found in the Antikythera mechanism . The lost work of Archimedes on the production of spheres, mentioned by Pappos, may be about the construction of planetariums.

He is also credited with inventing an odometer . A corresponding odometer with a counting mechanism with balls was described by Vitruvius. Vitruvius does not reveal the inventor (only that he was handed down by the ancients ), but Archimedes was also suspected to be the inventor here. A water clock mechanism that releases balls as a counting aid, described in an Arabic manuscript, has also been ascribed to him.

Leonardo da Vinci and Petrarch (who referred to a Cicero manuscript) attributed the invention of a steam cannon to Archimedes . Leonardo also made reconstruction sketches for the machine he called Architronito. There were later attempts at replicas, such as by the Greek Ioannis Sakas in 1981 and the Italian engineer Cesare Rossi from the University of Naples in 2010. Rossi also gave the burning mirrors a new interpretation - they would have provided the heat for the steam generation. In the traditional ancient writings by and about Archimedes, however, there are no references to this and experts such as Serafina Cuomo see it as just further evidence of the legendary reputation of Archimedes, to whom all possible inventions are ascribed. Steam power was known to the Greeks in principle ( Heronsball , 1st century AD).

## Lore

Knowledge of the works of Archimedes was not very widespread in antiquity, despite his legendary fame, in contrast to Euclid , for example , who compiled his book in the then scientific center of Alexandria. However, it is often mentioned by the mathematicians Heron , Pappos and Theon in Alexandria. The writings were systematically collected and commented on in Byzantium between the 6th and 10th centuries . The commentary by Eutokios (who lived from the end of the 5th century to the beginning of the 6th century) on the most important Archimedes' writings (on spheres and cylinders, circular measurements, equilibrium of flat surfaces) is well known, and was also widely known in Western Europe in the Middle Ages the works contributed and had a stimulating effect. The architects of Hagia Sophia Isidore of Miletus and Anthemios of Tralleis played an important role in the first compilation of the writings in Byzantium . Further writings were added, until in the 9th century Leon of Thessaloniki brought out the collection known as Codex A (Heiberg) of almost all archimedean writings (except Stomachion , Cattle Problem , About the Method and About Swimming Bodies ). This was one of the two sources for the Latin translations by Wilhelm von Moerbeke (completed in 1269). The other Greek manuscript of Archimedes available to him contained the equilibrium of flat surfaces , quadrature of the parabola , about floating bodies , perhaps also about spirals and was called Codex B by Heiberg. The Archimedes Palimpsest discovered by Heiberg in 1906 (Codex C, which was previously in Jerusalem, it contained On the Method , Stomachion and On Floating Bodies ) was unknown to translators in the Middle Ages and Renaissance. Codices A and B came from the possession of the Norman kings in Sicily to the Vatican, where Moerbeke used them for his translation. While Moerbeke's translation manuscript has been preserved in the Vatican, Codex B has been lost. On the other hand, several copies of Codex A have survived (nine are known), which were in the possession of Cardinal Bessarion (now in the Biblioteca Marciana ) and Giorgio Valla , for example . The original of Code A has also disappeared.

The translations of Wilhelm von Moerbeke particularly stimulated the scholars of the Paris School ( Nicole Oresme , Johannes de Muris ).

There is also an Arabic text tradition. Archimedes' most important works On Sphere and Cylinder and On Circular Measurement were translated into Arabic as early as the 9th century and reissued again and again at least until the 13th century. They also worked in the west from the 12th century. In particular, a translation of the circular measurement from Arabic into Latin, probably by Gerhard von Cremona (12th century), was influential in the Middle Ages. He also wrote a Latin translation of a treatise by the Banū Mūsā brothers, which contained further results from Archimedes: in addition to the measurement of circles and the Heron's theorem (which the Arabs often ascribed to Archimedes), parts from About Sphere and Cylinder . This manuscript, known as Verba filiorum , also suggested, for example, Leonardo Fibonacci and Jordanus Nemorarius . Both worked as mathematicians before the time when Moerbeke's translation was written.

Around 1460 Pope Nicholas V had Jacob von Cremona done a new translation into Latin, based on Codex A. It also contained the parts of the work not yet translated by Moerbeke (sand calculator and Eutokios commentary on circular measurements). Since Codex B was not available to him, the edition does not include About floating bodies . This translation was used by Nikolaus von Kues , among others .

The first printed edition (with the exception of excerpts that Giorgio Valla printed in 1501) were the Latin translations of the measurement of circles and squaring of the parable by Luca Gaurico in Venice in 1503 (based on a manuscript from Madrid). They were published again in 1543 by Nicolo Tartaglia , together with Moerbeke's translations of Equilibrium of Flat Surfaces and About Floating Bodies .

The first edition of the Greek text appeared in Basel in 1544 (edited by Thomas Venatorius , German Gechauff) together with a Latin translation by Jakob von Cremona (corrected by Regiomontanus ). The edition also contained Eutokios' comments. For the Latin text he used a copy of the translation by Jacob von Cremona (edited by Regiomontanus) brought to Germany by Regiomontanus around 1468, as well as for the Greek text a manuscript brought by Willibald Pirckheimer from Rome to Nuremberg. It was a copy of Codex A, which is why this Editio Princeps edition also lacks On Swimming Bodies . In 1558 a Latin translation of some of the main works by Federicus Commandinus was published in Venice. Other important editions before the Heiberg edition were by D´Rivault (Paris 1615), who only brings the propositions in Greek and the evidence in Latin, and by Giuseppe Torelli (Oxford 1794).

## Others

A portrait of Archimedes is embossed on the highest mathematician award, the Fields Medal .

In his honor, a moon crater on the Mare Imbrium was named Archimedes ; see Archimedes (moon crater) .

The asteroid (3600) Archimedes also bears his name.

István Száva wrote the novel The Giant of Syracuse (Prisma, Leipzig 1960, Corvina, Budapest 1960, 1968, 1978).

## Text output

• Archimedis Opera Omnia. Cum commentariis Eutocii , 3 volumes, Stuttgart, Teubner 1972 (Bibliotheca scriptorum Graecorum et Romanorum Teubneriana, reprint of the 2nd edition, Teubner, Leipzig 1910–1915, first edition 1880/81, edition by Heiberg , with the comments by Eutokios )
• Volume 4 of the 1972 reprint was published by Yvonne Dold-Samplonius , H. Hermelink, M. Schramm Archimedes: About circles touching each other , Stuttgart 1975
• Archimède (4 vol.), Ed. Charles Mugler, Paris 1971 (with French translation)

## Translations

Archimēdous Panta sōzomena , 1615
• Archimedes, Werke , Darmstadt, Wissenschaftliche Buchgesellschaft 1963, 1972 (translation by Arthur Czwalina based on the edition by Heiberg for Ostwald's classic in one volume)
• Archimedes, Werke , Verlag Harri Deutsch, 3rd edition 2009, ISBN 978-3-8171-3425-0 , (based on the translation by Arthur Czwalina), includes reprints of:
• About floating bodies and the number of sand , Ostwald's classic, Volume 213, Leipzig, Akademische Verlagsgesellschaft 1925
• The quadrature of the parabola and On the balance of flat surfaces or on the center of gravity of flat surfaces , Ostwald's classic, Volume 203, Leipzig, Akademische Verlagsgesellschaft 1923
• Sphere and cylinder , Ostwald's classics, volume 202, Leipzig, Academic Publishing Society 1922
• On paraboloids, hyberboloids and ellipsoids , Ostwald's classics, volume 210, Leipzig, Akademische Verlagsgesellschaft 1923
• About spirals , Ostwald's classics, Volume 201, Leipzig, Academic Publishing Society 1922
• Ferdinand Rudio : Archimedes, Huygens, Lambert, Legendre. Four treatises on circular measurement. Teubner, Leipzig 1892. (digitized version) (Archimedes' treatise on circular measurement)
• Heiberg Eine neue Archimedeshandschrift , Hermes: Zeitschrift für Philologie, Volume 42, 1907, pp 235-303 (Archimedes long-lost treatise on the method)
• English translation: Geometrical solutions derived from mechanics, a treatise of Archimedes, recently discovered and translated from the Greek by Dr. JL Heiberg , Chicago, the Open Court Publishing Company 1909 (introduced by David Eugene Smith ), online with Gutenberg
• The method of Archimedes - recently discovered by Heiberg. A supplement to the works of Archimedes 1897 , edited by Thomas L. Heath, Cambridge University Press 1912
• Thomas Little Heath (Ed.): The Works of Archimedes. Cambridge 1897, Dover Publications, Mineola NY 1953, 2002. ISBN 0-486-42084-1 . (in the Dover edition with the method)
• Reviel Netz (editor and translator): Works of Archimedes (with a critical edition of the diagrams and a translation of Eutocius commentary), Vol. 1, Cambridge University Press 2004 (with commentary, laid out in three volumes), ISBN 0-521- 66160-9 .
• Paul ver Eecke Les œuvres complètes d'Archimède, traduites du grec en français avec une introduction et des notes , Paris, Brussels 1921, 2nd edition, Paris 1960 with the translation of Eutokios' commentaries

## literature

Overview representations

Overall presentations and investigations

• Ivo Schneider : Archimedes. Engineer, scientist and mathematician. Wissenschaftliche Buchgesellschaft, Darmstadt 1979. ISBN 3-534-06844-0 , new edition Springer 2016
• Reviel Netz, William Noel: The Codex of Archimedes - the most famous palimpsest in the world is deciphered. CH Beck 2007, ISBN 3-406-56336-8 (English: The Archimedes Codex. Weidenfeld and Nicholson 2007)
• Günter Aumann : Archimedes. Mathematics in turbulent times. Scientific Book Society, 2013
• Klaus Geus : Mathematics and Biography: Notes on a Vita of Archimedes. In: Michael Erler, Stefan Schorn (eds.): The Greek biography in the Hellenistic period: files of the international congress from July 26th to 29th, 2006 in Würzburg. Walter de Gruyter, Berlin 2007. pp. 319–333 (contributions to antiquity; 245).
• Dennis Simms: Archimedes the Engineer. In: History of Technology. Volume 17, 1995, pp. 45-111.
• Sherman Stein: Archimedes. What did he do besides cry Eureka? Mathematical Association of America, 1999
• Andre Koch, Torres Assis: Archimedes, the Center of Gravity, and the First Law of Mechanics. Aperion Publishers, Montreal 2008 ( online )
• Chris Rorres: Completing Book 2 of Archimedes On Floating Bodies. In: Mathematical Intelligencer. Volume 26, No. 3, 2004 ( online )
• Eduard Jan Dijksterhuis : Archimedes. Groningen 1938 (Dutch), English translation Copenhagen 1956, Reprinted by Princeton University Press 1987 (with an overview of the more recent research by Wilbur Richard Knorr )
• Isabella Grigoryevna Bashmakowa : Les méthodes différentielles d'Archimède. Archive History Exact Sciences, Volume 2, 1962/66, pp. 87-107

reception

• Marshall Clagett : Archimedes in the Middle Ages. 5 volumes, Volume 1: University of Wisconsin Press 1964, Volumes 2 to 5: Memoirs of the American Philosophical Society 1976, 1978, 1980, 1984
• Volume 1: The Arabo-Latin tradition
• Volume 2: The translations from the Greek by William of Moerbeke (in two books, with English and Latin text)
• Volume 3: The fate of the medieval Archimedes 1300–1565, in three books (Part 1: The Moerbeke translations of Archimedes at Paris in the fourteenth century, Part 2: The Arabo-Latin and handbook traditions of Archimedes in the fourteenth and early fifteenth centuries, part 3: The medieval Archimedes in the renaissance, 1450–1565)
• Volume 4: A supplement on the medieval Latin traditions of conic sections (1150–1566), in two books
• Volume 5: Quasi-Archimedean geometry in the thirteenth century, in two books
• Diego De Brasi: Archimedes. In: Peter von Möllendorff , Annette Simonis, Linda Simonis (ed.): Historical figures of antiquity. Reception in literature, art and music (= Der Neue Pauly . Supplements. Volume 8). Metzler, Stuttgart / Weimar 2013, ISBN 978-3-476-02468-8 , Sp. 85-94.

Commons : Archimedes  - collection of images, videos and audio files
Wikisource: Archimedes  - Sources and full texts

Digital copies:

From Archimedes

## Individual evidence

1. ^ A b Sherman K. Stein: Archimedes: What Did He Do Besides Cry Eureka? MAA, 1999, ISBN 0-88385-718-9 , pp. 2–3 ( excerpt (Google) )
2. So reported Archimedes himself in his sand calculator .
3. This is how Plutarch reports in Life of Marcellus . Archimedes dedicated the sand calculator to Gelon.
4. Plutarch, Marcellus.
5. Plutarch, Marcellus 17:12.
6. ^ Cicero: Tusculan Conversations. Latin text, Latin Library Book 5, XXIII, 64, 65
7. ^ Heath: The works of Archimedes. Dover, S. XXXII. It goes back to Heiberg and Hultsch.
8. ^ Ivo Schneider: Archimedes. P. 32 gives the following sequence: 1. Equilibrium of flat surfaces, book 1, 2. Quadrature parabola, 3. Sphere and cylinder, 4. Spirals, 5. Conoids and spheroids, 6. Equilibrium of flat surfaces, book 2, 7th method, 8. Floating bodies
9. ^ Ivo Schneider: Archimedes. P. 33f. Ptolemy III was 241 BC Returned from the Syrian war. His wife Berenike consecrated her hair to Aphrodite as a thank you. Soon afterwards it disappeared, and the naming of a constellation ascribed to Konon after the lock of Berenike can be interpreted as the rediscovery of the lost hair in the sky. According to this, Konon, who died relatively young, 241 BC. Still lived.
10. ^ Ivo Schneider: Archimedes. Chapter 2.3
11. ^ AG Drachmann: Fragments of Archimedes in Heron's mechanics. Centaurus, Volume 8, 1963, pp. 91–146, further writings by Drachmann on the technology of antiquity and especially in Archimedes: The mechanical technology of greek and roman antiquity , Copenhagen 1963, Archimedes and the science of physics , Centaurus, Volume 12, 1967, pp. 1–11, Great Greek Inventors , Zurich 1967
12. Boris Rosenfeld : A history of non euclidean geometry , Springer Verlag 1988, p. 40 f.
13. Chris Rorres: The Lever. Courant Institute
14. ^ Ivo Schneider: Archimedes. 1979, chapter 3.3. On the interpretation of Archimedes' saying also Drachmann: How Archimedes expected to move the earth. Centaurus, Vol. 5, 1958, pp. 278-282
15. De Architectura IX , Foreword, Paragraph 9–12, German translation by Ivo Schneider Archimedes , Culture and Technology, 1979, pdf
16. Chris Rorres: The Golden Crown. Drexel University, 2009
17. Chris Rorres: The Golden Crown. Galileo's balance.
18. ^ John J. O'Connor, Edmund F. RobertsonArchimedes. In: MacTutor History of Mathematics archive .
19. NOVA | Infinite Secrets | TV Program Description | PBS
20. For example, in Proposition 2 there is the comparison of a spherical volume with that of a cylinder and a circular cone, Cut the knot, with Heath's translation
21. ^ Ivo Schneider: Archimedes. Knowledge Buchges. 1979, p. 39
22. ^
23. ^ JL Berggren: Mathematics in Medieval Islam. (PDF) §4 Abu Sahl on the regular heptagon. Spektrum.de, 2011, p. 85 , accessed on July 13, 2020 .
24. Archimedes: On floating bodies and the number of sands. In: Ostwald's classic of the exact sciences. No. 213. Leipzig 1925.
25. Rorres: Archimedean solids.
26. ^ Branko Grünbaum : An enduring error . Elements of Mathematics, 64 (3): 89–101, doi: 10.4171 / EM / 120, MR 2520469
27. ^ Aage Drachmann: The screw of Archimedes. Actes du VIIIe Congres International d´Histoire des Sciences, Florence 1958, Volume 3, p. 940.
28. ^ John Peter Oleson: Greek and Roman Mechanical Water-lifting Devices. Toronto 1984
29. ^ John Peter Oleson: Water lifting. In: Örjan Wikander (Ed.): Handbook of ancient water technology. Leiden 2000
30. After Stephanie Dalley, John Peter Oleson: Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World. In: Technology and Culture. Volume 44, 2003, pp. 1–26, the technology was possibly already known to the Assyrians in the 7th century BC. Known. Abstract
31. ^ Kurt von Fritz: Basic problems of ancient science. Verlag de Gruyter, Berlin 1971, ISBN 3-11-001805-5 . P. 114.
32. ^ Plutarch, Marcellus, German translation by Kaltwasser, Magdeburg 1801, p. 255, digitized
33. Chris Rorres: Archimedes' Claw - Illustrations and Animations - a range of possible designs for the claw. Courant Institute of Mathematical Sciences, accessed July 23, 2007 .
34. ^ Bradley W Carroll: Archimedes' Claw - watch an animation. Weber State University, archived from the original on August 13, 2007 ; accessed on August 12, 2007 .
35. ^ Thucydides, History of the Peloponnesian War, Part 1, Ed. Georg Peter Landmann, Tusculum Collection, Artemis / Winkler 1993, p. 525. According to Landmann's comment, this was the first mention of a grappling hook. According to Pliny, Pericles invented this.
36. ^ AA Mills, R. Clift: Archimedes Reflections of the 'Burning mirrors of Archimedes'. With a consideration of the geometry and intensity of sunlight reflected from plane mirrors. In: European Journal of Physics. Volume 13, Number 6, 1992
37. News Office 2005: Archimedes in a reflective mood. MITnews , October 5, 2005
38. ^ Gerhard Löwe, Heinrich Alexander Stoll: The antiquity in key words. Bassermann, Munich 1970, sv Archimedes
39. See Cicero: De re publica , Book I, chap. 21-22.
41. ^ Vitruvius: De Architectura. Book 10, Chapter 9, Bill Thayer, with commentary.
42. André Wegener Sleeswijk: Vitruvius' waywiser. Archives internationales d'histoire des sciences, Volume 29, 1979, pp. 11-22, Vitruvius Odometer , Scientific American, October 1981. Sleeswijk made a replica of the odometer described by Vitruvius and suggested that it went back to Archimedes
43. ^ DR Hill: On the Construction of Water Clocks: Kitâb Arshimídas fi`amal al ‑ binkamât. Turner & Devereux, London 1976
44. Lahanas on catapults and other war machines of the Greeks
45. ^ Jo Marchant: Reconstructed: Archimedes's flaming steam cannon. In: New Scientist. 2010.
46. A passage in Plutarch that the Romans were frightened during the siege by something stake-like protruding from the walls and ran away, can also be interpreted differently, e.g. B. by the also claw of Archimedes.
47. ^ Ivo Schneider: Archimedes. P. 160. The main sources for the transmission history are Heiberg and Claggett (see also his article Archimedes in Dictionary of Scientific Biography)
48. It was still listed in a catalog in the Vatican library in 1311.
49. Its use can be proven for the last time in 1544.
50. ^ Ivo Schneider: Archimedes. P. 164
51. De expedentis et fugiendis rebus opus. Venice 1501
52. Received from the estate of Regiomontanus in the Nuremberg City Library. Regiomontan's publisher's advertisement from 1473/74 ( Memento from December 3, 2013 in the Internet Archive )
53. Claggett: Archimedes. Dictionary of Scientific Biography
54. ^ Heath: The works of Archimedes. Dover, pp. Xxviii