Archimedes#Discoveries and inventions

File:Cicero Discovering the Tomb of Archimedes by Benjamin West.jpeg Discovering the Tomb of Archimedes (1805) by Benjamin West]]

The details of Archimedes life are obscure; a biography of Archimedes mentioned by Eutocius was allegedly written by his friend Heraclides Lembus, but this work has been lost, and modern scholarship is doubtful that it was written by Heraclides to begin with.Commentarius in dimensionem circuli (Archimedis opera omnia ed. Heiberg-Stamatis (1915), vol. 3, p. 228); Commentaria in conica (Apollonii Pergaei quae Graece exstant, ed. Heiberg (1893) vol. 2, p. 168: "Hērakleios"

Based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC, Archimedes is estimated to have been born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known; Plutarch wrote in his Parallel LivesPlutarch, Life of Marcellus that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero and Silius Italicus suggest he was of humble origin.{{sfn|Dijksterhuis|1987|p=10}} It is also unknown whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth;{{Cite book |last=Acerbi |first=Fabio |chapter=Archimedes |title=New Dictionary of Scientific Biography |volume=I |pages=85–91 |publisher=Scribner |location=Detroit |year=2008}} though his surviving written works, addressed to Dositheus of Pelusium, a student of the Alexandrian astronomer Conon of Samos, and to the head librarian Eratosthenes of Cyrene, suggested that he maintained collegial relations with scholars based there.{{sfn|Dijksterhuis|1987|pp=11-12}} In the preface to On Spirals addressed to Dositheus, 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.{{cn|date=April 2025}}

File:Displacement-measurement.svg

Another story of a problem that Archimedes is credited solving with in service of Hiero II is the "wreath problem."{{sfn|Dijksterhuis|1987|p=18}} According to Vitruvius, writing about two centuries after Archimedes' death, King Hiero II of Syracuse had commissioned a golden wreath for a temple to the immortal gods, and had supplied pure gold to be used by the goldsmith.Vitruvius, De Architectura, Book IX, 3 However, the king had begun to suspect that the goldsmith had substituted some cheaper silver and kept some of the pure gold for himself, and, unable to make the smith confess, asked Archimedes to investigate.{{sfn|Dijksterhuis|1987|p=19}} Later, while stepping into a bath, Archimedes allegedly noticed that the level of the water in the tub rose more the lower he sank in the tub and, realizing that this effect could be used to determine the golden crown's volume, was so excited that he took to the streets naked, having forgotten to dress, crying "Eureka!{{efn|{{langx|el|"εὕρηκα}}, heúrēka!)}}, meaning "I have found [it]!"{{sfn|Dijksterhuis|1987|p=19}} According to Vitruvius, Archimedes then took a lump of gold and a lump of silver that were each equal in weight to the wreath, and, placing each in the bathtub, showed that the wreath displaced more water than the gold and less than the silver, demonstrating that the wreath was gold mixed with silver {{sfn|Dijksterhuis|1987|p=19}}

A different account is given in the Carmen de Ponderibus,Metrologicorum Scriptorum reliquiae, ed. F. Hultsch (Leipzig 1864), II, 88 an anonymous 5th century Latin didactic poem on weights and measures once attributed to the grammarian Priscian.{{sfn|Dijksterhuis|1987|p=19}} In this poem, the lumps of gold and silver were placed on the scales of a balance, and then the entire apparatus was immersed in water; the difference in density between the gold and the silver, or between the gold and the crown, causes the scale to tip accordingly.Carmen de Ponderibus, lines 124-162 Unlike the more famous bathtub account given by Vitruvius, this poetic account uses the hydrostatics principle now known as Archimedes' principle that is found in his treatise On Floating Bodies, where a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.{{sfn|Dijksterhuis|1987|pp=20-21}} Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes' work, considered it "probable that this method is the same that Archimedes followed, since, besides being very accurate, it is based on demonstrations found by Archimedes himself."{{cite web |author=Van Helden, Al |title=The Galileo Project: Hydrostatic Balance |url=http://galileo.rice.edu/sci/instruments/balance.html |access-date=14 September 2007 |publisher=Rice University}}

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse.{{sfn|Dijksterhuis|1987|p=14}} Athenaeus of Naucratis in his Deipnosophistae quotes a certain Moschion for a description on how King Hiero II commissioned the design of a huge ship, the Syracusia, which is said to have been the largest ship built in classical antiquity and, according to Moschion's account, it was launched by Archimedes.Athenaeus, Deipnosophistae, V.40-45 Plutarch tells a slightly different account,Plutarch, Life of Marcellus 7-8 relating that Archimedes boasted to Hiero that he was able to move any large weight, at which point Hiero challenged him to move a ship.{{sfn|Dijksterhuis|1987|p=15}} These accounts contain many fantastic details that are historically implausible, and the authors of these stories provide conflicting about how this task was accomplished:{{sfn|Dijksterhuis|1987|p=15}} Plutarch states that Archimedes constructed a block-and-tackle pulley system, while Hero of Alexandria attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass. Heronis Opera Vol II, 1, 256, III 306 Pappus of Alexandria attributed this feat, instead, to Archimedes' use of mechanical advantage,{{sfn|Dijksterhuis|1987|p=15}} the principle of leverage to lift objects that would otherwise have been too heavy to move, attributing to him the oft-quoted remark: "Give me a place to stand on, and I will move the Earth."{{efn|{{langx|el|δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω}}}}Pappus of Alexandria, Synagoge Book VIII

Athenaeus, likely garbling the details of Hero's account of the baroulkos,{{sfn|Dijksterhuis|1987|p=16}} also mentions that Archimedes used a "screw" in order to remove any potential water leaking through the hull of the Syracusia. Although this device is sometimes referred to as Archimedes' screw, it likely predates him by a significant amount, and none of his closest contemporaries who describe its use (Philo of Byzantium, Strabo, and Vitruvius) credit him with its use.{{sfn|Dijksterhuis|1987|p=15}}

File:Archimedes Heat Ray conceptual diagram.svg to attack upcoming ships]]

The greatest reputation Archimedes earned during antiquity was for the defense of his city from the Romans during the Siege of Syracuse.{{sfn|Dijksterhuis|1987|pp=28-29}} According to Plutarch,Life of Marcellus, 25-27 Archimedes had constructed war machines for Hiero II, but had never been given an opportunity to use them during Hiero's lifetime. In 214 BC, however, during the Second Punic War, when Syracuse switched allegiances from Rome to Carthage, the Roman army under Marcus Claudius Marcellus attempted to take the city, Archimedes allegedly personally oversaw the use of these war machines in the defense of the city, greatly delaying the Romans, who were only able to capture the city after a long siege.{{sfn|Dijksterhuis|1987|pp=26,28}} Three different historians, Plutarch, Livy, and Polybius provide testimony about these war machines, describing improved catapults, cranes that dropped heavy pieces of lead on the Roman ships or which used an iron claw to lift them out of the water, dropping the back in so that they sank.{{efn|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.{{cite web |title=Archimedes' Claw: watch an animation |first=Bradley W |last=Carroll |publisher=Weber State University |url=http://physics.weber.edu/carroll/Archimedes/claw.htm |access-date=12 August 2007}}}}{{sfn|Dijksterhuis|1987|p=27}}

A much more improbable account, not found in any of the three earliest accounts (Plutarch, Polybius, or Livy) describes how Archimedes used "burning mirrors" to focus the sun's rays onto the attacking Roman ships, setting them on fire.{{sfn|Dijksterhuis|1987|pp=28-29}} The earliest account to mention ships being set on fire, by the 2nd century CE satirist Lucian of Samosata,Lucian, Hippias, [https://archive.org/details/lucianha01luciuoft/lucianha01luciuoft/page/36/ ¶ 2], in Lucian, vol. 1, ed. A. M. Harmon, Harvard, 1913, {{pgs|36–37}} does not mention mirrors, and only says the ships were set on fire by artificial means, which may imply that burning projectiles were used.{{sfn|Dijksterhuis|1987|pp=28-29}} The first author to mention mirrors is Galen, writing later in the same century.Galen, On temperaments 3.2 Nearly four hundred years after Lucian and Galen, Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.Anthemius of Tralles, On miraculous engines 153.{{Cite journal |last=Knorr |first=Wilbur |date=1983 |title=The Geometry of Burning-Mirrors in Antiquity |url=http://dx.doi.org/10.1086/353176 |journal=Isis |volume=74 |issue=1 |pages=53–73 |doi=10.1086/353176 |issn=0021-1753}} The purported device, sometimes called "Archimedes' heat ray", has been the subject of an ongoing debate about its credibility since the Renaissance. {{cite journal |last=Simms |first=D. L. |title=Archimedes and the Burning Mirrors of Syracuse |journal=Technology and Culture |year=1977 |volume=18 |number=1 |pages=1–24 |doi=10.2307/3103202 |jstor=3103202}} René Descartes rejected it as false,{{cite web |author=John Wesley |url=http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm |title=A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses |publisher=Online text at Wesley Center for Applied Theology |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012154432/http://wesley.nnu.edu/john_wesley/wesley_natural_philosophy/duten12.htm |archive-date=12 October 2007}} while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, with mixed results.{{cite book

| last = Jaeger | first = Mary

| editor-last = Rorres | editor-first = Chris

| contribution = Archimedes in the 21st century imagination

| doi = 10.1007/978-3-319-58059-3_8

| isbn = 9783319580593

| pages = 143–152

| publisher = Birkhäuser

| series = Trends in the History of Science

| title = Archimedes in the 21st Century: Proceedings of a World Conference at the Courant Institute of Mathematical Sciences

| year = 2017}} See p. 144.

File:Death of Archimedes (1815) by Thomas Degeorge.png]]

There are several divergent accounts of Archimedes' death during the sack of Syracuse after it fell to the Romans:{{sfn|Dijksterhuis|1987|pp=30-31}} The oldest account, from Livy,Livy, Ab Urbe Condita Book XXV, 31 says that, while drawing figures in the dust, Archimedes was killed by a Roman soldier who did not know he was Archimedes. According to Plutarch,Life of Marcellus, XIX, 1 the soldier demanded that Archimedes come with him, but Archimedes declined, saying that he had to finish working on the problem, and the soldier killed Archimedes with his sword. Another story from Plutarch has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items.{{sfn|Dijksterhuis|1987|pp=30-31}} Another Roman writer, Valerius Maximus (fl. 30 AD), wrote in Memorable Doings and Sayings that Archimedes' last words as the soldier killed him were "... but protecting the dust with his hands, said 'I beg of you, do not disturb this." which is similar to the last words now commonly attributed to him, "Do not disturb my circles,"{{efn|{{langx|la|Noli turbare circulos meos}}; {{langx|el|μὴ μου τοὺς κύκλους τάραττε}})}} which otherwise do not appear in any ancient sources.{{sfn|Dijksterhuis|1987|pp=30-31}}

Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometrical Briareus") and had ordered that he should not be harmed.Plutarch, Parallel LivesJaeger, Mary. Archimedes and the Roman Imagination. p. 113. Cicero (106–43 BC) mentions that Marcellus brought to Rome two planetariums Archimedes built,{{sfn|Dijksterhuis|1987|pp=23-25}} which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated to the Temple of Virtue in Rome, and the other he allegedly kept as his only personal loot from Syracuse."Cicero, [https://www.gutenberg.org/ebooks/14988 De republica] Pappus of Alexandria reports on a now lost treatise by Archimedes On Sphere-Making, which may have dealt with the construction of these mechanisms.{{sfn|Dijksterhuis|1987|p=27}} Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing, which was once thought to have been beyond the range of the technology available in ancient times, but the discovery in 1902 of the Antikythera mechanism, another device built {{circa|100}} BC designed with a similar purpose, has confirmed that devices of this kind were known to the ancient Greeks,{{cite web |last=Rorres |first=Chris |title=Spheres and Planetaria |url=http://www.math.nyu.edu/~crorres/Archimedes/Sphere/SphereIntro.html |access-date=23 July 2007 |publisher=Courant Institute of Mathematical Sciences}} with some scholars regarding Archimedes' device as a precursor.{{Cite journal |last=Freeth |first=Tony |date=2022 |title=Wonder of the Ancient World |url=https://www.scientificamerican.com/article/an-ancient-greek-astronomical-calculation-machine-reveals-new-secrets/ |journal=Scientific American |volume=32 |issue=1 |pages=24 |doi=10.1038/scientificamerican0122-24 |pmid=39016582 |issn=0036-8733}}{{cite web |title=The Antikythera Mechanism II |url=http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraII-0500/diff4.html |access-date=25 December 2013 |publisher=Stony Brook University |archive-date=12 December 2013 |archive-url=https://web.archive.org/web/20131212212956/http://www.math.sunysb.edu/~tony/whatsnew/column/antikytheraII-0500/diff4.html |url-status=dead}}

While serving as a quaestor in Sicily, Cicero himself found what was presumed to be Archimedes' tomb near the Agrigentine gate in Syracuse, in a neglected condition and overgrown with bushes. 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 tomb carried a sculpture illustrating Archimedes' favorite mathematical proof, that the volume and surface area of the sphere are two-thirds that of an enclosing cylinder including its bases.{{cite web |last=Rorres |first=Chris |title=Tomb of Archimedes: Sources |url=http://www.math.nyu.edu/~crorres/Archimedes/Tomb/Cicero.html |access-date=2 January 2007 |publisher=Courant Institute of Mathematical Sciences}}

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics, both in applying the techniques of his predecessors to obtain new results, and developing new methods of his own.

{{main|Method of exhaustion}}

File:PiArchimede4.svg and for each subsequent doubling of the sides of the regular polygon]]

In Quadrature of the Parabola, Archimedes states that a certain proposition in Euclid's Elements demonstrating that the area of a circle is proportional to its diameter was proven using a lemma now known as the Archimedean property, that “the excess by which the greater of two unequal regions exceed the lesser, if added to itself, can exceed any given bounded region.” Prior to Archimedes, Eudoxus of Cnidus and other earlier mathematicians{{efn|While Eudoxus is often credited for the whole of Euclid's Book XII, Archimedes explicitly credits him only with the proofs of XII.7 and XII.10, that the volume of a pyramid and a cone are, respectively, one-third of the volume of the rectangular prism and cone with the same base and height.{{harvnb|Acerbi|2018|p=279}}}} applied this lemma, a technique now referred to as the "method of exhaustion," to find the volume of a tetrahedron, cylinder, cone, and sphere, for which proofs are given in book XII of Euclid's Elements.{{sfn|Acerbi|2018|p=279}}

In Measurement of a Circle, Archimedes employed this method to show that the area of a circle is the same as a right triangle whose base and height are equal to its radius and circumference.{{sfn|Acerbi|2018|p=280}} He then approximated the ratio between the radius and the circumference, the value of {{pi}}, by drawing a larger regular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of each regular polygon, calculating the length of a side of each polygon at each step. As the number of sides increases, it becomes a more accurate approximation of a circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of {{pi}} lay between 3{{sfrac|1|7}} (approx. 3.1429) and 3{{sfrac|10|71}} (approx. 3.1408), consistent with its actual value of approximately 3.1416.{{cite web |title=The Computation of Pi by Archimedes |author=McKeeman, Bill |author-link=William M. McKeeman |website=Matlab Central |url=http://www.mathworks.com/matlabcentral/fileexchange/29504-the-computation-of-pi-by-archimedes/content/html/ComputationOfPiByArchimedes.html#37 |access-date=30 October 2012}} In the same treatise, he also asserts that the value of the square root of 3 as lying between {{sfrac|265|153}} (approximately 1.7320261) and {{sfrac|1351|780}} (approximately 1.7320512), which he may have derived from a similar method.{{Cite web |title=Of Calculations Past and Present: The Archimedean Algorithm |url=https://www.maa.org/programs/maa-awards/writing-awards/of-calculations-past-and-present-the-archimedean-algorithm |access-date=14 April 2021 |website=maa.org |publisher=Mathematical Association of America}}

File:Parabolic segment and inscribed triangle.svg segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure from Quadrature of the Parabola]]

In Quadrature of the Parabola, Archimedes used this technique to prove that the area enclosed by a parabola and a straight line is {{sfrac|4|3}} times the area of a corresponding inscribed triangle as shown in the figure at right, expressing the solution to the problem as an infinite geometric series with the common ratio {{sfrac|1|4}}:{{sfn|Netz|2022|p=139}}

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

If the first term in this series is the area of the triangle, then the second is the sum of the areas of two triangles whose bases are the two smaller secant lines, and whose third vertex is where the line that is parallel to the parabola's axis and that passes through the midpoint of the base intersects the parabola, and so on. This proof uses a variation of the series {{nowrap|1/4 + 1/16 + 1/64 + 1/256 + · · ·}} which sums to {{sfrac|1|3}}.

He also used this technique in order to measure the surface areas of a sphere and cone,On the Sphere and Cylinder 13-14, 33-34, 42, 44 to calculate the area of an ellipse,On Conoids and Spheroids 4 and to find the area contained within an Archimedean spiral.On Spirals, 24-25{{sfn|Acerbi|2018|p=280}}

=== Mechanical method ===

{{blockquote

|text=For it is more feasible, having already in one’s possession, through the method, of a knowledge of some sort of the matters under investigation, to provide the proof – rather than investigating it, knowing nothing.

|author=Archimedes

|title=The Method of Mechanical TheoremsNetz, Noel, Tchernetska, Wilson. "Archimedes Palimpsest" p. 71

}}

In addition to developing on the works of earlier mathematicians with the method of exhaustion, Archimedes also pioneered a novel technique using the law of the lever in order to measure the area and volume of shapes using physical means. He first gives an outline of this proof in Quadrature of the Parabola alongside the geometric proof, but he gives a fuller explanation in The Method of Mechanical Theorems.{{sfn|Netz|2022|p=139}} According to Archimedes, he proved the results in his mathematical treatises first using this method, and then worked backwards, applying the method of exhaustion only after he had already calculated an approximate value for the answer.{{sfn|Netz|2022|p=187}}

Archimedes also developed methods for representing large numbers.

In The Sand Reckoner, Archimedes devised a system of counting based on the myriad,{{efn|{{Langx|grc|μυριάς|translit=murias|label=none}}}} the Greek term for the number 10,000, in order to calculate a number that was greater than the grains of sand needed to fill the universe. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8 vigintillion, or 8{{e|63}}.{{cite web |title=The Sand Reckoner |first=Bradley W |last=Carroll |publisher=Weber State University |url=http://physics.weber.edu/carroll/Archimedes/sand.htm |access-date=23 July 2007}} In doing so, he demonstrated that mathematics could represent arbitrarily large numbers.

In the Cattle Problem, Archimedes challenges the mathematicians at the Library of Alexandria to count the numbers of cattle in the Herd of the Sun, which involves solving a number of simultaneous Diophantine equations. A more difficult version of the problem in which some of the answers are required to be square numbers, and the answer is a very large number, approximately 7.760271{{e|206544}}.{{cite web |first=Keith G |last=Calkins |url=http://www.andrews.edu/~calkins/profess/cattle.htm |title=Archimedes' Problema Bovinum |publisher=Andrews University |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012171254/http://andrews.edu/~calkins/profess/cattle.htm |archive-date=12 October 2007}}

{{main|Archimedean solids}}

In a lost work described by Pappus of Alexandria, Archimedes also proved that there are exactly thirteen semiregular polyhedra.{{sfn|Netz|2022|p=133}}

{{Primary sources|section|date=April 2025}}

File:Archimedes – Opere, 1615 – BEIC 9741168.jpg (1615)]]

Archimedes made his work known through correspondence with mathematicians in Alexandria,{{sfn|Acerbi|2018}} which were originally written in Doric Greek, the dialect of ancient Syracuse.{{cite book |last1=Wilson |first1=Nigel Guy |title=Encyclopedia of Ancient Greece |date=2006 |publisher=Psychology Press |page=77 |url=https://books.google.com/books?id=-aFtPdh6-2QC&pg=PA77 |access-date=29 April 2025 |language=en |isbn=978-0-7945-0225-6}}

The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).{{Cite journal |last=Knorr |first=W. R. |date=1978 |title=Archimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus |journal=Archive for History of Exact Sciences |volume=19 |issue=3 |pages=211–290 |doi=10.1007/BF00357582 |jstor=41133526}}{{Cite journal |last=Sato |first=T. |date=1986 |title=A Reconstruction of The Method Proposition 17, and the Development of Archimedes' Thought on Quadrature...Part One |journal=Historia scientiarum: International journal of the History of Science Society of Japan}}

{{Main|Measurement of a Circle}}

This is a short work consisting of three propositions. It is written in the form of a correspondence with Dositheus of Pelusium, who was a student of Conon of Samos. In Proposition II, Archimedes gives an approximation of the value of pi ({{pi}}), showing that it is greater than {{sfrac|223|71}} (3.1408...) and less than {{sfrac|22|7}} (3.1428...).

{{Main|The Sand Reckoner}}

In this treatise, also known as Psammites, Archimedes finds a number that is greater than the grains of sand needed to fill the universe. This book mentions the heliocentric theory of the Solar System proposed by Aristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between various celestial bodies, and attempts to measure the apparent diameter of the Sun.{{Cite journal |last=Osborne |first=Catherine |date=1983 |title=Archimedes on the Dimensions of the Cosmos |url=https://www.jstor.org/stable/233105 |journal=Isis |volume=74 |issue=2 |pages=234–242 |doi=10.1086/353246 |jstor=233105 |issn=0021-1753}}{{Citation |last1=Rozelot |first1=Jean Pierre |title=A brief history of the solar diameter measurements: a critical quality assessment of the existing data |date=2016 |arxiv=1609.02710 |last2=Kosovichev |first2=Alexander G. |last3=Kilcik |first3=Ali}} 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{{e|63}} in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy.{{cite web |year=2002 |title=English translation of The Sand Reckoner |publisher=University of Waterloo |url=http://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |archive-date=2002-06-01 |archive-url=https://web.archive.org/web/20020601231141/https://www.math.uwaterloo.ca/navigation/ideas/reckoner.shtml |url-status=dead}} Adapted from {{cite book |last=Newman |first=James R. |title=The World of Mathematics |volume=1 |publisher=Simon & Schuster |location=New York |year=1956}}

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner.{{Cite encyclopedia |last1=Toomer |first1=G. J. |last2=Jones |first2=Alexander |date=7 March 2016 |title=Astronomical Instruments |encyclopedia=Oxford Research Encyclopedia of Classics |doi=10.1093/acrefore/9780199381135.013.886 |isbn=9780199381135 |quote="Perhaps the earliest instrument, apart from sundials, of which we have a detailed description is the device constructed by Archimedes for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved."}} Without the use of either trigonometry or a table of chords, Archimedes determines the Sun's apparent diameter by first describing the procedure and instrument used to make observations (a straight rod with pegs or grooves),{{Cite journal |last=Evans |first=James |date=1 August 1999 |title=The Material Culture of Greek Astronomy |journal=Journal for the History of Astronomy |volume=30 |issue=3 |pages=238–307 |bibcode=1999JHA....30..237E |doi=10.1177/002182869903000305}} applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.{{Cite journal |last=Shapiro |first=A. E. |date=1975 |title=Archimedes's measurement of the Sun's apparent diameter. |journal=Journal for the History of Astronomy |volume=6 |issue=2 |pages=75–83 |bibcode=1975JHA.....6...75S |doi=10.1177/002182867500600201}}

Ptolemy, quoting Hipparchus, also references Archimedes' solstice observations in the Almagest. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.

{{Main|On the Equilibrium of Planes}}

There are two books to On the Equilibrium of Planes: the first contains seven postulates and fifteen propositions, while the second book contains ten propositions. In the first book, Archimedes proves the law of the lever,{{Cite journal |last=Goe |first=G. |date=1972 |title=Archimedes' theory of the lever and Mach's critique |journal=Studies in History and Philosophy of Science Part A |volume=2 |issue=4 |pages=329–345 |doi=10.1016/0039-3681(72)90002-7 |bibcode=1972SHPSA...2..329G}} which states that:

{{Blockquote|text=Magnitudes are in equilibrium at distances reciprocally proportional to their weights.}}

Earlier descriptions of the principle of the lever are found in a work by Euclid and in the Mechanical Problems, belonging to the Peripatetic school of the followers of Aristotle, the authorship of which has been attributed by some to Archytas.{{cite book |first=Marshall |last=Clagett |url=https://books.google.com/books?id=mweWMAlf-tEC&q=archytas%20lever&pg=PA72 |title=Greek Science in Antiquity |publisher=Dover Publications |isbn=978-0-486-41973-2 |year=2001}}

Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.{{cite book |author=Heath, T.L. |url=https://archive.org/details/worksofarchimede029517mbp |title=The Works of Archimedes |year=1897 |publisher=Cambridge University Press}}

{{Main|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 the area of a triangle with equal base and height. He achieves this by two different methods: first by applying the law of the lever, and by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

{{Main|On the Sphere and Cylinder}}

File:Esfera Arquímedes.svg

In this two-volume treatise addressed to Dositheus, 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. The volume is {{sfrac|4|3}}{{pi}}{{math|r}}³ for the sphere, and 2{{pi}}{{math|r}}³ for the cylinder. The surface area is 4{{pi}}{{math|r}}² for the sphere, and 6{{pi}}{{math|r}}² for the cylinder (including its two bases), where {{math|r}} is the radius of the sphere and cylinder.

{{Main|On Spirals}}

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in modern polar coordinates ({{math|r}}, {{math|θ}}), it can be described by the equation $\backslash ,\; r=a+b\backslash theta$ with real numbers {{math|a}} and {{math|b}}.

This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.

{{Main|On Conoids and Spheroids}}

This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.

{{Main|On Floating Bodies}}

There are two books of On Floating Bodies. In the first book, Archimedes spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity.{{Cite journal |last=Berggren |first=J. L. |date=1976 |title=Spurious Theorems in Archimedes' Equilibrium of Planes: Book I |journal=Archive for History of Exact Sciences |volume=16 |issue=2 |pages=87–103 |doi=10.1007/BF00349632 |jstor=41133463}}

This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round.{{cn|date=April 2025}} The fluids described by Archimedes are not {{nowrap|self-gravitating}} since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.{{cn|date=April 2025}}

Archimedes' principle of buoyancy is given in this work, stated as follows:{{cite book |last=Netz |first=Reviel |chapter=Archimedes' Liquid Bodies |title=ΣΩΜΑ: Körperkonzepte und körperliche Existenz in der antiken Philosophie und Literatur |year=2017 |pages=287–322 |editor1-first=Thomas |editor1-last=Buchheim |editor2-first=David |editor2-last=Meißner |editor3-first=Nora |editor3-last=Wachsmann |isbn=978-3-7873-2928-1 |place=Hamburg |publisher=Felix Meiner |chapter-url=https://books.google.com/books?id=rQ2KDwAAQBAJ&pg=PA287 |chapter-url-access=limited}}

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

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.{{cite book |last=Stein |first=Sherman |chapter=Archimedes and his floating paraboloids |editor1-first=David F. |editor1-last=Hayes |editor2-first=Tatiana |editor2-last=Shubin |title=Mathematical Adventures for Students and Amateurs |publisher=Mathematical Association of America |place=Washington |year=2004 |pages=219–231 |isbn=0-88385-548-8 |chapter-url=https://archive.org/details/mathematicaladve0000unse/page/219 |chapter-url-access=limited}} {{pb}} {{cite journal |last=Rorres |first=Chris |year=2004 |title=Completing Book II of Archimedes's on Floating Bodies |journal=The Mathematical Intelligencer |volume=26 |number=3 |pages=32–42 |doi=10.1007/bf02986750}} {{pb}} {{cite journal |last1=Girstmair |first1=Kurt |last2=Kirchner |first2=Gerhard |title=Towards a completion of Archimedes' treatise on floating bodies |journal=Expositiones Mathematicae |volume=26 |number=3 |year=2008 |pages=219–236 |doi=10.1016/j.exmath.2007.11.002 |doi-access=free}}

{{Main|Ostomachion}}

File:Stomachion.JPG is a dissection puzzle found in the Archimedes Palimpsest|200x200px]]

Also known as Loculus of Archimedes or Archimedes' Box, 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. Reviel Netz of Stanford University argued in 2003 that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Netz calculates that the pieces can be made into a square 17,152 ways.{{cite news |title=In Archimedes' Puzzle, a New Eureka Moment |author=Kolata, Gina |newspaper=The New York Times |date=14 December 2003 |url=https://query.nytimes.com/gst/fullpage.html?res=9D00E6DD133CF937A25751C1A9659C8B63&sec=&spon=&pagewanted=all |access-date=23 July 2007}} The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.{{cite web |title=The Loculus of Archimedes, Solved |author=Ed Pegg Jr. |publisher=Mathematical Association of America |date=17 November 2003 |url=http://www.maa.org/editorial/mathgames/mathgames_11_17_03.html |access-date=18 May 2008}} 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 that it is taken from the Ancient Greek word for "throat" or "gullet", stomachos ({{lang|grc|στόμαχος}}).{{cite web |first=Chris |last=Rorres |url=http://math.nyu.edu/~crorres/Archimedes/Stomachion/intro.html |title=Archimedes' Stomachion |publisher=Courant Institute of Mathematical Sciences |access-date=14 September 2007}} Ausonius calls the puzzle {{Langx|grc|Ostomachion|label=none|italic=yes}}, a Greek compound word formed from the roots of {{Langx|grc|osteon|label=none|italic=yes}} ({{Langx|grc|ὀστέον|label=none|lit=bone}}) and {{Langx|grc|machē|label=none|italic=yes}} ({{Langx|grc|μάχη|label=none|lit=fight}}).{{cite web |url=http://www.archimedes-lab.org/latin.html#archimede |title=Graeco Roman Puzzles |publisher=Gianni A. Sarcone and Marie J. Waeber |access-date=9 May 2008}}

{{Main|Archimedes's cattle problem|l1 = Archimedes' cattle problem}}

In this work, addressed to Eratosthenes and the mathematicians in Alexandria, Archimedes challenges them to count the numbers of cattle in the Herd of the Sun, which involves solving a number of simultaneous Diophantine equations. Gotthold Ephraim Lessing discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library in Wolfenbüttel, Germany in 1773. There is a more difficult version of the problem in which some of the answers are required to be square numbers. A. Amthor first solved this version of the problemKrumbiegel, B. and Amthor, A. Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift für Mathematik und Physik 25 (1880) pp. 121–136, 153–171. in 1880, and the answer is a very large number, approximately 7.760271{{e|206544}}.{{cite web |first=Keith G |last=Calkins |url=http://www.andrews.edu/~calkins/profess/cattle.htm |title=Archimedes' Problema Bovinum |publisher=Andrews University |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012171254/http://andrews.edu/~calkins/profess/cattle.htm |archive-date=12 October 2007}}

{{Main|The Method of Mechanical Theorems}}

As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.

In this work Archimedes uses a novel method, an early form of Cavalieri's principle,{{Cite web |last=Powers |first=J. |date=2020 |title=Did Archimedes do calculus? |url=https://old.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |access-date=14 April 2021 |website=maa.org}}; {{Citation |last=Jullien |first=V. |title=Archimedes and Indivisibles |date=2015 |work=Seventeenth-Century Indivisibles Revisited |volume=49 |pages=451–457 |editor-last=J. |editor-first=Vincent |series=Science Networks. Historical Studies |place=Cham |publisher=Springer International Publishing |doi=10.1007/978-3-319-00131-9_18 |isbn=978-3-319-00131-9}}; {{cite web |author1=O'Connor, J.J. |author2=Robertson, E.F. |date=February 1996 |title=A history of calculus |url=https://mathshistory.st-andrews.ac.uk/HistTopics/The_rise_of_calculus/ |access-date=7 August 2007 |publisher=University of St Andrews}}; {{Cite journal |last=Kirfel |first=Christoph |date=2013 |title=A generalisation of Archimedes' method |url=https://www.jstor.org/stable/24496758 |journal=The Mathematical Gazette |volume=97 |issue=538 |pages=43–52 |doi=10.1017/S0025557200005416 |issn=0025-5572 |jstor=24496758}} to rederive the results from the treatises sent to Dositheus (Quadrature of the Parabola, On the Sphere and Cylinder, On Spirals, On Conoids and Spheroids) that he had previously used the method of exhaustion to prove,{{sfn|Netz|2022|p=131}} using the law of the lever he applied in On the Equilbrium of Planes in order to find the center of gravity of an object first, and reasoning geometrically from there in order to more easily derive the volume of an object.{{sfn|Netz|2022|pp=187-193}} Archimedes states that he used this method to derive the results in the treatises sent to Dositheus before he proved them more rigorously with the method of exhaustion, stating that it is useful to know that a result is true before proving it rigorously, much as Eudoxus of Cnidus was aided in proving that the volume of a cone is one-third the volume of cylinder by knowing that Democritus had already asserted it to be true on the argument that this is true by the fact that the pyramid has one-third the rectangular prism of the same base.{{sfn|Netz|2022|p=150-151}}

This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906.{{cite book |last1=Smith |first1=David Eugene |title=Geometrical Solutions Derived from Mechanics: A Treatise of Archimedes |date=1909 |publisher=Open Court Publishing Company |url=https://archive.org/details/geometricalsolu00smitgoog/mode/2up |access-date=4 May 2025 |language=English}}

{{main|Pseudo-Archimedes}}

Archimedes' Book of Lemmas or Liber Assumptorum is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is in Arabic. T. L. Heath and Marshall Clagett argued that it cannot have been written by Archimedes in its current form,{{cn|date=April 2025}} since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.{{cn|date=April 2025}}

Other questionable attributions to Archimedes' work include the Latin poem Carmen de ponderibus et mensuris (4th or 5th century), which describes the use of a hydrostatic balance, to solve the problem of the crown, and the 12th-century text Mappae clavicula, which contains instructions on how to perform assaying of metals by calculating their specific gravities.Dilke, Oswald A. W. 1990. [Untitled]. Gnomon 62(8):697–99. {{JSTOR|27690606}}.Berthelot, Marcel. 1891. "Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques." Annales de Chimie et de Physique 6(23):475–85.

Many written works by Archimedes have not survived or are only extant in heavily edited fragments:{{sfn|Netz|2022|p=133-135}} Pappus of Alexandria mentions On Sphere-Making, as well as a work on semiregular polyhedra, and another work on spirals, while Theon of Alexandria quotes a remark about refraction from the {{nowrap|now-lost}} Catoptrica. Principles, addressed to Zeuxippus, explained the number system used in The Sand Reckoner; there are also On Balances; On Centers of Gravity.{{sfn|Netz|2022|p=133-135}}

Scholars in the medieval Islamic world also attribute to Archimedes a formula for calculating the area of a triangle from the length of its sides, which today is known as Heron's formula due to its first known appearance in the work of Heron of Alexandria in the 1st century AD, and may have been proven in a lost work of Archimedes that is no longer extant.{{sfn|Netz|2022|p=135-136}}

{{main|Archimedes Palimpsest}}

File:Archimedes Palimpsest.jpg

In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examine a 174-page goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Papadopoulos-Kerameus.{{Cite journal |last=Wilson |first=Nigel |date=2004 |title=The Archimedes Palimpsest: A Progress Report |journal=The Journal of the Walters Art Museum |volume=62 |pages=61–68 |jstor=20168629}}{{Cite journal |last1=Easton |first1=R. L. |last2=Noel |first2=W. |date=2010 |title=Infinite Possibilities: Ten Years of Study of the Archimedes Palimpsest |journal=Proceedings of the American Philosophical Society |volume=154 |issue=1 |pages=50–76 |jstor=20721527}} He confirmed that it was indeed a palimpsest, a document with text that had been written over an erased older work. Palimpsests were created by scraping the ink from existing works and reusing them, a common practice in the Middle Ages, as vellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes.{{cite magazine |title=Reading Between the Lines |author=Miller, Mary K. |magazine=Smithsonian |date=March 2007 |url=https://www.smithsonianmag.com/science-nature/reading-between-the-lines-148131057/}} 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 treatises in the Archimedes Palimpsest include:

• On the Equilibrium of Planes
• On Spirals
• Measurement of a Circle
• On the Sphere and Cylinder
• On Floating Bodies
• The Method of Mechanical Theorems
• Stomachion
• Speeches by the 4th century BC politician Hypereides
• A commentary on Aristotle's Categories
• Other works

The parchment spent hundreds of years in a monastery library in Constantinople before being sold to a private collector in the 1920s. On 29 October 1998, it was sold at auction to an anonymous buyer for a total of $2.2 million.{{cite news |title=Rare work by Archimedes sells for $2 million |publisher=CNN |date=29 October 1998 |url=http://edition.cnn.com/books/news/9810/29/archimedes/ |access-date=15 January 2008 |archive-url=https://web.archive.org/web/20080516000109/http://edition.cnn.com/books/news/9810/29/archimedes/ |archive-date=16 May 2008}}[https://www.christies.com/results/printauctionresults.aspx?saleid=8685&lid=1 Christie's (n.d). Auction results] The palimpsest was stored at the Walters Art Museum in Baltimore, Maryland, where it was subjected to a range of modern tests including the use of ultraviolet and {{nowrap|X-ray}} light to read the overwritten text.{{cite news |title=X-rays reveal Archimedes' secrets |publisher=BBC News |date=2 August 2006 |url=http://news.bbc.co.uk/1/hi/sci/tech/5235894.stm |access-date=23 July 2007}} It has since returned to its anonymous owner.{{Cite journal |last1=Piñar |first1=G. |last2=Sterflinger |first2=K. |last3=Ettenauer |first3=J. |last4=Quandt |first4=A. |last5=Pinzari |first5=F. |date=2015 |title=A Combined Approach to Assess the Microbial Contamination of the Archimedes Palimpsest |journal=Microbial Ecology |volume=69 |issue=1 |pages=118–134 |pmid=25135817 |doi=10.1007/s00248-014-0481-7 |pmc=4287661 |bibcode=2015MicEc..69..118P}}{{Cite journal |last=Acerbi |first=F. |date=2013 |title=Review: R. Netz, W. Noel, N. Tchernetska, N. Wilson (eds.), The Archimedes Palimpsest, 2001 |url=https://www.academia.edu/8016340 |journal=Aestimatio |volume=10 |pages=34–46}}

File:Gerhard Thieme Archimedes.jpg]]

{{further|List of things named after Archimedes}}

Sometimes called the father of mathematicsFather of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19. and mathematical physics,James H. Williams Jr., Fundamentals of Applied Dynamics, p 30., Carl B. Boyer, Uta C. Merzbach, A History of Mathematics, p 111., Stuart Hollingdale, Makers of Mathematics, p 67., Igor Ushakov, In the Beginning, Was the Number (2), p 114. historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.{{cite web |date=January 1999 |title=Archimedes of Syracuse |url=http://www-history.mcs.st-and.ac.uk/Biographies/Archimedes.html |access-date=9 June 2008 |publisher=The MacTutor History of Mathematics archive}}; {{cite book |last=Hirshfeld |first=Alan |url=https://books.google.com/books?id=zbcfLoZKDl8C&pg=PA206 |title=Eureka Man: The Life and Legacy of Archimedes |date=2009 |publisher=Bloomsbury Publishing |isbn=978-0-8027-1979-9 |pages=206 |language=en |quote="the Archimedes Palimpsest has ridden the roiling waves of circumstance to become the most celebrated link to antiquity's greatest mathematician-inventor"}}

The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented;{{Cite journal |last=Drachmann |first=A. G. |author-link=A. G. Drachmann |date=1968 |title=Archimedes and the Science of Physics |journal=Centaurus |volume=12 |issue=1 |pages=1–11 |doi=10.1111/j.1600-0498.1968.tb00074.x |bibcode=1968Cent...12....1D}} AthenaeusDeipnosophistae, V, 206d) recounts in his Deipnosophistae how Archimedes supervised the construction of the largest known ship in antiquity, the Syracusia, while ApuleiusApologia, 16 talks about his work in catoptrics.{{Cite journal |last=Russo |first=L. |date=2013 |title=Archimedes between legend and fact. |journal=Lettera Matematica |volume=1 |issue=3 |pages=91–95 |doi=10.1007/s40329-013-0016-y |doi-access=free}} PlutarchPlutarch, Parallel lives had claimed that Archimedes disdained mechanics and focused primarily on pure geometry, but this is generally considered to be a mischaracterization by modern scholarship, fabricated to bolster Plutarch's own Platonist values rather than to an accurate presentation of Archimedes,{{Cite thesis |title=Attitudes toward the natural philosopher in the early Roman empire (100 B.C. to 313 A.D.) |url=https://clio.columbia.edu/catalog/8602980 |date=2008 |first=Richard |last=Carrier |access-date=6 April 2021}} and, unlike his inventions, Archimedes' mathematical writings were little known in antiquity outside of the works of Alexandrian mathematicians.{{cn|date=April 2025}} The first comprehensive compilation was not made until {{circa|530{{nbsp}}AD}} by Isidore of Miletus in Byzantine Constantinople,{{cn|date=April 2025}} while Eutocius' commentaries on Archimedes' works earlier in the same century opened them to wider readership for the first time.{{cn|date=April 2025}}

{{Expand section|date=April 2025}}

Archimedes' work was translated into Arabic by Thābit ibn Qurra (836–901 AD), and into Latin via Arabic by Gerard of Cremona (c. 1114–1187). Direct Greek to Latin translations were later done by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).{{Cite journal |last=Clagett |first=Marshall |date=1982 |title=William of Moerbeke: Translator of Archimedes |journal=Proceedings of the American Philosophical Society |volume=126 |issue=5 |pages=356–36 6 |jstor=986212}}{{Cite journal |last=Clagett |first=Marshall |date=1959 |title=The Impact of Archimedes on Medieval Science |journal=Isis |volume=50 |issue=4 |pages=419–429 |doi=10.1086/348797}}

File:Archimedes bronze coin.jpg

During the Renaissance, the Editio princeps (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin,{{cite web |title=Editions of Archimedes's Work |publisher=Brown University Library |year=1999 |url=https://library.brown.edu/exhibits/archive/math/wholefr.html}} which were an influential source of ideas for scientists during the Renaissance and again in the 17th century.

{{cite journal |last=Høyrup |first=Jens |year=2017 |title=Archimedes: Knowledge and Lore from Latin Antiquity to the Outgoing European Renaissance |journal=Gaņita Bhāratī |volume=39 |number=1 |pages=1–22 |url=http://webhotel4.ruc.dk/~jensh/Publications/2017%7Bg%7D_Archimedes%20--%20%20Knowledge%20and%20Lore_S.pdf}} Reprinted in {{Cite book |last=Hoyrup |first=J. |year=2019 |title=Selected Essays on Pre- and Early Modern Mathematical Practice |pages=459–477 |doi=10.1007/978-3-030-19258-7_17}}{{Cite journal |last=Leahy |first=A. |date=2018 |title=The method of Archimedes in the seventeenth century. |journal=The American Monthly |volume=125 |issue=3 |pages=267–272 |doi=10.1080/00029890.2018.1413857}}

Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.{{cite news |url=http://paperspast.natlib.govt.nz/cgi-bin/paperspast?a=d&d=NENZC18420521.2.11 |title=The Steam-Engine |volume=I |issue=11 |date=21 May 1842 |work=Nelson Examiner and New Zealand Chronicle |publisher=National Library of New Zealand |page=43 |access-date=14 February 2011 |location=Nelson}}{{cite book |title=The Steam Engine |url=https://books.google.com/books?id=E1oFAAAAQAAJ&pg=RA1-PA104 |year=1838 |publisher=The Penny Magazine |page=104}}{{cite book |author=Robert Henry Thurston |title=A History of the Growth of the Steam-Engine |year=1996 |url=https://books.google.com/books?id=KCMUmXV1C1gC |publisher=Elibron |isbn=1-4021-6205-7 |page=12}} Galileo Galilei called him "superhuman" and "my master",Matthews, Michael. Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy. p. 96.{{Cite web |title=Archimedes – Galileo Galilei and Archimedes |url=https://exhibits.museogalileo.it/archimedes/section/GalileoGalileiArchimedes.html |access-date=16 June 2021 |website=exhibits.museogalileo.it}} while Christiaan Huygens said, "I think Archimedes is comparable to no one", consciously emulating him in his early work.{{Cite web |last=Yoder |first=J. |date=1996 |title=Following in the footsteps of geometry: the mathematical world of Christiaan Huygens |url=https://www.dbnl.org/tekst/_zev001199601_01/_zev001199601_01_0009.php |website=De Zeventiende Eeuw. Jaargang 12}} Gottfried Wilhelm Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times".Boyer, Carl B., and Uta C. Merzbach. 1968. A History of Mathematics. ch. 7.

Italian numismatist and archaeologist Filippo Paruta (1552–1629) and Leonardo Agostini (1593–1676) reported on a bronze coin in Sicily with the portrait of Archimedes on the obverse and a cylinder and sphere with the monogram ARMD in Latin on the reverse.{{Cite book |last1=Paruta |first1=Filippo |url=https://dn720400.ca.archive.org/0/items/lasiciliadifilip00paru/lasiciliadifilip00paru.pdf |title=La Sicilia descritta con medaglie |last2=Agostini |first2=Leonardo |publisher=Marco Maier |year=1697 |publication-date=1697 |pages=73, 326 |language=it |trans-title=Sicily described by medals |access-date=2025-01-20}} Although the coin is now lost and its date is not precisely known, Ivo Schneider described the reverse as "a sphere resting on a base – probably a rough image of one of the planetaria created by Archimedes," and suggested it might have been minted in Rome for Marcellus who "according to ancient reports, brought two spheres of Archimedes with him to Rome".{{Cite book |last=Schneider |first=Ivo |title=Archimedes. Ingenieur, Naturwissenschaftler und Mathematiker |publisher=Wissenschaftliche Buchgesellschaft |year=1979 |isbn=3-534-06844-0 |location=Darmstadt |pages=19, 23 |language=de |trans-title=Archimedes. Engineer, natural scientist and mathematician}}

File:FieldsMedalFront.jpg carries a portrait of Archimedes|190x190px]]

Gauss's heroes were Archimedes and Newton,Jay Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, p 88. and Moritz Cantor, who studied under Gauss in the University of Göttingen, reported that he once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein".E.T. Bell, Men of Mathematics, p 237 Likewise, Alfred North Whitehead said that "in the year 1500 Europe knew less than Archimedes who died in the year 212 BC."{{cite web |author=Alfred North Whitehead |title=The Influence of Western Medieval Culture Upon the Development of Modern Science |url=https://inters.org/Whitehead-Western-Development-Science |access-date=4 April 2022}} The historian of mathematics Reviel Netz,Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets of the World's Greatest Palimpsest echoing Whitehead's proclamation on Plato and philosophy, said that "Western science is but a series of footnotes to Archimedes," calling him "the most important scientist who ever lived." and Eric Temple Bell,E.T. Bell, Men of Mathematics, p 20. wrote that "Any list of the three "greatest" mathematicians of all history would include the name of Archimedes. The other two usually associated with him are Newton and Gauss. Some, considering the relative wealth—or poverty—of mathematics and physical science in the respective ages in which these giants lived, and estimating their achievements against the background of their times, would put Archimedes first."

The discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has provided new insights into how he obtained mathematical results.{{cite web |title=Works, Archimedes |date=23 June 2015 |publisher=University of Oklahoma |url=https://galileo.ou.edu/exhibits/works-archimedes |access-date=18 June 2019}}{{cite book |title=The Genius of Archimedes – 23 Centuries of Influence on Mathematics, Science and Engineering: Proceedings of an International Conference held at Syracuse, Italy |date=8–10 June 2010 |series=History of Mechanism and Machine Science |volume=11 |publisher=Springer |editor1=Paipetis, Stephanos A. |editor2=Ceccarelli, Marco |isbn=978-90-481-9091-1 |doi=10.1007/978-90-481-9091-1}}

The Fields Medal for outstanding achievement in mathematics carries a portrait of Archimedes, along with a carving illustrating his proof on the sphere and the cylinder. The inscription around the head of Archimedes is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri ("Rise above oneself and grasp the world").{{Cite journal |last=Riehm |first=C. |date=2002 |title=The early history of the Fields Medal |url=https://www.ams.org/notices/200207/comm-riehm.pdf |journal=Notices of the AMS |volume=49 |issue=7 |pages=778–782 |quote="The Latin inscription from the Roman poet Manilius surrounding the image may be translated 'To pass beyond your understanding and make yourself master of the universe.' The phrase comes from Manilius's Astronomica 4.392 from the first century A.D. (p. 782)."}}{{Cite web |date=5 February 2015 |title=The Fields Medal |url=http://www.fields.utoronto.ca/about/fields-medal |access-date=23 April 2021 |website=Fields Institute for Research in Mathematical Sciences}}{{cite web |title=Fields Medal |url=https://www.mathunion.org/imu-awards/fields-medal |access-date=23 April 2021 |publisher=International Mathematical Union}}

The world's first seagoing steamship with a screw propeller was the SS Archimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.{{cite web |title=SS Archimedes |publisher=wrecksite.eu |url=http://www.wrecksite.eu/wreck.aspx?636 |access-date=22 January 2011}}

Archimedes has also appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).{{cite web |first=Chris |last=Rorres |url=http://math.nyu.edu/~crorres/Archimedes/Stamps/stamps.html |title=Stamps of Archimedes |publisher=Courant Institute of Mathematical Sciences |access-date=25 August 2007}}

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.{{cite web |title=California Symbols |publisher=California State Capitol Museum |url=http://www.capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294 |access-date=14 September 2007 |archive-url=https://web.archive.org/web/20071012123245/http://capitolmuseum.ca.gov/VirtualTour.aspx?content1=1278&Content2=1374&Content3=1294 |archive-date=12 October 2007 |url-status=dead}}

There is a crater on the Moon named Archimedes ({{Coord|29.7|-4.0|display=}}) in his honor, as well as a lunar mountain range, the Montes Archimedes ({{Coord|25.3|-4.6|display=}}).{{cite web |title=Oblique view of Archimedes crater on the Moon |author1=Friedlander, Jay |author2=Williams, Dave |publisher=NASA |url=http://nssdc.gsfc.nasa.gov/imgcat/html/object_page/a15_m_1541.html |access-date=13 September 2007}}

{{Portal|Biography|Mathematics|Physics}}

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• Methods of computing square roots
• Salinon
• Steam cannon

• Zhang Heng

{{Reflist|35em|group=lower-alpha}}

{{Reflist}}

• Plutarch, Life of Marcellus
• {{Cite web |title=Athenaeus, Deipnosophistae |url=https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:2013.01.0003:book=5:chapter=pos=377 |access-date=7 March 2023 |website=perseus.tufts.edu}}

• {{Cite encyclopedia|last=Acerbi|first=Fabio |editor1-first=Paul T|editor1-last=Keyser|editor2-first=John| editor2-last=Scarborough| date=2018| title=Hellenistic Mathematics | encyclopedia = Oxford Handbook of Science and Medicine in the Classical World | url=https://www.academia.edu/36286615 |access-date=2021-05-26| pages=268–292| doi=10.1093/oxfordhb/9780199734146.013.69| isbn=978-0-19-973414-6}}
• {{cite book |last1=Dijksterhuis |first1=E. J. (Eduard Jan) |title=Archimedes |date=1987 |publisher=Princeton, N.J. : Princeton University Press |isbn=978-0-691-08421-3 |url=https://archive.org/details/archimedes0000dijk_m6s3 |access-date=30 April 2025}}
• {{cite book |last1=Netz |first1=Reviel |title=A New History of Greek Mathematics |date=2022 |publisher=Cambridge University Press |isbn=978-1-108-83384-4}}

{{EB1911 poster|Archimedes}}

• Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
• Clagett, Marshall. 1970. "Archimedes". In Charles Coulston Gillispie, ed. Dictionary of Scientific Biography. Vol. 1 (Abailard–Berg). New York: Charles Scribner's Sons. {{pgs|213–231}}.
• Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. {{ISBN|978-0-7660-2502-8}}.
• Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. {{ISBN|978-1-4042-0774-5}}.
• Netz, Reviel. 2004–2017. The Works of Archimedes: Translation and Commentary. 1–2. Cambridge University Press. Vol. 1: "The Two Books on the Sphere and the Cylinder". {{isbn|978-0-521-66160-7}}. Vol. 2: "On Spirals". {{isbn|978-0-521-66145-4}}.
• Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. {{ISBN|978-0-297-64547-4}}.
• Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. {{ISBN|978-0-19-533611-5}}.
• Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. {{ISBN|978-0-7201-2284-8}}.
• Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. {{ISBN|978-0-88385-718-2}}.

{{Sister project links|commons=Category:Archimedes|v=Ancient Innovations|n=Particle accelerator reveals long-lost writings of Archimedes|s=Author:Archimedes|b=FHSST Physics/Forces/Definition}}

• [https://www.wilbourhall.org/index.html#archimedes Heiberg's Edition of Archimedes]. Texts in Classical Greek, with some in English.
• {{In Our Time|Archimedes|b00773bv|Archimedes}}
• {{Gutenberg author | id=2545| name=Archimedes}}
• {{Internet Archive author}}
• {{InPho|thinker|2546}}
• {{PhilPapers|search|archimedes}}
• [http://www.archimedespalimpsest.org/ The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland]
• {{MathPages|id=home/kmath038/kmath038|title=Archimedes and the Square Root of 3}}
• {{MathPages|id=home/kmath343/kmath343|title=Archimedes on Spheres and Cylinders}}
• [http://web.mit.edu/2.009/www/experiments/steamCannon/ArchimedesSteamCannon.html Testing the Archimedes steam cannon] {{Webarchive|url=https://web.archive.org/web/20100329220142/http://web.mit.edu/2.009/www/experiments/steamCannon/ArchimedesSteamCannon.html |date=29 March 2010 }}

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