Archimedes of Syracuse
|Born||c. 287 BC|
|Died||c. 212 BC (aged approximately 75)|
Archimedes of Syracuse (
Archimedes' other mathematical achievements include deriving an
Archimedes died during the
Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Mathematicians from
Archimedes was born c. 287 BC in the seaport city of Syracuse, Sicily, at that time a self-governing colony in Magna Graecia. The date of birth is based on a statement by the Byzantine Greek historian John Tzetzes that Archimedes lived for 75 years before his death in 212 BC. In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known. A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving the details of his life obscure. It is unknown, for instance, whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth. From his surviving written works, it is clear that he maintained collegiate relations with scholars based there, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene.[b]
The standard versions of Archimedes' life were written long after his death by Greek and Roman historians. The earliest reference to Archimedes occurs in The Histories by Polybius (c. 200–118 BC), written about 70 years after his death. It sheds little light on Archimedes as a person, and focuses on the war machines that he is said to have built in order to defend the city from the Romans. Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign to take the city under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, which lasted from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved catapults, cranelike machines that could be swung around in an arc, and stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.
Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero 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 the cylinder including its bases. He also mentions that Marcellus brought to Rome two planetariums Archimedes built. The Roman historian Livy (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it.
Plutarch (45–119 AD) wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse. He also provides at least two accounts on how Archimedes died after the city was taken. According to the most popular account, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but he declined, saying that he had to finish working on the problem. This enraged the soldier, who killed Archimedes with his sword. Another story has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items. 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.
The last words attributed to Archimedes are "
Discoveries and inventions
The most widely known anecdote about Archimedes tells of how he invented a method for determining the volume of an object with an irregular shape. According to Vitruvius, a votive crown for a temple had been made for King Hiero II of Syracuse, who had supplied the pure gold to be used; Archimedes was asked to determine whether some silver had been substituted by the dishonest goldsmith. Archimedes had to solve the problem without damaging the crown, so he could not melt it down into a regularly shaped body in order to calculate its density.
In Vitruvius' account, Archimedes noticed while taking a bath that the level of the water in the tub rose as he got in, and realized that this effect could be used to determine the crown's volume. For practical purposes water is incompressible, so the submerged crown would displace an amount of water equal to its own volume. By dividing the mass of the crown by the volume of water displaced, the density of the crown could be obtained. This density would be lower than that of gold if cheaper and less dense metals had been added. Archimedes then took to the streets naked, so excited by his discovery that he had forgotten to dress, crying "Eureka!" (Greek: "εὕρηκα, heúrēka!, lit. 'I have found [it]!'). The test on the crown was conducted successfully, proving that silver had indeed been mixed in.
The story of the golden crown does not appear anywhere in Archimedes' known works. The practicality of the method it describes has been called into question due to the extreme accuracy that would be required while measuring the
A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of
Archimedes is said to have designed a claw as a weapon to defend the city of Syracuse. Also known as "the ship shaker", the claw consisted of a crane-like arm from which a large metal grappling hook was suspended. When the claw was dropped onto an attacking ship the arm would swing upwards, lifting the ship out of the water and possibly sinking it.
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.
Archimedes may have used mirrors acting collectively as a
This purported weapon has been the subject of an ongoing debate about its credibility since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, mostly with negative results. It has been suggested that a large array of highly polished bronze or copper shields acting as mirrors could have been employed to focus sunlight onto a ship, but the overall effect would have been blinding, dazzling, or distracting the crew of the ship rather than fire.
While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. There are several, often conflicting, reports regarding Archimedes' feats using the lever to lift very heavy objects. Plutarch describes how Archimedes designed block-and-tackle pulley systems, allowing sailors to use the principle of leverage to lift objects that would otherwise have been too heavy to move. According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth" (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω). Olympiodorus later attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass, rather than the lever.
Archimedes has also been credited with improving the power and accuracy of the catapult, and with inventing the odometer during the First Punic War. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner. Without the use of either trigonometry or a table of chords, Archimedes describes the procedure and instrument used to make observations (a straight rod with pegs or grooves), applies correction factors to these measurements, and finally gives the result in the form of upper and lower bounds to account for observational error. 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.
Hanc sphaeram Gallus cum moveret, fiebat ut soli luna totidem conversionibus in aere illo quot diebus in ipso caelo succederet, ex quo et in caelo sphaera solis fieret eadem illa defectio, et incideret luna tum in eam metam quae esset umbra terrae, cum sol e regione.
When Gallus moved the globe, it happened that the Moon followed the Sun by as many turns on that bronze contrivance as in the sky itself, from which also in the sky the Sun's globe became to have that same eclipse, and the Moon came then to that position which was its shadow on the Earth when the Sun was in line.
This is a description of a small planetarium. Pappus of Alexandria reports on a treatise by Archimedes (now lost) dealing with the construction of these mechanisms entitled On Sphere-Making. Modern research in this area has been focused on the Antikythera mechanism, another device built c. 100 BC that was probably designed for the same purpose. Constructing mechanisms of this kind would have required a sophisticated knowledge of differential gearing. This was once thought to have been beyond the range of the technology available in ancient times, but the discovery of the Antikythera mechanism in 1902 has confirmed that devices of this kind were known to the ancient Greeks.
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics. Plutarch wrote that Archimedes "placed his whole affection and ambition in those purer speculations where there can be no reference to the vulgar needs of life", though some scholars believe this may be a mischaracterization.
Method of exhaustion
Archimedes was able to use indivisibles (a precursor to infinitesimals) in a way that is similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay. This technique is known as the method of exhaustion, and he employed it to approximate the areas of figures and the value of π.
In On the Sphere and Cylinder, Archimedes postulates that any magnitude when added to itself enough times will exceed any given magnitude. Today this is known as the Archimedean property of real numbers.
Archimedes gives the value of the square root of 3 as lying between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512) in Measurement of a Circle. The actual value is approximately 1.7320508, making this a very accurate estimate. He introduced this result without offering any explanation of how he had obtained it. This aspect of the work of Archimedes caused John Wallis to remark that he was: "as it were of set purpose to have covered up the traces of his investigation as if he had grudged posterity the secret of his method of inquiry while he wished to extort from them assent to his results." It is possible that he used an iterative procedure to calculate these values.
The infinite series
In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4/3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1/4:
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
Myriad of myriads
In The Sand Reckoner, Archimedes set out to calculate the number of grains of sand that the universe could contain. In doing so, he challenged the notion that the number of grains of sand was too large to be counted. He wrote:
There are some, King Gelo (Gelo II, son of Hiero II), who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited.
To solve the problem, Archimedes devised a system of counting based on the myriad. The word itself derives from the Greek μυριάς, murias, for the number 10,000. 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 8×1063.
The works of Archimedes were written in Doric Greek, the dialect of ancient Syracuse. Many written works by Archimedes have not survived or are only extant in heavily edited fragments; at least seven of his treatises are known to have existed due to references made by other authors. Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra, while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica.[c]
Archimedes made his work known through correspondence with the mathematicians in Alexandria. The writings of Archimedes were first collected by the Byzantine Greek architect Isidore of Miletus (c. 530 AD), while commentaries on the works of Archimedes written by Eutocius in the sixth century AD helped to bring his work a wider audience. 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).
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 (π), showing that it is greater than 223/71 and less than 22/7.
The Sand Reckoner
In this treatise, also known as Psammites, Archimedes counts the number of
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 work, Archimedes proves the Law of the lever, which states that:
Magnitudes are in equilibrium at distances reciprocally proportional to their weights.
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 times the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio 1/4.
On the Sphere and Cylinder
In this two-volume treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a
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 polar coordinates (r, θ) it can be described by the equation with real numbers a and b.
On Conoids and Spheroids
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of
On Floating Bodies
In the first part of this two-volume treatise, Archimedes spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not self-gravitating since he assumes the existence of a point towards which all things fall in order to derive the spherical shape. Archimedes' principle of buoyancy is given in this work, stated as follows:
Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in sense 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.
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. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded. The puzzle represents an example of an early problem in combinatorics.
The origin of the puzzle's name is unclear, and it has been suggested that it is taken from the Ancient Greek word for "throat" or "gullet", stomachos (στόμαχος). Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον, 'bone') and machē (μάχη, 'fight').
The cattle problem
The Method of Mechanical Theorems
This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In this work Archimedes uses indivisibles, and shows how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume. He may have considered this method lacking in formal rigor, so he also used the method of exhaustion to derive the results. As with The Cattle Problem, The Method of Mechanical Theorems was written in the form of a letter to Eratosthenes in Alexandria.
It has also been claimed that the
The foremost document containing Archimedes' work is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examined a 174-page goatskin parchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier by Papadopoulos-Kerameus. 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. 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 $2 million.
The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. It is the only known source of The Method of Mechanical Theorems, referred to by Suidas and thought to have been lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than had been found in previous texts. The palimpsest 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 X-ray light to read the overwritten text. It has since returned to its anonymous owner.
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
- Speeches by the 4th century BC politician Hypereides
- A commentary on Aristotle's Categories
- Other works
Mathematics and physics
Historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity. Eric Temple Bell, for instance, wrote:
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.
... in the year 1500 Europe knew less than Archimedes who died in the year 212 BC ...
If we consider what all other men accomplished in mathematics and physics, on every continent and in every civilization, from the beginning of time down to the seventeenth century in Western Europe, the achievements of Archimedes outweighs it all. He was a great civilization all by himself.
And so, since Archimedes led more than anyone else to the formation of the calculus and since he was the pioneer of the application of mathematics to the physical world, it turns out that Western science is but a series of footnotes to Archimedes. Thus, it turns out that Archimedes is the most important scientist who ever lived.
Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes. Galileo called him "superhuman" and "my master", while Huygens said, "I think Archimedes is comparable to no one" and modeled his work after him. Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times." Gauss's heroes were Archimedes and Newton, 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."
The inventor Nikola Tesla praised him, saying:
Archimedes was my ideal. I admired the works of artists, but to my mind, they were only shadows and semblances. The inventor, I thought, gives to the world creations which are palpable, which live and work.
Honors and commemorations
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").
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.
- Archimedean point
- Archimedes' axiom
- Archimedes number
- Archimedes paradox
- Archimedean solid
- Archimedes' twin circles
- Methods of computing square roots
- Steam cannon
- Trammel of Archimedes
- In the preface to On Spirals addressed to Dositheus of Pelusium, Archimedes says that "many years have elapsed since Conon's death." Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.
- The treatises by Archimedes known to exist only through references in the works of other authors are: On Sphere-Making and a work on polyhedra mentioned by Pappus of Alexandria; Catoptrica, a work on optics mentioned by Theon of Alexandria; Principles, addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; On Balances and Levers; On Centers of Gravity; On the Calendar.
"To be sure, Pappus does twice mention the theorem on the tangent to the spiral [IV, 36, 54]. But in both instances the issue is Archimedes' inappropriate use of a 'solid neusis,' that is, of a construction involving the sections of solids, in the solution of a plane problem. Yet Pappus' own resolution of the difficulty [IV, 54] is by his own classification a 'solid' method, as it makes use of conic sections." (p. 48)
- "Archimedes". Collins Dictionary. n.d. Archived from the original on 3 March 2016. Retrieved 25 September 2014.
- "Archimedes (c. 287 – c. 212 BC)". BBC History. Archived from the original on 19 April 2012. Retrieved 7 June 2012.
- Powers, J (2020). "Did Archimedes do calculus?" (PDF). www.maa.org. Archived (PDF) from the original on 31 July 2020. Retrieved 14 April 2021.
- O'Connor, J.J.; Robertson, E.F. (February 1996). "A history of calculus". University of St Andrews. Archived from the original on 15 July 2007. Retrieved 7 August 2007.
- Heath, Thomas L. 1897. Works of Archimedes.
- Hoyrup, J. (2019). Archimedes: Knowledge and lore from Latin Antiquity to the outgoing European Renaissance. Selected Essays on Pre- and Early Modern Mathematical Practice. pp. 459–477.
- "Works, Archimedes". University of Oklahoma. 23 June 2015. Archived from the original on 15 August 2017. Retrieved 18 June 2019.
- "Archimedes – The Palimpsest". Walters Art Museum. Archived from the original on 28 September 2007. Retrieved 14 October 2007.
- Flood, Alison. "Archimedes Palimpsest reveals insights centuries ahead of its time". The Guardian. Archived from the original on 15 May 2021. Retrieved 10 February 2017.
- "The Death of Archimedes: Illustrations". math.nyu.edu. New York University. Archived from the original on 29 September 2015. Retrieved 13 December 2017.
- Acerbi, F. (2008). Archimedes. New Dictionary of Scientific Biography. pp. 85–91.
- Rorres, Chris. "Death of Archimedes: Sources". Courant Institute of Mathematical Sciences. Archived from the original on 10 December 2006. Retrieved 2 January 2007.
- Rorres, Chris. "Siege of Syracuse". Courant Institute of Mathematical Sciences. Archived from the original on 9 June 2007. Retrieved 23 July 2007.
- Rorres, Chris. "Tomb of Archimedes: Sources". Courant Institute of Mathematical Sciences. Archived from the original on 9 December 2006. Retrieved 2 January 2007.
- Rorres, Chris. "Tomb of Archimedes – Illustrations". Courant Institute of Mathematical Sciences. Archived from the original on 2 May 2019. Retrieved 15 March 2011.
- "The Planetarium of Archimedes". studylib.net. Archived from the original on 14 April 2021. Retrieved 14 April 2021.
- Plutarch (October 1996). Parallel Lives Complete e-text from Gutenberg.org. Project Gutenberg. Archived from the original on 20 September 2008. Retrieved 23 July 2007.
- Plutarch. Extract from Parallel Lives. fulltextarchive.com. Archived from the original on 7 March 2014. Retrieved 10 August 2009.
- Jaeger, Mary. Archimedes and the Roman Imagination. p. 113.
- Vitruvius (31 December 2006). De Architectura, Book IX, paragraphs 9–12. Project Gutenberg. Archived from the original on 6 November 2019. Retrieved 26 December 2018.
- "Incompressibility of Water". Harvard University. Archived from the original on 17 March 2008. Retrieved 27 February 2008.
- Rorres, Chris (ed.). "The Golden Crown: Sources". New York University. Archived from the original on 9 March 2021. Retrieved 6 April 2021.
- Morgan, Morris Hicky (1914). Vitruvius: The Ten Books on Architecture. Cambridge: Harvard University Press. pp. 253–254.
Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.
- Vitruvius (1567). De Architetura libri decem. Venice: Daniele Barbaro. pp. 270–271.
Postea vero repleto vase in eadem aqua ipsa corona demissa, invenit plus aquae defluxisse in coronam, quàm in auream eodem pondere massam, et ita ex eo, quod plus defluxerat aquae in corona, quàm in massa, ratiocinatus, deprehendit argenti in auro mixtionem, et manifestum furtum redemptoris.
- Morgan, Morris Hicky (1914). Vitruvius: The Ten Books on Architecture. Cambridge: Harvard University Press. pp. 253–254.
- Rorres, Chris. "The Golden Crown". Drexel University. Archived from the original on 11 March 2009. Retrieved 24 March 2009.
- Carroll, Bradley W. "Archimedes' Principle". Weber State University. Archived from the original on 8 August 2007. Retrieved 23 July 2007.
- Van Helden, Al. "The Galileo Project: Hydrostatic Balance". Rice University. Archived from the original on 5 September 2007. Retrieved 14 September 2007.
- Rorres, Chris. "The Golden Crown: Galileo's Balance". Drexel University. Archived from the original on 24 February 2009. Retrieved 24 March 2009.
- Dalley, Stephanie; Oleson, John Peter. "Sennacherib, Archimedes, and the Water Screw: The Context of Invention in the Ancient World". Technology and Culture Volume 44, Number 1, January 2003 (PDF). Archived from the original on 16 July 2015. Retrieved 23 July 2007.
- Rorres, Chris. "Archimedes' screw – Optimal Design". Courant Institute of Mathematical Sciences. Archived from the original on 22 July 2012. Retrieved 23 July 2007.
- "SS Archimedes". wrecksite.eu. Archived from the original on 2 October 2011. Retrieved 22 January 2011.
- Rorres, Chris. "Archimedes' Claw – Illustrations and Animations – a range of possible designs for the claw". Courant Institute of Mathematical Sciences. Archived from the original on 7 December 2010. Retrieved 23 July 2007.
- Carroll, Bradley W. "Archimedes' Claw – watch an animation". Weber State University. Archived from the original on 13 August 2007. Retrieved 12 August 2007.
- Hippias, 2 (cf. Galen, On temperaments 3.2, who mentions pyreia, "torches"); Anthemius of Tralles, On miraculous engines 153 [Westerman].
- "World's Largest Solar Furnace". Atlas Obscura. Archived from the original on 5 November 2016. Retrieved 6 November 2016.
- "Archimedes Death Ray: Testing with MythBusters". MIT. Archived from the original on 28 May 2013. Retrieved 23 July 2007.
- John Wesley. "A Compendium of Natural Philosophy (1810) Chapter XII, Burning Glasses". Online text at Wesley Center for Applied Theology. Archived from the original on 12 October 2007. Retrieved 14 September 2007.
- "TV Review: MythBusters 8.27 – President's Challenge". 13 December 2010. Archived from the original on 29 October 2013. Retrieved 18 December 2010.
- Finlay, M. (2013). Constructing ancient mechanics Archived 14 April 2021 at the Wayback Machine [Master's thesis]. University of Glassgow.
- Rorres, Chris. "The Law of the Lever According to Archimedes". Courant Institute of Mathematical Sciences. Archived from the original on 27 September 2013. Retrieved 20 March 2010.
- Dougherty, F.C.; Macari, J.; Okamoto, C. "Pulleys". Society of Women Engineers. Archived from the original on 18 July 2007. Retrieved 23 July 2007.
- Quoted by Pappus of Alexandria in Synagoge, Book VIII
- "Ancient Greek Scientists: Hero of Alexandria". Technology Museum of Thessaloniki. Archived from the original on 5 September 2007. Retrieved 14 September 2007.
- Cicero. "De re publica 1.xiv §21". thelatinlibrary.com. Archived from the original on 22 March 2007. Retrieved 23 July 2007.
- Cicero (9 February 2005). De re publica Complete e-text in English from Gutenberg.org. Project Gutenberg. Archived from the original on 20 September 2008. Retrieved 18 September 2007.
- Noble Wilford, John (31 July 2008). "Discovering How Greeks Computed in 100 B.C." The New York Times. Archived from the original on 24 June 2017. Retrieved 25 December 2013.
- "The Antikythera Mechanism II". Stony Brook University. Archived from the original on 12 December 2013. Retrieved 25 December 2013.
- Rorres, Chris. "Spheres and Planetaria". Courant Institute of Mathematical Sciences. Archived from the original on 10 May 2011. Retrieved 23 July 2007.
- "Ancient Moon 'computer' revisited". BBC News. 29 November 2006. Archived from the original on 15 February 2009. Retrieved 23 July 2007.
- Carrier, Richard (2008). Attitudes toward the natural philosopher in the early Roman empire (100 B.C. to 313 A.D.) (Thesis). Archived from the original on 14 April 2021. Retrieved 6 April 2021. "Hence Plutarch's conclusion that Archimedes disdained all mechanics, shop work, or anything useful as low and vulgar, and only directed himself to geometric theory, is obviously untrue. Thus, as several scholars have now concluded, his account of Archimedes appears to be a complete fabrication, invented to promote the Platonic values it glorifies by attaching them to a much-revered hero." (p.444)
- Heath, T.L. "Archimedes on measuring the circle". math.ubc.ca. Archived from the original on 3 July 2004. Retrieved 30 October 2012.
- Kaye, R.W. "Archimedean ordered fields". web.mat.bham.ac.uk. Archived from the original on 16 March 2009. Retrieved 7 November 2009.
- "Of Calculations Past and Present: The Archimedean Algorithm | Mathematical Association of America". www.maa.org. Archived from the original on 14 April 2021. Retrieved 14 April 2021.
- McKeeman, Bill. "The Computation of Pi by Archimedes". Matlab Central. Archived from the original on 25 February 2013. Retrieved 30 October 2012.
- Carroll, Bradley W. "The Sand Reckoner". Weber State University. Archived from the original on 13 August 2007. Retrieved 23 July 2007.
- "Editions of Archimedes' Work". Brown University Library. Archived from the original on 8 August 2007. Retrieved 23 July 2007.
- "English translation of The Sand Reckoner". University of Waterloo. Archived from the original on 11 August 2007. Retrieved 23 July 2007.
- Heath, T.L. (1897). The Works of Archimedes (1897). The unabridged work in PDF form (19 MB). Cambridge University Press. Archived from the original on 6 October 2007. Retrieved 14 October 2007.
- "Graeco Roman Puzzles". Gianni A. Sarcone and Marie J. Waeber. Archived from the original on 14 May 2008. Retrieved 9 May 2008.
- Kolata, Gina (14 December 2003). "In Archimedes' Puzzle, a New Eureka Moment". The New York Times. Archived from the original on 14 July 2021. Retrieved 23 July 2007.
- Ed Pegg Jr. (17 November 2003). "The Loculus of Archimedes, Solved". Mathematical Association of America. Archived from the original on 19 May 2008. Retrieved 18 May 2008.
- Rorres, Chris. "Archimedes' Stomachion". Courant Institute of Mathematical Sciences. Archived from the original on 26 October 2007. Retrieved 14 September 2007.
- Krumbiegel, 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.
- Calkins, Keith G. "Archimedes' Problema Bovinum". Andrews University. Archived from the original on 12 October 2007. Retrieved 14 September 2007.
- O'Connor, J.J.; Robertson, E.F. (April 1999). "Heron of Alexandria". University of St Andrews. Archived from the original on 9 May 2010. Retrieved 17 February 2010.
- 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.
- Miller, Mary K. (March 2007). "Reading Between the Lines". Smithsonian. Archived from the original on 19 January 2008. Retrieved 24 January 2008.
- "Rare work by Archimedes sells for $2 million". CNN. 29 October 1998. Archived from the original on 16 May 2008. Retrieved 15 January 2008.
- "X-rays reveal Archimedes' secrets". BBC News. 2 August 2006. Archived from the original on 25 August 2007. Retrieved 23 July 2007.
- Acerbi, F. (2013). "R. Netz, W. Noel, N. Tchernetska, N. Wilson (eds.), The Archimedes Palimpsest, 2 vols, Cambridge, Cambridge University Press 2011". Aestimatio. 10: 34–46.
- father of mathematics: Jane Muir, Of Men and Numbers: The Story of the Great Mathematicians, p 19.
- father of 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.
- E.T. Bell, Men of Mathematics, p 20.
- Alfred North Whitehead. "The Influence of Western Medieval Culture Upon the Development of Modern Science".
- George F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, p 43.
- Reviel Netz, William Noel, The Archimedes Codex: Revealing The Secrets Of The World's Greatest Palimpsest
- "The Steam-Engine". Nelson Examiner and New Zealand Chronicle. Vol. I, no. 11. Nelson: National Library of New Zealand. 21 May 1842. p. 43. Archived from the original on 24 July 2011. Retrieved 14 February 2011.
- The Steam Engine. The Penny Magazine. 1838. p. 104. Archived from the original on 7 May 2021. Retrieved 7 May 2021.
- Matthews, Michael. Time for Science Education: How Teaching the History and Philosophy of Pendulum Motion Can Contribute to Science Literacy. p. 96.
- "Archimedes - Galileo Galilei and Archimedes". exhibits.museogalileo.it. Archived from the original on 17 April 2021. Retrieved 16 June 2021.
- Yoder, J. (1996). "Following in the footsteps of geometry: the mathematical world of Christiaan Huygens". De Zeventiende Eeuw. Jaargang 12. Archived from the original on 12 May 2021.
- Boyer, Carl B., and Uta C. Merzbach. 1968. A History of Mathematics. ch. 7.
- Jay Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, p 88.
- E.T. Bell, Men of Mathematics, p 237
- W. Bernard Carlson, Tesla: Inventor of the Electrical Age, p 57
- Friedlander, Jay; Williams, Dave. "Oblique view of Archimedes crater on the Moon". NASA. Archived from the original on 19 August 2007. Retrieved 13 September 2007.
- Riehm, C. (2002). "The early history of the Fields Medal" (PDF). Notices of the AMS. 49 (7): 778–782. Archived (PDF) from the original on 18 January 2021. Retrieved 28 April 2021.
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).
- "The Fields Medal". Fields Institute for Research in Mathematical Sciences. 5 February 2015. Archived from the original on 23 April 2021. Retrieved 23 April 2021.
- "Fields Medal". International Mathematical Union. Archived from the original on 2 December 2017. Retrieved 23 April 2021.
- Rorres, Chris. "Stamps of Archimedes". Courant Institute of Mathematical Sciences. Archived from the original on 2 October 2010. Retrieved 25 August 2007.
- "California Symbols". California State Capitol Museum. Archived from the original on 12 October 2007. Retrieved 14 September 2007.
- ISBN 978-0-471-54397-8.
- Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 1–5. Madison, WI: University of Wisconsin Press.
- ISBN 978-0-691-08421-3.
- ISBN 978-0-7660-2502-8.
- Hasan, Heather. 2005. ISBN 978-1-4042-0774-5.
- ISBN 978-0-486-42084-4. Complete works of Archimedes in English.
- ISBN 978-0-297-64547-4.
- ISBN 978-0-19-533611-5.
- Simms, Dennis L. 1995. Archimedes the Engineer. ISBN 978-0-7201-2284-8.
- ISBN 978-0-88385-718-2.
- Heiberg's Edition of Archimedes. Texts in Classical Greek, with some in English.
- Archimedes on In Our Time at the BBC
- Works by Archimedes at Project Gutenberg
- Works by or about Archimedes at Internet Archive
- Archimedes at the Indiana Philosophy Ontology Project
- Archimedes at PhilPapers
- The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
- "Archimedes and the Square Root of 3". MathPages.com.
- "Archimedes on Spheres and Cylinders". MathPages.com.
- Testing the Archimedes steam cannon Archived 29 March 2010 at the Wayback Machine