Archimedes

2.1. Early life and education

Archimedes was born around 287 BC in the seaport city of Syracuse, Sicily, which was then a self-governing colony in Magna Graecia. His birth year is estimated based on a statement by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years before his death in 212 BC.

In his treatise The Sand Reckoner, Archimedes states his father's name as Phidias, an astronomer about whom nothing else is known. The historian Plutarch, in his Parallel Lives, suggests that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero implies he was of more humble origins. A biography of Archimedes was written by his friend Heracleides, but this work has been lost, leaving many details of his life obscure, such as whether he married, had children, or visited Alexandria, Egypt, during his youth. However, his surviving writings indicate that he maintained collegial relationships with scholars based in Alexandria, including his friend Conon of Samos and the head librarian Eratosthenes of Cyrene. In the preface to On Spirals, addressed to Dositheus of Pelusium, Archimedes notes that "many years have elapsed since Conon's death," suggesting he may have been an older man when writing some of his works.

2.2. Career and Scientific Pursuits

During the Second Punic War, Syracuse shifted its allegiance from Rome to Carthage, leading to a Roman military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. Polybius notes that the Romans underestimated Syracuse's defenses and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could swing in an arc, and other stone-throwers. Although the Romans eventually captured the city, they suffered considerable losses due to Archimedes' inventiveness.

Cicero (106-43 BC) also mentions Archimedes in some of his works. While serving as a quaestor in Sicily, Cicero discovered what was presumed to be Archimedes' tomb near the Agrigentine gate in Syracuse. The tomb was in a neglected condition, overgrown with bushes. Cicero had the tomb cleaned, allowing him to see the carving and read some of the verses inscribed. The tomb featured a sculpture illustrating Archimedes' favorite mathematical proof: that the volume and surface area of a sphere are two-thirds that of an enclosing cylinder, including its bases. Cicero also records that Marcellus brought two planetariums built by Archimedes to Rome. The Roman historian Livy (59 BC-17 AD) retells Polybius' account of the capture of Syracuse and Archimedes' role in it.

2.3. Death

Archimedes died during the siege of Syracuse in 212 BC. Plutarch (45-119 AD) provides at least two accounts of his death. According to the most popular version, Archimedes was deeply engrossed in a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet Marcellus, but Archimedes refused, stating he had to finish his problem. Enraged, the soldier killed Archimedes with his sword. Another story suggests Archimedes was carrying mathematical instruments and was killed because a soldier mistook them for valuable items. Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset, calling him "a geometrical Briareus", and had ordered that he should not be harmed.

The last words often attributed to Archimedes are "Do not disturb my circles" (Noli turbare circulos meos^Latin; μὴ μου τοὺς κύκλους τάραττε^{Greek, Modern}), referring to the mathematical drawing he was supposedly studying. However, there is no reliable evidence that Archimedes uttered these words, and they do not appear in Plutarch's account. A similar quotation is found in the work of Valerius Maximus (fl. 30 AD), who wrote in Memorable Doings and Sayings, "... sed protecto manibus puluere 'noli' inquit, 'obsecro, istum disturbare'^Latin", meaning "... but protecting the dust with his hands, said 'I beg of you, do not disturb this'."

Archimedes' tomb, as described by Cicero, was surmounted by a sphere and a cylinder, representing his proof that the volume and surface area of a sphere are two-thirds that of an enclosing cylinder. In 75 BC, 137 years after Archimedes' death, Cicero, serving as a quaestor in Sicily, sought out Archimedes' tomb. He found it near the Agrigentine gate in Syracuse, neglected and overgrown with bushes. Cicero had the tomb cleaned, revealing the carving and inscription. A tomb discovered in the 1960s in Syracuse was initially claimed to be Archimedes', but its true location remains unknown.

3.1. Method of Exhaustion and Infinitesimals

Archimedes utilized indivisibles, a precursor to infinitesimals, in a manner similar to modern integral calculus. Through proof by contradiction (reductio ad absurdum), he could provide solutions to problems with 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 -.

3.2. Calculation of Pi

In Measurement of a Circle, Archimedes used the method of exhaustion to approximate the value of pi. He did this by drawing a larger regular hexagon outside a circle and a smaller regular hexagon inside the circle. He then progressively doubled 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 increased, the polygons became more accurate approximations of the circle. After four such steps, when the polygons had 96 sides each, he was able to determine that the value of - lay between 223/71 (approximately 3.1408) and 22/7 (approximately 3.1429). This result is consistent with the actual value of approximately 3.1416.

3.3. Geometric Discoveries

Archimedes proved that the area of a circle was equal to - multiplied by the square of the radius of the circle (πr²). In On the Sphere and Cylinder, he obtained the result of which he was most proud: the relationship between a sphere and a circumscribed cylinder of the same height and diameter. He proved that the volume of a sphere is 4/3-r³, and its surface area is 4-r². For the circumscribing cylinder, its volume is 2-r³ and its surface area (including its two bases) is 6-r². Thus, the volume and surface area of a sphere are both two-thirds that of its circumscribing cylinder. This discovery was so significant to him that he requested a sphere and cylinder be placed on his tomb.

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. He expressed the solution to this problem as an infinite geometric series with a common ratio of 1/4:
:$\backslash sum\_\{n=0\}^\backslash infty\; 4^\{-n\}\; =\; 1\; +\; 4^\{-1\}\; +\; 4^\{-2\}\; +\; 4^\{-3\}\; +\; \backslash cdots\; =\; \{4\backslash over\; 3\}.\; \backslash ;$
If the first term in this series is the area of the initial inscribed triangle, then subsequent terms represent the sum of the areas of successively smaller triangles that fill the remaining parabolic segments. This proof effectively uses a variation of the series 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.

In Measurement of a Circle, Archimedes also gave a very accurate estimate for the square root of 3, stating it lay between 265/153 (approximately 1.7320261) and 1351/780 (approximately 1.7320512). The actual value is approximately 1.7320508. He introduced this result without explaining his method, leading John Wallis to remark that Archimedes seemed 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 he used an iterative procedure to calculate these values.

3.4. Archimedean Property

In On the Sphere and Cylinder, Archimedes postulated what is now known as the Archimedean property of real numbers: any magnitude, when added to itself a sufficient number of times, will exceed any given magnitude.

3.5. The Sand Reckoner

In The Sand Reckoner, also known as Psammites (Ψαμμίτης^{Greek, Ancient}), Archimedes set out to calculate a number greater than the grains of sand needed to fill the universe. He challenged the notion that the number of grains of sand was too large to be counted, stating:
:There are some, King Gelo, 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 this problem, Archimedes devised a system of counting based on the myriad. The word itself derives from the Greek μυριάς^{Greek, Ancient}, 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 8x10⁶³ 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.

3.6. Major Mathematical Treatises

Archimedes' mathematical works were written in Doric Greek, the dialect of ancient Syracuse. Many of his written works 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.

His major surviving mathematical works, ordered chronologically based on modern scholarly criteria, include:

• Measurement of a Circle (Κύκλου μέτρησις^{Greek, Ancient}): A short work consisting of three propositions, written as a correspondence with Dositheus of Pelusium. It provides an approximation of pi (π) between 223/71 and 22/7.
• The Sand Reckoner (Ψαμμίτης^{Greek, Ancient}): Also known as Psammites, this treatise calculates the number of grains of sand needed to fill the universe, introducing a system for denoting very large numbers. It also discusses heliocentrism and measurements of celestial bodies.
• On the Equilibrium of Planes (Περὶ ἐπιπέδων ἱσορροπιῶν^{Greek, Ancient}): This work, in two books, proves the law of the lever and calculates the areas and centers of gravity of various geometric figures including triangles, parallelograms, and parabolas.
• Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς^{Greek, Ancient}): In this work of 24 propositions addressed to Dositheus, Archimedes proves that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height, using an infinite series.
• On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου^{Greek, Ancient}): This two-volume treatise, also addressed to Dositheus, contains Archimedes' most celebrated result: the relationship between a sphere and its circumscribed cylinder of the same height and diameter, showing the sphere's volume and surface area are two-thirds that of the cylinder.
• On Spirals (Περὶ ἑλίκων^{Greek, Ancient}): This work of 28 propositions defines the Archimedean spiral, the locus of points corresponding to a point moving away from a fixed point with constant speed along a line rotating with constant angular velocity.
• On Conoids and Spheroids (Περὶ κωνοειδέων καὶ σϕαιοειδέων^{Greek, Ancient}): A work in 32 propositions addressed to Dositheus, where Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.
• On Floating Bodies (Περὶ τῶν ἐπιπλεόντων σωμάτων^{Greek, Ancient}): In two books, this treatise spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity. It also presents Archimedes' principle of buoyancy.
• Ostomachion (Στομάχιον, Ὀστομάχιον^{Greek, Ancient}): Also known as Loculus of Archimedes or Archimedes' Box, this is a dissection puzzle similar to a Tangram. The treatise, found in the Archimedes Palimpsest, calculates the areas of its 14 pieces. Scholars have debated whether Archimedes was trying to determine the number of ways the pieces could be assembled into a square (estimated at 17,152 ways, or 536 excluding rotations and reflections).
• The Cattle Problem: Discovered in a Greek manuscript in 1773, this work is a 44-line poem addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a set of simultaneous Diophantine equations. The answer to the more difficult version of the problem is a very large number, approximately 7.760271x10²⁰⁶⁵⁴⁴.
• The Method of Mechanical Theorems (Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος^{Greek, Ancient}): This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In it, Archimedes uses indivisibles to show 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 rigorously prove the results.

4.1. Principle of Buoyancy

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, King Hiero II of Syracuse commissioned a new crown for a temple, providing pure gold for its creation. The King suspected that the goldsmith had substituted some silver for gold without damaging the crown. Archimedes was tasked with determining this without melting the crown down.

The story goes that while taking a bath, Archimedes noticed that the water level in the tub rose as he got in. He realized that this effect could be used to determine the crown's volume. Overjoyed by this discovery, he reportedly ran naked through the streets, crying "Eureka!" (εὕρηκα!^{Greek, Modern}, heúrēka!, meaning "I have found [it]!"). Since water is practically incompressible, 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, its density could be obtained. If cheaper and less dense metals like silver had been added, the density would be lower than that of pure gold. Archimedes' test successfully proved that silver had been mixed in.

While this popular story is not found in Archimedes' own works and its practical accuracy has been questioned due to the extreme precision required for water displacement measurements, it is believed that Archimedes likely applied the hydrostatics principle known as Archimedes' principle, which he detailed in his treatise On Floating Bodies. This principle states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. Using this, he could have compared the crown's density to pure gold by balancing it on a scale with a pure gold reference sample of the same weight, then immersing the entire apparatus in water. A difference in density would cause the scale to tip. Galileo Galilei, who invented a hydrostatic balance in 1586 inspired by Archimedes' work, considered this method "probable... since, besides being very accurate, it is based on demonstrations found by Archimedes himself."

4.2. Law of the Lever

While Archimedes did not invent the lever, he provided a mathematical proof of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever principle are found in a work by Euclid and in the Mechanical Problems, attributed by some to Archytas.

There are several, often conflicting, reports regarding Archimedes' feats using the lever. 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 and his understanding of mechanical advantage led him to remark: "Give me a place to stand on, and I will move the Earth" (δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω^{Greek, Modern}). Olympiodorus later attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass, rather than the lever.

4.3. Statics and Hydrostatics

Archimedes was one of the first to apply mathematics to physical phenomena, working extensively on statics and hydrostatics. His achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of Archimedes' principle of buoyancy. In his treatise On Floating Bodies, he 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 to explain the theory of contemporary Greek astronomers like Eratosthenes that the Earth is round.

4.4. Inventions and Machines

Archimedes is credited with designing numerous innovative machines and mechanical devices, many of which arose from the practical needs of his home city of Syracuse.

4.4.1. Screw pump (Archimedes' screw)

Athenaeus of Naucratis describes how King Hiero II commissioned Archimedes to design a huge ship, the Syracusia, intended for luxury travel, carrying supplies, and as a display of naval power. The Syracusia is said to have been the largest ship built in classical antiquity, capable of carrying 600 people and featuring garden decorations, a gymnasium, and a temple dedicated to the goddess Aphrodite. To remove any potential water leaking through the hull, Archimedes designed a device with a revolving screw-shaped blade inside a cylinder.

This device, known as the Archimedes' screw, was turned by hand and could transfer water from a low-lying body of water into irrigation canals. The screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described by Vitruvius, Archimedes' device may have been an improvement on a screw pump used to irrigate the Hanging Gardens of Babylon. The world's first seagoing steamship with a screw propeller was the SS Archimedes, launched in 1839 and named in honor of Archimedes.

4.4.2. War Machines

Archimedes is said to have designed a claw as a weapon to defend 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 potentially sinking it. Modern experiments have tested 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 has also been credited with improving the power and accuracy of the catapult.

4.4.4. Other Inventions

Archimedes is also credited 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.

As legend has it, Archimedes arranged mirrors as a parabolic reflector to burn ships attacking Syracuse using focused sunlight. While there is no extant contemporary evidence of this feat and modern scholars believe it did not happen, Archimedes may have written a work on mirrors entitled Catoptrica. Lucian and Galen, writing in the second century AD, mentioned that during the siege of Syracuse Archimedes had burned enemy ships. Nearly four hundred years later, Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.

The purported device, sometimes called "Archimedes' heat ray", has been the subject of ongoing debate since the Renaissance. René Descartes rejected it as false, while modern researchers have attempted to recreate the effect using only the means available to Archimedes, mostly with negative results. In 1973, Greek scientist Ioannis Sakkas conducted an experiment at the Skaramagas naval base outside Athens. Using 70 copper-coated mirrors, each about 5 ft by 3 ft, he focused sunlight onto a plywood model of a Roman warship about 160 ft (164 ft ^{(50 m)}) away. The ship ignited in a few seconds, though it was coated with tar, which might have made it more flammable.

In October 2005, a group of students from the Massachusetts Institute of Technology (MIT) conducted an experiment using 127 one-square-foot mirrors, focusing sunlight onto a wooden model ship 100 ft (98 ft ^{(30 m)}) away. The ship caught fire, but only when the sky was clear and the ship remained stationary for about 10 minutes. The MIT team repeated the experiment for the television program MythBusters in San Francisco, targeting a wooden fishing boat. While some charring and small flames were produced, the overall conclusion was that the device was not a practical weapon due to the time and ideal weather conditions required. The program also noted that Syracuse faced east, meaning Roman fleets would have to attack in the morning for the mirrors to be most effective. Furthermore, conventional weapons like fire arrows or catapults would have been simpler and more effective at such close ranges. Wood typically ignites at around 572 °F ^{(300 °C)} (570 °F).

5.1. Surviving Works

The following are Archimedes' major surviving works, ordered chronologically based on new terminological and historical criteria:

• On the Equilibrium of Planes (Περὶ ἐπιπέδων ἱσορροπιῶν^{Greek, Ancient}): This work, in two books, contains seven postulates and fifteen propositions in the first book, and ten propositions in the second. Archimedes proves the law of the lever, stating that "Magnitudes are in equilibrium at distances reciprocally proportional to their weights." He uses these principles to calculate the areas and centers of gravity of various geometric figures, including triangles, parallelograms, and parabolas.
• Measurement of a Circle (Κύκλου μέτρησις^{Greek, Ancient}): A short work of three propositions, written as a correspondence with Dositheus of Pelusium. It provides an approximation of pi (π) between 223/71 and 22/7.
• On Spirals (Περὶ ἑλίκων^{Greek, Ancient}): This work of 28 propositions, also addressed to Dositheus, defines the Archimedean spiral, the locus of points corresponding to a point moving away from a fixed point with constant speed along a line rotating with constant angular velocity.
• On the Sphere and Cylinder (Περὶ σφαίρας καὶ κυλίνδρου^{Greek, Ancient}): In this two-volume treatise addressed to Dositheus, Archimedes obtains his most proud result: the relationship between a sphere and a circumscribed cylinder of the same height and diameter. He proves the volume of the sphere is 4/3-r³ and its surface area is 4-r², while for the cylinder, the volume is 2-r³ and the surface area is 6-r² (including its two bases).
• On Conoids and Spheroids (Περὶ κωνοειδέων καὶ σϕαιοειδέων^{Greek, Ancient}): This work of 32 propositions, addressed to Dositheus, calculates the areas and volumes of sections of cones, spheres, and paraboloids.
• On Floating Bodies (Περὶ τῶν ἐπιπλεόντων σωμάτων^{Greek, Ancient}): In two books, Archimedes spells out the law of equilibrium of fluids and proves that water will adopt a spherical form around a center of gravity. He also presents Archimedes' principle of buoyancy, stating:

: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, which was probably an idealization of the shapes of ships' hulls.

• Quadrature of the Parabola (Τετραγωνισμὸς παραβολῆς^{Greek, Ancient}): In this work of 24 propositions addressed to Dositheus, Archimedes proves that the area enclosed by a parabola and a straight line is 4/3 the area of a triangle with equal base and height, using a geometric series.
• Ostomachion (Στομάχιον, Ὀστομάχιον^{Greek, Ancient}): Also known as Loculus of Archimedes or Archimedes' Box, this is a dissection puzzle similar to a Tangram. The treatise, found in the Archimedes Palimpsest, calculates the areas of its 14 pieces. Scholars have debated whether Archimedes was trying to determine the number of ways the pieces could be assembled into a square (estimated at 17,152 ways, or 536 excluding rotations and reflections).
• The Cattle Problem: A 44-line poem in the form of a challenge to Eratosthenes and the mathematicians in Alexandria, presenting a set of simultaneous Diophantine equations to count the cattle of the Sun.
• The Sand Reckoner (Ψαμμίτης^{Greek, Ancient}): This treatise calculates the number of grains of sand needed to fill the universe, introducing a system for denoting very large numbers. It is the only surviving work in which Archimedes discusses his views on astronomy.
• The Method of Mechanical Theorems (Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος^{Greek, Ancient}): This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906. In it, Archimedes uses indivisibles to show how breaking up a figure into an infinite number of infinitely small parts can be used to determine its area or volume.

5.2. Lost Works

At least seven of Archimedes' treatises are known to have existed due to references made by other ancient authors. These include:

• 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 or On Levers.
• On Centers of Gravity.
• On the Calendar.

Some works are considered apocryphal or of questionable attribution. Archimedes' Book of Lemmas or Liber Assumptorum, a treatise with 15 propositions on the nature of circles, is known from an Arabic copy. Scholars like T. L. Heath and Marshall Clagett argue it cannot have been written by Archimedes in its current form, as it quotes Archimedes, suggesting modification by another author, possibly based on a lost earlier work. The formula for calculating the area of a triangle from the length of its sides, commonly known as Heron's formula, has also been claimed to be known to Archimedes, though its first reliable appearance is in the work of Heron of Alexandria in the 1st century AD. Other questionable attributions include the Latin poem Carmen de ponderibus et mensuris (4th or 5th century), which describes the use of a hydrostatic balance to solve the crown problem, and the 12th-century text Mappae clavicula, which contains instructions on how to perform assaying of metals by calculating their specific gravities.

5.3. Archimedes Palimpsest

The foremost document containing previously lost works by Archimedes is the Archimedes Palimpsest. In 1906, the Danish professor Johan Ludvig Heiberg visited Constantinople to examine a 174-page goatskin parchment of prayers, written in the 13th century. He confirmed that it was indeed a palimpsest, a document with text written over an erased older work. Palimpsests were created by scraping ink from existing works and reusing the expensive vellum, a common practice in the Middle Ages. The older works in this palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes.

The parchment remained in a monastery library in Constantinople for centuries before being sold to a private collector in the 1920s. On October 29, 1998, it was sold at auction to an anonymous buyer for 2.20 M USD. The palimpsest holds seven treatises, including the only surviving copy of On Floating Bodies in the original Greek. Crucially, it is the only known source of The Method of Mechanical Theorems, which had been referred to by Suidas and thought to be lost forever. Stomachion was also discovered in the palimpsest, with a more complete analysis of the puzzle than previously known texts. The palimpsest was stored at the Walters Art Museum in Baltimore, Maryland, where it underwent various 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 found 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

  The palimpsest also contains speeches by the 4th century BC politician Hypereides and a commentary on Aristotle's Categories, among other works.

6.1. Impact on 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.
Likewise, Alfred North Whitehead and George F. Simmons praised Archimedes, with Simmons stating:
: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.
Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at Stanford University and an expert in Archimedes, notes:
: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.
Many prominent figures in scientific history have expressed deep admiration for Archimedes. Leonardo da Vinci repeatedly showed admiration for Archimedes and attributed his invention of the Architonnerre (a steam cannon) to him. Galileo Galilei called him "superhuman" and "my master," while Christiaan Huygens said, "I think Archimedes is comparable to no one," consciously emulating him in his early work. Gottfried Wilhelm Leibniz stated, "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, reported that Gauss once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein." The inventor Nikola Tesla also 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."

6.2. Honors and Commemorations

Italian numismatist and archaeologist Filippo Paruta (1552-1629) and Leonardo Agostini (1593-1676) reported on a bronze coin found in Sicily with Archimedes' portrait on the obverse and a cylinder and sphere with the monogram ARMD in Latin on the reverse. 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," suggesting it might have been minted in Rome for Marcellus, who "according to ancient reports, brought two spheres of Archimedes with him to Rome."

A crater on the Moon named Archimedes and a lunar mountain range, the Montes Archimedes, are named in his honor. The minor planet 3600 Archimedes is also named after him.

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 Archimedes' head is a quote attributed to 1st century AD poet Manilius, which reads in Latin: Transire suum pectus mundoque potiri^Latin ("Rise above oneself and grasp the world").

Archimedes has appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963). The exclamation of Eureka! attributed to Archimedes is the state motto of California. In this instance, the word refers to the discovery of gold near Sutter's Mill in 1848, which sparked the California gold rush. The Archimedes Movement, a civil movement in Oregon, United States, aiming for universal health care access, was also named after him.