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Archimedes of Syracuse
Ἀρχιμήδης
A painting of an older man puzzling over geometric problems
Archimedes Thoughtful by Fetti (1620)
Bornc. 287 BC
Syracuse, Sicily
Diedc. 212 BC (aged approximately 75)
Syracuse, Sicily
Known for
List
Scientific career
FieldsMathematics
Physics
Astronomy
Mechanics
Engineering

Archimedes of Syracuse[a] (/ˌɑːrkɪˈmiːdiːz/ AR-kim-EE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily.

Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in Classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Archimedes' other mathematical achievements include deriving an approximation of pi (π), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' 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 the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion.

Archimedes died during the siege of Syracuse, when he was killed by a Roman soldier despite orders that he should not be harmed. Cicero describes visiting Archimedes' tomb, which was surmounted by a sphere and a cylinder that Archimedes requested be placed there to represent his most valued mathematical discovery.

Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Alexandrian mathematicians read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while Eutocius' commentaries on Archimedes' works in the same century opened them to wider readership for the first time. In the Middle ages, Archimedes' work was translated into Arabic in the 9th century and then into Latin in the 12th century, and were an influential source of ideas for scientists during the Renaissance and in the Scientific Revolution. The recent discovery in 1906 of previously lost works by Archimedes in the Archimedes Palimpsest has also provided new insights into how he obtained mathematical results.

Biography

Cicero 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.[1]

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.[2][better source needed] Plutarch wrote in his Parallel Lives that Archimedes was related to King Hiero II, the ruler of Syracuse, although Cicero suggests he was of humble origin.[3][4] In the Sand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known.[4] It is also unknown whether he ever married or had children, or if he ever visited Alexandria, Egypt, during his youth.[5] From his surviving written works, it is clear that he maintained collegial relations with scholars based there,[citation needed] including Conon of Samos and the head librarian Eratosthenes of Cyrene; 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.[citation needed]

The Histories by Polybius (c. 200–118 BC), written about 70 years after Archimedes' death,[4] provides an account of war machines that he is said to have built in order to defend the city from the Romans.[6] Polybius remarks how, during the Second Punic War, Syracuse switched allegiances from Rome to Carthage, resulting in a military campaign under the command of Marcus Claudius Marcellus and Appius Claudius Pulcher, who besieged the city from 213 to 212 BC. He notes that the Romans underestimated Syracuse's defenses, and mentions several machines Archimedes designed, including improved catapults, crane-like machines that could be swung around in an arc, and other stone-throwers. Although the Romans ultimately captured the city, they suffered considerable losses due to Archimedes' inventiveness.[7] The Roman historian Livy (59 BC–17 AD) retells Polybius' story of the capture of Syracuse and Archimedes' role in it.[6]

The Death of Archimedes (1815) by Thomas Degeorge

Plutarch (45–119 AD) provides at least two accounts on how Archimedes died after Syracuse was taken.[4] According to the most popular account,[citation needed] 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. The last words attributed to Archimedes are "Do not disturb my circles,"[b] a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier.[4] 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, "... but protecting the dust with his hands, said 'I beg of you, do not disturb this."[6]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.[8][9] Cicero (106–43 BC) also mentions that Marcellus brought to Rome two planetariums Archimedes built.>[4]

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.[10]

Discoveries and inventions

Archimedes' principle

Main article: Archimedes' principle
Measurement of volume by displacement, (a) before and (b) after an object has been submerged; the amount by which the liquid rises in the cylinder (∆V) is equal to the volume of the object

Archimedes has been credited since antiquity with the discovery of a principle for determining the density of an object with an irregular shape. According to Vitruvius,[11] writing about two centuries after Archimedes' death, a crown for a temple had been made for King Hiero II of Syracuse, who supplied the pure gold to be used.[12] Archimedes was asked to determine, without damaging the crown by melting it down, whether the goldsmith had substituted some cheaper silver and kept some of the pure gold for himself. In this account, which is likely apocryphal,[12] Archimedes noticed while taking a bath that the level of the water in the tub rose as he got in, 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![c], meaning "I have found [it]!"[11] For practical purposes water is incompressible,[13] 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, its density could be obtained; if cheaper and less dense metals had been added, the density would be lower than that of gold.[12][11], however, he practicality of the method described, which does not appear anywhere in Archimedes' known works, has been called into question due to the extreme accuracy that would be required to measure water displacement.[12]

Archimedes may have instead sought a solution that applied the hydrostatics principle known as Archimedes' principle, found in his treatise On Floating Bodies: a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.[14] Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing it on a scale with a pure gold reference sample of the same weight, then immersing the apparatus in water. The difference in density between the two samples would cause the scale to tip accordingly.[15] 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."[16][17]

Law of the lever

While Archimedes did not invent the lever, he gave a mathematical proof of the principle involved in his work On the Equilibrium of Planes.[18] 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.[19]

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.[20] According to Pappus of Alexandria, Archimedes' work on levers and his understanding of mechanical advantage caused him to remark: "Give me a place to stand on, and I will move the Earth,"[d][21] however, Olympiodorus later attributed the same boast to Archimedes' invention of the baroulkos, a kind of windlass, rather than the lever.[22]

Archimedes' screw

Main article: Archimedes' screw
The Archimedes' screw can raise water efficiently

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of Syracuse. Athenaeus of Naucratis quotes a certain Moschion in a description on how King Hiero II commissioned the design of a huge ship, the Syracusia, which could be used for luxury travel, carrying supplies, and as a display of naval power.[23] The Syracusia is said to have been the largest ship built in classical antiquity and, according to Moschion's account, it was launched by Archimedes.[24] The ship presumably was capable of carrying 600 people and included garden decorations, a gymnasium, and a temple dedicated to the goddess Aphrodite among its facilities.[25][non-primary source needed]

The account also mentions that, in order to remove any potential water leaking through the hull, Archimedes designed a device with a revolving screw-shaped blade inside a cylinder, , which could be turned by hand in order to 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. Archimedes' device as described by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon.[26][27]

Archimedes' claw

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.[28] 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.[29]

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.[30][31]

Heat ray

Main article: Archimedes' heat ray
Mirrors placed as a parabolic reflector to attack upcoming ships

As legend has it,[citation needed] Archimedes arranged mirrors as a parabolic reflector to burn ships attacking Syracuse using focused sunlight. However, Archimedes's contemporary Diocles made no mention of Archimedes or burning ships in his treatise about focusing reflectors,[32][non-primary source needed], 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,[2][better source needed] and Lucian[33] and Galen,[34] writing in the second century AD, mentioned that during the siege of Syracuse Archimedes had burned enemy ships.[non-primary source needed] Nearly four hundred years later, Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.[35][36]

The purported device, sometimes called "Archimedes' heat ray", has been the subject of an ongoing debate about its credibility since the Renaissance. [37] René Descartes rejected it as false,[38] while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, mostly with negative results.[39][non-primary source needed] 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.[40][better source needed] Using modern materials and larger scale, sunlight-concentrating solar furnaces can reach very high temperatures, and are sometimes used for generating electricity.[41][better source needed]

Astronomical instruments

Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well as Aristarchus' heliocentric model of the universe, in the Sand-Reckoner.[42] 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),[43] applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.[44]

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.[5]

Cicero provides a description of a planetarium allegedly constructed by Archimedes.[45][better source needed] After the capture of Syracuse in the Second Punic War, Marcellus is said to have taken back to Rome two mechanisms which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated the other to the Temple of Virtue in Rome, and the other he allegedly kept as his only personal loot from Syracuse. In Cicero's De re publica, he portrays a fictional conversation taking place in 129 BC between Gaius Sulpicius Gallus and Lucius Furius Philus that recounts this story, and provides a demonstration of the mechanism: "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."[46]

Pappus of Alexandria reports on a now lost treatise by Archimedes dealing with the construction of these mechanisms entitled On Sphere-Making.[47] 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 c. 100 BC designed with a similar purpose, has confirmed that devices of this kind were known to the ancient Greeks,[48] with some scholars regarding Archimedes' device as a precursor.[49][50]

Mathematics

While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field of mathematics.

Method of exhaustion

Archimedes calculates the side of the 12-gon from that of the hexagon and for each subsequent doubling of the sides of the regular polygon

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, a technique called the method of exhaustion.[51]

In Measurement of a Circle, he employed this method to approximate the value of π, 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 π lay between 3⁠1/7⁠ (approx. 3.1429) and 3⁠10/71⁠ (approx. 3.1408), consistent with its actual value of approximately 3.1416.[52][better source needed] He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle ().[citation needed]

Archimedean property

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.[53] In Measurement of a Circle, Archimedes gives the value of the square root of 3 as lying between ⁠265/153⁠ (approximately 1.7320261) and ⁠1351/780⁠ (approximately 1.7320512), which is remarkably close to the actual value, (approximately 1.7320508). Although Archimedes introduced this result without offering any explanation of how he had obtained it, it is possible that he used an iterative procedure to calculate these values using the Archimedean property.[54][55]

The infinite series

A proof that the area of the parabolic 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 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 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 1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to ⁠1/3⁠.

Myriad of myriads

In The Sand Reckoner, Archimedes set out to calculate a number that was greater than the grains of sand needed to fill the universe. 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, 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.[56]

Writings

Front page of Archimedes' Opera, in Greek and Latin, edited by David Rivault (1615)

Archimedes made his work known through correspondence with mathematicians in Alexandria,[57] which were originally written in Doric Greek, the dialect of ancient Syracuse.[58]

Surviving works

The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).[59][60]

Measurement of a Circle

Main article: 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 (3.1408...) and less than ⁠22/7⁠ (3.1428...).

The Sand Reckoner

Main article: 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.[61][62] By using a system of numbers based on powers of the myriad, Archimedes concludes that the number of grains of sand required to fill the universe is 8×1063 in modern notation. The introductory letter states that Archimedes' father was an astronomer named Phidias. The Sand Reckoner is the only surviving work in which Archimedes discusses his views on astronomy.[63]

On the Equilibrium of Planes

Main article: 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, which states that:

Magnitudes are in equilibrium at distances reciprocally proportional to their weights.

Archimedes uses the principles derived to calculate the areas and centers of gravity of various geometric figures including triangles, parallelograms and parabolas.[64]

Quadrature of the Parabola

Main article: 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 in one of his proofs by calculating the value of a geometric series that sums to infinity with the ratio 1/4.

On the Sphere and Cylinder

Main article: On the Sphere and Cylinder
A sphere has 2/3 the volume and surface area of its circumscribing cylinder including its bases

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 ⁠4/3⁠πr3 for the sphere, and 2πr3 for the cylinder. The surface area is 4πr2 for the sphere, and 6πr2 for the cylinder (including its two bases), where r is the radius of the sphere and cylinder.

On Spirals

Main article: 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 (r, θ), it can be described by the equation with real numbers a and b.

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

On Conoids and Spheroids

Main article: 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.

On Floating Bodies

Main article: 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.[65] 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:[15][66]

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.[67]

Ostomachion

Main article: Ostomachion
Ostomachion is a dissection puzzle found in the Archimedes Palimpsest

Also known as Loculus of Archimedes or Archimedes' Box,[68] 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.[69] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.[70] 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).[71] Ausonius calls the puzzle Ostomachion, a Greek compound word formed from the roots of osteon (ὀστέον, 'bone') and machē (μάχη, 'fight').[68]

The cattle problem

Main article: 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 problem[72] in 1880, and the answer is a very large number, approximately 7.760271×10206544.[73]

The Method of Mechanical Theorems

Main article: The Method of Mechanical Theorems

In this work Archimedes uses indivisibles,[51] 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,[citation needed] 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. This treatise was thought lost until the discovery of the Archimedes Palimpsest in 1906.

Apocryphal works

Main article: 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,[citation needed] since it quotes Archimedes, suggesting modification by another author. The Lemmas may be based on an earlier work by Archimedes that is now lost.[citation needed]

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.[74][75]

Lost works

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.[2][better source needed] Pappus of Alexandria mentions On Sphere-Making and another work on polyhedra,[non-primary source needed] while Theon of Alexandria quotes a remark about refraction from the now-lost Catoptrica.[non-primary source needed]; Principles was addressed to Zeuxippus and explaining the number system used in The Sand Reckoner; there are also On Balances or On Levers; On Centers of Gravity; On the Calendar.[non-primary source needed]

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,[76] and may have been proven in a lost work of Archimedes that is no longer extant.[77]

Archimedes Palimpsest

Main article: Archimedes Palimpsest
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost

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.[78][79] 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.[78][80] 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:

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.[81][82] 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.[83] It has since returned to its anonymous owner.[84][85]

Legacy

Bronze statue of Archimedes in Berlin
Further information: List of things named after Archimedes

Sometimes called the father of mathematics[86] and mathematical physics,[87] historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.[88]

Classical antiquity

The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented;[89] Athenaeus[90] recounts in his Deipnosophistae how Archimedes supervised the construction of the largest known ship in antiquity, the Syracusia, while Apuleius[91] talks about his work in catoptrics.[92] However, unlike his inventions, Archimedes' mathematical writings were little known in antiquity.[citation needed] Although Plutarch[93] claimed that Archimedes disdained mechanics and focused primarily on pure geometry, this outlier 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.[94] Alexandrian mathematicians read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople,[citation needed] while Eutocius' commentaries on Archimedes' works in the same century opened them to wider readership for the first time.[citation needed]

Middle ages

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).[95][96]

Renaissance and early modern Europe

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,[97] which were an influential source of ideas for scientists during the Renaissance and again in the 17th century.[98][99]

Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.[100][101][102] Galileo Galilei called him "superhuman" and "my master",[103][104] while Christiaan Huygens said, "I think Archimedes is comparable to no one", consciously emulating him in his early work.[105] Gottfried Wilhelm Leibniz said, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times".[106]

1612 drawing of a now-lost bronze coin depicting Archimedes

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.[107] 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".[108]

In modern mathematics

Gauss's heroes were Archimedes and Newton,[109] 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".[110] Likewise, Alfred North Whitehead said that "in the year 1500 Europe knew less than Archimedes who died in the year 212 BC."[111] The historian of mathematics Reviel Netz,[112] 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,[113] 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.[114][115]

The Fields Medal carries a portrait of Archimedes

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").[116][117][118]

Cultural influence

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.[119]

Archimedes has also appeared on postage stamps issued by East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).[120]

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.[121]

There is a crater on the Moon named Archimedes (29°42′N 4°00′W / 29.7°N 4.0°W / 29.7; -4.0) in his honor, as well as a lunar mountain range, the Montes Archimedes (25°18′N 4°36′W / 25.3°N 4.6°W / 25.3; -4.6).[122]

See also

Concepts

People

References

Notes

  1. ^ Doric Greek: Ἀρχιμήδης, pronounced [arkʰimɛːdɛ̂ːs].
  2. ^ Latin: Noli turbare circulos meos; Greek: μὴ μου τοὺς κύκλους τάραττε)
  3. ^ Greek: "εὕρηκα, heúrēka!)
  4. ^ Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω

Citations

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Further reading

Wikisource has the text of the 1911 Encyclopædia Britannica article "Archimedes".
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