Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and

inventor An invention is a unique or novel device, method, composition, idea or process. An invention may be an improvement upon a machine, product, or process for increasing efficiency or lowering cost. It may also be an entirely new concept. If an ...

from the ancient city of

Syracuse Syracuse may refer to: Places Italy *Syracuse, Sicily, or spelled as ''Siracusa'' *Province of Syracuse United States *Syracuse, New York **East Syracuse, New York **North Syracuse, New York *Syracuse, Indiana * Syracuse, Kansas *Syracuse, Miss ...

in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the

infinitely small In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

and the method of exhaustion to derive and rigorously prove a range of

geometrical Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ca ...

theorems. These include the area of a circle, the

surface area The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc ...

and volume of a sphere, the area of an

ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...

, 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 In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and investigating the

Archimedean spiral The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a con ...

, and devising a system using exponentiation for expressing

very large numbers Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical m ...

. He was also one of the first to apply mathematics to physical phenomena, founding hydrostatics and

statics Statics is the branch of classical mechanics that is concerned with the analysis of force and torque (also called moment) acting on physical systems that do not experience an acceleration (''a''=0), but rather, are in static equilibrium with ...

. Archimedes' achievements in this area include a proof of the principle of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy. He is also credited with designing innovative

machine A machine is a physical system using Power (physics), power to apply Force, forces and control Motion, movement to perform an action. The term is commonly applied to artificial devices, such as those employing engines or motors, but also to na ...

s, such as his screw pump, compound pulleys, and defensive war machines to protect his native

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 mathematical discoveries. Unlike his inventions, Archimedes' mathematical writings were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes by Eutocius in the 6th century opened them to wider readership for the first time. The relatively few copies of Archimedes' written work that survived through the Middle Ages were an influential source of ideas for scientists during the Renaissance and again in the 17th century, while the discovery in 1906 of previously lost works by Archimedes in the

Archimedes Palimpsest The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the ''Ostomachion'' and the ' ...

has provided new insights into how he obtained mathematical results.

Biography

Death of Archimedes (1815) by Thomas Degeorge

Archimedes was born c. 287 BC in the seaport city of

, Sicily, at that time a self-governing colony in

Magna Graecia Magna Graecia (, ; , , grc, Μεγάλη Ἑλλάς, ', it, Magna Grecia) was the name given by the Romans to the coastal areas of Southern Italy in the present-day Italian regions of Calabria, Apulia, Basilicata, Campania and Sicily; these re ...

. The date of birth is based on a statement by the Byzantine Greek historian

John Tzetzes John Tzetzes ( grc-gre, Ἰωάννης Τζέτζης, Iōánnēs Tzétzēs; c. 1110, Constantinople – 1180, Constantinople) was a Byzantine poet and grammarian who is known to have lived at Constantinople in the 12th century. He was able to p ...

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 Conon of Samos ( el, Κόνων ὁ Σάμιος, ''Konōn ho Samios''; c. 280 – c. 220 BC) was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices. Life and work Conon was born on 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."

lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works. 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 Polybius (; grc-gre, Πολύβιος, ; ) was a Greek historian of the Hellenistic period. He is noted for his work , which covered the period of 264–146 BC and the Punic Wars in detail. Polybius is important for his analysis of the mixed ...

( 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 The Second Punic War (218 to 201 BC) was the second of three wars fought between Carthage and Rome, the two main powers of the western Mediterranean in the 3rd century BC. For 17 years the two states struggled for supremacy, primarily in Ital ...

, 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 Discovering the Tomb of Archimedes by Benjamin West

Cicero Discovering the Tomb of Archimedes by Benjamin West

Cicero (106–43 BC) mentions Archimedes in some of his works. While serving as a

quaestor A ( , , ; "investigator") was a public official in Ancient Rome. There were various types of quaestors, with the title used to describe greatly different offices at different times. In the Roman Republic, quaestors were elected officials who ...

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 Plutarch's ''Lives of the Noble Greeks and Romans'', commonly called ''Parallel Lives'' or ''Plutarch's Lives'', is a series of 48 biographies of famous men, arranged in pairs to illuminate their common moral virtues or failings, probably writt ...

'' that Archimedes was related to King

Hiero II Hiero II ( el, Ἱέρων Β΄; c. 308 BC – 215 BC) was the Greek tyrant of Syracuse from 275 to 215 BC, and the illegitimate son of a Syracusan noble, Hierocles, who claimed descent from Gelon. He was a former general of Pyrrhus of Epirus a ...

, 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 "

Do not disturb my circles "''Nōlī turbāre circulōs meōs!''" is a Latin phrase, meaning "Do not disturb my circles!". It is said to have been uttered by Archimedes—in reference to a geometric figure he had outlined on the sand—when he was confronted by a Roman sold ...

" ( Latin, "''Noli turbare circulos meos''"; Katharevousa Greek, "μὴ μου τοὺς κύκλους τάραττε"), a reference to the mathematical drawing that he was supposedly studying when disturbed by the Roman soldier. 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).

Discoveries and inventions

Archimedes' principle

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 A votive crown is a votive offering in the form of a crown, normally in precious metals and often adorned with jewels. Especially in the Early Middle Ages, they are of a special form, designed to be suspended by chains at an altar, shrine or imag ...

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!" ( el, "εὕρηκα, ''heúrēka''!, ). 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 water displacement. Archimedes may have instead sought a solution that applied the principle known in hydrostatics as Archimedes' principle, which he describes in his treatise ''

On Floating Bodies ''On Floating Bodies'' ( el, Περὶ τῶν ἐπιπλεόντων σωμάτων) is a Greek-language work consisting of two books written by Archimedes of Syracuse (287 – c. 212 BC), one of the most important mathematicians, physicis ...

''. This principle states that a body immersed in a fluid experiences a

buoyant force Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...

equal to the weight of the fluid it displaces. Using this principle, it would have been possible to compare the density of the crown to that of pure gold by balancing the crown 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. Galileo Galilei, who in 1586 invented a hydrostatic balance for weighing metals in air and water inspired by the work of Archimedes, 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."

Archimedes' screw

A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city of

. The Greek writer Athenaeus of Naucratis described how King Hiero II commissioned Archimedes to design a huge ship, the '' Syracusia'', which could be used for luxury travel, carrying supplies, and as a

naval warship A naval ship is a military ship (or sometimes boat, depending on classification) used by a navy. Naval ships are differentiated from civilian ships by construction and purpose. Generally, naval ships are damage resilient and armed with w ...

. The ''Syracusia'' is said to have been the largest ship built in classical antiquity. According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, Archimedes' screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a body of water into irrigation canals. Archimedes' screw is still in use today for pumping liquids and granulated solids such as coal and grain. Described in Roman times by Vitruvius, Archimedes' screw may have been an improvement on a screw pump that was used to irrigate the

Hanging Gardens of Babylon The Hanging Gardens of Babylon were one of the Seven Wonders of the Ancient World listed by Hellenic culture. They were described as a remarkable feat of engineering with an ascending series of tiered gardens containing a wide variety of tre ...

. The world's first seagoing

steamship A steamship, often referred to as a steamer, is a type of steam-powered vessel, typically ocean-faring and seaworthy, that is propelled by one or more steam engines that typically move (turn) propellers or paddlewheels. The first steamships ...

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.

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 A grappling hook or grapnel is a device that typically has multiple hooks (known as ''claws'' or ''flukes'') attached to a rope; it is thrown, dropped, sunk, projected, or fastened directly by hand to where at least one hook may catch and hol ...

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.

Heat ray

Archimedes may have used mirrors acting collectively as a parabolic reflector to burn ships attacking Syracuse. The 2nd-century author

Lucian Lucian of Samosata, '; la, Lucianus Samosatensis ( 125 – after 180) was a Hellenized Syrian satirist, rhetorician and pamphleteer Pamphleteer is a historical term for someone who creates or distributes pamphlets, unbound (and therefore ...

wrote that during the siege of Syracuse (c. 214–212 BC), Archimedes destroyed enemy ships with fire. Centuries later, Anthemius of Tralles mentions burning-glasses as Archimedes' weapon. The device, sometimes called the "Archimedes heat ray", was used to focus sunlight onto approaching ships, causing them to catch fire. In the modern era, similar devices have been constructed and may be referred to as a

heliostat A heliostat (from ''helios'', the Greek word for ''sun'', and ''stat'', as in stationary) is a device that includes a mirror, usually a plane mirror, which turns so as to keep reflecting sunlight toward a predetermined target, compensating ...

or solar furnace. 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 Bronze is an alloy consisting primarily of copper, commonly with about 12–12.5% tin and often with the addition of other metals (including aluminium, manganese, nickel, or zinc) and sometimes non-metals, such as phosphorus, or metalloids such ...

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.

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 ''On the Equilibrium of Planes'' ( grc, Περὶ ἐπιπέδων ἱσορροπιῶν, translit=perí epipédōn isorropiôn) is a treatise by Archimedes in two volumes. The first book contains a proof of the Mechanical advantage#The law of ...

''. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to

Archytas Archytas (; el, Ἀρχύτας; 435/410–360/350 BC) was an Ancient Greek philosopher, mathematician, music theorist, astronomer, statesman, and strategist. He was a scientist of the Pythagorean school and famous for being the reputed founder ...

. 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 A pulley is a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable or belt, or transfer of power between the shaft and cable or belt. In the case of a pulley supported by a frame or shell that ...

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" ( el, δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω). Olympiodorus later attributed the same boast to Archimedes' invention of the ''baroulkos'', a kind of

windlass The windlass is an apparatus for moving heavy weights. Typically, a windlass consists of a horizontal cylinder (barrel), which is rotated by the turn of a crank or belt. A winch is affixed to one or both ends, and a cable or rope is wound arou ...

, rather than the lever. Archimedes has also been credited with improving the power and accuracy of the

catapult A catapult is a ballistic device used to launch a projectile a great distance without the aid of gunpowder or other propellants – particularly various types of ancient and medieval siege engines. A catapult uses the sudden release of stored p ...

, and with inventing the

odometer An odometer or odograph is an instrument used for measuring the distance traveled by a vehicle, such as a bicycle or car. The device may be electronic, mechanical, or a combination of the two (electromechanical). The noun derives from ancient Gr ...

during the

First Punic War The First Punic War (264–241 BC) was the first of three wars fought between Rome and Carthage, the two main powers of the western Mediterranean in the early 3rd century BC. For 23 years, in the longest continuous conflict and grea ...

. The odometer was described as a cart with a gear mechanism that dropped a ball into a container after each mile traveled.

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''. 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. Cicero's '' De re publica'' portrays a fictional conversation taking place in 129 BC, after the

. General Marcus Claudius Marcellus is said to have taken back to Rome two mechanisms after capturing Syracuse in 212 BC, which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets. Cicero also mentions similar mechanisms designed by Thales of Miletus and

Eudoxus of Cnidus Eudoxus of Cnidus (; grc, Εὔδοξος ὁ Κνίδιος, ''Eúdoxos ho Knídios''; ) was an ancient Greek astronomer, mathematician, scholar, and student of Archytas and Plato. All of his original works are lost, though some fragments are ...

. The dialogue says that Marcellus kept one of the devices as his only personal loot from Syracuse, and donated the other to the Temple of Virtue in Rome. Marcellus' mechanism was demonstrated, according to Cicero, by Gaius Sulpicius Gallus to

Lucius Furius Philus Lucius Furius Philus was a Roman statesman who became Roman consul, consul of ancient Rome in 136 BC. He was a member of the Scipionic Circle, and particularly close to Scipio Aemilianus. As proconsul, his allotted province was Spain. The consul of ...

, who described it thus: This is a description of a small

planetarium A planetarium ( planetariums or ''planetaria'') is a theatre built primarily for presenting educational and entertaining shows about astronomy and the night sky, or for training in celestial navigation. A dominant feature of most 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 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.

Mathematics

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

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that pl ...

(a precursor to

infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...

s) 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 '' Measurement of a Circle'', he did this by drawing a larger

regular hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

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 3 (approx. 3.1429) and 3 (approx. 3.1408), consistent with its actual value of approximately 3.1416. He also proved that the area of a circle was equal to π multiplied by the square of the radius of the circle (

\pi r^2

Archimedean property

In ''

On the Sphere and Cylinder ''On the Sphere and Cylinder'' ( el, Περὶ σφαίρας καὶ κυλίνδρου) is a work that was published by Archimedes in two volumes c. 225 BCE. It most notably details how to find the surface area of a sphere and the volume of the ...

'', 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 (approximately 1.7320261) and (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 John Wallis (; la, Wallisius; ) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal ...

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

Parabolic segment and inscribed triangle

In ''

Quadrature of the Parabola ''Quadrature of the Parabola'' ( el, Τετραγωνισμὸς παραβολῆς) is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions rega ...

'', Archimedes proved that the area enclosed by a parabola and a straight line is 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 : :

\sum_^\infty 4^ = 1 + 4^ + 4^ + 4^ + \cdots = . \;

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 which sums to .

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 A myriad (from Ancient Greek grc, μυριάς, translit=myrias, label=none) is technically the number 10,000 (ten thousand); in that sense, the term is used in English almost exclusively for literal translations from Greek, Latin or Sinospher ...

. The word itself derives from the Greek , 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.

Writings

The works of Archimedes were written in

Doric Greek Doric or Dorian ( grc, Δωρισμός, Dōrismós), also known as West Greek, was a group of Ancient Greek dialects; its varieties are divided into the Doric proper and Northwest Doric subgroups. Doric was spoken in a vast area, that included ...

, 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 ''Catoptrica''.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''. 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 Thābit ibn Qurra (full name: , ar, أبو الحسن ثابت بن قرة بن زهرون الحراني الصابئ, la, Thebit/Thebith/Tebit); 826 or 836 – February 19, 901, was a mathematician, physician, astronomer, and translator who ...

(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). During the Renaissance, the ''

Editio princeps In classical scholarship, the ''editio princeps'' (plural: ''editiones principes'') of a work is the first printed edition of the work, that previously had existed only in manuscripts, which could be circulated only after being copied by hand. For ...

'' (First Edition) was published in Basel in 1544 by Johann Herwagen with the works of Archimedes in Greek and Latin.

Surviving works

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

''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

. In Proposition II, Archimedes gives an

approximation An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else. Etymology and usage The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...

of the value of pi (), showing that it is greater than and less than .

''The Sand Reckoner''

In this treatise, also known as ''Psammites'', Archimedes counts the number of grains of sand that will fit inside the universe. This book mentions the

heliocentric Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth at ...

theory of the solar system proposed by

Aristarchus of Samos Aristarchus of Samos (; grc-gre, Ἀρίσταρχος ὁ Σάμιος, ''Aristarkhos ho Samios''; ) was an ancient Greek astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or ...

, as well as contemporary ideas about the size of the Earth and the distance between various

celestial bodies An astronomical object, celestial object, stellar object or heavenly body is a naturally occurring physical entity, association, or structure that exists in the observable universe. In astronomy, the terms ''object'' and ''body'' are often us ...

. By using a system of numbers based on powers of the

, Archimedes concludes that the number of grains of sand required to fill the universe is 8 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.

''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: Archimedes uses the principles derived to calculate the areas and

centers of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...

of various geometric figures including triangles,

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

s and parabolas.

''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 .

''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 sphere and a circumscribed cylinder of the same height and diameter. The volume is ³ for the sphere, and 2³ for the cylinder. The surface area is 4² for the sphere, and 6² for the cylinder (including its two bases), where is the radius of the sphere and cylinder. The sphere has a volume that of the circumscribed cylinder. Similarly, the sphere has an area that of the cylinder (including the bases).

''On Spirals''

This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the

. 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 In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...

. Equivalently, in polar coordinates (, ) it can be described by the equation

\, r=a+b\theta

with real numbers and . This is an early example of a mechanical curve (a curve traced by a moving

point Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland * Point ...

) considered by a Greek mathematician.

''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 Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...

of cones, spheres, and paraboloids.

''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 Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...

that the Earth is round. The fluids described by Archimedes are not 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.

''Ostomachion''

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 calculates the areas of the 14 pieces which can be assembled to form a square. Reviel Netz of

Stanford University Stanford University, officially Leland Stanford Junior University, is a private research university in Stanford, California. The campus occupies , among the largest in the United States, and enrolls over 17,000 students. Stanford is consider ...

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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

. 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'' (). Ausonius calls the puzzle , a Greek compound word formed from the roots of () and ().

The cattle problem

Gotthold Ephraim Lessing Gotthold Ephraim Lessing (, ; 22 January 1729 – 15 February 1781) was a philosopher, dramatist, publicist and art critic, and a representative of the Enlightenment era. His plays and theoretical writings substantially influenced the developmen ...

discovered this work in a Greek manuscript consisting of a 44-line poem in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is 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 number of simultaneous

Diophantine equation In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...

s. 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 in 1880, and the answer is a

very large number Large numbers are numbers significantly larger than those typically used in everyday life (for instance in simple counting or in monetary transactions), appearing frequently in fields such as mathematics, cosmology, cryptography, and statistical ...

, approximately 7.760271.

''The Method of Mechanical Theorems''

This treatise was thought lost until the discovery of the

in 1906. In this work Archimedes uses

, 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

in Alexandria.

Apocryphal works

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 Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath transla ...

and Marshall Clagett argued that it cannot have been written by Archimedes in its current form, since it quotes Archimedes, suggesting modification by another author. The ''Lemmas'' may be based on an earlier work by Archimedes that is now lost. It has also been claimed that the

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

for calculating the area of a triangle from the length of its sides was known to Archimedes, Boyer, Carl Benjamin. 1991. ''A History of Mathematics''. : "Arabic scholars inform us that the familiar area formula for a triangle in terms of its three sides, usually known as Heron's formula —

k = \sqrt

, where

s

is the semiperimeter — was known to Archimedes several centuries before Heron lived. Arabic scholars also attribute to Archimedes the 'theorem on the broken

chord Chord may refer to: * Chord (music), an aggregate of musical pitches sounded simultaneously ** Guitar chord a chord played on a guitar, which has a particular tuning * Chord (geometry), a line segment joining two points on a curve * Chord ( ...

' ... Archimedes is reported by the Arabs to have given several proofs of the theorem." though its first appearance is in the work of Heron of Alexandria in the 1st century AD. Other questionable attributions to Archimedes' work include the Latin poem ''

Carmen de ponderibus et mensuris ''Carmen'' () is an opera in four acts by the French composer Georges Bizet. The libretto was written by Henri Meilhac and Ludovic Halévy, based on the novella of the same title by Prosper Mérimée. The opera was first performed by the Opér ...

'' (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 The ''mappae clavicula'' is a medieval Latin text containing manufacturing recipes for crafts materials, including for metals, glass, mosaics, and dyes and tints for materials. The information and style in the recipes is very terse. Each recipe co ...

'', which contains instructions on how to perform assaying of metals by calculating their specific gravities. Dilke, Oswald A. W. 1990. ntitled ''

Gnomon A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ol ...

'' 62(8):697–99. .

Archimedes Palimpsest

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 Athanasios Papadopoulos-Kerameus ( el, Αθανάσιος Παπαδόπουλος-Κεραμεύς; 1856–1912) an Ottoman and subsequently Tsarist subject, Greek Orthodox in religious affiliation, was an Ottoman Greek scholar. Appointed secre ...

. 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 The ''Suda'' or ''Souda'' (; grc-x-medieval, Σοῦδα, Soûda; la, Suidae Lexicon) is a large 10th-century Byzantine encyclopedia of the ancient Mediterranean world, formerly attributed to an author called Soudas (Σούδας) or Souidas ...

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 Baltimore ( , locally: or ) is the most populous city in the U.S. state of Maryland, fourth most populous city in the Mid-Atlantic, and the 30th most populous city in the United States with a population of 585,708 in 2020. Baltimore was d ...

, Maryland, where it was subjected to a range of modern tests including the use of ultraviolet and light to read the overwritten text. It has since returned to its anonymous owner. The treatises in the Archimedes Palimpsest include: * ''

'' * ''

On Spirals ''On Spirals'' ( el, Περὶ ἑλίκων) is a treatise by Archimedes, written around 225 BC. Notably, Archimedes employed the Archimedean spiral in this book to square the circle and trisect an angle. Contents Preface Archimedes begins ''O ...

'' * '' Measurement of a Circle'' * ''

'' * ''

'' * '' The Method of Mechanical Theorems'' * '' Stomachion'' * Speeches by the 4th century BC politician Hypereides * A commentary on Aristotle's '' Categories'' * Other works

Legacy

Sometimes called the father of mathematics and mathematical physics, Archimedes had a wide influence on mathematics and science. * 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. James Henry Williams Jr. is a mechanical engineer, consultant, civic commentator, and teacher of engineering. He is currently Professor of Applied Mechanics in the Mechanical Engineering Department at the Massachusetts Institute of Technology (MIT ...

, 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.

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: Likewise,

Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...

and

George F. Simmons George Finlay Simmons (March 3, 1925 – August 6, 2019) was an American mathematician who worked in topology and classical analysis. He is known as the author of widely used textbooks on university mathematics. Life He was born on 3 March 1925 ...

said of Archimedes: Reviel Netz, Suppes Professor in Greek Mathematics and Astronomy at

and an expert in Archimedes notes: Leonardo da Vinci repeatedly expressed admiration for Archimedes, and attributed his invention Architonnerre to Archimedes.

Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...

called him "superhuman" and "my master", while

Huygens Huygens (also Huijgens, Huigens, Huijgen/Huygen, or Huigen) is a Dutch patronymic surname, meaning "son of Hugo". Most references to "Huygens" are to the polymath Christiaan Huygens. Notable people with the surname include: * Jan Huygen (1563– ...

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 Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton ( ...

, and Eisenstein." The inventor Nikola Tesla praised him, saying:

Honors and commemorations

There is a

crater Crater may refer to: Landforms *Impact crater, a depression caused by two celestial bodies impacting each other, such as a meteorite hitting a planet *Explosion crater, a hole formed in the ground produced by an explosion near or below the surfac ...

on the Moon named

Archimedes Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...

() in his honor, as well as a lunar mountain range, the Montes Archimedes (). The

Fields Medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of the International Mathematical Union (IMU), a meeting that takes place every four years. The name of the award ho ...

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 The gens Manilia was a plebeian family at ancient Rome. Members of this gens are frequently confused with the Manlii, Mallii, and Mamilii. Several of the Manilii were distinguished in the service of the Republic, with Manius Manilius obtaining ...

, which reads in Latin: ''Transire suum pectus mundoque potiri'' ("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 California Gold Rush (1848–1855) was a gold rush that began on January 24, 1848, when gold was found by James W. Marshall at Sutter's Mill in Coloma, California. The news of gold brought approximately 300,000 people to California fro ...

References

Notes

Citations

External links

*
Heiberg's Edition of Archimedes
'' Texts in Classical Greek, with some in English. * * * * *
The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
* *

{{DEFAULTSORT:Archimedes 3rd-century BC Greek people 3rd-century BC writers People from Syracuse, Sicily Ancient Greek engineers Ancient Greek inventors Ancient Greek geometers Ancient Greek physicists Hellenistic-era philosophers Doric Greek writers Sicilian Greeks Mathematicians from Sicily Scientists from Sicily Ancient Greeks who were murdered Ancient Syracusans Fluid dynamicists Buoyancy 280s BC births 210s BC deaths Year of birth uncertain Year of death uncertain 3rd-century BC mathematicians 3rd-century BC Syracusans