Archimedes died during thesiege 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 asphere and acylinder 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 untilc. 530AD byIsidore of Miletus inByzantineConstantinople, whileEutocius' commentaries on Archimedes' works in the same century opened them to wider readership for the first time. In theMiddle 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 theRenaissance and in theScientific Revolution. The discovery in 1906 of works by Archimedes in theArchimedes Palimpsest has provided new insights into how he obtained mathematical results.
The details of Archimedes's life are obscure; a biography of Archimedes mentioned byEutocius was allegedly written by his friendHeraclides 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 scholarJohn Tzetzes that Archimedes lived for 75 years before his death in 212BC, Archimedes is estimated to have been bornc. 287 BC in the seaport city ofSyracuse, Sicily, at that time a self-governing colony inMagna Graecia. In theSand-Reckoner, Archimedes gives his father's name as Phidias, an astronomer about whom nothing else is known;Plutarch wrote in hisParallel Lives[2] that Archimedes was related to KingHiero II, the ruler of Syracuse, althoughCicero andSilius Italicus suggest he was of humble origin.[3] It is also unknown whether he ever married or had children, or if he ever visitedAlexandria, Egypt, during his youth;[4] though his surviving written works, addressed to Dositheus of Pelusium, a student of the Alexandrian astronomerConon of Samos, and to the head librarianEratosthenes of Cyrene, suggested that he maintained collegial relations with scholars based there.[5] In the preface toOn Spirals addressed to Dositheus, Archimedes says that "many years have elapsed since Conon's death."Conon of Samos lived c. 280–220 BC, suggesting that Archimedes may have been an older man when writing some of his works.[citation needed]
Golden wreath
Measurement of volume (a) before and (b) after an object has been submerged, with (∆V) indicating the rising amount of liquid is equal to the volume of the object
Another story of a problem that Archimedes is credited solving with in service of Hiero II is the "wreath problem."[6] According toVitruvius, writing about two centuries after Archimedes' death,King Hiero II of Syracuse had commissioned a golden wreath for a temple to the immortal gods, and had supplied pure gold to be used by the goldsmith.[7] However, the king had begun to suspect that the goldsmith had substituted some cheaper silver and kept some of the pure gold for himself, and, unable to make the smith confess, asked Archimedes to investigate.[8] Later, while stepping into a bath, Archimedes allegedly noticed that the level of the water in the tub rose more the lower he sank in the tub and, realizing that this effect could be used to determine the golden crown'svolume, was so excited that he took to the streets naked, having forgotten to dress, crying "Eureka!"[b], meaning "I have found [it]!"[8] According to Vitruvius, Archimedes then took a lump of gold and a lump of silver that were each equal in weight to the wreath, and, placing each in the bathtub, showed that the wreath displaced more water than the gold and less than the silver, demonstrating that the wreath was gold mixed with silver.[8]
A different account is given in theCarmen de Ponderibus,[9] an anonymous 5th century Latin didactic poem on weights and measures once attributed to the grammarianPriscian.[8] In this poem, the lumps of gold and silver were placed on the scales of a balance, and then the entire apparatus was immersed in water; the difference in density between the gold and the silver, or between the gold and the crown, causes the scale to tip accordingly.[10] Unlike the more famous bathtub account given by Vitruvius, this poetic account uses thehydrostatics principle now known asArchimedes' principle that is found in his treatiseOn Floating Bodies, where a body immersed in a fluid experiences abuoyant force equal to the weight of the fluid it displaces.[11]Galileo Galilei, who invented ahydrostatic 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."[12]
Launching theSyracusia
A large part of Archimedes' work in engineering probably arose from fulfilling the needs of his home city ofSyracuse.[13]Athenaeus of Naucratis in hisDeipnosophistae quotes a certain Moschion for a description on how King Hiero II commissioned the design of a huge ship, theSyracusia, which is said to have been the largest ship built inclassical antiquity and, according to Moschion's account, it was launched by Archimedes.[14] Plutarch tells a slightly different account,[15] relating that Archimedes boasted to Hiero that he was able to move any large weight, at which point Hiero challenged him to move a ship.[16] These accounts contain many fantastic details that are historically implausible, and the authors of these stories provide conflicting about how this task was accomplished:[16] Plutarch states that Archimedes constructed ablock-and-tacklepulley system, whileHero of Alexandria attributed the same boast to Archimedes' invention of thebaroulkos, a kind ofwindlass.[17]Pappus of Alexandria attributed this feat, instead, to Archimedes' use ofmechanical advantage,[16] the principle ofleverage to lift objects that would otherwise have been too heavy to move, attributing to him the oft-quoted remark: "Give me a place to stand on, and I will move the Earth."[c][18]
Athenaeus, likely garbling the details of Hero's account of the baroulkos,[19] also mentions that Archimedes used a "screw" in order to remove any potential water leaking through the hull of theSyracusia. Although this device is sometimes referred to asArchimedes' screw, it likely predates him by a significant amount, and none of his closest contemporaries who describe its use (Philo of Byzantium,Strabo, andVitruvius) credit him with its use.[16]
The greatest reputation Archimedes earned during antiquity was for the defense of his city from the Romans during theSiege of Syracuse.[20] According to Plutarch,[21] Archimedes had constructed war machines for Hiero II, but had never been given an opportunity to use them during Hiero's lifetime. In 214 BC, however, during theSecond Punic War, when Syracuse switched allegiances fromRome toCarthage, the Roman army underMarcus Claudius Marcellus attempted to take the city, Archimedes allegedly personally oversaw the use of these war machines in the defense of the city, greatly delaying the Romans, who were only able to capture the city after a long siege.[22] Three different historians,Plutarch,Livy, andPolybius provide testimony about these war machines, describing improvedcatapults, cranes that dropped heavy pieces of lead on the Roman ships or which used an ironclaw to lift them out of the water, dropping them back in so that they sank.[d][24]
A much more improbable account, not found in any of the three earliest accounts (Plutarch, Polybius, or Livy) describes how Archimedes used "burning mirrors" to focus the sun's rays onto the attacking Roman ships, setting them on fire.[20] The earliest account to mention ships being set on fire, by the 2nd century CE satiristLucian of Samosata,[25] does not mention mirrors, and only says the ships were set on fire by artificial means, which may imply that burning projectiles were used.[20] The first author to mention mirrors isGalen, writing later in the same century.[26] Nearly four hundred years after Lucian and Galen,Anthemius, despite skepticism, tried to reconstruct Archimedes' hypothetical reflector geometry.[27][28] The purported device, sometimes called "Archimedes' heat ray", has been the subject of an ongoing debate about its credibility since theRenaissance.[29]René Descartes rejected it as false,[30] while modern researchers have attempted to recreate the effect using only the means that would have been available to Archimedes, with mixed results.[31]
There are several divergent accounts of Archimedes' death during the sack of Syracuse after it fell to the Romans:[32] The oldest account, fromLivy,[33] says that, while drawing figures in the dust, Archimedes was killed by a Roman soldier who did not know he was Archimedes. According to Plutarch,[34] the soldier demanded that Archimedes come with him, but Archimedes declined, saying that he had to finish working on the problem, and the soldier killed Archimedes with his sword. Another story from Plutarch has Archimedes carrying mathematical instruments before being killed because a soldier thought they were valuable items.[32] Another Roman writer,Valerius Maximus (fl. 30 AD), wrote inMemorable Doings and Sayings that Archimedes' last words as the soldier killed him were "... but protecting the dust with his hands, said 'I beg of you, do not disturb this." which is similar to the last words now commonly attributed to him, "Do not disturb my circles," which do not appear in any ancient sources.[32]
Marcellus was reportedly angered by Archimedes' death, as he considered him a valuable scientific asset (he called Archimedes "a geometricalBriareus") and had ordered that he should not be harmed.[35][36]Cicero (106–43 BC) mentions that Marcellus brought to Rome two planetariums Archimedes built,[37] which were constructed by Archimedes and which showed the motion of the Sun, Moon and five planets, one of which he donated to theTemple of Virtue in Rome, and the other he allegedly kept as his only personal loot from Syracuse."[38]Pappus of Alexandria reports on a now lost treatise by ArchimedesOn Sphere-Making, which may have dealt with the construction of these mechanisms.[24] Constructing mechanisms of this kind would have required a sophisticated knowledge ofdifferential 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 builtc. 100 BC designed with a similar purpose, has confirmed that devices of this kind were known to the ancient Greeks,[39] with some scholars regarding Archimedes' device as a precursor.[40][41]
While serving as aquaestor 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.[42]
Mathematics
While he is often regarded as a designer of mechanical devices, Archimedes also made contributions to the field ofmathematics, both in applying the techniques of his predecessors to obtain new results, and developing new methods of his own.
Archimedes calculates the side of the 12-gon from that of thehexagon and for each subsequent doubling of the sides of the regular polygon
InQuadrature of the Parabola, Archimedes states that a certain proposition inEuclid'sElements demonstrating that the area of acircle is proportional to its diameter was proven using a lemma now known as theArchimedean property, that “the excess by which the greater of two unequal regions exceed the lesser, if added to itself, can exceed any given bounded region.” Prior to Archimedes,Eudoxus of Cnidus and other earlier mathematicians[e] applied this lemma, a technique now referred to as the "method of exhaustion," to find the volume of atetrahedron,cylinder,cone, andsphere, for which proofs are given in book XII ofEuclid's Elements.[43]
InMeasurement of a Circle, Archimedes employed this method to show that the area of a circle is the same as a right triangle whose base and height are equal to its radius and circumference.[44] He then approximated the ratio between the radius and the circumference, the value ofπ, by drawing a largerregular hexagon outside a circle then a smaller regular hexagon inside the circle, and progressively doubling the number of sides of eachregular 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 ofπ lay between 31/7 (approx. 3.1429) and 310/71 (approx. 3.1408), consistent with its actual value of approximately 3.1416.[45] In the same treatise, he also asserts that the value of thesquare root of 3 as lying between265/153 (approximately 1.7320261) and1351/780 (approximately 1.7320512), which he may have derived from a similar method.[46]
A proof that the area of theparabolic segment in the upper figure is equal to 4/3 that of the inscribed triangle in the lower figure fromQuadrature of the Parabola
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 smallersecant 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 series1/4 + 1/16 + 1/64 + 1/256 + · · · which sums to 1/3.
He also used this technique in order to measure the surface areas of a sphere and cone,[48] to calculate the area of an ellipse,[49] and to find the area contained within anArchimedean spiral.[50][44]
Mechanical method
For it is more feasible, having already in one’s possession, through the method, of a knowledge of some sort of the matters under investigation, to provide the proof – rather than investigating it, knowing nothing.
In addition to developing on the works of earlier mathematicians with the method of exhaustion, Archimedes also pioneered a novel technique using thelaw of the lever in order to measure the area and volume of shapes using physical means. He first gives an outline of this proof inQuadrature of the Parabola alongside the geometric proof, but he gives a fuller explanation inThe Method of Mechanical Theorems.[47] According to Archimedes, he proved the results in his mathematical treatises first using this method, and then worked backwards, applying the method of exhaustion only after he had already calculated an approximate value for the answer.[52]
Large numbers
Archimedes also developed methods for representing large numbers.
InThe Sand Reckoner, Archimedes devised a system of counting based on themyriad,[f] the Greek term for the number 10,000, in order to calculate a number that was greater than the grains of sand needed to fill the universe. He proposed a number system using powers of a myriad of myriads (100 million, i.e., 10,000 x 10,000) and concluded that the number of grains of sand required to fill the universe would be 8vigintillion, or 8×1063.[53] In doing so, he demonstrated that mathematics could represent arbitrarily large numbers.
In a lost work described byPappus of Alexandria, Archimedes proved that there are exactly thirteen semiregular polyhedra.[55]
Writings
Front page of Archimedes'Opera, in Greek and Latin, edited byDavid Rivault (1615)
Archimedes made his work known through correspondence with mathematicians inAlexandria,[56] which were originally written inDoric Greek, the dialect of ancient Syracuse.[57]
Surviving works
The following are ordered chronologically based on new terminological and historical criteria set by Knorr (1978) and Sato (1986).[58][59]
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 ofConon of Samos. In Proposition II, Archimedes gives an approximation of the value of pi (π), showing that it is greater than223/71 (3.1408...) and less than22/7 (3.1428...).
In this treatise, also known asPsammites, Archimedes finds a number that is greater than thegrains of sand needed to fill the universe. This book mentions theheliocentric theory of theSolar System proposed byAristarchus of Samos, as well as contemporary ideas about the size of the Earth and the distance between variouscelestial bodies, and attempts to measure the apparent diameter of theSun.[60][61] By using a system of numbers based on powers of themyriad, 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.[62]
Archimedes discusses astronomical measurements of the Earth, Sun, and Moon, as well asAristarchus' heliocentric model of the universe, in theSand-Reckoner.[63] 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),[64] applying correction factors to these measurements, and finally giving the result in the form of upper and lower bounds to account for observational error.[65]
Ptolemy, quoting Hipparchus, also references Archimedes'solstice observations in theAlmagest. This would make Archimedes the first known Greek to have recorded multiple solstice dates and times in successive years.[4]
There are two books toOn the Equilibrium of Planes: the first contains sevenpostulates and fifteenpropositions, while the second book contains ten propositions. In the first book, Archimedes proves the law of thelever,[66] which states that:
Magnitudes are in equilibrium at distances reciprocally proportional to their weights.
In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by aparabola and a straight line is 4/3 the area of atriangle with equal base and height. He achieves this by two different methods: first by applying thelaw of the lever, and by calculating the value of ageometric series that sums to infinity with theratio 1/4.[69]
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 asphere and acircumscribedcylinder of the same height anddiameter. The volume is4/3πr3 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), wherer is the radius of the sphere and cylinder.[70][71]
This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called theArchimedean spiral.[72] It is thelocus 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 constantangular velocity. Equivalently, in modernpolar coordinates (r,θ), it can be described by the equation withreal numbersa andb.[73]
This is an early example of amechanical curve (a curve traced by a movingpoint) considered by a Greek mathematician.
This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes ofsections ofcones, spheres, and paraboloids.[74][75]
There are two books ofOn Floating Bodies. In the first book, Archimedes spells out the law ofequilibrium of fluids and proves that water will adopt a spherical form around a center of gravity.[76]
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.[78]
Also known asLoculus of Archimedes orArchimedes' Box,[79] this is adissection puzzle similar to aTangram, and the treatise describing it was found in more complete form in theArchimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form asquare.Reviel Netz ofStanford 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.[80] The number of arrangements is 536 when solutions that are equivalent by rotation and reflection are excluded.[81] The puzzle represents an example of an early problem incombinatorics.
The origin of the puzzle's name is unclear, and it has been suggested that it is taken from theAncient Greek word for "throat" or "gullet",stomachos (στόμαχος).[82]Ausonius calls the puzzleOstomachion, a Greek compound word formed from the roots ofosteon (ὀστέον, 'bone') andmachē (μάχη, 'fight').[79]
In this work, addressed to Eratosthenes and the mathematicians in Alexandria, Archimedes challenges them to count the numbers of cattle in theHerd of the Sun, which involves solving a number of simultaneousDiophantine equations.Gotthold Ephraim Lessing discovered this work in a Greek manuscript consisting of a 44-line poem in theHerzog August Library inWolfenbüttel, Germany, in 1773. There is a more difficult version of the problem in which some of the answers are required to besquare numbers. A. Amthor first solved this version of the problem[83] in 1880, and the answer is avery large number, approximately 7.760271×10206544.[84]
In this work Archimedes uses a novel method, an early form ofCavalieri's principle,[85] to rederive the results from the treatises sent to Dositheus (Quadrature of the Parabola,On the Sphere and Cylinder,On Spirals,On Conoids and Spheroids) that he had previously used themethod of exhaustion to prove,[86] using thelaw of the lever he applied inOn the Equilbrium of Planes in order to find thecenter of gravity of an object first, and reasoning geometrically from there in order to more easily derive the volume of an object.[87] Archimedes states that he used this method to derive the results in the treatises sent to Dositheus before he proved them more rigorously with the method of exhaustion, stating that it is useful to know that a result is true before proving it rigorously, much asEudoxus of Cnidus was aided in proving that the volume of a cone is one-third the volume of cylinder by knowing thatDemocritus had already asserted it to be true on the argument that this is true by the fact that the pyramid has one-third the rectangular prism of the same base.[88]
Archimedes'Book of Lemmas orLiber Assumptorum is a treatise with 15 propositions on the nature of circles. The earliest known copy of the text is inArabic.T. L. Heath andMarshall 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. TheLemmas may be based on an earlier work by Archimedes that is now lost.[citation needed]
Other questionable attributions to Archimedes' work include the Latin poemCarmen de ponderibus et mensuris (4th or 5th century), which describes the use of ahydrostatic balance, to solve the problem of the crown, and the 12th-century textMappae clavicula, which contains instructions on how to performassaying of metals by calculating their specific gravities.[90][91]
Lost works
Many written works by Archimedes have not survived or are only extant in heavily edited fragments:[92]Pappus of Alexandria mentionsOn Sphere-Making, as well as a work onsemiregular polyhedra, and another work on spirals, whileTheon of Alexandria quotes a remark aboutrefraction from thenow-lostCatoptrica.Principles, addressed to Zeuxippus, explained the number system used inThe Sand Reckoner; there are alsoOn Balances;On Centers of Gravity.[92]
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 asHeron's formula due to its first known appearance in the work ofHeron of Alexandria in the 1st century AD, and may have been proven in a lost work of Archimedes that is no longer extant.[93]
In 1906, the Archimedes Palimpsest revealed works by Archimedes thought to have been lost
In 1906, the Danish professorJohan Ludvig Heiberg visitedConstantinople to examine a 174-pagegoatskinparchment of prayers, written in the 13th century, after reading a short transcription published seven years earlier byPapadopoulos-Kerameus.[94][95] He confirmed that it was indeed apalimpsest, 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, asvellum was expensive. The older works in the palimpsest were identified by scholars as 10th-century copies of previously lost treatises by Archimedes.[94][96] The palimpsest holds seven treatises, including the only surviving copy ofOn Floating Bodies in the original Greek. It is the only known source ofThe Method of Mechanical Theorems, referred to bySuidas 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.[97][98] The palimpsest was stored at theWalters Art Museum inBaltimore,Maryland, where it was subjected to a range of modern tests including the use ofultraviolet andX-raylight to read the overwritten text.[99] It has since returned to its anonymous owner.[100][101]
Sometimes called the father of mathematics[102] andmathematical physics,[103] historians of science and mathematics almost universally agree that Archimedes was the finest mathematician from antiquity.[104]
Classical antiquity
The reputation that Archimedes had for mechanical inventions in classical antiquity is well-documented;[105]Athenaeus[106] recounts in hisDeipnosophistae how Archimedes supervised the construction of the largest known ship in antiquity, theSyracusia, whileApuleius[107] talks about his work incatoptrics.[108]Plutarch[109] had claimed that Archimedes disdained mechanics and focused primarily on puregeometry, but this is generally considered to be a mischaracterization by modern scholarship, fabricated to bolster Plutarch's ownPlatonist values rather than to an accurate presentation of Archimedes,[110] and, unlike his inventions, Archimedes' mathematical writings were little known in antiquity outside of the works ofAlexandrian mathematicians.[citation needed] The first comprehensive compilation was not made untilc. 530AD byIsidore of Miletus inByzantineConstantinople,[citation needed] whileEutocius' commentaries on Archimedes' works earlier in the same century opened them to wider readership for the first time.[citation needed]
Italian numismatist and archaeologist Filippo Paruta (1552–1629) andLeonardo 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.[123] 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".[124]
Gauss's heroes were Archimedes andNewton,[125] andMoritz Cantor, who studied under Gauss in theUniversity of Göttingen, reported that he once remarked in conversation that "there had been only three epoch-making mathematicians: Archimedes, Newton, andEisenstein".[126] Likewise,Alfred North Whitehead said that "in the year 1500 Europe knew less than Archimedes who died in the year 212 BC."[127] The historian of mathematicsReviel Netz,[128] echoing Whitehead's proclamation onPlato andphilosophy, said that "Western science is but a series of footnotes to Archimedes," calling him "the most important scientist who ever lived." andEric Temple Bell,[129] 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 theArchimedes Palimpsest has provided new insights into how he obtained mathematical results.[130][131]
TheFields 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 poetManilius, which reads in Latin:Transire suum pectus mundoque potiri ("Rise above oneself and grasp the world").[132][133][134]
Cultural influence
The world's first seagoingsteamship with ascrew propeller was theSSArchimedes, which was launched in 1839 and named in honor of Archimedes and his work on the screw.[135]
^There have been modern experiments to test the feasibility of the claw, and in 2005 a television documentary entitledSuperweapons of the Ancient World built a version of the claw and concluded that it was a workable device.[23]
^While Eudoxus is often credited for the whole of Euclid's Book XII, Archimedes explicitly credits him only with the proofs of XII.7 and XII.10, that the volume of a pyramid and a cone are, respectively, one-third of the volume of the rectangular prism and cone with the same base and height.Acerbi 2018, p. 279
^Commentarius in dimensionem circuli (Archimedis opera omnia ed. Heiberg-Stamatis (1915), vol. 3, p. 228); Commentaria in conica (Apollonii Pergaei quae Graece exstant, ed. Heiberg (1893) vol. 2, p. 168: "Hērakleios"
^Jaeger, Mary (2017). "Archimedes in the 21st century imagination". In Rorres, Chris (ed.).Archimedes in the 21st Century: Proceedings of a World Conference at the Courant Institute of Mathematical Sciences. Trends in the History of Science. Birkhäuser. pp. 143–152.doi:10.1007/978-3-319-58059-3_8.ISBN9783319580593. See p. 144.
^Knorr, W. R. (1978). "Archimedes and the Elements: Proposal for a Revised Chronological Ordering of the Archimedean Corpus".Archive for History of Exact Sciences.19 (3):211–290.doi:10.1007/BF00357582.JSTOR41133526.
^Sato, T. (1986). "A Reconstruction of The Method Proposition 17, and the Development of Archimedes' Thought on Quadrature...Part One".Historia scientiarum: International journal of the History of Science Society of Japan.
^Osborne, Catherine (1983). "Archimedes on the Dimensions of the Cosmos".Isis.74 (2):234–242.doi:10.1086/353246.JSTOR233105.
^Rozelot, Jean Pierre; Kosovichev, Alexander G.; Kilcik, Ali (2016).A brief history of the solar diameter measurements: a critical quality assessment of the existing data.arXiv:1609.02710.
^Toomer, G. J.; Jones, Alexander (7 March 2016). "Astronomical Instruments".Oxford Research Encyclopedia of Classics.doi:10.1093/acrefore/9780199381135.013.886.ISBN9780199381135.Perhaps the earliest instrument, apart from sundials, of which we have a detailed description is the device constructed by Archimedes for measuring the sun's apparent diameter; this was a rod along which different coloured pegs could be moved.
^Archimedes (1897). Heath (ed.).The works of Archimedes. Dover., page 151
^Archimedes (1897). Heath (ed.).The works of Archimedes. Dover., page 99
^Saito, Ken (2013). "Archimedes and double contradiction proof".Lettera Matematica International Edition.1:97–104.doi:10.1007/s40329-013-0017-x.
^Berggren, J. L. (1976). "Spurious Theorems in Archimedes' Equilibrium of Planes: Book I".Archive for History of Exact Sciences.16 (2):87–103.doi:10.1007/BF00349632.JSTOR41133463.
^Netz, Reviel (2017)."Archimedes' Liquid Bodies". In Buchheim, Thomas; Meißner, David; Wachsmann, Nora (eds.).ΣΩΜΑ: Körperkonzepte und körperliche Existenz in der antiken Philosophie und Literatur. Hamburg: Felix Meiner. pp. 287–322.ISBN978-3-7873-2928-1.
^Rorres, Chris."Archimedes' Stomachion". Courant Institute of Mathematical Sciences. Retrieved14 September 2007.
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^Berthelot, Marcel. 1891. "Sur l histoire de la balance hydrostatique et de quelques autres appareils et procédés scientifiques."Annales de Chimie et de Physique 6(23):475–85.
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^Riehm, C. (2002)."The early history of the Fields Medal"(PDF).Notices of the AMS.49 (7):778–782.The Latin inscription from the Roman poet Manilius surrounding the image may be translated 'To pass beyond your understanding and make yourself master of the universe.' The phrase comes from Manilius's Astronomica 4.392 from the first century A.D. (p. 782).
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