Archimedes

About Archimedes

By the standards of his own century, Archimedes of Syracuse was already legendary; by the standards of the nineteen that followed, he reads as the first scientist whose work the modern world can read on its own terms. He proved that the surface area of a sphere is two-thirds that of its circumscribing cylinder, and that the volume relation between the two is the same. That was the result he asked to be carved on his tomb. He pinned π between 3 10/71 and 3 1/7. He wrote down the law of the lever and the law of buoyancy in propositions a physics student today can still follow. He did this in a Greek city on the eastern coast of Sicily, in a Hellenistic world that did not yet have algebra or calculus or zero, working with geometry alone.

He was born around 287 BCE in Syracuse, then a major Greek polis under King Hiero II. The only direct biographical detail from his own pen comes in the dedication of The Sand Reckoner, where he names his father Phidias as an astronomer. He likely studied at Alexandria, the home of the Museum and the Library, where he formed the working friendships that produced his correspondence with Eratosthenes (the geometer who measured the Earth) and with Conon of Samos and Conon's student Dositheus, the recipients to whom most of his surviving treatises are addressed. After that Alexandrian period he returned to Syracuse and stayed there for the rest of his life.

He died in 212 BCE during the Roman sack of Syracuse, in the Second Punic War, when the city fell to the army of Marcus Claudius Marcellus after a long siege. Several ancient sources, including Plutarch's Life of Marcellus, Livy's History of Rome 25.31, and Valerius Maximus, report that Marcellus had ordered him spared and that a Roman soldier killed him anyway. The accounts disagree on the details (he was working on a geometric diagram in the sand; he was carrying instruments a soldier mistook for valuables; he refused to leave a problem unfinished) and converge on the bare fact: Archimedes died at the soldier's hand, and Marcellus was reportedly furious about it.

For a century and a half after his death his tomb in Syracuse fell into disuse. In 75 BCE, Cicero, then quaestor in Sicily, searched it out, found it overgrown with thorn-bushes near the Agrigentine gate, and identified it by the carved sphere-and-cylinder marker Archimedes had requested. Cicero tells the story himself in Tusculan Disputations V.64–66, with a small amount of self-congratulation about a Roman recovering for the Greeks the grave of their greatest geometer. That episode is part of why we still know where, roughly, to look.

His surviving corpus (about a dozen treatises, plus fragments) comes down through a tortuous textual history. The most consequential single transmission event was the rediscovery in 1906, by the Danish philologist Johan Ludvig Heiberg, of what is now called the Archimedes Palimpsest: a 10th-century Constantinople manuscript of seven Archimedes treatises that was scraped, washed, and overwritten in the 13th century with a Christian Euchologion (prayer book), then sat in a monastery library until it surfaced for sale in the early 20th century. The palimpsest contains the only known Greek text of The Method of Mechanical Theorems and the only known Greek text of On Floating Bodies. After the manuscript disappeared during the First World War, resurfaced at a Christie's auction in 1998, and was bought for two million dollars by an anonymous collector who lent it to the Walters Art Museum in Baltimore, a decade-long imaging project using multispectral and X-ray fluorescence techniques recovered most of the underwriting. The Walters' final transcription was published in 2011 by Reviel Netz, William Noel, Natalie Tchernetska, and Nigel Wilson. We can now read pages of Archimedes that no one had been able to read for eight hundred years.

Contributions

Archimedes' contributions split cleanly along the seam he himself drew between two genres of mathematical work: the heuristic, where he allowed himself to reason mechanically by treating geometric figures as physical bodies with weight and balance, and the rigorous, where he closed each result with a formal proof in the Euclidean style. He did the first kind to discover, the second kind to publish. The recovery of The Method of Mechanical Theorems from the palimpsest in 1906 is what made this division visible in his own words.

In pure geometry, his master result is the trio of propositions in On the Sphere and Cylinder: a sphere is two-thirds of its circumscribing cylinder in volume and in surface area, and the great circle of the sphere is two-thirds of the cylinder's total surface. He treated this as the work he most wanted remembered, and asked for the sphere-and-cylinder figure on his tomb. On the Measurement of a Circle proves that the area of a circle equals that of a right triangle whose legs are the radius and the circumference, and bounds π between 3 10/71 and 3 1/7 (between roughly 3.1408 and 3.1429) by inscribing and circumscribing 96-gons. On Spirals defines what is now called the Archimedean spiral (r = aθ) and proves rectification and area results that depend on a careful limiting argument.

In mechanics, On the Equilibrium of Planes sets out the law of the lever (equal weights at equal distances balance, unequal weights balance at distances inversely proportional to their weights) as a geometric theorem grounded in postulates about symmetry. The treatise then computes centers of gravity for triangles, parallelograms, and parabolic segments. Quadrature of the Parabola finds the area of a parabolic segment by two methods, one mechanical (balancing the segment against a triangle) and one purely geometric (an exhaustion-style sum of inscribed triangles whose total approaches 4/3 of the inscribed triangle as a limit). The two-method structure is the public face of the heuristic-rigorous split.

In hydrostatics, On Floating Bodies Books I and II establish the principle now taught as Archimedes' Principle (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces) and apply it to the equilibrium of solid paraboloid segments floating in water. The proofs are geometric, with no appeal to a bath or a crown; the popular eureka story does not appear anywhere in the treatise itself.

The Method of Mechanical Theorems, addressed to Eratosthenes and recovered only in 1906, opens the workshop door. Archimedes shows that he discovered the volume relation between the sphere and the cylinder, the area of the parabolic segment, the volume of a hyperboloid of revolution, the volume of a spherical segment, and other results, by mentally slicing each figure into indivisible cross-sections and balancing them on a lever against cross-sections of a known figure. He calls the technique a discovery procedure, not a proof, and he stresses that the formal demonstrations have to be done afterward in the geometric style. Reviel Netz and others have argued that Proposition 14 of The Method, in particular, performs a treatment of an actual infinity of cross-sections, which would put Archimedes closer to integration than any other surviving ancient mathematician.

The Sand Reckoner, addressed to King Gelon, builds a numeral system for very large numbers (up to what would now be written as 10^(8 × 10^16)) and uses it to estimate the number of grains of sand it would take to fill the universe under the cosmological model of Aristarchus, in which the Earth orbits the Sun. The treatise is one of the few surviving ancient sources to describe the Aristarchan heliocentric model.

His engineering record runs alongside the mathematics. The Archimedean screw, a helical screw inside a cylinder used to lift water, was reportedly invented by him to drain bilge water on a large Syracusan ship and was then adopted across the Mediterranean for irrigation. The compound pulley, in Plutarch's account, allowed him to single-handedly draw a fully laden ship across the beach as a demonstration for Hiero. The defensive engines of Syracuse during the Roman siege, the so-called Claw of Archimedes (a crane that grappled and capsized Roman ships under the city walls) and a battery of catapults of varying ranges, are described by Polybius (Histories 8.4–7), Livy, and Plutarch. The heat-ray of burning mirrors that supposedly set Roman ships on fire is later, less reliable, and treated under Controversies.

Works

Surviving treatises (in conventional Greek titles, with the standard Heiberg numbering retained by modern editors):

Lost or partly lost works mentioned in his extant treatises or in later sources include the Catoptrica on reflection, a treatise on polyhedra (the source for the thirteen Archimedean solids transmitted by Pappus), works on balances, on the construction of a celestial sphere, and on the regular heptagon. Cicero (De Republica I.21–22) describes a planetarium-like "sphere of Archimedes" Marcellus brought back from Syracuse and showed in Rome.

Standard editions and translations:

Controversies

Three of the famous stories about Archimedes do not survive scrutiny in the form they are usually told.

The first is the Eureka story. The earliest source is Vitruvius, De Architectura IX.preface 9–12, written about two centuries after Archimedes' death. In Vitruvius's telling, Hiero suspected a goldsmith of substituting silver for some of the gold in a votive crown, and asked Archimedes to determine the cheating without melting the crown down; Archimedes noticed in the bath that water displaced equaled the volume of the immersed body, leapt out shouting εὕρηκα, and ran home naked. The story is internally consistent and may be true, but it is not Archimedes' own account, and the actual proof in On Floating Bodies Book I (Propositions 5 through 7 in particular) uses density and weight in fluid, not displaced volume measured in a tub. The bath story dramatizes a different and less precise method than the one Archimedes published. Galileo, in La Bilancetta (1586), reconstructed what he thought was Archimedes' actual procedure: a hydrostatic balance comparing the apparent weight of the crown in air and in water against reference samples of pure gold and pure silver, which is more sensitive than volume by displacement. Whether or not Vitruvius got the bath right, he did not get the geometry right.

The second is the heat-ray of burning mirrors at the siege of Syracuse. The earliest surviving source for it is Lucian (Hippias, 2nd century CE), and it is elaborated by Anthemius of Tralles, Galen, and the 12th-century Byzantine historian Tzetzes. None of these sources is within four centuries of the event. Polybius, the closest contemporary military historian, describes the Claw and the catapults but says nothing about mirrors. Modern reconstructions have been mixed. René Descartes thought it physically impossible. A 1973 attempt by the Greek engineer Ioannis Sakkas reportedly ignited a wooden mock-up at fifty meters under ideal conditions. A 2005 MIT team set a stationary wooden ship section alight in still air at full sun, but a 2010 MythBusters revisitation, attempting more realistic siege conditions, failed to ignite a target. The current historiographic consensus, summarized by Mary Beard and others, treats the heat-ray as legendary embellishment of a real defensive operation that probably involved catapults and the Claw, not focused light.

The third is the relationship to the Antikythera Mechanism. The Antikythera device, dredged from a Roman shipwreck in 1901 and now dated to roughly 100–60 BCE, is a hand-cranked geared astronomical computer of a sophistication unmatched in the surviving record. Cicero (De Republica I.21–22) describes a planetarium that Marcellus brought from Syracuse to Rome after the city fell, attributing its construction to Archimedes; Pappus also reports that Archimedes wrote a treatise On Sphere-Making. The Antikythera Mechanism postdates Archimedes by roughly a century, and there is no direct evidence linking it to him. The most defensible statement is that the device sits within a Greek mechanical-astronomical tradition Archimedes helped found, and that some lineage of training, parts, or design conventions plausibly runs from his planetarium to it. Anything stronger than that overclaims. Tony Freeth's Antikythera Mechanism Research Project has been careful on this point.

A fourth, smaller controversy is the death scene. Plutarch gives at least three competing versions in Marcellus 19, and Valerius Maximus gives the famous noli turbare circulos meos ("do not disturb my circles") in Memorable Doings VIII.7.ext.7. That is a Latin phrase Archimedes did not speak in Latin, plausibly a later reconstruction of a Greek admonition. The historicity of the exact words is weak; the historicity of the killing is strong.

Notable Quotes

"Δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω." "Give me a place to stand and I will move the earth." On the lever; reported by Pappus of Alexandria, Synagoge VIII (4th century CE), and in slightly different form by Plutarch (Marcellus 14) and Simplicius. No Archimedean text contains the line; it is a later compression of his lever propositions.

"Noli turbare circulos meos." "Do not disturb my circles." Reported as his last words at the moment of his death, addressed to the Roman soldier who killed him; Valerius Maximus, Memorable Doings and Sayings VIII.7.ext.7. The Latin form is later; the underlying Greek admonition is plausibly historical, though Plutarch's parallel accounts vary.

"Εὕρηκα." "I have found it." On the bath and the crown; Vitruvius, De Architectura IX.preface 10. Vitruvius wrote about two centuries after the event, and the bath method does not match the proof Archimedes published in On Floating Bodies.

From the dedication of The Sand Reckoner to King Gelon: "There are some, King Gelon, 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. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude." — Archimedes, The Sand Reckoner, opening, Heath translation.

From The Method of Mechanical Theorems, addressed to Eratosthenes, in which Archimedes describes his discovery procedure: "Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge." — Archimedes, The Method, prefatory letter, Heiberg/Heath translation.

From Plutarch's Life of Marcellus 17, on Archimedes' attitude to his applied work: "He would not deign to leave behind him any commentary or writing on such subjects; but, repudiating as sordid and ignoble the whole trade of engineering, and every sort of art that lends itself to mere use and profit, he placed his whole ambition in those speculations the beauty and subtlety of which is untainted by any admixture of the common needs of life." — Plutarch, Marcellus 17, Perrin translation. Plutarch is editorializing more than reporting; the surviving treatises themselves include the mechanical work, so the dichotomy is sharper in Plutarch than in Archimedes.

Legacy

The first piece of his legacy is the survival itself. The Latin West largely lost direct access to Archimedes after the closing of the Alexandrian schools; the Arabic translators of the 9th and 10th centuries, Thābit ibn Qurra in particular, preserved On the Sphere and Cylinder, On the Measurement of a Circle, the Book of Lemmas, and other texts that fed back into Latin Europe through 12th- and 13th-century translations from Arabic. William of Moerbeke's 1269 Latin translation, made directly from two Greek manuscripts, is the version that circulated through the late medieval and early Renaissance West. The first printed Greek edition appeared in Basel in 1544. Heiberg's 1880–81 critical edition is the basis of all modern scholarship. The 1906 palimpsest discovery and the 2011 Walters publication added The Method and the Greek text of On Floating Bodies to the readable corpus.

The second is the direct line to early-modern mechanics. Galileo read Archimedes carefully and called him il divino Archimede; La Bilancetta (1586) reconstructs an Archimedean hydrostatic balance, and the proportional reasoning in Two New Sciences (1638) is openly Archimedean in style. Simon Stevin's hydrostatic work, Christiaan Huygens' centers of gravity, and Newton's geometric proofs in the Principia all reach back to On the Equilibrium of Planes and On Floating Bodies. Leibniz, who did not have The Method, nonetheless arrived independently at the indivisibles-and-balance heuristic Archimedes had used in private, and the calculus that he and Newton built absorbed the method-of-exhaustion limit arguments into the algebraic apparatus that finally made them routine.

Leonardo da Vinci's notebooks contain dozens of pages on Archimedean themes (water-lifting screws, lever-and-pulley systems, the buoyancy of ships, attempts at perpetual motion he correctly rejected on Archimedean grounds), plus a sustained but unfulfilled effort to track down a copy of the lost De Insidentibus Aquae (the Latin On Floating Bodies). The Renaissance encounter with Archimedes is documented across the workshops of Filippo Brunelleschi, Francesco di Giorgio Martini, and Leonardo himself.

The modern recovery has continued in three places. The Archimedes Palimpsest project at the Walters Art Museum, run by curator William Noel and scholars including Reviel Netz, Nigel Wilson, Natalie Tchernetska, Roger Easton, and Keith Knox, used multispectral imaging at multiple wavelengths plus X-ray fluorescence at the Stanford Synchrotron between 1999 and 2008 to lift the underwriting from the overwriting. The resulting transcriptions and digital images were released into the public domain in 2008 and the printed edition followed in 2011. The Antikythera Mechanism Research Project, led by Tony Freeth and Mike Edmunds with X-ray tomography by X-Tek Systems, has fed indirectly into Archimedean studies by clarifying what a Hellenistic geared astronomical device looked like in surviving form. The Bibliotheca Augustana and the Perseus Digital Library have made Heiberg's Greek text and Heath's English translation freely available online.

The Archimedean screw is still in service. Variants are used in modern wastewater plants in cities including London, Memphis, and Cairo, in archaeological and industrial water management, and in micro-hydro-electric installations where the screw is run in reverse as a turbine. The Fields Medal, awarded by the International Mathematical Union since 1936, carries on its obverse a profile of Archimedes and on its reverse a sphere inscribed in a cylinder with the legend transire suum pectus mundoque potiri, "to surpass one's own understanding and master the world." The medal was Cicero's tomb-marker re-cast for the 20th century.

Significance

Archimedes matters at three different scales, and conflating them is what produces both the cult and the deflationary backlash.

At the narrowest scale, he is the third-century-BCE Greek geometer who proved a specific cluster of results (sphere-cylinder, π-bounds, parabolic-segment area, lever law, buoyancy law, spiral rectification) that any history of Greek mathematics has to enumerate. The proofs are difficult, internally consistent, and verifiable by anyone who can read Heath's English text or Heiberg's Greek. This is the Archimedes of the textbook chapter on Hellenistic mathematics, sandwiched between Euclid and Apollonius.

At a middle scale, he is the figure in whom the Hellenistic split between theoretical mathematics and applied mechanics first becomes visible as a methodological position rather than a sociological accident. The Method of Mechanical Theorems, which we have only because of Heiberg's 1906 palimpsest discovery, shows him reasoning mechanically to discover and reasoning geometrically to prove, with explicit commentary on why the two registers cannot substitute for each other. That meta-awareness, heuristic versus demonstration, conjecture versus proof, is closer to the modern working mathematician's self-understanding than anything in Euclid. It is also why Reviel Netz has argued that Archimedes is the first Greek mathematician for whom "creative" is a useful word, in distinction from Euclid's encyclopedic compiler-of-results role.

At the broadest scale, he is the emblem of the case that there were periods in antiquity, such as Hellenistic Alexandria and its satellites, Syracuse among them, where mathematical and mechanical sophistication reached levels that the Latin West did not match again until the 16th and 17th centuries. The Antikythera Mechanism is the strongest material evidence for this; Archimedes' surviving treatises, especially with The Method restored, are the strongest textual evidence. The case has consequences for how the history of science is told, for what counts as a scientific revolution, and for whether the European "scientific revolution" of the 17th century is better described as a Hellenistic recovery.

For sacred geometry as Satyori uses the term, his work on the sphere and the cylinder, on the spiral, and on the conic sections of On Conoids and Spheroids sits at a particular junction. Archimedes did not frame these results in the cosmological-symbolic register that Plato had given to the regular polyhedra in the Timaeus; he was a working geometer, not a Pythagorean. But the figures he proved theorems about (the sphere with its inscribed cylinder, the parabolic mirror, the equiangular spiral) reappear in the symbolic vocabularies of Renaissance Hermeticism, in Kepler's Mysterium Cosmographicum, and in the contemporary visual canon of sacred geometry. The proofs are his; the symbolic afterlife is what later cultures did with the proofs.

Connections

Euclid stood a generation earlier in the Alexandrian tradition. Archimedes inherited and extended the geometric apparatus of the Elements, particularly the method of exhaustion that Euclid had codified from Eudoxus. Archimedes' citations of Euclidean propositions are routine and unannounced, the way a working mathematician cites a textbook he assumes the reader has read. See Euclid.

Eratosthenes of Cyrene was a working colleague and the recipient of The Method; the prefatory letter is one of the warmest pieces of personal address in the Greek mathematical corpus. Eratosthenes' measurement of the Earth's circumference and Archimedes' geometrical mechanics belong to the same Alexandrian generation and the same intellectual conversation.

Conon of Samos and his student Dositheus of Pelusium were Archimedes' most frequent correspondents in the surviving treatises. On the Sphere and Cylinder, On Spirals, and Quadrature of the Parabola are addressed to Dositheus after Conon's death; the Spiral was apparently a problem Conon had circulated.

Apollonius of Perga, a generation younger, took the conic sections, which Archimedes used as tools, and made them an object of systematic study in the Conics. Apollonius's work and Archimedes' work together define what "Hellenistic geometry" means.

Vitruvius, writing in the 1st century BCE, is the main Roman source for Archimedes' applied work (the gold-crown story, the screw, the planetarium) in De Architectura. Vitruvius's reliability is uneven, but his accounts are what shaped the Renaissance and modern popular image of Archimedes.

Plutarch, in Life of Marcellus 14–19, is the main narrative source for the siege of Syracuse and Archimedes' death. Plutarch's editorializing (the framing of Archimedes as too high-minded to write about engineering) colored centuries of reception, even though it conflicts with the surviving treatises.

Galileo Galilei made the explicit Archimedean revival programmatic. La Bilancetta (1586) reconstructs the gold-crown procedure as a hydrostatic balance; the proportional reasoning of Two New Sciences (1638) is openly Archimedean. Galileo called Archimedes il divino, and meant it as both compliment and method statement.

Leonardo da Vinci studied Archimedes carefully across the Codex Atlanticus and the Codex Madrid, sketched water-lifting screws and lever-pulley systems on Archimedean principles, repeatedly searched for a copy of the De Insidentibus Aquae (the Latin On Floating Bodies), and rejected perpetual-motion proposals on Archimedean grounds.

The Antikythera Mechanism sits about a century later than Archimedes but inside the Hellenistic mechanical-astronomical tradition that Cicero attributed to him via the "sphere of Archimedes" Marcellus brought to Rome. The mechanism is not Archimedes' work; it plausibly descends from a lineage he helped establish.

The figures and theorems Archimedes proved (the inscribed sphere and cylinder, the equiangular spiral, the parabolic mirror) sit inside the visual and symbolic canon of sacred geometry, even though Archimedes himself worked in a strictly demonstrative register that did not assign cosmological meaning to specific figures.