Archimedes

Archimedes of Syracuse ( AR-kih-MEE-deez; c. 287 – c. 212 BC), an Ancient Greek polymath originating from Syracuse, Sicily, distinguished himself as a mathematician, physicist, engineer, astronomer, and inventor. Despite the scarcity of biographical information, his extant works firmly establish him as a preeminent scientist of classical antiquity and one of history's most significant mathematicians. Archimedes notably foreshadowed modern calculus and analysis through his innovative application of infinitesimals and the method of exhaustion, which enabled him to rigorously derive and prove numerous geometric theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area beneath a parabola, the volume of a paraboloid of revolution segment, the volume of a hyperboloid of revolution segment, and the area of a spiral.

Archimedes of Syracuse ( AR-kih-MEE-deez; c. 287 – c. 212 BC) was an Ancient Greek mathematician, physicist, engineer, astronomer, and inventor from the city of Syracuse in Sicily. Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in classical antiquity, and one of the greatest mathematicians of all time. Archimedes anticipated modern calculus and analysis by applying the concept of the infinitesimals and the method of exhaustion to derive and rigorously prove many geometrical theorems, including the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral.

Further mathematical accomplishments by Archimedes encompass the derivation of an approximation for pi (π), the definition and exploration of the Archimedean spiral, and the creation of an exponential system for representing exceptionally large numbers. He was also among the first scholars to apply mathematical principles to physical phenomena, particularly in the fields of statics and hydrostatics. His contributions in this domain include a rigorous proof of the law of the lever, the widespread adoption of the center of gravity concept, and the articulation of the law of buoyancy, famously known as Archimedes' principle. In astronomy, he undertook measurements of the Sun's apparent diameter and estimations of the universe's scale. Tradition also attributes to him the construction of a planetarium that simulated the motions of known celestial bodies, potentially serving as an antecedent to the Antikythera mechanism. Moreover, he is credited with designing groundbreaking mechanical devices, such as his screw pump, compound pulleys, and defensive war machines engineered to safeguard Syracuse from military incursions.

Archimedes met his demise during the siege of Syracuse, killed by a Roman soldier despite explicit directives to ensure his safety. Cicero later recounted his

In contrast to the renown of his inventions, Archimedes' mathematical treatises received limited recognition during antiquity. While Alexandrian mathematicians engaged with and cited his work, the initial comprehensive compilation did not occur until c. 530AD, undertaken by Isidore of Miletus in Byzantine Constantinople. Concurrently, Eutocius' commentaries on Archimedes' works during the same century significantly broadened their accessibility. Throughout the Middle Ages, his writings were translated into Arabic in the 9th century and subsequently into Latin in the 12th century, becoming a pivotal intellectual resource for scholars during the Renaissance and the Scientific Revolution. The 1906 discovery of Archimedes' texts within the Archimedes Palimpsest has since offered unprecedented insights into his methodologies for achieving mathematical results.

Biography

The specifics of Archimedes' life remain largely enigmatic. Although Eutocius referenced a biography purportedly authored by Archimedes' associate, Heraclides Lembus, this work is no longer extant, and contemporary scholarship questions its original attribution to Heraclides.

Drawing upon the assertion by the Byzantine Greek scholar John Tzetzes that Archimedes lived for 75 years prior to his death in 212BC, his birth is estimated to have occurred c. 287 BC in Syracuse, Sicily, then a self-governing colony within Magna Graecia. In his treatise, Sand-Reckoner, Archimedes identifies his father as Phidias, an astronomer about whom no further information is available. While Plutarch, in his Parallel Lives, suggested a familial connection between Archimedes and King Hiero II of Syracuse, Cicero and Silius Italicus imply a more modest background. Details regarding his marital status, progeny, or any potential sojourn in Alexandria, Egypt, during his formative years remain unconfirmed. Nevertheless, his extant correspondence, addressed to Dositheus of Pelusium (a pupil of the Alexandrian astronomer Conon of Samos) and to the chief librarian Eratosthenes of Cyrene, indicates sustained collegial relationships with scholars in Alexandria. Specifically, in the preface to On Spirals, dedicated to Dositheus, Archimedes states that "many years have elapsed since Conon's death," with Conon of Samos having lived approximately 280–220 BC, suggesting Archimedes may have been of advanced age when composing certain works.

The Golden Wreath Problem

Among the problems Archimedes is credited with solving for Hiero II is the renowned "wreath problem." Vitruvius, writing approximately two centuries after Archimedes' demise, recounts that King Hiero II of Syracuse commissioned a golden wreath for a divine temple, providing the goldsmith with pure gold for its creation. The king, however, grew suspicious that the goldsmith had illicitly replaced some of the gold with cheaper silver and retained a portion of the pure metal. Unable to elicit a confession, Hiero II tasked Archimedes with the investigation. Subsequently, while entering a bath, Archimedes purportedly observed that the water level in the tub rose proportionally to his immersion. Recognizing that this phenomenon could ascertain the golden crown's volume, he was reportedly so elated that he ran naked through the streets, exclaiming "Eureka!" (meaning "I have found it!"), having forgotten to dress. Vitruvius further states that Archimedes proceeded to take a mass of gold and a mass of silver, each equivalent in weight to the wreath. By immersing each in the bathtub, he demonstrated that the wreath displaced more water than the pure gold but less than the pure silver, thereby proving that the wreath was an alloy of gold and silver.

An alternative narrative appears in the Carmen de Ponderibus, an anonymous 5th-century Latin didactic poem concerning weights and measures, which was formerly ascribed to the grammarian Priscian. According to this poem, masses of gold and silver were positioned on the pans of a balance, and the entire assembly was then submerged in water. The differential density between the gold and the silver, or between the gold and the crown, would consequently cause the balance to incline. In contrast to Vitruvius's more widely known bathtub anecdote, this poetic rendition employs the hydrostatic principle now recognized as Archimedes' principle. This principle, detailed in his treatise On Floating Bodies, posits that a body submerged in a fluid experiences an upward buoyant force equivalent to the weight of the fluid it displaces. Galileo Galilei, who in 1586 devised a hydrostatic balance influenced by Archimedes' contributions, deemed 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."

Launching the Syracusia

Much of Archimedes' engineering endeavors likely stemmed from addressing the requirements of his native city, Syracuse. Athenaeus of Naucratis, in his work Deipnosophistae, cites Moschion's description of King Hiero II's commission for the design of an immense vessel, the Syracusia. This ship is reputed to have been the largest constructed in classical antiquity and, as per Moschion's narrative, was launched by Archimedes. Plutarch presents a somewhat divergent account, recounting Archimedes' boast to Hiero that he possessed the capability to move any substantial weight, prompting Hiero to challenge him to move a ship. These narratives, however, incorporate numerous fantastical and historically improbable details. Furthermore, the authors offer conflicting explanations for how this feat was achieved: Plutarch asserts that Archimedes devised a block-and-tackle pulley system, whereas Hero of Alexandria attributed the same claim to Archimedes' invention of the baroulkos, a type of windlass. Pappus of Alexandria, conversely, ascribed this accomplishment to Archimedes' application of mechanical advantage, specifically the principle of leverage, to lift objects that would otherwise have been immovably heavy. He attributed to Archimedes the frequently cited declaration: "Give me a place to stand on, and I will move the Earth."

Athenaeus, possibly misinterpreting details from Hero's description of the baroulkos, also records Archimedes' use of a "screw" to extract any water potentially leaking into the hull of the Syracusia. While this apparatus is occasionally termed Archimedes' screw, it most probably predates him considerably. Notably, none of his immediate contemporaries who documented its application (including Philo of Byzantium, Strabo, and Vitruvius) attribute its invention or primary use to him.

War Machines

Archimedes' most significant ancient renown stemmed from his pivotal role in defending Syracuse against Roman forces during its siege. Plutarch recounts that Archimedes had engineered formidable war machines for Hiero II, though these devices remained unused during Hiero's lifetime. Nevertheless, in 214 BC, amidst the Second Punic War, Syracuse shifted its allegiance from Rome to Carthage. When the Roman army, led by Marcus Claudius Marcellus, subsequently attempted to capture the city, Archimedes reportedly directed the deployment of these war machines, substantially impeding the Roman advance. The city ultimately fell only after an extended siege. Accounts from three distinct historians—Plutarch, Livy, and Polybius—corroborate the existence of these military innovations, detailing enhanced catapults and cranes designed to either drop heavy lead projectiles onto Roman vessels or employ an iron claw to hoist ships from the water before submerging them.

A considerably less substantiated narrative, absent from the earliest historical records by Plutarch, Polybius, or Livy, posits that Archimedes employed "burning mirrors" to concentrate solar rays onto invading Roman ships, thereby igniting them. The initial mention of ships being set ablaze, attributed to the 2nd-century CE satirist Lucian of Samosata, makes no reference to mirrors, merely stating that the vessels were ignited through artificial methods, potentially suggesting the use of incendiary projectiles. Galen, writing later in the same century, is the first author to explicitly mention mirrors in this context. Approximately four centuries after Lucian and Galen, Anthemius, despite expressing skepticism, endeavored to reconstruct Archimedes' theoretical reflector geometry. This alleged apparatus, occasionally termed "Archimedes' heat ray," has been a subject of continuous scholarly debate regarding its veracity since the Renaissance. René Descartes dismissed the account as fictitious, whereas contemporary researchers have attempted to replicate the effect using only technologies available in Archimedes' era, yielding inconclusive outcomes.

Death

The circumstances surrounding Archimedes' death during the Roman sack of Syracuse are detailed in several disparate historical accounts. The earliest narrative, provided by Livy, states that Archimedes was killed by a Roman soldier, unaware of his identity, while engrossed in drawing geometric figures in the dust. Plutarch offers two distinct versions: in one, a soldier demanded Archimedes accompany him, but Archimedes refused, insisting on completing his mathematical problem, whereupon the soldier slew him with his sword. In Plutarch's alternative account, Archimedes was carrying mathematical instruments when he was killed by a soldier who mistook them for valuable possessions. Valerius Maximus, a Roman writer flourishing around 30 AD, recorded in his work Memorable Doings and Sayings that Archimedes' final utterance, as he was killed by the soldier, was "... but protecting the dust with his hands, said 'I beg of you, do not disturb this.'" This statement bears resemblance to the widely attributed, though historically unsubstantiated, last words, "Do not disturb my circles."

Marcellus was reportedly incensed by Archimedes' demise, having regarded him as an invaluable scientific resource—even referring to him as "a geometrical Briareus"—and had issued explicit orders for his protection. Cicero (106–43 BC) records that Marcellus transported two planetariums, constructed by Archimedes, to Rome. These devices depicted the movements of the Sun, Moon, and five planets; one was subsequently donated to the Temple of Virtue in Rome, while the other was purportedly retained by Marcellus as his sole personal acquisition from Syracuse. Pappus of Alexandria references a now-lost treatise by Archimedes, titled On Sphere-Making, which may have detailed the construction of such mechanisms. The engineering of these intricate devices would have necessitated an advanced understanding of differential gearing, a capability once believed to be beyond the technological scope of antiquity. However, the 1902 discovery of the Antikythera mechanism, another apparatus built around c. 100 BC with a comparable function, has substantiated that such sophisticated devices were indeed known to the ancient Greeks, leading some scholars to consider Archimedes' creations as precursors.

During his tenure as a quaestor in Sicily, Cicero located what was believed to be Archimedes' tomb near the Agrigentine Gate in Syracuse, in a state of disrepair and obscured by vegetation. He arranged for the tomb's restoration, which revealed a carving and legible inscribed verses. Notably, the tomb featured a sculpture depicting Archimedes' preferred mathematical proof: that the volume and surface area of a sphere constitute two-thirds of an encompassing cylinder, including its bases.

Mathematics

Although frequently recognized for his mechanical inventions, Archimedes also significantly advanced the field of mathematics by both extending the methodologies of his predecessors to derive novel outcomes and by pioneering his own innovative approaches.

Method of exhaustion

In Quadrature of the Parabola, Archimedes references a proposition from Euclid's Elements, which establishes that a circle's area is proportional to its diameter. This proposition was demonstrated using a lemma now termed the Archimedean property: “the excess by which the greater of two unequal regions exceed the lesser, if added to itself, can exceed any given bounded region.” Before Archimedes, Eudoxus of Cnidus and other ancient mathematicians utilized this lemma, a technique subsequently known as the "method of exhaustion," to determine the volumes of various geometric solids, including the tetrahedron, cylinder, cone, and sphere. The proofs for these calculations are detailed in Book XII of Euclid's Elements.

Within Measurement of a Circle, Archimedes utilized this method to demonstrate that a circle's area is equivalent to that of a right triangle with a base equal to the circle's radius and a height equal to its circumference. He subsequently approximated the ratio between the radius and the circumference, represented by π, by inscribing a regular hexagon within a circle and circumscribing another regular hexagon around it. He then iteratively doubled the number of sides of each regular polygon, meticulously calculating the side length of each polygon at every stage. This iterative process, increasing the number of sides, yielded progressively more accurate approximations of the circle. Following four such iterations, when the polygons reached 96 sides, he established that the value of π was bounded between 3⁠§89§/§1213§⁠ (approximately 3.1429) and 3⁠§1819§/71⁠ (approximately 3.1408), a range consistent with the actual value of approximately 3.1416. Furthermore, in the same work, he posited that the square root of 3 falls between ⁠265/153⁠ (approximately 1.7320261) and ⁠1351/780⁠ (approximately 1.7320512), likely derived through an analogous methodology.

Within Quadrature of the Parabola, Archimedes applied this method to demonstrate that the area bounded by a parabola and a straight line is ⁠4/§89§⁠ times the area of an equivalent inscribed triangle, as depicted in the accompanying figure. He articulated this solution as an infinite geometric series with a common ratio of ⁠§1415§/§1819§⁠:

∑<!-- ∑ --> n = §1516§ ∞ §2526§ − n = §3738§ + §4243§ − §4849§ + §5556§ − §6162§ + §6869§ − §7475§ + ⋯ = §8788§ §8990§ . {\displaystyle \sum _{n=0}^{\infty }4^{-n}=1+4^{-1}+4^{-2}+4^{-3}+\cdots ={4 \over 3}.\;}

The first term in this series represents the area of the initial triangle, while the second term corresponds to the sum of the areas of two smaller triangles. These smaller triangles have bases formed by the two smaller secant lines, and their third vertex is located at the intersection of the parabola with a line parallel to its axis, passing through the midpoint of the base. This iterative process continues. The proof utilizes a variation of the geometric series 1/4 + 1/16 + 1/64 + 1/256 + · · ·, which converges to ⁠§45§/§89§⁠.

Archimedes further applied this technique to determine the surface areas of spheres and cones, compute the area of ellipses, and ascertain the region enclosed by an Archimedean spiral.

Mechanical Method

It is more practical to furnish a proof when one already possesses some understanding of the subject matter, acquired through the method, than to undertake an investigation with no prior knowledge.

Beyond refining the method of exhaustion, which built upon the contributions of preceding mathematicians, Archimedes innovated a distinct technique that employed the principle of the lever to physically determine the areas and volumes of geometric figures. An initial outline of this proof appears in Quadrature of the Parabola, presented alongside the geometric demonstration, but a more comprehensive exposition is provided in The Method of Mechanical Theorems. Archimedes himself stated that he initially derived results in his mathematical works using this mechanical method, subsequently working in reverse to apply the method of exhaustion only after an approximate value for the solution had been established.

Large Numbers

Archimedes also devised methodologies for the representation of exceptionally large numbers.

In his treatise The Sand Reckoner, Archimedes developed a numerical system founded on the myriad (the Greek term for 10,000) to quantify a number exceeding the estimated grains of sand required to fill the cosmos. He posited a number system employing powers of a myriad of myriads (equivalent to 100 million, or 10,000 × 10,000) and determined that the quantity of sand grains necessary to fill the universe would be 8 vigintillion, or 8×10⁶³. Through this endeavor, he effectively illustrated the capacity of mathematics to represent arbitrarily vast quantities.

The Cattle Problem presents a challenge from Archimedes to the mathematicians of the Library of Alexandria, tasking them with enumerating the cattle in the Herd of the Sun, a task requiring the solution of multiple simultaneous Diophantine equations. A more complex variant of this problem mandates that certain solutions must be perfect squares, yielding an exceptionally large numerical answer, approximately 7.760271×10²⁰⁶⁵⁴⁴.

Archimedean Solid

In a now-lost treatise, documented by Pappus of Alexandria, Archimedes demonstrated the existence of precisely thirteen semiregular polyhedra.

Writings

Archimedes disseminated his mathematical findings through correspondence with scholars in Alexandria, with these original communications composed in Doric Greek, the dialect prevalent in ancient Syracuse.

Surviving Works

The subsequent list is arranged chronologically, adhering to the updated terminological and historical criteria established by Knorr (1978) and Sato (1986).

Measurement of a Circle

This concise treatise comprises three propositions. It is structured as a correspondence addressed to Dositheus of Pelusium, a pupil of Conon of Samos. In Proposition II, Archimedes provides an approximation for the value of pi (π), demonstrating that it lies between ⁠223/71⁠ (approximately 3.1408) and ⁠22/§1819§⁠ (approximately 3.1428).

The Sand Reckoner

Within this treatise, also referred to as Psammites, Archimedes calculates a number exceeding the estimated quantity of sand grains required to fill the universe. The work references the heliocentric model of the Solar System, as advanced by Aristarchus of Samos, alongside prevailing theories concerning Earth's dimensions, the distances between celestial objects, and efforts to ascertain the Sun's apparent diameter. Employing a numerical system founded on powers of the myriad, Archimedes deduces that the total number of sand grains needed to fill the universe amounts to 8×10⁶³ in contemporary scientific notation. The introductory epistle identifies Archimedes' father as Phidias, an astronomer. Notably, The Sand Reckoner stands as the sole extant work where Archimedes articulates his astronomical perspectives.

In the Sand-Reckoner, Archimedes examines astronomical measurements pertaining to the Earth, Sun, and Moon, alongside Aristarchus' heliocentric model of the universe. Lacking trigonometry or a table of chords, Archimedes ascertained the Sun's apparent diameter by first detailing the observational methodology and instrumentation (a straight rod with pegs or grooves), subsequently applying corrective factors to these empirical data, and finally presenting the outcome as a range defined by upper and lower bounds, thereby accommodating potential observational inaccuracies.

Ptolemy, citing Hipparchus, also alludes to Archimedes' solstice observations in the Almagest. Consequently, Archimedes is recognized as the earliest Greek scholar to document multiple solstice dates and times in successive years.

On the Equilibrium of Planes

The treatise On the Equilibrium of Planes comprises two volumes: the initial volume presents seven postulates and fifteen propositions, whereas the subsequent volume includes ten propositions. Within the first volume, Archimedes rigorously demonstrates the law of the lever, which states that:

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

Preceding formulations of the lever principle appear in a work by Euclid and in the Mechanical Problems, a text associated with the Peripatetic school, adherents of Aristotle, its authorship occasionally ascribed to Archytas.

Archimedes applies these derived principles to determine the areas and centers of gravity of diverse geometric configurations, such as triangles, parallelograms, and parabolas.

Quadrature of the Parabola

Comprising 24 propositions and dedicated to Dositheus, this treatise demonstrates through two distinct methodologies that the region bounded by a parabola and a secant line constitutes four-thirds the area of a triangle possessing an equivalent base and height. This achievement is realized through two approaches: initially, by employing the principle of the lever, and subsequently, by computing the sum of an infinite geometric series with a common ratio of 1/4.

On the Sphere and Cylinder

Within this two-volume treatise, also dedicated to Dositheus, Archimedes derives his most celebrated finding: the fundamental relationship between a sphere and its circumscribing cylinder, provided they share identical height and diameter. Specifically, the sphere's volume is calculated as ⁠4/§67§⁠πr§1617§, while the cylinder's volume is 2πr§2425§. The surface area of the sphere is determined to be 4πr§3233§, and for the cylinder (inclusive of its two bases), it is 6πr§4041§, where r denotes the common radius of both the sphere and the cylinder.

On Spirals

Comprising 28 propositions, this treatise is similarly dedicated to Dositheus. It formally introduces the curve now recognized as the Archimedean spiral. This spiral is characterized as the locus of points generated by a point that moves uniformly away from a fixed origin along a line simultaneously rotating at a constant angular velocity. In contemporary polar coordinates (r, θ), its mathematical representation is given by the equation ${\displaystyle \,r=a+b\theta }$, where a and b are real constants.

This represents an early instance of a mechanical curve (defined as a curve generated by a moving point) investigated by a Hellenic mathematician.

On Conoids and Spheroids

This treatise, comprising 32 propositions, is dedicated to Dositheus. Within this text, Archimedes computes the surface areas and volumes of various sections derived from cones, spheres, and paraboloids.

On Floating Bodies

The work On Floating Bodies is divided into two books. In the initial volume, Archimedes articulates the principles governing fluid equilibrium and demonstrates that water naturally assumes a spherical configuration around its center of gravity.

This treatise presents Archimedes' principle of buoyancy, articulated as follows:

Any body wholly or partially immersed in fluid experiences an upthrust equal to, but opposite in direction to, the weight of the fluid displaced.

The second part involves the calculation of equilibrium positions for various sections of paraboloids. This analysis likely served as an idealization for the forms of ship hulls. Certain sections are depicted floating with their base submerged and their apex above the water, analogous to the buoyancy observed in icebergs.

Ostomachion

Alternatively referred to as Loculus of Archimedes or Archimedes' Box, this constitutes a dissection puzzle resembling a Tangram. The associated treatise was discovered in a more comprehensive state within the Archimedes Palimpsest. Archimedes computed the areas of its 14 constituent pieces, which are capable of being arranged to construct a square. In 2003, Reviel Netz from Stanford University posited that Archimedes' objective was to ascertain the total number of configurations in which these pieces could be assembled to form a square. Netz's calculations indicate that 17,152 distinct arrangements of the pieces can result in a square. Excluding solutions deemed equivalent through rotation and reflection, the total number of unique arrangements is 536. This puzzle exemplifies an early challenge within the field of combinatorics.

The etymology of the puzzle's designation remains ambiguous; however, it has been proposed that its derivation stems from the Ancient Greek term for "throat" or "gullet," stomachos (στόμαχος). Ausonius referred to the puzzle as Ostomachion, a Greek compound term constructed from the lexical roots of osteon (ὀστέον, 'bone') and machē (μάχη, 'fight').

The cattle problem

Within this treatise, directed to Eratosthenes and the Alexandrian mathematicians, Archimedes presented them with the challenge of enumerating the cattle within the Herd of the Sun, a task necessitating the resolution of multiple simultaneous Diophantine equations. In 1773, Gotthold Ephraim Lessing identified this work within a Greek manuscript, comprising a 44-line poem, housed in the Herzog August Library in Wolfenbüttel, Germany. A more complex variant of the problem exists, wherein certain solutions must be perfect squares. A. Amthor provided the initial solution to this particular problem version in 1880, yielding an exceptionally large numerical result, approximately 7.760271×10²⁰⁶⁵⁴⁴.

The Method of Mechanical Theorems

Similar to The Cattle Problem, The Method of Mechanical Theorems was composed as an epistolary communication addressed to Eratosthenes in Alexandria.

Within this treatise, Archimedes employs an innovative methodology, an incipient manifestation of Cavalieri's principle, to re-establish the findings from the treatises dispatched to Dositheus (Quadrature of the Parabola, On the Sphere and Cylinder, On Spirals, On Conoids and Spheroids), which he had previously substantiated using the method of exhaustion. This involved applying the law of the lever, as detailed in On the Equilbrium of Planes, initially to determine an object's center of gravity, and subsequently employing geometric reasoning to facilitate the derivation of its volume. Archimedes explicitly indicates that he utilized this approach to deduce the results presented in the treatises sent to Dositheus prior to their more rigorous proof via the method of exhaustion, asserting the utility of knowing a result's veracity before undertaking its rigorous demonstration. This is analogous to how Eudoxus of Cnidus was assisted in demonstrating that a cone's volume is one-third that of a cylinder, by virtue of Democritus having previously affirmed this truth, based on the argument that a pyramid's volume is one-third that of a rectangular prism with an identical base.

This treatise was presumed lost until the 1906 discovery of the Archimedes Palimpsest.

Apocryphal works

Archimedes' Book of Lemmas, also known as Liber Assumptorum, comprises a treatise containing 15 propositions concerning the properties of circles. The earliest extant manuscript of this text is in Arabic. T. L. Heath and Marshall Clagett contended that its present form precludes Archimedean authorship, given that it cites Archimedes, thereby implying subsequent modification by a different author. It is plausible that the Lemmas are derived from a now-lost earlier work by Archimedes.

Additional works of dubious attribution to Archimedes encompass the 4th or 5th-century Latin poem Carmen de ponderibus et mensuris, which details the application of a hydrostatic balance for resolving the crown problem, and the 12th-century text Mappae clavicula, providing instructions for assaying metals through the computation of their specific gravities.

Lost works

Many of Archimedes' written works have either not survived or exist solely as heavily edited fragments. For instance, Pappus of Alexandria references On Sphere-Making, a treatise on semiregular polyhedra, and another on spirals. Similarly, Theon of Alexandria cites a comment on refraction from the currently lost work, Catoptrica. The treatise Principles, dedicated to Zeuxippus, elucidated the numerical system employed in The Sand Reckoner. Other notable works include On Balances and On Centers of Gravity.

Medieval Islamic scholars ascribed to Archimedes a formula for determining a triangle's area based on its side lengths. This formula is now recognized as Heron's formula, attributed to its initial documented appearance in the 1st-century AD writings of Heron of Alexandria. It is conjectured that Archimedes might have proven this formula in a now-lost treatise.

The Archimedes Palimpsest

In 1906, Danish professor Johan Ludvig Heiberg traveled to Constantinople to inspect a 174-page goatskin parchment containing prayers from the 13th century. His Heiberg verified that the document was a palimpsest, characterized by text inscribed over an earlier, erased work. The creation of palimpsests, involving the scraping of ink from existing manuscripts for reuse, was a prevalent practice during the Middle Ages due to the high cost of vellum. Scholars subsequently identified the underlying texts within this palimpsest as 10th-century copies of Archimedes' previously lost treatises. The palimpsest contains seven treatises, notably including the sole extant copy of On Floating Bodies in its original Greek. Furthermore, it represents the only known source for The Method of Mechanical Theorems, a work mentioned by Suidas and previously presumed irrevocably lost. The Stomachion was also found within the palimpsest, offering a more comprehensive analysis of the puzzle than earlier textual discoveries.

The Archimedes Palimpsest contains the following treatises:

• On the Equilibrium of Planes
• On Spirals
• Measurement of a Circle
• On the Sphere and Cylinder
• On Floating Bodies
• The Method of Mechanical Theorems
• Stomachion
• Orations by the 4th-century BCE politician Hypereides
• A critical commentary on Aristotle's Categories
• Additional Works

The parchment remained in a monastic library in Constantinople for centuries before its acquisition by a private collector in the 1920s. On October 29, 1998, it was auctioned to an undisclosed buyer for $2.2 million. Subsequently, the palimpsest was housed at the Walters Art Museum in Baltimore, Maryland, where it underwent various advanced examinations, including ultraviolet and X-ray imaging, to decipher the underlying text. It has since been returned to its anonymous proprietor.

Legacy

Often referred to as the progenitor of mathematics and mathematical physics, Archimedes is almost universally acknowledged by historians of science and mathematics as the preeminent mathematician of antiquity.

Classical Antiquity

Archimedes' renown for mechanical innovations during classical antiquity is extensively documented. Athenaeus, in his Deipnosophistae, details Archimedes' oversight of the construction of the Syracusia, the largest known ship of antiquity, while Apuleius discusses his contributions to catoptrics. Although Plutarch asserted that Archimedes held mechanics in disdain, prioritizing pure geometry, contemporary scholarship largely dismisses this as a misrepresentation. This perspective is believed to have been constructed to reinforce Plutarch's Platonist philosophical tenets rather than to accurately portray Archimedes. Furthermore, in contrast to his inventions, Archimedes' mathematical treatises received limited recognition in antiquity beyond the circles of Alexandrian mathematicians. The initial comprehensive compilation of his works was not undertaken until approximately c. 530AD by Isidore of Miletus in Byzantine Constantinople. Concurrently, Eutocius' commentaries on Archimedes' writings, produced earlier in the same century, significantly broadened their accessibility to a wider audience.

Middle Ages

Archimedes' corpus was translated into Arabic by Thābit ibn Qurra (836–901 AD) and subsequently into Latin from Arabic by Gerard of Cremona (c. 1114–1187). Later, direct translations from Greek into Latin were executed by William of Moerbeke (c. 1215–1286) and Iacobus Cremonensis (c. 1400–1453).

Renaissance and Early Modern Europe

The Editio princeps (First Edition) of Archimedes' works, published in Basel in 1544 by Johann Herwagen, presented his writings in both Greek and Latin. This publication served as a significant intellectual resource for scientists throughout the Renaissance and into the 17th century.

Leonardo da Vinci frequently voiced his admiration for Archimedes, even crediting him with the invention of the Architonnerre. Galileo Galilei lauded Archimedes as "superhuman" and "my master," while Christiaan Huygens declared, "I think Archimedes is comparable to no one," deliberately modeling his early endeavors after him. Gottfried Wilhelm Leibniz observed, "He who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times."

Italian numismatist and archaeologist Filippo Paruta (1552–1629), along with Leonardo Agostini (1593–1676), documented a bronze coin discovered in Sicily. This coin featured a portrait of Archimedes on its obverse and a cylinder and sphere, accompanied by the Latin monogram ARMD, on its reverse. Although the coin's current whereabouts are unknown and its precise minting date remains unestablished, Ivo Schneider characterized the reverse imagery as "a sphere resting on a base – probably a rough image of one of the planetaria created by Archimedes." Schneider further hypothesized that the coin might have been struck in Rome for Marcellus, who, "according to ancient reports, brought two spheres of Archimedes with him to Rome."

In Modern Mathematics

Carl Friedrich Gauss held Archimedes and Isaac Newton in high esteem; Moritz Cantor, a student of Gauss at the University of Göttingen, recounted Gauss's observation that "there had been only three epoch-making mathematicians: Archimedes, Newton, and Eisenstein." Similarly, Alfred North Whitehead asserted that "in the year 1500 Europe knew less than Archimedes who died in the year 212 BC." Reviel Netz, a historian of mathematics, echoed Whitehead's famous statement regarding Plato and philosophy by declaring that "Western science is but a series of footnotes to Archimedes," further designating him "the most important scientist who ever lived." Eric Temple Bell also noted 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 1906 discovery of Archimedes' previously lost works within the Archimedes Palimpsest has yielded novel insights into his methods for deriving mathematical results.

The Fields Medal, awarded for exceptional accomplishments in mathematics, features a portrait of Archimedes alongside an engraving depicting his proof concerning the sphere and the cylinder. Encircling Archimedes' head is a Latin inscription, attributed to the 1st-century AD poet Manilius, which states: Transire suum pectus mundoque potiri ("Rise above oneself and grasp the world").

Cultural Influence

The SS Archimedes, launched in 1839, holds the distinction of being the world's inaugural seagoing steamship equipped with a screw propeller, named in homage to Archimedes and his contributions to the understanding of the screw mechanism.

Archimedes has been featured on postage stamps issued by various nations, including East Germany (1973), Greece (1983), Italy (1983), Nicaragua (1971), San Marino (1982), and Spain (1963).

The exclamation "Eureka!", famously attributed to Archimedes, serves as the state motto of California. In this specific context, the term signifies the discovery of gold near Sutter's Mill in 1848, an event that precipitated the California Gold Rush.

A lunar crater, Archimedes (29.7°N 4.0°W / 29.7; -4.0), and a lunar mountain range, the Montes Archimedes (25.3°N 4.6°W / 25.3; -4.6), are named in his honor on the Moon.

Arbelos

• Arbelos
• Archimedean Point
• Archimedes Number
• Archimedes Paradox
• Methods for Computing Square Roots
• Salinon
• Steam Cannon
• Twin Circles
• Zhang Heng

Notes

Footnotes

Citations

References

Ancient Testimony

• Plutarch, *Life of Marcellus*
• "Athenaeus, Deipnosophistae". . Retrieved March 7, 2023.Modern Sources
  • Acerbi, Fabio (2018). "Hellenistic Mathematics". In Keyser, Paul T; Scarborough, John (eds.). Oxford Handbook of Science and Medicine in the Classical World. pp. 268–292. doi:10.1093/oxfordhb/9780199734146.013.69. ISBN 978-0-19-973414-6. Retrieved 26 May 2021.Dijksterhuis, E. J. (Eduard Jan) (1987). Archimedes. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-08421-3. Retrieved 30 April 2025.Netz, Reviel (2022). A New History of Greek Mathematics. Cambridge University Press. ISBN 978-1-108-83384-4.Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages, 5 volumes. Madison, WI: University of Wisconsin Press.
    • Clagett, Marshall. 1964–1984. Archimedes in the Middle Ages 5 volumes. Madison, WI: University of Wisconsin Press.
    • Clagett, Marshall. 1970. "Archimedes". In Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography, Vol. 1 (Abailard–Berg), pp. 213–231. New York: Charles Scribner's Sons.
    • Gow, Mary. 2005. Archimedes: Mathematical Genius of the Ancient World. Enslow Publishing. ISBN 978-0-7660-2502-8.
    • Hasan, Heather. 2005. Archimedes: The Father of Mathematics. Rosen Central. ISBN 978-1-4042-0774-5.
    • Netz, Reviel. 2004–2017. The Works of Archimedes: Translation and Commentary, 1–2. Cambridge University Press. Vol. 1: "The Two Books on the Sphere and the Cylinder". ISBN 978-0-521-66160-7. Vol. 2: "On Spirals". ISBN 978-0-521-66145-4.
    • Netz, Reviel, and William Noel. 2007. The Archimedes Codex. Orion Publishing Group. ISBN 978-0-297-64547-4.
    • Pickover, Clifford A. 2008. Archimedes to Hawking: Laws of Science and the Great Minds Behind Them. Oxford University Press. ISBN 978-0-19-533611-5.
    • Simms, Dennis L. 1995. Archimedes the Engineer. Continuum International Publishing Group. ISBN 978-0-7201-2284-8.
    • Stein, Sherman. 1999. Archimedes: What Did He Do Besides Cry Eureka?. Mathematical Association of America. ISBN 978-0-88385-718-2.
    • Heiberg's Edition of Archimedes. Texts in Classical Greek, with some in English.
    • Works by Archimedes at Project Gutenberg
    • Archimedes at the Indiana Philosophy Ontology Project
    • The Archimedes Palimpsest project at The Walters Art Museum in Baltimore, Maryland
    • "Archimedes on Spheres and Cylinders". MathPages.com.Source: TORIma Academy Archive