Archimedes

Greek mathematician and engineer

• Born: c. 287 b.c.e.
• Birthplace: Syracuse, Sicily (now in Italy)
• Died: 212 b.c.e.
• Place of death: Syracuse, Sicily (now in Italy)

The greatest mathematician of antiquity, Archimedes did his best work in geometry and also founded the disciplines of statics and hydrostatics.

Early Life

Few details are certain about the life of Archimedes (ar-kuh-MEED-eez). The birth date of 287 b.c.e. was established from a report, about fourteen hundred years after the fact, that he was seventy-five years old at his death in 212 b.c.e. Ancient writers agree in calling him a Syracusan by birth, and he himself provides the information that his father was the astronomer Phidias, the author of a treatise on the diameters of the sun and moon. His father’s profession suggests an explanation for the son’s early interest in astronomy and mathematics. Some scholars have characterized Archimedes as an aristocrat who actively participated in the Syracusan court and who may have been related to King Hiero II, the ruler of Syracuse. He certainly was friendly with Hiero and Hiero’s son Gelon, to whom he dedicated one of his works. (Original titles of Archimedes’ works are not known, but most of his books were first translated into English by Thomas L. Heath in 1897 in the volume The Works of Archimedes.)

Archimedes traveled to Egypt to study in Alexandria, then the center of the scientific world. Some of his teachers had, in their youth, been students of Euclid. He made two close friends in Alexandria: Conon of Samos, a gifted mathematician, and Eratosthenes of Cyrene, also a good mathematician. From the prefaces to his works, it is clear that Archimedes maintained friendly relations with several Alexandrian scholars, and he played an active role in developing the mathematical traditions of this intellectual center. It is possible that he visited Spain before returning to Syracuse, and a return trip to Egypt is also a possibility. This second visit would have been the occasion for his construction of dikes and bridges reported in some Arabian sources.

In Syracuse, Archimedes spent his time working on mathematical and mechanical problems. Although he was a remarkably ingenious inventor, his inventions were, according to Plutarch, merely diversions, the work of a geometer at play. He possessed such a lofty intellect that he considered these inventions of much less worth than his mathematical creations. Plutarch may have exaggerated Archimedes’ distaste for engineering, because there is evidence that he was fascinated by mechanical problems from a practical as well as a theoretical point of view.

In the stories that multiplied about him, Archimedes became a symbol of the learned man—absentminded and unconcerned with food, clothing, and the other necessities of life. In images created long after his death, he is depicted as the quintessential sage, with a heavily bearded face, massive forehead, and contemplative mien. He had a good sense of humor. For example, he often sent his theorems to Alexandria, but to play a trick on some conceited mathematicians there, he once slipped in a few false propositions, so that these individuals, who pretended to have discovered everything by themselves, would fall into the trap of proposing theorems that were impossible.

Life’s Work

The range of Archimedes’ interest was wide, encompassing statics, hydrostatics, optics, astronomy, and engineering, in addition to geometry and arithmetic. It is natural that stories should tell more about his engineering inventiveness than his mathematical ability, for clever machines appealed to the average mind more than abstract mathematical theorems. Unfortunately, many of these stories are doubtful. For example, Archimedes is supposed to have invented a hollow, helical cylinder that, when rotated, could serve as a water pump, but this device, now called the Archimedean screw, antedates its supposed inventor.

In another well-known story, Archimedes boasted to King Hiero that, if he had a place on which to stand, he could move the earth. Hiero urged him to make good this boast by hauling ashore a fully loaded, three-masted merchantman of the royal fleet. Using a compound pulley, Archimedes, with modest effort, pulled the ship out of the harbor and onto the shore. The compound pulley may have been Archimedes’ invention, but the story, told by Plutarch, is probably a legend.

The most famous story about Archimedes is attributed to Vitruvius, a Roman architect under Emperor Augustus. King Hieron, grateful for the success of one of his ventures, wanted to thank the gods by consecrating a golden wreath. On delivery, the wreath had the weight of the gold supplied for it, but Hiero suspected that it had been adulterated with silver. Unable to make the goldsmith confess, Hiero asked Archimedes to devise some way of testing the wreath. Because it was a consecrated object, Archimedes could not subject it to chemical analysis. He pondered the problem without success until one day, when he entered a full bath, he noticed that the deeper he descended into the tub, the more water flowed over the edge. This suggested to him that the amount of overflowed water was equal in volume to the portion of his body submerged in the bath. This observation gave him a way of solving the problem, and he was so overjoyed that he leapt out of the tub and ran home naked through the streets, shouting: “Eureka! Eureka!” Vitruvius then goes on to explain how Archimedes made use of his newly gained insight. By putting the wreath into water, he could tell by the rise in water level the volume of the wreath. He also dipped into water lumps of gold and silver, each having the same weight as the wreath. He found that the wreath caused more water to overflow than the gold and less than the silver. From this experiment, he determined the amount of silver admixed with the gold in the wreath.

As amusing and instructive as these legends are, much more reliable and interesting to modern historians of science are Archimedes’ mathematical works. These treatises can be divided into three groups: studies of figures bounded by curved lines and surfaces, works on the geometrical analysis of statical and hydrostatical problems, and arithmetical works. The form in which these treatises have survived is not the form in which they left Archimedes’ hand: They have all undergone transformations and emendations. Nevertheless, one still finds the spirit of Archimedes in the intricacy of the questions and the lucidity of the explanations.

In finding the areas of plane figures bounded by curved lines and the volumes of solid figures bounded by curved surfaces, Archimedes used a method originated by Eudoxus of Cnidus, unhappily called the “method of exhaustion.” This indirect proof involves inscribing and circumscribing polygons to approach a length, area, or volume. The name “exhaustion” is based on the idea that, for example, a circle would finally be exhausted by inscribed polygons with a growing number of sides. In Peri sphairas kai kylindron (c. 240 b.c.e.; On the Sphere and the Cylinder, 1897), Archimedes compares perimeters of inscribed and circumscribed polygons to prove that the volume of a sphere is two-thirds the volume of its circumscribed cylinder. He also proves that the surface of any sphere is four times the area of its greatest circle.

Having successfully applied this method to the sphere and cylinder, Archimedes went on to use the technique for many other figures, including spheroids, spirals, and parabolas. Peri konoeideon kai sphaireodeon (c. 240 b.c.e.; On Conoids and Spheroids, 1897) treats the figures of revolution generated by conics. His spheroids are what are now called oblate and prolate spheroids, which are figures of revolution generated by ellipses. Archimedes’ object in this work was the determination of volumes of segments cut off by planes from these conoidal and spheroidal solids. In Peri helikon (c. 240 b.c.e.; On Spirals, 1897), Archimedes studies the area enclosed between successive whorls of a spiral. He also defines a figure, now called Archimedes’ spiral: If a ray from a central point rotates uniformly about this point, like the hand of a clock, and if another point moves uniformly along this line (marked by the clock hand), starting at the central point, then this linearly moving and rotating point will trace Archimedes’ spiral.

Tetragonismos ten tou orthogonion konoy tomes (c. 250 b.c.e.; On the Quadrature of the Parabola, 1897), when translated, is not Archimedes’ original title for the treatise, as “parabola” was not used in the sense of a conic section in the third century b.c.e. On the other hand, quadrature is an ancient term: It denotes the process of constructing a square equal in area to a given surface, in this case a parabolic segment. Archimedes, in this treatise, proves the theorem that the area of a parabolic segment is four-thirds the area of its greatest inscribed triangle. He was so fond of this theorem that he gave different proofs for it. One proof uses a method of exhaustion in which the parabolic segment is exhausted by a series of triangles. The other consists of establishing the quadrature of the parabola by mechanically balancing elements of the unknown area against elements of a known area. This latter method gives an insight into how Archimedes discovered theorems to be proved. His most recently discovered work, Peri tōn mechanikon theorematon (c. 250 b.c.e.; On the Method of Mechanical Theorems, 1912), provides other examples of how Archimedes mathematically balanced geometrical figures as if they were on a weighing balance. He did not consider that this mechanical method constituted a demonstration, but it allowed him to find interesting theorems, which he then proved by more rigorous geometrical methods.

Archimedes also applied geometry to statics and hydrostatics successfully. In his Epipledon isorropion (c. 250 b.c.e.; On the Equilibrium of Planes, 1897), he proves the law of the lever geometrically and then puts it to use in finding the centers of gravity of several thin sheets of different shapes. By center of gravity, Archimedes meant the point at which the object can be supported so as to be in equilibrium under the pull of gravity. Earlier Greek mathematicians had made use of the principle of the lever in showing that a small weight at a large distance from a fulcrum would balance a large weight near the fulcrum, but Archimedes worked this principle out in mathematical detail. In his proof, the weights become geometrical magnitudes acting perpendicularly to the balance beam, which itself is conceived as a weightless geometrical line. In this way, he reduced statics to a rigorous discipline comparable to what Euclid had done for geometry.

Archimedes once more emphasizes geometrical analysis in Peri ochoymenon (c. 230 b.c.e.; On Floating Bodies, 1897). The cool logic of this treatise contrasts with his emotional discovery of the buoyancy principle. In this work, he proves that solids lighter than a fluid will, when placed in the fluid, sink to the depth where the weight of the solid will be equal to the weight of the fluid displaced. Solids heavier than the fluid will, when placed in the fluid, sink to the bottom, and they will be lighter by the weight of the displaced fluid.

Although Archimedes’ investigations were primarily in geometry and mechanics, he did perform some interesting studies in numerical calculation. For example, in Kykloy metresis (c. 230 b.c.e.; On the Measurement of the Circle, 1897) he calculated, based on mathematical principles rather than direct measurement, a value for the ratio of the circumference of a circle to its diameter (this ratio was not called pi until much later). By inscribing and circumscribing regular polygons of more and more sides within and around a circle, Archimedes found that the ratio was between 223/71 and 220/71, the best value for pi ever obtained in the classical world.

In Psiammites (c. 230 b.c.e.; The Sand-Reckoner, 1897), Archimedes devises a notation suitable for writing very large numbers. To put this new notation to a test, he sets down a number equal to the number of grains of sand it would take to fill the entire universe. Large numbers are also involved in his treatise concerned with the famous “Cattle Problem.” White, black, yellow, and dappled cows and bulls are grazing on the island of Sicily. The numbers of these cows and bulls have to satisfy several conditions. The problem is to find the number of bulls and cows of each of the four colors. It is unlikely that Archimedes ever completely solved this problem in indeterminate analysis.

Toward the end of his life, Archimedes became part of a worsening political situation. His friend Hiero II had a treaty of alliance with Rome and remained faithful to it, even after the Second Punic War began. After his death, however, his grandson Hieronymus, who became king, was so impressed by Hannibal’s victories in Italy that he switched sides to Carthage. Hieronymus was then assassinated, but Sicily remained allied with Carthage. Consequently, the Romans sent a fleet under the command of Marcellus to capture Syracuse. According to traditional stories, Archimedes invented devices for warding off the Roman enemy. He is supposed to have constructed large lenses to set the fleet on fire and mechanical cranes to turn ships upside down. He devised so many ingenious war machines that the Romans would flee if so much as a piece of rope appeared above a wall. These stories are grossly exaggerated, if not totally fabricated, but Archimedes may have helped in the defense of his city, and he certainly provided the Romans with a face-saving explanation for their frustratingly long siege of Syracuse.

Because of treachery by a cabal of nobles, among other things, Syracuse eventually fell. Marcellus ordered that the city be sacked, but he made it clear that his soldiers were to spare the house and person of Archimedes. Amid the confusion of the sack, however, Archimedes, while puzzling over a geometrical diagram drawn on sand in a tray, was killed by a Roman soldier. During his lifetime he had expressed the wish that on his tomb should be placed a cylinder circumscribing a sphere, together with an inscription giving the ratio between the volumes of these two bodies, a discovery of which he was especially proud. Marcellus, who was distressed by the great mathematician’s death, had Archimedes’ wish carried out. More than a century later, when Cicero was in Sicily, he found this tomb, overgrown with brush but with the figure of the sphere and cylinder still visible.

Significance

Some scholars rank Archimedes with Sir Isaac Newton (1642-1727) and Carl Friedrich Gauss (1777-1855) as one of the three greatest mathematicians who ever lived, and historians of mathematics agree that the theorems Archimedes discovered raised Greek mathematics to a new level of understanding. He tackled very difficult and original problems and solved them through boldness and vision. His skill in using mechanical ideas in mathematics was paralleled by his ingenious use of mathematics in mechanics.

The Latin West received its knowledge of Archimedes from two sources: Byzantium and Islam. His works were translated from the Greek and Arabic into Latin in the twelfth century and played an important role in stimulating the work of medieval natural philosophers. Knowledge of Archimedes’ ideas multiplied during the Renaissance, and by the seventeenth century his insights had been almost completely absorbed into European thought and had deeply influenced the birth of modern science. For example, Galileo was inspired by Archimedes and tried to do for dynamics what Archimedes had done for statics. More than any other ancient scientist, Archimedes observed the world in a way that modern scientists from Galileo to Albert Einstein admired and sought to emulate.

Bibliography

Aaboe, Asger. Episodes from the Early History of Mathematics. New York: Random House, 1964. After a brief account of Archimedes’ life and a survey of his works, the third chapter of this book presents three samples of Archimedean mathematics: the trisection of an angle, the construction of a regular heptagon, and the determination of a sphere’s volume and surface area.

Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1965. A collection of biographical essays on the world’s greatest mathematicians. Bell discusses Archimedes, along with Zeno of Elea and Eudoxus, in an early chapter on “Modern Minds in Ancient Bodies.”

Dijksterhuis, E. J. Archimedes. Princeton, N.J.: Princeton University Press, 1987. This edition of the best survey in English of Archimedes’ life and work also contains a valuable bibliographical essay by Wilbur R. Knorr.

Finley, Moses I. Ancient Sicily. Vol. 1 in A History of Sicily. New York: Viking Press, 1968. Finley’s account of the history of Sicily from antiquity to the Arab conquest has a section explaining how the politics of the Second Punic War led to Archimedes’ death.

Heath, T. L. A History of Greek Mathematics. 2 vols. 1921. Reprint. New York: Dover, 1981. A good general survey of ancient Greek mathematics that contains, in volume 2, a detailed account of the works of Archimedes.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1990. Kline’s treatment of Archimedes emphasizes the themes of his work rather than the events of his life.

Stein, Sherman K. Archimedes: What Did He Do Besides Cry Eureka? Washington, D.C.: Mathematical Association of America, 1999. An accessible account of Archimedes’ accomplishments, as well as Archimedes’ life, the 1906 discovery of his manuscript, and his methods. Includes bibliography and index.

Van der Waerden, B. L. Science Awakening. 4th ed. Princeton Junction, N.J.: Scholar’s Bookshelf, 1988. A survey of ancient Egyptian, Babylonian, and Greek mathematics. The chapter on the Alexandrian era (330-220 b.c.e.) contains a detailed account of Archimedes’ life, legends, and mathematical accomplishments.