Archimedes’ theoretical and practical discoveries led to innovations in mathematical theory as well as technological inventions.

By far the best-known scientist of the third century b.c.e. was Archimedes of Syracuse, a man revered in his own age for his skill as an inventor and since recognized as one of the greatest Greek mathematicians, ranking with Pythagoras (c. 580-c. 500 b.c.e.) and Euclid. Archimedes
Euclid
Hieron II of Syracuse
Marcellus, Marcus Claudius

Although tradition holds that Archimedes was born in Syracuse, virtually nothing is known of the scientist’s early life. No one thought of writing a biography of Archimedes during his era, and posterity has had to depend on legend, Roman historical accounts, and the inventor’s own works to piece together his life story. He may well have been of aristocratic descent, for the young Archimedes spent several years in study at Alexandria in Egypt, where he was introduced to the best mathematical and mechanical researchers. While there, he seems to have become such a close associate and admirer of the astronomers Conon of Samos (fl. c. 245 b.c.e.) and Eratosthenes of Cyrene (c. 285-c. 205 b.c.e.) that in later years he deferred to their judgment on the publication of his own mathematical treatises. Following his stay in Egypt, Archimedes spent most of his remaining life in Syracuse, where he enjoyed the patronage of King Hieron II. It is around this monarch that many of the legendary episodes of Archimedes’ life cluster, especially his development of a system of pulleys for drawing newly constructed ships into the water, his construction of military machinery, and his discovery of the fraudulent alloy in Hieron’s crown.

Archimedes, moments before his death.

(F. R. Niglutsch)

Contemporaries and later generations of ancient writers praised Archimedes more for his colorful technical ingenuity than for his significant mathematical formulations. His discovery of the “law” of hydrostatics, or water displacement, and his application of this theory to determine the actual gold content of Hieron’s crown may be true, but the exact methodology, if indeed he pursued any, is not at all clear in the ancient accounts. Similar vagueness surrounds the development of the cochlias, or Archimedian screw, a device by which water could be raised from a lower level to a higher level by means of a screw rotating inside a tube. Supposedly, Archimedes developed this invention in Egypt, but he may well have taken an existing mechanism and improved it. Other pieces of apparatus he either invented or constructed include a water organ and a model planetarium, the latter being the sole item of booty that the conqueror of Syracuse, Marcus Claudius Marcellus, took back to Rome.

The great historians of the Roman Republican period, such as Polybius (c. 200-c. 118 b.c.e.), Livy (59 b.c.e.-17 c.e.), and Plutarch (c. 46-after 120 c.e.), give accounts of Archimedes’ genius in inventing military weapons. In his life of Marcellus in Bioi paralleloi (c. 105-115 b.c.e.; Parallel Lives, 1579), Plutarch emphasized Archimedes’ dramatic role in the defense of Syracuse. In constructing military weapons, Archimedes seems to have put to use all the laws of physics at his disposal. His knowledge of levers and pulleys was applied to the construction of ballistic weapons, cranes, grappling hooks, and other devices, so that the Roman siege of Syracuse was stalemated for two years, from 213 to 211 b.c.e. Even the improbable use of large mirrors for directing sharply focused rays of sunlight in order to ignite the Roman fleet is credited to Archimedes. Doubtless, he was the mind behind the defense of Syracuse, and the Romans respected his ability. Although Marcellus wished to capture Archimedes alive, the scientist was killed by a Roman legionnaire when the city of Syracuse fell.

Archimedes preferred to be remembered for his theoretical achievements rather than his discoveries in mechanics. In the third century b.c.e., Greek mathematical thought had advanced as far as it could in terms of geometric models of reasoning without algebraic notation, and the mathematical work of Archimedes appears as the culmination of Hellenistic mathematics. His work on plane curves represented an extension of Euclid’s geometry, and it predicted integral calculus. Archimedes’ studies included conic sections, the ratio of the volume of a cylinder to its inscribed square, and some understanding of pure numbers as opposed to the then prevalent notion of infinity. Through his sand-reckoner, Archimedes supposedly could express any integer up to 8 ‘ 10¹⁶.

In his own lifetime, Archimedes’ works were forwarded to Alexandria, where they were studied and dispersed. Two major Greek collections of Archimedes’ works made by the mathematical schools of Constantinople were later passed on to Sicily and Italy, and then to northern Europe, where they were translated into Latin and widely published after the sixth century c.e. Because none of the Greek collections is complete, Arabic collections and associated commentaries have been used to tabulate the works attributed to Archimedes. Through these legacies, modern scholars have been able to study Archimedes’ work, and some modern scholars consider him to be the greatest mathematician of antiquity.

• Archimedes. The Works of Archimedes. Translated by Sir Thomas Heath. 1897. Reprint. New York: Dover, 2002. Contains nineteenth century translations of Archimedes’ work along with more contemporary commentary.
• Clagett, Marshall, ed. Archimedes in the Middle Ages. 4 vols. Madison: University of Wisconsin Press, 1964. A series of texts in Latin, with English translation, that illustrate the continuing influence (and reinterpretation) of Archimedes in the Christian and Islamic worlds during the medieval period.
• Dijksterhuis, Eduard Jan. Archimedes. Translated by C. Dikshoorn, with a new bibliographic essay by Wilbur R. Knorr. Princeton, N.J.: Princeton University Press, 1987. The most thorough treatment of Archimedes available. Also provides an extensive look at the development of Greek mathematics.
• Netz, Reviel. The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. New York: Cambridge University Press, 2003. An advanced discussion of the subject that provides the intellectual context for understanding Archimedes and his work.
• Stein, Sherman. Archimedes: What Did He Do Beside Cry Eureka? Washington, D.C.: The Mathematical Association of America, 1999. Uses high-school-level math to explain Archimedes’ accomplishments.
• Tuplin, C. J., and T. E. Rihll, eds. Science and Mathematics in Ancient Greek Culture. New York: Oxford University Press, 2002. A collection of essays that cover many specific topics touching on Archimedes’ mathematics and inventions.

Apollonius of Perga; Archimedes; Diophantes; Euclid; Eratosthenes of Cyrene; Eudoxus of Cnidus; Hero of Alexandria; Hypatia; Pappus; Pythagoras. Archimedes