ARCHIMEDES (c. 287-212 B.C.), Greek mathematician and inventor, was horn at Syracuse, in Sicily. He was the son of Pheidias, an astronomer, and was on intimate terms with, if not related to, Hieron, king of Syracuse, and Gelon his son. He studied at Alexandria and doubtless met there Conon of Samos, whom he admired as a mathematician and cherished as a friend. On his return to his native city he devoted himself to mathemati cal research. He himself set no value on the ingenious mechanical contrivances which made him famous, regarding them as beneath the dignity of pure science and even declining to leave any written record of them except in the case of the Q4aepoiroita (sphere making), as to which see below. As, however, these machines impressed the popular imagination, they naturally figure largely in the traditions about him. Thus he devised for Hieron engines of war which almost terrified the Romans, and which protracted the siege of Syracuse for three years. There is a story that he constructed a burning mirror which set the Roman ships on fire when they were within a bow-shot of the wall. It is probable that Archimedes had constructed some such burning instrument, though the connection of it with the destruction of the Roman fleet is more than doubtful. More important is the story of Hieron's reference to him of the question whether a crown made for him and purporting to be of gold, did not actually contain a proportion of silver. According to one story, Archimedes was puzzled till one day, as he was stepping into a bath and observed the water running over, it occurred to him that the excess of bulk occasioned by the introduction of alloy could be measured by putting the crown and equal weights of gold and of silver sepa rately into a vessel of water,and noting the differences of overflow. He was so overjoyed when this happy thought struck him that he ran home without his clothes, shouting ei prlKa, €i piKa (gener ally anglicized as Eureka—"I have found it, I have found it") . Similarly his pioneer work in mechanics is illustrated by the story o'f his having said 661 µoc T-ov QTW Ka Ktv& Trio 7riv (or as another version has it in his dialect, ire j3&) Kai KevW Tav "give me a place to stand and I (will) move the earth"). Hieron asked him to give an illustration of his contention that a very great weight could be moved by a very small force. He is said to have fixed on a large and fully laden ship and to have used a mechan ical device by which Hieron was enabled to move it by himself ; but accounts differ as to the particular mechanical powers em ployed. The water-screw which he invented (see below) was probably devised in Egypt for the purpose of irrigating fields.
Archimedes died at the capture of Syracuse by Marcellus, 212 B.C. In the general massacre which followed the fall of the city, Archimedes, while engaged in drawing a mathematical figure on the sand, was run through the body by a Roman soldier. No blame attaches to the Roman general, Marcellus, since he had given orders to his men to spare the house and person of the sage, and, in the midst of his triumph he lamented the death of so illustrious a person, directed an honourable burial to be given him, and be friended his surviving relatives. In accordance with the expressed desire of the philosopher, his tomb was marked by a sphere in scribed in a cylinder, the discovery of the relation between the surface and volume of a sphere and its circumscribing cylinder being regarded by him as his most valuable achievement. When Cicero was quaestor in Sicily (7 5 B.c.) , he found the tomb of Archimedes, near the Agrigentine gate, overgrown with thorns and briers. "Thus," says Cicero (Tusc. Disp. v. c. 23, §64), "would this most famous and once most learned city of Greece have re mained a stranger to the tomb of one of its most ingenious citi zens. had it not been discovered by a man of Arpinum." Works.—The range and importance of the scientific labours of Archimedes will be best understood from a brief account of those writings which have come down to us; and it need only he added that his greatest work was in geometry, where he so extended the method of exhaustion as originated by Eudoxus, and followed by Euclid, that it became in his hands, though purely geometrical in form, actually equivalent in several cases to integration, as expounded in the first chapters of our text-books on the integral calculus. This remark applies to the finding of the area of a parabolic segment (mechanical solution) and of a spiral, the surface and volume of a sphere and of a segment thereof, and the volume of any segments of the solids of revolution of the second degree.
The extant treatises are as follows: (I) On the Sphere and Cylinder (Hopi ocaipas Kai KvXivSpov). This treatise is in two books, dedicated to Dositheus, and deals with the dimensions of spheres, cones, "solid rhombi" and cyl inders, all demonstrated in a strictly geometrical method.
(2) The Measurement of the Circle (K'KXov µtrmiens) is a short book of three propositions, the main result being obtained in Prop. 2, which shows that the circumference of a circle is less than 34 and greater than 34-4 times its diameter.
(3) On Conoids and Spheroids (IIEpi KwvoeCSEwv Kai o-4acpoFaoEwv) is a treatise in 32 propositions, on the solids generated by the revolution of the conic sections about their axes, the main results being the comparisons of the volume of any segment cut off by a plane with that of a cone having the same base and axis (Props. 21, 22 for the paraboloid, 25, 26 for the hyperboloid, and 27-32 for the spheroid) .
(4) On Spirals (Ilepi ALKWP) is a book of 28 propositions. Propositions 1-11 are preliminary, 13-20 contain tangential prop erties of the curve now known as the spiral of Archimedes, and 21-28 show how to express the area included between any por tion of the curve and the radii vectores to its extremities.
(5) On Plane Equilibria or Centres of Gravity of Planes (IIEpi E7rc7rESWv vvri KEVrpa 13apc.w E7rcirESWv.) This con sists of two books, and may be called the foundation of theoreti cal mechanics, for the previous contributions of Aristotle were comparatively vague and unscientific. In the first book there are 15 propositions, with seven postulates; and demonstrations are given, much the same as those still employed, of the centres of gravity (I) of any two weights, (2) of any parallelogram, (3) of any triangle, (4) of any trapezium. The second book in 10 propositions is devoted to the finding the centres of gravity (I) of a parabolic segment, (2) of the area included between any two parallel chords and the portions of the curve intercepted by them.
(6) The Quadrature of the Parabola 7rapa (3oMijs) is a book in 24 propositions, containing two demonstra tions that the area of any segment of a parabola is of the tri angle which has the same base as the segment and equal height.
(7) On Floating Bodies (I Bpi. 6xovµEvwv) is a treatise in two books, the first of which establishes the general principle of hy drostatics, and the second discusses with the greatest complete ness the positions of rest and stability of a right segment of a paraboloid of revolution floating in a fluid.
(8) The Psammites ('Yappirns, Lat. Arenarius, or sand reck oner), a small treatise addressed to Gelon, the eldest son of Hieron, expounding, as applied to reckoning the number of grains of sand that could be contained in a sphere of the size of our "universe," a system of naming large numbers according to "or ders" and "periods" which would enable any number to be ex pressed up to that which we should write with 1 followed by 8o,000 ciphers! (9) The Method, addressed to Eratosthenes, is a treatise of vital interest, since in it Archimedes explains how he first arrived at many of his important results by means of mechanical considerations, namely, by weighing an indefinite number of ele ments of one figure against similar elements of another. This treatise, formerly supposed to be lost, was discovered in 1906 by J. L. Heiberg in a palimpsest at Constantinople, and now forms part of Heiberg's Greek text of Archimedes.
(io) A Collection of Lemmas, consisting of 15 propositions in plane geometry. This has come down to us through a Latin ver sion of an Arabic manuscript ; it cannot, however, have been written by Archimedes in its present form, as his name is quoted in it more than once.
Lastly, Archimedes is credited with the famous Cattle-Problem enunciated in the epigram edited by G. E. Lessing in 1773, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a letter to Eratosthenes. Of lost works by Archi medes we find references to six: (1) investigations on poly hedra mentioned by Pappus; (2) 'Apxat, Principles, a book ad dressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) On balances or levers; (4) KEVrpo(3apcdi On centres of gravity; (5) Karoirrpooi, an optical work from which Theon of Alexandria quotes a remark about refraction; (6) IIEpi I4acpoirotas, Oy Sphere-making, in which Archimedes explained the construc• tion of the sphere which he made to imitate the motions of the sun, the moon and the five planets in the heavens. Cicero actually saw this contrivance and describes it (De Rep. i. c. 14 §§ 21-22) BIBLIOGRAPHY.--The editio princeps of the works of ArchimedesBibliography.--The editio princeps of the works of Archimedes with the commentaries of Eutocius, is that printed at Basle (1544) it Greek and Latin, by Hervagius. A Latin version was published by Isaac Barrow in 1675. Torelli's edition (1792) remained the best Greek text until the definitive text edited, with Eutocius' commentaries Latin trans. etc., by J. L. Heiberg (188e-81, 2nd ed. 1910-15) super seded it. T. L. Heath edited The Works of Archimedes in modern notation, with Introduction etc. (1897) and also, as a Supplement, the newly discovered Method (1912) . Modern translations are those of Peyrard (Paris, 1808) ; E. Nizze, with notes (German, 1824) ; P. ver Eecke, Les Oeuvres Completes (1921) ; A. Czwalina Allenstein, Kugel and Zylinder (1922); Uber Spiralen (1922) ; Die Quadratur der Parabel (1923) ; Ober Paraboloide, Hyperboloide una Ellipsoide (1923), and fiber Schwimmende Korper and die Sandzabei (1925). See Plutarch's Life of Marcellus (in Plutarch's Lives, Eng. trans. Sir T. North, 1579, A. Stewart and E. Long, 1914-23) ; J. Heiberg, Geometrical Solutions Derived from Mechanics (trans. of the Method, Chicago, 1909), and Mathematics and Physical Science in Classical Antiquity (Eng. trans. 1922) ; F. Jansen, De Cirkelquad ratuur bij de Ericken (Haarlem, 1909) ; P. Midolo, Archimede e suo tempo (with useful bibliography, Syracuse, 1912) ; T. L. Heath, Archimedes (192o)- and F. Winter, Der Tod des Archimedes (1924) for a possible though by no means certain portrait. (T. L. H.)