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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 an equal See also:

WEIGHT

weight of gold separately into a See also:

VESSEL (O. Fr. vaissel, from a rare Lat. vascellum, dim. of vas, vase, urn)

vessel filled with water, and observing the difference of overflow. He was so overjoyed when this happy thought struck him that he ran See also:

home without his clothes, shouting euprlKa, euprKa, " I have found it, I have found it." Similarly his See also:

PIONEER

pioneer See also:

WORK

work in See also:

MECHANICS

mechanics is illustrated by the story of his having said 56s lam roil See also:

ARIA (Ital. for " air ")

aria Kal Kwio -ry~lv (or as another version has it, in his See also:

DIALECT (from Gr. Sia stcror, conversation, manner of speaking, &aXeyec6at, to converse)

dialect, era /3ia Kai See also:

LAVA

lava) rav 'yap), " Give me a See also:

PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place to stand and I (will) move the See also:

earth." Hiero asked him to give an See also:

ILLUSTRATION

illustration of his contention that a very See also:

GREAT

great weight could be moved by a very small force.

He is said to have fixed on a large and fully laden See also:

SHIP

ship and to have used a mechanical See also:

DEVICE

device by which Hiero was enabled to move it by himself: but accounts differ as to the particular mechanical See also:

POWERS, HIRAM (1805-1873)

powers employed. The water-See also:

SCREW (O.E. scrue, from O. Fr. escroue, mod. ecrou; ultimate origin uncertain; the word, or a similar one, appears in Teutonic languages, cf. Ger. Schraube, Dan. skrue, but Skeat, following Diaz, finds the origin in Lat. scrobs, a ditch, hole, particularl

screw which he invented (see below) was probably devised in See also:

EGYPT

Egypt for the purpose of irrigating See also:

FIELDS, JAMES THOMAS (1817-1881)

fields. Archimedes died at the See also:

CAPTURE (from Lat. capere, to take; Fr. prise maritime; Ger. Wegnahme)

capture of Syracuse by See also:

MARCELLUS

Marcellus, 212 B.C. In the See also:

general See also:

MASSACRE

massacre which followed the fall of the city, Archimedes, while engaged in See also:

DRAWING

drawing a mathematical figure on the See also:

sand, was run through the See also:

BODY

body by a Roman soldier. No blame attaches to the Roman general, Marcellus, since he had given orders to his men to spare the See also:

house and See also:

PERSON, OFFENCES AGAINST THE

person of the See also:

SAGE, RUSSELL (1816-1906)

sage; and in the midst of his See also:

TRIUMPH (triumphus)

triumph he lamented the See also:

DEATH

death of so illustrious a person, directed an See also:

HONOURABLE (Fr. honorable, from Lat. honorabilis, worthy of honour)

honourable See also:

BURIAL

burial to be given him, and befriended his surviving relatives. In accordance with the expressed See also:

DESIRE

desire of the philosopher, his See also:

TOMB (Gr. Tuµ/3a, Tuµ(3os, probably allied to Lat. tumulus, literally a swelling, tumere, to swell)

tomb was marked by the figure of a sphere inscribed in a See also:

CYLINDER (Gr. KvAwSpos, from KvXivaety, to roll)

cylinder, the discovery of the relation between the volumes of a sphere and its circumscribing cylinder being regarded by him as his most valuable achievement. When See also:

CICERO

Cicero was See also:

QUAESTOR (from Lat. quaero, investigate)

quaestor in Sicily (75 B.C.), he found the tomb of Archimedes, near the Agrigentine See also:

GATE

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 See also:

GREECE

Greece have remained a stranger to the tomb of one of its most ingenious citizens, had it not been discovered by a See also:

man of Arpinum." See also:

WORKS

Works.—The range and importance of the scientific labours of Archimedes will be best understood from a brief See also:

account of those writings which have come down to us; and it need only be added that his greatest work was in See also:

GEOMETRY

geometry, where he so extended the method of exhaustion as originated by See also:

EUDOXUS

Eudoxus, and followed by See also:

Euclid, that it became in his hands, though purely geometrical in See also:

FORM (Lat. forma)

form, actually See also:

EQUIVALENT

equivalent in several cases to integration, as expounded in the first chapters of our See also:

TEXT (Lat. textum, woven fabric, from texere, to weave)

text-books on the integral calculus. This remark applies to the finding of the See also:

AREA

area of a parabolic segment (mechanical See also:

SOLUTION (from Lat. solvere, to loosen, dissolve)

solution) and of a See also:

SPIRAL

spiral, the See also:

SURFACE

surface and See also:

VOLUME

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 See also:

TREATISES

treatises are as follows: (1) On the Sphere and Cylinder (17epi a43atpas Kal KvXivSpou).

This See also:

TREATISE

treatise is in two books, dedicated to Dositheus, and deals with the dimensions of See also:

SPHERES, MUSIC OF THE

spheres, cones, " solid rhombi " and cylinders, all demonstrated in a strictly geometrical method. The first See also:

BOOK

book contains See also:

FORTY

forty-four propositions, and those in which the most important results are finally obtained are: 13 (surface of right cylinder), 14, 15 (surface of right See also:

CONE (Gr. Kwvos)

cone), 33 (surface of sphere), 34 (volume of sphere and its relation to that of circumscribing cylinder), 12, 43 (surface of segment of sphere), 44 (volume of sector of sphere). he second book is in nine propositions, eight of which See also:

DEAL

deal with segments of spheres and include the problems of cutting a given sphere by a See also:

PLANE

plane so that (a) the surfaces, (b) the volumes, of the segments are in a given ratio (Props. 3, 4), and of constructing a segment of a sphere similar to one given segment and having (a) its volume, (b) its surface, equal to that of another (5, 6). (2) The Measurement of the Circle (KkXov pieprivts) is a See also:

SHORT, FRANCIS JOB (1857– )

short book of three propositions, the See also:

main result being obtained in Prop. 2, which shows that the circumference of a circle is less than 3i and greater than 3;; times its See also:

DIAMETER (from the Cr. &a, through, µErpov, measure)

diameter. Inscribing in and circumscribing about a circle two polygons, each of ninety-six sides, and assuming that the perimeter of the circle See also:

lay between those of the polygons, he obtained the limits he has assigned by sheer calculation, starting from two See also:

close approximations to the value of d 3, which he assumes as known (265/153 < sl 3 < 1351/780). (3) On Conoids and Spheroids (IIEpi KWVOELaEWV Kai ?r4JaipoeiIiwv) is a treatise in See also:

THIRTY

thirty-two 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 See also:

BASE

base and See also:

AXIS (Lat. for " axle ")

axis (Props. 21, 22 for the paraboloid, 25, 26 for the hyperboloid, and 27-32 for the See also:

SPHEROID (Gr. c4aipa-etbis, like a sphere)

spheroid). (4) On Spirals (Heal EXIKWV) is a book of twenty-eight propositions. Propositions 1-II are preliminary, 1-20 contain tangential properties of the See also:

CURVE (Lat. curvus, bent)

curve now known as the spiral of Archimedes, and 21-28 show how to See also:

EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the area included between any portion of the curve and the radii vectores to its extremities. (5) On the See also:

EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

Equilibrium of Planes or Centres of Gravity of Planes (See also:

MAI, ANGELO (1782—1854)

Mai irLAESWv iooppoatiav 4 son-pa 19apeoe E,rnrkawe). This See also:

con- sists of two books, and may be called the foundation of theoretical mechanics, for the previous contributions of See also:

Aristotle were comparatively vague and unscientific.

In the first book there are fifteen propositions, with seven postulates; and demonstrations are given, much the same as those still employed, of the centres of gravity (1) of any two weights, (2) of any parallelogram, (3) of any triangle, (4) of any trapezium. The second book in ten propositions is devoted to the finding the centres of gravity (1) 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 See also:

QUADRATURE (from Lat. quadratura, a making square)

Quadrature of the See also:

PARABOLA

Parabola (Terpaywyu,µbs irapa/3oX4s) is a book in twenty-four propositions, containing two demonstrations that the area of any segment of a parabola is s of the triangle which has the same base as the segment and equal height. The first (a mechanical See also:

PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof) begins, after some preliminary propositions on the parabola, in Prop. 6, ending with an integration in Prop. 16. The second (a geometrical proof) is expounded in Props. 17-24. (7) On Floating Bodies (Mai bxov z vwv) is a treatise in two books, the first of which establishes the general principles of hydro-See also:

STATICS (from Gr. root am-, stand, or cause to stand)

statics, and the second discusses with the greatest completeness the positions of See also:

REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest and stability of a right segment of a paraboloid of revolution floating in a fluid. (8) The Psammites (gYaµµ(r>,s, See also:

Lat. A renarius, or sand reckoner), a small treatise, addressed to Gelo, the eldest son of Hiero, expounding, as applied to reckoning the number of grains of sand that could be contained in a sphere of the See also:

SIZE

size of our " universe," a See also:

SYSTEM

system of naming large See also:

numbers according to " orders " and " periods " which would enable any number to be expressed up to that which we should write with 1 followed by 8o,000 ciphers! (9) A Collection of Lemmas, consisting of fifteen propositions in plane geometry.

This has come down to us through a Latin version of an Arabic See also:

MANUSCRIPT

manuscript; it cannot, however, have been written by Archimedes in its See also:

PRESENT

present form, as his name is quoted in it more than once. Lastly, Archimedes is credited with the famous See also:

CATTLE (Norman Fr. catel, from Late Lat. capitale, wealth or property, a word applied in the feudal system to movable property and particularly to live stock, and surviving in its wider meaning as " chattel " or " chattle ")

Cattle-Problem. enunciated in the See also:

EPIGRAM

epigram edited by G. E. See also:

LESSING, GOTTHOLD EPHRAIM (1729–1781)

Lessing; in 177, which purports to have been sent by Archimedes to the mathematicians at Alexandria in a See also:

LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter to Eratosthenes. Of lost works by Archimedes we can identify the following: (I) investigations on polyhedra mentioned by Pappus; (2) 'Apxat, Principles, a book addressed to Zeuxippus and dealing with the naming of numbers on the system explained in the Sand Reckoner; (3) IIspl j'vyi v, On balances or levers; (4) Kevrpo,3aptxb., On centres of gravity; (5) Ka.roirrpith, an See also:

OPTICAL

optical work from which See also:

Theon of Alexandria quotes a remark about See also:

refraction; (6) 'E464tov, a Method, mentioned by Suidas; (7) llEpl o¢aipo,rotias, On Sphere-making, in which Archimedes explained the construction of the sphere which he made to imitate the motions of the See also:

sun, the See also:

moon and the five See also:

PLANETS, MINOR

planets in the heavens. Cicero actually saw this contrivance and describes it (De See also:

REP, REPP, or REPS

Rep. i. c. 14, §§ 21-22). (See also:

Venice, 1543) ; Trojanus See also:

Curtius published the two books on Floating Bodies in 1565 after See also:

TARTAGLIA, or TARTALEA, NICCOLO (c. 1506—1559)

Tartaglia's death; See also:

FREDERIC, HAROLD (1856-1898)

Frederic Cornmandine edited the Aldine edition of 1558, 4to, which contains Circuli Dimensio, De Lineis Spiralibus, Quadratura Paraboles, De Conoidibus et Spheroidibus, and De numeeo Arenae; and in 1565 the same mathematician published the two books De its quae vehuntur in aqua. J. Torelli's monumental edition of the works with the commentaries of Eutocius, published at See also:

Oxford in 1792, See also:

FOLIO (properly the ablative case of the Lat. folium, leaf, but also frequently an adaptation of the Ital. foglio)

folio, remained the best Greek text until the definitive text edited, with Eutocius' commentaries, Latin See also:

TRANSLATION

translation, &c., by J. L. See also:

HEIBERG, JOHAN LUDVIG (1791-186o)

Heiberg (See also:

LEIPZIG

Leipzig, 188o—1881) superseded it.

The Arenarius and Dimensio Circuli, with Eutocius' commentary on the latter, were edited by See also:

WALLIS, JOHN (1616-1703)

Wallis with Latin translation and notes in 1678 (Oxford), and the A renarius was also published in See also:

ENGLISH

English by See also:

George See also:

Anderson (See also:

LONDON

London, 1784), with useful notes and illustrations. The first See also:

MODERN

modern translation of the works is the See also:

French edition published by F. Peyrard (See also:

Paris, 1808, 2 vols. 8vo.). A valuable See also:

German translation with notes, by E. Nizze, was published at See also:

STRALSUND

Stralsund in 1824. There is a See also:

COMPLETE

complete edition in modern notation by T. L. See also:

Heath (The Works of Archimedes, See also:

Cambridge, 1897). On Archimedes himself, see Plutarch's See also:

LIFE

Life of Marcellus. (T. L.

End of Article: ARCHIMEDES (c. 287–212 B.C.)

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