# Archimedes

**ARCHIMEDES** (c.287-212 B.C.), Greek mathematician and inventor, was born at Syracuse, in Sicily. He was the son of Pheidias, an astronomer, and was on intimate terms
with, if not related to, Hiero, king of Syracuse, and Gelo his son. He studied at Alexandria and doubtless met there Conon of Samos, whom he admired as a mathematician and cherished as a friend, and
to whom he was in the habit of communicating his discoveries before publication. On his return to his native city he devoted himself to mathematical 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
(*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 Hiero
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 bowshot of the wall. This has been discredited because it is not mentioned by Polybius, Livy or Plutarch; but it is probable that Archimedes had constructed some such burning
instrument, though the connexion of it with the destruction of the Roman fleet is more than doubtful. More important, as being doubtless connected with the discovery of the principle in hydrostatics
which bears his name and the foundation by him of that whole science, is the story of Hiero'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 an equal weight of gold separately into a vessel filled with water, and observing the difference of
overflow. He was so overjoyed when this happy thought struck him that he ran home without his clothes, shouting , "I have found it, I have found it." Similarly his pioneer work in mechanics is
illustrated by the story of his having said (or as another version has it, in his dialect, ), "Give me a place to stand and I (will) move the earth." Hiero 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 mechanical device by which Hiero was enabled to
move it by himself: but accounts differ as to the particular mechanical powers employed. 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 befriended his surviving relatives. In accordance with the expressed desire of the
philosopher, his tomb was marked by the figure of a Sphere inscribed in a 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 Cicero was quaestor in Sicily (75 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 remained a stranger to the tomb of one of its most ingenious citizens, 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 be 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: -

(1) *On the Sphere and Cylinder* . This treatise is in two books, dedicated to Dositheus, and deals with the dimensions of spheres, cones, "solid rhombi" and cylinders, all demonstrated in a
strictly geometrical method. The first book contains 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 cone), 33 (surface of sphere), 34 (volume of Sphere and its relation to that of circumscribing cylinder), 42, 43 (surface of segment of sphere), 44 (volume of sector of sphere). The second book
is in nine propositions, eight of which deal with segments of spheres and include the problems of cutting a given Sphere by a 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* 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 31/7 and greater than 310/71 times its diameter. Inscribing in and
circumscribing about a circle two polygons, each of ninety-six sides, and assuming that the perimeter of the circle lay between those of the polygons, he obtained the limits he has assigned by sheer
calculation, starting from two close approximations to the value of √3, which he assumes as known (265/153 < √3 < 1351/780).

(3) *On Conoids and Spheroids* is a treatise in 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 base and axis (Props. 21, 22 for the paraboloid, 25, 26 for the hyperboloid, and 27-32 for the
spheroid).

(4) *On Spirals* is a book of twenty-eight propositions. Propositions 1-11 are preliminary, 13-20 contain tangential properties of the curve now known as the spiral of Archimedes, and 21-28
show how to express the area included between any portion of the curve and the radii vectores to its extremities.

(5) *On the Equilibrium of Planes or Centres of Gravity of Planes* . This consists of two books, and may be called the foundation of theoretical mechanics, for the previous contributions of
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 Quadrature of the Parabola* is a book in twenty-four propositions, containing two demonstrations that the area of any segment of a parabola is 4/3 of the triangle which has the same base as the segment and equal height. The first (a mechanical 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* is a treatise in two books, the first of which establishes the general principles of hydrostatics, and the second discusses with the greatest completeness the
positions of rest and stability of a right segment of a paraboloid of revolution floating in a fluid.

(8) The *Psammites* (Gr., Lat. *Arenarius*, 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 size of our "universe," a system of naming large 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 80,000 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 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 Archimedes we can identify the following: (1) investigations on *polyhedra* mentioned by Pappus; (2) , *Principles*, a book
addressed to Zeuxippus and dealing with the *naming of numbers* on the system explained in the *Sand Reckoner*; (3) , *On balances or levers*; (4) , *On centres of gravity*; (5) ,
an optical work from which Theon of Alexandria quotes a remark about refraction; (6) , a *Method*, mentioned by Suidas; (7) , *On Sphere-making*, in which Archimedes explained the
construction 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 Archimedes, with the commentary of Eutocius, is that printed at Basel, in 1544, in Greek and Latin, by Hervagius. D. Rivault's
edition (Paris, 1615) gave the enunciations in Greek and the proofs in Latin somewhat retouched. A Latin version of them was published by Isaac Barrow in 1675 (London, 4to); Nicolas Tartaglia
published in Latin the treatises on *Centres of Gravity*, on the *Quadrature of the Parabola*, on the *Measurement of the Circle*, and on *Floating Bodies*, i. (Venice, 1543);
Trojanus Curtius published the two books on *Floating Bodies* in 1565 after Tartaglia's death; Frederic Commandine edited the Aldine edition of 1558, 4to, which contains *Circuli Dimensio*,
*De Lineis Spiralibus*, *Quadratura Paraboles*, *De Conoidibus et Spheroidibus*, and *De numero Arenae*; and in 1565 the same mathematician published the two books *De iis quae
vehuntur in aqua*. J. Torelli's monumental edition of the works with the commentaries of Eutocius, published at Oxford in 1792, folio, remained the best Greek text until the definitive text edited,
with Eutocius' commentaries, Latin translation, etc., by J.L. Heiberg (Leipzig, 1880-1881) superseded it. The *Arenarius* and *Dimensio Circuli*, with Eutocius' commentary on the latter,
were edited by Wallis with Latin translation and notes in 1678 (Oxford), and the *Arenarius* was also published in English by George Anderson (London, 1784), with useful notes and illustrations.
The first modern translation of the works is the French edition published by F. Peyrard (Paris, 1808, 2 vols. 8vo.). A valuable German translation with notes, by E. Nizze, was published at Stralsund
in 1824. There is a complete edition in modern notation by T.L. Heath (*The Works of Archimedes*, Cambridge, 1897). On Archimedes himself, see Plutarch's *Life of Marcellus*.

(T. L. H.)

*Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)*