#### Archime'des

（**Ἀρχιμήδης**), of Syracuse, the most famous of ancient mathematicians, was born B. C. 287, if the statement of Tzetzes, which makes him 75 years old at his death, be correct.

Of his family little is known. Plutarch calls him a relation of king Hiero; but Cicero (*Tusc. Disp.* 5.23), contrasting him apparently not with Dionysius (as Torelli suggests in order to avoid the contradiction), but with Plato and Archytas, says, " humilem homunculum a pulvere et radio excitabo."
At any rate, his actual condition in life does not seem to have been elevated (Silius Ital. 14.343), though he was certainly a friend, if not a kinsman, of Hiero.
A modern tradition makes him an ancestor of the Syracusan virgin martyr St. Lucy. (Rivaltus, *in vit. Archim. Mazzuchelli,* p. 6.)
In the early part of his life he travelled into Egypt, where he is said, on the authority of Proclus, to have studied under Conon the Samian, a mathematician and astronomer (mentioned by Virg. *Eel.* 3.40), who lived under the Ptolemies, Philadelphus and Euergetes, and for whom he testifies his respect and esteem in several places of his works. (See the introductions to the Quadratura Paraboles and the De Helicibus.)
After visiting other countries, he returned to Syracuse. (**Diod. 5.37**.) Livy (**24.34**) calls him a distinguished astronomer, "unicus spectator coeli siderumque;" a description of which the truth is made sufficiently probable by his treatment of the astronomical questions occurring in the Arenarius. (See also Macrob. *Somn. Scip.* 2.3.)
He was popularly best known as the inventor of several ingenious machines; but Plutarch (**Plut. Marc. 100.14**), who, it should be observed, confounds the application of geometry to mechanics with the solution of geometrical problems by mechanical means, represents him as despising these contrivances, and only condescending to withdraw himself from the abstractions of pure geometry at the request of Hiero. Certain it is, however, that Archimedes did cultivate not only pure geometry, but also the mathematical theory of several branches of physics, in a truly scientific spirit, and with a success which placed him very far in advance of the age in which he lived. His theory of the lever was the foundation of statics till the discovery of the composition of forces in the time of Newton, and no essential addition was made to the principles of the equilibrum of fluids and floating bodies, established by him in his treatise " De Insidentibus," till the publication of Stevin's researches on the pressure of fluids in 1608. (Lagrange, *Méc. Anal.* vol. i. pp. 11, 176.)

He constructed for Hiero various engines of war, which, many years afterwards, were so far effectual in the defence of Syracuse against Marcellus, as to convert the siege into a blockade, and delay the taking of the city for a considerable time. (**Plut. Marc. 15**-**18**; **Liv. 24.34**; **Plb. 8.5**-**9**.)
The accounts of the performances of these engines are evidently exaggerated; and the story of the burning of the Roman ships by the reflected rays of the sun, though very current in later times, is probably a fiction, since neither Polybius, Livy, nor Plutarch gives the least hint of it.
The earliest writers who speak of it are Galen (*De Temper.* 3.2) and his contemporary Lucian (*Hippias,* 100.2), who (in the second century) merely allude to it as a thing well known. Zonaras (about A. D. 1100) mentions it in relating the use of a similar apparatus, contrived by a certain Proclus, when Byzantium was besieged in the reign of Anastasius ; and gives Dion as his authority, without referring to the particular passage.
The extant works of Dion contain no allusion to it. Tzetzes (about 1150) gives an account of the principal inventions of Archimedes (*Chil.* 2.103-156), and amongst them of this burning machine, which, he says, set the Roman ships on fire when they came within a bow-shot of the walls; and consisted of a large hexagonal mirror with smaller ones disposed round it, each of the latter being a polygon of 24 sides.
The subject has been a good deal discussed in modern times, particularly by Cavalieri (in cap. 29 of a tract entitled " Del Specchio Ustorio," Bologna, 1650), and by Buffon, who has left an elaborate dissertation upon it in his introduction to the history of minerals. (*Oeuvres,* tom. v. p. 301, &c.)
The latter author actually succeeded in igniting wood at a distance of 150 feet, by means of a combination of 148 plane mirrors.
The question is also examined in vol. ii. of Pevrard's Archimedes ; and a prize essay upon it by Capelle is translated from the Dutch in Gilbert's " Annalen der Physik," vol. liii. p. 242.
The most probable conclusion seems to be, that Archimedes had on some occasion set fire to a ship or ships by means of a burning mirror, and that later writers falsely connected the circumstance with the siege of Syracuse. (See Ersch and Gruber's *Cyclop.* art. *Archim.* note, and Gibbon, chap. 40.)

The following additional instances of Archimedes' skill in the application of science have been collected from various authors by Rivaltus (who edited his works in 1615) and others.

He detected the mixture of silver in a crown which Hiero had ordered to be made of gold, and determined the proportions of the two metals, by a method suggested to him by the overflowing of the water when he stepped into a bath. When the thought struck him he is said to have been so much pleased that, forgetting to put on his clothes, he ran home shouting εὕρηκα, εὕρηκα.
The particulars of the calculation are not preserved, but it probably depended upon a direct comparison of the weights of certain volumes of silver and gold with the weight and volume of the crown; the volumes being measured, at least in the case of the crown, by the quantity of water displaced when the mass was immersed.
It is not likely that Archimedes was at this time acquainted with the theorems demonstrated in his hydrostatical treatise concerning the *loss of weight* of bodies immersed in water, since he would hardly have evinced such lively gratification at the obvious discovery that they might be applied to the problem of the crown ; his delight must rather have arisen from his now first catching sight of a line of investigation which led immediately to the solution of the problem in question, and ultimately to the important theorems referred to. (**Vitr. 9.3**.; Proclus. *Comm. in lib.* i. *Eucl.* 2.3.)

He superintended the building of a ship of extraordinary size for Hiero, of which a description is given in Athenaeus (v. p. 206, D), where he is also said to have moved it to the sea by the help of a screw. According to Proclus, this ship was intended by Hiero as a present to Ptolemy; it may possibly have been the occasion of Archimedes' visit to Egypt.

He invented a machine called, from its form, Cochlea, and now known as the water-screw of Archimedes, for pumping the water out of the hold of this vessel; it is said to have been also used in Egypt by the inhabitants of the Delta in irrigating their lands. (**Diod. 1.34**; **Vitr. 10.11**.)
An investigation of the mathematical theory of the water screw is given in Ersch and Gruber. The Arabian historian Abulpharagius attributes to Archimedes the raising of the dykes and bridges used as defences against the overflowing of the Nile. (Pope-Blount, *Censura,* p. 32.) Tzetzes and Oribasius (*de Mach.* xxvi.) speak of his __Trispast,__ a machine for moving large weights; probably a combination of pulleys, or wheels and axles. A *hydraulic organ* (a musical instrument) is mentioned by Tertullian (*de Anima,* cap. 14), but Pliny (**7.37**) attributes it to Ctesibius. (See also Pappus, *Math. Coll.* lib. 8, introd.)
An apparatus called *loculus.* apparently somewhat resembling the *Chinese puzzle,* is also attributed to Archimedes. (Fortunatianus, *de Arte Metrica,* p. 2684.) His most celebrated performance was the construction of a *spbere :* a kind of orrery, representing the movements of the heavenly bodies, of which we have no particular description. (Claudian, *Epigr.* xxi. *in Sphaeram Archimedis;* Cic. *Nat. Deor.* 2.35, *Tutsc. Disp.* 1.25; Sext. Empir. *ad v. Math.* 9.115 ; Lactant. *Div. Inst.* 2.5; **Ov. Fast. 6.277**.)

When Syracuse was taken, Archimedes was killed by the Roman soldiers, ignorant or careless who he might be.
The accounts of his death vary in some particulars, but mostly agree in describing him as intent upon a mathematical problem at the time.
He was deeply regretted by Marcellus, who directed his burial, and befriended his surviving relations. (**Liv. 25.31**; Valer. Max. 8.7.7; **Plut. Marc. 19**; Cic. *de fin.* 5.19.) Upon his tomb was placed the figure of a sphere inscribed in a cylinder, in accordance with his known wish, and in commemoration of the discovery which he most valued. When Cicero was quaestor in Sicily (B. C. 75) he found this tomb near one of the gates of the city, almost hid amongst briars, and forgotten by the Syracusans. (*Tusc. Disp.* 5.23.)

Of the general character of Archimedes we have no direct account.
But his apparently disinterested devotion to his friend and admirer Hiero, in whose service he was ever ready to exercise his ingenuity upon objects which his own taste would not have led him to choose (for there is doubtless some truth in what Plutarch says on this point) ; the affectionate regret which he expresses for his deceased master Conon, in writing to his surviving friend Dositheus (to whom most of his works are addressed); and the unaffected simplicity with which he announces his own discoveries, seem to afford probable grounds for a favourable estimate of it.
That his intellect was of the very highest order is unquestionable.
He possessed, in a degree never exceeded unless by Newton, the inventive genius which discovers new provinces of inquiry, and finds new points of view for old and familiar objects; the clearness of conception which is essential to the resolution of complex phaenomena into their constituent elements; and the power and habit of intense and persevering thought, without which other intellectual gifts are comparatively fruitless. (See the introd. to the treatise " De Con. et Sphaer.")
It maybe noticed that he resembled other great thinkers, in his habit of complete abstraction from outward things, when reflecting on subjects which made considerable demands on his mental powers.
At such times he would forget to eat his meals, and require compulsion to take him to the bath. (Plut. *l.* c.) Compare the stories of Newton sitting great part of the day half dressed on his bed, while composing the Principia; and of Socrates standing a whole day and night, thinking, on the same spot. (Plat. Symp. p. 220c. d.)
The success of Archimedes in conquering difficulties seems to have made the expression πρόβλημα Ἀρχιμήδειυν proverbial. (See **Cic. Att. 13.28**, *pro Cluent.* 32.)

#### Works

#### Equiponderants and Centres of Gravity

The following works of Archimedes have come down to us: A treatise on __Equiponderants and Centres of Gravity,__in which the theory of the equilibrium of the straight lever is demonstrated, both for commensurable and incommensurable weights; and various properties of the centres of gravity of plane surfaces bounded by three or four straight lines, or by a straight line and a parabola, are established.

#### The Quadrature of the Parabola,

__The Quadrature of the Parabola,__in which it is proved, that the area cut off from a parabola by any chord is equal to two-thirds of the parallelogram of which one side is the chord in question, and the opposite side a tangent to the parabola. This was the first real example of the quadrature of a curvilinear space; that is, of the discovery of

*a rectilinear*figure equal to an area not bounded entirely by straight lines.

A treatise on *the Sphere and Cylinder,* in which various propositions relative to the surfaces and volumes of the sphere, cylinder, and cone, were demonstrated for the first time. Many of them are now familiarly known; for example, those which establish the ratio (2/3) between the volumes, and also between the surfaces, of the sphere and circumscribing cylinder; and the ratio (1/4) between the area of a great circle and the surface of the sphere. They are easily demonstrable by the modern analytical methods, but the original discovery and geometrical proof of them required the genius of Archimedes. Moreover, the legitimacy of the modern applications of analysis to questions concerning curved lines and surfaces, can only be proved by a kind of geometrical reasoning, of which Archimedes gave the first example. (See Lacroix, *Diff. Cal.* vol. i. pp. 63 and 431; and compare De Morgan, *Diff. Cal.* p. 32.)

#### Dimension of the Circle

The book on the __Dimension of the Circle__consists of three propositions. 1st. Every circle is equal to a right-angled triangle of which the sides containing the right angle are equal respectively to its radius and circumference. 2nd. The ratio of the area of the circle to the square of its diameter is nearly that of 11 to 14. 3rd. The circumference of the circle is greater than three times its diameter by a quantity greater than 10/71 of the diameter but less than 1/7 of the same. The last two propositions are established by comparing the circumference of the circle with the perimeters of the inscribed and circumscribed polygons of 96 sides.

#### Spirals

The treatise on __Spirals__contains demonstrations of the principal properties of the curve, now known as the Spiral of Archimedes, which is generated by the uniform motion of a point along a straight line revolving uniformly in one plane about one of its extremities. It appears from the introductory epistle to Dositheus that Archimedes had not been able to put these theorems in a satisfactory form without long-continued and repeated trials; and that Conon, to whom he had sent them as problems along with various others, had died without accomplishing their solution.

#### Conoids and Spheroids

The book on __Conoids and Spheroids__relates chiefly to the volumes cut off by planes from the solids so called; those namely which arc generated by the rotation of the Conic Sections about their principal axes. Like the work last described, it was the result of laborious, and at first unsuccessful, attempts. (See the introduction.)

#### Arenarius

The __Arenarius__(ὁ Ψαμμίτης) is a short tract addressed to Gelo, the eldest son of Hiero, in which Archimedes proves, that it is possible to assign a number greater than that of the grains of sand which would fill the sphere of the fixed stars. This singular investigation was suggested by an opinion which some persons had expressed, that the sands on the shores of Sicily were either infinite, or at least would exceed any numbers which could be assigned for them; and the success with which the difficulties caused by the awkward and imperfect notation of the ancient Greek arithmetic are eluded by a device identical in principle with the modern method of logarithms, affords one of the most striking instances of the great mathema tician's genius. Having briefly discussed the opinions of Aristarchus upon the constitution and extent of the Universe [ARISTARCHUS], and described his own method of determilling the apparent diameter of the sun, and the magnitude of the pupil of the eye, he is led to assume that the diameter of the sphere of the fixed stars may be taken as not exceeding 100 million of millions of stadia; and that a sphere, one δάκτυλος in diameter, cannot contain more than 640 millions of grains of sand; then, taking the stadinm, in round numbers, as not greater than 10,000 δάκτυλοι, he shews that the number of grains in question could not be so great as 1000 myriads multiplied by the eighth term of a geometrical progression of which the first term was unity and the common ratio a myriad of myriads; a number which in our notation would be expressed by unity with 63 ciphers annexed.

The Arenarius, having been little meddled with by the ancient commentators, retains the Doric dialect, in which Archimedes, like his countryman Theocritus, wrote. (See Wallis, *Op.* vol. iii. pp. 537, 545. Tzetzes says, ἔλεγε δὲ καὶ δωριστὶ, φωνῇ Συρακουσίἁ, Πᾶ Βῶ, καὶ χαριστίωνι τὰν γὰν κινήσω πᾶσαν.)

#### On Floating Bodies

The two books __On Floating Bodies__(Περὶ τῶν Ὀχουμένων) contain demonstrations of the laws which determine the position of bodies immersed in water; and particularly of segments of spheres and parabolic conoids. They are extant only in the Latin version of Commandine, with the exception of a fragment Περὶ τῶν Ὕδατι ἐφισταμένων in Ang. Mai's Collection, vol. i. p. 427.

#### Lemmata

The treatise entitled __Lemmata__is a collection of 15 propositions in plane geometry. It is derived from an Arabic MS. and its genuineness has been doubted. (See Torelli's preface.)

#### Eutocius' Commentary

Eutocius of Ascalon, about A. D. 600, wrote a commentary on the Treatises on the Sphere and Cylinder, on the Dimension of the Circle, and on Centres of Gravity.

#### Editions

*All the works above mentioned, together with Eutocius' Commentary, were found on the taking of Constantinople, and brought first into Italy and then into Germany. They were printed at Basle in 1544, in Greek and Latin, by Hervagius. Of the subsequent editions by far the best is that of Torelli, " Archim. quae supers. omnia, cum Eutocii Ascalonitae commentariis. Ex recens. Joseph. Torelli, Veronensis," Oxon. 1792.*

*It was founded upon the Basle edition, except in the case of the Arenarius, the text of which is taken from that of Dr. Wallis, who published this treatise and the Dimensio Circuli, with a translation and notes, at Oxford, in 1679. (They are reprinted in vol. iii. of his works.)*

#### Translations

*A French translation of the works of Archimedes, with notes, was published by F. Peyrard, Paris, 1808, 2 vols. 8vo.*,

*and an English translation of the Arenarius by G. Anderson, London, 1784.*

#### Further Information

G. M. Mazuchlelli,*Notizie istoriche e critiche intorno alla rita, alle imenzioni, ed agli scritti di Archimede,*Brescia, 1737, 4to.; C. M. Brandelii,

*Dissertatio sistens Archimedis vitam, ejusque in Mathesin merita,*Gryphiswald. 1789, 4to.; Märtens, in Ersch und Gruber,

*Allgemeine Encyclopädie*art.

*Archimedes ;*Quarterly Review, vol. iii. art.

*Pcyrard's Archimedes ;*Rigaud,

*The Arenarins of Archimedes,*Oxford, 1837, printed for the Ashmolean Society; Fabric.

*Bibl. Graec.*vol. ii. p. 544 ; Pope-Blount,

*Censura celebriorum Authorum,*Lond. 1690, fol.

[W.F.D]