ARCHIMEDES

, one of the most celebrated mathematicians among the ancients, who flourished about 250 years before Christ, being about 50 years later than Euclid. He was born at Syracuse in Sicily, and was related to Hiero, who was then king of that city. The mathematical genius of Archimedes set him with such distinguished excellence in the view of the world, as rendered him both the honour of his own age, and the admiration of posterity. He was indeed the prince of the ancient mathematicians, being to them what Newton is to the moderns, to whom in his genius and character he bears a very near resemblance. He was frequently lost in a kind of reverie, so as to appear hardly sensible; he would study for days and nights together, neglecting his food; and Plutarch tells us that he used to be carried to the baths by force. Many particulars of his life, and works, mathematical and mechanical, are recorded by several of the ancients, as Polybius, Livy, Plutarch, Pappus, &c. He was equally skilled in all the sciences, astronomy, geometry, mechanics, hydrostatics, optics, &c, in all of which he excelled, and made many and great inventions.

Archimedes, it is said, made a sphere of glass, of a most furprising contrivance and workmanship, exhibiting the motions of the heavenly bodies in a very pleasing manner. Claudian has an epigram upon this invention, which has been thus translated: When in a glass's narrow space confin'd, Iove saw the fabric of th' almighty mind, He smil'd, and said, Can mortals' art alone, Our heavenly labours mimic with their own? The Syracusian's brittle work contains Th' eternal law, that through all nature reigns. Fram'd by his art, see stars unnumber'd burn, And in their courses rolling orbs return: His sun through various signs describes the year; And every month his mimic moons appear. Our rival's laws his little planets bind, And rule their motions with a human mind. Salmoneus could our thunder imitate, But Archimedes can a world create.

Many wonderful stories are told of his discoveries, and of his very powerful and curious machines, &c. Hiero once admiring them, Archimedes replied, these effects are nothing, “But give me, said he, some other place to fix a machine on, and I shall move the earth.” He fell upon a curious device for discovering the deceit which had been practiced by a workman, employed by the said king Hiero to make a golden crown. Hiero, having a mind to make an offering to the gods of a golden crown, agreed for one of great value, and weighed out the gold to the artificer. After some time he brought the crown home of the full weight; but it was afterwards discovered or suspected that a part of the gold had been stolen, and the like weight of silver substituted in its stead. Hiero, being angry at this imposition, desired Archimedes to take it into consideration, how such a fraud might be certainly discovered. While engaged in the solution of this difficulty, he happened to go into the bath; where observing that a quantity of water overflowed, equal to the bulk of his body, it presently occurred to him, that Hiero's question might be answered by a like method: upon which he leaped out, and ran homeward, crying out eu(\rhka! eu(\rhka! I have found it! I have found it! He then made two masses, each of the same weight as the crown, one of gold and the other of silver: this done, he filled a vessel to the brim with water, and put the silver mass into it, upon which a quantity of water overflowed equal to the bulk of the mass; then taking the mass of silver out he filled up the vessel again, measuring the water exactly, which he put in; this shewed him what measure of water answered to a certain quantity of silver. Then he tried the gold in like manner, and sound that it caused a less quantity of water to overflow, the gold being less in bulk than the silver, though of the same weight. He then filled the vessel a third time, and putting in the crown itself, he found that it caused more water to overflow than the golden mass of the same weight, but less than the silver one; so that, finding its bulk between the two masses of gold and silver, and that in certain known proportions, he hence computed the real quantities of gold and silver in the crown, and so manifestly discovered the fraud.

Archimedes also contrived many machines for useful and beneficial purposes: among these, engines for launching large ships; screw pumps, for exhausting the water out of ships, marshes or overflowed lands, as Egypt, &c, which they would do from any depth.

But he became most famous by his curious contrivances, by which the city of Syracuse was so long defended, when besieged by the Roman consul Marcellus; showering upon the enemy sometimes long darts, and stones of vast weight and in great quantities; at other times lifting their ships up into the air, that had come near | the walls, and dashing them to pieces by letting them fall down again; nor could they find their safety in removing out of the reach of his cranes and levers, for there he contrived to fire them with the rays of the sun reflected srom burning glasses.

However, notwithstanding all his art, Syracuse was at length taken by storm, and Archimedes was so very intent upon some geometrical problem, that he neither heard the noise, nor minded any thing else, till a soldier that found him tracing of lines, asked him his name, and upon his request to begone, and not disorder his figures, slew him. “What gave Marcellus the greatest concern, says Plutarch, was the unhappy fate of Archimedes, who was at that time in his museum; and his mind, as well as his eyes, so sixed and intent upon some geometrical sigures, that he neither heard the noise and hurry of the Romans, nor perceived the city to be taken. In this depth of study and contemplation, a soldier came suddenly upon him, and commanded him to follow him to Marcellus; which he refusing to do, till he had finished his problem, the soldier, in a rage, drew his sword, and ran him through.” Livy says he was slain by a soldier, not knowing who he was, while he was drawing schemes in the dust: that Marcellus was grieved at his death, and took care of his funeral; and made his name a protection and honour to those who could claim a relationship to him. His death it seems happened about the 142 or 143 Olympiad, or 210 years before the birth of Christ.

When Cicero was questor for Sicily, he discovered the tomb of Archimedes, all overgrown with bushes and brambles; which he caused to be cleared, and the place set in order. There was a sphere and cylinder cut upon it, with an inscription, but the latter part of the verses quite worn out.

Many of the works of this great man are still extant, though the greatest part of them are lost. The pieces remaining are as follow: 1. Two books on the Sphere and Cylinder.—2. The Dimension of the Circle, or proportion between the diameter and the circumference.— 3. Of Spiral lines.—4. Of Conoids and Spheroids.— 5. Of Equiponderants, or Centres of Gravity.—6. The Quadrature of the Parabola.—7. Of Bodies floating on Fluids.—8. Lemmata.—9. Of the Number of the Sand.

Among the works of Archimedes which are lost, may be reckoned the descriptions of the following inventions, which may be gathered from himself and other ancient authors. 1. His account of the method which he employed to discover the mixture of gold and silver in the crown, mentioned by Vitruvius.—2. His description of the Cochleon, or engine to draw water out of places where it is stagnated, still in use under the name of Archimedes's Screw. Athenæus, speaking of the prodigious ship built by the order of Hiero, says, that Archimedes invented the cochleon, by means of which the hold, notwithstanding its depth, could be drained by one man. And Diodorus Siculus says, that he contrived this machine to drain Egypt, and that by a wonderful mechanism it would exhaust the water from any depth.—3. The Helix, by means of which, Athenæus informs us, he launched Hiero's great ship.—4. The Trispaston, which, according to Tzetzes and Oribasius, could draw the most stupendous weights.—5. The machines, which, according to Polybius, Livy, and Plutarch, he used in the defence of Syracuse against Marcellus, consisting of Tormenta, Balistæ, Catapults, Sagittarii, Scorpions, Cranes, &c.—6. His Burning Glasses, with which he set fire to the Roman gallies.—7. His Pneumamatic and Hydrostatic engines, concerning which subjects he wrote some books, according to Tzetzes, Pappus, and Tertullian.—8. His Sphere, which exhibited the celestial motions. And probably many others.

A whole volume might be written upon the curious methods and inventions of Archimedes, that appear in his mathematical writings now extant only. He was the first who squared a curvilineal space; unless Hypocrates must be excepted on account of his lunes. In his time the conic sections were admitted into geometry, and he applied himself closely to the measuring of them, as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whose quadratures had long before been determined by Euclid. He has left us also his attempts upon the circle: he proved that a circle is equal to a right-angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently, that its area is equal to the rectangle of half the diameter and half the circumference; thus reducing the quadrature of the circle to the determination of the ratio between the diameter and circumference; which determination however has never yet been done. Being disappointed of the exact quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to assign an useful approximation to it: this he effected by the numeral calculation of the perimeters of the inscribed and circumscribed polygons: from which calculation it appears that the perimeter of the circumscribed regular polygon of 192 sides, is to the diameter, in a less ratio than that of 3 1/7 or 3 10/70 to 1; and that the perimeter of the inscribed polygon of 96 sides, is to the diameter, in a greater ratio than that of 3 10/71 to 1; and consequently that the ratio of the circumference to the diameter, lies between these two ratios. Now the first ratio, of 3 1/7 to 1, reduced to whole numbers, gives that of 22 to 7, for 3 1/7 : 1 :: 22 : 7; which therefore is nearly the ratio of the circumference to the diameter. From this ratio between the circumference and the diameter, Archimedes computed the approximate area of the circle, and he found that it is to the square of the diameter, as 11 is to 14. He determined also the relation between the circle and ellipse, with that of their similar parts. And it is probable that he likewise attempted the hyperbola; but it is not to be expected that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented.

Beside these sigures, he determined the measures of the spiral, described by a point moving unisormly along a right line, the line at the same time revolving with a uniform angular motion; determining the proportion of its area to that of the circumscribed circle, as also the proportion of their sectors.

Throughout the whole works of this great man, we every where perceive the deepest design, and the finest invention. He seems to have been, with Euclid, exceedingly careful of admitting into his demonstrations | nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them; yet to determine those relations, he does not increase the number, and diminish the magnitude, of the sides of the polygon ad infinitum; but from this plain fundamental principle, allowed in Euclid's Elements, (viz, that any quantity may be so often multiplied, or added to itself, as that the result shall exceed any proposed finite quantity of the same kind,) he proves that to deny his figures to have the proposed relations, would involve an absurdity. And when he demonstrated many geometrical properties, particularly in the parabola, by means of certain progressions of numbers, whose terms are similar to the inscribed figures; this was still done without considering such series as continued ad infinitum, and then collecting or summing up the terms of such infinite series.

There have been various editions of the existing writings of Archimedes. The whole of these works, together with the commentary of Eutocius, were found in their original Greek language, on the taking of Constantinople, from whence they were brought into Italy; and here they were found by that excellent mathematician John Muller, otherwise called Regiomontanus, who brought them into Germany: where they were, with that Commentary, published long afterwards, viz, in 1544, at Basil, being most beautifully printed in folio, both in Greek and Latin, by Hervagius, under the care of Thomas Gechauff Venatorius.—A Latin translation was published at Paris 1557, by Pascalius Hamellius.—Another edition of the whole, in Greek and Latin, was published at Paris 1615, in folio, by David Rivaltus, illustrated with new demonstrations and commentaries: a life of the author is presixed; and at the end of the volume is added some account, by way of restoration, of our author's other works, which have been lost; viz, The Crown of Hiero; the Cochleon or Water Screw; the Helicon, a kind of endless screw; the Trispaston, consisting of a combination of wheels and axles; the Machines employed in the defence of Syracuse; the Burning Speculum; the Machines moved by Air and Water; and the Material Sphere.—In 1675, Dr. Isaac Barrow published a neat edition of the works, in Latin, at London, in 4to; illustrated, and succinctly demonstrated in a new method.—But the most complete of any, is the magnisicent edition, in folio, lately printed at the Clarendon press, Oxford, 1792. This edition was prepared ready for the press by the learned Joseph Torelli, of Verona, and in that state presented to the University of Oxford. The Latin translation is a new one. Torelli also wrote a preface, a commentary on some of the pieces, and notes on the whole. An account of the life and writings of Torelli is prefixed, by Clemens Sibiliati. And at the end a a large appendix is added, in two parts; the first being a Commentary on Archimedes's paper upon Bodies that float on Fluids, by the Rev. Abram Robertson of Christ Church College; and the latter is a large collection of various readings in the Manuscript works of Archimedes, found in the library of the late king of France, and of another at Florence, as collated with the Basil edition above mentioned.

There are also extant other editions of certain parts of the works of Archimedes. Thus, Commandine published, in 4to, at Bologna 1565, the two books concerning Bodies that Float upon Fluids, with a Commentary. Commandine published also a translation of the Arenarius. And Borelli published, in folio, at Florence 1661, Archimedes's Liber Assumplorum, translated into Latin from an Arabic manuscript copy. This is accompanied with the like translation, from the Arabic, of the 5th, 6th, and 7th books of Apollonius's Conics. Mr. G. Anderson published (in 8vo. Lond. 1784) an English translation of the Arenarius of Archimedes, with learned and ingenious notes and illustrations. Dr. Wallis published a translation of the Arenarius. And there may be other editions beside the above, but these are all that I have got, or know of.

Archimedes's Screw. See Screw of Archimedes.

Archimedes's Burning-glass. See Burning-glass.