But he became most famous by his en rious coutrivances, by which the city of Syracuse was so long defended, when be sieged 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 nearthe walls, and dashing them to picots 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 set fire to diem with the rays of the sun reflected from 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 regarded any thing else, till a soldier that found him tracing lines asked his name, and upon his re quest to begone, and not disorder his figures, slew him. " What gave Marcel lus 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 fixed and in tent upon some geometrical figures, that he neither heard the noise and hurry of the Romans, nor perceived the city to be taken. In the depth of study and contem plation,a soldier came suddenly upon him, and commanded him to follow him to MareElIns which he refusing to do, till he had finished his problem, the soldier in adrew his sword, and ran him through." Livy says lie was slain by a soldier, not knowing who lie was, while he was drawing schemes in the dust; that .larccllns 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. Ills death it seems happened about the 142d or 143d olympiad, or 210 years be fore the birth of Christ.
When Cicero was qutestor for Sicily, he discovered the tomb of Archimedes, all overgrown with bushes and brambles; which lie caused to he cleared, and the place set in order. There were a sphere and cylinder cot upon it, with an inscrip tion, but the latter part of the verses were quite worn out.
Many of the works of this great man are still extant, thoughthe greatest parts of them are lost. The 'pieces remaining are as follow: 1. Two books °tithe Sphere and Cylinder. —2. The Dimension of the Circle, or Proportion between the Diame ter and thc Circumference.-3. Of Spiral lines.-4. Of Colloids and Spheroids.-5. Of Equiponderants, or Centres of Gravity. —6. The Quadrature of the Parabola.—. 7. Of Bodies floating on Fluids.-8. Lem insta.-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 Mix ture of Cold and Silver in the crown men tioned 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 Archinicdes's Screw. Atheraus, speaking of the pro digious ship buih by the order of Hier°, says, that Archimedes invented the cock icon, by tneans ofwhitli the hold, notwith standing its depth, could be drained by one naan. And Diodorus Siculus says, that he contrived this machine to drain Egy pt, and that, by a wonderful mechanism, it would exhalist the water from any depth. 3. The Helix, by rneans of which, Athe nxtis informs tis,he launchedthero's great ship.--4.;The Trispaston, which, accord ing to Tzetzes an4 Oribasius, could draw the most stupendous weights.-5. Machines, winch, according to Polybius, Lilly, and Plutarch, !lensed in the defence of' Syracuse against Marcellus, consisting of Tormenta, Balistm, Catapults, Sagitta rii, Scorpions, Cranes, &c.-6. Ills Burn ing Glasses with which he set fire to the Boman gallies.-7. His Pneumatic and Hydrostatic Engines, concerning which subjects he wrote some books, according to Tzetzes, Pappus, and Turtulfian.-8. His Sphere, which exhibited the celestial motions. And probably many others.
A considerable volume might be writ ten upon the curious methods and inven tion of Archimedes, that appear in his mathematical writings now extant only. fle was the first who squared a curvilineal space; unless I lipocrates be excepted on account of his hines. In his time the conic sections were admitted into geometry, and he applied himself closely to the measu ring of them, as well as other figures. Accordingly he determined the relation's of spheres, spheroids, and conoids, to cy linders and cones ; and the relations of parabolas to rectilineal planes,whose qu ad. Tatures had long before been determined by Euclid. He has leftus also his attempts upon the circle : he proved that a circle is equal to aright-angled triangle whose base is equalto the eiretunferenee, and its altitude equal to the radius; and conse quently, that its area is equal to the rec tangle of half the diameter and half the circumference ; thus reducing the quad rature of tlfe circle to the determination of the ratio between the diameter and cir cumference; which determination how ever has never yet been done. Beingdisap Pointed of the exact quadrapre of the ci rele, for want ofthe rectification of its cir cumference, which all his methods would: not effect, lie proceeded to assign an useful approximation to it: this he effecat edby the numeral calculation of the peri inetees of the inscribed an cl circumscribed polygons: from which caleitlation it ap pears that the perimeter of the circum scribed regular polygon of 192 sides is to the diarneterin a less ratio than that of 3+ or 34105 to 1; and that the perimeter of the inscribed polygon of% sidesis to the di ameter in a greater ratio than•-that of '1a to 1 • and consequently that the ra tio of the circumference to the diameter lies between these two ratios. Now the first ratio, of 34- to 1, reduced to whole num_ bers, gives that of 22 to 7, for 3 : 1 : : 22 : 7; which therefore is neatly the ratio of the circumference to the diame ter. Fiom this ratio between the erum ference and the diameter, Arch' edes computed the approximate area of the cir cle, and he found that it is to the square of the diameter :Ls 11 is to 14. lle de terrained also the relation between the circle and eclipse with that of theinipimi lar parts. And it it probable that lierlike wise atterapted 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 varieus methods that havt since been invented.
Besides these figures,he determined the measures of the spiral, desaibecl by a point moving uniformly :dung 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 propor tion 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, ex ceedingly careful of admitting into his de monstrations nothing but principles per fectly geometrical and unexceptionable : and although his most general method of denionstniting the relations of curved 11 8/Te At Atrawbt, relations, he does not increase the num ber, and diminish the magnitude, of the sides of the polygon ad iv rutum ; but from this plain fundamental principle, al lowed in Euclid's Elements, (•iz. 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 pro perties, particularly in the parabola, by means of certain progressions of numbers, whose terms are similar to the inscribed figures ; this was still done without con sidering such series as continued ad iln nitum, and then collecting or summing up the 'terms of such infinite series.
Thum have been various editions of