ARCHIMEDES. 



are still extant, though the greatest parts 

 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 Diame- 

 ter 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. Lem- 

 mata. 9. Of the Number of the Sand. 



Among the works of Archimedes which 

 are lost maybe 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 Gold 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 Archimedes's 

 Screw. Athenxus, speaking of the pro- 

 digious ship built by the order of Hiero, 

 says, that Archimedes invented the coch- 

 leon, by means of which the hold, notwith- 

 standing 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, Athe- 

 nxus informs us. he launched Hiero's great 

 ship. 4. The Trispaston, which, accord- 

 ing 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, Balistae, Catapults, Sagitta- 

 rii, Scorpions, Cranes, &c. 6. His Burn- 

 ing Glasses, with which he set fire to the 

 Roman gallies. 7. His Pneumatic and 

 Hydrostatic Engines, concerning which 

 subjects he wrote some books, according 

 to Tzetzes, Pappus, and Turtullian. 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- 

 tions of Archimedes, that appear in his 

 mathematical writings now extant only. 

 He was the first who squared a curvilineal 

 space; unless Hippocratesbe exceptedon 

 account of his lunes. In his time the conic 

 sections were admitted into geometry ,and 

 he applied himself closely to the measur- 

 ing of them, as weli as other figures. 

 Accordingly he determined the relations 

 of spheres, spheroids, arid conoids, to cy- 

 linders and cones ; and the relations of 

 parabolas to rectilinealplanes, whose quad- 

 ratures 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 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 the circle to the determination 

 of the ratio between the diameter and cir- 

 cumference ; which determination how. 

 ever has neveryetbeendone.Beingdisap- 

 pointed of the exact quadrature of the 

 circle, for want of the rectification of its cir* 

 cumference, which all his methods would 

 not effect, he proceeded to assign an. 

 useful approximation to it : this he effect- 

 ed by the numeral calculation of the peri- 

 meters of the inscribed and circumscribed 

 polygons : from which calculation it ap- 

 pears that the perimeter of the circum- 

 scribed regular polygon of 192 sides is to 

 the diameter in a less ratio than that of 3*. 

 or 3j.o to 1 ; and that the perimeter of the 

 inscribed polygon of 96 sides is to the di- 

 ameter in a greater ratio than that of 

 3.1.0 to I- anc j consequently that the ra- 

 tio of the circumference to the diameter 

 lies;between these two ratios. Now the first 

 ratio, of 3^. to 1, reduced to whole num. 

 bers, gives that of 22 to 7, for 3| : 1 : : 

 22 : 7 ; which therefore is nearly the 

 ratio of the circumference to the diame- 

 ter. From this ratio between the circum- 

 ference and the diameter, Archimedes 

 computed the approximate area of the cir- 

 cle, and he found that it is to the square 

 of the diameter as 11 is to 14. He de- 

 termined also the relation between the 

 circle and eclipse with that of their simi- 

 lar parts. And it is probable that he like- 

 wise 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. 



Besides these figures, he determined 

 the measures of the spiral, described by a 

 ' point moving uniformly 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 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 : 



