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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 tJie 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 (f) between the volumes, and also between the surfaces, of the sphere and circumscribing cylinder; and the ratio (5) 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, Dif. Col. vol. i. pp. 63 and 431; and compare De Morgan, Diffi. Cal. p. 32.)
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 ff of the diameter but less than f 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.
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.
The book on Conoids and Splieroids relates chiefly to the volumes cut off by planes from the solids so called; those namely which are 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.)
The Arenarius (6 Tafi/uiTTjs) 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 mathematician's genius. Having briefly discussed the opinions of Aristarchus upon the constitution and extent of the Universe [aristarchus], and described his own method of determining 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 Sa/rruAos in diameter, cannot contain more than 640 millions of grains of sand ; then, taking the stadium, in round numbers, as not greater than 10,000 5a/cruAoj, 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 two books On Floating Bodies (Ilepl T<av 'Oxou/xez/wi/) 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 Ilepl twv "Ydcm e^io-ra-l^kvuv in Ang. Mai's Collection, vol. i. p. 427.
The treatise entitled Lemmata is a collection oi 15 propositions in plane geometry. It is derived from an Arabic MS. and its genuineness lias beer doubted. (See Torelli's preface.)
Eutocius of Ascalon, about a. d. 600, wrote j commentary on the Treatises on the Sphere anc Cylinder, on the Dimension of the Circle, and 01 Centres of Gravity. All the works above men tioned, together with this Commentary, were fount on the taking of Constantinople, and brought firs into Italy and then into Germany. They wer printed at Basle in 1544, in Greek and Latin, b; Hervagius. Of the subsequent editions by far th best is that of Torelli, "Archim. quae super! omnia, cum Eutocii Ascalonitae commentarii: Ex recens. Joseph. Torelli, Veronensis," Oxoi 1792. It was founded upon the Basle editioi except in the case of the Arenarius, the text < which is taken from that of Dr. Wallis, who pul lished this treatise and the Dimensio Circuli, wit a translation and notes, at Oxford, in 1679. (The are reprinted in vol. iii. of his works.)
The Arenarius, having been little meddled wil by the ancient commentators, retains the Doi dialect, in which Archimedes, like his countrynif Theocritus, wrote. (See Wallis, Op. vol. iii. p 537, 545. Tzetzes says, eAeye Se ical Swptcn cbwvrj 'Svpa.Kovo'ict, Ha /8o>, Kal %api<rTia>fi r&v y< Kivrjffta Tratrai/.) A French translation of t works of Archimedes, with notes, was publish' by F. Peyrard, Paris, 1808, 2 vols. 8vo., and ; English translation of the Arenarius by G. And( son, London, 1784.
(G. M. Mazuchelli, Notizie istoriche e critu intorno alia vita^ alle invenzioni, ed agli scritti Archimede, Brescia, 1737, 4to.; C. M. Brande Dissertatio sistens Archimedis vitam, ejusque Mathesin merita, Gryphiswald. 1789,4to.; Marte in Ersch und Gruber, Allyemeine Encydop'dc art. Archimedes; Quarterly Review, vol. iii. i PeyranVs Archimedes; Rigaucl, The Arenarius Archimedes, Oxford, 1837, printed for the Ashr