ir. ARCHIMEDES. nay chord is equal to two-thirds of the parallelo- gram 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 Sp/tere 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 (|) between the volumes, and also between the surfaces, of the sphere and circumscribing cylinder ; and the ratio {) 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 dis- covery and geometrical proof of them required the genius of Archimedes. Moreover, the legitimacy of the modem 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, Dif. Cat. p. 32.) The book on the Dimension nfthe Circle consists of three propositions. 1st. Every circle is equal to a right-angled triangle of which the sides con- taining 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 proposi- tions are established by comparing the circum- ference of the circle with the perimeters of the inscribed and circumscribed polygons of QQ 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 tliat Conon, to whom he had sent them as pro- blems 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 unsuccess- ful, attempts. (See the introduction.) The Arenarius (6 Waixiiir-qs) 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 in- finite, 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 dence identical in principle with AIICIIIMEDES. the modem 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 determining the ap- parent 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 haKrvKos in diame- ter, cannot contain more than 640 millions of grains of sand ; then, taking the stadium, in round numbers, as not greater than 10,000 5a«TuAoz, 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 nota- tion would be expressed by unity with 63 ciphers annexed. The two books On Floating Bodies (iTcpl rwv 'OxovfjLevuv) 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 ex- ception of a fragment Uepl rwv "TSari ecpiaTa^ ixfvoiv in Ang. Mai's Collection, vol. i. p. 427. 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 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. All the works above men- tioned, together with this 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, "Archira. 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 pub- lished this treatise and the Dimensio Circuli, with a translation and notes, at Oxford, in 1679. (They are reprinted in vol. iii. of his works.) 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, eA676 Se Kal ^wpiarrl^ (pavrj 'ZvpaKovaia, Hd )8co, Kal xaptCTiwi/t rdv yav Kivriaa irdaau.) 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. Ander- son, London, 1784. (G. M. Mazuchelli, Notizie istoricke e critiche intorno alia vita, alle invenzioni, ed agli scritti di Ai-chimede, Brescia, 1737, 4to. ; C. M. Brandelii, Dissertatio sistens Archimedis vitam^ ejusque in Mathesiyimerita, Grj'phiswald. 1789,4to.; Martens, in Ersch und Gruber, Allgemeine Enct/clopadiCf art. Archimedes; Quarterly Review, vol. iii. art. Pe>/rards Archimedes; Rigaud, The Arenarius of ArchimedcSy Oxford, 1837, printed for the Aslimo-
