3rd. De Sectione Determinata resolved the problem: Given two, three or four points on a straight line, to find another point on it such that its distances from the given points satisfy the condition that the square on one or the rectangle contained by two has to the square on the remaining one or the rectangle contained by the remaining two, or to the rectangle contained by the remaining one and another given straight line, a given ratio. Several restorations of the solution have been attempted, one by W. Snellius (Leiden, 1698), another by Alex. Anderson of Aberdeen, in the supplement to his Apollonius Redivivus (Paris, 1612), but by far the best is by Robert Simson, Opera quaedam reliqua (Glasgow, 1776).
4th. De Tactionibus embraced the following general problem: Given three things (points, straight lines or circles) in position, to describe a circle passing through the given points, and touching the given straight lines or circles. The most difficult case, and the most interesting from its historical associations, is when the three given things are circles. This problem, which is sometimes known as the Apollonian Problem, was proposed by Vieta in the 16th century to Adrianus Romanus, who gave a solution by means of a hyperbola. Vieta thereupon proposed a simpler construction, and restored the whole treatise of Apollonius in a small work, which he entitled Apollonius Gallus (Paris, 1600). A very full and interesting historical account of the problem is given in the preface to a small work of J.W. Camerer, entitled Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c. (Gothae, 1795, 8vo).
5th. De Inclinationibus had for its object to insert a straight line of a given length, tending towards a given point, between two given (straight or circular) lines. Restorations have been given by Marino Ghetaldi, by Hugo d’Omerique (Geometrical Analysis, Cadiz, 1698), and (the best) by Samuel Horsley (1770).
6th. De Locis Planis is a collection of propositions relating to loci which are either straight lines or circles. Pappus gives somewhat full particulars of the propositions, and restorations were attempted by P. Fermat (Oeuvres, i., 1891, pp. 3-51), F. Schooten (Leiden, 1656) and, most successfully of all, by R. Simson (Glasgow, 1749).
Other works of Apollonius are referred to by ancient writers, viz. (1) Περὶ τοῦ πυρίου, On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus); (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere; (4) Ή καθόλου πραγματεία, perhaps a work on the general principles of mathematics in which were included Apollonius’ criticisms and suggestions for the improvement of Euclid’s Elements; (5) Ώκυτόκιον (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of π than the 31⁄7 and 310⁄71 of Archimedes; (6) an arithmetical work (as to which see [Pappus]) on a system of expressing large numbers in language closer to that of common life than that of Archimedes’ Sand-reckoner, and showing how to multiply such large numbers; (7) a great extension of the theory of irrationals expounded in Euclid, Book x., from binomial to multinomial and from ordered to unordered irrationals (see extracts from Pappus’ comm. on Eucl. x., preserved in Arabic and published by Woepcke, 1856). Lastly, in astronomy he is credited by Ptolemy with an explanation of the motion of the planets by a system of epicycles; he also made researches in the lunar theory, for which he is said to have been called Epsilon (ε).
The best editions of the works of Apollonius are the following: (1) Apollonii Pergaei Conicorum libri quatuor, ex versione Frederici Commandini (Bononiae, 1566), fol.; (2) Apollonii Pergaei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo (Oxoniae, 1710), fol. (this is the monumental edition of Edmund Halley); (3) the edition of the first four books of the Conics given in 1675 by Barrow; (4) Apollonii Pergaei de Sectione, Rationis libri duo: Accedunt ejusdem de Sectione Spatii libri duo Restituti: Praemittitur, &c., Opera et Studio Edmundi Halley (Oxoniae, 1706), 4to; (5) a German translation of the Conics by H. Balsam (Berlin, 1861); (6) the definitive Greek text of Heiberg (Apollonii Pergaei quae Graece exstant Opera, Leipzig, 1891-1893); (7) T.L. Heath, Apollonius, Treatise on Conic Sections (Cambridge, 1896); see also H.G. Zeuthen, Die Lehre van den Kegelschnitten im Altertum (Copenhagen, 1886 and 1902).
(T. L. H.)
APOLLONIUS OF RHODES (Rhodius), a Greek epic poet and grammarian, of Alexandria, who flourished under the Ptolemies Philopator and Epiphanes (222-181 B.C.). He was the pupil of Callimachus, with whom he subsequently quarrelled. In his youth he composed the work for which he is known—Argonautica, an epic in four books on the legend of the Argonauts. When he read it at Alexandria, it was rejected through the influence of Callimachus and his party. Disgusted with his failure, Apollonius withdrew to Rhodes, where he was very successful as a rhetorician, and a revised edition of his epic was well received. In recognition of his talents the Rhodians bestowed the freedom of their city upon him—the origin of his surname. Returning to Alexandria, he again recited his poem, this time with general applause. In 196, Ptolemy Epiphanes appointed him librarian of the Museum, which office he probably held until his death. As to the Argonautica, Longinus’ (De Sublim. p. 54, 19) and Quintilian’s (Instit, x. 1, 54) verdict of mediocrity seems hardly deserved; although it lacks the naturalness of Homer, it possesses a certain simplicity and contains some beautiful passages. There is a valuable collection of scholia. The work, highly esteemed by the Romans, was imitated by Virgil (Aeneid, iv.), Varro Atacinus, and Valerius Flaccus. Marianus (about A.D. 500) paraphrased it in iambic trimeters. Apollonius also wrote epigrams; grammatical and critical works; and Κτίσεις (the foundations of cities).
Editio Princeps (Florence, 1496); Merkel-Keil (with scholia, 1854); Seaton (1900). English translations: Verse, by Greene (1780); Fawkes (1780); Preston (1811); Way (1901); Prose by Coleridge (1889); see also Couat, La Poésie alexandrine; Susemihl, Geschichte der griech. Lit. in der alexandnnischen Zeit.