APOLLONIUS OF PERGA (PERGAEUS), Greek geometer of the Alexandrian school, was probably born some twenty-five years later than Archimedes, i.e. about 262 B.C. He flourished in the reigns of Ptolemy Euergetes and Ptolemy Philopator 205 B.e.). His treatise on Conics gained him the title of the Great Geometer, and is that by which his fame has been trans mitted to modern times. All his numerous other treatises have perished, save one, and we have only their titles handed down, with general indications of their contents, by later writers, es pecially Pappus. After the Conics in eight Books had been written in a first edition, Apollonius brought out a second edition, con siderably revised as regards Books i.–ii., at the instance of one Eudemus of Pergamum ; the first three books were sent to Eudemus at intervals, as revised, and the later books were dedi cated (after Eudemus's death) to King Attalus I. (241-197 B.c. ). Only four Books have survived in Greek ; three more are extant in Arabic ; the eighth has never been found. Books v.–vii. were translated into Latin by Giacomo Alfonso Borelli and Abraham Ecchellensis from the free version in Arabic made in 983 by Abu'l Fath of Ispahan and preserved in a Florence ms. But the best Arabic translation is that made as regards Books i.–iv. by Hilal b. Abi Hilal (d. about 883), and as regards Books v.–vii. by Thabit b. Qurra (826-901). Halley used the latter version for his translation of Books v.–vii., but the best ms. (Bodl. 943) is still unpublished except for a fragment of Book v., published by L. Nix with German translation (Drugulin, Leipzig, 1889). Halley added in his edition (1710) a restoration of Book viii.
The degree of originality of the Conics can best be judged from Apollonius's own prefaces. Books i.–iv. form an "elementary introduction," i.e. contain the essential principles; the rest are specialized investigations in particular directions. For Books i.–iv. he claims only that the generation of the curves and their fundamental properties in Book i. are worked out more fully and generally than they were in earlier treatises, and that a number of theorems in Book iii. and the greater part of Book iv. are new. That he made the fullest use of his predecessors' works, such as Euclid's four books on Conics, is clear from his allusions to Euclid, Conon and Nicoteles. The generality of treatment is indeed remarkable ; he gives as the fundamental property of all the conics the equivalent of the Cartesian equation referred to oblique axes (consisting of a diameter and the tangent at its extremity) obtained by cutting an oblique circular cone in any manner, and the axes appear only as a particular case after he has shown that the property of the conic can be expressed in the same form with reference to any new diameter and the tangent at its extremity. On the basis of the form of the funda mental property (expressed in the terminology of the "application of areas") Apollonius called the curves for the first time by the names parabola, ellipse, hyperbola. Books v.–vii. are clearly original. Apollonius's genius takes its highest flight in Book v., where he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent properties), discusses how many normals can be drawn from particular points, finds their feet by construction, and gives propositions determining the centre of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.
The other treatises of Apollonius (each in two Books) men tioned by Pappus are-1 st, A6 yov airoroili Cutting off a Ratio; 2nd, Xwpiov airoroµil Cutting off an Area; 3rd, 0LcDpccT/s€vfl roµ7 Determinate Section; 4th, 'Eiracpai, Tangencies ; 5th, Nei Bets In clinations; 6th, T67roc EirilrEbot Plane Loci.
An Arabic version of the first was found towards the end of the 17th century in the Bodleian library by Dr. Edward Bernard, who began a translation of it ; Halley finished it and published it with a restoration of the second (1706). A restoration of the third was given by Robert Simson, Opera quaedam reliqua (Glasgow, 1776).
Tangencies 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 touch ing the given straight lines or circles. The most difficult case, and the most interesting historically, 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 hyper bola. Vieta himself solved it by elementary methods and restored the whole treatise of Apollonius in Apollonius Gallus (Paris, 160o) ; an interesting account of the problem is given by J. W. Camerer in Apollonii Pergaei de tactionibus quae supersunt, ac maxime Lemmata Pappi in hos Libros, cam Observationibus, etc. (Gothae, 1795.) A restoration of the fifth has been given by Samuel Horsley (1770), and one of the sixth by Robert Simson (Glasgow, Other works of Apollonius are referred to by ancient writers, viz. (I) 'WI Tov irvpiov On the Burning-Glass, where the focal properties of the parabola probably found a place; (2) Ili pi. Tov KoXXLov On the Cylindrical Helix (mentioned by Proclus) ; (3) a comparison of the dodecahedron and the icosahedron inscribed in the same sphere ; (4) `H 'Kaz,oXov upavy,uareia, perhaps a work on the general principles of mathematics in which were included Apollonius's criticisms and suggestions for the improve ment of Euclid's Elements; (5) 'SlKvTOKCoV (quick bringing-to birth), in which, according to Eutocius, he showed how to find closer limits for the value of than the 34 and 3+1 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) extensions of the theory of irrationals expounded in Euclid, Book x. (see extracts from Pappus's 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 means of epicycles and eccentric circles; he also made researches in the lunar theory, for which he is said to have been called Epsilon (€).
The editions of the Conics include, besides Halley's monu mental edition of all seven Books (Oxford, 1710), Commandinus's Latin Translation (1566), and Barrow's edition (1675), of the first four Books, and Heiberg's definitive text of the same Books, with Eutocius's commentary, etc. (Leipzig, 1891-93) . There is a German translation by H. Balsam (I 861), and an edition in mod ern notation, with introduction, etc., by T. L. Heath (Cambridge, 1896) ; see also H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886 and 1902) . (T. L. H.)