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Encyclopedia Britannica - Main :: ANC-APO |
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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
Ptolemy
the first three books were sent to Eudemus at intervals, as revised, and the later books were dedicated (after Eudemus' death) to King Attalus I. (241197 B.C.). Only four Books have survived in Greek; three more are extant in Arabic; the eighth has never been found. Although a fragment has been found of a Latin translation from the Arabic made in the 13th century, it was not until 1661 that a Latin translation of Books v.-vii. was available. This was made by Giovanni Alfonso Borelli and Abraham Ecchellensis from the free version in Arabic made in 983 by Abu '1-Fath of Ispahan and preserved in a Florence MS. But the best Arabic translation is that made as regards Books i.-iv. by Hilal ibn Abi Hilal (d. about 883), and as regards Books v.-vii. by Tobit ben Korra (836901). Halley used for his translation an Oxford MS. of this translation of Books v.-vii., but the best MS. (Bodl. 943) he only referred to in order to correct his translation, and it is still unpublished except for a fragment of Book v. published by L. Nix with German translation (Drugulin, Leipzig
The degree of originality of the Conics can best be judged from Apollonius' 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
parabola
original
drawn
drawn
The other treatises of Apollonius mentioned by Pappus are -1st, AOyov aaoro a', Cutting off a Ratio; 2nd, Xwplou airoroil, Cutting off an Area; 3rd, t.twplvpEVq ropil, Determinate Section; 4th,'Eiraai, Tangencies; 5th, Nebo-es, Inclinations; 6th, Tolroi E7rl7rEb01, Plane Loci. Each of these was divided into two books, and, with the Data, the Porisms and Surface-Loci of Euclid and the Conics of Apollonius were, according to Pappus, included in the body
1st. De Rationis Sectione had for its subject the resolution of the following problem: Given two straight lines and a point in each, to draw through a third given point a straight line cutting the two fixed lines, so that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio. 2nd. De Spatii Sectione discussed the similar problem which requires the rectangle contained by the two intercepts to be equal to a given rectangle. 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 along with a restoration of the second treatise in r7o6. 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, 1622), 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, 2600). A very full and interesting historical account of the problem is given in the preface to a small work of J. W. Camerer, entitled A pollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libros, 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, 2698), 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 (Euvres, i., 2891, 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) HEpi Tor) 7rvplou, On the Burning-Glass, where the focal properties of the parabola
quick
The best editions of the works of Apollonius are the following : (I) Apollonii Pergaei Conicorum libri quatuor, ex version Frederici Commandini (Bononiae, 1566), fol. ;. (2) A pollonii Pergaei Conicorum libri octo, et Sereni Antissensis de Sectione Cylindri et Coni libri duo (Oxoniae, 171o), 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 (A pollonii Pergaeiquae Graece exstant Opera, Leipzig
Cambridge , 1896) ; see also H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886 and 1902). (T. L. H.)End of Article: APOLLONIUS OF PERGA [PERGAEUS] If you wish, you can link directly to this article.
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