Apollonius of Perga

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Apollonius of Perga [Pergaeus] (c. 262 BC–c. 190 BC) was a Greek geometer and astronomer, of the Alexandrian school, noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Isaac Newton, and René Descartes. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets and the varying speed of the Moon, are also attributed to him. Apollonius' theorem demonstrates that the two models are equivalent given the right parameters. Ptolemy describes this theorem in the Almagest XII.1. Apollonius also researched the lunar theory, for which he is said to have been called Epsilon (ε). The Apollonius crater on the Moon was named in his honour.

Contents

[edit] Life and major work

Apollonius was probably born some twenty-five years later than Archimedes, i.e. about 262 BC He flourished in the reigns of Ptolemy Euergetes and Ptolemy Philopator (247-205 BC). His treatise on Conics gained him the title of 'The Great Geometer', and is that by which his fame has been transmitted 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, especially Pappus. After the Conics in eight Books had been written in a first edition, Apollonius brought out a second edition, considerably revised as regards Books i.-ii., at the suggestion of one Eudemus of Pergamum; 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. (241-197 BC). 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 'l-Fath of Ispahan and preserved in a Florence manuscript. 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 Thabit ibn Qurra (836-901).

Halley used for his translation an Oxford manuscript of this translation of Books v.-vii., but the best manuscript (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, 1889). Halley added in his edition (1710) a restoration of Book viii., in which he was guided by the fact that Pappus of Alexandria gives lemmas "to the seventh and eighth books" under that one heading, as well as by the statement of Apollonius himself that the use of the seventh book was illustrated by the problems solved in the eighth.

[edit] Conics

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 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. It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names parabola, ellipse, hyperbola. Books v.-vii. are clearly original.

Apollonius' 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 center of curvature at any point and leading at once to the Cartesian equation of the evolute of any conic.

[edit] Other works

The other treatises of Apollonius mentioned by Pappus are

  1. Λογου αποτομη, De Rationis Sectione ("Cutting off a Ratio")
  2. Χωριου αποτομη, De Spatii Sectione ("Cutting of an Area")
  3. Διωρις μενη τομη, De Sectione Determinata ("Determinate Section")
  4. Επαφαι, De Tactionibus ("Tangencies")
  5. Νευσεις, De Inclinationibus ("Inclinations")
  6. Τοποι επιπεδοι, De Locis Planis ("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 of the ancient analysis.

[edit] De Rationis Sectione

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.

[edit] De Spatii Sectione

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 De Rationis Sectione was found towards the end of the 17th century in the Bodleian library by Edward Bernard, who began a translation of it; Halley finished it and published it along with a restoration of the De Spatii Sectione in 1706.

[edit] De Sectione Determinata

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 Snellius (Willebrord Snell, Leiden, 1698), another by Alexander 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).

[edit] De Tactionibus

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).

[edit] De Inclinationibus

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).

[edit] De Locis Planis

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).

[edit] Additional works

Other works of Apollonius are referred to by ancient writers, viz.

  1. Περι του πυριου, On the Burning-Glass, where the focal properties of the parabola probably found a place
  2. Περι του κοχλιου, On the Cylindrical Helix (mentioned by Proclus)
  3. a comparison of the dodecahedron and the icosahedron inscribed in the same sphere
  4. Ἡ καθολου πραγματεια, 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. Ωκυτοκιον (quick bringing-to-birth), in which, according to Eutocius, he showed how to find closer limits for the value of π than the 3-1/7 and 3-10/71 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).

[edit] Published editions

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 Isaac 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)
  8. A translation of the Books v-vii from the Arabic to English was published in two volumes by Springer Verlag in 1990 (ISBN 0-387-97216-1), volume 9 in the "Sources in the history of mathematics and physical sciences" series. The translation, by G. J. Toomer, features English and Arabic on facing pages.

[edit] References

  • Apollonius. Apollonii Pergaei quae Graece exstant cum commentariis antiquis. Edited by I. L. Heiberg. 2 volumes. (Leipzig: Teubner, 1891/1893).
  • Apollonius. Apollonius of Perga Conics Books I-III. Translated by R. Catesby Taliaferro. (Santa Fe: Green Lion Press, 1998).
  • Apollonius. Apollonius of Perga Conics Book IV. Translated with introduction and notes by Michael N. Fried. (Santa Fe: Green Lion Press, 2002).
  • Fried, Michael N. and Unguru, Sabetai. Apollonius of Perga’s Conica: Text, Context, Subtext. (Leiden: Brill, 2001).

[edit] See also

[edit] External links

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