Apollonius of Perga
Encyclopedia
Apollonius of Perga
[Pergaeus] (ca. 262 BC – ca. 190 BC) was a Greek
geometer and astronomer
noted for his writings on conic section
s. 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 orbit
s, or equivalently, deferent and epicycle
s, to explain the apparent motion of the planets and the varying speed of the Moon
, is 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 history, for which he is said to have been called Epsilon
(ε). The crater Apollonius
on the Moon
is named in his honor.
's four Books on Conics, show a debt not only to Euclid but also to Conon
and Nicoteles.
The generality of Apollonius's treatment is indeed remarkable. He defines the fundamental conic property as the equivalent of the Cartesian equation applied to oblique axes—i.e., axes consisting of a diameter and the tangent at its extremity—that are obtained by cutting an oblique circular cone. The way the cone is cut does not matter. He shows that the oblique axes are only a particular case after demonstrating that the basic conic property can be expressed in the same form with reference to any new diameter and the tangent at its extremity. It is the form of the fundamental property (expressed in terms of the "application of areas") that leads him to give these curves their names: parabola
, ellipse
, and hyperbola
. Thus Books v–vii are clearly original.
Apollonius's genius reaches its highest heights in Book v. Here 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 that both determine the center of curvature
at any point and lead at once to the Cartesian equation of the evolute
of any conic.
Apollonius in the Conics further developed a method that is so similar to analytic geometry
that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
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.
In the late 17th century, Edward Bernard
discovered an Arabic version of De Rationis Sectione in the Bodleian Library
. Although he began a translation, it was Halley who finished it and included it in a 1706 volume with his restoration of De Spatii Sectione.
of Aberdeen
, in the supplement to his Apollonius Redivivus (Paris, 1612); and Robert Simson
in his Opera quaedam reliqua (Glasgow, 1776), by far the best attempt.
De Tactionibus embraced the following general problem: Given three things (points, straight lines, or circles) in position, describe a circle passing through the given points and touching the given straight lines or circles. The most difficult and historically interesting case arises when the three given things are circles. In the 16th century, Vieta presented this problem (sometimes known as the Apollonian Problem) to Adrianus Romanus, who solved it with a hyperbola
. Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work Apollonius Gallus (Paris, 1600). The history of the problem is explored in fascinating detail in the preface to J. W. Camerer's brief Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c (Gothae, 1795, 8vo).
and Hugo d'Omerique (Geometrical Analysis, Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).
Perga
Perga was an ancient Greek city in Anatolia and the capital of Pamphylia, now in Antalya province on the southwestern Mediterranean coast of Turkey. Today it is a large site of ancient ruins east of Antalya on the coastal plain. Located there is an acropolis dating back to the Bronze Age...
[Pergaeus] (ca. 262 BC – ca. 190 BC) was a Greek
Greeks
The Greeks, also known as the Hellenes , are a nation and ethnic group native to Greece, Cyprus and neighboring regions. They also form a significant diaspora, with Greek communities established around the world....
geometer and astronomer
Astronomer
An astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
noted for his writings on conic section
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
s. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy
Ptolemy
Claudius Ptolemy , was a Roman citizen of Egypt who wrote in Greek. He was a mathematician, astronomer, geographer, astrologer, and poet of a single epigram in the Greek Anthology. He lived in Egypt under Roman rule, and is believed to have been born in the town of Ptolemais Hermiou in the...
, Francesco Maurolico
Francesco Maurolico
Francesco Maurolico was a Greek mathematician and astronomer of Sicily. Throughout his lifetime, he made contributions to the fields of geometry, optics, conics, mechanics, music, and astronomy...
, Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, and René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
. It was Apollonius who gave the ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, the parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, and the hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
the names by which we know them. The hypothesis
Hypothesis
A hypothesis is a proposed explanation for a phenomenon. The term derives from the Greek, ὑποτιθέναι – hypotithenai meaning "to put under" or "to suppose". For a hypothesis to be put forward as a scientific hypothesis, the scientific method requires that one can test it...
of eccentric orbit
Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System...
s, or equivalently, deferent and epicycle
Deferent and epicycle
In the Ptolemaic system of astronomy, the epicycle was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, Sun, and planets...
s, to explain the apparent motion of the planets and the varying speed of the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
, is also attributed to him. Apollonius' theorem demonstrates that the two models are equivalent given the right parameters. Ptolemy describes this theorem in the Almagest
Almagest
The Almagest is a 2nd-century mathematical and astronomical treatise on the apparent motions of the stars and planetary paths. Written in Greek by Claudius Ptolemy, a Roman era scholar of Egypt,...
XII.1. Apollonius also researched the lunar history, for which he is said to have been called Epsilon
Epsilon
Epsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a close-mid front unrounded vowel . In the system of Greek numerals it has a value of 5. It was derived from the Phoenician letter He...
(ε). The crater Apollonius
Apollonius (crater)
Apollonius is a lunar crater located near the eastern limb of the Moon. It lies in the region of uplands to the west of Mare Undarum and northeast of the Sinus Successus on the Mare Fecunditatis...
on the Moon
Moon
The Moon is Earth's only known natural satellite,There are a number of near-Earth asteroids including 3753 Cruithne that are co-orbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasi-satellites and not true moons. For more...
is named in his honor.
Conics
The degree of originality of the Conics can best be judged from Apollonius's own prefaces. Books i–iv he describes as an "elementary introduction" containing essential principles, while the other books are specialized investigations in particular directions. He then claims that, in Books i–iv, he only works out the generation of the curves and their fundamental properties presented in Book i more fully and generally than did earlier treatises, and that a number of theorems in Book iii and the greater part of Book iv are new. Allusions to predecessor's works, such as EuclidEuclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's four Books on Conics, show a debt not only to Euclid but also to Conon
Conon of Samos
Conon of Samos was a Greek astronomer and mathematician. He is primarily remembered for naming the constellation Coma Berenices.-Life and work:...
and Nicoteles.
The generality of Apollonius's treatment is indeed remarkable. He defines the fundamental conic property as the equivalent of the Cartesian equation applied to oblique axes—i.e., axes consisting of a diameter and the tangent at its extremity—that are obtained by cutting an oblique circular cone. The way the cone is cut does not matter. He shows that the oblique axes are only a particular case after demonstrating that the basic conic property can be expressed in the same form with reference to any new diameter and the tangent at its extremity. It is the form of the fundamental property (expressed in terms of the "application of areas") that leads him to give these curves their names: parabola
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
, ellipse
Ellipse
In geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
, and hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
. Thus Books v–vii are clearly original.
Apollonius's genius reaches its highest heights in Book v. Here he treats of normals as minimum and maximum straight lines drawn from given points to the curve (independently of tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
properties); discusses how many normals can be drawn from particular points; finds their feet by construction; and gives propositions that both determine the center of curvature
Center of curvature
In geometry, center of curvature of a curve is found at a point that is at a distance equal to the radius of curvature lying on the normal vector. It is the point at infinity if the curvature is zero. The osculating circle to the curve is centered at the center of curvature....
at any point and lead at once to the Cartesian equation of the evolute
Evolute
In the differential geometry of curves, the evolute of a curve is the locus of all its centers of curvature. Equivalently, it is the envelope of the normals to a curve....
of any conic.
Apollonius in the Conics further developed a method that is so similar to analytic geometry
Analytic geometry
Analytic geometry, or analytical geometry has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties...
that his work is sometimes thought to have anticipated the work of Descartes by some 1800 years. His application of reference lines, a diameter and a tangent is essentially no different than our modern use of a coordinate frame, where the distances measured along the diameter from the point of tangency are the abscissas, and the segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. He further developed relations between the abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves. However, although Apollonius came close to developing analytic geometry, he did not manage to do so since he did not take into account negative magnitudes and in every case the coordinate system was superimposed upon a given curve a posteriori instead of a priori. That is, equations were determined by curves, but curves were not determined by equations. Coordinates, variables, and equations were subsidiary notions applied to a specific geometric situation.
Other works
Pappus mentions other treatises of Apollonius:- Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio")
- Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area")
- Διωρισμένη τομή, De Sectione Determinata ("Determinate Section")
- Ἐπαφαί, De Tactionibus ("Tangencies")
- Νεύσεις, De Inclinationibus ("Inclinations")
- Τόποι ἐπίπεδοι, 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.
De Rationis Sectione
De Rationis Sectione sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.De Spatii Sectione
De Spatii Sectione discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.In the late 17th century, Edward Bernard
Edward Bernard
Edward Bernard was an English scholar and Savilian professor of astronomy at the University of Oxford, from 1673 to 1691.-Life:He was born at Paulerspury, Northamptonshire. He was educated at Merchant Taylors' School and St John's College, Oxford, where he was a scholar in 1655; he became a Fellow...
discovered an Arabic version of De Rationis Sectione in the Bodleian Library
Bodleian Library
The Bodleian Library , the main research library of the University of Oxford, is one of the oldest libraries in Europe, and in Britain is second in size only to the British Library...
. Although he began a translation, it was Halley who finished it and included it in a 1706 volume with his restoration of De Spatii Sectione.
De Sectione Determinata
De Sectione Determinata deals with problems in a manner that may be called an analytic geometry of one dimension; with the question of finding points on a line that were in a ratio to the others. The specific problems are: Given two, three or four points on a straight line, 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 a given ratio either (1) to the square on the remaining one or the rectangle contained by the remaining two or (2) to the rectangle contained by the remaining one and another given straight line. Several have tried to restore the text to discover Apollonius's solution, among them Snellius (Willebrord Snell, Leiden, 1698); Alexander AndersonAlexander Anderson (mathematician)
Alexander Anderson was a Scottish mathematician. He was the son of David Anderson of Finshaugh. His sister was Janet Anderson, the mother of the celebrated James Gregory...
of Aberdeen
Aberdeen
Aberdeen is Scotland's third most populous city, one of Scotland's 32 local government council areas and the United Kingdom's 25th most populous city, with an official population estimate of ....
, in the supplement to his Apollonius Redivivus (Paris, 1612); and Robert Simson
Robert Simson
Robert Simson was a Scottish mathematician and professor of mathematics at the University of Glasgow. The pedal line of a triangle is sometimes called the "Simson line" after him.-Life:...
in his Opera quaedam reliqua (Glasgow, 1776), by far the best attempt.
De Tactionibus
- For more information, see Problem of ApolloniusProblem of ApolloniusIn Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...
.
De Tactionibus embraced the following general problem: Given three things (points, straight lines, or circles) in position, describe a circle passing through the given points and touching the given straight lines or circles. The most difficult and historically interesting case arises when the three given things are circles. In the 16th century, Vieta presented this problem (sometimes known as the Apollonian Problem) to Adrianus Romanus, who solved it with a hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...
. Vieta thereupon proposed a simpler solution, eventually leading him to restore the whole of Apollonius's treatise in the small work Apollonius Gallus (Paris, 1600). The history of the problem is explored in fascinating detail in the preface to J. W. Camerer's brief Apollonii Pergaei quae supersunt, ac maxime Lemmata Pappi in hos Libras, cum Observationibus, &c (Gothae, 1795, 8vo).
De Inclinationibus
The object of De Inclinationibus was to demonstrate how a straight line of a given length, tending towards a given point, could be inserted between two given (straight or circular) lines. Though Marin GetaldićMarin Getaldic
Marin Getaldić was a scientist from the Republic of Ragusa. A mathematician and physicist who studied in Italy, England and Belgium, his best results are mainly in physics, especially optics, and mathematics. He was one of the few students of François Viète....
and Hugo d'Omerique (Geometrical Analysis, Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).
De Locis Planis
De Locis Planis is a collection of propositions relating to loci that are either straight lines or circles. Since Pappus gives somewhat full particulars of its propositions, this text has also seen efforts to restore it, not only by P. Fermat (Oeuvres, i., 1891, pp. 3–51) and F. Schooten (Leiden, 1656) but also, most successfully of all, by R. Simson (Glasgow, 1749).Additional works
Ancient writers refer to other works of Apollonius that are no longer extant:- Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola
- Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus)
- A comparison of the dodecahedron and the icosahedron inscribed in the same sphere
- Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's ElementsEuclid's ElementsEuclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the Greek mathematician Euclid in Alexandria c. 300 BC. It is a collection of definitions, postulates , propositions , and mathematical proofs of the propositions...
- Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of π (pi) than those of Archimedes, who calculated 3+1/7 as the upper limit (3.1428571, with the digits after the decimal point repeating) and 3+10/71 as the lower limit (3.1408456338028160, with the digits after the decimal point repeating)
- an arithmetical work (see PappusPappus of AlexandriaPappus of Alexandria was one of the last great Greek mathematicians of Antiquity, known for his Synagoge or Collection , and for Pappus's Theorem in projective geometry...
) on a system both for expressing large numbers in language more everyday than that of Archimedes' The Sand ReckonerThe Sand ReckonerThe Sand Reckoner is a work by Archimedes in which he set out to determine an upper bound for the number of grains of sand that fit into the universe. In order to do this, he had to estimate the size of the universe according to the then-current model, and invent a way to talk about extremely...
and for multiplying these large numbers - 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).
Published editions
The best editions of the works of Apollonius are the following:- Apollonii Pergaei Conicorum libri quatuor, ex versione Frederici Commandini (Bononiae, 1566), fol.
- 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)
- the edition of the first four books of the Conics given in 1675 by Isaac BarrowIsaac BarrowIsaac Barrow was an English Christian theologian, and mathematician who is generally given credit for his early role in the development of infinitesimal calculus; in particular, for the discovery of the fundamental theorem of calculus. His work centered on the properties of the tangent; Barrow was...
- 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
- a German translation of the Conics by H. Balsam (Berlin, 1861)
- The definitive Greek text is the edition of Heiberg (Apollonii Pergaei quae Graece exstant Opera, Leipzig, 1891–1893)
- T. L. HeathT. L. HeathSir Thomas Little Heath was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College...
, Apollonius, Treatise on Conic Sections (Cambridge, 1896) - The Arabic translation of the Books V–VII was first 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 edition was produced by G. J. Toomer and provided with an English translation and various commentaries.
- Conics: Books I–III translated by R. Catesby TaliaferroR. Catesby TaliaferroRobert Catesby Taliaferro was an American mathematician, philosopher, and translator of ancient Greek and Latin works into English. He translated Ptolemy's Almagest, a 2nd-century book on astronomy, the 13 books of Euclid's Elements, Apollonius' works on conic sections, and some works of Plato,...
, published by Green Lion Press (ISBN 1-888009-05-5). (An English translation of Book IV by Michael N. Fried is also available from the same publisher. ISBN 1-888009-20-9) - Apollonius de Perge, Coniques: Texte grec et arabe etabli, traduit et commenté (De Gruyter, 2008–2010), eds. R. Rashed, M. Decorps-Foulquier, M. Federspiel. (This is a new edition of the surviving Greek text (Books I–IV), a full edition of the surviving Arabic text (Books I–VII) with French translation and commentaries.)
- Apollonius of Perga's Conica: Text, Context, Subtext. By Michael N. Fried and Sabetai Unguru (Brill).
- Edmund Halley's Reconstruction of the Lost Book of Apollonius' Conics. By Michael N. Fried (ISBN 1461401453).
See also
- Apollonian circlesApollonian circlesApollonian circles are two families of circles such that every circle in the first family intersects every circle in the second family orthogonally, and vice versa. These circles form the basis for bipolar coordinates...
- Apollonian gasketApollonian gasketIn mathematics, an Apollonian gasket or Apollonian net is a fractal generated from triples of circles, where each circle is tangent to the other two. It is named after Greek mathematician Apollonius of Perga.-Construction:...
- Apollonian networkApollonian networkIn combinatorial mathematics, an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar 3-trees, the maximal planar chordal graphs, the uniquely 4-colorable...
- Circles of ApolloniusCircles of ApolloniusThe term circle of Apollonius is used to describe several types of circles associated with Apollonius of Perga, a renowned Greek geometer. Most of these circles are found in planar Euclidean geometry, but analogs have been defined on other surfaces; for example, counterparts on the surface of a...
- Descartes' theoremDescartes' theoremIn geometry, Descartes' theorem, named after René Descartes, establishes a relationship between four kissing, or mutually tangent, circles. The theorem can be used to construct a fourth circle tangent to three given, mutually tangent circles.-History:...
- Problem of ApolloniusProblem of ApolloniusIn Euclidean plane geometry, Apollonius' problem is to construct circles that are tangent to three given circles in a plane . Apollonius of Perga posed and solved this famous problem in his work ; this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived...
- Apollonius' theoremApollonius' theoremIn geometry, Apollonius' theorem is a theorem relating the length of a median of a triangle to the lengths of its side.Specifically, in any triangle ABC, if AD is a median, thenAB^2 + AC^2 = 2\,...
The Works of Apollonius of Perga online
- Text in Classical Greek: PDF scans of Heiberg's edition of the Conic Sections of Apollonius of Perga, now in the public domain
- In English translation: Treatise on the Conic Sections, trans. T.L. Heath