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'''Apollonius of [[Perga]]''' ['''Pergaeus'''] (ca. 262 B.C.E.–ca. 190 B.C.E.) was a [[Greeks|Greek]] [[geometer]] and [[astronomer]], of the [[Alexandrian school]], 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 [[planet]]s 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)|Apollonius crater]] on the Moon was named in his honor.
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[[Image:Conicas1.PNG|right|thumb|150px|An ellipse (shaded green) was one of the conic sections studied and named by Apollonius.]]
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'''Apollonius of [[Perga]]''' '''(Pergaeus)''' (ca. 262 B.C.E. – ca. 190 B.C.E.) was a [[Greeks|Greek]] [[geometer]] and [[astronomer]] of the [[Alexandrian school]], 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]].
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[[Image:Conicas2.PNG|right|thumb|150px|A parabola (shaded green) is another [[conic section]] described by Apollonius.]]
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[[Image:Conicas3.PNG|right|thumb|150px|A hyperbola (shaded green) is a third [[conic section]] studied by Apollonius.]]
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It was Apollonius who gave the [[ellipse]], the [[parabola]], and the [[hyperbola]] the names by which they are now known. The [[hypothesis]] of eccentric [[orbit]]s, or [[deferent and epicycle]]s, to explain the apparent motion of the [[planet]]s and the varying speed of the [[Moon]], are also attributed to him. [[Apollonius' theorem]] demonstrates that two models can be equivalent, given the right parameters. Ptolemy describes this theorem in the ''[[Almagest]]'' 12.1. Apollonius also researched the lunar theory, which he termed [[Epsilon]] (ε). The [[Apollonius (crater)|Apollonius crater]] on the Moon was named in his honor.
  
 
==Life and major work==
 
==Life and major work==
Apollonius was born circa 262 B.C.E., some 25 years after [[Archimedes]]. He flourished in the reigns of [[Ptolemy Euergetes]] and [[Ptolemy Philopator]] ([[247 B.C.E.|247]]-[[205 B.C.E.|205]] B.C.E.). His treatise on [[conics]] earned him fame as "The Great Geometer," an achievement that has assured his fame ever since.
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Apollonius was born circa 262 B.C.E., some 25 years after [[Archimedes]]. He flourished under the reigns of [[Ptolemy Euergetes]] and [[Ptolemy Philopator]] ([[247 B.C.E.|247]]-[[205 B.C.E.|205]] B.C.E.). His treatise on [[conics]] earned him his name, "The Great Geometer," an achievement that assured his fame.
  
Of all his treatises, only ''Conics'' survives. Of the others, we have titles and some indication of their contents thanks to later writers, especially [[Pappus of Alexandria|Pappus]]. After the first edition of the eight-book ''Conics'', Apollonius brought out a second edition at the suggestion of one [[Eudemus of Pergamum]]. As he revised each of the first three books, Appolonius sent [[Eudemus]] a copy; the most considerable changes came in the first two books. Eudemus died before the completion of the rest of the revision, so Appolonius dedicated the last five books to King [[Attalus]] I (241-197 B.C.E.). Only four books have survived in Greek; three more are extant in Arabic; the eighth has never been discovered. Although a fragment has been found of a 13th-century Latin translation from the Arabic, it was not until 1661 that [[Giovanni Alfonso Borelli]] and [[Abraham Ecchellensis]] made a translation of Books v-vii into Latin. Although they used [[Abu 'l-Fath of Ispahan]]'s Arabic version of 983, which was preserved in a Florentine manuscript, most scholars now agree that the best Arabic renderings are those of [[Hilal ibn Abi Hilal]] for Books i-iv and [[Thabit ibn Qurra]] for Books v-vii.
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Of all his treatises, only ''Conics'' survives. Of the others, historians have titles and some indication of their contents thanks to later writers, especially [[Pappus of Alexandria|Pappus]]. After the first edition of the eight-book ''Conics,'' Apollonius brought out a second edition at the suggestion of [[Eudemus of Pergamum]]. As he revised each of the first three books, Apollonius sent [[Eudemus]] a copy; the most considerable changes came in the first two books. Eudemus died before the completion of the rest of the revision, so Apollonius dedicated the last five books to King [[Attalus]] I (241-197 B.C.E.). Only four books have survived in Greek; three more are extant in Arabic; the eighth has never been discovered.
  
In 1710, [[Halley]] used for his version of ''Conics'' an [[Oxford University]] copy of the Borelli-Ecchellensis translation; only for correcting his version did he look at the best (Arabic) manuscript (Bodl. 943).{{Fact|date=May 2007}} Thus the only part of the Arabic manuscript for Books v-vii to be published remains the 1889 [[L. Nix]] Arabic and German edition (publ. Drugulin, Leipzig) of a fragment of Book v. However, Halley also tried to reconstruct Book viii, his task guided partly by lemmas "to the seventh and eighth books" that Pappus included in his own writings and partly by Apollonius's statement that problems solved in the eighth book illustrated the content of the seventh.
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Although a fragment has been found of a thirteenth century Latin translation from the Arabic, it was not until 1661, that [[Giovanni Alfonso Borelli]] and [[Abraham Ecchellensis]] made a translation of Books 5-7 into Latin. Although they used [[Abu 'l-Fath of Ispahan]]'s Arabic version of 983, which was preserved in a Florentine manuscript, most scholars now agree that the best Arabic renderings are those of [[Hilal ibn Abi Hilal]] for Books 1-4 and [[Thabit ibn Qurra]] for Books 5-7.
  
Apollonius was concerned with [[pure mathematics]]. When he was asked about the usefulness of some of his theorem in Book IV of ''Conics'' he proudly asserted that "they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in [[mathematics]] for this and for no other reason." And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that "the subject is one of those which seems worthy of study for their own sake."<ref name="Apollonius">{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=Apollonius of Perga|pages=152|quote=It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, as in ours, there were narrow-minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: "They are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." (Heath 1961, p.lxxiv).<BR>The preface to Book V, relating to maximum and minimum straight lines drawn to a conic, again argues that "the subject is one of those which seems worthy of study for their own sake." While one must admire the author for his lofty intellectual attitude, it may be pertinently pointed out that s day was beautiful theory, with no prospect of applicability to the science or engineering of his time, has since become fundamental in such fields as terrestrial dynamics and celestial mechanics.}}</ref>
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Apollonius was concerned with [[pure mathematics]]. When he was asked about the usefulness of some of his theorem in Book 4 of ''Conics'' he proudly asserted that "they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in [[mathematics]] for this and for no other reason." And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of ''Conics'' that "the subject is one of those which seems worthy of study for their own sake."<ref>Carl B. Boyer (1991), pg. 152.</ref>
  
 
==Conics==
 
==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 [[Euclid]]'s four ''Books on Conics'', show a debt not only to Euclid but also to [[Conon of Samos|Conon]] and [[Nicoteles]].
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Apollonius states that in Books 1-4, he works out the generation of the curves and their fundamental properties presented in Book 1 more fully than did earlier treatises, and that a number of theorems in Book 3 and the greater part of Book 4 are new. Allusions to predecessor's works, such as [[Euclid]]'s four ''Books on Conics,'' show a debt not only to Euclid but also to [[Conon of Samos|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.  
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The generality of Apollonius's treatment is remarkable. He defines and names the conic sections, ''[[parabola]],'' ''[[ellipse]],'' and ''[[hyperbola]].'' He sees each of these curves as a fundamental conic property that is the equivalent of an equation (later called the [[Cartesian equation]]) applied to ''oblique'' axes—for example, axes consisting of a diameter and the tangent at its extremity—that are obtained by cutting an [[oblique circular cone]]. (An oblique circular cone is one in which the axis does not form a 90-degree angle with the directrix. By contrast, a right circular cone is one in which the axis forms a 90-degree angle with the directrix.) The way the cone is cut, he affirms, 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. Thus, Books 5-7 are clearly original.
  
Apollonius's genius reaches its highest heights in Book v. Here he treats of [[normal (mathematics)|normal]]s 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.
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Apollonius's genius reaches its greatest heights in Book 5. Here he treats mathematical [[normal (mathematics)|normal]]s (a ''normal'' is a straight line drawn perpendicular to a surface or to another straight line) 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 determine the [[center of curvature]] at any point and also leads to the Cartesian equation of the [[evolute]] of any conic section.
  
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.<ref name="Boyer Apollonius">{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=Apollonius of Perga |pages=156-157|quote=The method of Apollonius in the ''Conics'' in many respects are so similar to the modern approach that his work sometimes is judged to be an analytic geometry anticipating that of Descartes by 1800 years. The application of references lines in general, and of a diameter and a tangent at its extremity in particular, is, of course, not essentially different from the use fo a coordinate frame, whether rectangular or, more generally, oblique. Distances measured along the diameter from the point of tangency are the abscissas, and segments parallel to the tangent and intercepted between the axis and the curve are the ordinates. The Apollonian relationship between these abscissas and the corresponding ordinates are nothing more nor less than rhetorical forms of the equations of the curves. However, Greek geometric algebra did not provide for negative magnitudes; moreover, the coordinate system was in every case superimposed ''a posteriori'' upon a given curve in order to study its properties. There appear to be no cases in ancient geometry in which a coordinate frame of reference was laid down ''a priori'' for purposes of graphical representation of an equation or relationship, whether symbolically or rhetorically expressed. Of Greek geometry we may say that equations are determined by curves, but not that curves are determined by equations. Coordinates, variables, and equations were subsidiary notions derived from a specific geometric situation; [...] That Apollonius, the greatest geometer of antiquity, failed to develop analytic geometry, was probably the result of a poverty of curves rather than of thought. General methods are not necessary when problems concern always one of a limited number of particular cases.}}</ref>
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In ''Conics,'' Apollonius further developed a method that is so similar to [[analytic geometry]] that his work is sometimes regarded as anticipating the work of [[Descartes]] by some 1800 years. His application of reference lines (such as a diameter and a tangent) is essentially the same as our modern use of a coordinate frame. However, unlike modern analytic geometry, he did not take into account negative magnitudes. Also, he superimposed the coordinate system on each curve after the curve had been obtained. Thus, he derived equations from the curves, but he did not derive curves from equations.<ref>Boyer, pg. 156-157.</ref>
  
 
==Other works==
 
==Other works==
Pappus mentions other treatises of Apollonius:
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Pappus mentions other treatises of Apollonius. 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.
# Λογου αποτομη, ''De Rationis Sectione'' ("Cutting of a [[Ratio]]")
 
# Χωριου αποτομη, ''De Spatii Sectione'' ("Cutting of an Area")
 
# Διωρις μενη τομη, ''De Sectione Determinata'' ("Determinate Section")
 
# Επαφαι, ''De Tactionibus'' ("Tangencies")
 
# Νευσεις, ''De Inclinationibus'' ("Inclinations")
 
# 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.
 
 
 
=== ''De Rationis Sectione'' ===
 
''De Rationis Sectione'' sought to resolve a certain 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]] 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''.
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===''De Rationis Sectione''===
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''De Rationis Sectione'' ''(Cutting of a Ratio)'' sought to resolve a certain 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 Sectione Determinata'' ===
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===''De Spatii Sectione''===
''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.<ref>{{cite book|first=Carl B. |last=Boyer |authorlink=Carl Benjamin Boyer |title=A History of Mathematics |edition=Second Edition |publisher=John Wiley & Sons, Inc. |year=1991 |isbn=0471543977|chapter=Apollonius of Perga |pages=142|quote=The Apollonian treatise ''On Determinate Section'' dealt with what might be called an analytic geometry of one dimension. It considered the following general problem, using the typical Greek algebraic analysis in geometric form: Given four points A, B, C, D on a straight line, determine a fifth point P on it such that the rectangle on AP and CP is in a given ratio to the [[rectangle]] on BP and DP. Here, too, the problem reduces easily to the solution of a quadratic; and, as in other cases, Apollonius treated the question exhaustively, including the limits of possibility and the number of solutions.}}</ref> 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 Anderson (mathematician)|Alexander Anderson]] 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.{{Fact|date=May 2007}}
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''De Spatii Sectione'' ''(Cutting of an Area)'' discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.  
  
=== ''De Tactionibus'' ===
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===''De Sectione Determinata''===
''De Tactionibus'' embraced the following general problem: Given three things (points, straight lines, or [[circle]]s) 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).
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''De Sectione Determinata'' ''(Determinate Section)'' 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.
  
=== ''De Inclinationibus'' ===
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===''De Tactionibus''===
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 [[Marino Ghetaldi]] and [[Hugo d'Omerique]] (''Geometrical Analysis'', Cadiz, 1698) attempted restorations, the best is by Samuel Horsley (1770).{{Fact|date=May 2007}}
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''De Tactionibus'' ''(Tangencies)'' embraced the following general problem: Given three things (points, straight lines, or [[circle]]s) 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 sixteenth 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.''
  
=== ''De Locis Planis'' ===
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===''De Inclinationibus''===
''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).
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The object of ''De Inclinationibus'' ''(Inclinations)'' 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.  
  
===Additional works===
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===''De Locis Planis''===
Ancient writers refer to other works of Apollonius that are no longer extant:
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''De Locis Planis'' ''(Plane Loci)'' is a collection of propositions relating to loci that are either straight lines or circles.
# Περι του πυριου, ''On the Burning-Glass'', a treatise probably exploring the focal properties of the parabola
 
# Περι του κοχλιου, ''On the Cylindrical Helix'' (mentioned by Proclus)
 
# a comparison of the dodecahedron and the icosahedron inscribed in the same sphere
 
# Ἡ καθολου πραγματεια, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's ''[[Euclid's Elements|Elements]]''
 
# Ωκυτοκιον ("quick bringing-to-birth"), in which, according to Eutocius, Appolonius 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 [[Pappus]]) on a system both for expressing large numbers in language more everyday than that of Archimedes' ''[[The Sand Reckoner]]'' 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 ==
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==Legacy==
The best editions of the works of Apollonius are the following:
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Known as "The Great Geometer," Apollonius' works greatly influenced the development of mathematics. His famous book, ''Conics,'' introduced the terms parabola, ellipse, and hyperbola. He conceived the [[hypothesis]] of eccentric [[orbit]]s to explain the apparent motion of the [[planet]]s and the varying speed of the [[Moon]]. A further contribution to the field of mathematics is [[Apollonius' theorem]], which demonstrates that two models can be equivalent given the right parameters.
# ''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 Barrow]]
 
# ''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 of Heiberg (''Apollonii Pergaei quae Graece exstant Opera'', Leipzig, 1891-1893)
 
# [[T. L. Heath]], ''Apollonius, Treatise on Conic Sections'' (Cambridge, 1896)
 
# 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.
 
# ''Conics: Books I-III'' translated by R. Catesby Taliaferro, published by Green Lion Press (ISBN 1-888009-05-5).
 
 
 
== See also ==
 
*[[Apollonian circles]]
 
*[[Apollonian gasket]]
 
*[[Circles of Apollonius]]
 
*[[Descartes' theorem]]
 
  
 
==Notes==
 
==Notes==
 
<references/>
 
<references/>
  
== References ==
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==References==
* Apollonius. ''Apollonii Pergaei quae Graece exstant : cum commentariis antiquis''.  Edited by I. L. Heiberg.  2 volumes.  (Leipzig: [[Teubner]], 1891/1893).
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* Boyer, Carl B. ''A History of Mathematics''. John Wiley & Sons, 1991. ISBN 978-0471543977
* Apollonius. ''Apollonius of Perga Conics Books I-III''. Translated by R. Catesby Taliaferro. (Santa Fe: Green Lion Press, 1998).
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* Fried, Michael N. and Sabetai Unguru. ''Apollonius of Perga’s Conica: Text, Context, Subtext''. Brill, 2001. ISBN 978-9004119779
* Apollonius. ''Apollonius of Perga Conics Book IV''.  Translated with introduction and notes by Michael N. Fried.  (Santa Fe: Green Lion Press, 2002).
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* Heath, T.L. ''Treatise on Conic Sections''. W. Heffer & Sons, 1961.  
* Fried, Michael N. and Unguru, Sabetai. ''Apollonius of Perga’s Conica: Text, Context, Subtext''. (Leiden: Brill, 2001).
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* {{cite encyclopedia
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==External links==
  | last = Toomer
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All links retrieved April 8, 2016.
  | first = G.J.
 
  | title = Apollonius of Perga
 
  | encyclopedia = [[Dictionary of Scientific Biography]]
 
  | volume = 1
 
  | pages = 179-193
 
}} (New York: Charles Scribner's Sons, 1970). ISBN 0684101149
 
  
* [[H. G. Zeuthen|Zeuthen, H.G.]], ''Die Lehre von den Kegelschnitten im Altertum'' (Copenhagen, 1886 and 1902). [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath&amp;idno=AAT2765 University of Michigan Historical Math Collection]. Retrieved October 11, 2007.
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*[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html Apollonius of Perga]. ''www-groups.dcs.st-and.ac.uk''. Apollonius Summary.
* {{1911}}
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*[http://agutie.homestead.com/files/circle/apollonius_problem_circle_1.htm Apollonius' Tangency Problem, Circles]. ''agutie.homestead.com''.
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*[http://www.wilbourhall.org PDF scans of Heiberg's edition of Apollonius of Perga's Conic Sections (public domain)]. ''www.wilbourhall.org''.
  
== External links ==
 
All links retrieved October 11, 2007.
 
*[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=201 Can You Really Derive Conic Formulae from a Cone?] by Gary S. Stoudt, ''Convergence''
 
*[http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html Apollonius of Perga] - Apollonius Summary
 
*[http://agutie.homestead.com/files/circle/apollonius_problem_circle_1.htm Apollonius' Tangency Problem, Circles] by Antonio Gutierrez
 
*[http://www.wilbourhall.org PDF scans of Heiberg's edition of Apollonius of Perga's Conic Sections (public domain)] by Joseph Leichter, J.D., Ph.D
 
  
{{Greek mathematics}}
 
  
 
[[Category:Biography]]
 
[[Category:Biography]]

Revision as of 18:51, 8 April 2016


An ellipse (shaded green) was one of the conic sections studied and named by Apollonius.

Apollonius of Perga (Pergaeus) (ca. 262 B.C.E. – ca. 190 B.C.E.) 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.

A parabola (shaded green) is another conic section described by Apollonius.
A hyperbola (shaded green) is a third conic section studied by Apollonius.

It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which they are now known. The hypothesis of eccentric orbits, or 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 two models can be equivalent, given the right parameters. Ptolemy describes this theorem in the Almagest 12.1. Apollonius also researched the lunar theory, which he termed Epsilon (ε). The Apollonius crater on the Moon was named in his honor.

Life and major work

Apollonius was born circa 262 B.C.E., some 25 years after Archimedes. He flourished under the reigns of Ptolemy Euergetes and Ptolemy Philopator (247-205 B.C.E.). His treatise on conics earned him his name, "The Great Geometer," an achievement that assured his fame.

Of all his treatises, only Conics survives. Of the others, historians have titles and some indication of their contents thanks to later writers, especially Pappus. After the first edition of the eight-book Conics, Apollonius brought out a second edition at the suggestion of Eudemus of Pergamum. As he revised each of the first three books, Apollonius sent Eudemus a copy; the most considerable changes came in the first two books. Eudemus died before the completion of the rest of the revision, so Apollonius dedicated the last five books to King Attalus I (241-197 B.C.E.). Only four books have survived in Greek; three more are extant in Arabic; the eighth has never been discovered.

Although a fragment has been found of a thirteenth century Latin translation from the Arabic, it was not until 1661, that Giovanni Alfonso Borelli and Abraham Ecchellensis made a translation of Books 5-7 into Latin. Although they used Abu 'l-Fath of Ispahan's Arabic version of 983, which was preserved in a Florentine manuscript, most scholars now agree that the best Arabic renderings are those of Hilal ibn Abi Hilal for Books 1-4 and Thabit ibn Qurra for Books 5-7.

Apollonius was concerned with pure mathematics. When he was asked about the usefulness of some of his theorem in Book 4 of Conics he proudly asserted that "they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." And since many of his results were not applicable to the science or engineering of his day, Apollonius further argued in the preface of the fifth book of Conics that "the subject is one of those which seems worthy of study for their own sake."[1]

Conics

Apollonius states that in Books 1-4, he works out the generation of the curves and their fundamental properties presented in Book 1 more fully than did earlier treatises, and that a number of theorems in Book 3 and the greater part of Book 4 are new. Allusions to predecessor's works, such as Euclid'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 remarkable. He defines and names the conic sections, parabola, ellipse, and hyperbola. He sees each of these curves as a fundamental conic property that is the equivalent of an equation (later called the Cartesian equation) applied to oblique axes—for example, axes consisting of a diameter and the tangent at its extremity—that are obtained by cutting an oblique circular cone. (An oblique circular cone is one in which the axis does not form a 90-degree angle with the directrix. By contrast, a right circular cone is one in which the axis forms a 90-degree angle with the directrix.) The way the cone is cut, he affirms, 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. Thus, Books 5-7 are clearly original.

Apollonius's genius reaches its greatest heights in Book 5. Here he treats mathematical normals (a normal is a straight line drawn perpendicular to a surface or to another straight line) 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 determine the center of curvature at any point and also leads to the Cartesian equation of the evolute of any conic section.

In Conics, Apollonius further developed a method that is so similar to analytic geometry that his work is sometimes regarded as anticipating the work of Descartes by some 1800 years. His application of reference lines (such as a diameter and a tangent) is essentially the same as our modern use of a coordinate frame. However, unlike modern analytic geometry, he did not take into account negative magnitudes. Also, he superimposed the coordinate system on each curve after the curve had been obtained. Thus, he derived equations from the curves, but he did not derive curves from equations.[2]

Other works

Pappus mentions other treatises of Apollonius. 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 (Cutting of a Ratio) sought to resolve a certain 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 (Cutting of an Area) discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.

De Sectione Determinata

De Sectione Determinata (Determinate Section) 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.

De Tactionibus

De Tactionibus (Tangencies) 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 sixteenth 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.

De Inclinationibus

The object of De Inclinationibus (Inclinations) 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.

De Locis Planis

De Locis Planis (Plane Loci) is a collection of propositions relating to loci that are either straight lines or circles.

Legacy

Known as "The Great Geometer," Apollonius' works greatly influenced the development of mathematics. His famous book, Conics, introduced the terms parabola, ellipse, and hyperbola. He conceived the hypothesis of eccentric orbits to explain the apparent motion of the planets and the varying speed of the Moon. A further contribution to the field of mathematics is Apollonius' theorem, which demonstrates that two models can be equivalent given the right parameters.

Notes

  1. Carl B. Boyer (1991), pg. 152.
  2. Boyer, pg. 156-157.

References
ISBN links support NWE through referral fees

  • Boyer, Carl B. A History of Mathematics. John Wiley & Sons, 1991. ISBN 978-0471543977
  • Fried, Michael N. and Sabetai Unguru. Apollonius of Perga’s Conica: Text, Context, Subtext. Brill, 2001. ISBN 978-9004119779
  • Heath, T.L. Treatise on Conic Sections. W. Heffer & Sons, 1961.

External links

All links retrieved April 8, 2016.

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