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EspañolAPOLLONIUS OF PERGA
'Apollonius of Perga' ['Pergaeus'] (ca. 262 BC–ca. 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.
Apollonius was born circa 262 BC, some 25 years after Archimedes. He flourished in the reigns of Ptolemy Euergetes and Ptolemy Philopator (247-205 BC). His treatise on conics earned him fame as "The Great Geometer", an achievement that has assured his fame ever since.
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. 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 BC). 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.
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). 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.
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."[1]
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 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.[2]
Pappus mentions other treatises of Apollonius:
# 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'' 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'' 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''.
''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.[3] 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 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).
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).
''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).
Ancient writers refer to other works of Apollonius that are no longer extant:
# 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 ''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).
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 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).
★ 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).
★
★ Zeuthen, H.G., ''Die Lehre von den Kegelschnitten im Altertum'' (Copenhagen, 1886 and 1902). University of Michigan Historical Math Collection
★
1. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
2. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
3. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
★ Conic Sections of Apollonius at Convergence
★
★ Apollonian Problem Interactive illustration.
★ Apollonian circles
★ Apollonian gasket
★ Circles of Apollonius
★ Descartes' theorem
Life and major work
Apollonius was born circa 262 BC, some 25 years after Archimedes. He flourished in the reigns of Ptolemy Euergetes and Ptolemy Philopator (247-205 BC). His treatise on conics earned him fame as "The Great Geometer", an achievement that has assured his fame ever since.
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. 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 BC). 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.
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). 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.
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."[1]
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 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.[2]
Other works
Pappus mentions other treatises of Apollonius:
# 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''.
''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.[3] 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 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''
''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).
''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 Marino Ghetaldi 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:
# 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 ''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
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 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).
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).
★
★ Zeuthen, H.G., ''Die Lehre von den Kegelschnitten im Altertum'' (Copenhagen, 1886 and 1902). University of Michigan Historical Math Collection
★
1. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
2. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
3. A History of Mathematics, , Carl B., Boyer, John Wiley & Sons, Inc., 1991,
External links
★ Conic Sections of Apollonius at Convergence
★
★ Apollonian Problem Interactive illustration.
See also
★ Apollonian circles
★ Apollonian gasket
★ Circles of Apollonius
★ Descartes' theorem
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