Difference between revisions of "Apollonius of Perga" - New World Encyclopedia

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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.
+
'''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]]'' 12.1. 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.
  
 
==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.
 
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.
  
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.
+
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 thirteenth-century Latin translation from the Arabic, it was not until 1661 that [[Giovanni Alfonso Borelli]] and [[Abraham Ecchellensis]] made a translation of Books v-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.
 +
 
 +
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 8, 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 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 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).<BR>The preface to Book 5, 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 in his day it was a 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>
  
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.
 
  
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>
 
  
 
==Conics==
 
==Conics==
Line 53: Line 59:
  
  
== 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)
 
# ''Sources in the history of mathematics and physical sciences''. volume 9. Berlin ; New York : Springer-Verlag, 1976- ISSN 0172-6315 
 
# ''Conics: Books I-III'' translated by R. Catesby Taliaferro, published by Green Lion Press. ISBN 1-888009-05-5.
 
  
== See also ==
+
 
*[[Apollonian circles]]
+
==Legacy==
*[[Apollonian gasket]]
+
Known as "The Great Geometer," Apollonius' works had a very great influence on the development of mathematics and his famous book ''Conics'' introduced the terms parabola, ellipse, and hyperbola.
*[[Circles of Apollonius]]
 
*[[Descartes' theorem]]
 
  
 
==Notes==
 
==Notes==
 
<references/>
 
<references/>
  
== References ==
+
==References==
* Apollonius. ''Apollonii Pergaei quae Graece exstant : cum commentariis antiquis''.  Edited by I. L. Heiberg.  2 volumes.  (Leipzig: [[Teubner]], 1891/1893).
+
*Boyer, Carl B. ''Apollonius of Perga, 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).
+
*Fried, Michael N. and Unguru, Sabetai. ''Apollonius of Perga’s Conica: Text, Context, Subtext'', Brill, 2001. ISBN 978-9004119779, ISBN 978-9004119772
* Apollonius. ''Apollonius of Perga Conics Book IV''.  Translated with introduction and notes by Michael N. Fried.  (Santa Fe: Green Lion Press, 2002).
+
*Toomer, G.J. "Apollonius of Perga," ''Dictionary of Scientific Biography'' (Vol. I, pp. 179-193), Charles Scribner's Sons, 1970. ISBN 978-0684101149
* Fried, Michael N. and Unguru, Sabetai. ''Apollonius of Perga’s Conica: Text, Context, Subtext''.  (Leiden: Brill, 2001). ISBN 9004119779 ISBN 9789004119772
 
* {{cite encyclopedia
 
  | last = Toomer
 
  | 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.
 
* {{1911}}
 
  
== External links ==
+
==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://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://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Apollonius.html Apollonius of Perga] - Apollonius Summary
Line 98: Line 78:
 
*[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
 
*[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 01:33, 3 December 2007


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. 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 12.1. 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 honor.

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 (247-205 B.C.E.). 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 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 v-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.

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 8, 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 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

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:

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

Each of these was divided into two books, and—with the Data, the Porisms, and Surface-Loci of Euclid and the Conics of Apollonius—were, according to Pappus, included in the body of the ancient analysis.

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.[citation needed]

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).[citation needed]

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).


Legacy

Known as "The Great Geometer," Apollonius' works had a very great influence on the development of mathematics and his famous book Conics introduced the terms parabola, ellipse, and hyperbola.

Notes

  1. Boyer, Carl B. (1991). "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 152. ISBN 0471543977. “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).
    The preface to Book 5, 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 in his day it was a 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.”
     
  2. Boyer, Carl B. (1991). "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 156-157. ISBN 0471543977. “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.” 
  3. Boyer, Carl B. (1991). "Apollonius of Perga", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 142. ISBN 0471543977. “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.” 

References
ISBN links support NWE through referral fees

  • Boyer, Carl B. Apollonius of Perga, A History of Mathematics, John Wiley & Sons, 1991. ISBN 978-0471543977
  • Fried, Michael N. and Unguru, Sabetai. Apollonius of Perga’s Conica: Text, Context, Subtext, Brill, 2001. ISBN 978-9004119779, ISBN 978-9004119772
  • Toomer, G.J. "Apollonius of Perga," Dictionary of Scientific Biography (Vol. I, pp. 179-193), Charles Scribner's Sons, 1970. ISBN 978-0684101149

External links

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