Apollonius of Perga
Apollonius of Perga (Greek: Ἀπολλώνιος ο Περγαῖος; Latin: Apollonius Pergaeus; late 3rd - early 2nd centuries BC) was a Greek geometer and astronomer known for being the canonical theorist on the topic of Conic sections. There are from time to time in the history of mathematics theoreticians whose theories on their topic of interest are so coherent and complete that they can be taught in modern times with but little alteration. Geometry, for example, as taught in the schools, follows the writings of Euclid so closely that his name is synonymous with his topic. All geometry is understood to be “Euclidean” unless otherwise designated.
Apollonius believed himself a Euclidean. He saw himself as bringing to perfection certain lines of geometric thought already begun by both Euclid and Archimedes but not developed. Specifically, some geometric relationships in nature do not conform to a circular or spherical model. Alternatives are obviously required. Apollonius developed a unified theory that explains many of them as cross-sections of a double cone. There are three types (excluding circles), to which Apollonius gave their modern names: ellipse, parabola, hyperbola.
Rene Descartes adopted this sectional theory in his application of Algebra to Euclidean space; that is, the conceptual space in which the figures of Euclid, Apollonius, and all the other ancient Greek geometers are presumed to exist. They took it to be the real space, but that epistemological "given" has become less certain in modern times. Descartes applied a grid system of location, now called after him the Cartesian coordinate system, to account for every possible location of Euclidean space. The topic is taught as analytic geometry.
Isaac Newton made use of Descartes’ system, and therefore of Euclid’s and Apollonius’ figures, in developing his Method of Fluxions, a theory mathematically accounting for the flux (“flow”) of changing figures. The topic is taught today as calculus. Newton and Gottfried Wilhelm Leibniz are its twin canons. By that time all natural phenomena were presumed representable by graphs in the form of geometric figures. Calculus thus became the study of natural quantities. Its special topics are only accessible after long study; nevertheless, they typically rely heavily on the canonship of the Greek geometers.
- 1 Life of Apollonius of Perga
- 2 Documented works of Apollonius
- 2.1 Conics
- 2.2 Lost and reconstructed works described by Pappus
- 2.3 Lost works mentioned by other ancient writers
- 2.4 Early printed editions
- 3 Ideas attributed to Apollonius by other writers
- 4 Honors accorded by history
- 5 See also
- 6 Notes
- 7 References
- 8 External links
Life of Apollonius of Perga
For such an important contributor to the field of mathematics, scant biographical information remains. The 6th century Palestinian commentator, Eutocius of Ascalon, on Apollonius’ major work, Conics, states:
“Apollonius, the geometrician, ... came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes ....”
Perga at the time was a Hellenized city of Pamphylia in Anatolia. The ruins of the city yet stand. It was a center of Hellenistic culture. Euergetes, “benefactor,” identifies Ptolemy III Euergetes, third Greek dynast of Egypt in the diadochi succession. Presumably, his “times” are his regnum, 246-222/221 BC. Times are always recorded by ruler or officiating magistrate, so that if Apollonius was born earlier than 246, it would have been the “times” of Euergetes’ father. The identity of Herakleios is uncertain. The approximate times of Apollonius are thus certain, but no exact dates can be given. Specific birth and death years stated by the various scholars are only speculative.
Eutocius appears to associate Pergamon with the Ptolemaic dynasty of Egypt. Never under Egypt, Pergamon was an independent diadochi state ruled by the Attalid dynasty. The time of Euergetes at Alexandria was covered at Pergamum by Eumenes I and Attalus I, not mentioned by Eutocius. Someone designated “of Perga” might well be expected to have lived and worked there and to have been a director at the Library of Pergamum. To the contrary, if Apollonius was later identified with Perga, it was not on the basis of his residence. The remaining autobiographical material implies that he lived, studied and wrote in Alexandria.
The material is located in the surviving false “Prefaces” of the books of his Conics. These are letters delivered to influential friends of Apollonius asking them to review the book enclosed with the letter. The Preface to Book I, addressed to one Eudemus, reminds him that Conics was initially requested by a house guest at Alexandria, the geometer, Naucrates, otherwise unknown to history. Naucrates had the first draft of all eight books in his hands by the end of the visit. Apollonius refers to them as being “without a thorough purgation” (ou diakatharantes in Greek, ea non perpurgaremus in Latin). He intended to verify and emend the books, releasing each one as it was completed.
Hearing of this plan from Apollonius himself on a subsequent visit of the latter to Pergamon, Eudemus had insisted Apollonius send him each book before release. The circumstances imply that at this stage Apollonius was a young geometer seeking the company and advice of established professionals. Pappus states that he was with the students of Euclid at Alexandria. Euclid was long gone. This stay had been, perhaps, the final stage of Apollonius’ education. Eudemus was perhaps a senior figure in his earlier education at Pergamon; in any case, there is reason to believe that he was or became the head of the Library and Research Center (Museum) of Pergamon. Apollonius goes on to state that the first four books were concerned with the development of elements while the last four were concerned with special topics.
There is something of a gap between Prefaces I and II. Apollonius has sent his son, also Apollonius, to deliver II. He speaks with more confidence, suggesting that Eudemus use the book in special study groups, which implies that Eudemus was a senior figure, if not the headmaster, in the research center. Research in such institutions, which followed the model of the Lycaeum of Aristotle at Athens, due to the residency of Alexander the Great and his companions in its northern branch, was part of the educational effort, to which the library and museum were adjunct. There was only one such school in the state. Owned by the king, it was under royal patronage, which was typically jealous, enthusiastic, and participatory. The kings bought, begged, borrowed and stole the precious books whenever and wherever they could. Books were of the highest value, affordable only to wealthy patrons. Collecting them was a royal obligation. Pergamon was known for its parchment industry, where “parchment” is derived from “Pergamon.”
Apollonius brings to mind Philonides of Laodicea, a geometer whom he introduced to Eudemus in Ephesus. Philonides became Eudemus' student. He lived mainly in Syria during the 1st half of the 2nd century BC. Whether the meeting indicates that Apollonius now lived in Ephesus is unresolved. The intellectual community of the Mediterranean was international in culture. Scholars were mobile in seeking employment. They all communicated via some sort of postal service, public or private. Surviving letters are abundant. They visited each other, read each other’s works, made suggestions to each other, recommended students and accumulated a tradition termed by some “the golden age of mathematics.”
Preface III is missing. During the interval Eudemus passed away, says Apollonius in IV, again supporting a view that Eudemus was senior over Apollonius. Prefaces IV-VII are more formal, omitting personal information and concentrating on summarizing the books. They are all addressed to a mysterious Attalus, a choice made “because”, as Apollonius writes to Attalus, “of your earnest desire to possess my works.” By that time a good many people at Pergamum had such a desire. Presumably, this Attalus was someone special, receiving copies of Apollonius’ masterpiece fresh from the author’s hand. One strong theory is that Attalus is Attalus II Philadelphus, 220-138 BC, general and defender of his brother’s kingdom (Eumenes II), co-regent on the latter’s illness in 160 BC, and heir to his throne and his widow in 158 BC. He and his brother were great patrons of the arts, expanding the library into international magnificence. The dates are consonant with those of Philonides, while Apollonius’ motive is consonant with Attalus' book-collecting initiative.
Apollonius sent to Attalus Prefaces V-VII. In Preface VII he describes Book VIII as “an appendix” ... “which I will take care to send you as speedily as possible.” There is no record that it was ever sent or ever completed. It may be missing from history because it was never in history, Apollonius having died before its completion.
Documented works of Apollonius
Apollonius was a prolific geometer, turning out a large number of works. Only one survives, Conics. Of its eight books, only the first four have a credible claim to descent from the original texts of Apollonius. Books 5-7 have been translated from the Arabic into Latin. The original Greek has been lost. The status of Book 8 is unknown. A first draft existed. Whether the final draft was ever produced is not known. A "reconstruction" of it by Edmond Halley exists in Latin. There is no way to know how much of it, if any, is verisimilar to Apollonius. Halley also reconstructed De Rationis Sectione and De Spatii Sectione. Beyond these works, except for a handful of fragments, documentation that might in any way be interpreted as descending from Apollonius ends.
Many of the lost works are described or mentioned by commentators. In addition are ideas attributed to Apollonius by other authors without documentation. Credible or not, they are hearsay. Some authors identify Apollonius as the author of certain ideas, consequently named after him. Others attempt to express Apollonius in modern notation or phraseology with indeterminate degrees of fidelity.
The Greek text of Conics uses the Euclidean arrangement of definitions, figures and their parts; i.e., the “givens,” followed by propositions “to be proved.” Books I-VII present 387 propositions. This type of arrangement can be seen in any modern geometry textbook of the traditional subject matter. As in any course of mathematics, the material is very dense and consideration of it, necessarily slow. Apollonius had a plan for each book, which is partly described in the Prefaces. The headings, or pointers to the plan, are somewhat in deficit, Apollonius having depended more on the logical flow of the topics.
An intellectual niche is thus created for the commentators of the ages. Each must present Apollonius in the most lucid and relevant way for his own times. They use a variety of methods: annotation, extensive prefatory material, different formats, additional drawings, superficial reorganization by the addition of capita, and so on. There are subtle variations in interpretation. The modern English speaker encounters a lack of material in English due to the preference for New Latin by English scholars. Such intellectual English giants as Edmund Halley and Isaac Newton, the proper descendants of the Hellenistic tradition of mathematics and astronomy, can only be read and interpreted in translation by populations of English speakers unacquainted with the classical languages; that is, most of them.
Presentations written entirely in native English begin in the late 19th century. Of special note is Heath’s Treatise on Conic Sections. His extensive prefatory commentary includes such items as a lexicon of Apollonian geometric terms giving the Greek, the meanings, and usage. Commenting that “the apparently portentious bulk of the treatise has deterred many from attempting to make its acquaintance,” he promises to add headings, changing the organization superficially, and to clarify the text with modern notation. His work thus references two systems of organization, his own and Apollonius’, to which concordances are given in parentheses.
Heath’s work is indispensible. He taught throughout the early 20th century, passing away in 1940, but meanwhile another point of view was developing. St. John's College (Annapolis/Santa Fe), which had been a military school since colonial times, preceding the United States Naval Academy at Annapolis, Maryland, to which it is adjacent, in 1936 lost its accreditation and was on the brink of bankruptcy. In desperation the board summoned Stringfellow Barr and Scott Buchanan from the University of Chicago, where they had been developing a new theoretical program for instruction of the Classics. Leaping at the opportunity, in 1937 they instituted the “new program” at St. John's, later dubbed the Great Books program, a fixed curriculum that would teach the works of select key contributors to the culture of western civilization. At St. John’s, Apollonius came to be taught as himself, not as some adjunct to Analytic Geometry.
The “tutor” of Apollonius was R. Catesby Taliaferro, a new PhD in 1937 from the University of Virginia. He tutored until 1942 and then later for one year in 1948, supplying the English translations by himself, translating Ptolemy’s Almagest and Apollonius’ Conics. These translations became part of the Encyclopedia Britannica’s Great Books of the Western World series. Only Books I-III are included, with an appendix for special topics. Unlike Heath, Taliaferro did not attempt to reorganize Apollonius, even superficially, or to rewrite him. His translation into modern English follows the Greek fairly closely. He does use modern geometric notation to some degree.
Contemporaneously with Taliaferro's work, Ivor Thomas an Oxford don of the World War II era, was taking an intense interest in Greek mathematics. He planned a compendium of selections, which came to fruition during his military service as an officer in the Royal Norfolk Regiment. After the war it found a home in the Loeb Classical Library, where it occupies two volumes, all translated by Thomas, with the Greek on one side of the page and the English on the other, as is customery for the Loeb series. Thomas' work has served as a handbook for the golden age of Greek mathematics. For Apollonius he only includes mainly those portions of Book I that define the sections.
Heath, Taliaferro, and Thomas satisfied the public demand for Apollonius in translation for most of the 20th century. The subject moves on. More recent translations and studies incorporate new information and points of view as well as examine the old.
Conics Book I
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He defines the fundamental conic property[clarification needed] 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.[clarification needed] 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.[clarification needed] Thus Books v–vii are clearly original.
Conics Book V-VIII
In Book v, Apollonius treats 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[clarification needed] 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.
Lost and reconstructed works described by Pappus
Pappus mentions other treatises of Apollonius:
- Λόγου ἀποτομή, De Rationis Sectione ("Cutting of a Ratio")
- Χωρίου ἀποτομή, De Spatii Sectione ("Cutting of an Area")
- Διωρισμένη τομή, De Sectione Determinata ("Determinate Section")
- Ἐπαφαί, De Tactionibus ("Tangencies")
- Νεύσεις, De Inclinationibus ("Inclinations")
- Τόποι ἐπίπεδοι, De Locis Planis ("Plane Loci").
Each of these was divided into two books, and—with the Data, the Porisms, and Surface-Loci of Euclid and the Conics of Apollonius—were, according to Pappus, included in the body of the ancient analysis. Descriptions follow of the six works mentioned above.
De Rationis Sectione
De Rationis Sectione sought to resolve a simple problem: Given two straight lines and a point in each, draw through a third given point a straight line cutting the two fixed lines such that the parts intercepted between the given points in them and the points of intersection with this third line may have a given ratio.
De Spatii Sectione
De Spatii Sectione discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle.
In the late 17th century, Edward Bernard discovered a 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. 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.
- For more information, see Problem of Apollonius.
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 Marin Getaldić 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).
Lost works mentioned by other ancient writers
Ancient writers refer to other works of Apollonius that are no longer extant:
- Περὶ τοῦ πυρίου, On the Burning-Glass, a treatise probably exploring the focal properties of the parabola
- Περὶ τοῦ κοχλίου, On the Cylindrical Helix (mentioned by Proclus)
- A comparison of the dodecahedron and the icosahedron inscribed in the same sphere
- Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements
- Ὠκυτόκιον ("Quick Bringing-to-birth"), in which, according to Eutocius, Apollonius demonstrated how to find closer limits for the value of π (pi) than those of Archimedes, who calculated 3+1/7 as the upper limit (3.1428571, with the digits after the decimal point repeating) and 3+10/71 as the lower limit (3.1408456338028160, with the digits after the decimal point repeating)
- an arithmetical work (see 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 Woepke, 1856).
Early printed editions
The early printed editions began for the most part in the 16th century. At that time, scholarly books were expected to be in Latin, today's New Latin. As almost no manuscripts were in Latin, the editors of the early printed works translated from the Greek or Arabic to Latin. The Greek and Latin were typically juxtaposed, but only the Greek is original, or else was restored by the editor to what he thought was original. Critical apparatuses were in Latin. The ancient commentaries, however, were in ancient or medieval Greek. Only in the 18th and 19th centuries did modern languages begin to appear. A representative list of early printed editions is given below. The originals of these printings are rare and expensive. For modern editions in modern languages see the references.
- Pergaeus, Apollonius (1566). Conicorum libri quattuor: una cum Pappi Alexandrini lemmatibus, et commentariis Eutocii Ascalonitae. Sereni Antinensis philosophi libri duo ... quae omnia nuper Federicus Commandinus Vrbinas mendis quampluris expurgata e Graeco conuertit, & commentariis illustrauit (in Ancient Greek and Neo-Latin). Bononiae: Ex officina Alexandri Benatii. A presentation of the first four books of Conics in Greek by Fredericus Commandinus with his own translation into Latin and the commentaries of Pappus of Alexandria, Eutocius of Ascalon and Serenus of Antinouplis.
- Apollonius; Barrow, I (1675). Apollonii conica: methodo nova illustrata, & succinctè demonstrata (in Neo-Latin). Londini: Excudebat Guil. Godbid, voeneunt apud Robertum Scott, in vico Little Britain. Translation by Barrow from ancient Greek to Neo-Latin of the first four books of Conics. The copy linked here, located in the Boston Public Library, once belonged to John Adams.
- Apollonius; Pappus; Halley, E. (1706). Apollonii Pergaei de sectione rationis libri duo: Ex Arabico ms. Latine versi. Accedunt ejusdem de sectione spatii libri duo restituti (in Neo-Latin). Oxonii. A presentation of two lost but reconstructed works of Apollonius. De Sectione Rationis comes from an unpublished manuscript in Arabic in the Bodleian Library at Oxford originally partially translated by Edward Bernard but interrupted by his death. It was given to Edmond Halley, professor, astronomer, mathematician and explorer, after whom Halley's Comet later was named. Unable to decipher the corrupted text, he abandoned it. Subsequenty David Gregory (mathematician) restored the Arabic for Henry Aldrich, who gave it again to Halley. Learning Arabic, Halley created De Sectione Rationis and as an added emolument for the reader created a Neo-Latin translation of a version of De Sectione Spatii reconstructed from Pappus Commentary on it. The two Neo-Latin works and Pappus' ancient Greek commentary were bound together in the single volume of 1706. The author of the Arabic manuscript is not known. Based on a statement that it was written under the "auspices" of Al-Ma'mun, Latin Almamon, astronomer and Caliph of Baghdad in 825, Halley dates it to 820 in his "Praefatio ad Lectorem."
- Apollonius; Alexandrinus Pappus; Halley, Edmond; Eutocius; Serenus (1710). Apollonii Pergaei Conicorum libri octo, et Sereni Antissensis De sectione cylindri & coni libri duo (PDF) (in Neo-Latin and Ancient Greek). Oxoniae: e Theatro Sheldoniano. Encouraged by the success of his translation of David Gregory’s emended Arabic text of de Sectione rationis, published in 1706, Halley went on to restore and translate into Latin Apollonius’ entire elementa conica. Books I-IV had never been lost. They appear with the Greek in one column and Halley’s Latin in a parallel column. Books V-VI came from a windfall discovery of a previously unappreciated translation from Greek to Arabic that had been purchased by the antiquarian scholar Jacobus Golius in Aleppo in 1626. On his death in 1696 it passed by a chain of purchases and bequests to the Bodleian Library (originally as MS Marsh 607, dated 1070). The translation, dated much earlier, comes from the branch of Almamon’s school entitled the Banū Mūsā, “sons of Musa,” a group of three brothers, who lived in the 9th century. The translation was performed by writers working for them. In Halley’s work, only the Latin translation of Books V-VII is given. This is its first printed publication. Book VIII was lost before the scholars of Almamon could take a hand at preserving it. Halley’s concoction, based on expectations developed in Book VII, and the lemmas of Pappus, is given in Latin. The commentaries of Pappus, Serenus and Eutocius are included as a guide to the interpretation of the Conics.
Ideas attributed to Apollonius by other writers
Apollonius' failed astronomy
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, is also attributed to him. Ptolemy describes Apollonius' theorem in the Almagest XII.1.
Methods of Apollonius
According to Heath, “The Methods of Apollonius” were not his and were not personal. Whatever influence he had on later theorists was that of geometry, not of his own innovation of technique. Heath says,
“As a preliminary to the consideration in detail of the methods employed in the Conics, it may be stated generally that they follow steadily the accepted principles of geometrical investigation which found their definitive expression in the Elements of Euclid.”
Heath goes on to use the term geometrical algebra for the methods of the entire golden age. The term is “not inappropriately” called that, he says. Today the term has been resurrected for use in other senses (see under geometric algebra). Heath was using it as it had been defined by Henry Burchard Fine in 1890 or before. Fine applies it to La Géométrie of René Descartes, the first full-blown work of analytic geometry. Establishing as a precondition that “two algebras are formally identical whose fundamental operations are formally the same,” Fine says that Descartes’ work “is not ... mere numerical algebra, but what may for want of a better name be called the algebra of line segments. Its symbolism is the same as that of numerical algebra; ....”
For example, in Apollonius a line segment AB (the line between Point A and Point B) is also the numerical length of the segment. It can have any length. AB therefore becomes the same as an algebraic variable, such as x (the unknown), to which any value might be assigned; e.g., x=3.
Varables are defined in Apollonius by such word statements as “let AB be the distance from any point on the section to the diameter,” a practice that continues in algebra today. Every student of basic algebra must learn to convert “word problems” to algebraic variables and equations, to which the rules of algebra apply in solving for x. Apollonius had no such rules. His solutions are geometric.
Relationships not readily amenible to pictorial solutions were beyond his grasp; however, his repertory of pictorial solutions came from a pool of complex geometric solutions generally not known (or required) today. One well-known exception is the indispensible Pythagorean Theorem, even now represented by a right triangle with squares on its sides illustrating an expression such as a2 + b2 = c2. The Greek geometers called those terms “the square on AB,” etc. Similary, the area of a rectangle formed by AB and CD was "the rectangle on AB and CD."
These concepts gave the Greek geometers algebraic access to linear functions and quadratic functionss, which latter the conic sections were. They contain powers of 1 or 2 respectively. Apollonius had not much use for cubes (featured in solid geometry), even though a cone is a solid. His interest was in conic sections, which are plane figures. Powers of 4 and up were beyond visualization, requiring a degree of abstraction not available in geometry, but ready at hand in algebra.
Honors accorded by history
- Eutocius, Commentary on Conica, Book I, Lines 5-10, to be found translated in Apollonius of Perga & Thomas 1953, p. 277
- Fried, Michael N.; Unguru, Sabatai (2001). "Introduction". Apollonius of Perga's Conica: text, context, subtext. Leiden: Brill.
- Apollonius of Perga & Heath 1896, pp. clvii-clxx
- Apollonius of Perga, & Heath 1896, p. vii
- Dana Mackenzie. "A Tisket, a Tasket, an Apollonian Gasket". American Scientist. 98, January–February 2010 (1): 10–14.
- Boyer, Carl B. (1991). "Apollonius of Perga". A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 142. ISBN 0-471-54397-7.
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.
- Chisholm, Hugh, ed. (1911). "Apollonius of Perga". Encyclopædia Britannica (11th ed.). Cambridge University Press.
- He said in his Praefatio of 1710, that although Apollonius was second only (in his opinion) to Archimedes, a large part of his elementa conica was “truncated” and the remaining part “less faithful;” consequently he was now going to emend it. The question of exactly what items are to be regarded as “faithful” pervades today's literature.
- For a more precise version of the chain see Wakefield, Colin. "Arabic Manuscripts in the Bodleian Library" (PDF). pp. 136–137.
- Apollonius of Perga & Heath 1896, p. ci
- Fine, Henry B (1902). The number-system of algebra treated theoretically and historically. Boston: Leach. pp. 119–120.
- Alhazen; Hogendijk, JP (1985). Ibn al-Haytham's Completion of the "Conics". New York: Springer Verlag.
- Apollonius of Perga; Halley, Edmund; Balsam, Paul Heinrich (1861). Des Apollonius von Perga sieben Bücher über Kegelschnitte Nebst dem durch Halley wieder hergestellten achten Buche; dabei ein Anhang, enthaltend Die auf die Geometrie der Kegelschnitte bezüglichen Sätze aus Newton's "Philosophiae naturalis principia mathematica." (in German). Berlin: De Gruyter.
- Apollonius of Perga; Halley, Edmund; Fried, Michael N (2011). Edmond Halley's reconstruction of the lost book of Apollonius's Conics: translation and commentary. Sources and studies in the history of mathematics and physical sciences. New York: Springer. ISBN 1461401453.
- Apollonius of Perga; Heath, Thomas Little (1896). Treatise on conic sections. Cambridge: University Press.
- Apollonius of Perga; Heiberg, JL (1891). Apollonii Pergaei quae Graece exstant cum commentariis antiquis (in Ancient Greek and Neo-Latin). Volume I. Leipzig: Teubner.
- Apollonius of Perga; Heiberg, JL (1893). Apollonii Pergaei quae Graece exstant cum commentariis antiquis (in Ancient Greek and Neo-Latin). Volume II. Leipzig: Teubner.
- Apollonius of Perga; Densmore, Dana (2010). Conics, books I-III. Santa Fe (NM): Green Lion Press.
- Apollonius of Perga; Fried, Michael N (2002). Apollonius of Perga’s Conics, Book IV: Translation, Introduction, and Diagrams. Santa Fe, NM: Green Lion Press.
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- The Editors of Encyclopædia Britannica (2006). "Apollonius of Perga". Encyclopaedia Britannica.
- Kunkel, Paul (2016). "Conics of Apollonius". Whistler Alley Mathematics. whistleralley.com. Retrieved 15 February 2017.
- O'Connor, John J.; Robertson, Edmund F., "Apollonius of Perga", MacTutor History of Mathematics archive, University of St Andrews.
- "Mathematics and Mathematical Astronomy". Brown University.
- "Apollonii Pergaei Conicorum". Linda Hall Library Digital Collection.