In this step, we will. see how Apollonius defined the conic sections, or conics. learn about several beautiful properties of conics that have been known for over. Conics: analytic geometry: Elementary analytic geometry: years with his book Conics. He defined a conic as the intersection of a cone and a plane (see. Apollonius and Conic Sections. A. Some history. Apollonius of Perga (approx. BC– BC) was a Greek geometer who studied.
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Apollonius of Perga Greek: Beginning from the theories of Euclid and Archimedes on the topic, he brought them to the state they were in just before the invention of analytic geometry. His definitions of the terms ellipseparabolaand hyperbola are the ones in use today.
Conics of Apollonius
Apollonius worked on many other topics, apollobius astronomy. Most of the work has not survived apollonijs in fragmentary references in other authors. His hypothesis of eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance. For such an important contributor to the field of mathematics, scant biographical information remains.
Perga at the time was a Hellenized city of Pamphylia in Anatolia. The ruins of the city yet stand.
Apollonius of Perga – Wikipedia
It was a center of Hellenistic culture. The identity of Herakleios is uncertain. The approximate times of Apollonius are thus certain, but no exact dates can be given. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Kingdom of Pergamon to the north, ruled by the Attalid dynasty.
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. A letter by the Greek mathematician and astronomer Hypsicles was originally part of the supplement taken from Euclid’s Book XIV, part of the thirteen books of Euclid’s Elements. And on one occasion, when looking into the tract written by Apollonius about the comparison of the dodecahedron and icosahedron inscribed in one and the same sphere, that is to say, on the question what ratio they bear to one another, they came to the conclusion that Apollonius’ treatment of it in this book was not correct; accordingly, as I understood from my father, they proceeded to amend and rewrite it.
But I myself afterwards came across another book published by Apollonius, containing a demonstration of the matter in question, and I was greatly attracted by his investigation of the problem.
Now the book published by Apollonius is accessible to all; for it has a large circulation in a form which seems to have been the result of later careful elaboration.
But it is time to have done with the preamble and to begin my treatise itself. Apollonius donics toward the end of a historical period now termed the Hellenistic Periodcharacterized by the superposition of Hellenic culture over extensive non-Hellenic regions to various depths, radical in some places, hardly at all in others. The change was initiated by Philip II of Macedon and his son, Alexander the Greatwho, subjecting all of Greece is a series of stunning victories, went on to conquer the Persian Empirewhich ruled territories from Egypt to Pakistan.
Philip was assassinated in BC. Alexander went on to fulfill his plan by conquering the vast Persian empire. 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 spollonius a house guest at Alexandria, the geometer, Naucrates, otherwise unknown to history.
Apollonius, Conics Book IV
Naucrates had the first draft of all eight books in his hands by the end of the visit. 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. 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. Apollonius brings to mind Philonides of Laodiceaa 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 apollonisu service, public or private.
Surviving letters are abundant. Preface III is missing. During the interval Eudemus passed away, says Apollonius in IV, again supporting a view that Eudemus was apolloniuss over Apollonius. Prefaces IV—VII are more formal, omitting personal information and concentrating on summarizing the books. He and his brother were great patrons of the arts, expanding the library into international magnificence.
It may be missing from history because it was never in history, Apollonius having died before its completion. Apollonius was a prolific geometer, turning out a large number of works. Only one survives, Conics. It is a dense and extensive reference work on the topic, even by today’s standards, serving as a repository of now little known geometric propositions as well as a vehicle for some new ones devised by Apollonius.
Its audience was not the general population, which could not read or write. It was always intended for savants of mathematics and their small number of educated readers associated with the state schools and their associated libraries. It always was, in other words, a library reference work.
Its basic definitions have become an important mathematical heritage. For the most part its methods and conclusions have been superseded by Analytic Geometry.
Of its eight books, only the first four have a credible claim to descent from the original texts of Apollonius. Books have been translated from the Arabic into Latin. The original Greek has been lost. 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. 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. This type ocnics 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 apolloinus 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 apollonous 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: 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 conixs.
Such intellectual English giants as Edmund Halley and Isaac Newton, the proper descendants of the Cpnics 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 apolloius 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. Heath’s work is indispensable.
He taught throughout the early 20th century, passing away inbut meanwhile another point of view was developing. In desperation the board summoned Stringfellow Barr and Scott Buchanan from the University of Chicagowhere they had been developing a new theoretical program for instruction of the Classics.
John’s, later cobics the Great Books program, a fixed curriculum that would teach the works of select key contributors apolloniys the culture of western civilization. John’s, Apollonius came to be taught as himself, not as some adjunct to analytic geometry. 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.