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Ancient Greek Mathematician Apollonius and His Insoluble Problem

Apollonius Crater
Apollonius Crater on Mars. The ancient Greek mathematician’s brilliance in the realm of geometry was so appreciated by NASA that this Moon feature was named for him. Credit: NASA/Public Domain

Apollonius of Perga (Greek: Ἀπολλώνιος ὁ Περγαῖος) who lived from 240 BC to c. 190 BC, was a brilliant ancient Greek mathematician, geometer and astronomer known for his work on conic sections. He was born in Perga, an ancient Greek city of Pamphylia, what is now Murtina, Turkey.

Tragically, we know almost nothing from the life of this brilliant man; but the most famous mathematical problem he posed has lived on through the centuries, taunting researchers until recent times, when it was finally solved by Joseph Gergonne.

Beginning from the contributions of Euclid and Archimedes on the topic, he brought them up to the point of the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today — especially by those in space exploration.

Planetary orbits having such shapes are used in space travel and in sending satellites and probes into space, as they use the orbits to propel them further into the great beyond.

Gottfried Wilhelm Leibniz stated that “he who understands Archimedes and Apollonius will admire less the achievements of the foremost men of later times.”

All of history’s most brilliant mathematicians, including Sir Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Rene Descartes, worked on this problem in one form or another.

In the end, it was only the French mathematician Joseph Gergonne who finally solved the problem, using Circle Inversion.

Apollonius’ major work, parts of which thankfully still exist, concerned Conics, regarding the mathematics of such shapes. His thinking on the geometry of these shapes influenced many later scholars throughout Western intellectual history.

Apollonius of Perga worked at Great Library of Alexandria

Conic Sections deals with the shapes that are seen when a cone is intersected by a plane, i.e., when it is sliced open. You can get one of four shapes when that happens, including a circle, an ellipse, a parabola, or a hyperbola. It is believed to have been Apollonius who gave the last three shapes the names that we use today.

Apollonius worked on numerous other topics, including astronomy. Most of this work has not survived, however. 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, but aspects of his other astronomical observations have been borne out through modern research.

The sixth-century Greek commentator Eutocius of Ascalon states, regarding Conics, “Apollonius, the geometrician,…came from Perga in Pamphylia in the times of Ptolemy Euergetes, so records Herakleios the biographer of Archimedes.”

The remaining autobiographical material implies that he lived, studied, and wrote in Alexandria, Egypt at the research institution that developed around the Great Library of Alexandria, where he would have had contact with all the other great minds of the day.

Research in such institutions, which followed the model of the Lycaeum of Aristotle at Athens, where Alexander the Great and his companions learned from the greatest thinkers of the ancient world, was part of the educational effort to which the library and museum were adjunct.

Ancient scholars communicated by means of postal system

The intellectual community of the Mediterranean was international in culture. Incredibly, historians know that scholars of the time all communicated with one another via some sort of postal service, public or private, because the number of surviving letters is considerable.

They visited each other, read each other’s works, made suggestions for each other, recommended students, and accumulated a tradition that has been termed by some “The Golden Age of Mathematics.”

A letter referencing Apollonius of Perga 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:

Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of the bond between them due to their common interest in mathematics. 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, they came to the conclusion that Apollonius’ treatment of it in this book was not correct…

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.

The material is located in the surviving “Prefaces” of the books of Apollonius’ Conics in the form of letters delivered to influential friends of his, 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 —who otherwise is unknown to history.

Documented works of Apollonius are many, though only one survives

Apollonius was a prolific geometer, turning out a large number of works although only Conics survives. Tragically, the original Greek has been lost.

Many of the lost works are described or mentioned by commentators, however, in the many letters that flew between them and do survive.

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.

Other treatises written by ancient Greek mathematician Apollonius are:

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”)[27]
Νεύσεις, De Inclinationibus (“Inclinations”)
Τόποι ἐπίπεδοι, De Locis Planis (“Plane Loci”).

Other works of the great man, according to ancient writer’s references, include:

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 than those of Archimedes.

Apollonius’ contribution to astronomy led to crater on Moon being named after him

Apollonius’ equivalence of two descriptions of planet motions, one using excentrics and another deferent and epicycles, is just one concept that can be attributed to the great thinker. Ptolemy describes this equivalence as “Apollonius’ theorem” in the Almagest XII.1.

“Apollonius,” a lunar impact crater located near the eastern limb of the Moon, has the distinction of being named after the brilliant ancient Greek mathematician, making note of his name despite the fact that so many of his works have been lost. It lies in the region of uplands to the west of Mare Undarum and northeast of the Sinus Successus on the Mare Fecunditatis. It is southwest of the crater Firmicus and north of Condon.

The outer rim of Apollonius is somewhat worn and is overlain by a pair of small craters (including Apollonius E) across the western wall. The nearly flat interior floor has a low albedo and has been covered by lava.

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