Apollonius of Perga

Apollonius of Perga (Ancient Greek: Ἀπολλώνιος ὁ Περγαῖος Apollṓnios ho Pergaîos; c. 240 BC – c. 190 BC) was an ancient Greek geometer and astronomer known for his work on conic sections. Beginning from the earlier contributions of Euclid and Archimedes on the topic, he brought them to the state prior to the invention of analytic geometry. His definitions of the terms ellipse, parabola, and hyperbola are the ones in use today. With his predecessors Euclid and Archimedes, Apollonius is generally considered among the greatest mathematicians of antiquity.

Aside from geometry, Apollonius worked on numerous other topics, including astronomy. Most of this work has not survived, where exceptions are typically fragments referenced by other authors like Pappus of Alexandria. 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eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance. The Apollonius crater on the Moon is named in his honor.

Life

Despite his momentous contributions to the field of mathematics, scant biographical information on Apollonius remains. The 6th century Greek commentator Eutocius of Ascalon, writing on Apollonius's Conics, states:

Apollonius, the geometrician, ... came from Perga in Pamphylia in the times of Ptolemy III Euergetes, so records Herakleios the biographer of Archimedes ....

From this passage Apollonius can be approximately dated, but specific birth and death years stated by modern scholars are only speculative. Ptolemy III Euergetes ("benefactor") was third Greek dynast of Egypt in the Diadochi succession, who reigned 246–222/221 BC. "Times" are always recorded by ruler or officiating magistrate, so Apollonius was likely born after 246. The identity of Herakleios is uncertain.

Perga was a Hellenized city in Pamphylia, Anatolia, whose ruins yet stand. It was a center of Hellenistic culture. Eutocius appears to associate Perga with the Ptolemaic dynasty of Egypt. Never under Egypt, Perga in 246 BC belonged to the Seleucid Empire, an independent diadochi state ruled by the Seleucid dynasty. During the last half of the 3rd century BC, Perga changed hands a number of times, being alternatively under the Seleucids and under the Attalids of Pergamon to the north.

Someone designated "of Perga" might be expected to have lived and worked there; 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 a pseudepigraphic work transmitted as Book XIV 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, 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.

Autobiographical prefaces

Some autobiographical material can be found in the surviving prefaces to the books of Conics. These are letters Apollonius addressed to influential friends asking them to review the book enclosed with the letter. The first two prefaces are addressed to Eudemus of Pergamon.

Eudemus likely was or became the head of the research center of the Museum of Pergamon, a city known for its books and parchment industry from which the name parchment is derived. Research in Greek mathematical institutions, which followed the model of the Athenian Lycaeum, was part of the educational effort to which the library and museum were adjunct.

There was only one such school in the state, under royal patronage. Books were rare and expensive and collecting them was a royal obligation.

Apollonius's preface to Book I tells Eudemus that the first four books were concerned with the development of elements while the last four were concerned with special topics. Apollonius reminds Eudemus that Conics was initially requested by Naucrates, a geometer and house guest at Alexandria otherwise unknown to history.

Apollonius provided Naucrates the first draft of all eight books, but he refers to them as being "without a thorough purgation", and intended to verify and correct the books, releasing each one as it was completed.

Having heard this plan from Apollonius himself, who visited Pergamon, Eudemus insisted Apollonius send him each book before release. At this stage Apollonius was likely still a young geometer, who according to Pappus stayed at Alexandria with the students of Euclid (long after Euclid's time), perhaps the final stage of his education. Eudemus may have been a mentor from Apollonius's time in Pergamon.

There is a gap between the first and second prefaces. Apollonius has sent his son, also named Apollonius, to deliver the second. He speaks with more confidence, suggesting that Eudemus use the book in special study groups. Apollonius mentions meeting Philonides of Laodicea, a geometer whom he introduced to Eudemus in Ephesus, and who became Eudemus's student.

Philonides 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 cosmopolitan and scholars in this "golden age of mathematics" sought employment internationally, visited each other, read each other's works and made suggestions, recommended students, and communicated via some sort of postal service. Surviving letters are abundant.

The preface to Book III is missing, and during the interval Eudemus died, says Apollonius in the preface to Book IV. Prefaces to Books IV–VII are more formal, mere summaries omitting personal information. All four are addressed to a mysterious Attalus, a choice made, Apollonius says, "because of your earnest desire to possess my works".

Presumably Attalus was important to be sent Apollonius's manuscripts. One theory is that Attalus is Attalus II Philadelphus (220–138 BC), general and defender of Pergamon whose brother Eumenes II was king, and who became co-regent after his brother's illness in 160 BC and acceded to the throne in 158 BC. Both brothers were patrons of the arts, expanding the library into international magnificence.

Attalus was a contemporary of Philonides and Apollonius's motive is consonant with Attalus's book-collecting initiative.

In Preface VII Apollonius 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, and Apollonius might have died before finishing it. Pappus of Alexandria, however, provided lemmas for it, so it must have been in circulation in some form.

Writings

Apollonius was a prolific geometer, turning out a large number of works. Only one survives, Conics. Of its eight books, only the first four persist as untranslated original texts of Apollonius. Books 5-7 are only preserved via an Arabic translation by Thābit ibn Qurra commissioned by the Banū Mūsā; the original Greek is lost.

The status of Book 8 is unknown. A first draft existed, but whether the final draft was ever produced is not known. A "reconstruction" of it by Edmond Halley exists in Latin, but there is no way to know how much of it, if any, is verisimilar to Apollonius.

Other than a single other work surviving in Arabic translation, De Rationis Sectione, The rest of Apollonius's works are fragmentary or lost. Many of the lost works are described or mentioned by commentators, especially Pappus of Alexandria, who provides epitomes and lemmas for many of Apollonius's lost works in book 7 of his collection. Based on Pappus's summaries, Edmond Halley reconstructed De Spatii Sectione.

Conics

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.

Book I

Book I presents 58 propositions. Its most salient content is all the basic definitions concerning cones and conic sections. These definitions are not exactly the same as the modern ones of the same words. Etymologically the modern words derive from the ancient, but the etymon often differs in meaning from its reflex.

A conical surface is generated by a line segment rotated about a bisector point such that the end points trace circles, each in its own plane. A cone, one branch of the double conical surface, is the surface with the point (apex or vertex), the circle (base), and the axis, a line joining vertex and center of base.

A section (Latin sectio, Greek tome) is an imaginary "cutting" of a cone by a plane.

• Proposition I.3: “If a cone is cut by a plane through the vertex, the section is a triangle.” In the case of a double cone, the section is two triangles such that the angles at the vertex are vertical angles.
• Proposition I.4 asserts that sections of a cone parallel to the base are circles with centers on the axis.
• Proposition I.13 defines the ellipse, which is conceived as the cutting of a single cone by a plane inclined to the plane of the base and intersecting the latter in a line perpendicular to the diameter extended of the base outside the cone (not shown). The angle of the inclined plane must be greater than zero, or the section would be a circle. It must be less than the corresponding base angle of the axial triangle, at which the figure becomes a parabola.
• Proposition I.11 defines a parabola. Its plane is parallel to a side in the conic surface of the axial triangle.
• Proposition I.12 defines a hyperbola. Its plane is parallel to the axis. It cut both cones of the pair, thus acquiring two distinct branches (only one is shown).

The "application of areas" implicitly asks, given an area and a line segment, does this area apply; that is, is it equal to, the square on the segment? If yes, an applicability (parabole) has been established. Apollonius followed Euclid in asking if a rectangle on the abscissa of any point on the section applies to the square of the ordinate.

If it does, his word-equation is the equivalent of ${\textstyle y^{2}=kx}$ which is one modern form of the equation for a parabola.

The rectangle has sides ${\displaystyle k}$ and ${\displaystyle x}$. It was he who accordingly named the figure, parabola, "application".

The "no applicability" case is further divided into two possibilities. Given a function, ${\textstyle f(x)}$, such that, in the applicability case, ${\textstyle y^{2}=g(x)}$, in the no applicability case, either ${\textstyle y^{2}>g(x)}$ or ${\textstyle y^{2}<g(x)}$.

In the former, ${\displaystyle g(x)}$ falls short of ${\textstyle y^{2}}$ by a quantity termed the ellipsis, "deficit".

In the latter, ${\displaystyle g(x)}$ overshoots by a quantity termed the hyperbole, "surfeit".

Applicability could be achieved by adding the deficit, ${\textstyle y^{2}=f(x)=g(x)+d,}$ or subtracting the surfeit, ${\displaystyle g(x)-s.}$ The figure compensating for a deficit was named an ellipse; for a surfeit, a hyperbola. The terms of the modern equation depend on the translation and rotation of the figure from the origin, but the general equation for an ellipse,

${\displaystyle Ax^{2}+By^{2}=C}$

can be placed in the form

${\displaystyle y^{2}=\left|{\frac {A}{B}}x^{2}\right|+{\frac {C}{B}}}$

where ${\displaystyle C/B}$ is the deficit, while an equation for the hyperbola,

${\displaystyle Ax^{2}-By^{2}=C}$

becomes

${\displaystyle y^{2}=\left|{\frac {A}{B}}x^{2}\right|-{\frac {C}{B}}}$

where ${\displaystyle C/B}$ is the surfeit.

Book II

Book II contains 53 propositions. Apollonius says that he intended to cover "the properties having to do with the diameters and axes and also the asymptotes and other things ... for limits of possibility." His definition of "diameter" is different from the traditional, as he finds it necessary to refer the intended recipient of the letter to his work for a definition.

The elements mentioned are those that specify the shape and generation of the figures. Tangents are covered at the end of the book.

Book III

Book III contains 56 propositions. Apollonius claims original discovery for theorems "of use for the construction of solid loci ... the three-line and four-line locus ...." The locus of a conic section is the section. The three-line locus problem (as stated by Taliafero's appendix to Book III) finds "the locus of points whose distances from three given fixed straight lines ... are such that the square of one of the distances is always in a constant ratio to the rectangle contained by the other two distances." This is the proof of the application of areas resulting in the parabola.

The four-line problem results in the ellipse and hyperbola. Analytic geometry derives the same loci from simpler criteria supported by algebra, rather than geometry, for which Descartes was highly praised. He supersedes Apollonius in his methods.