Conics

Apollonius of Perga ( ; ) 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eccentric orbits to explain the apparently aberrant motion of the planets, commonly believed until the Middle Ages, was superseded during the Renaissance.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' ''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''.

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 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 Appolonius' 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' 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' 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' motive is consonant with Attalus' 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' lost works in book 7 of his collection. Based on Pappus' 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 , Greek ) 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 $y^2 = kx$ which is one modern form of the equation for a parabola. The rectangle has sides $k$ and $x$. It was he who accordingly named the figure, parabola, "application".

The "no applicability" case is further divided into two possibilities. Given a function, $f(x)$, such that, in the applicability case, $y^2 = g(x)$, in the no applicability case, either $y^2 > g(x)$ or $y^2 < g(x)$. In the former, $g(x)$ falls short of $y^2$ by a quantity termed the , "deficit". In the latter, $g(x)$ overshoots by a quantity termed the , "surfeit".

Applicability could be achieved by adding the deficit, $y^2 = f(x) = g(x) + d,$ or subtracting the surfeit, $g(x) - s.$ 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,

:$Ax^2 + By^2 = C$

can be placed in the form

:$y^2 = \left, \frac AB x^2 \ + \frac CB$

where $C/B$ is the deficit, while an equation for the hyperbola,

:$Ax^2 - By^2 = C$

becomes

:$y^2 = \left, \frac AB x^2 \ - \frac CB$

where $C/B$ 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.

Book IV

Book IV contains 57 propositions. The first sent to Attalus, rather than to Eudemus, it thus represents his more mature geometric thought. The topic is rather specialized: "the greatest number of points at which sections of a cone can meet one another, or meet a circumference of a circle, ...." Nevertheless, he speaks with enthusiasm, labeling them "of considerable use" in solving problems (Preface 4).

Book V

In contrast to Book I, Book V contains no definitions and no explanation. It contains 77 propositions, the most of any book, dealing with maximum and minimum lines. Propositions 4 to 25 deal with maximum and minimum lines "from a point on the axis to the section," propositions 27 to 52 deal with maximum and minimum lines "in a section" and "drawn from the section" while propositions 53 to 77 deal with maximum and minimum lines "cut off between the section and axis" and "cut off by the axis." Thomas Heath interpreted these "maximum and minimum lines" as normals to the sections, which exerted a great deal of influence on the interpretation of the ''Conics'' in the 20th century. However, more recent scholarship has shown that these are standard terms used in Ancient Greek mathematics to refer to maximum and minimum distances.

Book VI

Book VI, known only through translation from the Arabic, contains 33 propositions, the least of any book. It also has large lacunae, or gaps in the text, due to damage or corruption in the previous texts.

Preface 1 states that the topic is “equal and similar sections of cones.” Apollonius extends the concepts of congruence and similarity presented by Euclid for more elementary figures, such as triangles, quadrilaterals, to conic sections. Preface 6 mentions “sections and segments” that are “equal and unequal” as well as “similar and dissimilar,” and adds some constructional information.

Book VI features a return to the basic definitions at the front of the book. “Equality” is determined by an application of areas. If one figure; that is, a section or a segment, is “applied” to another, they are “equal” if they coincide and no line of one crosses any line of the other. In Apollonius' definitions at the beginning of Book VI, similar right cones have similar axial triangles. Similar sections and segments of sections are first of all in similar cones. In addition, for every abscissa of one must exist an abscissa in the other at the desired scale. Finally, abscissa and ordinate of one must be matched by coordinates of the same ratio of ordinate to abscissa as the other. The total effect is as though the section or segment were moved up and down the cone to achieve a different scale.

Book VII

Book VII, also a translation from the Arabic, contains 51 Propositions. In Preface I, Apollonius does not mention them, implying that, at the time of the first draft, they may not have existed in sufficiently coherent form to describe. The topic of Book VII is stated in Preface VII to be diameters and “the figures described upon them.” The 51 propositions of Book VII define the relationships between sections, diameters, and conjugate diameters.

Apollonius uses obscure language, that they are “peri dioristikon theorematon” , which Halley translated as “de theorematis ad determinationem pertinentibus,” and Heath as “theorems involving determinations of limits.”

''Cutting off a Ratio''

''Cutting off a ratio'' 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.

''Cutting off a Ratio'' survives in an unpublished manuscript in Arabic in the Bodleian Library at Oxford originally discovered and 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. Subsequently, David Gregory (mathematician) restored the Arabic for Henry Aldrich, who gave it again to Halley. 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."

Lost works described by Pappus

Besides ''Conics'' and ''Cutting of a Ratio'' Pappus mentions other treatises of Apollonius:
* ''Cutting off an Area'' (, )
* ''Determinate Section'' (, )
* ''Tangencies'' (, )
* ''Neusis'' (, )
* ''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.

''Cutting off an Area''

''Cutting of an Area'' discussed a similar problem requiring the rectangle contained by the two intercepts to be equal to a given rectangle. Although the work is lost, Edmund Halley, having translated ''Cutting off a Ratio'', attempted a Neo-Latin translation of a version of ''Cutting off an Area'' reconstructed from Pappus' summary of it in his ''Collection''.

''Determinate Section''

''Determinate Section'' 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

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