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63-656: He gave a better Arabic edition of the Conics of Apollonius and commented on the first books. The Conics had been translated a century before by Hilal al-Himsi (books 1–4) and Thabit ibn Qurra (books 5–7). This article about an Iranian scientist is a stub . You can help Misplaced Pages by expanding it . This article about an Asian mathematician is a stub . You can help Misplaced Pages by expanding it . Apollonius of Perga Apollonius of Perga ( ‹See Tfd› Greek : Ἀπολλώνιος ὁ Περγαῖος Apollṓnios ho Pergaîos ; c.  240 BC  – c.  190 BC )

126-412: A centroid , serving as a center of symmetry in two directions. These figures are the circle, ellipse, and two-branched hyperbola. There is only one centroid, which must not be confused with the foci . A diameter is a chord passing through the centroid, which always bisects it. For the circle and ellipse, let a grid of parallel chords be superimposed over the figure such that the longest is a diameter and

189-510: A plane . The Greek geometers were interested in laying out select figures from their inventory in various applications of engineering and architecture, as the great inventors, such as Archimedes, were accustomed to doing. A demand for conic sections existed then and exists now. The development of mathematical characterization had moved geometry in the direction of Greek geometric algebra , which visually features such algebraic fundamentals as assigning values to line segments as variables. They used

252-459: A central point for mathematicians at the times. Basilides letter is part of the supplement taken from Euclid's Book XIV, written by Hypsicles . 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 (Apollonius of Perga) about

315-468: A change of view. In 2001, Apollonius scholars Fried & Unguru, granting all due respect to other Heath chapters, balked at the historicity of Heath's analysis of Book V, asserting that he “reworks the original to make it more congenial to a modern mathematician ... this is the kind of thing that makes Heath’s work of dubious value for the historian, revealing more of Heath’s mind than that of Apollonius.” Some of his arguments are in summary as follows. There

378-463: A closed figure; e.g., a parabola has a diameter. A parabola has symmetry in one dimension. If you imagine it folded on its one diameter, the two halves are congruent, or fit over each other. The same may be said of one branch of a hyperbola. Conjugate diameters (Greek suzugeis diametroi, where suzugeis is “yoked together”), however, are symmetric in two dimensions. The figures to which they apply require also an areal center (Greek kentron), today called

441-453: 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, known only through translation from the Arabic, contains 77 propositions, the most of any book. They cover the ellipse (50 propositions), the parabola (22), and the hyperbola (28). These are not explicitly

504-435: 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

567-548: A coordinate system intermediate between a grid of measurements and the Cartesian coordinate system . The theories of proportion and application of areas allowed the development of visual equations. (See below under Methods of Apollonius). 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

630-455: 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. 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

693-657: 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. For my part, I determined to dedicate to you what I deem to be necessary by way of commentary, partly because you will be able, by reason of your proficiency in all mathematics and particularly in geometry, to pass an expert judgment upon what I am about to write, and partly because, on account of your intimacy with my father and your friendly feeling towards myself, you will lend

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756-402: A point on the axis to the section,” which is exactly the opposite of a hypothetical “from a point on the section to the axis.” The former does not have to be normal to anything, although it might be. Given a fixed point on the axis, of all the lines connecting it to all the points of the section, one will be longest (maximum) and one shortest (minimum). Other phrases are “in a section,” “drawn from

819-411: 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 = k x {\textstyle y^{2}=kx} which is one modern form of the equation for a parabola . The rectangle has sides k {\displaystyle k} and x {\displaystyle x} . It

882-565: 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 (Halley's si applicari possit altera super alteram ), they are “equal” (Halley's aequales ) if they coincide and no line of one crosses any line of the other. This is obviously a standard of congruence following Euclid, Book I, Common Notions, 4: “and things coinciding ( epharmazanta ) with one another are equal ( isa ).” Coincidence and equality overlap, but they are not

945-459: A section,” “cut off between the section and its axis,” cut off by the axis,” all referring to the same image. In the view of Fried and Unguru, the topic of Book V is exactly what Apollonius says it is, maximum and minimum lines. These are not code words for future concepts, but refer to ancient concepts then in use. The authors cite Euclid, Elements, Book III, which concerns itself with circles, and maximum and minimum distances from interior points to

1008-561: 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

1071-420: 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

1134-406: 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

1197-412: 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

1260-452: Is no mention of maxima/minima being per se normals in either the prefaces or the books proper. Out of Heath's selection of 50 propositions said to cover normals, only 7, Book V: 27–33, state or imply maximum/minimum lines being perpendicular to the tangents. These 7 Fried classifies as isolated, unrelated to the main propositions of the book. They do not in any way imply that maxima/minima in general are normals. In his extensive investigation of

1323-487: 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. 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

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1386-406: Is the deficit, while an equation for the hyperbola, becomes where C / B {\displaystyle C/B} is the surfeit. 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

1449-504: Is the language of definition, but no definitions are forthcoming. Whether the reference might be to a specific kind of definition is a consideration but to date nothing credible has been proposed. The topic of Book VII, completed toward the end of Apollonius’ life and career, is stated in Preface VII to be diameters and “the figures described upon them,” which must include conjugate diameters , as he relies heavily on them. In what way

1512-452: Is the point on the axis. Rotating a ruler around it, one discovers the distances to the section, from which the minimum and maximum can be discerned. The technique is not applied to the situation, so it is not neusis. The authors use neusis-like, seeing an archetypal similarity to the ancient method. 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

1575-571: 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. The Greek text of Conics uses the Euclidean arrangement of definitions, figures and their parts; i.e.,

1638-639: The Middle Ages , was superseded during the Renaissance . The Apollonius crater on the Moon is named in his honor. 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

1701-458: The evolute of the section. Such a figure, the edge of the successive positions of a line, is termed an envelope today. Heath believed that in Book V we are seeing Apollonius establish the logical foundation of a theory of normals, evolutes, and envelopes. Heath's was accepted as the authoritative interpretation of Book V for the entire 20th century, but the changing of the century brought with it

1764-472: 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

1827-517: 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

1890-902: 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

1953-431: The circumference. Without admitting to any specific generality they use terms such as “like” or “the analog of.” They are known for innovating the term “neusis-like.” A neusis construction was a method of fitting a given segment between two given curves. Given a point P, and a ruler with the segment marked off on it. one rotates the ruler around P cutting the two curves until the segment is fitted between them. In Book V, P

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2016-422: 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

2079-570: The credit for them. Basilides of Tyre Basilides of Tyre ( Ancient Greek : Βασιλείδης ) was a mathematician , mentioned by Hypsicles in his prefatory letter of Euclid's Elements , Book XIV. Barnes and Brunschwig suggested that Basilides of Tyre and Basilides the Epicurean could be the same Basilides. From Hypsicles letter it appears plausible that a Basilides of Tyre has met Hypsicles and his father, perhaps in Alexandria ,

2142-615: The diameter cut both branches of the hyperbola. These lines are chord-like except that they do not terminate on the same continuous curve. A conjugate diameter can be drawn from the centroid to bisect the chord-like lines. These concepts mainly from Book I get us started on the 51 propositions of Book VII defining in detail the relationships between sections, diameters, and conjugate diameters. As with some of Apollonius other specialized topics, their utility today compared to Analytic Geometry remains to be seen, although he affirms in Preface VII that they are both useful and innovative; i.e., he takes

2205-427: 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

2268-439: 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 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

2331-402: The foot. The distance from the foot to the center is the radius of curvature . The latter is the radius of a circle, but for other than circular curves, the small arc can be approximated by a circular arc. The curvature of non-circular curves; e.g., the conic sections, must change over the section. A map of the center of curvature; i.e., its locus, as the foot moves over the section, is termed

2394-554: The former, g ( x ) {\displaystyle g(x)} falls short of y 2 {\textstyle y^{2}} by a quantity termed the ellipsis , "deficit". In the latter, g ( x ) {\displaystyle g(x)} overshoots by a quantity termed the hyperbole , "surfeit". Applicability could be achieved by adding the deficit, y 2 = f ( x ) = g ( x ) + d , {\textstyle y^{2}=f(x)=g(x)+d,} or subtracting

2457-636: 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

2520-417: The major and minor axes are, the elongation destroying the perpendicularity in all other cases. Conjugates are defined for the two branches of a hyperbola resulting from the cutting of a double cone by a single plane. They are called conjugate branches. They have the same diameter. Its centroid bisects the segment between vertices. There is room for one more diameter-like line: let a grid of lines parallel to

2583-409: The original. 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

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2646-406: The other 43 propositions, Fried proves that many cannot be. Fried and Unguru counter by portraying Apollonius as a continuation of the past rather than a foreshadowing of the future. First is a complete philological study of all references to minimum and maximum lines, which uncovers a standard phraseology. There are three groups of 20-25 propositions each. The first group contains the phrase “from

2709-648: 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, also a translation from the Arabic, contains 51 Propositions. These are the last that Heath considers in his 1896 edition. 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. 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.” This

2772-426: The others are successively shorter until the last is not a chord, but is a tangent point. The tangent must be parallel to the diameter. A conjugate diameter bisects the chords, being placed between the centroid and the tangent point. Moreover, both diameters are conjugate to each other, being called a conjugate pair. It is obvious that any conjugate pair of a circle are perpendicular to each other, but in an ellipse, only

2835-447: The plan, are somewhat in deficit, Apollonius having depended more on the logical flow of the topics. 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

2898-511: The same and do not share aspects that are different. Intuitively the geometricians had scale in mind; e.g., a map is similar to a topographic region. Thus figures could have larger or smaller versions of themselves. The aspects that are the same in similar figures depend on the figure. Book 6 of Euclid's Elements presents similar triangles as those that have the same corresponding angles. A triangle can thus have miniatures as small as you please, or giant versions, and still be “the same” triangle as

2961-417: The same plane” is a circle, ellipse or parabola, while “two curves in the same plane” is a hyperbola. A chord is a straight line whose two end points are on the figure; i.e., it cuts the figure in two places. If a grid of parallel chords is imposed on the figure, then the diameter is defined as the line bisecting all the chords, reaching the curve itself at a point called the vertex. There is no requirement for

3024-403: The same: the application of areas used to define the sections depends on quantitative equality of areas but they can belong to different figures. Between instances that are the same (homos), being equal to each other, and those that are different , or unequal , are figures that are “same-ish” (hom-oios), or similar . They are neither entirely the same nor different, but share aspects that are

3087-405: The section and the axis. Heath is led into his view by consideration of a fixed point p on the section serving both as tangent point and as one end of the line. The minimum distance between p and some point g on the axis must then be the normal from p. In modern mathematics, normals to curves are known for being the location of the center of curvature of that small part of the curve located around

3150-429: The sections. A normal in this case is the perpendicular to a curve at a tangent point sometimes called the foot. If a section is plotted according to Apollonius’ coordinate system (see below under Methods of Apollonius), with the diameter (translated by Heath as the axis) on the x-axis and the vertex at the origin on the left, the phraseology of the propositions indicates that the minima/maxima are to be found between

3213-425: The surfeit, g ( x ) − s . {\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, can be placed in the form where C / B {\displaystyle C/B}

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3276-464: The term “limits” or “determinations” might apply is not mentioned. Diameters and their conjugates are defined in Book I (Definitions 4–6). Not every diameter has a conjugate. The topography of a diameter (Greek diametros) requires a regular curved figure . Irregularly-shaped areas, addressed in modern times, are not in the ancient game plan. Apollonius has in mind, of course, the conic sections, which he describes in often convolute language: “a curve in

3339-514: The text, due to damage or corruption in the previous texts. The topic is relatively clear and uncontroversial. Preface 1 states that it 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

3402-498: 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

3465-479: The topic, which in Prefaces I and V Apollonius states to be maximum and minimum lines. These terms are not explained. In contrast to Book I, Book V contains no definitions and no explanation. The ambiguity has served as a magnet to exegetes of Apollonius, who must interpret without sure knowledge of the meaning of the book's major terms. Until recently Heath's view prevailed: the lines are to be treated as normals to

3528-460: 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

3591-441: 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 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

3654-466: 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

3717-399: 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

3780-583: Was he who accordingly named the figure, parabola, "application". The "no applicability" case is further divided into two possibilities. Given a function, f ( x ) {\textstyle f(x)} , such that, in the applicability case, y 2 = g ( x ) {\textstyle y^{2}=g(x)} , in the no applicability case, either y 2 > g ( x ) {\textstyle y^{2}>g(x)} or y 2 < g ( x ) {\textstyle y^{2}<g(x)} . In

3843-483: 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

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3906-639: 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

3969-492: 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

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