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Apollonius  

Michael N. Fried

Apollonius of Perga (fl. 200 bce) was the youngest of the three leading mathematicians of the Hellenistic age, the other two being Euclid and Archimedes. His most significant work was Conics. Conics, comprising eight books, remained the most comprehensive treatment of the subject of conic sections until the 17th century. It was because of Conics that Apollonius was known in antiquity as the “Great Geometer.” Its first four books formed a “course in the elements” of conics, as Apollonius describes it. In these books, which are the only works of Apollonius extant in Greek, he shows how the parabola, ellipse, hyperbola, and opposite sections (what in the early 21st century are called the two branches of a hyperbola) could be obtained by sectioning an oblique double-napped cone; he specified their distinct properties, diameters, axes, tangents, and asymptotes. These books also include Apollonius’s solution to the three- and four-line locus. The later four books, of which only three survive in an Arabic translation, consider “special topics”: “maximum and minimum lines”; similarity and equality of conic sections; special theorems related to limits of possibility (diorismoi), involving diameters, conjugate diameters and the “figures” related to them; determinate problems in conics.

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conic sections  

Michael N. Fried

The curves known as conic sections, the ellipse, hyperbola, and parabola, were investigated intensely in Greek mathematics. The most famous work on the subject was the Conics, in eight books by Apollonius of Perga, but conics were also studied earlier by Euclid and Archimedes, among others. Conic sections were important not only for purely mathematical endeavors such as the problem of doubling the cube, but also in other scientific matters such as burning mirrors and sundials. How the ancient theory of conics is to be understood also played a role in the general development of the historiography of Greek mathematics.The term conic sections, familiar to all students in modern mathematics classrooms, is a direct translation of the Greek, tomai tōn kōnōn. For the Greek mathematicians, conic sections were, true to their name, those curves produced by cutting the surface of cone with a plane not containing the vertex of the cone. The exact cutting procedure—indeed, the definition of the cone itself—changed in significant ways from the end of the 4th to the beginning of the 2nd century .