Apollonius

Abstract

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. The eighth book—containing problems in conics—was lost sometime in late antiquity, but reconstructions were attempted by Ibn al-Haytham and Edmond Halley. Other works by Apollonius include Cutting Off of a Ratio, Cutting Off of an Area, Determinate Section, Tangencies, Vergings, and Plane Loci. These are mentioned in Pappus’s Collection, Book VII and are central to the tradition of Greek analysis. Apollonius, apparently, also did work in astronomy.