Conic Sections

Abstract

We now turn our attention to another of the great treatises of the classical period, the Conics of Apollonius of Perga. Apollonius wrote eight books on conic sections; the first four have survived in the original Greek text (a critical edition was published by Heiberg 1893), books V, VI and VII were reconstructed from arabic texts by E.Halley (1710), the last volume is lost. We base our quotations on the French translation by Ver Eecke (1923). An authoritative English edition, slightly arranged and adapted to modern notation, was published by Heath (1896). The theory of conics was taken up again by Kepler (1604), who included a short section on conics in his book on astronomy and optics. He emphasised the two particularly important points of a conic and called them foci (”Nos lucis causˆa, et oculis in Mechanicam intensis ea puncta Focos appellabimus”). Many of Apollonius’ proofs were later simplified with the use of analytic methods, see Chap. 7. Even the most elegant geometric idea in this field, Dandelin spheres, had to wait another 2000 years before being discovered by a Belgian army engineer (G.P.Dandelin, 1794–1847). This discovery turned the presentation of conics upside down.

“The cream of the classical period’s contributions are Euclid’s Elements and Apollonius’ Conica.” (M.Kline, 1972, p. 27)

“Quotusquisque Mathematicorum est, qui tolerat laborem per legendi Appollonii Pergaei Conica? [How few mathematicians would endure the effort of reading the entire Conics of Apollo-nius of Perga?]” (J.Kepler, 1609, from the introduction)

“… i libri di Apollonio, … delle quali sole siamo bisogni nel presente trattato [the books of Apollonius, the only tools which we requirein the present treatise]” (Galilei, Discorsi 1638, fourth day)

“A peine la G′eom′etrie sortoit-elle de l’enfance, qu’elle s’occupa des Sections coniques, … [Barely out of infancy, geometry devoted itself to conic sections …]” (G.Cramer, 1750, p. vi)