VI.4 Apollonius

b. Perge, Pamphylia (now Perga, Turkey), ca. 262 B.C.E.;

d. Alexandria, Egypt?, ca. 190 B.C.E.

Conic sections; diorism; locus problems

The Conics, in eight books, only seven of which have come down to us, has had fewer modern readers than other recognized masterpieces of Greek mathematics: it is complex, difficult to summarize, and easy to mistranslate into modern algebraic notation. Apollonius of Perga also wrote about arithmetic and astronomy, but none of these works survive. The letters prefacing six of the surviving books indicate that he was a highly esteemed member of a network of mathematicians, to whom he sent his results. He refers to the fact that various versions of his Conics were circulated, the latest probably incorporating his correspondents’ feedback. Knowledge of the parabola, hyperbola, and ellipse predates Apollonius (we find conics in Archimedes), but his is the first known systematic account of these curves, which were of interest both in themselves and because they could be used as auxiliary lines for the solution of problems such as the trisection of an angle or the duplication of the cube.

Apollonius himself declares that the first four books of the Conics are an introduction to the subject, and indeed he starts with definitions of the cone and its various parts. The parabola, hyperbola, and ellipse are not introduced until later, so that their origin (from a plane cutting a cone or a conic surface at different angles) is already accompanied by a statement of their properties, which are further and fully explored in the next three books. These include theorems on tangents, asymptotes, and axes; constructions of conic sections on the basis of certain data; and an account of the conditions under which conics can intersect in the same plane.

The nonelementary books, which exist only in Arabic, contain treatments of maximum and minimum lines within the sections, the construction of conic sections equal or similar to a given conic section (including the theorem that all parabolas are similar), and “diorismic theorems.” These are propositions that set the limits of possibility of a construction, or the limits of validity of some property of a geometrical configuration, given a certain number of known positions or known objects at the outset. Indeed, several of the propositions in the Conics are about loci, i.e., geometrical configurations consisting of all the points sharing a certain family of properties. Apollonius criticizes EUCLID [VI.2] for not having provided an exhaustive solution to the construction of the three-line and four-line locus (configurations of three or four lines, arranged so that they have specific properties).

In terms of demonstrative methods, Apollonius is in the axiomatic-deductive mold: general enunciations, lettered diagrams, each step justified by appeal to undemonstrated premises or previous proofs. Instead of indirect methods, we find a real mastery of the intricacies (and power) of proportion theory. At the same time, his propositions easily lend themselves to the consideration of different subcases: when, for instance, a certain line falls inside, outside, or on the vertex of a conical surface. Apollonius, in other words, combines a systematic approach with an almost playful fascination with exploring the possibilities of mathematical objects and their properties under varying circumstances.

Further Reading

Apollonius. 1990. Conics, books V–VII. Arabic Translation of the Lost Greek Original in the Version of the Banu Musa, edited with translation and commentary by G. J. Toomer, two volumes. New York: Springer.

Fried, M. N., and S. Unguru. 2001. Apollonius of Perga’s Conica: Text, Context, Subtext. Leiden: Brill.

Serafina Cuomo