Elements XIII), as well as a study in the cylindrical helix (a response to Archime ̄de ̄s’ Spiral
Lines?), a study of “unordered irrationals” (a response to Euclid’s Elements X?), and a “gen-
eral treatise” of unknown significance.
The Conics. Apollo ̄nios himself considered the opus, or at least its first four books, as
“Elements of Conics.” It was certainly not the first of its kind, and Apollo ̄nios’ originality is
never certain. While, in part, “Elementary,” Conics – not an axiomatic sequence in the
manner of Euclid’s Elements – instead forms a diverse collection of nearly independent
treatises, with certain unifying themes and goals. Book I is the most obviously “Elementary”
in character, defining the main conic sections, deriving their most useful proportions and
leading up to their construction from given points and lines. This appears also to be the least
original to Apollo ̄nios. Furthermore, this book inspires the main modern scholarly debates
concerning the Conics: did Apollo ̄nios define the conic sections primarily as (a) the results of
a geometrical cut, or as (b) the locus satisfying a certain proportion? (a) is consistent with a
thoroughly geometrical interpretation of Greek mathematics, (b) – with a more modern-
izing and algebraic one. Books II–III form a certain continuity (Book III is the only extant
book not to carry any introduction), building up various surprising equalities arising from
conic sections. The great historical significance of that sequence is that it provides the tools
(as asserted by Apollo ̄nios himself) for the problem of a three and four line locus, a major
inspiration for the algebraization of geometry down to Descartes and beyond. Book IV
studies the intersections of conic sections. Its more qualitative character allows for less
spectacular results than those of Books II–III, and it has therefore been relatively neglected
by scholarship. Fried makes the case for its importance. Book V, the most ambitious among
the extant books, studies the shortest and longest lines drawn to conic sections from given
points. It can be taken as an example of the precise, systematic and advanced character of
Apollo ̄nios’ geometry. Once again, the question whether Apollo ̄nios’ shortest and longest
lines should be considered as “normals” impinges on the question of the overall interpret-
ation of the Conics as geometric or algebraic in character. Book VI studies the similarities
between conic sections, and suffers the same critical fate as Book IV, for similar reasons.
Book VII returns to the spirit of Books II–III, producing a remarkable succession of
theorems concerning, specifically, conjugate diameters. It may have served as preparation
for the problems of Book VIII, now lost.
For survey and study of the Conics, see Fried and Unguru (2001). Zeuthen (1886), usually
rejected today for its extreme modernizing interpretations, remains a classic.
Ed.: G.J. Toomer, Conics: Books V to VII/Apollonius: the Arabic Tradition (1990); M. Fried, Apollonius of
Perga: Conics Book IV (2002).
H.G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum (1886); M. Fried and S. Unguru, Apollonius of
Perga’s Conica: Text, Context, Subtext (2001); NDSB 1.83–85, F. Acerbi.
Reviel Netz
Apollo ̄nios of Pergamon (Agric.) (325 – 90 BCE)
Wrote a work on agriculture which may have treated cereals, livestock, poultry, viticulture,
and arboriculture (cf. P, 1.ind.8, 10, 14–15, 17–18), and was excerpted by C
D (V, RR 1.1.8–10, cf. C, 1.1.9). In Wellman’s opinion he is to be
distinguished from the homonymous medical writer.
RE 2.1 (1895) 150 (#104), M. Wellmann.
Philip Thibodeau
APOLLO ̄NIOS OF PERGAMON (AGRIC.)