Preface, Introduction & Prefatory Letters¶
1. From Three Right Cones to One Oblique Scalene Cone¶
Prior to Apollonius of Perga, Greek geometers generated the three conic sections by cutting three distinct species of right circular cones with a plane strictly perpendicular to one generator of the cone (Heath 1896, pp. xvii–xxx):
Menaechmus (c. 350 BCE), seeking two mean proportionals to solve the Delian cube-duplication problem (, , ), cut a right-angled, acute-angled, and obtuse-angled right cone perpendicular to a slant side, calling the curves simply the section of a right-angled cone (orthotome), section of an acute-angled cone (oxytome), and section of an obtuse-angled cone (amblytome).
Aristaeus the Elder (in his Five Books on Solid Loci) and Euclid (in his lost four-book Conics) preserved this three-cone triad, while Archimedes (Quadrature of the Parabola, On Conoids and Spheroids) already recognized that elliptical sections arise when an arbitrary plane cuts all generators of a cone or cylinder (Archimedes, ed. Heath 1897).
Apollonius revolutionized the science by proving in Book I that every circular cone whatsoever—whether right or oblique (scalene)—contains all three species of conic sections together with the two-branched opposite sections, solely by varying the inclination of the cutting plane relative to the axial triangle:
Plate 1 — Apollonius’s Oblique Double Cone (Book I.1–14). From apex A over circular base BC, a plane parallel to generator AC cuts the parabola; a plane meeting all generators of one nape cuts the ellipse; and a plane meeting both napes cuts the hyperbola and its opposite branch.
2. Apollonius’s Own Prefatory Letters (Books I–VII)¶
Unlike Euclid’s impersonal axiomatic treatise, Apollonius prefaced each Book of the Conics with an epistolary cover letter explaining its genesis, scope, and mathematical novelty (Heiberg 1891, Halley 1710, Heath 1896, pp. lxix–lxxv).
General Preface (Book I — To Eudemus of Pergamum)¶
“Apollonius to Eudemus, greeting. If you are in good health and spirits, it is well; I too am moderately well. During the time I spent with you at Pergamum I observed your eagerness to become acquainted with my work in conics; I am therefore sending you the first book, which I have corrected, and I will forward the remaining books when I have finished them to my satisfaction. For I daresay you have not forgotten my telling you that I undertook the investigation of this subject at the request of Naucrates the geometer, at the time when he came to Alexandria and stayed with me, and that, when I had worked it out in eight books, I gave them to him at once, too hurriedly, because he was on the point of sailing, and therefore they were not thoroughly revised—indeed I had put down everything just as it occurred to me, postponing revision till the end.”
Apollonius then divides the entire eight-book architecture into two grand divisions:
Books I–IV — Elementary Introduction (πέπτωκεν εἰς ἀγωγὴν στοιχειώδη): - Book I: Modes of producing the three sections and the opposite branches, and their fundamental properties (symptōmata) worked out more fully and generally than in earlier writers. - Book II: Properties of diameters, axes, and asymptotes, together with constructions of tangents. - Book III: Many remarkable theorems useful for the synthesis of solid loci and diorismoi, the majority and prettiest being new—providing at last the complete synthesis of the Locus with respect to Three and Four Lines, of which Euclid had worked out only a fortuitous fragment. - Book IV: In how many ways conic sections meet one another and the circumference of a circle, besides the behavior of double-branch hyperbolas left unproven by Conon of Samos and Nicoteles of Cyrene.
Books V–VIII — Advanced Special Investigations (περιουσιαστικώτερα): - Book V (to Attalus, preserved via Thābit ibn Qurra's Arabic): Minimum and maximum straight lines (normals), treated for their own intrinsic beauty rather than mere tangent corollaries. - Book VI (to Attalus): Equal and unequal, similar and dissimilar conic sections and segments, and how to cut a given conic section inside a prescribed right cone. - Book VII (to Attalus): New propositions concerning conjugate diameters and the rectilinear figures described upon them, serving as diorismoi for the determinate conic problems solved in Book VIII (now lost).
Preface to Book II (To Eudemus & Philonides)¶
“Apollonius to Eudemus, greeting… I am sending my son Apollonius to you with the second book of my conics. Read it carefully and communicate it to such others as are worthy of it. And if Philonides the geometer, whom I introduced to you in Ephesus, should ever happen to visit the neighbourhood of Pergamum, communicate it to him also.”