Apollonius of Perga

Introduction: a unified theory of conics

The only major work of Greek geometry to survive in written form that studies the conic sections in detail is the Conics of Apollonius of Perga (262? - 190?BCE).  Even so, it survives only partially.  We have at present only the first seven of the eight books that Apollonius wrote.  As we have seen, the conics were used by Menaechmus in dealing with the problem of the duplication of the cube in around 350BC, and we have references in other works to treatises on the conics written by Aristaeus, a contemporary of Menaechmus, and by Euclid, but these are now lost.  In any event, the work by Apollonius was extremely well-received by geometers of the ancient world, so much so that it seems to have displaced all other writings in the subject.  As Carl Boyer, a noted historian of mathematics, puts it, "If survival is a measure of quality, the Elements of Euclid and the Conics of Apollonius were clearly the best works in their field."
About Apollonius we know very little.  He was born in Perga, in the region of Pamphylia in modern-day Turkey, wrote portions of the Conics under the patronage of the king of Pergamum, and having lived for many years in Alexandria, died there.
Euclid studies the properties of cones in the 11th book of the Elements.  In the definitions that open this book, he speaks of three types of cone: right-angled, acute-angled, and obtuse-angled.  These concepts are not explored in the Elements, but testify to the understanding at the time that the conic sections were obtained by cutting one of these different types of cone by a plane perpendicular to an edge of the cone.  The orthotome was the section of a right-angled cone, the oxytome the section of an acute-angled cone, and the amblytome was the section of an obtuse-angled cone.

Apollonius discovered, and published in his Conics, that if one allowed the cutting plane to vary its angle with respect to the side of the cone, then any cone would produce all three types of section.  He was able to identify characteristic geometric properties of the curves, which he called their symptoms, that are equivalent in modern algebraic language to the equations of these curves.  These symptomatic forms will be explained more carefully in the text and commentary below.  He also provided solutions to the tangent and normal problems for each of these curves.  (The normal to a curve at a point is the line perpendicular to the tangent there.)