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.)

Read the text

Return to the calendar

last modified 10/7/00

Copyright (c) 2000. Daniel E. Otero