Subject: Re: [HM] History of the Conics
From: Bill Everdell (Everdell@aol.com)
Date: Tue Mar 21 2000 - 23:53:04 EST
In a message dated 3/6/00 11:01:50 AM, Michael N. Fried quotes Euclid on
conics:
<<It may be that thought about conic sections came together with thought
about cylindrical sections, in which case, one could point to the shape
produced by a column broken at a slant. In one of the early references to
conic sections, Euclid does, in the same breath, mention such a cylindrical
section: "If a cone or cylinder is cut by a plane not parallel to the base,
the section produced will be a section of an acute-angled cone [i.e. that
produced by cutting such a cone by a plane perpendicular to one of the cone's
generating lines], which is similar to a shield" (Eucl. ed. Heiberg-Menge,
viii.6).>>
This reminded me of a fragment of Democritus (Democritus #9 in Wheelwright's
Presocratics anthology) where Democritus asks an interlocutor to imagine a
cone sliced parallel to its base. The question he then asks is whether the
circle on the base of the resulting shorter cone and the circle on the top of
the resulting frustum are equal, which gets him (and my 6th-grade class this
week) into the irreconcilability of continuity and finite differences. The
cone-slicing, though, is taken for granted in the passage as if it were a
common thought experiment in D's time, which current wisdom would put at ca.
400 BC, perhaps a bit before Plato but long before Euclid.
Bill Everdell
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