Subject: Re: [HM] History of the Conics
From: Michael Fried (mfried@ramat-negev.org.il)
Date: Mon Mar 06 2000 - 05:14:34 EST
Victor Steinbok wrote:
> ------------------------- Original Message -------------------------
>
> Subject: History of the Conics
> Date: 5 Mar 00 14:18:16 -0500 (EST)
> From: Jim Totten <jtotten@saintanns.k12.ny.us>
> To: geometry-pre-college@mathforum.com
>
> I'm wondering how Apollonius (or someone else?) first made the
> connection between the shapes--parabola, ellipse, hyperbola--and the
> slicing of a cone to generate those shapes.
> Also, can anyone recommend an informative book on the subject?
> Thanks.
Dear Jim,
Like the origins of so many things, we shall probably never know for sure
how or why the curves produced by sectioning a cone became a subject for
inquiry among the Greeks. Relying on Geminus as his authority, Eutocius
(6th cent. A.D.) describes the conic sections before Apollonius as
sections of a right cone where the cutting plane is perpendicular to the
generating line of the cone. Neugebauer suggested that the origin of the
conic sections, according to this particular formulation, lay in the
construction of sundials ("The Astronomical Origin of conic Sections,"
The Proceedings of the American Astronomical Society, vol. 92, July
1948). In particular, he described a sundial whose gnomon is adjusted so
as to point directly at the sun at noon and whose shadow plane is set
perpendicular to the gnomon. The shadow curve arising in such a sundial
would clearly be a conic section produced in the manner described by
Eutocius. But, as Neugebauer himself admits, there are no known Greek
sundials of this type, and, in general, the whole discussion seems
somewhat factitious. That said, the idea that the investigation of conic
sections arose in connection with shadow curves does not seem to me
completely far-fetched. Indeed, in the lower latitudes, where Greece is,
a simple gnomon perpendicular to the ground produces maybe the oddest of
the conic sections, the hyperbola, and in such a way that its geometric
origin, as a section of a cone, is quite evident.
Of course, there are other observations that, one could speculate, gave
rise to thought about conic sections: one need only think of the way
light shining through a chink in the wall or the ceiling looks when it is
cast on the floor. In a Greek temple, which was otherwise a very dark
place, such, I can imagine, would have been a very impressive sight! 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). Naturally, there is a
problem with this explanation since the section of a cylinder is
described in terms of a section of a cone, and not the other way around.
In fact, the only work that I know treating the sections of a cylinder
explicitly is a very late work, namely, that by Serenus (4th cent. A. D.
(?)), and the main theorem of Serenus' work shows that an oblique section
of a cylinder is an ellipse by showing how the very same section can be
produced by a cone!
Before ending this already over long note (I hope I shall be
forgiven...), I should say something about the form of your question: it
is put as if the conic sections were first known in some abstract way and
only later conceived of as curves generated by sectioning a cone. Knorr
suggests something to this effect in his fairly well-known piece,
"Observations on the Early History of the Conics" (Centaurus 26, (!982),
253-291). Knorr sees "the triads of Menaechmus," usually associated with
the three conic sections, the parabola, hyperbola, and ellipse, in the
context of problems involving application of areas (the kind of
procedures one has in Euclid VI.28-29). In Knorr's view, the theory of
conic sections developed according to the following sequence:
"...Menaechmus and the geometers in the decades immediately before and
after him initiated not a theory of the conic sections, but a body of
geometric problems solved according to a form of the method of analysis.
When the resolving loci turned out to be curves which were later known as
conic sections, these were at first constructed, if at all, via
point-wise procedures. Only late in the fourth century, near the time of
Euclid, did one conceive of generating a class of curves via the
sectioning of cones and begin their geometric investigation" (p.7).
Knorr is probably right about Menaechmus' thinking more about parabolic,
elliptic and hyperbolic application of areas, than about parabolas,
ellipses, and hyperbolas, and he is probably right about the chronology,
i.e., that the theory of conic sections, as such, developed sometime in
the late 4th -- early 3rd century. I am far less convinced, however,
that the course of this development was as linear as Knorr would have it.
I think it is more likely that in the time of Apollonius there was a
convergence of ideas, that the curves produced by sectioning a cone were
first being studied for their own interest (of course, this begs your
original question...) and only afterwards were seen to be connected to
the three kinds of area application. I have two reasons for thinking
this.
The first is that before Apollonius, as I already mentioned, the names of
the conic sections were related immediately to sectioning
cones--"orthogwniou kwnou tomh" (section of a right-angled cone),
"amblugwniou kwnou tomh" (section of a obtuse-angled cone), "oxugwniou
kwnou tomh" (section of an acute-angled cone). Only later, with
Apollonius, are names given that reflect the connection between these
sections and parabolic, hyperbolic, and elliptic application of areas.
Indeed, early references to conic sections, as in the quotation from
Euclid above, speak only in a general way about the shape of conic
sections (that the section of an acute-angled cone is like a shield)
rather than about their specific "symptomata," their main properties.
The second reason (and this, I admit, is slightly more controversial
point, but, still, not very much so) is that, in general, curves in Greek
mathematics were not seen as objects summarizing properties, but as
objects that have properties. In Greek mathematics, in other words, one
does not begin with a property, for example, squares on lines being equal
to rectangles contained by other lines and a given lines, and, from that,
define a curve. Rather, one finds a curve by some geometrically sensible
means and shows that certain lines in it have the property that the
square on the one is equal to a rectangle contained by the other and a
fixed line. Thus, it seems to me that the nexus in Knorr's account is
far too dependent on this (I should say, modern way of) defining curves
by a property.
Yours,
Michael N. Fried
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