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### The Origin of Conic Sections

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Date: 10/02/98 at 16:04:42
From: Kevin Gray
Subject: Origin of Conic Sections

Hello,

We are studying conic sections in math right now. Our teacher told us
about each one, but failed to tell us why you have 2 right cones. When
we asked, she told us that she didn't know. My question is: Why do you
use 2 right cones? Where did that originate from? It just seems that
if you take any shape, you can make other shapes by intersecting them
with a plane, but who thought of taking two right cones?

I am basically looking for a historical background. I tried looking on
the Web, but it is too complicated for me to understand the things that
I found.

Thanks,
Kevin Gray
```

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Date: 10/02/98 at 17:12:01
From: Doctor Rob
Subject: Re: Origin of Conic Sections

Kevin,

A right circular cone is formed by taking a circle (called the
generatrix of the cone), and a line (called the axis of the cone)
through its center perpendicular to the plane of the circle. Then you
pick a point (called the vertex of the cone) on the axis different
from the center of the circle. Now you look at the set of all lines
through the vertex and a point on the circle. All the points on all
these lines form the cone. It is an infinite surface, consisting of
two "sheets" touching only at the vertex. (Remove the vertex, and the
surface is no longer "connected.")

You can see that, since all these lines extend infinitely in two
directions, both sheets of the cone are included. One might ask who
thought of ignoring one of the sheets? To ignore the part of the cone
farther from the vertex than the plane of the circle is also somewhat
artificial.

What most people think of as a cone is actually one of two different
things: either a region of space bounded by a cone (the surface
described above) and a plane perpendicular to its axis (a 3-dimensional
figure) or else the curved part of the boundary of that region (a
2-dimensional figure). Of course this plane only intersects one of the
sheets, so the other sheet is not part of the boundary of that region.

For a similar discussion, see What is a three-dimensional figure? on
this Web page:

http://mathforum.org/dr.math/faq/formulas/faq.figuredef.html

Probably the Greeks at first only thought of one sheet of a cone, and
one half of the hyperbola. (Of course with the ellipse and parabola,
the second sheet is not an issue.) Apollonius of Perga was the Greek
who left us with the greatest amount of information about their
knowledge of the conic sections. For Apollonius' biography, see the
MacTutor Math History Archive:

http://www-groups.dcs.st-and.ac.uk:80/~history/Indexes/A.html

For a page from one of his books, choose Apollonius3.gif from:

http://www-groups.dcs.st-and.ac.uk:80/~history/Bookpages/

Note the single half of the hyperbola and the single sheet of the cone
in the diagram.

With the advent of Cartesian coordinate geometry, invented by Rene
Descartes, he and others analyzed the conic sections and found their
equations. These turned out to be quadratic equations in x and y. It
was probably at this point that it was realized that the hyperbola had
two parts, not connected to each other, and that each represented an
intersection of the same plane with a separate sheet of the full cone.

http://www-groups.dcs.st-and.ac.uk:80/~history/Indexes/D.html

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School History/Biography

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