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Bob C
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May-30-05, 09:24 PM (EST) |
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"Conics"
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I have a math history question that you may be able to answer for my Algebra 2 class. After our study of the proof of ellipses at https://www.cut-the-knot.org/proofs/conics.shtml, one of my students asked me if there are other examples in mathematics where mathemeticians construct a geometric or other mathematical concept using different definitions, only to find later that there is a connection that relates them in a elegent proof like the one above? It'seems remarkable that some mathemeticians defined conic sections using the intersection of a plane and a right circular cone, while others used the focal definition, not realizing that there was a connection between them until G.P.Dandelin's proof. In fact, until G.P.Dandelin's proof, how could mathemeticians be certain that the two definitions always produced the same geometric shape? Thank you. |
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alexb
Charter Member
1541 posts |
May-31-05, 12:16 PM (EST) |
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1. "RE: Conics"
In response to message #0
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>I have a math history question that you may be able to
>answer for my Algebra 2 class. After our study of the proof
>of ellipses at
>https://www.cut-the-knot.org/proofs/conics.shtml, one of my
>students asked me if there are other examples in mathematics
>where mathemeticians construct a geometric or other
>mathematical concept using different definitions, only to
>find later that there is a connection that relates them in a
>elegent proof like the one above? I'll have to think about this to give you a good example. But, say, Euclid knew that the ratio of the circumference of the circle to its diameter is constant across all circles as is the ratio of the area and the square of the radius. However, the Elements give no indication that the two constants are related, let alone coincide. Archimedes, though, knew about that. >It'seems remarkable that some mathemeticians defined conic
>sections using the intersection of a plane and a right
>circular cone, while others used the focal definition, not
>realizing that there was a connection between them until
>G.P.Dandelin's proof. In fact, until G.P.Dandelin's proof,
>how could mathemeticians be certain that the two definitions
>always produced the same geometric shape? Strange as it may appear, Apollonius was aware of the focal properties of ellipse. He knew of the existence of two points C and on the big axis AB such that for any point E on the ellipse EC + ED = AB (Conics III, Proposition 52). He also knew how to construct these points. Dandelin came up with a simple graphical explanation of the origins of that property. |
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alexb
Charter Member
1541 posts |
May-31-05, 12:36 PM (EST) |
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3. "RE: Conics"
In response to message #1
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>I'll have to think about this to give you a good example. An elementary example will have to wait its turn, but look at Wiles' proof of the FLT. The Tanayama-Shimura conjecture that every elliptic equation is related to a modular form was a bombshell. Later on, Gerhard Frey conjectured that Tanayama-Shimura implies the FLT. Ken Ribet proved Frey conjecture with the help from Barry Masur during a coffee break. Ribet, who spent 18 months trying to prove the conjecture, complained to Masur that he was getting nowhere. Masur saw otherwise: he almost immedately pointed to Ribet that the latter in fact had already a solution. Ribet kept wondering how he could have missed that. |
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alexb
Charter Member
1541 posts |
Jun-03-05, 00:51 AM (EST) |
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4. "RE: Conics"
In response to message #0
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I got a message from Douglas Rogers from the U. of Hawaii: Alex, Perhaps it is too innocuous an example to cause much stir, but the
entries in Pascal's triangle can be defined in several different ways,
and it is then a matter of reconciling them to be reassured that one
is indeed in the presence of one and the same object. This is the opening theme of the book, ``Pascal's Arithmetical
Triangle'', by A.W.F. Edwards <awfe@cam.ac.uk>, newly republished by
The Johns Hopkins University Press. I believe that you would enjoy this
work, if you have not already encountered it, along with the same
author's more recent book, ``Cogwheels of the Mind'', which tells the
story of Venn diagrams. This message is shared with AWFE for ease and speed of communication. Douglas Rogers.
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alexb
Charter Member
1541 posts |
Jun-03-05, 00:52 AM (EST) |
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5. "RE: Conics"
In response to message #4
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And more: Alex, Those different ways of looking at the binomial coefficients certainly were not entertained all at the same time. But I do not suppose that anyone thought that only with one new interpretation was understanding finally achieved. Similarly, it was possible for matrix theory to be redeveloped
unwittingly for quantum mechanics, but it was quickly observed that what was being developed was matrix theory after all. Bose showed that certain families of mutually orthogonal Latin squares were equivalent to projective planes - that tied together two familiar objects in a surprising way. GPD's result is not really so much about sections of the cone being
conic sections but rather about how those sections arise in terms of
intersetings with the cone of tangents to a sphere sitting inside the
cone, which gives greater unity to these results. By the bye, you might enjoy visiting at https://lostlecture.host.sk/JDandelinEn.htm if you have not done so already. Douglas Rogers. |
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