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Recently, I came to know that ancient Greeks had already studied conic sections. I find myself wondering if they knew about things like directrix or eccentricity. (I mean familiar with these concepts in the spirit not in terminology).

This is just the appetizer. What I really want to understand is what will make someone even think of these (let me use the word) contrived constructions for conic sections.

I mean let us pretend for a while that we are living in the $200$ BC period. What will motivate the mathematicians of our time (which is $200$ BC) study the properties of figures that are obtained on cutting a cone at different angles to its vertical?

Also, what will lead them to deduce that if the angle of cut is acute then the figure obtained has the curious property that for any point on that figure the sum of its distances from some $2$ fixed points a constant.

And in the grand scheme of things how do our friend mathematicians deduce the concepts of directrix and eccentricity (I am not sure if this was an ancient discovery, but in all, yes I will find it really gratifying to understand the origin of conic sections).

Please shed some light on this whenever convenient. I will really find it helpful.

Thanks

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    $\begingroup$ I believe Apollonius of Perga was the first to seriously write about conic sections. He wrote several volumes at least on conics and their properties. I bet you could find a lot of interesting and intuitive constructions just by reading those. As far as motivation, I would say simple mathematics curiosity is one reason, with astronomy a very likely reason as well. Maybe that can get you started $\endgroup$ – Brent J Mar 29 '13 at 23:06
  • $\begingroup$ @BrentJ Thanks for your suggestion. Following your lead, I have something I need to study over the weekend I was wondering if you could suggest something else I could go through. As for the motivation part, I find the curiosity a surprising reason (one, i respectfully find myself disagreeing with). Given my limited experience with math, I have difficulties accepting that a mathematician will cut a cone (why just cone, go cut up some more fancy objects) and then study the figures obtained. I agree that they do create abstract objects. $\endgroup$ – Akash Kumar Mar 29 '13 at 23:39
  • $\begingroup$ [cont'd] Things like circles and lines were also probably once abstract. But I find ellipses, parabolas and hyperbolas fancier (and more so when I stand in 200 BC). Is there something else, that I am missing which historians of math can help with? In case I am being too demanding in which case I apologize. $\endgroup$ – Akash Kumar Mar 29 '13 at 23:39
  • $\begingroup$ (disclaimer: I'm not a historian) Before the popularization of algebra, a lot of time was devoted to geometry. People were more concerned with things like ruler and compas constructions (I suggest you read up on "squaring the circle" for a famous example of this type of problem). I suspect it was this line of thinking that inspired much of the research done. $\endgroup$ – hasnohat Mar 29 '13 at 23:57
  • $\begingroup$ @AkashKumar First, Julien's mentioning of "squaring the circle" has reminded me of another classic problem: "doubling the cube." I would look into that as well, because I believe that is how some early study of conics arose. Keep in mind that cutting through the double cone might also be a more modern construction. I don't know if the Greeks considered the conics like we do today. I suspect you a thorough study of conics will answer that question $\endgroup$ – Brent J Mar 30 '13 at 0:17
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As others noted, Apollonius wrote 8 volumes on conics.

For further info, try this web site or this book by Coolidge.

One very early motivation was apparently the design of mirrors for burning things.

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  • $\begingroup$ Thanks for the book by Coolidge. $\endgroup$ – Akash Kumar Mar 30 '13 at 17:08
  • $\begingroup$ The mentionned website has moved to www.hellenicaworld.com $\endgroup$ – Jean Marie Mar 2 '18 at 23:02
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I'm not a historian. However, my guess is that ellipses, parabolas, and hyperbolas were initially defined as certain loci, and then only later discovered to be cross-sections of a cone. At least, if I were living back in 200 BC (as you say), this is how I would've gone about it.

Let's be slightly more concrete. I hope you can agree that circles are intrinsically important. Now, a circle is really just the set of points at a fixed distance from a given fixed point (the center). So, as ancient geometers, we should use this "locus" idea to find other important shapes.

So, we are led to think about the set of points which are equidistant between a fixed point and a fixed line. This turns out to be a parabola. (The fixed line we chose is called the "directrix.") Or, we could ask about the set of points whose distances from two fixed points add up to a given fixed number: this is an ellipse. And if we replace "add up to" with "subtract to," we get a hyperbola.

To me, these definitions are simple and natural. The truly amazing thing is that every one of these constructions can be described as a cross section of a cone. Wow!

(Again, though, I should emphasize that I'm really just guessing how these shapes were initially constructed. So where the actual history is concerned, I could be totally wrong.)

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  • $\begingroup$ Thanks for your response. Like you said, I am not sure if that is historically correct as it seems somewhat unnatural to my untrained mathematical senses. Assuming it is, I am wondering how earthshaking would it have been to realize that these curves can be obtained by cutting a cone. Really, WOW! $\endgroup$ – Akash Kumar Mar 30 '13 at 17:06
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The initial motivation seems to have been the solution of the Duplication of the Cube which is with the Quadrature of the Circle one of the most famous problems of Antiquity.

See Sir Thomas Heath's modern adaptation of Apollonius' Treatise on Conic Sections with a detailed history of the matters concerning you.

It can be downloaded freely from

http://www.wilbourhall.org/pdfs/Treatise_on_Conic_Sections.pdf

Dr R J-M Grognard

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