Skip to 0 minutes and 12 seconds So our story really beings more than 2,000 years ago with the ancient Greeks, who were very interested in geometry. And in Euclid’s geometry, the line and the circle are considered the most perfect figures. Apollonius, who was a successor of Euclid and considered himself a rival to Archimedes, was the greatest geometer of the ancient world. And he took this idea further and said, well, why don’t we look in three dimensions at what we get when we combine a line with a circle? We can form a cone. So here’s an example of a cone. There’s the apex there. But the ancient Greeks actually considered the cone as actually extending also below, so there was a top part and also a bottom part.

Skip to 0 minutes and 59 seconds So here we have one of these cones that Apollonius studied. So this is a three-dimensional figure obtained by starting with a point and a circle, which is above that point, and connecting each point of the circle with the given point here with a straight line. This is supposed to be an infinite straight line, so this cone actually extends up and below as well. So it has these two parts of it– the top part and the bottom part. And Apollonius realised it was interesting to think about what happens when you slice this cone with a plane. That’s called a conic section.

Skip to 1 minute and 37 seconds So for example, if we slice it with a plane, perhaps, in a direction like that, then we would get a curve of intersection that looks like this. That’s an ellipse.

Skip to 2 minutes and 0 seconds On the other hand, if we take a plane and slice the cone in such a way that we’re touching both the top and the bottom parts of the cone, then we get something that’s quite different. For example, if we take a plane like this and slice the cone, then we get a curve that has a branch in this direction and also another branch in this direction. And that’s a hyperbola.

Skip to 2 minutes and 34 seconds And then there’s a third case, which is intermediate between these two. When we take a plane which is just touching one of the branches but almost touching the other one but not quite, the way we do that is we take a plane– say this one here– which is parallel to one of the sides or one of the lines in the cone. So if we take this line in the cone and take a plane which is parallel to that, then it will slice this bottom part but not touch the top part. And that will give us a curve that looks something like that, and that’s a parabola.

Skip to 3 minutes and 19 seconds These are the names given to these curves by Apollonius of Perga. He proved many things about these curves. He knew hundreds of interesting facts about them. And it was really only 2,000 years later the Europeans were able to go beyond Apollonius’s understanding. And that happened with the discoveries of Descartes and his introduction of the Cartesian coordinates that we talked about last week. So in Descartes’s framework, it turned out that you could also describe ellipses or hyperbolas or parabolas in a very precise algebraic way. So these conic sections as they’re called, because they’re sections of a cone, to Descartes were second degree curves.

Skip to 4 minutes and 21 seconds That means they were curves that had an equation that maybe was like this– x squared minus 2y squared plus 3x minus 5y equals 7. That’s an equation of degree 2 in x and y. And it turns out that that’s exactly one of these conic sections. So this remarkable correspondence between this classical subject from ancient Greece and this purely algebraic formulation of things discovered by Descartes. And it was a very, very powerful tool, and that’s really going to be a central tool for us in understanding the geometry of the parabola. So it will turn out that quadratic relations that come up in practise are often mostly conveniently described by this parabola.

Skip to 5 minutes and 13 seconds So we have to turn to understanding this parabola next, which is poised very delicately between the cases of the ellipse and the hyperbola. That’s actually very interesting both geometrically and practically.

# Conic sections

Conic sections go back to the ancient Greek geometer Apollonius of Perga around 200 B.C. He viewed these curves as slices of a cone and discovered many important properties of ellipses, parabolas and hyperbolas.

Apollonius considered the cone to be a two-sided one, and this is quite important. It means that if we slice the cone with a plane, then we can get not just a *circle* or *ellipse*, but also more subtle curves with two separate branches called *hyperbolas*, and balanced between these two, a very special kind of curve called a *parabola*.

The parabola will turn out to be particularly important for us, as this is exactly the shapes assumed by the graphs of quadratic functions of the form

\[\Large{y=ax^2+bx+c}.\]So in this video we span several thousand years, from the ancient world’s greatest geometer, to one of the most innovative and influential geometers of the modern era. If you are interested in the historical aspects of the subject (and all students of mathematics really ought to be!) then check out Norman’s YouTube lecture on Greek geometry.

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