 The greatest geometer of ancient times was Archimedes. Probably. In the fourth century, Pappas of Alexandria put together a recommended reading list of advanced mathematical works, the domain of analysis. We have the authors and titles of these works. Unfortunately, most of them have been lost to history. And if you look over these works, there are two things that are immediately apparent. Notably absent are any works of Archimedes. And most of these works are by Apollonius, Archimedes' contemporary, who lived around 250 BC. And in fact, the most advanced work on the list that has not been lost is Apollonius' conics. So if we view the domain of analysis as the required reading for an advanced study of mathematics, Apollonius' conics, one of the few works that still survive, would correspond to a text you'd read in your second or third year of graduate school. Most of the propositions of the conics require a full paragraph to state and several pages to prove. We won't go into much detail, but focus on some important ideas. To understand the role of Apollonius' conics, we begin with Euclid. Euclid defined a cone as a solid of revolution, formed when a right triangle is revolved around one of its legs. The type of cone depended on the type of triangle. If the legs had equal length, the cone was right-angled. If the base was longer, the cone was obtuse-angled. But if the base was shorter, the cone was acute-angled. Now we know the Euclidean definition of the cone because it appears in the elements. And we also know that Monochmas at Euclid both wrote about the conic sections, but we don't know what they wrote because Apollonius' work replaced them. We do know they defined the sections by intersecting the cone with the plane perpendicular to a side of the generating triangle. And depending on the type of cone, this gave us three sections. The section of a right-angled cone, the section of an obtuse-angled cone, and the section of an acute-angled cone. And Apollonius took a different view. Given a circle on a plane and a point not on the plane, the lines through the given point and the points on the circle, if extended, form a conic surface. And what's worth noting here is that we produce a double cone. Apollonius went further and allowed any plane to cut the cone at any angle. Now in many cases, the cutting plane only cuts through one part of the cone. And we produce a conic section that is not dissimilar from those described by Monochmas. But it is possible for the cutting plane to pass through both sides. And if that happens, we get a curve that is totally unprecedented in the history of mathematics, a curve with two completely disconnected parts. So let's introduce a few definitions. Given a curve at a line, a diameter of the curve is a line where the diameter bisect lines drawn parallel to the given line. So if I were to bisect this line, and if I were to take another line parallel to it and bisect it, the diameter would bisect both lines. The bisected lines are said to be drawn ordinate-wise. The diameter meets the curve at a vertex. And we have a special case. If the lines drawn ordinate-wise meet the diameter at a right angle, the diameter is the axis. And it's worth noting that these are slightly different from our own definition of diameter, vertex, and axis. These definitions allow Apollonias to identify certain properties of the conic sections. These are known as the symptoms. They are relationships between lines drawn ordinate-wise and other fixed lines that hold for all points on a conic section. We'll give somewhat simplified versions of Apollonias as a result using the following notation. The length of a line drawn ordinate-wise will be x, the length of the line drawn from the vertex to the intersection of the diameter, and the line drawn ordinate-wise will be y, and our fixed lengths will be p, q, and so on. So let's start with a parabola. Suppose we have a parabola with a diameter and a vertex. And remember Apollonias' definitions are different from our own. So our parabola with diameter and vertex might look like this. We have x, the length of the line drawn ordinate-wise. Then y, the length of the line from the vertex to where the diameter meets the line drawn ordinate-wise. Then part of conics, book 1, Proposition 11, is the symptom of a parabola. In any parabola there is a length p, so p, y equals x squared. It's possible to understand Apollonias' results on the symptom of a parabola without a thorough understanding of Euclid's elements. It's far more difficult to understand Apollonias' results on the symptoms of hyperbola and ethylips, so we'll present them in a very modernized form. As before, x is the length of the line drawn ordinate-wise, not necessarily horizontal. y is the length of the line drawn along the diameter, not necessarily perpendicular. And p is the length of some other line. And if we do that, we'll give the Apollonian result, conics, book 1, Proposition 12, as for any hyperbola, there's a length p, where x squared is y times p plus py. Similarly, Apollonias gives the symptom of an ellipse in conics, book 1, Proposition 13, again, letting x be our line drawn ordinate-wise, y being our line along the vertex, and p being some other line, and we'll state that as for any ellipse, there is a length p, where x squared equals y times p minus py. So Apollonias is responsible for the modern names of the conic sections. The terms come from Euclidean geometry. And these come from the problems known as the application of areas, where the goal is to produce a figure with a specific area with a specific property. In parable, a figure is produced that equals another. And so we might have x squared equals py. A square is equal to a rectangle with a specified side. In hyperbola, a figure is produced that exceeds another by a figure similar to a given figure. So if we look at the Apollonian symptom of hyperbola, x squared equals y times p plus py, we have that's also py plus py squared. And so we have a figure py, and we are exceeding it by py squared. And in ellipsis, a figure is produced that falls short by a figure similar to a given figure. So from the Apollonian symptom, we see that our ellipse has a figure py minus py squared, another figure similar to a given figure.