PART II. CONIC SECTIONS

finally: the meat and bones of conics

1. Entirely New Shapes?!

In this next cluster of propositions, Apollonius reveals that, by using cutting planes at specific angles and locations within the cone, one can not only create triangles and circles, but one can create sections with figures entirely unique to the cone. These shapes are the parabola (I.11), the hyperbola (I.12), and the ellipse (I.13). Additionally, he also constructs a figure on "vertically opposite surfaces," (I.14) which are simply two hyperbolae on two opposite cones. These propsitions also use the transverse diameter and upright side for the first time (as defined in Part I and seen on the diagrams below).

Make a point to remember these shapes, as well as the upright and transverse sides. We will be analyzing them for the remainder of the book.

Pictured Above: The parabola, hyperbola, and the ellipse, all created within the cone. On each figure, the upright is the line sticking out from the top of the section, and the transverse is the line down the middle of each.

Not Pictured: Opposite sections. :(

2. ...And Some Other Propositions Which Are Important, Too

These propositions are no doubt important, but they are in a bit of an odd spot relative to the rest of the "parts" which I have divided the book into, for they neither reveal new sections, like the prior four propositions, nor do they deal with the properties of lines, like the following ones. Instead, I.15 and I.16 seem to explain concepts relating to the new sections. They also directly tie into the next set of definitions, further down this page.

3. Second Definitions

As mentioned earlier on, there would be additional definitions. Here, Apollonius states:

• The center of an ellipses and a hyperbola can be found (on the midpoint of the diameter), as well as that of an opposite section (the midpoint of its transverse diamater). Additionally, any straight line drawn from the center of one of these figures, which meets the section, is a radius. These terms do not apply to the parabola. (Def. I. 9-10)
• A straight line drawn from the center, parallel to an ordinate, is the "mean proportional" to the sides, as well as the second diameter. (Def. I. 11)

While these concepts will not be as widely-used as, say, the ordinate line, it is important to keep in mind the properties of the second diameter, especially as Book I moves to focus on ratios.