PART I. THE CONE

just getting started

1. Definitions

No good mathematical treatise can begin without establishing its own language and framework, and Conics is no exception! Apollonius lays out, in eight definitions...

• The parts of a conic surface and a cone (axis, vertex, base) and how to construct both figures (a conic surface is generated through; a cone is the figure within the surface and the circumference). (Def. I. 1, 2)
• The types of cones (right and oblique). (Def. I. 3)
• What the diameter of the cone is (a line in the cone, drawn from a curved line, which bisects all parallel straight lines within the curved line) and what ordinate lines are (the aforementioned parallel lines). (Def. I. 4)
• What the transverse diameter is (a line between two curved lines in the same plane with the same properties as the diameter mentioned above) and the upright diameter (). (Def. I. 5)
• The parts of a conic surface and a cone (axis, vertex, base) and how to construct both figures (a conic surface is generated through; a cone is the figure within the surface and the circumference). (Def. I. 6-8)

It is worth noting, however, that the definitions do not cover all of the important terms or figures that are needed to understand conics. Some of these aforesaid figures will instead be described through either propositions or another, later set of definitions.

2. Lines Within and Outside the Cone

Using the definitions set out by Apollonius, our first conic surface (of many!) is constructed. The first two propositions place points and lines on the cone, which, as we discover lines have to sit flat on the conic surface (I. 1), and a line drawn inside the conic surface have to be contained within the conic surface, unless they’re made to go outside (I. 2). Pretty straightforward stuff... for now. >:)

3. Creating Cutting Planes

Propositions I. 3-9 focus on the use of cutting planes and how, with them, new figures and properties can be discovered. For example, by cutting the cone vertically, one can create a triangle (I. 3), and, cut horizontally at the correct angle, one can create a cone within a cone! (I. 4 (pictured to the right), 5). Further propositions detail other concepts, such as the properties of lines within cutting planes (I. 6) cutting planes intersecting one another (I. 7).

And so we venture forth into the next propositions...