PART V. CONSTRUCTION

solving all of our problems

The remaining propositions in Book I are not discovering new concepts or proving new things, but they are rather applying what was established over the last 51 props. Now, instead of using a cone to construct the conic section, the conic section is being used to construct a cone. How novel!

1. Constructing Cones

Using a couple of lines and an angle, we can create a parabola on which to construct a right or oblique cone (I. 52, 53)...

...which can also be done on the hyperbola with two perpendicular lines and some creative use of the circle (I. 54, 55)...

...and on the ellipse, too! (I. 56-58)

I wish I had something more profound to say, but I don't. I just think all of our accumulated knowledge allowing us to reverse engineer the cone is really neat :)

2. Constructing Opposite Sections

The only propositions in this last part of the book not creating cones are props. I. 59 and 60. Instead, Apollonius creates opposite sections. In particular I. 60 is important, as he creates conjugate opposite sections, which will later be known as asymptotes, a figure which will appear throughout Book II.

3. Čau!

And so my summary/guide is complete. If you want to learn more about Apollonius' Conics, I wish you luck. I, for one, am getting outta here.