Notes — Book I: Geometric Algebra, Order of Curves & Projective Frames

1. Why Apollonius Orders the Curves Parabola → Hyperbola → Ellipse

Modern textbooks routinely list ellipse, parabola, hyperbola by eccentricity (, , ). In Book I.11–13, however, Apollonius proves Parabola (`I.11`), then Hyperbola (`I.12`), then Ellipse (`I.13`):

• In I.11, the cutting plane is parallel to a side of the axial triangle, giving the simplest one-parameter application .

• In I.12 and I.13, the cutting plane is non-parallel to both sides; when the intersection P with the second generator falls above the apex A on the opposite nape (I.12, hyperbola), the auxiliary parallel AF falls outside the axial triangle, whereas for I.13 (ellipse) it falls inside, and both proofs pave the immediate transition to I.14 (opposite branches).

2. Connection to Euclid VI.27–29 & Rational Trigonometry

Apollonius’s naming triad translates Euclid's Pythagorean application of areas ([I.44], [VI.27], [VI.28], [VI.29] in geometor/euclid) directly into curve theory:

• Because defines the conjugate diameter for transverse diameter (so ), shifting origin from the vertex P () to the center C converts Apollonius’s two-vertex proportion immediately into the central quadratic form

  \[\frac{X^2}{a^2} \pm \frac{Y^2}{b^2} = 1,\]

  purely in quadrances (Wildberger 2005) without a single trigonometric function even when are measured along oblique conjugate diameters!

3. Tangent Harmonicity as 1D Projective Polarity

Propositions 12–14 (I.33–37) reveal why the parabola’s is the exact projective limit of : as the second vertex P recedes to infinity along the diameter, the ratio , leaving P as the midpoint of TV—that is, .