Notes — Book II: Asymptotes as Tangents at Infinity & Menaechmus’s Curve

1. Menaechmus’s Rectangular Hyperbola Recaptured (II. 12)

In Proposition 34 (`II. 12`), Apollonius proves that along the two asymptotes as oblique coordinate axes, every hyperbola satisfies the constant-rectangle law When the asymptotes are perpendicular (, the equilateral or rectangular hyperbola), this reproduces Menaechmus's cube-duplication curve (Heath 1896, pp. xxi–xxvi), demonstrating that the asymptotic equation is simply the conic referred to its self-conjugate isotropic directions.

2. Asymptotes as Tangents at Infinity

Heath remarks in his footnote to Book III (p. 102) and Introduction (pp. cxxx–cxxxvii) how closely Apollonius approaches the projective doctrine of Desargues and Kepler:

• For a finite tangent at P, and for a parallel secant, .

• As the point of contact P moves indefinitely far along the branch, the bisected tangent LL approaches the asymptote itself—making the asymptotes the two real tangents to the hyperbola at its points on the line at infinity.

3. Constructibility Over Quadratic Extensions

Every problem in II.44–53 (Props. 47–53) uses only straightedge and compass once a conic arc is drawn on the plane. In geometor.model, recovering the principal axes of a rotated conic amounts to intersecting it with a concentric circle (II.47), reducing the eigenvalue problem of the quadratic form to a single quadratic radical over the base field!