Book V — Normals, Minimum & Maximum Lines, Subnormals & the Evolute

[Folio Heath 1896, Props. 84–113 [V. 1–77], pp. 139–196 (PDF 319–376)]

Preserved solely through Thābit ibn Qurra's ninth-century Arabic translation (Halley 1710, Nix 1889), Book V is universally acclaimed by historians of mathematics as the supreme masterpiece of ancient synthetic geometry—anticipating differential curvature, centers of osculation, and the evolute of a conic nearly two millennia before Huygens, Leibniz, and Newton.

1. Normals as Maxima and Minima & the Subnormal Laws [Props. 84–94 / V. 1–34]

[Folio Heath 1896, pp. 139–159 (PDF 319–339)]

Extending Euclid's theorems on the shortest and longest segments from an eccentric point to a circle ([III.7], [III.8]), Apollonius investigates the minimum and maximum straight lines GP drawn to a conic from a point G on its principal axis:

  • Constant Subnormal of the Parabola [Props. 87, 90 / V. 8, 13, 27]: If G lie on the axis of a parabola with , the minimum straight line from G to the curve is GP whose ordinate foot N cuts off a constant subnormal:

    \[NG = \tfrac{1}{2}\, p_a,\]

    and GP is strictly perpendicular to the tangent at P (hence a normal).

  • Proportional Subnormal of Central Conics [Props. 88–91 / V. 9–15, 28–29]: For an ellipse or hyperbola with center C, transverse axis AA, and principal parameter , a segment GP from an axis point G is a minimum or maximum line (and perpendicular to the tangent at P) if and only if its subnormal NG satisfies:

    \[NG : CN = p_a : AA' \quad \bigl(= b^2 : a^2\bigr).\]

2. Determination of the Evolute Diorismos [Props. 99–100 / V. 51–52]

[Folio Heath 1896, pp. 168–179 (PDF 348–359)]

Apollonius next poses the profound diorismos problem for an arbitrary point O off the axis: how many normals can be drawn through O that cut the axis between O and the vertex?

V.51–62: Envelope of Normals (Astroidal Evolute) & Four Normals from Interior Point O N₁ N₂ N₃ N₄ O Evolute Envelope Inside Evolute: 4 Normals (ON₁, ON₂, ON₃, ON₄) · Outside Evolute: 2 Normals

Plate 9 — Normals & the Astroidal Evolute Envelope (Book V.51–63 [Props. 99–106]). Inside the four-cusped evolute , an interior point O admits four distinct normals OP1OP2OP3OP4 to the ellipse; on the evolute two normals coalesce at a center of curvature; outside the evolute only two real normals exist.

Proposition 99 [V. 51, 52] — Exact Evolute Threshold. If AM measured along the axis from vertex A be greater than (and in the ellipse less than AC), and if MO be drawn perpendicular to the axis, then a critical length can be assigned by pure cubic/mean-proportional division along the axis such that:

  • (a) if , no normal can be drawn through O cutting the axis in that quadrant (only 2 normals exist globally on an ellipse);

  • (b) if , one unique coalesced normal can be so drawn through O (hence O lies directly on the evolute!);

  • (c) if , two distinct normals can be so drawn in that quadrant (yielding 4 real normals globally on an ellipse, or 3 on the evolute!).

3. Constructing Normals via Apollonius’s Rectangular Hyperbola [Props. 105–106 / V. 58–63]

[Folio Heath 1896, pp. 180–186 (PDF 360–366)]

Because finding the feet Pi of the four normals from Ox0y0 is a solid (degree-4 / cubic-resolvent) problem, Apollonius constructs the normal feet synthetically (V.62–63 [Prop. 106]) by intersecting the given conic with an auxiliary rectangular hyperbola (the Hyperbola of Apollonius) whose asymptotes are parallel to the conic’s axes!