Book III — Chord Rectangles, Poles & Polars, Foci & the 4-Line Locus¶
In the General Preface (Unit 0), Apollonius singles out Book III with special pride: it contains “many remarkable theorems useful for the syntheses of solid loci and for diorismoi, the majority and prettiest of which are new”, culminating in the long-sought synthesis of the Three-Line and Four-Line Locus.
1. Equality of Tangent Triangles & Intersecting Chord Rectangles [Props. 54–61 / III. 1–29]¶
Equal Tangent-Diameter Triangles [Prop. 54 / III. 1, 4, 11]: If tangents at P and Q meet at O, and diameters through P, Q meet the opposite tangents in E, T, the triangles OPT and OQE have equal area—the foundational lemma behind every chord-product theorem.
Generalization of Euclid III.35–36 to All Conics [Props. 57–61 / III. 16–29]: If two chords PQ, pq parallel to two fixed directions intersect at any interior or exterior point O, the ratio of the rectangles under their segments is invariant with respect to the position of O, being equal to the ratio of the squares on the parallel tangents TP, Tp (or parallel diameters):
\[PO \cdot OQ : pO \cdot Oq = TP^2 : Tp^2.\]When the conic is a circle, by symmetry, immediately recovering [III.35] and [III.36]!
2. Harmonic Properties of Poles & Polars [Props. 62–66 / III. 30–40]¶
Starting with transversals parallel to an asymptote (Prop. 62 [III. 30–34]) and chords parallel to an asymptote (Prop. 63 [III. 35–36]), Apollonius ascends in Propositions 64–66 (`III. 37–40`) to the universal theorem of projective polarity:
Plate 5 — Harmonic Pole & Polar Duality (Book III.37–40 [Props. 64–66]). From exterior pole T, tangents touch the conic at QQ defining the chord of contact (polar) QQ; every secant through T meets the conic in RR and the polar in M in harmonic cross-ratio .
Proposition 64 [III. 37–40] — Universal Harmonic Polar Theorem. If TQ, Tq be two tangents to any conic section (parabola, ellipse, single-branch or opposite-branch hyperbola) and any straight line drawn through T meet the conic in R, R and the chord of contact Qq in M (or conversely if M lie inside the conic on Qq and a secant through M meet the curve in RR and the parallel through T in N), the transversal is divided harmonically:
3. Focal Properties of Central Conics [Props. 69–73 / III. 45–52]¶
Apollonius introduces the two foci S, S on the principal axis AA without using a word for “burning point” (coined by Kepler in 1604), naming them strictly from his area-application geometry τὰ ἐκ τῆς παραβολῆς γινόμενα σημεῖα ("the points arising out of the application"): apply along the axis AA a rectangle (falling short by a square on the ellipse so , exceeding by a square on the hyperbola so ).
Plate 6 — Focal Properties of Central Conics (Book III.45–52 [Props. 69–73]). For foci SS determined by , the focal radii SPSP make equal angles with the tangent at P (III.48), the perpendicular feet YY lie on the auxiliary circle of diameter AA (III.49), and (III.52).
Optical Reflection Law [Prop. 71 / III. 48]: The focal distances SP, SP of any point P on an ellipse or hyperbola make equal angles with the tangent at P.
Auxiliary Pedal Circle [Prop. 72 / III. 49–50]: If from either focus S a perpendicular SY be dropped upon the tangent at P, the locus of Y is the circle on the principal axis AA as diameter ().
Focal Sum & Difference [Prop. 73 / III. 51–52]: In a hyperbola , and in an ellipse .
4. The Locus with Respect to Three and Four Lines [Props. 74–76 / III. 53–56]¶
Plate 7 — The Three-Line and Four-Line Locus (Book III.53–56 [Props. 74–76]). For a point P on a conic inscribed with quadrilateral ABCD, the product of oblique distances to opposite side-pairs maintains a constant ratio (degenerating when two sides coalesce into a tangent and chord of contact to the Three-Line Locus ).
Propositions 74–76 (III.53–56) prove that if from two fixed points QQ on any conic rays QRQR from a variable curve point R intercept segments on the tangents at QQ (or parallel lines), the product of those intercepts is constant—from which (Zeuthen 1886, Heath 1896, pp. cxxxviii–cl) the Three-Line and Four-Line Locus follows immediately by compound ratios!