Appendix — The Lost Analytic Treasury in Pappus Book VII

[Folio Heath 1896, Intro pp. lxviii, cxxv–clvi (PDF 76, 133–164); Pappus Synagoge VII]

Beyond the Conics, Apollonius of Perga authored six major two-book treatises forming the core of the ancient Domain of Analysis (Τόπος ἀναλυόμενος) preserved for posterity in Book VII of Pappus of Alexandria's Mathematical Collection (Synagoge, ed. Hultsch 1877; geometor/pappus). Five of these treatises were lost in Greek, while one (Cutting Off of a Ratio) survived in Arabic (Halley 1706).

1. On Tangencies (Περὶ ἐπαφῶν) — The Problem of Apollonius (CCC)

Pappus summarizes the famous ten contact problems of Apollonius’s two-book treatise On Tangencies: given any three geometric elements in a plane—each being a Point (`P`), a Straight Line (`L`), or a Circle (`C`)—construct by straightedge and compass a circle tangent to all three (PPP, PPL, PPC, PLL, PLC, PCC, LLL, LLC, LCC, and CCC). While PPP (IV.5) and LLL (IV.4) belong to Euclid's Elements, the crowning tenth problem `CCC` (Three Given Circles) admits up to eight distinct tangent circles:

Lost Tangencies (Pappus VII): The Problem of Apollonius (CCC — All 8 Tangent Circles) C₁ C₂ C₃ Given C₁, C₂, C₃ (bold): 1 externally tangent (green) · 1 enclosing (gold) · 6 mixed-contact pairs (blue/rose)

Plate 11 — The Problem of Apollonius (`CCC`: All 8 Solutions). Given three mutually exterior circles C1C2C3 (bold), exact algebraic solution yields 8 tangent circles: the inner externally-tangent Soddy/Apollonius circle (gold solid), the outer internally-enclosing Apollonius circle (gold dashed, defining ETC X(181)), and three conjugate pairs of mixed-contact circles (teal and rose).

Reconstructed by François Viète in Apollonius Gallus (1600) by concentric radius dilation (reducing CCC to PCC) and by Gergonne via the radical center and pole-polar duality of the four similarity axes, the CCC configuration connects directly to geometor/elements and geometor/etc!

2. Plane Loci (Τόποι ἐπίπεδοι) — The Circle of Apollonius

In Book II of Plane Loci, Apollonius proved one of the most fertile theorems of Euclidean and triangle geometry:

Locus of Apollonius (Plane Loci II). Whereas the locus of a point P whose distances to two fixed points AB are equal () is the perpendicular bisector of AB, the locus of P whose distances bear any given unequal ratio () is a circle (the Circle of Apollonius), whose diameter HmathrminHmathrmex divides AB internally and externally in the ratio ([VI.3]).
Plane Loci II: Locus PA : PB = k A B I₁ I₂ M P PA : PB = AI₁ : I₁B = AI₂ : BI₂ = k (∠I₁PI₂ = 90°) ETC Bridge: Three Apollonius Circles ⇒ X(15), X(16) A B C X(15) X(16) a·PA = b·PB = c·PC at Isodynamic Points X(15), X(16)

Plate 12 — Circle of Apollonius & Coaxal Triangle Pencil (`X(15)`, `X(16)`). Left: Harmonic division generates the Apollonius circle . Right: On any scalene triangle ABC, the three Apollonius circles (, , ) belong to a coaxal pencil orthogonal to the circumcircle and intersect at the First and Second Isodynamic Points X(15) and X(16).

3. The Remaining Four Analytic Treatises in Pappus VII

  1. Cutting Off of a Ratio (Περὶ λόγου ἀποτομῆς, 2 Books — extant in Arabic, ed. Halley 1706): Through a given point draw a straight line cutting off segments on two given lines (measured from given base points) bearing a prescribed ratio—exhaustively classified across 21 loci and diorismoi.

  2. Cutting Off of an Area (Περὶ χωρίου ἀποτομῆς, 2 Books — restored by Halley 1706): Same construction except the two intercepted segments must contain a given rectangle (hyperbolic/quadratic counterpart of Cutting Off of a Ratio).

  3. Determinate Section (Περὶ διωρισμένης τομῆς, 2 Books): Given four collinear points ABCD, find P on the line such that (or )—a complete 1D projective involution theory with exact extrema!

  4. Inclinations (Περὶ νεύσεων, 2 Books — Neusis): Construct by pure straightedge and compass a segment of given length between two given lines/circles verging through a given pole (e.g., inserting a given chord in a circle or rhombus).