Chapter I: Geometric Constructions — Commentary

Constructive Analysis, Historical Provenance & GEOMETOR Notes (0020–0052, pp. 1–33)

Constructive Architecture & Symbolic Modeling (§§ 1–22)

Chapter I lays down the operational grammar of the entire book. Every ruler-and-compass construction is governed by Pappus' four-stage analytic method (Collection, Book VII):

  1. Analysis — Assume the required configuration already exists and trace algebraic/synthetic incidences back to known data.

  2. Construction — Execute elementary straightedge (line(P, Q)) and compass (circle(A, r)) steps in forward topological order.

  3. Proof — Verify deductively that the constructed object satisfies every hypothesis.

  4. Discussion (Diorismos) — Determine exact existence inequalities (s < 2 OE, s^2 > 2a^2), count multiplicity of solutions (0, 1, 2, 3, 4, 6, or \(\infty\)), and identify poristic degeneracies (as in § 9 when \(AC \perp BD\) and \(AC = BD\)).

Mapping Chapter I to geometor.model & geometor.elements

  • Symbolic DAG Representation: Altshiller-Court’s § 1 notation—circles \((A, r)\) and line intersections \(M = (PQ, RS)\)—maps one-to-one onto geometor.model primitives, where every constructed point retains its exact SymPy algebraic coordinates in a quadratic extension tower of \(\mathbb{Q}\).

  • The Twelve Fundamental Loci (§ 11, Folios `0030`–`0036`): - Linear Loci: Parallel pair (Locus 3), mediator (Locus 4, Euclid I.10–11), angle bisectors (Locus 5), and constant difference of squares \(AM^2 - BM^2 = d^2\) (Locus 12 — the point-circle precursor of the radical axis in Chapter VIII, § 422). - Circular Loci: Circle \((A, r)\) (Locus 1), concentric tangent-length & subtended-angle circles (Loci 2, 6, 9), capable arc & Thales’ circle (Locus 7, Euclid III.21, III.31), chord-midpoint circle on \(OP\) (Locus 8), Apollonius’ median circle \(OM^2 = \tfrac{1}{2}s^2 - a^2\) (Locus 10), and the Apollonian circle \(AM : MB = p : q\) (Locus 11, Apollonius).

  • Transformational & Unfolding Previews: § 8 previews parallel translation (Petersen), § 9 previews \(90^\circ\) rotational congruence (Castillon), and § 15 introduces indirect elements by unfolding perimeter \(2p\) into auxiliary triangle \(AEF\) with vertex angle \(90^\circ + \tfrac{1}{2}A\).

Folio-by-Folio Diplomatic & Constructive Notes

Folio 0020 (Page 1) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0019 (Page [xx], Blank Verso)

  • Current Folio (0020.rst): Page 1, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0021 (Page 2, § 2 concluded, Exercises 1–24, and § 3)

Structural & Diplomatic Observations

  • Overview: Opening recto of Chapter 1 (printed page 1; tall chapter numeral 1 above GEOMETRIC CONSTRUCTIONS, section A. PRELIMINARIES), defining Altshiller-Court’s global symbolic conventions (§ 1) and beginning the catalog of Euclidean elementary constructions (§ 2).

  • Document Headings on Folio: 1 GEOMETRIC CONSTRUCTIONS, A. PRELIMINARIES, 1. Notation., 2. Basic Constructions.

  • Line-Break Hyphenations Resolved: circum-center, ex-ternally

Mathematical & Constructive Commentary

Note two signature conventions of Altshiller-Court (§ 1): (1) \(2p\) denotes the perimeter so that \(p\) is the semiperimeter (where many British texts write \(2s\) and \(s\)); (2) \((A, r)\) and \(M = (PQ, RS)\) correspond directly to model.construct_circle_by_center_and_point and line-line intersection nodes in geometor.model. The three § 2 items on this page invoke Euclid VI.9, VI.10, and VI.12.

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Folio 0021 (Page 2) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0020 (Page 1, §§ 1–2)

  • Current Folio (0021.rst): Page 2, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0022 (Page 3, Section B, §§ 4–5)

Structural & Diplomatic Observations

  • Overview: Page 2 (running header 2 / GEOMETRIC CONSTRUCTIONS / [Ch. I, §§ 2, 3), completing § 2 (Basic Constructions), presenting the 24 Preliminary EXERCISES, and adding § 3 (Suggestion).

  • Document Headings on Folio: EXERCISES, 3. Suggestion.

  • Line-Break Hyphenations Resolved: tri-angle, hypote-nuse, con-ventional

Mathematical & Constructive Commentary

Raw OCR omitted Exercise 7 (\(A, h_a, t_a\)) and garbled Exercise 5 (\(m_a\)), Exercise 14 (\(\angle ABD\)), Exercise 21 (\(AB^2 : AC^2 = BD : DC\)), and Exercise 23 (\(x^2 : a^2 = p : q\)); all verified against 0021.png. The § 2 constructions correspond to Euclid VI.13, II.14, I.47, and III.17.

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Folio 0022 (Page 3) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0021 (Page 2, §§ 2–3)

  • Current Folio (0022.rst): Page 3, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0023 (Page 4, Fig. 1, solution of § 5, and statement of § 6)

Structural & Diplomatic Observations

  • Overview: Page 3 (running header Ch. I, §§ 4, 5] / GENERAL METHOD OF SOLUTION / 3), opening Section B with § 4 (Analytic Method — defining Analysis, Construction, Proof, Discussion) and stating § 5 (Problem).

  • Document Headings on Folio: B. GENERAL METHOD OF SOLUTION OF CONSTRUCTION PROBLEMS, 4. Analytic Method., 5. Problem.

  • Line-Break Hyphenations Resolved: applica-tions, follow-ing, approxi-mately, con-struction, re-quirements, possi-bility

Mathematical & Constructive Commentary

Altshiller-Court’s four-part schema in § 4 restates for modern collegiate instruction the classical Greek method of analysis and synthesis preserved by Pappus of Alexandria in Collection, Book VII (with Discussion corresponding to Greek diorismos, and the opening example invoking Euclid I.1).

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Folio 0023 (Page 4) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0022 (Page 3, §§ 4–5)

  • Current Folio (0023.rst): Page 4, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0024 (Page 5, Fig. 2 and solution of § 6)

Structural & Diplomatic Observations

  • Overview: Page 4 (running header 4 / GEOMETRIC CONSTRUCTIONS / [Ch. I, §§ 5, 6), presenting FIG. 1 (0023.g1.png), the four-part solution of § 5, and the italicized enunciation of § 6 (Problem).

  • Document Headings on Folio: 6. Problem.

  • Line-Break Hyphenations Resolved: None

Calibrated Diagrams on Folio

  • 0023.g1.png (bbox [383, 162, 664, 611], white-point 238): Fig. 1. Secant CA’B’ through C meeting parallel lines x, y so that AA’ : BB’ = p : q (§ 5).

Mathematical & Constructive Commentary

In § 5, \(O\) and \(O'\) are harmonic conjugates with respect to \(A\) and \(B\) (anticipating Chapter II, § 59). When \(p = q\), the internal division point \(O'\) is the midpoint of \(AB\), while the external division point \(O\) recedes to infinity along \(AB\), so the second solution becomes the line through \(C\) parallel to \(AB\)!

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Folio 0024 (Page 5) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0023 (Page 4, §§ 5–6)

  • Current Folio (0024.rst): Page 5, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0025 (Page 6, conclusion of § 6 Discussion, Fig. 3, and § 7)

Structural & Diplomatic Observations

  • Overview: Page 5 (running header Ch. I, § 6] / GENERAL METHOD OF SOLUTION / 5), presenting the Analysis, Construction, Proof, FIG. 2 (0024.g1.png), and Discussion of § 6.

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: None

Calibrated Diagrams on Folio

  • 0024.g1.png (bbox [407, 1018, 608, 488], white-point 238): Fig. 2. Secant MBA from external point M cutting circle (O) so that central angle AOB equals angle M (§ 6).

Mathematical & Constructive Commentary

Remarkable economy of construction in geometor.model: given circle \((O, R)\) and exterior point \(M\), a single compass operation circle(M, O) immediately constructs both solution secants \(MA\) and \(MA'\) in two steps!

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Folio 0025 (Page 6) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0024 (Page 5, § 6)

  • Current Folio (0025.rst): Page 6, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0026 (Page 7, § 8, Figs. 4–5)

Structural & Diplomatic Observations

  • Overview: Page 6 (running header 6 / GEOMETRIC CONSTRUCTIONS / [Ch. I, §§ 6, 7), concluding the Discussion of § 6 and presenting FIG. 3 (0025.g1.png) and the complete four-part solution of § 7.

  • Document Headings on Folio: 7. Problem.

  • Line-Break Hyphenations Resolved: tri-angle

Calibrated Diagrams on Folio

  • 0025.g1.png (bbox [380, 530, 663, 513], white-point 238): Fig. 3. Points D, E on sides AB, AC of triangle ABC such that AD = DE = EC (§ 7).

Mathematical & Constructive Commentary

Problem 7 is a classic gem of synthetic analysis (also solvable by homothecy in Chapter II): by projecting the unknown broken equal chain \(AD = DE = EC\) from \(A\) onto \(B\), the scale ambiguity vanishes and \(K, L\) are constructed by two compass transfers of \(BA\). When \(\angle A = 90^\circ\), \(K\) coincides with \(A\).

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Folio 0026 (Page 7) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0025 (Page 6, §§ 6–7)

  • Current Folio (0026.rst): Page 7, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0027 (Page 8, § 9, Fig. 6)

Structural & Diplomatic Observations

  • Overview: Page 7 (running header Ch. I, § 8] / GENERAL METHOD OF SOLUTION / 7), presenting the complete solution of § 8 together with side-by-side diagrams FIG. 4 (0026.g1.png) and FIG. 5 (0026.g2.png).

  • Document Headings on Folio: 8. Problem.

  • Line-Break Hyphenations Resolved: direc-tion, con-structed

Calibrated Diagrams on Folio

  • 0026.g1.png (bbox [95, 1248, 585, 424], white-point 238): Fig. 4. Points P, Q on circles (A), (B) at given distance m and given direction — Analysis (§ 8).

  • 0026.g2.png (bbox [720, 1225, 606, 439], white-point 238): Fig. 5. Four solutions for points on circles (A), (B) at given distance and direction (§ 8).

Mathematical & Constructive Commentary

Problem 8 exemplifies Julius Petersen's Method of Parallel Translation (Methods and Theories): translating circle \((A)\) by \(\pm\vec{m}\) yields two circles centered at \(R, R'\) whose intersections with \((B)\) give up to four solutions \(P_1Q_1, P_2Q_2, P_1'Q_1', P_2'Q_2'\) (Fig. 5).

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Folio 0027 (Page 8) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0026 (Page 7, § 8)

  • Current Folio (0027.rst): Page 8, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0028 (Page 9, Fig. 7, Proof & Discussion of § 9, and § 10)

Structural & Diplomatic Observations

  • Overview: Page 8 (running header 8 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 9), stating § 9 (circumscribing a square through four given points), giving two distinct proofs in the Analysis (one by \(90^\circ\) rotation, one by congruent right triangles), FIG. 6 (0027.g1.png), and the Construction.

  • Document Headings on Folio: 9. Problem.

  • Line-Break Hyphenations Resolved: as-sumption, paral-lel

Calibrated Diagrams on Folio

  • 0027.g1.png (bbox [313, 668, 801, 722], white-point 238): Fig. 6. Construction of a square PQRS whose sides pass through four given points A, B, C, D (§ 9).

Mathematical & Constructive Commentary

Notice how Altshiller-Court pairs a transformational argument (\(90^\circ\) rotation around the center of the square) with a pure Euclidean congruence argument (\(\triangle PQL \cong \triangle QRK\)). Compare Castillon's and Catalan's classical incidence problems.

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Folio 0028 (Page 9) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0027 (Page 8, § 9)

  • Current Folio (0028.rst): Page 9, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0029 (Page 10, conclusion of § 10 and Exercises 1–13)

Structural & Diplomatic Observations

  • Overview: Page 9 (running header Ch. I, §§ 9, 10] / GENERAL METHOD OF SOLUTION / 9), displaying FIG. 7 (0028.g1.png), the Proof and full combinatoric/poristic Discussion of § 9, and the opening paragraph of § 10 (Suggestions).

  • Document Headings on Folio: 10. Suggestions.

  • Line-Break Hyphenations Resolved: tri-angles, equi-angular, sym-metric, ac-cordingly

Calibrated Diagrams on Folio

  • 0028.g1.png (bbox [327, 160, 774, 596], white-point 238): Fig. 7. General configuration of six squares whose sides pass through four given points A, B, C, D (§ 9).

Mathematical & Constructive Commentary

The poristic case in the Discussion (\(F = B\), when \(AC \perp BD\) and \(AC = BD\), an orthodiagonal equidiagonal quadruple) yields a 1-parameter infinity of circumscribed squares—a striking preview of Chapter IV (Other Quadrilaterals, §§ 271–282).

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Folio 0029 (Page 10) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0028 (Page 9, §§ 9–10)

  • Current Folio (0029.rst): Page 10, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0030 (Page 11, Exercise 14 and Section C: Geometric Loci, § 11)

Structural & Diplomatic Observations

  • Overview: Page 10 (running header 10 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 10), completing § 10 (Suggestions on exact ruler-and-compass execution and exhaustive Discussion branching) and listing Exercises 1–13.

  • Document Headings on Folio: EXERCISES

  • Line-Break Hyphenations Resolved: construc-tion, opera-tions, inter-cept, com-mon

Mathematical & Constructive Commentary

Notice how Exercises 6 and 7 quietly anticipate Chapter VIII (Circles): Exercise 6 constructs a circle bisecting a given circle (\(R_1^2 = d^2 + R^2\)), while Exercise 7 constructs a circle whose radical axis with a given circle has a prescribed direction.

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Folio 0030 (Page 11) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0029 (Page 10, § 10, Exercises 1–13)

  • Current Folio (0030.rst): Page 11, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0031 (Page 12, Fig. 8 and Loci 1–7)

Structural & Diplomatic Observations

  • Overview: Page 11 (running header Ch. I, §§ 10, 11] / GEOMETRIC LOCI / 11), concluding Section B with Exercise 14 and opening Section C (Geometric Loci) with the theoretical introduction to § 11 (Important Loci).

  • Document Headings on Folio: C. GEOMETRIC LOCI, 11. Important Loci.

  • Line-Break Hyphenations Resolved: geo-metric, dis-carded, equidis-tant, bi-sectors

Mathematical & Constructive Commentary

Altshiller-Court’s exposition of the Method of Loci (relaxing one constraint at a time to intersect two 1-dimensional straightedge-and-compass loci) directly follows Plato's school, Simon L'Huilier (1809), and Julius Petersen (1866/1879). In Exercise 14, the hypotenuse of each right subtriangle is a side (\(b = 2R_1, c = 2R_2\))!

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Folio 0031 (Page 12) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0030 (Page 11, § 11 introduction)

  • Current Folio (0031.rst): Page 12, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0032 (Page 13, proof of Locus 7, Fig. 9, and Locus 8)

Structural & Diplomatic Observations

  • Overview: Page 12 (running header 12 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 11), displaying FIG. 8 (0031.g1.png) at top and enunciating LOCUS 1 through LOCUS 7 (introducing the term mediator in LOCUS 4). Ends mid-sentence across the page turn.

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: dis-tance

Calibrated Diagrams on Folio

  • 0031.g1.png (bbox [433, 162, 553, 673], white-point 238): Fig. 8. Arc of circle AMB as the locus of points on one side of AB subtending a given angle, with tangent AT (§ 11, Locus 7).

Mathematical & Constructive Commentary

The term mediator (French médiatrice, coined by Joseph Neuberg) introduced here in Locus 4 is used systematically throughout College Geometry (e.g., § 15, § 68).

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Folio 0032 (Page 13) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0031 (Page 12, Loci 1–7: ‘…Pass a circle through the three points’)

  • Current Folio (0032.rst): Page 13, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0033 (Page 14, exterior case of Locus 8, Locus 9, Fig. 10, and Locus 10)

Structural & Diplomatic Observations

  • Overview: Page 13 (running header Ch. I, § 11] / GEOMETRIC LOCI / 13), completing the double-inclusion proof and tangent construction of LOCUS 7 (including Thales’ right-angle corollary), and presenting LOCUS 8 with FIG. 9 (0032.g1.png).

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: semi-circles

Calibrated Diagrams on Folio

  • 0032.g1.png (bbox [451, 1291, 518, 514], white-point 238): Fig. 9. Circular locus of the midpoints M of chords AB of circle (O) passing through a fixed point P (§ 11, Locus 8).

Mathematical & Constructive Commentary

Notice Altshiller-Court’s rigor on every locus proof: he establishes both inclusiveness (every point on the path satisfies the condition) and exclusiveness (no point off the path satisfies it, or conversely every point on the path belongs to the locus). Locus 7 rests on Euclid III.21, III.31, III.32, and III.33.

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Folio 0033 (Page 14) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0032 (Page 13, Loci 7–8)

  • Current Folio (0033.rst): Page 14, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0034 (Page 15, conclusion of Locus 10, Fig. 11, and Locus 11: Apollonian circle)

Structural & Diplomatic Observations

  • Overview: Page 14 (running header 14 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 11), treating the exterior-point restriction of LOCUS 8, stating LOCUS 9, displaying FIG. 10 (0033.g1.png), and deriving LOCUS 10 via Apollonius’ median theorem. Ends mid-sentence.

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: dis-tances

Calibrated Diagrams on Folio

  • 0033.g1.png (bbox [406, 488, 612, 386], white-point 238): Fig. 10. Locus of a point M the sum of the squares of whose distances from A and B is constant (§ 11, Locus 10).

Mathematical & Constructive Commentary

Raw OCR dropped both \(\frac{1}{2}\) fractions in \(OA = OB = \frac{1}{2} AB\) and \(OM^2 = \frac{1}{2} s^2 - a^2\); restored from 0033.png. Note the implicit diorismos of equation (1): the locus is a real circle iff \(s^2 > 2a^2 = \frac{1}{2} AB^2\), degenerates to the midpoint \(O\) when \(s^2 = 2a^2\), and is empty when \(s^2 < 2a^2\).

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Folio 0034 (Page 15) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0033 (Page 14, Loci 8–10: ‘…for its other leg half the’)

  • Current Folio (0034.rst): Page 15, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0035 (Page 16, conclusion of Locus 11, Note, Fig. 12, and Locus 12)

Structural & Diplomatic Observations

  • Overview: Page 15 (running header Ch. I, § 11] / GEOMETRIC LOCI / 15), finishing the compass-and-straightedge radius construction of LOCUS 10 and introducing LOCUS 11 (circle of Apollonius / Apollonian circle) with FIG. 11 (0034.g1.png) and the first half of its CONVERSELY proof.

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: there-fore, re-quired

Calibrated Diagrams on Folio

  • 0034.g1.png (bbox [318, 1005, 790, 485], white-point 238): Fig. 11. Circle of Apollonius: locus of a point M whose distances from fixed points A, B have a constant ratio p : q (§ 11, Locus 11).

Mathematical & Constructive Commentary

Locus 11 (Apollonius of Perga, Plane Loci II) is one of the pillars of College Geometry: \((A, B; C, D) = -1\) is a harmonic range, reappearing in Chapter II (§ 61), Chapter VIII (coaxal limiting points, § 448), and Chapter X (the three Apollonian circles of a triangle meeting in the isodynamic points \(X(15), X(16)\), §§ 597–606).

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Folio 0035 (Page 16) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0034 (Page 15, Locus 11: ‘therefore: ME = MF.’)

  • Current Folio (0035.rst): Page 16, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0036 (Page 17, conclusion of Locus 12, Exercises 1–4, and § 12)

Structural & Diplomatic Observations

  • Overview: Page 16 (running header 16 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 11), completing the converse proof and Socratic query of LOCUS 11 (plus Note on the twin Apollonian circles), displaying FIG. 12 (0035.g1.png), and beginning LOCUS 12.

  • Document Headings on Folio: None

  • Line-Break Hyphenations Resolved: seg-ment, state-ment, tri-angles

Calibrated Diagrams on Folio

  • 0035.g1.png (bbox [419, 1020, 585, 368], white-point 238): Fig. 12. Perpendicular line locus of a point M where AM² - BM² = d² is constant (§ 11, Locus 12).

Mathematical & Constructive Commentary

Answer to Altshiller-Court’s Socratic question on LOCUS 11: the hypothesis that \(M\) lies on the circle with diameter \(CD\) is used strictly in asserting that \(\angle CMD\) is a right angle (by Thales’ locus, Locus 7)! And LOCUS 12 (\(AM^2 - BM^2 = d^2\)) is precisely the point-circle case of the radical axis of two circles (Chapter VIII, § 422).

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Folio 0036 (Page 17) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0035 (Page 16, Locus 12: ‘(AC - BC)(AC + BC) = d^2,’)

  • Current Folio (0036.rst): Page 17, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0037 (Page 18, Fig. 13, Construction of § 12, and § 13)

Structural & Diplomatic Observations

  • Overview: Page 17 (running header Ch. I, §§ 11, 12] / GEOMETRIC LOCI / 17), concluding LOCUS 12 (constructing \(AC - BC = d^2 \div a\) via fourth proportional and sum-and-difference of segments, § 2), listing Locus Exercises 1–4, and presenting the enunciation and Analysis of § 12 (Problem).

  • Document Headings on Folio: EXERCISES, 12. Problem.

  • Line-Break Hyphenations Resolved: con-sider, con-stant

Mathematical & Constructive Commentary

Raw OCR corrupted \(AC - BC = d^2 \div a\) to dl-, a; verified against 0036.png. Note Exercise 1 (the median through \(A\) is the locus of \(M = (BE, CD)\) by harmonic properties of trapezoids, Ch. VII) and Exercise 3 (an Apollonian circle on \(AC\) passing through \(B\), since \(MB\) bisects \(\angle AMC\) iff \(MA : MC = AB : BC\)!).

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Folio 0037 (Page 18) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0036 (Page 17, §§ 11–12)

  • Current Folio (0037.rst): Page 18, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0038 (Page 19, Figs. 14–15, Discussion of § 13, and § 14)

Structural & Diplomatic Observations

  • Overview: Page 18 (running header 18 / GEOMETRIC CONSTRUCTIONS / [Ch. I, §§ 12, 13), displaying FIG. 13 (0037.g1.png), the Construction of § 12, and the Analysis, Construction, and Proof of § 13 (Problem).

  • Document Headings on Folio: 13. Problem.

  • Line-Break Hyphenations Resolved: re-spectively, trape-zoid

Calibrated Diagrams on Folio

  • 0037.g1.png (bbox [403, 162, 618, 466], white-point 238): Fig. 13. Circle (O) passing through A and B and subtending a given angle at external point C (§ 12).

Mathematical & Constructive Commentary

In § 12, notice that \(O\) also lies on the mediator of \(AB\), which is simultaneously the radical axis of the two Apollonian circles \((EF)\) and \((GH)\)—giving at most two centers \(O, O'\)! In § 13, Locus 9 and Locus 1 intersect cleanly at \(F\).

Scan Quality & Verification

  • Archival Scan: 01_geometric_constructions/0037.png (300 DPI, 1421 × 2225 px)

  • Verification Status: ok (visually verified against raster scan)

Folio 0038 (Page 19) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0037 (Page 18, §§ 12–13)

  • Current Folio (0038.rst): Page 19, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0039 (Page 20, Fig. 16, conclusion of § 14, and Exercises 1–16)

Structural & Diplomatic Observations

  • Overview: Page 19 (running header Ch. I, §§ 13, 14] / GEOMETRIC LOCI / 19), presenting the Discussion of § 13 with side-by-side FIG. 14 (0038.g1.png) and FIG. 15 (0038.g2.png), and starting § 14 (Problem).

  • Document Headings on Folio: 14. Problem.

  • Line-Break Hyphenations Resolved: intersec-tion

Calibrated Diagrams on Folio

  • 0038.g1.png (bbox [107, 458, 607, 619], white-point 238): Fig. 14. Parallel chords AC, BD through A, B in circle (O) having a given sum 2s (§ 13).

  • 0038.g2.png (bbox [732, 543, 593, 530], white-point 238): Fig. 15. Case where tangent crosses chord AB, yielding difference of bases 2EF (§ 13).

Mathematical & Constructive Commentary

Raw OCR misread 2 OE as 2 OR and FIG. 15 as FIG. 16; verified against 0038.png. Note the topological transition at \(s = EG\) in § 13: when \(s < EG\), the parallel chords cross inside the circle so that \(2\,EF = |AC - BD|\) instead of \(AC + BD\), immediately solving the companion difference problem!

Scan Quality & Verification

  • Archival Scan: 01_geometric_constructions/0038.png (300 DPI, 1421 × 2225 px)

  • Verification Status: ok (visually verified against raster scan)

Folio 0039 (Page 20) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0038 (Page 19, §§ 13–14)

  • Current Folio (0039.rst): Page 20, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0040 (Page 21, Exercises 17–23 and Section D: Indirect Elements, § 15, Fig. 17)

Structural & Diplomatic Observations

  • Overview: Page 20 (running header 20 / GEOMETRIC CONSTRUCTIONS / [Ch. I, § 14), displaying FIG. 16 (0039.g1.png), completing § 14 via Locus 7, and listing Locus Exercises 1–16.

  • Document Headings on Folio: EXERCISES

  • Line-Break Hyphenations Resolved: dis-tances

Calibrated Diagrams on Folio

  • 0039.g1.png (bbox [409, 162, 608, 553], white-point 238): Fig. 16. Point M on circle (A, B) whose joins AM, BM cut line FPQ in given ratio FP : FQ from F (§ 14).

Mathematical & Constructive Commentary

Look at Exercises 4–6 (\(a, h_a, b:c\); \(a, m_a, b:c\); \(a, t_a, b:c\)): each intersects the Apollonian circle for \(b:c\) on base \(BC\) with a second locus (parallel line at height \(h_a\) in Ex. 4, circle \((M_a, m_a)\) in Ex. 5, and a neat circle locus in Ex. 6). Exercise 16 intersects two Apollonian circles—foreshadowing the isodynamic points \(X(15), X(16)\) in Chapter X (§ 606)!

Scan Quality & Verification

  • Archival Scan: 01_geometric_constructions/0039.png (300 DPI, 1421 × 2225 px)

  • Verification Status: ok (visually verified against raster scan)

Folio 0040 (Page 21) — Diplomatic & Mathematical Notes

Summary & Folio Continuity

  • Preceding Folio: Folio 0039 (Page 20, § 14, Exercises 1–16)

  • Current Folio (0040.rst): Page 21, Chapter I [01_geometric_constructions] — Status: ok

  • Following Folio: Folio 0041 (Page 22, conclusion of § 15: ‘and therefore the angle EAF is known. The altitude AD of the triangle ABC is also the altitude of AEF…’)

Structural & Diplomatic Observations

  • Overview: Page 21 (running header Ch. I, §§ 14, 15] / INDIRECT ELEMENTS / 21), completing Locus Exercises 17–23, introducing Section D (Indirect Elements), and presenting § 15 (Problem: constructing \(\triangle ABC\) from \((2p, A, h_a)\)) with FIG. 17 (0040.g1.png). Ends with a comma at the bottom of page 21, continuing onto folio 0041 (page 22).

  • Document Headings on Folio: D. INDIRECT ELEMENTS, 15. Problem.

  • Line-Break Hyphenations Resolved: adja-cent, dif-ference, neces-sary, op-posite

Calibrated Diagrams on Folio

  • 0040.g1.png (bbox [247, 1400, 923, 330], white-point 238): Fig. 17. Construction of triangle ABC given perimeter 2p, vertical angle A, and altitude h_a via indirect triangle AEF (§ 15).

Mathematical & Constructive Commentary

In § 15, unfolding \(BA\) and \(CA\) onto line \(BC\) turns the indirect perimeter \(2p\) into the single segment \(EF = 2p\) of auxiliary triangle \(AEF\), which is constructed from base \(EF = 2p\), capable arc for \(\angle EAF = 90^\circ + \frac{1}{2}A\) (Locus 7), and parallel line at height \(h_a\) (Locus 3); on folio 0041 (p. 22), vertices \(B, C\) are recovered as the traces on \(EF\) of the mediators of \(AE\) and \(AF\)!

Scan Quality & Verification

  • Archival Scan: 01_geometric_constructions/0040.png (300 DPI, 1421 × 2225 px)

  • Verification Status: ok (visually verified against raster scan)