Unit I: Preface & The Anatomy of a Geometric Construction — Notes

Historical, Mathematical & Cross-Repository Commentary (pp. 1–6 (with Problem 42, p. 15))

Historical, Mathematical & GEOMETOR Commentary

1. From Pappus’s Analyomenos to Petersen (1866) & Pólya (1945)

The heuristic method of analysis—“assuming that which is sought as if it were already admitted” (ὡς γεγονός)—descends from Pappus of Alexandria (Synagoge Book VII), refined in the seventeenth century by Viète and Descartes (La Géométrie). Yet prior to Julius Petersen's Methoder og Theorier (1866 Danish 1st ed.; 1879 English ed.), school and university textbooks presented geometric constructions as isolated tricks.

Petersen’s 1866 monograph revolutionized mathematical pedagogy across Europe:

  • Simultaneity of translation in 1879–1880—into English by Sophus Haagensen, German by R. von Fischer-Benzon, and French by Octave Chemin (Méthodes et théories, Gauthier-Villars, 1880)—made “Petersen” a household verb among French agrégation, German Gymnasium, and British Tripos candidates.

  • George Pólya, who worked through Petersen’s 410 problems as a student in Budapest, directly modeled the “Pattern of Two Loci” in How to Solve It and Volume I, Chapter 1 of Mathematical Discovery on Petersen’s First Fundamental Rule (“keep only a part of the condition, drop the other part”).

  • Nathan Altshiller-Court structured Chapter I (Geometric Constructions) of College Geometry directly around Petersen’s four-stage schema (Analysis, Construction, Proof, Discussion) and locus/transformation taxonomy.

2. Exact geometor.model Realization: discussion_mean_proportional

In geometor.petersen.constructions.discussion_mean_proportional(), we instantiate Petersen’s Problem 42 (p. 15: dividing \(BC = a\) into two segments having \(h\) as mean proportional) over \(\mathbb{Q}\) with auto_intersect=False:

  • Given: \(B = (0, 0)\), \(C = (10, 0)\) (\(a = 10\)), and altitude \(h = 4\).

  • Two Loci: Thales circle \((M, B)\) with \(M = (5, 0), R = 5\) (Locus e for \(90^\circ\)) intersected explicitly with parallel line \(y = 4\) (Locus b).

  • Exact Algebraic Verification: The discriminant \((a/2)^2 - h^2 = 25 - 16 = 9 = 3^2\) splits over \(\mathbb{Q}\), yielding two rational right-angle apices \(A_1 = (2, 4)\) and \(A_2 = (8, 4)\) with pedal \(D_1 = (2, 0)\). Both Rational Trigonometry invariants hold identically: line_spread([B, A1], [C, A1]) == 1 and quadrance(A1, D1) == 16 == 2 * 8.