Unit I Commentary & Notes: Epistemology of Constructive Complexity

Scholarly Apparatus, Cross-Corpus Concordance & Algorithmic Interpretation

1. Problems vs. Theorems: Proclus, Hoenen, and Lemoine

In his Commentary on Book I of Euclid’s Elements (proclus-in-euclidem), Proclus records the ancient controversy between the schools of Speusippus (who held that all geometric propositions are timeless contemplative theorems) and Menaechmus (who held that all geometric propositions are constructive problems bringing figures into genesis within spatial imagination, phantasia). In De Noetica Geometriae (hoenen-1954; see ), Peter Hoenen S.J. analyzes how materia intelligibilis bridges ideal geometric essence and imaginative construction.

Émile Lemoine discovers an overlooked third dimension of this classical distinction (pp. 60–65): Greek geometers drew schematic figures freehand in sand or wax dust, using postulates (aitemata) solely as existential licenses inside deductive syllogisms. Consequently, Greek synthesis optimized logical call-depth—reducing Problem \(P_n\) to \(P_{n-1}\)—which behaves exactly like an un-inlined recursive subroutine in computer science: concise source code, heavy runtime instruction count! Géométrographie is the historical birth of concrete computational complexity theory for geometric algorithms, counting exact primitive instructions rather than asymptotic big-\(O\) steps.

2. Critical Nuances in Counting \(C_1\) and \(C_2\)

Students reading lemoine-1902 for the first time should note three subtle rules verified across Lemoine’s sixty-three problems:

  1. Circle :math:`O(A)` Costs :math:`2C_1 + C_3`, Whereas Circle :math:`O(BC)` Costs :math:`3C_1 + C_3`: When drawing \(O(A)\) (Operation VII), one compass leg is set on \(O\) (\(C_1\)), the other on \(A\) (\(C_1\)), and the circle is swept immediately around \(O\) (\(C_3\))—total \(2C_1 + C_3\) (\(S = 3\)). When transferring a detached distance \(BC\) to center \(O\), spanning \(BC\) costs \(2C_1\), lifting and setting the pivot on \(O\) costs a third \(C_1\), and sweeping costs \(C_3\)—total \(3C_1 + C_3\) (\(S = 4\)).

  2. Retaining Compass Aperture Across Ruler Operations: Lemoine explicitly permits keeping a fixed radius in the compass while picking up the straightedge to draw a line. Thus in Problem XVIII (perpendicular bisector), choosing an arbitrary sufficient aperture costs \(0\) to open, \(C_1 + C_3\) around \(A\), and \(C_1 + C_3\) around \(B\) (\(2C_1 + 2C_3\) for both circles, versus \(3C_1 + 2C_3\) in Euclid I.10 where the aperture is forced to equal \(AB\)).

  3. Why Arbitrary Plane Points Cost :math:`0`, Locus Points Cost :math:`C_2`, and Intersection Points Cost :math:`C_1`: The cost of placing a point equals the codimension of the constraint in \(\mathbb{R}^2\):

    • Open 2D plane (codimension \(0\)): \(0\) alignment cost;

    • 1D curve—traced line or circle (codimension \(1\)): \(C_2\) (or \(R_1\) for ruler);

    • 0D determined point (codimension \(2\)): \(C_1\) (or \(R_1\) for ruler).

3. Reception in Palermo (1901), Edinburgh (1893), and Vienna (1890–1906)

Within a decade of Pau (lemoine-1892), Géométrographie became an international standard:

  • John Sturgeon Mackay introduced the system in Britain in The Geometrography of Euclid’s Problems (mackay-1893), followed by Lemoine’s own Edinburgh memoir (lemoine-1895).

  • In Palermo, Gaetano Fazzari opened his critical 1901 edition of Mascheroni’s La Geometria del Compasso (fazzari-1901, Preface p. xi) by comparing Mascheroni’s compass-only fourth proportional against Euclidean ruler-and-compass methods via “il metodo della Geometrografia del Lemoine”.

  • In Vienna and Prague, August Adler (adler-1890) integrated Lemoine’s operation counts directly into his unified treatise on Mascheroni, Steiner, and set-square constructions.