Unit I Summary: The Operational Calculus of Géométrographie

Executive Synthesis of Genesis, Axiomatic Idealizations & Elementary Cost Vectors

1. Historical Stemma (1888 → 1892 → 1902)

  • Oran (30 March 1888): Émile Lemoine introduces quantitative simplicity measurement (lemoine-1888) using an experimental eight-operator system (\(\rho_1..\gamma'_2\)) and right-angle set-square (équerre) primitives.

  • Pau (September 1892 / Gauthier-Villars 1893): Consolidates the five-operator ruler-and-compass canon \((R_1, R_2, C_1, C_2, C_3)\), defines \(S\) and \(E\), and benchmarks Problems I–LXIII (lemoine-1892).

  • Paris, Collection *Scientia* No. 18 (1902): Unifies standard Géométrographie with Géométrographie de l’équerre, Géométrographie Mascheronienne (mascheroni-1797), and Géométrographie Steinerienne (steiner-1833).

2. The Five-Dimensional Vector Space of Constructions

Every construction maps to a non-negative integer vector \(\mathbf{v} = (l_1, l_2, m_1, m_2, m_3) \in \mathbb{Z}_{\ge 0}^5\):

Canonical Géométrographie Cost Primitives

Symbol

Instrument

Physical / Incidence Action

\(\Delta S\)

\(\Delta E\)

R₁

Straightedge

Make straightedge pass through 1 determined point (\(2R_1\) for 2 points)

\(+1\)

\(+1\)

R₂

Straightedge

Trace straight line along aligned straightedge

\(+1\)

\(0\)

C₁

Compass

Set one compass point on a determined point (\(2C_1\) to span \(AB\))

\(+1\)

\(+1\)

C₂

Compass

Set one compass point at an indeterminate point of a traced locus

\(+1\)

\(+1\)

C₃

Compass

Describe circle or circular arc

\(+1\)

\(0\)

Linear functionals on \(\mathbf{v}\) yield:

  • Simplicity: \(S(\mathbf{v}) = l_1 + l_2 + m_1 + m_2 + m_3\) (total elementary steps).

  • Exactitude: \(E(\mathbf{v}) = l_1 + m_1 + m_2\) (total point/locus alignments; smaller \(E\) = higher precision).

  • Locus Complexity: \(l_2\) straight lines and \(m_3\) circles drawn on the plane.

3. Why Euclid I.2–I.3 Costs \(S = 16\) vs. Lemoine VIII \(S = 4\)

In Euclid’s Elements I.2–I.3 (euclid-heath; see VIII — Laying Off a Given Segment Along a Traced Line), transferring a segment \(AB\) to a line without assuming a persistent-aperture divider compass requires constructing an equilateral triangle (\(I.1\)) and two concentric transfer circles: \(4R_1 + 2R_2 + 5C_1 + 5C_3\) (\(S = 16, E = 9\), 2 lines, 5 circles). With a modern divider compass, Lemoine’s Operation VIII executes the transfer in \(2C_1 + C_2 + C_3\) (or \(3C_1 + C_3\) from a marked origin), i.e., \(S = 4, E = 3\), 0 lines, 1 circle—immediately illustrating the chasm between axiomatic minimalism of postulates and operational minimalism of execution.