Unit IV: Restricted & Alternative Instruments — Équerre, Mascheroni Compass-Only & Poncelet–Steiner Fixed-Circle

Émile Lemoine (Scientia No. 18, 1902, pp. 55–88) · Lorenzo Mascheroni (1797) · Jakob Steiner (1833)

[Folio L'Équerre | p. 55]

§ 1. Géométrographie de l’équerre (Draftsman’s Set-Square)

Already in his Oran communication of 30 March 1888 (lemoine-1888, p. 9; recalled on p. 55 of lemoine-1892 and expanded in Scientia No. 18, lemoine-1902), Émile Lemoine observed that practical geometers and engineers continually replace multi-circle parallel and perpendicular constructions with the équerre (draftsman’s right-angled set-square) sliding against a straightedge. Extending Géométrographie to the set-square introduces dedicated alignment and sliding primitives:

  • Placing one edge of the équerre along a traced line (or against the guiding ruler) and sliding the ruler-guided set-square until an edge passes through a given point \(A\) reduces Problem XVII (Parallel through :math:`A` to :math:`BC`) and Problem XX (Perpendicular through :math:`A` to :math:`BC`) from nine ruler-and-compass operations (and 2–3 circles) to a pure linear translation without a single compass arc!

  • Conversely, because the équerre introduces sliding contact between two rigid edges, Lemoine carefully separates its operational ledger from pure Euclidean ruler-and-compass symbols so that constructions are compared only within identical instrument postulates.

[Folio Géométrographie Mascheronienne | p. 66]

§ 2. Géométrographie Mascheronienne: Compass Alone (La Geometria del Compasso, 1797)

When the straightedge is banished altogether (\(l_1 = l_2 = 0\)), every construction reduces to a pure compass symbol:

\[\mathrm{Op.}_{\text{Masch.}} : \left(m_1 C_1 + m_2 C_2 + m_3 C_3\right), \qquad S = m_1 + m_2 + m_3, \quad E = m_1 + m_2, \quad N_{\text{lines}} = 0.\]

Unknown to eighteenth-century Europe, Georg Mohr had proved compass-only Euclidean completeness in Euclides Danicus (mohr-1672). Independently in Pavia in 1797, Lorenzo Mascheroni published La Geometria del Compasso (mascheroni-1797; French translation by Antoine-Michel Carette, carette-1798), dedicated to Napoléon Bonaparte. Because no continuous straight line can be drawn without a ruler, Mascheroni represents every straight line implicitly by two distinct points \((A, B)\) and proves across twelve books that every point constructible with ruler and compass is constructible by intersecting circular arcs alone:

  1. Collinear Doubling & Multiplication (Libro III, § 64): Stepping three chords of radius \(AB\) around semicircle \(B(BA)\) from \(A\) produces \(E\) collinear with \(A, B\) such that \(AE = 2\,AB\) in \(5C_1 + 4C_3\) (\(S = 9\), 4 circles).

  2. Bisecting an Un-Drawn Segment (Libro III, § 66, Sol. V): Having doubled \(AB\) to \(E\), sweep \(E(EB)\) cutting \(B(BA)\) at \(P, p\); then \(P(PB)\) and \(p(PB)\) intersect at the exact midpoint \(M\) of \(AB\) in 7 circles: \(10C_1 + 7C_3\) (\(S = 17, E = 10\)).

  3. Reflection, Perpendicular & Parallel Point (Libro IV, §§ 76–80): Reflecting \(A\) across \((B, C)\) takes \(4C_1 + 2C_3\) (\(S = 6\)); constructing \(D\) so \((C, D) \parallel (A, B)\) via BAC and CAB takes \(5C_1 + 2C_3\) (\(S = 7, E = 5\)).

  4. Concentric Fourth Proportional (Libro V, § 93): Highlighted by Gaetano Fazzari (fazzari-1901, p. xi) as simpler in Lemoine’s Géométrographie than standard ruler-and-compass Thales constructions: draw concentric circles \(O(a)\) and \(O(b)\), place chord \(BC = c\) in \(O(a)\), and step equal arcs \(BD = CE\) onto \(O(b)\); then chord \(DE = bc/a\) in just 4 circles (\(S = 14\) when \(c \le 2a\))!

  5. Circle Inversion & Compass Completeness (Libro V, § 86; Libro VII, §§ 110–112): Intersecting a circle with a point-pair line (§§ 110–111, by reflecting the center across \(AB\)) and intersecting two point-pair lines \(AB \cap CD\) (§ 112, via symmetric reflections \(c, d\) and fourth proportional) complete Euclidean geometry. In 1890, August Adler (adler-1890) unified all of Libro VII into a single conceptual stroke: Lemoine’s Problem XXXVIII (\(5C_1 + 3C_3\) compass-only inversion) maps lines not through \(O\) into circles through \(O\), reducing line–line and line–circle intersections directly to circle–circle intersections!

A B C D Mascheroni (1797) — Compass Alone 5C₁ + 2C₃ (S = 7, 0 lines): (C, D) ∥ (A, B) M a b S P P' K Poncelet–Steiner (1833) — Straightedge + Fixed Circle M 11R₁ + 6R₂ (S = 17, 0 circles): PP' ∥ diameter aMb
[Folio Géométrographie Steinerienne | p. 74]

§ 3. Géométrographie Steinerienne: Straightedge Alone with One Fixed Circle (Die geometrischen Constructionen, 1833)

At the opposite pole, banishing the compass (\(m_1 = m_2 = m_3 = 0\)) leaves pure straightedge operations:

\[\mathrm{Op.}_{\text{Steiner}} : \left(l_1 R_1 + l_2 R_2\right), \qquad S = l_1 + l_2, \quad E = l_1, \quad N_{\text{circles}} = 0.\]

By projective invariance, a straightedge alone cannot bisect a segment, draw a parallel, or solve quadratic metric equations—because non-affine projective collineations preserve straight lines while moving the line at infinity and the circular points. In 1822 Jean-Victor Poncelet (poncelet-1822) announced, and in 1833 Jakob Steiner (steiner-1833) constructively proved across eight fundamental problems (Drittes Kapitel, § 18, [68]–[85]), that one single permanently drawn circle Omega together with its given center \(M\) (fester Hülfskreis) restores the entire power of the compass:

  1. Erste Aufgabe (Parallel Line, ``[68]``–``[71]``): Any line through center \(M\) cuts Omega in a diameter ab bisected at \(M\). Joining \(a, M, b\) to an auxiliary vertex \(S\) and completing the quadrangle through \(P\) constructs the parallel PQpar in \(11R_1 + 6R_2\) (\(S = 17, E = 11\), 6 lines, 0 circles); for an arbitrary line not through \(M\), two secant chords through \(M\) first supply a parallel diameter.

  2. Zweite Aufgabe (Segment Multiplication & Harmonic Bisection, ``[71]``): Once a parallel line is drawn, complete-quadrangle diagonals bisect or multiply any segment \(AB\) by straightedge alone.

  3. Dritte Aufgabe (Perpendicular Line, ``[71]``–``[75]``): Inscribed angles on diameter \(ab\) supply right-angle chords in Omega (orthocenter construction in \(10R_1 + 5R_2, S = 15\)), transferred to any point by Problem 1.

  4. Vierte & Fünfte Aufgabe (Copying & Bisecting Angles, ``[75]``–``[76]``): Drawing diameters of Omega parallel to the arms of any angle transports the angle to center \(M\), where parallel chords copy it (Problem 4) and the parallel to chord \(ad\) bisects it (Problem 5)!

  5. Sechste Aufgabe (Segment Transfer, ``[76]``–``[78]``): Given a segment \(M_1 a_1\) (defining a virtual un-drawn circle centered at \(M_1\)), parallel radii in Omega and \(M_1\) determine their external and internal centers of similitude \(A, I\), transferring length \(M_1 a_1\) to any new origin \(M_2\) in any direction!

  6. Siebente & Achte Aufgabe (Intersecting a Line with a Virtual Circle & Two Virtual Circles, ``[78]``–``[85]``): Homothetic projection through a center of similitude maps any secant line of un-drawn circle \((M_1, r_1)\) onto a secant of the drawn fixed circle Omega (Problem 7); finally, constructing the radical axis (Linie der gleichen Potenzen) of two un-drawn circles \((M_1, r_1)\) and \((M_2, r_2)\) via auxiliary homothetic chords reduces circle–circle intersection to Problem 7 (Problem 8)!

Why Center :math:`M` Cannot Be Omitted: If circle Omega is given without its center \(M\), any hyperbolic projective transformation (Möbius/Klein automorphic collineation of the disk interior) maps Omega onto itself and straight lines onto straight lines, yet moves the Euclidean center \(M\) to another interior point—proving that \(M\) itself cannot be found by straightedge alone from a single circle!