Unit IV Summary: The Three Restricted-Instrument Regimes

Comparative Architecture of Ruler+Compass, Set-Square, Mascheroni, and Poncelet–Steiner

1. Instrument Regimes & Allowed Operation Subspaces

Constructibility & Operation Subspaces Across Classical and Minimal Instruments

Regime

Primary Source

Operation Symbol

Algebraic Field & Invariance

Canonical Ruler + Compass

lemoine-1892, lemoine-1902

\(l_1 R_1 + l_2 R_2 + m_1 C_1 + m_2 C_2 + m_3 C_3\)

Full Euclidean quadratic tower (\(\sqrt{x^2 \pm y^2}\))

Ruler + Équerre (Set-Square)

lemoine-1888, lemoine-1902

\(R_1, R_2 + E_1, E_2\) (sliding right angle)

Affine + orthogonal projections (pythagorean field when segment transfer added)

Mascheroni Compass Alone

mohr-1672, mascheroni-1797

\(m_1 C_1 + m_2 C_2 + m_3 C_3\) (\(l_1 = l_2 = 0\))

Full Euclidean quadratic tower (lines implicit as point-pairs \((A, B)\))

Poncelet–Steiner Straightedge + Fixed Circle :math:`M`

poncelet-1822, steiner-1833

\(l_1 R_1 + l_2 R_2\) (\(m_1 = m_2 = m_3 = 0\))

Full Euclidean quadratic tower (circles implicit as center + radius endpoint)

Hilbert Straightedge + Eichmaß (Unit Gauge)

hilbert-1899 (§§ 36–39)

\(R_1, R_2\) + unit-segment transferrer

Strictly pythagorean subfield (\(\sqrt{1 + \omega^2}\), smaller than Euclidean!)

2. Steiner’s Eight Fundamental Problems (Drittes Kapitel, § 18, [68]–[85])

  1. Erste Aufgabe ([68]–[71]): Parallel through external point to a diameter (\(11R_1 + 6R_2, S = 17\)) or general line.

  2. Zweite Aufgabe ([71]): Harmonic bisection and rational multiplication/\(n\)-section of a segment.

  3. Dritte Aufgabe ([71]–[75]): Perpendicular to a line (orthocenter on fixed-circle diameter: \(10R_1 + 5R_2, S = 15\)).

  4. Vierte Aufgabe ([75]): Copying a given angle to a given vertex and ray via parallel diameters of \(M\).

  5. Fünfte Aufgabe ([75]–[76]): Bisecting a given angle via parallel diameters and subtended chord.

  6. Sechste Aufgabe ([76]–[78]): Transferring a given segment to a given point along a given ray via centers of similitude \(A, I\).

  7. Siebente Aufgabe ([78]–[82]): Intersecting a given line with an un-drawn circle \((M_1, r_1)\) via homothetic secant on fixed circle \(M\).

  8. Achte Aufgabe ([82]–[85]): Intersecting two un-drawn circles \((M_1, r_1)\) and \((M_2, r_2)\) via their radical axis (Linie der gleichen Potenzen) reduced to Problem 7.