Mascheroni (1797) & Poncelet–Steiner (1833)

Euclidean geometry traditionally postulates both the ungraduated straightedge (Postulates 1–2) and the compass (Postulate 3). Between 1672 and 1890, synthetic and projective geometers discovered that either instrument can dispense with the other—provided a subtle asymmetry between circles and lines is respected:

1. Compass Alone — Georg Mohr (1672) & Lorenzo Mascheroni (1797): First proved by Georg Mohr in Euclides Danicus (Amsterdam, 1672) and independently developed into a twelve-book constructive masterpiece by Lorenzo Mascheroni in La Geometria del Compasso (Pavia, 1797), every Euclidean point-determination problem can be solved using only the compass (\(m_1 C_1 + m_2 C_2 + m_3 C_3\)). Straight lines cannot be continuously drawn, but any line is completely determined by two of its points \((A, B)\), and circular inversion (\(5C_1 + 3C_3\),August Adler, 1890) maps lines and circles into circles.

2. Straightedge Alone with One Fixed Circle — Poncelet (1822) & Jakob Steiner (1833): Conversely, a straightedge alone in an empty plane can execute only projectively invariant constructions (such as harmonic conjugates and polars) and cannot solve quadratic metric problems. Jean-Victor Poncelet (1822) observed, and Jakob Steiner rigorously demonstrated in Die geometrischen Constructionen, ausgeführt mittelst der geraden Linie und Eines festen Kreises (Berlin, 1833), that a single permanently drawn circle together with its center \(M\) suffices to execute every ruler-and-compass construction using the straightedge alone (\(l_1 R_1 + l_2 R_2\)) via eight canonical problems (§ 18, [68]–[85]) built from complete quadrangles, centers of similitude (Aehnlichkeitspunkte), and radical axes (Linie der gleichen Potenzen).