" *Although the class of constructions that can be carried out with a straightedge alone was not defined until the 19th century, straightedge geometry itself can be said to date back much further. From very early times, craftspeople such as carpenters and stonemasons passed down trade techniques in which the straightedge was the only, or the principal, instrument used, because of the very nature of the objects on which the lines were drawn. These craftspeople were not equipped to recognize that their work embodied geometrical propositions, so to speak*. "
This is how the French mathematician Henri Lebesgue (1875–1941) introduces Chapter III of his Leçons sur les constructions géométriques, professées au Collège de France en 1940–1941 (Gauthier-Villars, 1950). It is therefore not so surprising to ask whether "all" geometrical constructions—or at least those traditionally known as straightedge-and-compass constructions—can be carried out with a straightedge alone.
Projective geometry and artists' perspective ------------------------------------------------
In 1833, the Swiss mathematician Jacob Steiner (1796–1863) published a short booklet whose German title might be translated as Geometrical Constructions with a Straightedge, Given a Fixed Circle and Its Center. Several mathematicians, he notes, had drawn attention to the many problems requiring only a straightedge. They included François-Joseph Servois (1767–1847), with his 1804 work Solutions peu connues de différens problèmes de Géométrie pratique, pour servir de supplément aux traités connus de cette science, and Jean-Victor Poncelet (1788–1867), with his 1822 Traité des propriétés projectives des figures. Steiner might also have cited Jean-Henri Lambert (1728–1777), in the 1774 reissue of Perspective affranchie de l’embarras du plan géométral, and indeed several others. All these works are based on projective geometry, whose deeper roots lie in the mathematical treatment of artists' perspective (see our feature "Projective geometry", Tangente 162, 2015).
This straightedge geometry thus represents a kind of alliance between practical requirements and the highly innovative geometry of the 19th century, whose founders included Jean-Victor Poncelet and Jacob Steiner.