Reading: Poncelet, geometry written in prison

Jean-Victor Poncelet (1788–1867) was born in Metz, in eastern France. He studied at the Polytechnic School in Paris, where he was a student of Gaspard Monge, the creator of descriptive geometry, and then at the military engineering school in Metz. In 1812, as an engineering lieutenant, he accompanied Napoleon’s army in the invasion of Russia.

Jean-Victor Poncelet.Smithsonian Libraries · Wikimedia Commons · Public domain

During his retreat from Moscow, he was presumed dead in the battle of Krasnoi and ended up a prisoner. Taken on foot in winter to Saratov, on the banks of the Volga, he spent around eighteen months in captivity. Without books, he reconstructed from memory what he had learned with Monge and went further: he began to develop projective geometry, the study of the properties of figures that do not change when they are projected from one plane to another — the same properties that Desargues and Pascal had begun to explore in the 17th century.

Released in 1814, he returned to France and began publishing his results, first on inscribed polygons circumscribed by conics. The great work, the Traité des propriétés projectives des figures (“Treatise on the projective properties of figures”), came out in 1822. In it, Poncelet showed that properties such as the alignment of points or tangency are preserved through projection, and systematically used the relationship between pole and polar of a conic, the basis of the duality principle: in flat projective geometry, every theorem about points and lines remains true when the words “point” and “line” are exchanged.

These ideas paved the way for the study of projective invariants, such as the double ratio (or anharmonic) of four points, developed by Möbius and Chasles, and for the program of Felix Klein, which in 1872 classified geometries by the groups of transformations that preserve them. Poncelet was also an important engineer: he designed hydraulic wheels and taught applied mechanics. Died in Paris, in 1867.

Sources: Poncelet (MacTutor) · J.-V. Poncelet, Traité des proprietés projectives des figures (1822).

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Reading: Poncelet, geometry written in prison

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