As a lieutenant of engineers in 1812, he took part in Napoleon’s Russian campaign, in which he was abandoned as dead at Krasnoy and imprisoned at Saratov; he returned to France in 1814. During his imprisonment Poncelet studied projective geometry and wrote Applications d’analyse et de géométrie, 2 vol. (1862–64; “Applications of Analysis and Geometry”). This work was originally planned as an introduction to his celebrated Traité des propriétés projectives des figures (1822; “Treatise on the Projective Properties of Figures”), for which Poncelet is regarded as one of the greatest projective geometers. His development of the pole and polar lines associated with conic sections led to the principle of duality (exchanging “dual” elements, such as points and lines, along with their corresponding statements, in a true theorem produces a true “dual statement”) and a dispute over priority with the German mathematician Julius Plücker for its discovery. His principle of continuity, a concept designed to add generality to synthetic geometry (limited to geometric arguments), led to the introduction of imaginary points (see complex numbers) and the development of algebraic geometry.
From 1815 to 1825 Poncelet was occupied with military engineering at Metz, and from 1825 to 1835 he was a professor of mechanics at the École d’Application there. He applied mathematics to the improvement of turbines and waterwheels. Although the first inward-flow turbine was not built until 1838, he proposed such a turbine in 1826. In Paris from 1838 to 1848 he was a professor at the Faculty of Sciences, and from 1848 to 1850 he was commandant of the École Polytechnique, with the rank of general.