Essay on Conic Sections

Abstract

Blaise Pascal’s two papers on mathematics, Essay on Conic Sections and The generation of conic sections, are considered basic texts in the history of projective geometry. The two essays are not only important from the perspective of the history of science but are also significant from the perspective of Pascal’s subsequent thinking. When Pascal interpreted conic sections projectively, he encountered the problem of the mathematical infinite in several places.

In projective geometry one needs to presuppose that parallel lines cross each other in the infinite, which is not evident in Euclidean geometry. Also, while generating conic sections projectively, often the picture of a finite form will be infinite, like a parabola or a hyperbola, while they are images of the cone’s base, the circle. Pascal had to handle mathematical paradoxes connected to the infinite at an early age, and he tried to integrate these problems into his work rather than reject them. This attitude to the infinite would characterize his subsequent mathematical and philosophical works.