Foundations

Abstract

In mathematics Pascal was a child prodigy. By the age of fourteen or fifteen, having mastered Desargues’s bafflingly abstruse work on synthetic projective geometry, he had become a fervent admirer of the Brouillon Projet. In February 1640, when he was sixteen years old, he published his own Essay on Conic Sections.¹ Starting with the intersections of circles, he seems (in accordance with the methods of Desargues’s projective geometry) to have extended his proof of the Mystic Hexagon to ellipses (or antobolas), parabolas and hyperbolas. The Mystic Hexagon, discovered but unpublished, soon became the key to his understanding of this area of mathematics. From the two lemmas of the Essay on Conic Sections he is said to have deduced over 400 corollaries (58), including most of the propositions contained in the Conics of Apollonius of Perga. He believed, therefore, that the analytical co-ordinate geometry of which Descartes gave his first published exposition in 1637² was by no means the only way of elaborating a complete theory of conics: the same could be done by means of the Mystic Hexagon and the projective methods which Desargues had pioneered.³ But he never achieved this unified theory and it was left to Poncelet and Chasles to carry the study of conics to its fullest development, thereby producing a general theory of transformations.