Here Bernard Cache provides a detailed analysis of a paper written in 1636 by the French mathematician, architect and engineer, Girard Desargues. Desargues is best known as the founder of projective geometry. Cache explains how he initally developed this significant concept in response to the very practical problems of producing a perspectival drawing. The introduction of projective geometry, though, had potentially more far-reaching implications on philosophical thought, informing the theory of monads developed by the German philosopher and mathematician Gottfried Leibniz in 1714 to explain the metaphysics of simple substances. © 2011 John Wiley & Sons Ltd.