dis that they can not be doubted." Descartes attached less importance to the geometrical and mathematical methods of which he was the inventor than to his moral and metaphysical speculations. But, while the latter have been sifted and riddled in discussion, and have suffered under the revolutions of thought, the mathematical principles he established and the methods he introduced remain. In geometry he gave demonstrations of general principles, under which solutions adapted to one problem could be applied to all like it. In algebra, for the old clumsy notation and nomenclature, always suggesting material relations, he substituted the beautiful, convenient system, purely abstract, by the aid of which that branch of the science has marched to almost universal application and perfection. And in the application of algebra to geometry, he introduced the method of abscissas and ordinates, by means of which any curve and any condition of form can be computed by a process as beautiful as it is direct.
The Meditations on the First Philosophy, which appeared in manuscript in 1640, consists of six parts, in the first of which the author expounds his philosophy of doubt; in the second, he reaches the certainty of his own being, through the use of his famous maxim, cogito, ergo sum (I think, therefore I am); in the third, he deduces an argument to prove the existence of God from the idea of an infinite and sovereignly perfect being; in the fourth, he draws a distinction between speculative reasoning, for which the light of nature is sufficient, and doctrines of faith and the conduct of life, which rest on another foundation; in the fifth, he explains the corporeal nature, and brings forward another argument for the existence of God; and in the sixth he treats of the distinctions between intellect and imagination, the difference yet intimate connection of soul and body, errors of the senses and the means of avoiding them, and the reasons upon which we can conclude concerning the existence of material things, which he, however, regarded as inferior to the evidence on which we predicate the existence of God and the soul. The book in this form was submitted to the criticisms of a number of distinguished students, whose objections were printed and bound with the main treatise when it was published in 1641, and with them the replies of the author, considerably swelling the bulk of the volume.
The Principles of Philosophy, 1644, contained an exposition of the principles of knowledge as developed in the Meditations; an explanation of the primary laws of nature, the properties of matter, space, motion, etc.; the system of the world, the sky, and celestial bodies; and a treatise on the Earth. The statement of the three laws of nature, the seven secondary laws of impact (which are pronounced by later science to a large extent incorrect), and the famous theory of Vortices, by which Descartes at-