One of the most fundamental genera] principles of the philosophical system of Descartes is the essential difference between spirit and matter thinking and extended substances—a differ ence so great, according to Descartes. that they can exert no influence upon each other. hence, in order to account for the correspondence be tween material and spiritual phenomena. he was obliged to have recourse to a constant co operation I concurcus) on the part of God — doctrine which gave rise subsequently to the system called occasionalism (q.v.). the principle of which was that body and mind do not really affect each other. (hod being always the true cause of the apparent or occasional influence of one on the other. This doctrine received its com plete development in the harmony of Leibnitz (q.v.). In Descartes's. thought it re sulted merely in a strenuous insistence upon the differences between primary and secondary quali ties (q.v.). Descartes maintained also that the lower animals belong merely to the world of ex tension, being unconscious automata.
Descartes did not confine his attention to mental philosophy, hut devoted himself syste matically to the explanation of the properties of the bodies composing the material universe. In this department his reforms amounted to a revolution, though many of his explanations of physical phenomena arc purely a priori and quite absurd. His corpuscular philosophy—in which he endeavored to explain all the appearances of the material world simply by the motion of the Intimate particles of bodies—was a great advance on the system held up to that time. according to which special qualities and powers were assumed to account for every phenomenon. It was in pure mathematics, however, that Descartes achieved the greatest and most lasting results, especially by his invention of the analytic geome try, which is known front his name as Cartesian. Tn developing this branch of mathematics he had in mind. not the revolutionizing of geometry, hut the elucidation of algebra by means of geometric intuition and concepts. He intended to estah lish a universal mathematic, to which algebra, arithmetic., and geometry (with its applications) should be entirely subordinate. Ile discarded Virta's improvements in algebraic symbolism. introduced the present plan of representing known and unknown quantities, gave standing to the present system of exponents, placed the theory of negative quantities on a satisfactory basis, and set forth without demonstration the well-known rule for finding the limit of the num ber of positive and roots of an equation through inspection of the variations in the signs.
While his expectations were, in one sense. not fulfilled. he nevertheless succeeded in imparting a powerful impulse to the progress of mathe matics, and in giving to the science its modern trend. The establishment of a correspondence between geometry and analysis has been of in ealenla,ble assistance to both, and Deseartes's invention may he said to constitute the point of departure of modern mathematics. In 1637 he published at Loy&11 a treatise entitled niSCOUrS de la niOhofic pour Inert cominire sr, raison it (hcreher lu •critt: lbws s sciences (Eng. trans. 1850: recent ed., Chicago. 1899). This was fol lowed in the same year by three appendices en titled, La (Hopi rig lor, 1,cs inOlores, and La g(Mactric. The new was set forth entirely in the GeomOrir. a book of only about a hundred pages. obscurely written. The first part shows how arithmetical operations may be represented geometrically by taking a certain unit of length. in which lay the sole novelty of the plan. The second part shows how to trace algebraic (which he calls geometric) and transcendental lwhiclh he calls mechanical) curves, explaining the use of eM)rdinates, and setting forth the general scheme (now discar• ed) of classification of curves according to the order of their equations. The third part treats of the theory of equations. showing how their roots may be found by the intersection of the corresponding curves. It was in this part that he set forth his improvements in algebra. The appearance of the placed Descartes foremost among the mathematicians of his time.
His works in Latin were published at Am sterdam (1650). Modern editions are those by Cousin (Paris, 182-1-2G), by Gamier (Paris. 1834-35). by Martin (Paris, 1882). Eng lish translations of portions of his works have been made by Veiteh (Edinburgh. 1880), by Lowndes (London. 1878), and by Torrey (New York, 1892). The publication of his complete works was begun under the auspices of the French Minister of Public Instruction in 18!17. Consult: Slahaffy. Descartes (Edinburgh, 1880) ; Millet, Descartes. sir vie, .sex Ira onus, etc. (Paris, 1867) : FouillC, Deseu•trs (Paris, 1893) ; Fischer, Geschichte (1, r ncucrn. Philosophic. Vol. (Heidelberg. 1897) P,ontroux. 7;i/tory/nation it les mathnnatiql«'s scion Descartes (Paris, 1900).