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* DESCARTES. 153 DESCENDANTS. guarantee of the proviously deterniinod ground of ii'rlitude, for God the perfect being cannot deceive, and therefore wliatever our conscious- ness clearly testitics may be implicitly believed. This Cartesian position that the truth of a projiosition is tested by its clear and distinct intelligibility is the principle of rationalism (q.v. ). One of the most fundamental general principles of the philosophical system of Descartes is the essential ditVcreiicc between spirit and matter — thinking and extended substances — a differ- ence so great, according to Descartes, that they can exert no inllucnce 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 icoiieiirsiis) on the part of God — a doctrine which gave rise subsequently to the system called occasionalism (q.v.K the principle of which was that body and mind do not really alTect each other. God 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 preestablished harmony of Leibnitz (q.v.). Tn Descartes's thought it re- sulted mciely in a strenuous insistence u]>on the ditfcrenees 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 r|)uscular philosophy — in which he endeavored to explain all the appearances of the material world simply by the motion of the Ultimate 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 from his name as Cartesian. Tn developing this branch of mathematics he had in mind, not the revolutionizing of geometry, but the elucidation of algebra by means of geometric intuition and concepts. He intended to estab- lish a universal mathematie. to which algebra, arithmetic, and geometry (with its applications) should be entirely subordinate. He discarded Victa'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-knowTi rtile for finding the limit of the num- ber of positive and negative 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 modem trend. The establishment of a correspondence between geometry and analysis has been of in- calcul,T,ble assistance to both, and Descartes's invention may he said to constitute the point of departure of modern mathematics. In 1C37 he ]>ublished at l.cydcn a treatise entitled Discours dc la mcthodc pour bicii conduire sa raison et ihcrclter la vcritc dans les sciences (Eng. trans. 1850: recent ed., Chicago, 1899). This was fol- lowed in the same year by three appendices en- titled, La dioptriijuc. I.cs nu'tcores, and La (iconu'lric. The new mathematical discipline was -set forth entirely in the (Icomrtric, 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 (which he calls mechanical) curves, explaining the use of coiirdinates. and setting forth tlic general scheme (now discard- 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 Geometric pliiced Descartes foremost among the mathematicians of his time. His works in Latin were published at Am- sterdam (1050). Modern editions are those by- Cousin (Paris, 1821-20). bv Garnier (Paris. 1834.35), by Aime ilartin (Paris, 1882). Eng- lish translations of portions of his works have been made by Veitch (Edinburgh. 1880), by I.o^vndes (London. 1878), and by Torrey (New York, 1892). The publication of his complete works was begun luider the auspices of the French itinister of Public Instruction in 1897. Consult: Jlahatfy, Dcscarfcs (Edinburgh, 1880); Millet, Descartes, sa vie, ses traeatix, etc. (Paris, 1867) : Fouillc, Descartes (Paris, 1893) ; Fischer, Geschichte der ncucrti Philosophie, vol. i. (Heidelberg. 1897) ; Bontroux. L'imnijination et les mathcmatiques scion Descartes (Paris, 1900). DESCARTES'S RULE OF SIGNS. Through the early attempts to determine the nature and value of the roots of numerical algebraic equa- tions (q.v.), many useful properties were dis- covered. Among these is Descartes's rule of signs, according to which no equation can have more positive roots than it has changes of sign from + to —, and from — to +. in the terms of the first member. By substituting ( — x) for (x) in f{x) = 0, the law may be applied to the case of negative roots. It is often possible to detect the existence of imag- inary' roots in equations by the application of this rule; for if it should happen that the sum of the greatest jmssible number of ])ositive roots, added to the greatest possible number of negative roots, is less than the degree of the equation, we are sure of the existence of imaginary roots. The rule also bears Harriot's name, being given in his Artis Anahitica' I'raxis (London. l03I). Consult: Burnsidc and Panton, Theoni of Equa- tions (Dublin, 1,8991901): Matthiessen, Grund- siige der aiitil;cn and modrrnen Alfiebra der lit- teralen Gleichunfirn (2d ed., Leipzig, 1890). DESCENDANTS (from Lat. descendcre. to de-s(ciic|. from <?', ilown + scandere, to climb. Skt. slcnnd. to spring). The issue of an individual, including all who have proceeded or descended from the issue of his body in any generation or degree. They take precedence over collatenil relatives and ascendents (q.v.) in the inheritance