Descartes was a philosopher and mathematician. In his Discourse on Method and his Rules for the Direction of the Mind (1628) he laid emphasis on deduction rather than on induction. In the subordination of particulars to general principles he experienced a satisfaction akin to the sense of beauty or the joy of artistic production. He speaks enthusiastically of that pleasure which one feels in truth, and which in this world is about the only pure and unmixed happiness.
At the same time he shared Bacon's distrust of the Aristotelian logic and maintained that ordinary dialectic is valueless for those who desire to investigate the truth of things. There is need of a method for finding out the truth. He compares himself to a smith forced to begin at the beginning by fashioning tools with which to work.
In his method of discovery he determined to accept nothing as true that he did not clearly recognize to be so. He stood against assumptions, and insisted on rigid proof. Trust only what is completely known. Attain a certitude equal to that of arithmetic and geometry. This attitude of strict criticism is characteristic of the scientific mind.
Again, Descartes was bent on analyzing each difficulty in order to solve it; to neglect no intermediate steps in the deduction, but to make the enumeration of details adequate and methodical. Preserve a certain order; do not attempt to jump from the ground to the gable, but rise gradually from what is simple and easily understood.
Descartes' interest was not in the several branches of mathematics; rather he wished to establish a universal mathematics, a general science relating to order and measurement. He considered all physical nature, including the human body, as a mechanism, capable of explanation on mathematical principles. But his immediate interest lay in numerical relationships and geometrical proportions.
Recognizing that the understanding was dependent on the other powers of the mind, Descartes resorted in his mathematical demonstrations to the use of lines, because he could find no method, as he says, more simple or more capable of appealing to the imagination and senses. He considered, however, that in order to bear the relationships in memory or to embrace several at once, it was essential to explain them by certain formulæ, the shorter the better. And for this purpose it was requisite to borrow all that was best in geometrical analysis and algebra, and to correct the errors of one by the other.
Descartes was above all a mathematician, and as such he may be regarded as a forerunner of Newton and other scientists; at the same time he developed an exact scientific method, which he believed applicable to all departments of human thought. "Those long chains of reasoning," he says, "quite simple and easy, which geometers are wont to employ in the accomplishment of their most difficult demonstrations, led me to think that everything which might fall under the cognizance of the human mind might be connected together in the same manner, and that, provided only one should take care not to receive anything as true which was not so, and if one were always careful to preserve the order necessary for deducing one truth from another, there would be none so remote at which he might not at last arrive, or so concealed which he might not discover."
REFERENCES
Francis Bacon, Philosophical Works (Ellis and Spedding edition), vol. IV, Novum Organum.