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## DESCARTES, R.(1596-1650)

 Rene Descartes was borm mear Tours in 1596.  At the age of eight. he was sent to the Jesuit school at La Fleche.  It was there that he deceloped (at first because of delicate health)his lifelong habit of lying in bed till late in morning.  These meditative periods.  In 1612, Descartes left school and shortly after went to Paris, where, with Mersenne and Mydorge (see Section 10-6), he devoted some time to the study of mathematics.  In 1617, he commenced several years of soldiering by joining the army of Prince Maurice of Orange. Upon quitting military life, he spent four or five years traveling through Germany, Demnark, Holland, Switzerland, and Italy.  After resettling for a couple of years in Paris, where he continued his mathematical studies and his philosophical contemplations and where for a while he took up the construction of optical instruments, he decided to meve to Holland, than at the height of its power.  There he lived for twenty years, devoting his time to philiosophy, mathematics, and science.  In 1649, he reluctantly went to Sweden at the invitation of Queen Christina.  A few months later, he contracted inflammation of the lungs and died in Stockholm early in 1650.  The great philosopher-mathematician was entombed in Sweden, and efforts to have his rematins transported to France failed.  Then, seventeen years after Descarted' death, his bones, except for those of his right hand, were returned to France and reinterred in Paris.  The bones of the right hand were secured, as a souvenir, by the French Treasurer-General who had arranged the transportation of the bones.     It was during his stay of twenty years in Holland that Descartes accomplisheed his writing.  He spent the first four years writing Le monde, a physical account of the universe, but this was prudently abandoned and left incomplete when Descartes heard of Galileo's condemnation by the Church.  He turned to the writing of a philosophical treatise on universal science under the litle of Discours de la methode pour bien conduire sa raison et chercher la verite dans les sciences(A Discourse on the Method of Rightly Conducting the Reason and Seeking Truth in the Sciences); this was accompanied by three appendices entitled La dioptrique, Les metieores, and La geometrie.   The Discours, with the appendices, was published in 1637; it is in the last of the three appendices that Descartes' contributions to analytic geometry appear.     La geometrie, the famous third appendix of the Discours, occupies about one hundred pages of the complete work and is itself divided into three parts.  It is the only mathematical writing published by Descartes.  The first part contains an explanation of some of the principles of algebraic geometry and shows a real advance over the Greeks.  To the Greeks, a variable corresponded to the lenght of some line segment, the produvt of two variables to the area of some rectangle, and the product of three variables to the volume of some rectangular parallelepiped.  Beyond this the Greeks could not go.   To Descartes, on the other hand, $x2$ did not suggest an area, but rather the fourth term in the proportion 1:x=x:$x2$, and as such is representable by an appropriate line length that can easily be constructed when x is known.  Using a unit segment, we can, in this way, represent any power of a variable, or the product of any number of variables, by a line length, and actually construct the line length with Euclidean tools when the values of the variables are assigned.     There are a couple of legends describing the initial flash that led Descartes to the contemplation of analytic geometry.  According to one story, it came to him in a dream.  On St.  Martin's Eve, November 10, 1616, while encamped in the army's winter quarters on the banks of the Danube, Descartes experienced three singularly vivid and coherent dreams that, he claimed, changed the whole course of his life.  The dreams, he said, clarified his purpose in life and determined his future endeabors by revealing to him "a marvelous science" and "a wondergul discovery."   Descarles never explicitly disclosed just what were the marvelous science and the wonderful discovery, but some believe them to have beem analytic geometry, or the application of algebra to geometry, and then the reduction of all science to geometry.  It was eighteen years later that he expounded some of his ideas inn his Discours.     Another story, perhaps on a par with the story of Isaac Newton and the falling apple, stys that the initial flash of analytic geometry came to Descartes when watching a fly erawling about on the ceiling near a corner of his room.  It struck him that the path of the fly on the ceilig could be described if only on knew the relation connecting the fly's distances from two adjacent walls.  Even though this second stroy maty be apocryphal, it has good pedagogic value.