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,
did not
suggest an area, but rather the fourth term in the proportion
1:x=x:,
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.
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