Ted Urbanek

Ellsworth High School

Calculus

December 5, 2000

René Descartes

René Descartes was one of the greatest philosophers and mathematicians of his time. His works today are still very influential in the teachings of mathematics. He gave several laws and theories that to this day are still held true and proven. Without his discoveries in the field of analytical geometry, much of the advanced calculus that we perform today would not be possible.

René Descartes was born on March 31, 1596 in La Haye (later named Descartes in his honor), France. He was born to a well-to-do noble, and he was the second son in the family of 2 boys and a girl. His mother died shortly after his birth. He was a sickly, frail child, which affected him throughout his life.

Descartes entered a Jesuit "college" at the age of 8 to begin his formal education. Due to his frail and weakened condition, the head master of the school allowed Descartes to sleep late into the morning (around 11 o’clock), and he kept up this "tradition" until his death. Later, he stated that the quiet mornings to himself allowed him to think clearly and in depth; this is also where he began developing his theories on philosophy and mathematics.

By the age of 18, he began living the "gentlemen’s" life in Paris. It is also during this time period that he joined the army. In his leisure time, Descartes spent most of his time with mathematical studies. During the night of November 10, 1619, he had three dreams which inspired his new philosophies and ideas about analytical geometry. (During this time, he also developed "his first knowable fact," which pertained to his very existence, "Cogito ergo sum" (I think, therefore I am").) In 1621, he gave up the military life to pursue his philosophical and mathematical studies by traveling and studying abroad for 5 years.

In 1629, Descartes began living in Holland and writing his greatest works. The first of his writings Le Monde, tried to give a physical theory to the shape of the universe. The second of his great works was Discours de la méthode pour bien conduire sa raison et chercher la vérité dans les sciences. This piece contained three appendices "La Dioptrique," "Les Météores," and "La Géométrie."

In the latter of the aforementioned appendices, Descartes fully develops his theories of analytical geometry. It is in this work that Descartes creates the Cartesian plane, or the modern day coordinate graph. This discovery also came about with his astronomical hypothesis that everything in space lay on the same plane as the sun. Furthermore, he also develops equations for graphs and tangent lines (i.e.-y = mx + c) that are very fundamental today in understanding the principles of geometry and calculus. Along with the "new" equations, he performed studies on tangent and secant lines. Today, these developments in mathematics are useful in finding and applying the functions of calculus. Also in this book, he made use of the rules for finding the limit to the amount of negative and positive roots an algebraic equation has. This is known as Descartes’ Rule of Signs. Also, he was one of the first mathematicians to understand the benefits of having all of the variables of the equation on one side. Truly, this book is one of the most important works in math history.

Along with his workings in math, he also formulated ideas on the physical world. He hypothesized that the earth is not in a vacuum because he did not believe in action at a distance. He believed that forces must work through contact. Therefore, he believed that the matter in the universe was out there acting on other matter causing reactions.

Descartes studied in Holland until 1649 when 19 year old Queen Christina of Sweden asked and pleaded with him to go to Stockholm and teach her his theories of philosophy and mathematics. The Queen was a very demanding person. She would have Descartes break his rule of not awaking before 11:00 a.m. so that she could be tutored at 5 o’clock in the morning. Since Descartes had always been a frail man, it didn’t take too many trips in the cold early morning air to contract a fatal case of pneumonia. René Descartes died on February 11, 1650 in Stockholm, Sweden.

Descartes lived a life indulged in his love of mathematics. His belief that "only mathematics is certain, so all must be based on mathematics" led him come up with many important principles of mathematics that are still in use today. Without the discoveries and advances made by Descartes, modern day physics, geometry, and calculus would be impossible.

Bibliography

scidiv.bcc.ctc.edu/Math/Descartes.html (12-1-00)

www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Descartes.html

(12-1-00)

www.maths.tcd.ie/pub/HistMath/People/Descartes/RousBall/RB_Descartes.html (12-1-00)