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René Descartes and La Géometrie

René Descartes was born on March 31, 1596, in La Haye, France. As a young man he travelled around Europe; he eventually settled in Holland, where he spent most of his life.

Descartes started out being interested in physics. But while he was writing his first book about physics, he heard that Galileo had been arrested for claiming that the Earth rotates around the sun. Descartes decided not to publish his book, and to concentrate on mathematics instead. But his friends wanted him to publish his ideas, so in 1637 Descartes published a book about methods of studying science called Discours de la méthod pour bien conduire sa raison et chercher la vérité dans les sciences, or Discourse on methods for conducting reason and seeking truth in the sciences. The Discourse had three appendices. One was about optics (light and lenses), one was about weather, and one was about mathematics.

Descartes' appendix on mathematics was called La Géometrie. Although its title means geometry, it focussed on the connections between geometry and algebra. Many modern algebraic conventions come from this book: for example, Descartes used letters from the beginning of the alphabet for constants and known quantities, and letters from the end of the algebra for variables. So Descartes is the reason we solve for x, and not some other symbol!

La Géometrie was divided into several books. The first book explained connections betweeen algebra and geometry, and gave both algebraic and geometric ways to solve equations. The second and third books investigated more complicated subjects including conic sections and the more general problem of plane loci, which involves finding curves at a fixed distance from different lines in a plane. These books also dealt with connections between algebra and geometry. For example, Descartes found general equations for certain kinds of parabolas and hyperbolas. Descartes also classified curves based on their algebraic representations: he put first- and second-degree equations in one category, third- and fourth-degree equations in another category, and so on.

Descartes used reference lines to analyze the curves he studied. He also used algebra to investigate curves. But Descartes still did not graph curves in the coordinate plane the way we do. Like Apollonius, he usually drew his curves first, then drew reference lines to analyze them with. Descartes often used reference lines that were tilted, not at right angles. Also, Descartes thought that negative numbers did not represent "real" physical quantities, so he ignored negative roots of equations, and he avoided measuring in more than one direction on a line whenever possible.

Descartes had gathered all the tools for coordinate graphing. Because of this accomplishment, he is often given credit for inventing the coordinate plane, even though he never graphed an equation.

La Géometrie was soon recognized as an important work of mathematics. But it still took several years for its ideas to become well known. The information moved slowly for two reasons. First, Descartes wrote his book in French, so scholars who didn't know French could not read it; and second, he discussed many difficult problems without giving examples or explaining simpler cases. Eventually, Descartes' friends and students published guides and commentaries for La Géometrie. Many of these guides were in Latin, since almost all educated people knew Latin at that time. The guides also gave examples and explained simpler problems.

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