René Descartes: His mathematical legacy By Nicolas Synnott

René Descartes ( ) French philosopher, mathematician, scientist and writer Westerner/European (lived most of his adulthood in the Dutch Republic) 17th century (Early Modern period of Europe, Dutch Golden Age, Baroque cultural movement, French Grand Siècle, Scientific revolution, The General Crisis) Revolutionized philosophy and mathematics (Father of Modern Philosophy and first of the Modern School of Mathematics) Scientific method: foundation of all his works

Summarized Biography Born in conservative France Finished his studies in law Engaged as a soldier in diverse army Met Isaac Beeckman (Dutch philosopher and scientist; sparked his interest in mathematics and the New Physics) After several dreams, he quitted army career for the study of philosophy and mathematics Died in liberal Netherlands

The Mathematician Before Descartes, the different mathematical fields were considered as separated entities. Descartes had a pioneering approach: he merged geometry and algebra. Le Discours de la Méthode Explain everything in terms of matter and motion La Géométrie (one of three appendices) Presented analytic geometry Three main contributions Notation Cartesian coordinate system Analytic or Cartesian geometry (most important legacy)

Father of Standard Notation Letters at the beginning of the alphabet: data or known quantities (a, b, c) Letters at the end of the alphabet: variables or unknown quantities (x, y, z) The number of distinct roots of an equation is equal to the degree of the equation (ax+bx^2+cx^3=y) Use of superscripts to show powers or exponents (4x4x4x4x4=4 5)

Cartesian Coordinate System Descartes defined analytic geometry as “a way of visualizing algebraic formulas”. To do so, he first had to develop the coordinates system: (a “device to locate points on a plane”). Composition Two perpendicular lines: the vertical (y) and horizontal (x) axis The intersection of the two axis: the origin (0,0) Axis are both positive and negative 4 quadrants: (x,y), (-x,y), (-x,-y), (x,-y) Location of a point on the Cartesian coordinates: Indicated by its two coordinates (x,y) The coordinated are determined by how far perpendicularly is the point from each axis in term of units