# René Descartes

## Founder of Analytic Geometry

Descartes was born on March 31, 1596, in La Haye, France. His mother died when he was an infant, leaving him with sufficient funds to live comfortably for the rest of his life. His father, a judge, sent him to a school run by Jesuits at La Flëche, where he received a rigorous secondary education in languages, humanities, logic, philosophy, physics, and mathematics. Mathematics, with its deductive precision, was the only subject he truly enjoyed.

At age 16 Descartes went to Paris and became reacquainted with Father Marin Mersenne, an advocate of scholarship whom he had known at school. Together they spent two years investigating mathematics in quiet retirement. Descartes returned to school, earning a law degree from the University of Poitiers in 1616.

Deciding to experience the world firsthand, from 1617 to 1628 Descartes served as an army volunteer under Dutch and Bavarian command. He did not participate in battle; his time in the army was spent mostly in studies of philosophy and mathematics.

In 1628 Descartes quit the army; moved to Holland, and began 20 years of meditation and writing. The first six years were spent compiling a physics treatise that he did not publish because he feared the Catholic Church would condemn him for believing in the Earth’s movement around the Sun (he knew the church had forced GALILEO to recant the same view in 1632).

In 1637 Descartes published Discourse on Method, a philosophical work containing three scientific appendixes. The first two, “Dioptrics” and “Meteorology,” address the properties of light and atmospheric phenomena. The third, “Geometry;” represents the foundation of analytic geometry.

With “Geometry” Descartes became the first to represent geometric shapes on a coordinate system of two axes. He defined the positions of points in a plane by their distance from two fixed axes, one horizontal and one crossing diagonally (as opposed to the perpendicular axes used today); he was thus able to describe lines and curves with algebraic equations. He introduced the notation of x, y, and z to signify variables (unknown quantities) and a, b, and c to designate constants. He systematized the use of negative and imaginary roots and of exponential variables. Finally; he addressed numerous problems in algebra, particularly those relating to polynomial equations.

Descartes continued to write, and his Principles of Philosophy, appearing in 1644, included some general scientific theory. He advocated the search for a single mechanical description of all natural phenomena (within the fields of physics, chemistry, and physiology), to be based on mathematical methods.

In 1649 Descartes accepted an invitation to live in Stockholm, Sweden, to teach philosophy to Queen Christina. Unaccustomed to the icy climate and the earlymorning lessons demanded by the queen, he caught pneumonia and died on February 11, 1650.

**René Descartes’s Legacy**

Descartes was among the most influential figures in the history of western science. He initiated both the modern approach to mathematics and the systematic application of mathematical methods to other fields of science.

Descartes’s Discourse on Method, in which “Geometry” appeared, was written in French (rather than Latin) and thus had popular appeal. It was widely read during the author’s lifetime and was responsible for his fame as a philosopher and mathematician.

Descartes’s analytic geometry provided the tools for a revolution in mathematics. Analytic geometry made it possible to apply algebraic calculation—whose properties were fairly well understood—to geometrical shapes such as lines and curves. Descartes’s twoaxis system, named the Cartesian coordinate system by GOTTFRIED WILHELM LEIBNIZ in 1692, was the key to analytic geometry. The coordinate system is used in most branches of modern mathematics; it is the basis of all graphs and an important element in cartography. Descartes’s notation for constants and variables also survives to this day.

Analytic geometry’s main offshoot was the calculus, the use of algebra to study changing quantities. It was invented independently by ISAAC NEWTON in England and by Leibniz in Germany in the 1660s and 1670s. Descartes’s “Geometry” was the spark that began Newton’s interest in mathematics and it continued to inspire him as he worked out elements of the calculus. The calculus provided invaluable methods for solving problems in physics and astronomy and survives as a basic tool in higher mathematics and physics.

Descartes’s influence on Newton is also evident in the latter’s Principia, published in 1687. The work was a quintessential example of Descartes’s idea that nature should be studied by applying mathematical laws to physical phenomena. Principia laid the foundations of modern celestial mechanics and the mathematical physics of the following two centuries.

René Descartes – 1596-1650