## Descartes and His Coordinate System

## Descartes and His Coordinate System

# Descartes and His Coordinate System

Every time you graph an equation on a Cartesian coordinate system, you are using the work of René Descartes. Descartes, a French mathematician and philosopher, was born in La Haye, France (now named in his honor) on March 31, 1596. His parents taught him at home until he was 8 years old, when he entered the Jesuit college of La Flèche. There he continued his studies until he graduated at age 18.

Descartes was an outstanding student at La Flèche, especially in mathematics. Because of his delicate health, his teachers allowed him to stay in bed until late morning. Despite missing most of his morning classes, Descartes was able to keep up with his studies. He would continue the habit of staying late in bed for his entire adult life.

After graduating from La Flèche, Descartes traveled to Paris and eventually enrolled at the University of Poitiers. He graduated with a law degree in 1616 and then enlisted in a military school. In 1619, he joined the Bavarian army and spent the next nine years as a soldier, touring throughout much of Europe in between military campaigns. Descartes eventually settled in Holland, where he spent most of the rest of his life. There Descartes gave up a military career and decided on a life of mathematics and philosophy.

Descartes attempted to provide a philosophical foundation for the new mechanistic physics that was developing from the work of Copernicus and Galileo. He divided all things into two categories—mind and matter—and developed a dualistic philosophical system in which, although mind is subject to the will and does not follow physical laws, all matter must obey the same mechanistic laws.

The philosophical system that Descartes developed, known as Cartesian philosophy, was based on **skepticism** and asserted that all reliable knowledge must be built up by the use of reason through logical analysis. Cartesian philosophy was influential in the ultimate success of the Scientific Revolution and provides the foundation upon which most subsequent philosophical thought is grounded.

Descartes published various treatises about philosophy and mathematics. In 1637 Descartes published his masterwork, *Discourse on the Method of Reasoning Well and Seeking Truth in the Sciences.* In *Discourse,* Descartes sought to explain everything in terms of matter and motion. *Discourse* contained three appendices, one on optics, one on meteorology, and one titled *La Géometrie* (The Geometry). In *La Géometrie,* Descartes described what is now known as the system of Cartesian Coordinates, or coordinate geometry. In Descartes's system of coordinates, **geometry** and **algebra** were united for the first time to create what is known as **analytic geometry** .

## The Cartesian Coordinate System

Cartesian coordinates are used to locate a point in space by giving its relative distance from **perpendicular** intersecting lines. In coordinate geometry, all points, lines, and figures are drawn in a **coordinate plane** . By reference to the two coordinate axes, any point, line, or figure may be precisely located.

In Descartes's system, the first coordinate value (*x* -coordinate) describes where along the horizontal axis (the *x* -axis) the point is located. The second coordinate value (*y* -coordinate) locates the point in terms of the vertical axis (the *y* -axis). A point with coordinates (4, −2) is located four units to the right of the intersection point of the two axes (point O, or the origin) and then two units below the vertical position of the origin. In example (a) of the figure, point D is at the coordinate location (4, −2). The coordinates for point A are (3, 2); for point B, (2, −4); and for point C, (−2, −5).

The coordinate system also makes it possible to exactly duplicate geometric figures. For example, the triangle shown in (b) has coordinates A (3, 2), B (4, 5), and C (−2, 4) that make it possible to duplicate the triangle without reference to any drawing.

The triangle may be reproduced by using the coordinates to locate the position of the three **vertex** points. The vertex points may then be connected with segments to replicate triangle ABC. More complex figures may likewise be described and duplicated with coordinates.

A straight line may also be represented on a coordinate grid. In the case of a straight line, every point on the line has coordinate values that must

satisfy a specific equation. The line in (c) may be expressed as *y* = 2*x*. The coordinates of every point on the line will satisfy the equation *y* = 2*x*, as for example, point A (1, 2) and point B (2, 4). More complex equations are used to represent circles, ellipses, and curved lines.

## Other Contributions

*La Géometrie* made Descartes famous throughout Europe. He continued to publish his philosophy, detailing how to acquire accurate knowledge. His philosophy is sometimes summed up in his statement, "I think, therefore I am."

Descartes also made a number of other contributions to mathematics. He discovered the Law of Angular Deficiency for all **polyhedrons** and was the first to offer a quantifiable explanation of rainbows. In *La Géometrie,* Descartes introduced a familiar mathematics symbol, a raised number to indicate an **exponent** . The expression 4 × 4 × 4 × 4 × 4 may be written as 4^{5} using Descartes's notation. He also instituted using *x*, *y*, and *z* for unknowns in an equation.

In 1649, Descartes accepted an invitation from Queen Christina to travel to Sweden to be the royal tutor. Unfortunately for Descartes, the queen expected to be tutored while she did her exercises at 5:00 a.m. in an unheated library. Descartes had been used to a lifetime of sleeping late, and the new routine was much too rigorous for him. After only a few weeks of this regimen, Descartes contracted pneumonia and died on February 11, 1650.

see also Computer Animation; Coordinate System, Polar; Coordinate System, Three-Dimensional; Global Positioning System; Navigation.

*Arthur V. Johnson II* and

*J. William Moncrief*

## Bibliography

Boyer, Carl B. *History of Mathematics.* New York: John Wiley & Sons, Inc., 1968.

Burton, David. *The History of Mathematics: An Introduction.* New York: Allyn and Bacon, 1985.

Eves, Howard. *In Mathematical Circles.* Boston: Prindle, Webber & Schmidt, Inc., 1969.

Johnson, Art. *Classic Math: History Topics for the Classroom.* Palo Alto, CA: Dale Seymour Publications, 1994.

### Internet Resources

*MacTutor History of Mathematics Archive.* University of St Andrews. <http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians.html>.

## GEOMETRY AND THE FLY

Some mathematics historians claim it may be that Descartes's inspiration for the coordinate system was due to his lifelong habit of staying late in bed. According to some accounts, one morning Descartes noticed a fly walking across the ceiling of his bedroom. As he watched the fly, Descartes began to think of how the fly's path could be described without actually tracing its path. His further reflections about describing a path by means of mathematics led to *La Géometrie* and Descartes's invention of coordinate geometry.

## WHO USES COORDINATES?

The system of coordinates that Descartes invented is used in many modern applications. For example, on any map the location of a country or a city is usually given as a set of coordinates. The location of a ship at sea is determined by longitude and latitude, which is an application of the coordinate system to the curved surface of Earth. Computer graphic artists create figures and computer animation by referencing coordinates on the screen.