The path to Analytic Geometry (Or a few of the many geniuses it took to learn 5th grade math)

Analytic geometry is the study of geometry using a coordinate system. Basically it’s the idea of expressing geometric objects such a as a line or a plane as an algebraic equation, think y=mx+b or ax+by+cz=k. This may be done by use of the more familiar Cartesian coordinates, by something such as polar coordinates or by just about any system for defining coordinates in a Euclidean space. The Common Core has the concept of graphing introduced in 5th grade, and graphing simple functions in the 8th grade. It’s quite interesting that something which took brilliant men so long to develop is now introduced to ten year olds.

The earliest evidence of anything resembling analytic geometry was by the Geek mathematician Menaechmus (380–320 BC), who was a student of Eudoxus and a tutor of Alexander the Great. Proclus and Eutocius both report that Menaechmus discovered the ellipse, hyperbola and parabola and that these were initially called the “Menaechmian triad”. These were used along with something resembling analytic geometry to solve the Delian problem, which is to, given the edge of a cube to construct the edge of a cube with double the volume. Though most of what we know of Menaechmus and his exact solution is second hand as his original work was lost, it appears as though he argued his solution for doubling the cube with proportions of a side length to the area of a side which fairly quickly leads to conics.

Another early manifestation of analytic geometry was by Omar Khayyám, whom we have mentioned in class. He drew a connection between algebra and geometry in his solution of general cubic equations. His idea to do this was to create a geometrical construction of a cubic equation by considering the variable to be the edge of a cube and constructing a set of curves from which a solution could be discerned. While it might seem far flung from Cartesian coordinates it was a significant leap in connecting the separate concepts of algebra and geometry.

Analytic geometry was more or less formalized in the early 17th century independently by René Descartes and Pierre de Fermat. Descartes published first and so he is commonly credited as the sole creator which leads to analytic geometry often being call Cartesian geometry. As Fermat has already been much discussed, I’ll skip his background and instead jump to Descartes. René Descartes was a French mathematician and philosopher who is most well known as the (co-)creator of analytic geometry and as the father of modern philosophy. He is the origin of the well-known quote “Je pense, donc je suis” or “I think, therefore I am” which appeared in in Discours de la methode (Discourse on the Method).

While the Fermat and Descartes constructions are equivalent, they did differ in several ways which primarily stem from which direction their creator worked. Fermat started with the algebraic equation and described the analogous geometric curve while Descartes worked in reverse, starting with the curve and finding the equation. To contrast the methods, the way most of us learn analytic geometry is much more similar to Fermat than to Descartes, where we learn to recognize that a degree 1 polynomial will represent a straight line then we learn how to find that line, next that quadratic function represents a parabola and so on. Whereas if we were to learn as Descartes’ work, we would take a straight line then learn that it represented a degree 1 polynomial which is similar to Fermat.  But then working further in this direction, it doesn’t make sense to jump to parabolas and instead to talk about conics and all degree 2 polynomials with no reason to talk specifically about parabolas.

In 1637, Descartes published his method of connecting arithmetic, algebra, and geometry in the appendix La géométrie (The Geometry) of Discourse on the Method. However, given Descartes’s opaque writing style (to discourage “dabblers”) as well as The Geometry being written in French rather than in the more common (for academic purposes) Latin, the book was not very well received until it was translated into Latin in 1649, by Frans van Schooten, with the addition of commentary clarifying certain arguments. Interestingly, though Descartes is credited with the invention of the coordinate plane, since he describes all necessary concepts, no equations are in fact graphed in The Geometry and his examples used only one axis. It was not until its translation into Latin that the concept of 2 axes was introduced in Schooten’s commentary.

One of the most important early uses for analytic geometry was to help prove the validity of the heliocentric theory of planetary motion, the (then) theory that the planets orbited around the Sun. As analytic geometry was one of the first methods one could use to actually make computations about curves, it was used to model elliptical orbits so as to demonstrate the correctness of this theory. Analytical geometry, and particularly Cartesian coordinates, were instrumental in the creation of calculus. Just consider how you might calculate something like the “area under the curve” without the concept of the curve being described by some algebraic equation. Similarly, the idea of rate of change of as function of time at a particular time becomes much clearer when thought of as the slope of the tangent line, but to do this, we need to think of the function as having some representation in the plane for which we need analytic geometry.

Sources :

Mathematics: Its Content, Methods and Meaning (Dover Books on Mathematics) Jul 7, 1999

by A. D. Aleksandrov and A. N. Kolmogorov