La Géométrie

La Géométrie was published in 1637 as an appendix to Discours de la méthode (Discourse on the Method), written by René Descartes. In the Discourse, Descartes describes his method for finding clear answers to any question. La Géométrie and two other appendices, La Dioptrique (Optics) and Les Météores (Meteorology), were included to show examples of what could be achieved using his method.

This important work was the first to suggest joining algebra and geometry into one subject. It introduced an idea called analytic geometry, which means using algebra to solve geometry problems. This was a new and exciting idea at the time. The work helped shape the ideas of later mathematicians like Leibniz and Newton and played a role in the development of calculus.

The text

This appendix is split into three parts.

In the first part, called Problems Which Can Be Constructed by Means of Circles and Straight Lines Only, Descartes introduced new ways to write math problems. He used letters like x, y, z for unknown numbers and a, b, c for known numbers. He also introduced a way to show powers, like squares and cubes, using modern notation. This helped him link numbers to lengths that could be drawn with simple tools like a ruler and compass. Most of this part shows how to solve special geometry problems suggested by Pappus.

The second part, On the Nature of Curved Lines, talks about different types of curves. Descartes described curves that can be defined by equations with two variables and called them "geometrical." He also mentioned other curves, like the quadratrix and spiral, which he called "mechanical" and thought were not suitable for detailed study. He showed a way to find important points on these curves using algebra.

The third part, On the Construction of Solid and Supersolid Problems, focuses more on solving equations. Descartes suggested a way to rearrange equations to make them easier to solve and talked about roots of equations, including negative and imaginary ones. He also shared a rule to estimate the number of positive roots in an equation.

Aftermath

Descartes wrote La Géométrie in French instead of Latin, which was commonly used for scholarly works at the time. His writing style was not very clear, and the material was not organized in a systematic way. He often only gave hints of proofs, leaving many details for readers to figure out on their own. Descartes explained that he left out much of the information so that others could enjoy discovering it themselves.

Although Descartes is often credited with creating the coordinate plane, the modern rectangular coordinate system does not actually appear in La Géométrie. Later mathematicians worked to clarify and explain Descartes' ideas. This work was mainly done by Frans van Schooten, a mathematics professor, and his students. Van Schooten published a Latin version of La Géométrie in 1649, followed by several more editions. These editions included extra explanations and examples, helping to establish analytic geometry in the seventeenth century. One of van Schooten's students, Johannes Hudde, introduced a simpler method for finding double roots of polynomials, known as Hudde's rule.