Other contributions and legacy

Other contributions

Optics: in addition to the law of refraction, he studied the ideal shape of lenses, which should have a profile of ellipse or hyperbola to focus light on a point.
Descartes’ Folio: the curve $x^3 + y^3 = 3axy$, which he proposed as a challenge to Fermat to find tangent lines.
Physics: formulated a version of the principle of inertia, that a body in motion continues in straight line if nothing gets in your way.
Philosophy: the Meditations and the Principles of Philosophy made it the starting point of modern philosophy.

Descartes lived between France, Holland and Sweden, and wrote the Discourse on the Method in French to be read by everyone.
He created analytical geometry: points became pairs $(x, y)$ and figures became equations.
He showed how to construct $\sqrt{a}$ with a ruler and compass and fixed the notation $x$, $y$, $z$ and $x^3$.
The rule of signs tells how many positive roots a polynomial can have.
The sum of the vertex defects of a polyhedron is 720°, which leads to the formula $V - A + F = 2$.
He explained the rainbow: the rays accumulate at 42° from the point opposite the Sun.
With it comes the idea of function: a rule that “listens” to one number and “speaks” another. The word comes from Latin functus, and the symbol $f(x)$ is Euler (1734).
There is no $1 \div 0$: close to zero, $\frac{1}{x}$ runs away to $+\infty$ on one side and to $-\infty$ on the other.