The friar and mathematician Marin Mersenne was Descartes’ great friend. It was through letters Mersenne that Descartes talked to the wise men of Europe, including Fermat. In the next generation, the Englishman John Wallis created the symbol $\infty$ for infinity in 1655. Marin Mersenne (1588–1648), Descartes’ friend and main correspondent.Euler’s Eyes Collection John Wallis (1616–1703), creator of the ∞ symbol.Euler’s Eyes Collection At school we learned a canon, a rule: you will never divide by zero. But it’s almost never explained why. Let’s build the thought, gradually reducing the divisor:
Other contributions Descartes’ vortices: for him, the planets were dragged by whirlpools of matter around each star.René Descartes, Principles of Philosophy · public domain 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. To remember 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. All topics by René Descartes · ← Infinity and division by zero · Image credits →
Images from Wikimedia Commons, all in the public domain: Portrait of Descartes: Frans Hals. Method Discourse, 1637. Princess Elisabeth of Bohemia: Gerard van Honthorst. Descartes and Queen Christina: Nils Forsberg, according to Pierre-Louis Dumesnil. Page Geometry e book I of Geometry: René Descartes, 1637. Rainbow Figure: René Descartes, The Meteors, 1637. Vortices: René Descartes, Principles of Philosophy. The images in the part “Descartes and the idea of function” are from the Euler’s Eyes collection. The figures interactives were made for this site.