History of Mathematics

Why does the rainbow appear at 42°

In The Meteors, Descartes explained the rainbow using the law of refraction, which he published in The Dioptric. When light passes from air to water, it bends $$\sin i = n \cdot \sin r,$$ where $i$ is the entry angle, $r$ the angle into the water and $n \approx \frac{4}{3}$ for the water. (The Dutchman Willebrord Snell had found the same law before, but did not publish it. That is why it is called Snell-Descartes law.)

September 30, 2026 Read

The Königsberg bridges and the birth of graphs

Königsberg at 1652, with the river Pregel and its islands. You can count the bridges.Merian-Erben · public domain The city of Königsberg (today Kaliningrad, Russia) was crossed by the Pregel River, which formed two islands. Seven bridges connected the four regions of the city. Residents wondered if it was possible take a walk across each bridge exactly once.

September 29, 2026 Read

The integral in the complex plane

At the end of the eighteenth century, Euler and Laplace computed hard real integrals with a daring trick: they replaced the real variable with a complex one, did the calculations and returned to the real. Laplace called this “the passage from the real to the imaginary”. The results checked out, but nobody could say when the trick was legitimate. It was to answer this question that Cauchy, at 25, submitted a memoir on definite integrals to the Academy in 1814. It was only published in 1827, and it was only the beginning.

October 31, 2026 Read

The Riemann hypothesis

In the middle of the 1859 paper on prime numbers, Riemann writes a sentence almost in passing. Speaking of a function linked to zeta, whose real roots correspond to the zeros of zeta on the critical line, he says: It is very probable that all the roots are real. One would of course wish for a rigorous proof of this; but I have, after some fleeting vain attempts, provisionally put aside the search for one, since it did not seem necessary for the next objective of my investigation. — Bernhard Riemann, On the Number of Primes Less Than a Given Magnitude (1859)

October 31, 2026 Read

The central limit theorem

Roll one die: each face has the same chance, and the graph of the probabilities is flat. Roll two and add: the sum 7 is the most likely, and the graph becomes a triangle. Roll ten, thirty, a hundred: the graph of the sum gradually takes the shape of a bell. The same happens if, instead of dice, we add coins, roulette wheels or measurement errors. This surprising fact is the central limit theorem.

October 30, 2026 Read

The spiral of Archimedes

In the treatise On Spirals, also dedicated to Dositheus, Archimedes studies a new curve, defined by a motion. If a straight line in a plane revolves at a uniform rate about one of its extremities, which remains fixed, and returns to the position from which it started, and if, at the same time, a point moves at a uniform rate along the line, starting from the fixed extremity, the point will describe a spiral in the plane. — Archimedes, On Spirals, definition

October 29, 2026 Read

A woman at the university: the speech of 1727 and the chair at Bologna

Oratio qua ostenditur artium liberalium studia a femineo sexu neutiquam abhorrere. — Title of Agnesi's speech, 1727 The Latin sentence means: “A speech in which it is shown that the study of the liberal arts is by no means unsuited to the female sex”. Agnesi was nine when she recited it before her father’s guests, in 1727. The text had been written in Italian by one of her teachers; she translated it into Latin and learned it by heart.

October 29, 2026 Read

The Cayley–Hamilton theorem

Every quaternion $q = a + v$, with scalar part $a$ and vector part $v$, satisfies a quadratic equation with real coefficients. Since $v^2 = -\lvert v \rvert^2$ (see vectors), $$q^2 = a^2 + 2av + v^2 = a^2 - \lvert v \rvert^2 + 2av,$$and so $$q^2 - 2a\,q + \lvert q \rvert^2 = 0, \qquad \lvert q \rvert^2 = a^2 + \lvert v \rvert^2.$$A quaternion is a root of a real polynomial of degree 2, just as a complex number $z$ is a root of $z^2 - 2\operatorname{Re}(z)\,z + \lvert z \rvert^2 = 0$. In the Lectures on Quaternions of 1853 Hamilton went further and showed that certain linear functions of quaternions also satisfy a polynomial equation of their own, which he used to compute their inverses.

October 27, 2026 Read

The biased coin that becomes fair

Suppose you have a crooked coin. It lands heads with a probability $p$ that you do not know, perhaps $0.7$, perhaps $0.55$. Can you use it for a perfectly fair, fifty-fifty draw? In the same 1951 paper in which he spoke of the “state of sin” of those who manufacture random numbers, von Neumann gave a short and surprising answer. The trick Toss the coin twice: if it comes up heads, tails, say “heads”; if it comes up tails, heads, say “tails”; if it comes up heads, heads or tails, tails, forget it and toss twice more. Why is it fair? The two tosses are independent, so

October 27, 2026 Read

The imitation game: can a machine think?

I propose to consider the question, “Can machines think?” — Alan Turing, “Computing Machinery and Intelligence” (1950) So begins the paper Turing published in 1950 in the philosophy journal Mind. In the very next sentence he gives up on the question. To answer it, one would have to define “machine” and “think”, and any definition would turn into an endless argument about words. Turing proposes replacing it with another question, which has the advantage that it can be tested.

October 18, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of John Napier, unknown artist, 1616, National Galleries of Scotland: source page · public domain. Title page of the Mirifici logarithmorum canonis descriptio, 1614: source page · public domain. A page of the 1614 tables: source page · public domain. Title page of Briggs’s Arithmetica logarithmica, 1624, photo by Phidus: source page · CC BY-SA 4.0. Napier’s bones in the Whipple Museum, photo by Marsupium: source page · public domain. Merchiston Tower, photo by Kim Traynor: source page · CC BY-SA 3.0. Slide rule, photo by Jacek Halicki: source page · CC BY-SA 4.0. Blog figures The map, the journey graph and the interactive figures (the two progressions, the two points, Briggs’s scale, the bones and the slide rule) are original and work in every language of the site. The map uses OpenStreetMap data.

October 17, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Galois at 15, drawing by an unknown artist: source page · public domain. Galois drawn by his brother Alfred, published in 1848: source page · public domain. The letter to Auguste Chevalier, 29 May 1832: source page · public domain. The Louis-le-Grand college, Paris, photo by Celette: source page · CC BY-SA 4.0. Blog figures The figure of the six symmetries of the triangle is an original illustration. The table in the post was checked by a program that composes the permutations of the corners. The figure has no text and serves every language of the site.

October 16, 2026 Read