History of Mathematics

Image credits for Lagrange

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Joseph-Louis Lagrange, engraving by an unknown artist: source page · public domain. Medallion with Lagrange’s profile, bronze by Pierre-Jean David d’Angers, Musées d’Angers: source page · CC0. Title page of the Méchanique analitique, Paris, 1788: source page · public domain. Statue of Lagrange in Turin, photo by Twice25, 2009: source page · CC BY-SA 3.0. Blog figures The cover, the diagram of the five Lagrange points and the graph of curves of equal area with the constraint line are original illustrations, with versions in each of the site’s languages. The diagram of the points is not to scale: L1 and L2 are in reality a hundred times closer to the Earth than the Sun is.

October 14, 2026 Read

Pascal's wager: a calculation, not a proof

Many people have heard that Pascal “proved that God exists”. He did not, and he said plainly that it could not be done. What he proposed is more curious: since reason cannot settle the question, change the question. Instead of “does God exist?”, ask “what is the better bet?”. And answer this second question with the mathematics he had just invented with Fermat, the mathematics of fair division. A sheet full of scribbles Autograph sheets of the Pensées, pasted into a large album after Pascal’s death, at an exhibition at the National Library of France in 2016. Pascal wrote on scraps of paper, in several directions, and crossed out a great deal.ActuaLitté · Wikimedia Commons · CC BY-SA 2.0 In the last years of his life, Pascal gathered notes for a book in defence of Christianity. He never wrote it. After his death, in 1662, his friends found hundreds of fragments, some cut out with scissors and strung together on threads, and published a selection in 1670 under the title Pensées (Thoughts).

October 11, 2026 Read

Fermat's legacy

The father of number theory Title page of the 1670 edition of Diophantus’s Arithmetica, printed in Toulouse “with the observations of P. de Fermat”. It was through this book, published by his son five years after his death, that the margin notes reached Euler.Bernard Bosc, Toulouse (1670) · Wikimedia Commons · public domain Before Fermat, problems about whole numbers were seen as riddles, Diophantus’s puzzles. Fermat turned them into an area of mathematics, with deep questions and methods of its own. For almost a century hardly anyone followed him. It was Euler, from 1730 onwards, who read Fermat’s notes, proved many of them and made number theory a central branch of mathematics. Then came Lagrange, Legendre and Gauss.

October 11, 2026 Read

Image credits

Historical portrait The portrait of Leibniz painted by Christoph Bernhard Francke around 1695 comes from Wikimedia Commons and is in the public domain. We use the 960-pixel thumbnail provided by Commons, without editing. Source page. Public domain mark. Blog figures The cover and the sketch of the multiplication $23\times14$ are original illustrations. The sketch explains the calculation and does not reproduce the mechanism of the historical machine. The interactive figures for the eight bits, the terms of the product rule and the sums of the series for π were created for this trail. The examples in the text let you follow the calculations even without JavaScript.

October 11, 2026 Read

Image credits

Historical images The two historical images come from Wikimedia Commons and are in the public domain. We use the thumbnails provided by Commons, without editing. Portrait of Isaac Newton, Godfrey Kneller, 1689: source page · public domain. Title page of the Principia, 1687: source page · public domain. Blog figures The cover and the prism sketch are original illustrations. The interactive figures for the secant, the approximations of √2 and the orbits were created for this trail. They are teaching aids, not reproductions of manuscripts or simulations of Newton’s experiments.

October 11, 2026 Read

Systems of equations: the determinant and Gaussian elimination

To calculate orbits using least squares, Gauss needed to solve systems of equations with many unknowns. Two names that every student knows come from this story: the determinant and the Gaussian elimination. The determinant. Take a system with two equations: $$\begin{cases} a\,x + b\,y = e \\ c\,x + d\,y = f \end{cases} \quad\Longrightarrow\quad x = \frac{e\,d - b\,f}{a\,d - b\,c}, \qquad y = \frac{a\,f - e\,c}{a\,d - b\,c}.$$ It all depends on the number $ad - bc$ in the denominator: if it is not zero, the system has a single solution. This number “determines” the system.

October 10, 2026 Read

Descartes and the problem of cannons

Cannons and artillerymen in an old engraving.Euler’s Eyes Collection Descartes experienced war up close. In 1618 he served as a volunteer in the army of Maurice of Nassau and, according to his first biographer, Adrien Baillet, he was in 1628 at the siege of La Rochelle, led by Cardinal Richelieu. On the battlefields, a practical problem occupied the artillerymen: with what angle tilt the cannon to hit a target at a certain distance?

October 10, 2026 Read

Fermat's number that was not prime

Pierre de Fermat believed that all numbers of the form $2^{2^n} + 1$ were prime. For $n = 0, 1, 2, 3, 4$ works: 3, 5, 17, 257 and 65 537 are primes. The next one is $$2^{32} + 1 = 4\,294\,967\,297.$$ Testing all dividers by hand is impractical. In 1732, Euler showed that this number is divisible by 641. An elegant way to look at this uses two ways of writing 641:

October 10, 2026 Read

Image credits

Historical images and photographs All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, with no editing beyond compression. Turing at 16, passport photo attributed to Arthur Reginald Chaffin: source page · public domain. Turing in 1951, portrait by the Elliott & Fry studio: source page · public domain. King’s College chapel, photo by Martinvl: source page · CC BY-SA 4.0. The Bletchley Park mansion, photo by DeFacto: source page · CC BY-SA 4.0. A three-rotor Enigma, National Security Agency: source page · public domain. The bombe rebuild, photo by Mike Peel: source page · CC BY-SA 4.0. The Pilot ACE, photo by Philafrenzy: source page · CC BY 2.0. Replica of the Manchester Baby, photo by Parrot of Doom: source page · CC BY-SA 3.0. The Sackville Park statue, photo by David Dixon: source page · CC BY-SA 2.0. The skin of the giant pufferfish, photo by Chiswick Chap: source page · CC BY-SA 3.0. Turing’s patterns and the cheetah, figure by Philip Ball: source page · CC BY 4.0. Figures made for the site The cover, the interactive Turing machine and the reaction–diffusion simulation were made for this trail. The simulation uses the Gray–Scott model, computed on a 200 by 200 grid of cells, and contains no text, so it serves all the languages of the site. The interactive machine works in all four languages.

October 16, 2026 Read

Pascal's legacy

The heart has its reasons, which reason does not know. — Blaise Pascal, Pensées Pascal left few finished books. Almost everything he wrote on mathematics came out as loose sheets, in letters or after his death. Even so, few authors turn up in so many different places: in the triangle at school, in the weather forecast, in a car’s brakes, in an insurance calculation and even in the name of a programming language.

October 11, 2026 Read

Image credits

Historical portrait The portrait attributed to Rolland Lefebvre comes from Wikimedia Commons and is in the public domain. We use the thumbnail provided by Commons, without editing, and the caption keeps the attribution given by the collection. Source page. Public domain mark. Blog figures The cover and the refraction sketch are original illustrations. The interactive figures for remainders of powers, the area of rectangles and the division of the interrupted game were created for this trail. They are teaching aids and do not reproduce Fermat’s manuscripts. The division figure, for example, assumes independent rounds and players with equal chances.

October 11, 2026 Read

Gaussian quadrature: integrate with two points

Quadratura circuli: approximating the area of the circle by polygons, the idea that leads to integral calculus.Euler’s Eyes Collection “Squaring” is the old name for calculating an area. The Greeks approximated the area of the circle by polygons with increasingly more sides, and integral calculus was born. The practical problem is: how to calculate the area under a curve doing few math?

October 10, 2026 Read