History of Mathematics

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 7, 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 7, 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 6, 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 1, 2026 Read

Descartes and the problem of cannons

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? A man in front of a board on the table.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection Two men next to the cannon and the bullets.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection An angel next to a balanced scale.Crop from the original engraving; improved sharpness for the Euler’s Eyes collection It is a relationship between two numbers: the distance to the target (what the gunner “hears”) and the angle of the gun. (what he “says”). Galileo showed in 1638 that, without air resistance, the bullet describes a parabola, and the range $R$ depends on the angle $\theta$ and the exit speed $v$:

September 30, 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:

September 30, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the files or thumbnails provided by Commons, reduced to at most 800 pixels wide, with no other editing. Portrait of Augustin-Louis Cauchy, lithograph by Rudolf Hoffmann (1856) after a photograph by Bertsch: source page · public domain. Oil portrait of Cauchy, around 1840, attributed to Jean Roller, Château de Sceaux: source page · public domain. Lithograph of Cauchy by Louis Grégoire and Deneux, around 1840, Dibner collection of the Smithsonian Libraries: source page · public domain. Bronze medallion by Pierre-Jean David d’Angers (1843): source page · CC0. Title page of the Cours d’analyse (1821), cleaned version (converted from GIF to JPEG): source page · CC0. Cauchy’s name on the Eiffel Tower, photo by Juanedc: source page · CC BY 2.0. Cauchy’s grave in the cemetery of Sceaux, photo by Antoine Rex: source page · CC0. Figures made for the blog The cover and the four interactive figures (bisection, the band of width ε in Cauchy’s criterion, the integral of 1/(1 + z²) around a circle and the stress vector on a rotating cut) are original illustrations, with text in each of the site’s languages.

October 31, 2026 Read

The speed of sound

Sound is a compression wave: the air is pushed, compresses, pushes the next air and decompresses. The speed of this wave depends on two properties of the air: how much it resists being compressed and how heavy it is. Newton’s formula In the Principia, of 1687, Newton calculated the speed of sound for the first time. Assuming that, when compressed, air obeys Boyle’s law, that is, that pressure is proportional to density, he obtained

October 30, 2026 Read

The legacy of Archimedes

Whoever understands Archimedes and Apollonius will admire less the discoveries of the greatest men of today. — Gottfried Wilhelm Leibniz The manuscripts What remains of Archimedes reached us by a thread. Almost all of his work depended on three Byzantine Greek manuscripts. Two of them, which scholars call codices A and B, were lost after being copied or translated in the Renaissance; the third is the palimpsest, which stayed hidden under a prayer book. In 1269 the Dominican William of Moerbeke translated the Greek texts into Latin, and through this translation Archimedes came to be read in European universities. The first printed edition of the Greek text came out in Basel in 1544.

October 29, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons; the Kirkwood etching was reduced in size and the conical refraction diagram had its colour palette reduced, with no other editing. Portrait of Hamilton, Library of Congress: source page · public domain. Oil portrait of Hamilton, unknown author: source page · public domain. Etching by J. Kirkwood, Wellcome Collection: source page · CC BY 4.0. Irish stamp of 1943: source page · public domain. The plaque on Broom Bridge, photo by Cone83: source page · CC BY-SA 4.0. Broom Bridge, photo by Johnflan: source page · CC BY-SA 4.0. Dunsink Observatory, photo by David Malone: source page · CC BY-SA 2.5. Trinity College Dublin, photo by JoachimKohler-HB: source page · CC BY-SA 4.0. Wordsworth on Helvellyn, painting by Benjamin Robert Haydon: source page · public domain. Internal conical refraction, by Thomas Preston and William Edward Thrift: source page · public domain. Replica of the Icosian game at the Matemateca IME-USP, Estes Objethos Atelier, photo by Alisson Ricardo: source page · CC BY-SA 4.0. Figures made for the blog The cover and the interactive figures for the pendulum’s phase space, rotations with quaternions and the Icosian game were made for this trail and work in all four languages of the site.

October 27, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, with no editing beyond compression. Von Neumann’s Los Alamos badge photo, Los Alamos National Laboratory: source page · free use with attribution to the laboratory. Von Neumann in New Mexico, Los Alamos National Laboratory: source page · free use with attribution to the laboratory. The plaque on his birthplace in Budapest, photo by Misibacsi: source page · CC BY-SA 3.0. The Lutheran gymnasium of Budapest, photo by VinceB: source page · CC BY-SA 3.0. Fuld Hall, Princeton, photo by Zeete: source page · CC BY-SA 4.0. Two programmers at the ENIAC, United States Army: source page · public domain. The title page of the EDVAC report: source page · public domain. The EDVAC at Aberdeen, United States Army: source page · public domain. Oppenheimer and von Neumann in front of the IAS machine, Jeremy Norman Collection of Images: source page · CC BY 4.0. Ulam with the FERMIAC, Los Alamos National Laboratory: source page · public domain. Ulam’s Los Alamos badge: source page · public domain. Figures made for the blog The cover and the interactive figures for the minimax theorem, the Monte Carlo method and merge sort were made for this trail and work in all four languages of the site.

October 27, 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 18, 2026 Read