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Hero Image
The cycloid: the toothache curve

Paint a dot on a bicycle tyre and push the bike in a straight line. The dot rises, makes an arch and comes back down to touch the ground, once for every turn of the wheel. The curve it traces is called the cycloid. In the 17th century it was the fashionable curve. The French called it the roulette. So many mathematicians quarrelled over discoveries about it that it earned a nickname: the Helen of geometers, after Helen of Troy, who started a war.

October 8, 2026 Read
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The legacy of John Nash

Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts, to add much to his or her previous achievements. However I am still making the effort. — John Nash, autobiographical note for the Nobel Foundation (1994) John Nash is the only person to have received both the Nobel Prize and the Abel Prize, the two highest honours in economics and in mathematics. The two halves of his work seem to have nothing in common: one is about human decisions, the other about surfaces and equations. What unites them is the style: attacking problems everyone thought impossible with ideas nobody had had.

October 15, 2026 Read
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The fifth postulate and the other geometries

Out of nothing I have created a strange new universe. — János Bolyai, letter to his father (3 November 1823) Euclid’s first four postulates are short and obvious: draw a line, extend it, draw a circle, all right angles are equal. The fifth is different: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side.

October 12, 2026 Read
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Pascal's principle: multiplying forces with water

An elephant and a woman in high heels: who sinks deeper into soft ground? It may be the woman. The elephant weighs fifty times more, but its weight spreads over four broad feet. Hers is concentrated on the thin tip of a heel. What sinks into the ground is not force but pressure: the force divided by the area it rests on. $$p=\frac FA.$$The unit of pressure is the pascal: $1\ \text{Pa}=1\ \text{N/m}^2$, a force of one newton spread over one square metre. It is a tiny pressure; at sea level the atmosphere exerts about $100{,}000$ Pa.

October 8, 2026 Read
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Image credits

Photographs and images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of John Nash, photo by Peter Badge: source page · CC BY-SA 3.0. John Nash in 2006, in Cologne, photo by Elke Wetzig: source page · CC BY-SA 3.0. Isometric embedding of the flat torus, Hevea project, uploaded by Romainbehar: source page · CC0. Blog figures The cover, the Hex board and the graph of the product of gains in bargaining are original illustrations, with versions in each language of the site. The board shows an invented game, in which red has completed its chain.

October 15, 2026 Read
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The legacy of Euclid

He studied and nearly mastered the six books of Euclid, since he was a member of Congress. — Abraham Lincoln, autobiographical note (1860) After the Bible, the Elements are one of the most copied, translated and printed books in history. Since the first printed edition, in Venice in 1482, there have been more than a thousand editions. For more than two thousand years, “studying geometry” meant “studying Euclid”. The first printed edition of the Elements, made by Erhard Ratdolt in Venice in 1482, in a Latin translation. The figures are in the margins, beside the text; an early reader filled the pages with notes.Folger Shakespeare Library · Wikimedia Commons · CC BY-SA 4.0 The many faces of a book This site’s old trigonometry course gathered some editions that changed the way Euclid was presented:

October 12, 2026 Read
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The weight of air: the Puy de Dôme experiment

For two thousand years scholars repeated a saying of Aristotle’s: nature abhors a vacuum. If you suck the air out of a straw, the liquid rises because nature will not let the space stay empty. It seemed to explain everything. But the well-diggers of Florence had a problem. Suction pumps could not raise water more than about 10 metres, however hard they tried. Why would nature’s horror of the vacuum stop at 10 metres?

October 8, 2026 Read
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Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. No portrait of Euclid was made in his lifetime; the images of him in circulation are inventions by much later artists. Oxyrhynchus papyrus with Proposition II.5, c. AD 100, University of Pennsylvania Museum: source page · public domain. Detail of the School of Athens, by Raphael, 1509–1511: source page · public domain. Proposition I.47 in Oliver Byrne’s edition, London, 1847, scan by Mirko Junge: source page · public domain. First printed edition of the Elements, Erhard Ratdolt, Venice, 1482, Folger Shakespeare Library: source page · CC BY-SA 4.0. Arabic edition of the Elements, version of Nasir al-Din al-Tusi, Rome, 1594, Houghton Library: source page · public domain. Blog figures The cover, the construction of the equilateral triangle, the figure for Pythagoras’s theorem and the one for completing the square are original illustrations, with versions in each of the site’s languages. The completing-the-square example, $x^2+10x=39$, and the exercise $x^2+8x=65$ come from this site’s old course on the quadratic formula, whose figures were redrawn; the examples 3, 4, 5 and 6, 8, 10 and the table of editions come from the old geometry and trigonometry courses.

October 12, 2026 Read
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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
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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 8, 2026 Read
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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 7, 2026 Read
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Pascal image credits

Historical images All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Blaise Pascal, unknown artist, around 1690, Versailles: source page · public domain. Yang Hui’s triangle, in Zhu Shijie’s book, 1303: source page · public domain. First page of the Triangulus arithmeticus, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Pascaline at the Musée des Arts et Métiers, photo by David Monniaux, 2005: source page · CC BY-SA 3.0. Marker of the experiment on the Puy de Dôme, Agence Rol, 1923, Bibliothèque nationale de France: source page · public domain. Plaque of the grave in Saint-Étienne-du-Mont, Agence Meurisse, 1923, Bibliothèque nationale de France: source page · public domain. Autograph manuscript of the Pensées, exhibition at the Bibliothèque nationale de France, photo by ActuaLitté, 2016: source page · CC BY-SA 2.0. First page of the Pensées, counterfeit of the 1670 edition, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Blog figures The cover, the fountain of bowls, the carry diagram and the cycloid arch are original illustrations. The controls for the triangle, the split of the stakes and the hydraulic press were made for this trail. They are teaching aids, not reproductions of manuscripts or of the internal mechanism of the Pascaline.

October 8, 2026 Read
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