Geometry

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 14, 2026 Read

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 14, 2026 Read

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 11, 2026 Read

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 14, 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

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

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 11, 2026 Read

Sine, leg and hypotenuse: trigonometry and its lost names

Almost every student learns to say “sine”, “cosine”, “leg” (or cathetus) and “hypotenuse” without knowing what these words mean. They are labels to memorise, like street names in a strange city. But each of them was once a description, and some hide unexpected stories. The sine, for example, is the result of a translation that went wrong. Pythagoras in Latin The most famous theorem of geometry was for a long time a sentence about real squares, drawn on the sides of a triangle. The old texts on this site wrote it in the medieval manner:

October 13, 2026 Read

Euclid: the figure and the proof

Around 300 BC, in Alexandria, Euclid gathered Greek geometry into thirteen books, the Elements. For more than two thousand years it was the most widely read mathematics book in the world, and its first proposition shows how it works: On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, book I, proposition 1 The recipe is short. Let $AB$ be the given line. With centre $A$ and radius $AB$, draw a circle; with centre $B$ and the same radius, draw another. Call $C$ a point where the two circles cut each other. Then $AC=AB$, because both are radii of the first circle, and $BC=AB$, because both are radii of the second. Things equal to the same thing are equal to each other, so $AC=BC=AB$, and triangle $ABC$ is equilateral.

October 13, 2026 Read

FUVEST 2022 – 1st phase: Mathematics solved

Source: FUVEST – General Knowledge test, version V and rectified official template of the 1st phase (applied in 12/12/2021). The 1st phase of FUVEST 2022 had 90 questions from all subjects. This post brings the 12 mathematics questions, which in test V are questions 30 to 41. In the others versions (K, Q, X and Z) the same questions appear with different numbering and another order of alternatives, so always check the version of your test.

September 26, 2026 Read

Right Triangles

A triangle is a three-sided polygon, in which we use straight segments for the sides. It is a particular case of quadrilaterals, as it is a quadrilateral divided by 2. Vertices: can be defined as meeting points between the sides of a triangle. Generally speaking, vertices are meeting points between sides of any polygon. In particular, a triangle is formed by 3 vertices. Diagonal: is the straight line segment that connects two non-consecutive vertices of any flat and spatial polygon. One of the important properties of triangles is that they have diagonals. An important observation is that there is no triangle that has two non-consecutive vertices. The diagonal of a quadrilateral, for example, is one of the sides of a triangle.

May 4, 2024 Read