History of Mathematics | Euler's Eyes

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.

October 10, 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 16, 2026 Read

Image credits

Historical images and photographs All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Abel, drawing by Johan Gørbitz, Paris, 1826: source page · public domain. First volume of Crelle’s Journal, Berlin, 1826: source page · public domain. Abel’s grave at Froland, photo by Karl Ragnar Gjertsen: source page · CC BY 4.0. Abel monument in Oslo, sculpture by Gustav Vigeland, photo by Wolfmann: source page · CC BY-SA 4.0. Blog figures The figure of the lemniscate divided into eight equal arcs is an original illustration. The points were computed with the formula $r=\sqrt{\sqrt2-1}$ derived in the post. The figure has no text and serves every language of the site.

October 15, 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 15, 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 15, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons; the largest were reduced to 1,200 pixels wide, with no other editing. Portrait of Einstein in 1921, possibly by Herman Mishkin, used by the Nobel Foundation: source page · public domain. Einstein around 1904, in Bern, photo by Lucien Chavan: source page · public domain. Einstein at the National Museum, Rio de Janeiro, 7 May 1925, unknown author: source page · public domain. Einstein in the Botanical Garden of Rio de Janeiro, May 1925, unknown author: source page · public domain. The instruments of the expedition in Sobral, 1919, photo by Charles Davidson: source page · public domain. Plate of the 1919 eclipse, from the report by Dyson, Eddington and Davidson: source page · public domain. Blog figures The cover, the diagram of the light clock and the 2,000-step random walk are original illustrations, with versions in each of the site’s languages. The light-clock diagram was redrawn from a figure in this site’s old course on the theory of relativity, which is also the source of the examples of the astronaut and of the cosmic-ray particles. The random walk was generated by computer, with randomly drawn directions.

October 14, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Johann Bernoulli, engraving in the Rijksmuseum collection: source page · CC0. Portrait of Guillaume de l’Hôpital: source page · public domain. Title page of the Analyse des infiniment petits, Paris, 1696: source page · public domain. Portrait of Daniel Bernoulli, engraving by Johann Jakob Haid after Johann Rudolf Huber: source page · public domain. The chain and the arch, drawings by Giovanni Poleni, Memorie istoriche della gran cupola del Tempio Vaticano, 1748: source page · public domain. Blog figures The cover and the diagram of the cycloid and the straight line between the same points are original illustrations, with versions in each of the site’s languages. The diagram is to scale: both axes use the same unit. The table of function notations, in the post on the legacy, was redone from a table in this site’s old course on functions.

October 14, 2026 Read

Lagrange multipliers: optimising under a constraint

Almost every real decision is a maximum or minimum problem with a constraint. A factory wants the greatest profit but has a limited amount of raw material. A can must hold 350 ml using as little aluminium as possible. A pendulum moves however it likes, as long as the bob stays attached to the string. Lagrange found a single method for all these problems. It appears in the Analytical Mechanics of 1788, to deal with the ties of mechanical systems, and in the Theory of Analytic Functions of 1797, as a general rule of calculation.

October 14, 2026 Read

Sophie Germain's legacy

Algebra is nothing but written geometry; geometry is nothing but drawn algebra. — Sophie Germain, General Considerations on the State of the Sciences and Letters (1833) Sophie Germain worked almost always alone, without a degree, a post or students. She had no access to the lectures, journals or discussions through which the mathematicians of her time learned from one another, and this shows in her work: highly original ideas, sometimes with gaps a teacher would have pointed out in time. Even so, she left her name in two fields that barely touch, number theory and the theory of elasticity.

October 14, 2026 Read

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 14, 2026 Read

The legacy of James Bernoulli

James Bernoulli lived only fifty years and published little in his lifetime, almost all of it in short papers in the Acta Eruditorum. Even so, few mathematicians left so many words, symbols and ideas that we still use every day: integral, Bernoulli trial, binomial distribution, Bernoulli numbers, law of large numbers, stochastic. The word integral Leibniz called the operation inverse to the derivative calculus summatorius, the calculus of sums, and wrote it with $\int$, an elongated S for summa. In May 1690, solving the isochrone problem, Jakob wrote that the two expressions had “equal integrals”. It was the first time the word integral appeared in print in a mathematical text. Johann adopted the term, and in 1696 Leibniz agreed to swap his “summatory calculus” for the Bernoullis’ integral calculus. The symbol stayed Leibniz’s; the name, Jakob’s.

October 14, 2026 Read

The legacy of James Clerk Maxwell

This change in the conception of reality is the most profound and the most fruitful that physics has experienced since the time of Newton. — Albert Einstein, at Maxwell's centenary (1931) Maxwell died at 48, little known to the general public. In his lifetime, hardly anyone outside a small group of British physicists accepted his field theory. Today anyone using a mobile phone depends on it dozens of times a second.

October 14, 2026 Read

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