History of Mathematics

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 16, 2026 Read

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

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

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

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 13, 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 13, 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 12, 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 12, 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 11, 2026 Read

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

Infinite descent: a smaller solution leads to a contradiction

Imagine a staircase where each step has a positive whole number. You can go down: 10, 7, 4, 2, 1… but at some point the staircase ends. There is no infinite sequence of ever-smaller positive integers. It seems obvious. Fermat turned this obvious fact into one of the most powerful tools in number theory: infinite descent. How it works To prove that an equation has no solution in positive integers:

October 7, 2026 Read

Calculemus: the idea of calculating with reasoning

The only way to rectify our reasonings is to make them as tangible as those of the mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: let us calculate!, without further ado, to see who is right. — Leibniz, The Art of Discovery (1685) Leibniz was a lawyer, diplomat and philosopher. He spent his life watching arguments that never ended, between religions, between countries, between schools of philosophy. A mathematician, on the other hand, does not argue about whether $2+2$ is 4: he does the sum. What if arguments could be calculated like numbers?

October 7, 2026 Read