History of Mathematics

When an equation can be solved by radicals

With the idea of the group of an equation, Galois answered the question left open after Abel: which equations can be solved by radicals? The answer does not depend on any trick. It depends only on the shape of the group. Each root cuts the group Solving an equation by radicals means adding roots, one at a time, to the numbers already known. At the start we know only the rational numbers, and the group of the equation is whole. Each time we add a root, such as $\sqrt3$ or $\sqrt[3]{5}$, new true relations appear, and fewer permutations respect them. The group shrinks. When the group becomes just the identity, all the roots of the equation are written with known numbers: the equation is solved.

October 15, 2026 Read

Mixed strategies: the penalty kick and odds and evens

Every finite game has an equilibrium point. — John Nash, Non-Cooperative Games, doctoral thesis (1950) Some games have no equilibrium at all if each player picks a fixed move. The simplest is odds and evens. Ana calls “evens”, Bruno calls “odds”, and each shows one or two fingers. If the sum is even, Ana wins a point; if it is odd, Bruno wins. Reducing each choice to “one” or “two” fingers, the payoffs are:

October 14, 2026 Read

Pythagoras's theorem, Euclid's way

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. — Euclid, Elements, Proposition I.47 Long before the Greeks, the surveyors of Egypt, who re-marked the fields after each Nile flood, were called rope-stretchers. The story goes that they used a rope with 12 equally spaced knots: stretched into a triangle with sides 3, 4 and 5, it gave a right angle. The Babylonians knew much more: a clay tablet from around 1800 BC, Plimpton 322, lists right triangles with whole-number sides, some of them huge, such as 3,367, 3,456 and 4,825.

October 14, 2026 Read

Sophie Germain primes and cryptography

A prime number $p$ is a Sophie Germain prime when $2p+1$ is also prime. 5 is one, because 11 is prime. 7 is not, because $15=3\cdot5$. The list begins: $$2,\ 3,\ 5,\ 11,\ 23,\ 29,\ 41,\ 53,\ 83,\ 89,\ 113,\ 131,\ldots$$The prime $2p+1$ that goes with each is called a safe prime: $5, 7, 11, 23, 47, 59, 83, \ldots$ Sophie Germain needed them for her theorem on Fermat. Two centuries later, they protect internet connections.

October 14, 2026 Read

Special relativity: a clock made of light

If I pursue a beam of light with the velocity c, the velocity of light in a vacuum, I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating. There seems to be no such thing, however. — Albert Einstein, Autobiographical Notes (1949) At 16, at school in Aarau, Einstein asked himself a question: what would I see if I ran alongside a beam of light, at the same speed? According to Newton’s physics, the light would seem to stand still, like a car beside ours on the motorway. But Maxwell’s equations do not allow light at rest: for them, light always travels at $c\approx300{,}000$ km/s. Someone was wrong. Einstein took ten years to find out who.

October 14, 2026 Read

The brachistochrone: the challenge of 1696

Given two points A and B in a vertical plane, to find the curve AMB along which a moving body M, descending under its own weight, passes from A to B in the shortest possible time. — Johann Bernoulli, Acta Eruditorum (June 1696) In June 1696, in the Acta Eruditorum, Johann Bernoulli challenged “the most acute mathematicians of the whole world”. The question sounds like something from a funfair: what is the shape of the fastest slide between two points? The name of the problem comes from Greek: brachistos, shortest, and chronos, time. Brachistochrone, the curve of least time.

October 14, 2026 Read

The Lagrange points: where gravity balances

On 25 December 2021 a rocket lifted off from French Guiana carrying the James Webb Space Telescope. It was not placed in orbit around the Earth, like Hubble. It travelled for a month to an empty point in space, 1.5 million kilometres from here, always on the side opposite the Sun, and stayed there. The address of that point comes from a prize-winning paper by Lagrange, from 1772. It is called L2.

October 14, 2026 Read

The marvellous spiral and the lemniscate

With Leibniz’s calculus in hand, the Bernoulli brothers went hunting for curves. For each one they asked: what is the tangent, what is the length, what is the area, what is the curvature? Many of these curves had been known since the Greeks, but only now could these questions be answered by a general rule. Two of them kept Jakob’s name forever. The spiral that does not change shape The logarithmic spiral had already been studied by Descartes and by Torricelli. In polar coordinates, where each point is given by its distance $r$ from the centre and by the angle $\theta$, it is

October 14, 2026 Read

The speeds of molecules: the Maxwell distribution

I have been occupied lately with some questions about the motions and collisions of perfectly elastic spheres, as an exercise in mechanics. — James Clerk Maxwell, letter to George Stokes (30 May 1859) A litre of air contains about $2.5\times10^{22}$ molecules, all moving and colliding billions of times a second. Following each one is impossible. In 1859 Maxwell had an idea that changed physics: not to ask what each molecule does, but what fraction of them has each speed. It was the first time probabilities had been used in a law of physics. He presented the result at a meeting in Aberdeen in 1859 and published it in 1860.

October 14, 2026 Read

The Chevalier de Méré's dice and the fair split

Antoine Gombaud, who styled himself the Chevalier de Méré, was a writer at the French court and a passionate gambler. Around 1654 he brought Pascal two questions about bets. Pascal found them so good that he wrote to Pierre de Fermat, and probability theory was born from those letters. About his friend, Pascal remarked to Fermat: He has a very good mind, but he is not a geometer; and that, as you know, is a great defect. — Blaise Pascal, letter to Fermat (29 July 1654)

October 11, 2026 Read

Fermat's little theorem: powers and remainders

Take the number 2 and raise it to larger and larger powers. Divide each result by 7 and note the remainder: Power $2^1$ $2^2$ $2^3$ $2^4$ $2^5$ $2^6$ Value 2 4 8 16 32 64 Remainder mod 7 2 4 1 2 4 1 At $2^6$ the remainder is 1. Now try 3 and the prime 5: $3^4=81$, which leaves remainder 1 when divided by 5. And 10 and the prime 13: $10^{12}=1{,}000{,}000{,}000{,}000$, which also leaves remainder 1 when divided by 13.

October 11, 2026 Read

The ∫ sign: from sum to accumulation

On 29 October 1675 Leibniz was in Paris working on a problem about areas. Until then he had written omn., short for the Latin omnia, “all”, to mean the sum of all the lines of a figure, as the Italian Cavalieri did. Halfway down the page he wrote: It will be useful to write ∫ instead of omn. — Leibniz, manuscript of 29 October 1675 The symbol $\int$ is a long S, for summa, sum. At the time the long s was an ordinary letter in handwriting. It has never been replaced since.

October 11, 2026 Read