History of Mathematics

Quantum mechanics in a Hilbert space

In 1925 and 1926 two versions of quantum mechanics were born almost at once. Werner Heisenberg’s used infinite matrices; Erwin Schrödinger’s, a wave equation. Both gave the same results in experiments, but they seemed to speak of different things, and physicists used tricks that mathematicians found unacceptable, such as Dirac’s delta “function”. In Göttingen, working with Hilbert, von Neumann showed that the two were the same theory written in different coordinates, and rewrote everything in a single language. The book Mathematical Foundations of Quantum Mechanics, of 1932, is still the reference for anyone who wants the theory with full rigour.

October 26, 2026 Read

Computable numbers: almost every number is impossible to calculate

The title of the 1936 paper does not talk about machines. It talks about numbers: “On computable numbers”. Turing defined them like this: a real number is computable when there is a machine that writes out, one after another, all of its digits, without ever getting any wrong. Who is computable Almost all the numbers that appear at school are computable. Fractions: $\frac17=0.142857142857\ldots$ The machine does the long division and repeats the cycle forever. Roots: $\sqrt2=1.41421356\ldots$ To find the next digit, the machine tests from 0 to 9 which is the largest that, when appended, still gives a number whose square does not exceed 2. $\pi$ and $e$: there are series that approximate them as closely as we like, such as $e=1+\frac1{1!}+\frac1{2!}+\frac1{3!}+\cdots$, and with them a machine produces any digit, sooner or later. It looks as if every “real” number is computable. And here comes the surprise: almost none is.

October 18, 2026 Read

Briggs and base-10 logarithms

These numbers were first invented by the most illustrious John Napier, Baron of Merchiston; they were then changed, following his own opinion, by Henry Briggs. — Title page of Henry Briggs's Arithmetica logarithmica (1624) Henry Briggs was professor of geometry at Gresham College in London and one of the greatest calculators in England. In March 1615 he wrote to a friend that Napier had set his “head and hands a work” with his new logarithms, and that he hoped to meet him that summer. He did. In the conversation at Merchiston the two agreed that the tables would be far more practical with two changes: the logarithm of 1 would be 0, and that of 10 would be 1.

October 17, 2026 Read

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 16, 2026 Read

Inverting the integral: elliptic functions

Start with a familiar function, seen in an odd way. The integral $$\arcsin x=\int_0^x\frac{dt}{\sqrt{1-t^2}}$$gives the length of an arc of a circle of radius 1, measured from the lowest point, as a function of the height $x$. It is a complicated function. But its inverse, which gives the height as a function of the arc, is the sine, the simplest function in trigonometry. And the sine has a property the integral hides: it is periodic. Since a quarter turn measures

October 16, 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 15, 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 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 13, 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 13, 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 12, 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 12, 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 11, 2026 Read