Physics

Combinations: counting without listing

In Brazil’s Mega-Sena lottery you choose 6 numbers out of 60. How many different tickets are there? Listing them all would take a lifetime: there are more than fifty million. But you can count without listing, and Pascal showed how. Choosing without order Start with a small case: in how many ways can you choose 2 people out of a group of 5 to form a committee? If order mattered, there would be $5$ options for the first person and $4$ for the second: $5\cdot4=20$. But on a committee “Ana and Bruno” is the same as “Bruno and Ana”, and every pair has been counted twice. There are $20/2=10$ committees.

October 8, 2026 Read

Hamilton's mechanics

In 1788 Lagrange had turned Newton’s mechanics into a set of equations valid in any coordinates. Hamilton admired that book, which he called “a kind of scientific poem”. In 1834 and 1835, in the two essays On a General Method in Dynamics, he took the idea further, bringing into mechanics what he had learned from rays of light. Position and momentum Lagrange described a system by its positions $q$ and velocities $\dot q$. Hamilton replaced the velocity with the momentum $p$ (for a simple particle, mass times velocity, $p = m\dot q$) and put at the centre of everything a single function, the Hamiltonian $H(q, p)$, which in most cases is the total energy, kinetic plus potential:

October 27, 2026 Read

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

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

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

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 8, 2026 Read

Maxwell's equations: from Faraday's lines of force to vectors

I was at first almost frightened when I saw such mathematical force made to bear upon the subject, and then wondered to see that the subject stood it so well. — Michael Faraday, letter to Maxwell (25 March 1857) Until 1820, electricity and magnetism were separate subjects. Electricity made sparks and raised the hair on your head; magnetism made compasses point north. That year, during a lecture at the University of Copenhagen, the Dane Hans Christian Oersted connected a battery to a wire and saw a nearby magnetic needle move. Electric current produced magnetism.

October 14, 2026 Read

E = mc²: the energy hidden in mass

One more consequence of the paper on electrodynamics has occurred to me. The principle of relativity, together with Maxwell’s equations, requires that mass be a direct measure of the energy contained in a body. The idea is amusing and enticing; but for all I know, the Lord may be laughing over the whole thing and leading me around by the nose. — Albert Einstein, letter to Conrad Habicht (summer 1905) In September 1905, three months after the relativity paper, Einstein sent the Annalen der Physik a supplement of just three pages, with a title in the form of a question: “Does the inertia of a body depend upon its energy content?”. The answer was yes, and the conclusion, written in words and with other letters, is the most famous formula in physics:

October 12, 2026 Read

The Pascaline: a sum made of gears

In 1640 Étienne Pascal was appointed tax commissioner for Normandy. The job meant adding up column after column of figures late into the night, and his son Blaise, then 17, helped. To make matters worse, French money was not decimal: $$1\ \text{livre}=20\ \text{sols},\qquad 1\ \text{sol}=12\ \text{deniers}.$$Adding $3$ livres, $15$ sols and $8$ deniers to $2$ livres, $7$ sols and $9$ deniers needs three kinds of carry: deniers: $8+9=17$, which is $1$ sol and $5$ deniers; sols: $15+7+1=23$, which is $1$ livre and $3$ sols; livres: $3+2+1=6$. Result: $6$ livres, $3$ sols and $5$ deniers. Do that a thousand times a night and you will make mistakes. In 1642 Blaise began building a machine that would not.

October 8, 2026 Read

The displacement current: the missing term

Of Maxwell’s four equations, three were already known in some form before him: Gauss’s law, the magnetic law and Faraday’s law. The fourth, Ampère’s, was known too, but it was incomplete. What Maxwell added to it was a single term, $\mu_0\varepsilon_0\frac{\partial\mathbf E}{\partial t}$. It looks like little. It was this term that turned electricity and magnetism into light. Ampère’s law In 1826 the Frenchman André-Marie Ampère found the rule linking a current to the magnetic field around it. In today’s language: if you go round a closed path around a wire and add up the component of the magnetic field along the path, the result is proportional to the current $I$ crossing any surface bounded by that path:

October 14, 2026 Read

General relativity: from the lift to the Sobral eclipse

I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me: if a person falls freely, he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me towards a theory of gravitation. — Albert Einstein, on an idea of 1907 The relativity of 1905 had a flaw: it did not include gravity. Newton’s law says the Sun pulls the Earth instantly, across 150 million kilometres, and relativity says nothing travels faster than light. Einstein spent eight years trying to fix this. His starting point was what he called “the happiest thought of my life”.

October 12, 2026 Read

The mystic hexagram: six points on a conic

In February 1640 a sheet printed on one side only, like a poster, circulated in Paris under the title Essay on conics. It had no proofs, only definitions, three lemmas and a list of results. The author was 16 years old and his name was Blaise Pascal. When Father Mersenne showed the sheet to Descartes, the reaction was suspicion: Descartes thought the young man had taken everything from Girard Desargues, the architect and geometer Pascal admired. Desargues’s influence was real, and Pascal himself acknowledged it. But the central result on the sheet was new.

October 8, 2026 Read