History of Mathematics

Napier's two points: the definition of the logarithm

Napier did not have the exponent notation we use today. To define logarithms he used an image of motion, long before Newton and Leibniz created the calculus. The definition Imagine two points that start together. The first, $P$, moves along a segment $AB$ of length $10^7$, and its speed at each instant equals the distance still left to $B$: it starts fast and keeps slowing down, never arriving. The second, $Q$, moves along an endless line, always at $P$’s initial speed.

October 17, 2026 Read

What a group is: the six symmetries of the triangle

Cut an equilateral triangle out of paper, number its corners 1, 2 and 3 and place it over its outline drawn on the table. In how many ways can you pick the triangle up and put it back so that it covers exactly the same outline? There are six. You can do nothing; turn it by $120°$ or $240°$; or flip the triangle over about one of its three axes, each passing through a corner and the midpoint of the opposite side.

October 16, 2026 Read

The binomial series and the devil's series

Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever. — Niels Henrik Abel, letter to Holmboe (January 1826) In 1665 the young Newton discovered that the binomial formula also held for exponents that are not whole numbers: $$(1+x)^m=1+mx+\frac{m(m-1)}{2}x^2+\frac{m(m-1)(m-2)}{6}x^3+\cdots$$For a positive integer $m$ the sum stops after $m+1$ terms. For $m=\frac12$ or $m=-1$ it never stops: it is a series, a sum with infinitely many terms. Newton, Euler and all the mathematicians of the eighteenth century used this series with enormous success. Almost nobody asked when it works.

October 16, 2026 Read

The Nash equilibrium: nobody does better alone

An equilibrium point is an n-tuple such that each player’s mixed strategy maximizes his payoff if the strategies of the others are held fixed. — John Nash, Proceedings of the National Academy of Sciences (1950) In 1944 John von Neumann and the economist Oskar Morgenstern published Theory of Games and Economic Behavior, a huge book that turned strategic decisions into mathematics: card games, wars, markets. Von Neumann’s great result applied to two-player zero-sum games, in which what one wins the other loses, like chess or poker. But most of life is not like that: in a trade deal, a negotiation or traffic, both sides can win together, or lose together.

October 15, 2026 Read

Saturn's rings cannot be solid

The only system of rings which can exist is one composed of an indefinite number of unconnected particles, revolving round the planet with different velocities according to their respective distances. — James Clerk Maxwell, On the Stability of the Motion of Saturn's Rings (1859) Galileo looked at Saturn through his telescope in 1610 and thought the planet had “ears”. In 1655 Huygens realised they were a thin, flat ring that does not touch the planet. Two hundred years later, nobody knew what the ring was made of. Was it a solid disc, like a vinyl record? A liquid? A cloud of stones?

October 14, 2026 Read

The Analytical Mechanics: a book without figures

Open Newton’s Principia of 1687 and you will see figures on almost every page: triangles, ellipses, arrows, tangents. Open Lagrange’s Analytical Mechanics, published a century later in 1788, and you will see none. The author warns of this himself in the preface, with pride, and adds: Those who love analysis will be pleased to see mechanics become a new branch of it, and will be grateful to me for having thus extended its domain. — Joseph-Louis Lagrange, preface to the Analytical Mechanics (1788)

October 13, 2026 Read

L'Hôpital's rule: the lessons sold to a marquis

I acknowledge that I owe much to the insights of Messrs Bernoulli, above all to those of the younger, now professor at Groningen. I have made free use of their discoveries and of those of Mr Leibniz. — Marquis de l'Hôpital, preface to the Analysis of the Infinitely Small (1696) In 1691 the Marquis Guillaume de l’Hôpital was 30 years old and had a title, a castle and a passion: to understand Leibniz’s calculus, which had barely been published and which almost nobody in France could read. When the young Swiss Johann Bernoulli, who knew the calculus better than almost anyone, turned up in Paris, the marquis hired him as a private tutor.

October 13, 2026 Read

Sophie Germain's theorem and Fermat's Last Theorem

Around 1637 Pierre de Fermat noted in the margin of a book that the equation $$x^n+y^n=z^n$$has no solutions in positive whole numbers when $n$ is greater than 2, and that he had a marvellous proof which the margin was too narrow to contain. This is Fermat’s Last Theorem. Nobody ever found his proof, and the problem was only solved by Andrew Wiles in 1994. Until 1800 the situation was modest. Fermat had proved the case $n=4$, Euler the case $n=3$. It was enough to study prime exponents $n=p$, but each prime needed its own trick, and nobody could get past 3. Sophie Germain was the first person to prove something for infinitely many exponents at once.

October 12, 2026 Read

The first proposition: the equilateral triangle

On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, Proposition I.1 After the definitions, postulates and common notions, the Elements begin with a small and beautiful problem. You have a segment $AB$. You want to construct, with straightedge and compass only, a triangle whose three sides are equal to $AB$. It is the first of the book’s 465 propositions, and it shows in a few lines how all the others will work.

October 12, 2026 Read

Brownian motion: counting the atoms

If the motion discussed here can actually be observed, then classical thermodynamics can no longer be looked upon as strictly valid for spaces distinguishable under the microscope, and an exact determination of actual atomic dimensions is then possible. — Albert Einstein, Annalen der Physik (May 1905) In 1905 many physicists and chemists still doubted that atoms existed. Chemistry worked very well with the hypothesis, but nobody had seen an atom, and important scientists, such as the Austrian physicist Ernst Mach and the German chemist Wilhelm Ostwald, thought atoms were just a convenient way of doing calculations. Einstein found a way to see them, or almost: through their effect on things big enough to show up under a microscope.

October 11, 2026 Read

The harmonic series and the call for help

Even as the infinite series is contained by the finite sum, and in the unbounded a bound is discerned, so the soul of immensity dwells in the minute, and in the narrowest limits there are no limits. — Jakob Bernoulli, Tractatus de seriebus infinitis (1689) Between 1689 and 1704, Jakob Bernoulli published in Basel five dissertations on infinite series, sums with infinitely many terms. It was a new and dangerous subject. Some of these sums give a number, like

October 11, 2026 Read

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