Algebra

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

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

Quaternions: the formula on the bridge

After showing that complex numbers are couples of real numbers and that their multiplication rotates the plane, Hamilton wanted to do the same in space. He was looking for triplets $a + bi + cj$ with a multiplication that respected the law of the moduli: the size of the product should be the product of the sizes. He tried for years, above all in 1843, so much so that his children asked him every morning whether he could multiply triplets yet.

October 27, 2026 Read

Galois imaginaries: finite fields

In June 1830, aged 18, Galois published a short paper, On the theory of numbers, in a Paris scientific bulletin. In it he invented new numbers, which he called imaginaries, for a world in which only finitely many numbers exist. For more than a century the paper looked like a curiosity. Today it is inside every mobile phone. Counting with remainders In clock arithmetic we count only with the remainders of division by a fixed number. With the number 5 there are only $0, 1, 2, 3, 4$, and the sums “wrap around”:

October 16, 2026 Read

Dividing the lemniscate with ruler and compass

In 1694 Jakob Bernoulli studied a curve shaped like a figure eight lying on its side, which he called lemniscus, a ribbon bow. With its foci at the points $(\pm\frac1{\sqrt2},0)$, it is the set of points whose distances to the two foci have product $\frac12$. Its equation is $$(x^2+y^2)^2=x^2-y^2,$$or, in polar coordinates, $r^2=\cos2\theta$. A century and a half later Abel proved about it a theorem that seems to have come out of Euclid’s book: for which numbers $n$ can the lemniscate be divided into $n$ arcs of equal length using only ruler and compass?

October 16, 2026 Read

Four squares are enough

Take any positive whole number and try to write it as a sum of perfect squares, $0, 1, 4, 9, 16, 25, \ldots$, using as few terms as possible: $$\begin{aligned} 5&=4+1,\\ 6&=4+1+1,\\ 7&=4+1+1+1,\\ 30&=25+4+1,\\ 2026&=2025+1=45^2+1^2. \end{aligned}$$Some numbers are squares, others need two, others three. The number 7 needs four: there is no other way. Could some number need five? In 1770 Lagrange proved that no number does: Every positive whole number is a sum of four squares of integers.

October 14, 2026 Read

Cauchy's inequality

The 1821 Cours d’analyse ends with nine notes, a kind of appendix with results Cauchy did not want to put in the main text. The second, on inequalities, contains two gems that every olympiad student knows. The inequality Cauchy’s theorem says that, for any real numbers $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$, $$(a_1 b_1 + a_2 b_2 + \cdots + a_n b_n)^2 \le (a_1^2 + a_2^2 + \cdots + a_n^2)(b_1^2 + b_2^2 + \cdots + b_n^2),$$with equality only when the two lists are proportional.

October 31, 2026 Read

Doubling the cube and trisecting the angle: the impossible constructions

The Greeks left three construction problems that nobody managed to solve with ruler and compass, the tools of Euclid’s Elements: Doubling the cube: constructing the edge of a cube with twice the volume of a given cube. Trisecting the angle: dividing any angle into three equal parts. Squaring the circle: constructing a square with the same area as a given circle. Legend has it that, during a plague, the oracle of Delos told the Athenians to double the altar of Apollo, which had the shape of a cube. They doubled the edge, the volume became eight times larger, and the plague went on. For two thousand years mathematicians tried. The answer only came in the nineteenth century, with the ideas of Gauss and Galois: it is impossible.

October 16, 2026 Read

Abelian groups: when order does not matter

Putting on your sock and then your shoe does not give the same result as putting on your shoe and then your sock. Adding 3 and then 5 gives the same as adding 5 and then 3. When the order of two operations does not change the result, we say they commute. The word mathematicians use for structures in which everything commutes is a tribute to Abel: abelian. The last memoir After proving that the general equation of the fifth degree has no formula, Abel wanted to know which equations do have one. In 1829, in one of the last papers he published in Crelle’s Journal, he studied a special family: equations in which all the roots can be obtained from one of them, $x$, by rational functions $\theta_1(x), \theta_2(x),\dots$ that commute with each other:

October 16, 2026 Read

Shuffling the roots: Lagrange's theorem

In 1770 algebra had a problem that had been stuck for more than two centuries. There were formulas for solving equations of degree 2, the quadratic formula, and also of degrees 3 and 4, published by Cardano in 1545. For degree 5, nobody could find anything. The best mathematicians had tried, and every method that worked up to degree 4 jammed at 5. Lagrange did not try yet another method. He asked a different question: why do the old methods work?

October 14, 2026 Read

The legacy of Évariste Galois

This letter, if judged by the novelty and profundity of ideas it contains, is perhaps the most substantial piece of writing in the whole literature of mankind. — Hermann Weyl, Symmetry (1952) Galois left some sixty pages of mathematics, much of it written in haste. It took half a century for them to be understood. When they were, they changed the very idea of what mathematics is: instead of studying numbers and figures, studying the transformations that leave them unchanged.

October 16, 2026 Read

The legacy of Niels Henrik Abel

Abel has left mathematicians enough work for a hundred and fifty years. — Saying attributed to Charles Hermite Abel worked as a mathematician for less than eight years. In that time he settled the oldest question in algebra, brought rigour to series, created elliptic functions and wrote a memoir that was only fully understood decades after his death. His name is now found in more places in mathematics than that of almost anyone else: abelian groups, abelian integrals, abelian functions, abelian varieties, Abel’s theorem, Abel summation, Abel’s test.

October 16, 2026 Read