History of Mathematics

Laplace's expansion of determinants

Determinants were born to solve systems of linear equations. In 1750 Gabriel Cramer published the rule that bears his name, which gives each unknown as a ratio of two expressions formed from the coefficients. But an organized way of computing those expressions when there were many unknowns was missing. In 1772, almost at the same time, Alexandre-Théophile Vandermonde and Laplace studied these expressions as objects in their own right. The word determinant, in its present sense, was fixed only by Cauchy in 1812, and the vertical bars we use today came from Arthur Cayley in 1841.

October 30, 2026 Read

The sphere and the cylinder

Greek geometers knew how to compute the volume of prisms, pyramids, cylinders and cones. The sphere resisted. In the two-book treatise On the Sphere and Cylinder, Archimedes solved the problem and found relations of surprising simplicity. The theorems Call a great circle the circle that passes through the centre of the sphere, like the equator. Archimedes proved that: the surface of the sphere is four times the area of the great circle: $S = 4\pi r^2$; the volume of the sphere is four times that of a cone whose base is the great circle and whose height is the radius: $V = 4 \cdot \frac{1}{3}\pi r^2 \cdot r = \frac{4}{3}\pi r^3$. And from this he drew the consequence that delighted him most. Consider the cylinder that exactly encloses the sphere, with radius $r$ and height $2r$. Its volume is $\pi r^2 \cdot 2r = 2\pi r^3$, and its total surface, counting both lids, is $2\pi r \cdot 2r + 2\pi r^2 = 6\pi r^2$. So

October 29, 2026 Read

The inverse method of tangents

Differential calculus answers a question: given a curve, what is its tangent at each point? The mathematicians of the seventeenth century soon realized that the inverse question was harder and more useful: given a property the tangents must have, what is the curve? This problem was called the inverse method of tangents, and it is the subject of the fourth and last book of the Instituzioni analitiche. Today we call it solving a differential equation.

October 29, 2026 Read

Rotating in space with quaternions

Complex numbers of modulus 1 rotate the plane. Hamilton hoped his numbers would do the same in space, and they do, but in a subtler way than he first imagined. The recipe Write a point of space $(x, y, z)$ as the pure quaternion $v = xi + yj + zk$. To rotate by an angle $\theta$ about an axis given by a unit vector $u = u_1 i + u_2 j + u_3 k$, take

October 27, 2026 Read

Merge sort: sorting by joining

Sorting a list is one of the tasks computers do most: names in a register, prices in a search, election results. In 1945, when the EDVAC was still on paper, von Neumann wanted to show that the machine in his report was good for work that was not just arithmetic, and wrote by hand a program to sort and merge lists. It is one of the first programs written for a stored-program computer, and the method he used, now called merge sort, is still one of the most widely used.

October 26, 2026 Read

The Enigma: 159 quintillion keys and one flaw

A three-rotor Enigma machine on display at the National Cryptologic Museum of the American security agency. At the top, the rotors; in the middle, the lamps; below, the keyboard; at the front, the plugboard.National Security Agency · Wikimedia Commons · public domain The Enigma looked like a typewriter inside a wooden box. The operator pressed a letter on the keyboard and another letter lit up on a panel of lamps. A second operator wrote down the letters that lit up, and that sequence was sent by radio, in Morse code. At the other end, an operator with an identical machine, set up the same way, typed in the enciphered message and read the original text on the lamps.

October 18, 2026 Read

The slide rule: logarithms that slide

A table of logarithms still requires looking numbers up, adding and looking up again. A few years after Napier, the English realised that the logarithms could be put on a ruler and the addition done with the hands. Gunter’s scale In 1620 the astronomer Edmund Gunter, Briggs’s colleague at Gresham College, drew a ruler on which each number $x$ stood at a distance proportional to $\log x$ from the mark for 1:

October 17, 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

Crumpling without stretching: the embedding theorem

Take a sheet of paper. You can roll it into a cylinder without tearing or stretching it: distances measured on the sheet, with a flexible ruler, stay the same. But you cannot wrap a ball with it without making folds. The sheet and the sphere have different intrinsic geometries, and Gauss proved in 1827 that this difference, the curvature, does not change when the surface is bent without stretching. In the nineteenth century Bernhard Riemann had an even bolder idea: to describe a surface, or a space of any dimension, by its internal measurements alone, without drawing it anywhere. All it takes is a rule, the metric, that says how to measure the length of any curve. This is how general relativity describes spacetime: from inside, without a larger space around it.

October 15, 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

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