History of Mathematics

The derivative and the integral

Two years after the Cours d’analyse, Cauchy published the Résumé des leçons données à l’École royale polytechnique sur le calcul infinitésimal (1823). In forty short lessons he rebuilt differential and integral calculus on the idea of limit. It is, in essence, the outline that calculus textbooks still follow today. The derivative For Lagrange, the derivative of $f$ was the coefficient of $i$ in the power series of $f(x + i)$. Cauchy defined it as a limit:

October 31, 2026 Read

Counting the prime numbers

The prime numbers seem to appear at random: 2, 3, 5, 7, 11, 13, then a gap until 17, 19, 23, another until 29. But seen from afar, they obey a rule. Call $\pi(x)$ the number of primes less than or equal to $x$. While still a teenager, Gauss noticed, looking at tables, that near a number $x$ the density of primes is about $\frac{1}{\ln x}$, and proposed the estimate $$\pi(x) \approx \operatorname{li}(x) = \int_0^x \frac{dt}{\ln t},$$the logarithmic integral. Nobody knew how to prove that it was good, nor how to explain the deviations.

October 31, 2026 Read

Classical probability and the Gaussian integral

Mathematical probability began with games of chance, in the letters between Pascal and Fermat in 1654, and gained its first great theorem with Jakob Bernoulli’s law of large numbers. Laplace turned it into a complete theory. His first work on the subject dates from 1774; the Analytic Theory of Probabilities, of 1812, gathered almost forty years of research, and the Philosophical Essay on Probabilities, of 1814, explained it all to the public without formulas.

October 30, 2026 Read

The Method and the palimpsest

Archimedes’s books present perfect proofs, but they do not say how he reached the results. For centuries, mathematicians suspected he had a secret method. They were right, and the method was found in the most unlikely place. The letter to Eratosthenes Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards, because their investigation by this method did not furnish an actual proof. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge. — Archimedes, The Method, letter to Eratosthenes

October 29, 2026 Read

The witch in statistics: the Cauchy distribution

Agnesi died in 1799 without imagining that her curve would have a second life, far from geometry. In the nineteenth century it reappeared in probability theory, as one of the most treacherous examples in statistics. A probability density We saw in The area under the curve that the total area under $y = \frac{1}{1 + x^2}$ is $\pi$. Dividing by $\pi$, the area becomes 1, and the curve becomes a probability density:

October 29, 2026 Read

Vectors and scalars: Hamilton's words

In a quaternion $q = a + bi + cj + dk$ there are two parts of different kinds. The number $a$ stays put on the line of real numbers; the part $bi + cj + dk$ points in a direction in space. In 1846 Hamilton named them: the first is the scalar, because it can be read off a scale, from less to more; the second is the vector, from the Latin vector, “carrier”, because it carries one point to another. Both words caught on, and today they are used by people who have never heard of quaternions.

October 27, 2026 Read

The Monte Carlo method: calculating by rolling dice

In 1946 the Polish mathematician Stanisław Ulam, von Neumann’s colleague at Los Alamos, was recovering from encephalitis and passing the time playing solitaire. He wondered what the probability was that a game would come out. Working it out by combinatorics was hopelessly complicated. But there was another way: play hundreds of games and count how many came out. With a computer, one could play thousands. Ulam told von Neumann the idea, and he saw its reach at once: the same trick served for the problem occupying them at Los Alamos, the path of neutrons in a bomb or a reactor, where each neutron is deflected, absorbed or multiplied at random. Instead of solving the equations, simulate thousands of neutrons, drawing lots for what happens to each one, and take averages. Nicholas Metropolis named the method after the Monte Carlo casino, where an uncle of Ulam’s used to gamble.

October 27, 2026 Read

Banburismus: weighing evidence in decibans

The bombe was fast, but not fast enough to test every rotor order of the naval Enigma. In 1940 Turing invented a pencil-and-paper method to rule out most of them before switching on the machines. Because the sheets used in the method were printed in the town of Banbury, his colleagues named it Banburismus. The method is statistical. And to devise it, Turing reinvented one of the central ideas of modern statistics.

October 18, 2026 Read

The legacy of John Napier

By shortening the labours, the invention of logarithms doubled, so to speak, the life of the astronomer. — Saying attributed to Pierre-Simon Laplace Few inventions spread so quickly. In less than ten years logarithms were in use in England, Holland, Germany and Italy. For three and a half centuries they were humanity’s main calculating tool. Kepler The first great user was Johannes Kepler, who saw Napier’s tables in 1617 and was delighted. In 1619 he wrote a letter of thanks to the Scottish baron, not knowing he had been dead for two years, and dedicated his Ephemerides of 1620 to him. Kepler computed his own tables of logarithms and used them in the Rudolphine Tables of 1627, the most accurate tables of planetary positions of the time, which confirmed the elliptical orbits.

October 17, 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

The heat equation and the regularity of De Giorgi and Nash

In 1900, at the International Congress of Mathematicians in Paris, David Hilbert presented a list of 23 problems that, in his view, would guide twentieth-century mathematics. Problem 19 asked, in simple terms: are the solutions of the “well-behaved” problems of the calculus of variations always smooth functions? Half a century later, the question was still open for an important class of equations. In 1956 Nash heard about it from a colleague and decided to attack it.

October 15, 2026 Read