History of Mathematics

The life of René Descartes

Timeline 1596Born on March 31, in La Haye, in the Touraine region, France. 1607Enter the Jesuit school of La Flèche, one of the best in Europe. 1616He graduated in Law from the University of Poitiers. 1618He joins the army of Maurice of Nassau, in the Netherlands, and meets Isaac Beeckman. 1619On the night of 10 November, he has three dreams that he decides to follow as a mission. 1628He moves to the Netherlands, where he lives for about 20 years. 1633He gave up publishing The World upon learning of Galileo’s conviction. 1637Publishes the Method Discourse, with the appendix Geometry. 1641Publishes the Meditations on First Philosophy. 1643He begins exchanging letters with Princess Elisabeth of Bohemia. 1649He goes to Stockholm, at the invitation of Queen Christina of Sweden. 1650Dies in 11 February, in Stockholm. ◀ Previous Follow the journey Next ▶ 1596 La HayeBorn in a small town in Touraine, now called Descartes in his honour. 1607–1614 La FlècheHe studies at the Jesuit college of La Flèche, one of the best in Europe, and is allowed to stay in bed in the mornings. 1616 PoitiersHe graduates in law from the University of Poitiers. 1618 BredaHe joins the army of Maurice of Nassau and meets Isaac Beeckman, who challenges him with problems in physics and mathematics. 1619 near UlmOn a winter night, in a room heated by a stove, he has the three dreams he decides to follow as a mission. 1628–1649 HollandHe lives for about twenty years in the United Provinces, moving house many times. In Leiden he publishes the Discourse on Method (1637), with La Géométrie. 1649–1650 StockholmHe goes at the invitation of Queen Christina, who wants lessons at five in the morning. He dies of pneumonia on 11 February 1650. Enable JavaScript to see the map. The stages are listed below. Descartes’s path, from Touraine to Holland and, at the end of his life, to Sweden. Map: © OpenStreetMap contributors

September 30, 2026 Read

The life of Leonhard Euler

Timeline 1707Born on April 15, in Basel, Switzerland. 1720Enters the University of Basel, at 13 years old. He began studying with Johann Bernoulli. 1727Moves to the Academy of Sciences in Saint Petersburg, Russia. 1735Solve the Basel problem and analyze the Königsberg bridges. 1738He loses almost all vision in his right eye. 1741He goes to the Berlin Academy, at the invitation of Frederick II of Prussia. 1748Publishes the Introductio in analysin infiniterum, where $e^{ix} = \cos x + i \sin x$ appears. 1750In a letter to Goldbach, he states the formula for polyhedra $V - A + F = 2$. 1766Returns to Saint Petersburg, at the invitation of Catherine the Great. A cataract affects the left eye. 1771Cataract surgery fails and he is left practically blind. Keep working. 1783Dies in 18 September, in Saint Petersburg. ◀ Previous Follow the journey Next ▶ 1707–1727 BaselBorn and raised in Switzerland. At 13 he enters the university and starts Saturday lessons with Johann Bernoulli. 1727 LübeckHe travels down the Rhine by boat, crosses Germany by coach and sets sail from Lübeck for Russia. 1727–1741 Saint PetersburgHe works at the Academy of Sciences. He solves the Basel problem, studies the Königsberg bridges and loses the sight of his right eye. 1741–1766 BerlinAt the invitation of Frederick II he moves to the Berlin Academy. He publishes the Introductio in analysin infinitorum and the formula $e^{ix}=\cos x+i\sin x$. 1766–1783 Saint PetersburgHe returns at the invitation of Catherine the Great. He goes almost blind but keeps dictating papers until his death in 1783. Enable JavaScript to see the map. The stages are listed below. Euler’s cities, from Basel to Saint Petersburg. Click a stage or use Follow the journey. Map: © OpenStreetMap contributors

September 29, 2026 Read

The Riemann integral

Newton and Leibniz invented integral calculus in the seventeenth century, but for a long time nobody worried about defining precisely what an integral is. For ordinary functions, the integral was simply the inverse of the derivative, or the area under the curve, and that was enough. Strange functions In the early nineteenth century the situation changed. Fourier showed that very irregular functions, with jumps and corners, could be written as sums of sines and cosines, and the coefficients of those sums were given by integrals. But what sense did it make to integrate a function full of jumps? Cauchy gave a first definition for continuous functions. Dirichlet, Riemann’s teacher, found in 1829 a function that could not be integrated at all: the one that equals 1 at the rationals and 0 at the irrationals.

October 31, 2026 Read

Jupiter, Saturn and the stability of the Solar System

Newton explained the orbits of the planets with a single law, but he knew it brought a problem with it. Each planet is pulled by the Sun and also, a little, by the other planets. These small attractions pile up over time. Newton suspected that, in the long run, the Solar System would fall into disorder and that divine intervention would be needed, from time to time, to set it right again.

October 30, 2026 Read

Eureka: the king's crown and Archimedes's principle

The most famous story about Archimedes does not appear in any of his books. It was told two centuries later by the Roman architect Vitruvius, in book IX of his treatise On Architecture. Hiero’s crown King Hiero had commissioned a votive crown to offer to the gods and had given the goldsmith an exact quantity of gold. The crown was finished and weighed the same as the gold supplied. But a rumour spread that the goldsmith had kept part of the gold and put the same mass of silver in its place. Hiero asked Archimedes to find out the truth without damaging the crown.

October 29, 2026 Read

The curve of Agnesi

Agnesi did not discover the curve that bears her name. But it was through her book that the curve became known, because she used it as an example of how to pass from a geometric construction to an equation, which was exactly what Descartes’s new method promised. The construction Start with a circle of diameter $OM = a$, with $O$ at the bottom and $M$ at the top, and draw the line tangent to the circle at $M$. Now:

October 28, 2026 Read

The polygonal number theorem

The Greeks already arranged pebbles into figures. With 1, 3, 6, 10 pebbles you make triangles; with 1, 4, 9, 16, squares; with 1, 5, 12, 22, pentagons. These are the polygonal numbers. The formula The $n$-th number with $k$ sides is obtained by adding up an arithmetic progression that starts at 1 and grows in steps of $k - 2$: $$P_k(n) = 1 + (k-1) + (2k-3) + \cdots = \frac{n\,\big((k-2)n - (k-4)\big)}{2}.$$For $k = 3$ we get the triangular numbers $\frac{n(n+1)}{2}$: 1, 3, 6, 10, 15, … For $k = 4$, the squares $n^2$. For $k = 5$, the pentagonal numbers $\frac{n(3n-1)}{2}$: 1, 5, 12, 22, 35, … We will also count zero, $P_k(0) = 0$.

October 27, 2026 Read

Systems of rays and conical refraction

At the beginning of the nineteenth century optics was a collection of cases: the plane mirror, the curved mirror, the lens, the prism, each with its own formulas. Hamilton, still a student, asked himself whether there was not a single idea behind them all. The result was the Theory of Systems of Rays, read at the Royal Irish Academy in 1827 and extended over the following years. One function to govern all rays The starting point is a principle of Fermat: light goes from one point to another along the path that takes the least time, or, more precisely, a stationary time. Hamilton took that time as a function of the endpoints. For two points $A$ and $B$, let $V(A, B)$ be the “optical length” of the light’s path between them: the distance travelled in each medium, multiplied by the refractive index of that medium. He called $V$ the characteristic function.

October 27, 2026 Read

Ordinals: numbers made of empty sets

At the beginning of the twentieth century mathematicians wanted to build the whole of mathematics from a single idea, that of a set. But what, in that language, is the number $3$? Georg Cantor had spoken of “ordinals”, numbers that measure positions in a queue: first, second, third, and after all the finite ones, a first infinite one, $\omega$. What was missing was to say what these numbers are, as sets.

October 26, 2026 Read

The Turing machine: calculating with pencil, tape and rules

In 1936 there was no computer. The English word computer meant a person who did calculations for a living, with pencil and paper, in insurance offices, observatories and government departments. Turing started from that person. He asked what they really do when they calculate and stripped away everything that is not essential. What is left of someone calculating Turing’s reasoning, in the paper “On computable numbers”, is almost a list of simplifications:

October 18, 2026 Read

Multiplying by adding: the two progressions

Write the powers of 2 in a row and, below them, the exponents: $$\begin{array}{c|ccccccccccc} 2^n & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024\\ \hline n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \end{array}$$The top row is a geometric progression: each number is the previous one times 2. The bottom row is an arithmetic progression: each number is the previous one plus 1. Now try to multiply $16\times64$. Under 16 is 4; under 64, 6. Add them: $4+6=10$. Above 10 is 1024. And indeed

October 16, 2026 Read

Shuffling the roots: the group of an equation

To jump with both feet over the calculations; to group the operations, to classify them according to their difficulties and not according to their forms: such is, in my view, the mission of future geometers. — Évariste Galois, preface to a manuscript (1831) Sixty years before Galois, Lagrange had looked at expressions made from the roots of an equation and counted how many values they take when the roots change places. Galois went one step further. Instead of looking at the expressions, he looked at the swaps. And not at all of them: only at those the equation allows.

October 16, 2026 Read