Number Theory

Euclid of Alexandria: the author without a face

Timeline c. 530 BCPythagoras and his followers study numbers and figures in southern Italy. c. 387 BCPlato founds the Academy in Athens. Over the door, according to tradition: “Let no one ignorant of geometry enter”. 331 BCAlexander the Great founds Alexandria, in the Nile delta. c. 300 BCEuclid teaches in Alexandria, in the time of King Ptolemy I, and writes the Elements. c. 250 BCArchimedes, in Syracuse, cites the Elements. c. AD 100A papyrus found at Oxyrhynchus, in Egypt, preserves a passage from Book II. c. 370Theon of Alexandria prepares the edition that will be copied over the following centuries. c. 450Proclus writes his commentary on Book I, with the little news we have of Euclid. c. 800The Elements are translated into Arabic in Baghdad. c. 1120Adelard of Bath translates the Arabic text into Latin. 1482First printed edition, in Venice, by Erhard Ratdolt. ◀ Previous Follow the journey Next ▶ c. 387 a.C. AthensPlato founds the Academy. The geometry Euclid will organise grows up in the Greek schools. c. 300 a.C. AlexandriaEuclid teaches in the time of Ptolemy I and writes the thirteen books of the Elements. c. 250 a.C. SyracuseArchimedes already cites the Elements in his works. c. 100 OxyrhynchusA papyrus found in the town’s rubbish heaps preserves a passage from Book II, with a figure. c. 370 AlexandriaTheon of Alexandria prepares the edition that will be copied for a thousand years. c. 450 AthensProclus writes his commentary on Book I, with the little news we have about Euclid. c. 800 BaghdadIn the House of Wisdom the Elements are translated into Arabic. c. 1120 BathAdelard of Bath translates the Arabic text into Latin, and the book returns to Europe. 1482 VeniceErhard Ratdolt prints the first edition, with the figures in the margins. More than a thousand editions followed. Enable JavaScript to see the map. The stages are listed below. The journey of the Elements, from Athens and Alexandria to the first printed edition in Venice. Map: © OpenStreetMap contributors

October 14, 2026 Read

The life of Joseph-Louis Lagrange

Timeline 1736Born on 25 January in Turin, capital of the Kingdom of Sardinia, as Giuseppe Lodovico Lagrangia. 1753At 17, reads an essay by Edmond Halley and decides to devote himself to mathematics. 1755Writes to Euler with the method that will give rise to the calculus of variations. In September he is appointed professor at the Artillery School in Turin, aged 19. 1757Helps found the scientific society that will become the Turin Academy of Sciences. 1764Wins the Paris Academy prize by explaining why the Moon always shows the same face. 1766Moves to Berlin to take Euler’s place at the Academy of Frederick the Great. 1770Proves that every whole number is a sum of four squares and begins studying the permutations of the roots of an equation. 1772Shares the Paris prize with Euler for the three-body problem, where the Lagrange points appear. 1787After Frederick’s death, moves to Paris, where he lives in the Louvre. 1788Publishes the Analytical Mechanics. 1790Joins the commission that will create the metric system. 1794Lavoisier is guillotined. Lagrange becomes a professor at the new École Polytechnique. 1797Publishes the Theory of Analytic Functions, with the notation $f'(x)$. 1808Napoleon makes him a count of the Empire. 1813Dies on 10 April in Paris and is buried in the Panthéon. ◀ Previous Follow the journey Next ▶ 1736–1766 TurinAt 19 he is a professor at the Artillery School. He writes to Euler the method that will give rise to the calculus of variations. 1766–1787 BerlinHe takes Euler’s place at the Academy of Frederick the Great. He proves the four-square theorem and finds the Lagrange points. 1787–1813 ParisHe lives in the Louvre, publishes the Analytical mechanics, helps create the metric system and teaches at the École Polytechnique. He is buried in the Panthéon. Enable JavaScript to see the map. The stages are listed below. Lagrange in three capitals, Turin, Berlin and Paris. Map: © OpenStreetMap contributors

October 14, 2026 Read

The life of Sophie Germain

Timeline 1776Born on 1 April in Paris, the daughter of a wealthy silk merchant. 1789At 13, during the Revolution, she reads the story of Archimedes’ death and decides to study mathematics. 1794The École Polytechnique is founded and does not admit women. Sophie obtains the lecture notes and sends Lagrange a paper under the name Monsieur Le Blanc. 1801Gauss publishes the Disquisitiones Arithmeticae. 1804She begins writing to Gauss, again as Monsieur Le Blanc. 1806French troops occupy Braunschweig. Sophie asks a general who is a friend of the family to protect Gauss, who thus discovers who she is. 1809The Academy of Sciences announces a prize for the mathematical theory of vibrating plates. 1816On her third attempt, she wins the prize. She is the first woman to receive a prize from the Academy. 1823Legendre makes public her theorem on Fermat’s Last Theorem. 1829She learns she has breast cancer. 1831Dies on 27 June in Paris, aged 55. ◀ Previous Follow the journey Next ▶ 1776–1794 Rue Saint-DenisShe grows up in the house of her father, a rich silk merchant. At 13, during the Revolution, she decides to study mathematics. 1794 École PolytechniqueThe school does not admit women. She obtains the lecture notes and sends a paper to Lagrange as Monsieur Le Blanc. 1804–1806 BrunswickShe writes to Gauss about number theory. In 1806, with the city occupied, she asks a French general to protect Gauss, who thus learns who she is. 1816 Institut de FranceAt the third attempt she wins the Academy’s prize on vibrating plates, the first woman to receive a prize from the Academy. 1831 Père-Lachaise cemeteryShe dies on 27 June, aged 55. It is said that her death certificate recorded her as a rentière, a person of private means, and not as a mathematician. Enable JavaScript to see the map. The stages are listed below. Sophie Germain’s Paris, with a line for the letters sent to Gauss in Brunswick. Map: © OpenStreetMap contributors

October 14, 2026 Read

The calculus of variations: a letter from a 19-year-old

The calculus of Newton and Leibniz can find the highest point of a curve: it is where the derivative vanishes. But there are questions of another kind. What is the shortest path between two points? What shape should a slide have so that a ball runs down it fastest? What curve does a hanging chain make? In these questions the answer is not a number but a whole curve, chosen from infinitely many possible curves.

October 14, 2026 Read

The Elements: definitions, postulates and common notions

A point is that which has no part. A line is breadthless length. — Euclid, Elements, Book I, Definitions 1 and 2 Imagine you want to convince someone, beyond any possible doubt, that the angles of a triangle always add up to 180°. Measuring is not enough: the protractor makes errors, and you can only measure a few triangles. You need an argument. But every argument starts from something. Where should you start? If every claim must be justified by another, and that one by another, the chain never ends.

October 14, 2026 Read

The letters of Monsieur Le Blanc: the arithmetic of remainders

When Sophie Germain wrote to Gauss in 1804, the two spoke a language that had just been born. Three years earlier, in the Disquisitiones Arithmeticae, Gauss had introduced a new sign, $\equiv$, and a new way of thinking about whole numbers: not by what they are, but by the remainder they leave on division. Everything Sophie later did on Fermat was written in that language. It is worth learning. Clock arithmetic If it is 9 o’clock now, what time will it be in 5 hours? Nobody answers 14 on a clock with hands: they answer 2. The clock keeps only the remainder on division by 12. Gauss wrote this as

October 14, 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 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 14, 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 14, 2026 Read

Pythagoras's theorem, Euclid's way

In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle. — Euclid, Elements, Proposition I.47 Long before the Greeks, the surveyors of Egypt, who re-marked the fields after each Nile flood, were called rope-stretchers. The story goes that they used a rope with 12 equally spaced knots: stretched into a triangle with sides 3, 4 and 5, it gave a right angle. The Babylonians knew much more: a clay tablet from around 1800 BC, Plimpton 322, lists right triangles with whole-number sides, some of them huge, such as 3,367, 3,456 and 4,825.

October 14, 2026 Read

Sophie Germain primes and cryptography

A prime number $p$ is a Sophie Germain prime when $2p+1$ is also prime. 5 is one, because 11 is prime. 7 is not, because $15=3\cdot5$. The list begins: $$2,\ 3,\ 5,\ 11,\ 23,\ 29,\ 41,\ 53,\ 83,\ 89,\ 113,\ 131,\ldots$$The prime $2p+1$ that goes with each is called a safe prime: $5, 7, 11, 23, 47, 59, 83, \ldots$ Sophie Germain needed them for her theorem on Fermat. Two centuries later, they protect internet connections.

October 14, 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 14, 2026 Read