History of Mathematics

Image credits for Sophie Germain

Historical images and photos All come from Wikimedia Commons. We use the thumbnails provided by Commons; the portrait was only converted from PNG to JPEG. Portrait of Sophie Germain, drawing by Berthe Chégaray of a medal in her memory, published in 1896: source page · public domain. Figures from the Research on the Theory of Elastic Surfaces, Sophie Germain, 1821, copy in the ETH-Bibliothek Zürich: source page · public domain. Chladni plate excited by a bow, engraving from 1879: source page · public domain. Grave in the Père-Lachaise cemetery, photo by Pierre-Yves Beaudouin, 2015: source page · CC BY-SA 4.0. Blog figures The cover, the nodal lines of a square plate and the drawing of the curvatures of the sphere, cylinder and saddle are original illustrations. The nodal lines were computed with the simplified model described in the text and do not reproduce photos of real plates.

October 14, 2026 Read

Lagrange's legacy

Lagrange is the lofty pyramid of the mathematical sciences. — Napoleon Bonaparte, according to his contemporaries Lagrange left no anecdotes of precocious genius like Pascal, nor a body of work the size of Euler’s. He left something else: a way of doing mathematics. Where others drew, he calculated; where others found a trick for each problem, he looked for the rule that explained all the tricks. Almost everything he did is still in the textbooks under his name: the Euler–Lagrange equation, the Lagrangian, the Lagrange points, Lagrange’s theorem, the Lagrange remainder, Lagrange multipliers.

October 14, 2026 Read

The legacy of Euclid

He studied and nearly mastered the six books of Euclid, since he was a member of Congress. — Abraham Lincoln, autobiographical note (1860) After the Bible, the Elements are one of the most copied, translated and printed books in history. Since the first printed edition, in Venice in 1482, there have been more than a thousand editions. For more than two thousand years, “studying geometry” meant “studying Euclid”. The first printed edition of the Elements, made by Erhard Ratdolt in Venice in 1482, in a Latin translation. The figures are in the margins, beside the text; an early reader filled the pages with notes.Folger Shakespeare Library · Wikimedia Commons · CC BY-SA 4.0 The many faces of a book This site’s old trigonometry course gathered some editions that changed the way Euclid was presented:

October 14, 2026 Read

The weight of air: the Puy de Dôme experiment

For two thousand years scholars repeated a saying of Aristotle’s: nature abhors a vacuum. If you suck the air out of a straw, the liquid rises because nature will not let the space stay empty. It seemed to explain everything. But the well-diggers of Florence had a problem. Suction pumps could not raise water more than about 10 metres, however hard they tried. Why would nature’s horror of the vacuum stop at 10 metres?

October 11, 2026 Read

Fermat's last theorem: from the margin to Wiles's proof

Everyone knows Pythagoras’s theorem: $3^2+4^2=5^2$. There are infinitely many triples of whole numbers like this, such as $5,12,13$ and $8,15,17$. What about cubes? Are there positive integers with $a^3+b^3=c^3$? And with fourth powers? The note in the margin The page of problem II.8 in the 1670 edition of Diophantus’s Arithmetica. Just below the problem comes the “Observatio Domini Petri de Fermat”: the margin note, printed by Fermat’s son.Diophantus, ed. Samuel de Fermat (1670) · Wikimedia Commons · public domain Around 1637, reading Diophantus’s problem about splitting a square into two squares, Fermat wrote in the margin of the book, in Latin:

October 11, 2026 Read

Leibniz's legacy

The calculus war Leibniz published calculus in 1684 and 1686. Newton had reached the same ideas in 1665–1666 but kept everything in manuscripts. For fifteen years nobody disputed anything. In 1699 the Swiss mathematician Nicolas Fatio de Duillier, a friend of Newton, hinted in a book that Leibniz had drawn on Newton. In 1708 the Scot John Keill went further and accused Leibniz of plagiarism. The argument: in 1676 Leibniz had visited London and seen a copy of Newton’s De analysi.

October 11, 2026 Read

Newton's legacy

Newton was the greatest genius who ever lived, and the most fortunate, for there is only one universe to discover, and it can be discovered only once. — Joseph-Louis Lagrange, as reported by his contemporaries The calculus war Newton created the method of fluxions in 1665–1666 but did not publish it. Leibniz reached calculus independently in 1675 and published in 1684. For some years nobody quarrelled. In 1676 the two even exchanged cordial letters through Henry Oldenburg, and Newton hid his ideas in an anagram.

October 11, 2026 Read

The Egregium Theorem and curvature

Measuring Hannover, Gauss thought a lot about how to make maps of a curved Earth. In 1827 published the result he called Theorema Egregium, the “remarkable theorem”: the curvature of a surface can be measured by those who live inside it, without going out to look from the outside. A consequence: in a plane, the angles of a triangle add up to 180°. In a sphere, they add up to more. Take the triangle with one vertex at the north pole and the other two at the equator:

October 10, 2026 Read

A reading of the Cogito: listening and speaking

Cogito ergo sum drawn on the Cartesian plane.Euler’s Eyes Collection Here at Euler’s Eyes we like to read “I think, therefore I am” as a little machine. The thought hears something (in Latin, audire) and speaks another (fabulari). The rule that binds what What you hear is the function. Method words physis: nature; methode: the reasoning, the path; graphiko’s: the lines, the drawing; máthema: what is learned, the act of deciphering; functus: who fulfills a task; dominus: the lord, where the word dominion comes from. An example: “say what you hear twice”. Empty parentheses act as a waiting ear a number:

October 10, 2026 Read

Infinitely many prime numbers from a sum

Euclid had already proven, around 300 BC, that there are infinitely many prime numbers. In 1737 Euler found a completely different proof, which linked the primes to calculus. For each prime $p$, the sum of a geometric progression gives $$\frac{1}{1 - \frac{1}{p}} = 1 + \frac{1}{p} + \frac{1}{p^2} + \frac{1}{p^3} + \cdots$$ Multiply these expressions to $p = 2, 3, 5, 7, \dots$ Each product of terms has the form $\frac{1}{2^a 3^b 5^c \cdots}$. As every natural number is written in a single way as a product of primes, each fraction $\frac{1}{n}$ appears exactly once:

October 10, 2026 Read

The legacy of Alan Turing

Turing believes machines think. Turing lies with men. Therefore machines do not think. — Alan Turing, letter to Norman Routledge (1952) The bitter syllogism Turing wrote to a friend, just after his conviction, sums up his life: a man who thought with logic and was judged by people who did not understand him. The work he left behind, however, is everywhere. In computer science The Turing machine is still the official definition of what it means to calculate. When a computer scientist says a problem “has no algorithm”, they mean no Turing machine solves it; when they say a problem is “hard”, they count the steps a Turing machine needs.

October 16, 2026 Read

Image credits

Historical images All come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. No portrait of Euclid was made in his lifetime; the images of him in circulation are inventions by much later artists. Oxyrhynchus papyrus with Proposition II.5, c. AD 100, University of Pennsylvania Museum: source page · public domain. Detail of the School of Athens, by Raphael, 1509–1511: source page · public domain. Proposition I.47 in Oliver Byrne’s edition, London, 1847, scan by Mirko Junge: source page · public domain. First printed edition of the Elements, Erhard Ratdolt, Venice, 1482, Folger Shakespeare Library: source page · CC BY-SA 4.0. Arabic edition of the Elements, version of Nasir al-Din al-Tusi, Rome, 1594, Houghton Library: source page · public domain. Blog figures The cover, the construction of the equilateral triangle, the figure for Pythagoras’s theorem and the one for completing the square are original illustrations, with versions in each of the site’s languages. The completing-the-square example, $x^2+10x=39$, and the exercise $x^2+8x=65$ come from this site’s old course on the quadratic formula, whose figures were redrawn; the examples 3, 4, 5 and 6, 8, 10 and the table of editions come from the old geometry and trigonometry courses.

October 14, 2026 Read