Mathematical Notation

Euler and functions: giving a rule a name

If anyone deserves the title of “architect of modern notation”, it is Leonhard Euler. He wrote more pages of mathematics than anyone in history, and he wrote to be read: his books were used for teaching, and the conventions he adopted spread along with them. Many symbols that seem to have always existed passed through his hands. Leonhard Euler (1707–1783), portrait by Jakob Emanuel Handmann, 1753.Jakob Emanuel Handmann · public domain 1706William Jones uses $\pi$ for the ratio of circumference to diameter. Hardly anyone notices. 1727In a manuscript, Euler calls $e$ the number whose logarithm is $1$. 1734Euler writes $f\!\left(\frac{x}{a}+c\right)$: the birth of the notation $f(x)$ for functions. 1736In the Mechanica, Euler adopts Jones’s $\pi$; with the Introductio (1748) the symbol becomes standard. 1755Euler uses $\Sigma$ to denote a sum. 1777Euler uses $i$ for $\sqrt{-1}$, in a text published only after his death. What these symbols have in common is that each turns a description into a name. Before Euler, to talk about a function you had to describe the rule (“the quantity formed from $x$ by squaring and adding one”). After $f(x)$, the rule gets a short name, and a name can go into other calculations: $f(x+h)-f(x)$, $f(g(x))$, $f'(x)$. The function becomes an object to be handled the way a number was handled. For more on each symbol, see the language Euler left us.

October 13, 2026 Read

From exhaustion to the integral: Σ, ∫ and the number π

Working out the area of a rectangle is easy: base times height. A triangle’s is too: half of that. The trouble begins when the figure is curved. There is no way to tile a circle with little squares without leaving or overshooting a piece. For two thousand years mathematicians attacked this problem with the same idea: approximate the curve by straight figures that get smaller and smaller. What changed was the way of writing that idea.

October 13, 2026 Read

Leibniz against the dots: active and sterile notation

On 29 October 1675, in Paris, Leibniz was adding up infinitely many thin slices under a curve and got tired of writing omnia, “all”, in front of each sum. He noted in his notebook: Utile erit scribi ∫ pro omnia. (It will be useful to write ∫ instead of “all”.) — Leibniz, manuscript of 29 October 1675 The $\int$ is a stretched S, for summa. Two weeks later, in the same notebook, the $d$ appeared, for differentia, to denote infinitely small differences. With these two signs Leibniz had in his hands a language for the whole of calculus. Isaac Newton had reached the same ideas ten years earlier, in Cambridge, but wrote them differently. This is history’s best case for seeing how the shape of a sign changes what can be done with it.

October 13, 2026 Read

Napier and the logarithm: the marvellous abbreviation

Today, for many students, the logarithm is a bogeyman: a strange word, a $\log$ that appears out of nowhere and rules to memorise. In the seventeenth century it was the opposite. It was Europe’s calculator, the tool that saved astronomers, navigators and accountants months of work. The old texts on this site said the logarithm was the purest example of mirabilis abbreviatio, the marvellous abbreviation. And its inventor agreed: the title of his book begins with the word marvellous.

October 13, 2026 Read

+, −, × and =: a short history of the signs

The simplest signs in mathematics are also the youngest. Euclid never wrote a $+$. Al-Khwarizmi never wrote an $=$. Viète, who already used letters, still wrote aequatur in full. Almost all the operation signs a child learns in the first year of school were born in Europe between the end of the fifteenth century and the end of the seventeenth, in a span of about two hundred years. 1489Johannes Widmann prints $+$ and $-$ in a book of commercial arithmetic in Leipzig. 1525Christoff Rudolff uses $\surd$ for the square root in his algebra book, the Coss. 1557Robert Recorde invents $=$ in The Whetstone of Witte. 1591François Viète uses letters for known and unknown quantities. 1631William Oughtred uses $\times$ for multiplication; in the same year Thomas Harriot’s posthumous book appears with $<$ and $>$. 1637René Descartes writes powers as exponents: $x^3$, $x^4$. 1659Johann Rahn uses $\div$ for division. 1698Leibniz proposes the dot $\cdot$ for multiplication, so as not to confuse $\times$ with the letter $x$. Plus and minus: warehouse marks The signs $+$ and $-$ first appear in print in 1489, in an arithmetic manual for merchants written by Johannes Widmann. There they do not quite mean an operation yet: they mark excess and shortfall. A crate of goods that weighed more than it should was marked $+$; one that weighed less, $-$. The most accepted hypothesis is that $+$ is an abbreviation of et (“and”, in Latin), written quickly until it became a cross.

October 13, 2026 Read

From words to letters: algebra learns to write

In Baghdad, around the year 820, Muhammad ibn Musa al-Khwarizmi worked at the House of Wisdom, the great library of the caliph al-Ma’mun. There he wrote a book with a long title, The Compendious Book on Calculation by Restoration and Balancing. “Restoration”, in Arabic, is al-jabr. From that word came algebra. Opening page of al-Khwarizmi’s book on al-jabr and al-muqābala, in a manuscript copy.Muhammad ibn Musa al-Khwarizmi · Wikimedia Commons · public domain The whole book contains not a single symbol. No $x$, no $+$, no $=$. Even the numbers are written out in words. One of the most famous problems is this:

October 13, 2026 Read

Sine, leg and hypotenuse: trigonometry and its lost names

Almost every student learns to say “sine”, “cosine”, “leg” (or cathetus) and “hypotenuse” without knowing what these words mean. They are labels to memorise, like street names in a strange city. But each of them was once a description, and some hide unexpected stories. The sine, for example, is the result of a translation that went wrong. Pythagoras in Latin The most famous theorem of geometry was for a long time a sentence about real squares, drawn on the sides of a triangle. The old texts on this site wrote it in the medieval manner:

October 13, 2026 Read

Euclid: the figure and the proof

Around 300 BC, in Alexandria, Euclid gathered Greek geometry into thirteen books, the Elements. For more than two thousand years it was the most widely read mathematics book in the world, and its first proposition shows how it works: On a given finite straight line, to construct an equilateral triangle. — Euclid, Elements, book I, proposition 1 The recipe is short. Let $AB$ be the given line. With centre $A$ and radius $AB$, draw a circle; with centre $B$ and the same radius, draw another. Call $C$ a point where the two circles cut each other. Then $AC=AB$, because both are radii of the first circle, and $BC=AB$, because both are radii of the second. Things equal to the same thing are equal to each other, so $AC=BC=AB$, and triangle $ABC$ is equilateral.

October 13, 2026 Read

Zero and place value: numbers that calculate

Try multiplying $\text{XLVII}$ by $\text{XII}$. Without converting to our digits. It is hard to know even where to start. The $\text{X}$ in $\text{XLVII}$ is worth ten, but it comes before the $\text{L}$, so it is subtracted: $\text{XL}=40$. The $\text{V}$ is worth five wherever it stands. There is no ones column and no tens column; each symbol carries its value hanging from its own shape. The same sum, written as $47\times 12$, is done in two lines:

October 13, 2026 Read

Radix quadrati: mathematics has a script too

Ask a student what the square root of 9 is and they will probably say “3”. Ask why it is called a root, and why square, and the conversation usually ends there. The name has become an empty label, and the symbol $\sqrt{\phantom{9}}$ a strange hook that “pulls” something out of the number. But the name once told the whole story. Greek geometers simply called the number they were looking for the side (pleurá): the side of a square of area 9 measures 3. Arab algebraists called it jidhr, the root from which the square “grows”, and medieval Latin translators wrote radix. A sentence such as

October 13, 2026 Read