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The word "function"

The word comes from the Latin functus, participle of the verb fungor: fulfill a task, exercise an activity. The function is what “does the work” of taking one number to another. Already defunctus is someone who no longer performs any function, and hence the word defunct. A Leibniz notation for functions, from letters to Johann Bernoulli.Euler’s Eyes Collection The name and symbol took more than a century to establish itself:

October 10, 2026 Read
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Euler's theorem and cryptography

In 1763, Euler studied the function $\varphi(n)$ (the symbol $\varphi$ came later, with Gauss): the number of numbers from 1 to $n$ that have no common factor with $n$. For example, $\varphi(10) = 4$, because from 1 to 10 only 1, 3, 7 and 9 are suitable. He proved that if $a$ did not has a common factor with $n$, so $$a^{\varphi(n)} \equiv 1 \pmod{n}.$$ See the idea with $n = 10$ and $a = 3$. Multiply the numbers 1, 3, 7 and 9 by 3:

October 10, 2026 Read
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Pascal image credits

Historical images All of them come from Wikimedia Commons. We use the thumbnails provided by Commons, unedited. Portrait of Blaise Pascal, unknown artist, around 1690, Versailles: source page · public domain. Yang Hui’s triangle, in Zhu Shijie’s book, 1303: source page · public domain. First page of the Triangulus arithmeticus, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Pascaline at the Musée des Arts et Métiers, photo by David Monniaux, 2005: source page · CC BY-SA 3.0. Marker of the experiment on the Puy de Dôme, Agence Rol, 1923, Bibliothèque nationale de France: source page · public domain. Plaque of the grave in Saint-Étienne-du-Mont, Agence Meurisse, 1923, Bibliothèque nationale de France: source page · public domain. Autograph manuscript of the Pensées, exhibition at the Bibliothèque nationale de France, photo by ActuaLitté, 2016: source page · CC BY-SA 2.0. First page of the Pensées, counterfeit of the 1670 edition, copy held by Clermont Auvergne Métropole: source page · CC BY-SA 4.0. Blog figures The cover, the fountain of bowls, the carry diagram and the cycloid arch are original illustrations. The controls for the triangle, the split of the stakes and the hydraulic press were made for this trail. They are teaching aids, not reproductions of manuscripts or of the internal mechanism of the Pascaline.

October 11, 2026 Read
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Other contributions and legacy

Other contributions Physics: Gauss’s law of electromagnetism, the study of Earth’s magnetism and the Gauss unit. Complex numbers: Gaussian integers, numbers $a + bi$ with $a$ and $b$ integers. Astronomy and geodesy: orbits of planets and comets, the heliotrope and the Hannover measurement. Algorithms: a method for calculating the date of Easter and, in a manuscript by 1805, a version of the fast Fourier transform, rediscovered in 1965 and used today in every cell phone. To remember Gauss was born poor in Braunschweig, studied under a duke and spent his life in Göttingen. Added from 1 to 100 by joining pairs: $1 + 2 + \cdots + n = \frac{n(n+1)}{2}$. At 18 years he showed how to construct a polygon with 17 sides, using only square roots. He created the congruence notation, $a \equiv b \pmod n$, the mathematics of the clock and nines-out. He proved that every polynomial of degree $n$ has $n$ complex roots. Estimated the number of primes: $\pi(x) \approx x / \ln x$. He rediscovered Ceres with the least squares method, which shows why the average is the best estimate. He gave the name to the normal curve, the Gaussian bell. He showed that curvature can be measured from within the surface: this is why there is no perfect flat map. He named the determinant and organized the elimination that solves systems of equations. With two well-chosen points, $\pm 1/\sqrt{3}$, he calculated exact integrals for polynomials up to degree 3. All topics by Carl Friedrich Gauss · ← Gaussian quadrature: integrate with two points · Image credits →

October 10, 2026 Read
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*Physis*: deciphering the traits of nature

Nothing happens in the universe without some rule of maximum or minimum appearing. — Leonhard Euler, 1744 For Leibniz and Euler, nature is made in the best possible way, and therefore follows rules. In our language: the reasoning (méthode) of the creator is in the traits (graphiko’s) of nature (physis), and it is up to us to decipher them (máthema). The same drawings appear in the most different places: Armadillo carapace: hexagonal plates.Euler’s Eyes Collection Earth’s crust: basalt columns in hexagons, formed when lava cools.Euler’s Eyes Collection Tenon: again, the hexagonal pattern.Euler’s Eyes Collection Nautilus shell: a logarithmic spiral.Euler’s Eyes Collection Egg: The oval curve distributes pressure like an arc.Euler’s Eyes Collection Starfish: pentagonal symmetry.Euler’s Eyes Collection Why so many hexagons? Only three regular polygons cover the plane without leaving holes: the triangle, the square and the hexagon. Of these, the hexagon is the one that encloses the most area with the least border. It is the most economical way to divide a space in equal parts, and that is what mathematician Thomas Hales proved in 1999.

October 10, 2026 Read
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Other contributions and legacy

Other contributions The complete list would take up an entire book. Some highlights: Euler’s straight line: in every triangle, orthocenter, centroid and circumcenter are aligned. Euler-Mascheroni constant: $\gamma \approx 0{,}5772$, the difference between the harmonic series and the natural logarithm. Mechanics and fluids: Euler’s equations for fluids and for the rotation of rigid bodies. Calculation of variations: the Euler-Lagrange equation, the basis of modern physics. Special functions: the gamma function, which extends the factorial to non-integer numbers. To remember Euler lived between Basel, Saint Petersburg and Berlin, and produced half of the work while almost blind. He gave mathematics the notation $f(x)$, $e$, $i$, $\Sigma$ and popularized $\pi$. Found that $\sum \frac{1}{n^2} = \frac{\pi^2}{6}$ and that $e^{i\pi} + 1 = 0$. He named the number $e = \lim (1 + 1/n)^n$, which arises from compound interest and is the base of the natural logarithm. Solved the spring equation with $e^{rt}$ and its formula: the solution is a sinusoid. With the Königsberg bridges, he created graph theory. With $V - A + F = 2$, he explained why every classic football has 12 pentagons. He linked prime numbers to calculation and created the theorem that protects the internet today. All topics by Leonhard Euler · ← Euler’s theorem and cryptography · Image credits →

October 10, 2026 Read
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Image credits

Images from Wikimedia Commons, all in the public domain: Gauss at 1840: Christian Albrecht Jensen. Gauss at 1803: Johann Christian August Schwartz. Gauss's birthplace, photo by 1914. Gauss's diary: Göttinger Digitalisierungszentrum. Disquisitiones Arithmeticae, 1801. Giuseppe Piazzi: F. Bordiga. Sophie Germain: Berthe Chégaray. Gaussian heliotrope. Triangle network: Reinhold Gerstetter for the Deutsche Bundesbank. Theoria motus corporum coelestium, 1809. Wilhelm Weber: Rudolph Hoffmann. The images of the 10 mark note, the squaring of the circle, and the determinant and matrix notations are from Euler’s Eyes collection. The interactive figures were made for this website.

October 10, 2026 Read
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Infinity and division by zero

The friar and mathematician Marin Mersenne was Descartes’ great friend. It was through letters Mersenne that Descartes talked to the wise men of Europe, including Fermat. In the next generation, the Englishman John Wallis created the symbol $\infty$ for infinity in 1655. Marin Mersenne (1588–1648), Descartes’ friend and main correspondent.Euler’s Eyes Collection John Wallis (1616–1703), creator of the ∞ symbol.Euler’s Eyes Collection At school we learned a canon, a rule: you will never divide by zero. But it’s almost never explained why. Let’s build the thought, gradually reducing the divisor:

October 10, 2026 Read
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Image credits

Images from Wikimedia Commons, all in the public domain (or CC0): Portrait of Euler: Jakob Emanuel Handmann, 1753. Basel at 1615: Matthäus Merian. Johann Bernoulli: Rijksmuseum (CC0). Saint Petersburg Academy of Sciences: G. Kachalov, after M. Makhaev, 1753. Frederick II: Antoine Pesne's studio, 1745. Introduction to analysin infinityum, 1748. Lettres à une princesse d'Allemagne, 1768. Catherine the Great: Alexander Roslin. Euler at 1778: Joseph Friedrich August Darbes. Königsberg at 1652: Merian-Erben. Figure of bridges: Leonhard Euler, 1736. The images in the sections on the square root, the ages of mathematics, the number $e$ and the spring are from the collection of Euler’s Eyes. The interactive figures were made for this website.

October 10, 2026 Read
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Other contributions and legacy

Other contributions Descartes’ vortices: for him, the planets were dragged by whirlpools of matter around each star.René Descartes, Principles of Philosophy · public domain Optics: in addition to the law of refraction, he studied the ideal shape of lenses, which should have a profile of ellipse or hyperbola to focus light on a point. Descartes’ Folio: the curve $x^3 + y^3 = 3axy$, which he proposed as a challenge to Fermat to find tangent lines. Physics: formulated a version of the principle of inertia, that a body in motion continues in straight line if nothing gets in your way. Philosophy: the Meditations and the Principles of Philosophy made it the starting point of modern philosophy. To remember Descartes lived between France, Holland and Sweden, and wrote the Discourse on the Method in French to be read by everyone. He created analytical geometry: points became pairs $(x, y)$ and figures became equations. He showed how to construct $\sqrt{a}$ with a ruler and compass and fixed the notation $x$, $y$, $z$ and $x^3$. The rule of signs tells how many positive roots a polynomial can have. The sum of the vertex defects of a polyhedron is 720°, which leads to the formula $V - A + F = 2$. He explained the rainbow: the rays accumulate at 42° from the point opposite the Sun. With it comes the idea of ​​function: a rule that “listens” to one number and “speaks” another. The word comes from Latin functus, and the symbol $f(x)$ is Euler (1734). There is no $1 \div 0$: close to zero, $\frac{1}{x}$ runs away to $+\infty$ on one side and to $-\infty$ on the other. All topics by René Descartes · ← Infinity and division by zero · Image credits →

October 10, 2026 Read
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Image credits

Images from Wikimedia Commons, all in the public domain: Portrait of Descartes: Frans Hals. Method Discourse, 1637. Princess Elisabeth of Bohemia: Gerard van Honthorst. Descartes and Queen Christina: Nils Forsberg, according to Pierre-Louis Dumesnil. Page Geometry e book I of Geometry: René Descartes, 1637. Rainbow Figure: René Descartes, The Meteors, 1637. Vortices: René Descartes, Principles of Philosophy. The images in the part “Descartes and the idea of ​​function” are from the Euler’s Eyes collection. The figures interactives were made for this site.

October 10, 2026 Read
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Chapter 1: references and credits

Further reading Cajori, Florian. A History of Mathematical Notations. 2 vols. Chicago: Open Court, 1928–1929. The reference work on the origin of each symbol; almost every date in this chapter can be checked in it. Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton: Princeton University Press, 2014. An accessible history on the same theme as this book. Ifrah, Georges. The Universal History of Numbers. London: Harvill Press, 1998. On number systems and zero. Klein, Jacob. Greek Mathematical Thought and the Origin of Algebra. Cambridge, MA: MIT Press, 1968. On the passage from Diophantus to Viète. Boyer, Carl B.; Merzbach, Uta C. A History of Mathematics. 3rd ed. Hoboken: Wiley, 2011. Crowe, Michael J. A History of Vector Analysis. Notre Dame: University of Notre Dame Press, 1967. On the dispute between quaternions and vectors. Maor, Eli. Trigonometric Delights. Princeton: Princeton University Press, 1998. On the history of the sine and the names of trigonometry. Grabiner, Judith V. The Origins of Cauchy’s Rigorous Calculus. Cambridge, MA: MIT Press, 1981. On Berkeley, Cauchy and the limit. Miller, Jeff. Earliest Uses of Various Mathematical Symbols. A page maintained since the 1990s, with the earliest known occurrences of each sign. MacTutor History of Mathematics, University of St Andrews: biographies of al-Khwarizmi, Fibonacci, Recorde, Viète, Harriot, Oughtred, Cayley, Boole, Frege and Peano. The old texts on this site about geometry, trigonometry, the square root, quadratic equations, logarithms, derivatives, the chain rule, integrals, limits, functions, differential equations, groups, matrices, vectors and quaternions were the basis for rewriting this chapter. From them come the story of radix quadrati, the completing-the-square figure, Pythagoras’ theorem in Latin, the etymology of cathetus, hypotenuse, minute and second, the passage from $\Sigma$ to $\int$, the proof that $1=2$, Hooke’s law, the “canon of heat” and the propagation of errors, the abbreviation of logarithmus to $\log$, the etymology of derivative and function, reading a function as a sentence with a hole, the group table of the triangle’s moves, the dispute between quaternions and vectors, and the ideas of active and sterile notation.

October 13, 2026 Read