Gauss

The life of Bernhard Riemann

Timeline 1826Born on 17 September in Breselenz, in the Kingdom of Hanover, the son of a Lutheran pastor. 1846Enters the University of Göttingen to study theology, but soon moves to mathematics. 1847Goes to study in Berlin, with Dirichlet, Jacobi, Steiner and Eisenstein. 1851Doctorate in Göttingen, supervised by Gauss, with the thesis on complex functions. 1854On 10 June, gives his habilitation lecture on the foundations of geometry. 1857Publishes the theory of abelian functions and becomes an associate professor. 1859Succeeds Dirichlet in the chair at Göttingen and is elected to the Berlin Academy. Publishes the paper on prime numbers. 1862Marries Elise Koch. Falls ill with tuberculosis. 1866Dies on 20 July in Selasca, Italy, aged 39. The pastor’s son Georg Friedrich Bernhard Riemann was born on 17 September 1826 in Breselenz, a village in the Kingdom of Hanover, in northern Germany. His father, also named Friedrich Bernhard, was a Lutheran pastor who had fought in the wars against Napoleon. The family was poor, and Bernhard was the second of six children. It was his father who taught him until he was ten.

October 30, 2026 Read

The life of Carl Friedrich Gauss

Timeline 1777Born on 30 April, in Braunschweig, son of a gardener and bricklayer. 1786Around 9 years, sum of 1 to 100 in seconds at school. 1791Duke Carl Wilhelm Ferdinand starts paying for his studies. 1795Enter the University of Göttingen. 1796In March 30, discover how to construct the regular polygon with 17 sides. Start your math journal. 1799Doctorate from Helmstedt, with proof of the fundamental theorem of algebra. 1801Publishes the Disquisitiones Arithmeticae and calculates the orbit of the dwarf planet Ceres. 1807He became director of the Göttingen Observatory, where he remained until his death. 1809Publishes the Theoria motus, with the least squares method and the normal curve. 1818Start measuring the Kingdom of Hannover with triangles. In 1821 he invents the heliotrope. 1827Publishes the Theorema Egregium, on the curvature of surfaces. 1833With Wilhelm Weber, he set up one of the first electric telegraphs. 1855Dies on 23 February, in Göttingen. ◀ Previous Follow the journey Next ▶ 1777–1795 BrunswickThe son of a gardener and bricklayer, he adds 1 to 100 in seconds at school. Duke Carl Wilhelm Ferdinand starts paying for his studies. 1795–1798 GöttingenHe studies at the university. On 30 March 1796 he discovers how to construct the regular 17-gon and begins his diary. 1799 HelmstedtHe takes his doctorate with a proof of the fundamental theorem of algebra. 1799–1807 BrunswickHe publishes the Disquisitiones Arithmeticae (1801) and computes the orbit of Ceres. Here he receives the letters of “Monsieur Le Blanc”. 1807–1855 Göttingen ObservatoryHe directs the observatory for almost fifty years. He surveys the Kingdom of Hanover with triangles, publishes the Theorema Egregium and builds a telegraph with Weber. He dies in 1855. Enable JavaScript to see the map. The stages are listed below. Gauss hardly travelled. He spent his life within a hundred kilometres of Braunschweig and Göttingen. Map: © OpenStreetMap contributors

October 1, 2026 Read

The sum of 1 to 100

Gauss’s most famous story is from when he was about 9 years old. To keep the class busy, the Professor Büttner told the students to add up all the numbers from 1 to 100. In a few seconds, Gauss placed the blackboard on the table with the correct answer: 5050. (The story was told by the first biographer of Gauss, and the exact numbers in the account may have been different, but the idea is the same.)

October 1, 2026 Read

The 17 sided polygon

Since the Greeks, it was known how to construct regular polygons with 3, 4, 5 and 15 sides with a ruler and compass, and all that come out of them doubling the number of sides (6, 8, 10, 12…). For 2000 years, no one found another. Gauss showed that constructing the $n$ sided polygon is the same as calculating $\cos\frac{2\pi}{n}$ using only integers, the four operations and square roots. And ruler and compass know how to take root square: it is exactly the construction that we saw in the chapter on Descartes. For 17 sides, he reached

October 1, 2026 Read

Clock arithmetic

If it is 15h now, what time will it be in 10 hours? It’s not “25 h”, but 1 am: on the clock, after 24 the count starts again. Gauss gave a name and symbol to this idea in Disquisitiones: $$25 \equiv 1 \pmod{24}$$ It reads “25 is congruous to 1, modulo 24”: both leave the same remainder when divided by 24. The symbol $\equiv$ works almost like an equals sign: you can add and multiply on both sides.

October 1, 2026 Read

The fundamental theorem of algebra

In his doctoral thesis, in 1799, Gauss proved that every polynomial of degree $n$ has $n$ roots, as long as complex numbers are accepted. For example, $x^2 + 1 = 0$ has no real solution, but has two complex ones: $$x^2 + 1 = (x - i)(x + i).$$ Gauss also popularized the way of drawing complex numbers as points on a plane, the plane complex: $a + bi$ is the point $(a, b)$. Norwegian Caspar Wessel (1797) and Swiss Jean-Robert Argand (1806) had the same idea, but almost no one read their work; was the fame of Gauss, from 1831, who spread the drawing. That’s why it is also called Argand-Gauss plane. And then a nice connection appears with the polygon with 17 sides. The three roots of $x^3 = 1$ are

October 1, 2026 Read

Counting prime numbers

Euler proved that the primes never end (see his chapter). But how many are there? At 15 years old, Gauss He spent his free time counting cousins in tables, one thousand at a time. Did you notice that, close to a number $x$, more or less 1 in every $\ln x$ number is prime. Hence the estimate: the number of primes up to $x$, which we call $\pi(x)$, is similar to

October 1, 2026 Read

Ceres and the least squares method

On January 1, 1801, astronomer Giuseppe Piazzi, in Palermo, discovered a tiny dot that moved among the stars: it was Ceres, today classified as a dwarf planet. Piazzi accompanied him for 41 days, until it disappeared in the Sun’s glare. With so few observations, no one knew how to calculate the orbit, and Ceres seemed lost forever. Gauss, at 24 years old, created a new method and calculated the orbit in a few weeks. On the night of 31 December 1801, astronomer Franz von Zach pointed the telescope where Gauss indicated, and there it was Ceres.

October 1, 2026 Read

The normal curve, Gauss's bell

If measurement errors are random, how are they distributed? In Theoria motus, Gauss started from the idea that the mean is the best estimate and arrived at the shape of the error curve: $$f(x) = \frac{1}{\sigma\sqrt{2\pi}}\, e^{-\frac{(x - \mu)^2}{2\sigma^2}},$$ where $\mu$ is the mean and $\sigma$ is the standard deviation. It is the normal curve, or Gaussian bell: high in medium, symmetrical, and increasingly lower away from the mean. It appears everywhere, from the heights of the people to the errors of a machine.

October 1, 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 1, 2026 Read

Systems of equations: the determinant and Gaussian elimination

To calculate orbits using least squares, Gauss needed to solve systems of equations with many unknowns. Two names that every student knows come from this story: the determinant and the Gaussian elimination. The determinant. Take a system with two equations: $$\begin{cases} a\,x + b\,y = e \\ c\,x + d\,y = f \end{cases} \quad\Longrightarrow\quad x = \frac{e\,d - b\,f}{a\,d - b\,c}, \qquad y = \frac{a\,f - e\,c}{a\,d - b\,c}.$$ It all depends on the number $ad - bc$ in the denominator: if it is not zero, the system has a single solution. This number “determines” the system.

October 1, 2026 Read

Gaussian quadrature: integrate with two points

Quadratura circuli: a vector-redrawn cover featuring geometric constructions for squaring the circle.Euler’s Eyes redraw “Squaring” is the old name for calculating an area. The Greeks approximated the area of the circle by polygons with increasingly more sides, and integral calculus was born. The practical problem is: how to calculate the area under a curve doing few math?

October 1, 2026 Read