Euler

The life of Leonhard Euler

Timeline 1707Born on April 15, in Basel, Switzerland. 1720Enters the University of Basel, at 13 years old. He began studying with Johann Bernoulli. 1727Moves to the Academy of Sciences in Saint Petersburg, Russia. 1735Solve the Basel problem and analyze the Königsberg bridges. 1738He loses almost all vision in his right eye. 1741He goes to the Berlin Academy, at the invitation of Frederick II of Prussia. 1748Publishes the Introductio in analysin infiniterum, where $e^{ix} = \cos x + i \sin x$ appears. 1750In a letter to Goldbach, he states the formula for polyhedra $V - A + F = 2$. 1766Returns to Saint Petersburg, at the invitation of Catherine the Great. A cataract affects the left eye. 1771Cataract surgery fails and he is left practically blind. Keep working. 1783Dies in 18 September, in Saint Petersburg. ◀ Previous Follow the journey Next ▶ 1707–1727 BaselBorn and raised in Switzerland. At 13 he enters the university and starts Saturday lessons with Johann Bernoulli. 1727 LübeckHe travels down the Rhine by boat, crosses Germany by coach and sets sail from Lübeck for Russia. 1727–1741 Saint PetersburgHe works at the Academy of Sciences. He solves the Basel problem, studies the Königsberg bridges and loses the sight of his right eye. 1741–1766 BerlinAt the invitation of Frederick II he moves to the Berlin Academy. He publishes the Introductio in analysin infinitorum and the formula $e^{ix}=\cos x+i\sin x$. 1766–1783 Saint PetersburgHe returns at the invitation of Catherine the Great. He goes almost blind but keeps dictating papers until his death in 1783. Enable JavaScript to see the map. The stages are listed below. Euler’s cities, from Basel to Saint Petersburg. Click a stage or use Follow the journey. Map: © OpenStreetMap contributors

October 10, 2026 Read

The language that Euler left us

Part of Euler’s influence is in things that seem to have always existed. He created or popularized: $f(x)$ to indicate a function of $x$; the letter $e$ for the base of natural logarithms, $e \approx 2{,}71828$; the letter $\pi$ for the ratio between the length of the circumference and the diameter (the symbol already existed, but it was Euler who spread it); $i$ for the imaginary unit, with $i^2 = -1$; $\Sigma$ to indicate a sum. The equals sign and the letter π Symbols have birthdays. The sign $=$ was invented by the Welshman Robert Recorde in 1557, in the book The Whetstone of Witte, so as not to have to write “is equal to” all the time. He chose two parallel strokes of the same length because, as he explained, no two things can be more equal. The letter $\pi$, the initial of the Greek word periphéreia, the circumference, was first used in today’s sense by another Welshman, William Jones, in 1706. Euler began using it in the 1730s, and his books made it standard.

October 10, 2026 Read

The Basel problem

In 1650, Pietro Mengoli asked how much the sum of the inverse squares is worth: $$1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2} = \ ?$$ The Bernoullis tried and failed. The sum grows slowly: with 10 terms it gives $1{,}5498$, with 100 terms gives $1{,}6350$, with 1000 terms gives $1{,}6439$. No one recognized the number to which she walked. See the figure: each bar is the sum up to $N$ terms.

October 10, 2026 Read

The number $e$ and compound interest

Banca, trapeza, mensa: the table of the first bankers, between 1450 and 1650. Hence the words bank and banking come from.Euler’s Eyes Collection Before reaching Euler, the number $e$ passed through the banks. The vocabulary of money comes from Latin: depositum (what was put, from ponere), creditum (what was trusted, from credere, to believe) and juro (from jus, juris, the right).

October 10, 2026 Read

Euler's formula and the "most beautiful equation"

The number $e$ can be written as a series: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + \cdots$$ Euler had the courage to put an imaginary number in the exponent. Changing $x$ to $i\theta$ and using $i^2 = -1$, $i^3 = -i$, $i^4 = 1$: $$e^{i\theta} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + i\frac{\theta^5}{5!} - \cdots$$ Separating terms without $i$ from terms with $i$:

October 10, 2026 Read

The spring and differential equations

A differential equation is an equation in which the unknown is a function, and the equation speaks of its derivatives. Euler organized this path into a scheme that we still use today: observe a phenomenon, write the equation, find the function that solves it and draw the graph. Euler’s scheme: phenomenon → differential equation → solution function → graph. The derivative also points to the maximum and minimum points.Euler’s Eyes Collection The phenomenon. In 1678, Robert Hooke summarized the behavior of a spring in Latin: ut tensio, sic vis, “as extension, so strength.” A spring of constant $k$, stretched by $x$, pulls back with strength $F = -kx$.

October 10, 2026 Read

The Königsberg bridges and the birth of graphs

Königsberg at 1652, with the river Pregel and its islands. You can count the bridges.Merian-Erben · public domain The city of Königsberg (today Kaliningrad, Russia) was crossed by the Pregel River, which formed two islands. Seven bridges connected the four regions of the city. Residents wondered if it was possible take a walk across each bridge exactly once.

October 10, 2026 Read

$V - A + F = 2$ and the soccer ball

In 1750, in a letter to mathematician Christian Goldbach, Euler noted a relationship between the number of vertices ($V$), edges ($A$) and faces ($F$) of a convex polyhedron: $$V - A + F = 2.$$ Rotate the polyhedra with your finger or mouse and check the calculation: V, A and F are counted by the program from the drawing itself. In all cases it gives 2. Ative o JavaScript para ver esta figura interativa.

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

Fermat's number that was not prime

Pierre de Fermat believed that all numbers of the form $2^{2^n} + 1$ were prime. For $n = 0, 1, 2, 3, 4$ works: 3, 5, 17, 257 and 65 537 are primes. The next one is $$2^{32} + 1 = 4\,294\,967\,297.$$ Testing all dividers by hand is impractical. In 1732, Euler showed that this number is divisible by 641. An elegant way to look at this uses two ways of writing 641:

October 10, 2026 Read

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

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