If you have ever written \(f(x)\), used the letter \(e\), summed with \(\Sigma\) or called \(\sqrt{-1}\) by the name \(i\), you have used notation made standard by one man: Leonhard Euler. In a career spanning more than 50 years, much of it partly or fully blind, he produced more mathematics than anyone before or since.
From Basel to St Petersburg
Euler was born on 15 April 1707 in Basel, Switzerland. His father, a Protestant pastor, hoped he would join the church, but the boy's talent caught the attention of Johann Bernoulli, then one of Europe's leading mathematicians. Bernoulli gave the teenage Euler private guidance on Saturday afternoons and persuaded his father that Leonhard was destined for mathematics.
In 1727, aged 20, Euler moved to the new Imperial Academy of Sciences in St Petersburg, Russia, invited by Johann's sons Daniel and Nicolaus Bernoulli. There he worked on everything from ship design to cartography. Around 1738 he lost the sight of his right eye after an illness, yet his output barely slowed.
In 1741 King Frederick the Great invited him to the Berlin Academy, where he stayed 25 years, writing hundreds of papers and popular works such as Letters to a German Princess, explaining science to a general reader. In 1766 he returned to St Petersburg. Soon after, a cataract left him almost totally blind. Astonishingly, nearly half of his work was produced after this, dictated to his sons and assistants and carried out with a phenomenal memory and mental arithmetic. He died on 18 September 1783 in St Petersburg; as one tribute put it, he "ceased to calculate and to live."
Just How Prolific?
Euler's collected works, the Opera Omnia, have been published since 1911 and fill more than 80 large volumes. The standard catalogue of his writings, compiled by Gustaf Eneström, lists 866 items. The St Petersburg Academy was still publishing his unpublished papers decades after his death.
His contributions cover almost every area of mathematics and physics:
- Analysis: the function concept, infinite series, the exponential and logarithmic functions, the gamma function.
- Number theory: proofs of many of Fermat's claims, the totient function \(\varphi(n)\), quadratic reciprocity patterns.
- Graph theory and topology: the Königsberg bridges and the polyhedron formula.
- Mechanics and fluids: the Euler equations of fluid flow, rigid-body rotation, buckling of columns.
- Notation: \(e\), \(i\), \(\pi\) (popularised), \(f(x)\), \(\Sigma\) and the modern trigonometric functions as ratios.
The Basel Problem
In 1650 Pietro Mengoli asked for the exact value of
$$ 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots = \sum_{n=1}^{\infty} \frac{1}{n^2} $$
The Bernoulli brothers tried and failed. Around 1734–1735 the young Euler announced the answer:
$$ \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6} $$
His original argument boldly treated \(\frac{\sin x}{x}\) as an "infinite polynomial" factorised by its roots \(\pm\pi, \pm 2\pi, \ldots\), and compared coefficients. It was not fully rigorous by modern standards, but Euler later gave sounder proofs, and the result made him famous across Europe.
Worked Example 1: Checking the Basel Sum
Why was the answer so hard to guess? Because the series converges slowly. Let us add the first 10 terms:
$$ 1 + 0.25 + 0.11111 + 0.0625 + 0.04 + 0.02778 + 0.02041 + 0.01563 + 0.01235 + 0.01 $$
Adding step by step: \(1.25 \to 1.36111 \to 1.42361 \to 1.46361 \to 1.49139 \to 1.51180 \to 1.52743 \to 1.53977 \to 1.54977\).
Now compute Euler's value:
$$ \frac{\pi^2}{6} = \frac{9.86960}{6} = 1.64493 $$
After 10 terms we are still about 0.095 short. Even after 1,000 terms the error is about 0.001. Euler did not just add terms; he invented acceleration methods (later called Euler–Maclaurin summation) to compute the sum to around 20 decimal places, which helped him recognise \(\pi^2/6\).
The Seven Bridges of Königsberg
The Prussian city of Königsberg (now Kaliningrad, Russia) sat on the river Pregel, with two islands joined to the banks and each other by seven bridges. Residents wondered whether one could walk through the city crossing every bridge exactly once.
In 1736 Euler showed it was impossible, and in doing so founded graph theory. He realised that only the connections matter: represent each land mass by a point (vertex) and each bridge by a line (edge).
Worked Example 2: Euler's Bridge Argument
Step 1: Count the degree of each vertex (the number of bridges touching each land mass). In Königsberg:
- the central island (Kneiphof): 5 bridges
- the north bank: 3 bridges
- the south bank: 3 bridges
- the eastern island: 3 bridges
Check: the sum of degrees is \(5 + 3 + 3 + 3 = 14 = 2 \times 7\), since each bridge has two ends.
Step 2: Apply Euler's reasoning. On a walk, every time you pass through a land mass you use one bridge to enter and one to leave, so each intermediate land mass must have an even degree. Only the starting and finishing points may have odd degree.
Step 3: Conclude. A walk using every edge once can exist only if 0 or 2 vertices have odd degree. Königsberg has 4 odd vertices, so no such walk exists.
Euler's Polyhedron Formula
Around 1750, Euler noticed that for any convex polyhedron with \(V\) vertices, \(E\) edges and \(F\) faces,
$$ V - E + F = 2 $$
Worked Example 3: Testing the Formula
Cube: \(V = 8\), \(E = 12\), \(F = 6\), so \(8 - 12 + 6 = 2\). Correct.
Square pyramid: 4 base corners plus 1 apex gives \(V = 5\); 4 base edges plus 4 slanted edges gives \(E = 8\); 1 square plus 4 triangles gives \(F = 5\). Then \(5 - 8 + 5 = 2\). Correct.
Icosahedron: 20 triangular faces, each with 3 edges, each edge shared by 2 faces, so \(E = \tfrac{20 \times 3}{2} = 30\). Each vertex meets 5 faces, so \(V = \tfrac{20 \times 3}{5} = 12\). Then \(12 - 30 + 20 = 2\). Correct.
The number 2 is now called the Euler characteristic and is one of the founding ideas of topology.
The Most Beautiful Equation
In his 1748 textbook Introductio in analysin infinitorum, Euler presented the formula
$$ e^{ix} = \cos x + i \sin x $$
Putting \(x = \pi\) gives \(\cos\pi = -1\) and \(\sin\pi = 0\), so
$$ e^{i\pi} + 1 = 0 $$
This links the five most important constants in mathematics, \(0, 1, e, i\) and \(\pi\), in one short line. Surveys of mathematicians regularly name it the most beautiful equation ever written.
Did You Know?
- In 1732 Euler disproved Fermat's guess that all numbers \(2^{2^n} + 1\) are prime, showing \(2^{32} + 1 = 4294967297 = 641 \times 6700417\).
- Euler had 13 children, of whom 5 survived childhood, and reportedly wrote papers with a baby on his lap.
- Pierre-Simon Laplace advised students: "Read Euler, read Euler, he is the master of us all."
Key Takeaways
- Euler (1707–1783) worked in Basel, St Petersburg and Berlin and produced over 800 works, many while blind.
- He solved the Basel problem, showing \(\sum 1/n^2 = \pi^2/6\).
- His Königsberg bridges solution (1736) founded graph theory, and \(V - E + F = 2\) helped found topology.
- Much of our standard notation, including \(e\), \(i\), \(f(x)\) and \(\Sigma\), comes from Euler.
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