Seven Bridges of Königsberg problem (1736)
Proved no walk crosses each bridge exactly once, founding graph theory
Science · Maths, astronomy, medicine & more
Leonhard Euler
1707–1783 · Swiss
Mathematics — analysis, number theory and mechanics
Leonhard Euler, born in Basel and taught by Johann Bernoulli, was the most prolific mathematician in history, working mainly at the St Petersburg Academy and, under Frederick the Great, in Berlin. He founded graph theory with the Königsberg bridges problem, solved the Basel problem, gave us the notation e, i, f(x) and Σ and popularised π, and made fundamental contributions to analysis, number theory, mechanics, optics and astronomy. He kept producing work at an astonishing rate even after becoming almost totally blind.
How to spot them
- ▸Seven Bridges of Königsberg — birth of graph theory
- ▸Euler's identity e^(iπ) + 1 = 0; the number e
- ▸V − E + F = 2 for polyhedra; the Euler line; Euler's totient function
- ▸Basel, St Petersburg and Berlin; blind but still calculating
Key discoveries & works
Solution of the Basel problem (sum of reciprocal squares equals π²/6) (1734–35)
Made his name by summing 1 + 1/4 + 1/9 + … exactly
e to the power iπ plus 1 equals 0 (1748 (formula e^(ix) = cos x + i sin x))
The identity linking e, i, π, 1 and 0, often voted the most beautiful equation
Introductio in analysin infinitorum (1748)
Made the function the central concept of analysis and set out the exponential function and e
Polyhedron formula V − E + F = 2 (1750s)
Vertices minus edges plus faces equals two for any convex polyhedron
Letters to a German Princess (1768–72)
Popular letters on science and philosophy written to a Prussian princess
Mechanica (1736)
Recast Newtonian mechanics in the language of analysis
Key ideas
- Introduced or standardised the notation f(x), e, i, Σ for summation, and π for the circle ratio
- Founded graph theory and anticipated topology through the Königsberg problem and the polyhedron formula
- Euler–Lagrange equation: basis of the calculus of variations
- Euler's totient function φ(n) and the generalisation of Fermat's little theorem
Quick-fire facts
- •Born in Basel; studied under Johann Bernoulli
- •Worked at the St Petersburg Academy (1727–41), the Berlin Academy under Frederick the Great (1741–66), then St Petersburg again until his death
- •Lost the sight of his right eye in the 1730s and became almost totally blind around 1771, yet his output increased
- •Published over 800 works; his collected Opera Omnia is still not fully complete
- •Condorcet wrote that on 18 September 1783 'he ceased to calculate and to live'
- •The Euler line passes through a triangle's orthocentre, centroid and circumcentre
- •Appeared on the Swiss ten-franc note
- •Euler's number e is about 2.71828
Drill it — 9 cards
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Sources
- MacTutor History of Mathematics, University of St Andrews — Leonhard Euler
- William Dunham — Euler: The Master of Us All