Euler's Unnamed Innovations: 26 Significant Contributions Beyond His Legacy

1. Twenty Other Ideas Countdown of two dozen of Euler’s big ideas that don’t have his name on them

2. # 26 - Laplace transform • In his 1769 Integral Calculus book, Euler wrote the Laplace Transform integral • Didn’t follow through, like Laplace did • Did Laplace really say “Read Euler. Read Euler. He is the Master of us all!” • No

3. #25 – Fourier series • 1770s • Odd functions only • Elliptical orbits • Also an early use of subscript-like notation • [0], [4], [8], etc.

4. #24 - Paddle wheel, Screw propeller • Described for 1753 Paris Prize • Propulsion of ships without wind • 2nd place • Actually built about 80 years later

5. # 23 - Centrifugal pump • Invented at the command of Frederick the Great • Developed about a hundred years later • New patents, often for nautical applications

6. # 22 – Differential equationsof fluid dynamics • Conservation of mass in a stream line • Equation of continuity

7. # 21 – Knight’s tour • “… and sufficient” part of Koenigsburg Bridge Problem

8. # 20 - Statistics of observational data • Best fit equations for observation of a comet • Used absolute value, not least squares

9. # 19 – Partition numbers • Naude’s problem • How many ways can you write n as a sum? • Ramanujan

10. # 18 – Generating functions • Invented them to solve the partition problem in 1741 • Using the coefficients of a power series to count something • Relations with recursive calculations

11. # 17 – Zeta function • Sum of reciprocals of nth powers • Riemann extended it from positive reals to complex plane • Sum-Product formula -

12. # 16 – Gamma function • First letter to Goldbach • Generalized n! • Suggested fractional derivatives

13. # 15 – FLT n = 4 • First published proof • Fermat probably did it • Also had a false general proof, never published

14. # 14 – Density of primes • Showed diverges

15. # 13 – continued fractions • Unless you are a specialist, you don’t know anything about continued fractions that isn’t in Euler’s first paper. • And you probably don’t know all of that, either.

16. # 12 – elliptic integrals • Summation formula for elliptic integrals • Generalizes trigonometric functions • Also series for arc length of an ellipse

17. # 11 - Derangements • Permutations that move every element • Showed probability approaches 1/e • Genoese lottery • Command of Frederick II

18. # 10 – integrating factor • Reduces order of a differential equation • Often attributed to Clairaut • Euler was 2 years earlier

19. # 9 – E = edges • Before Euler, nobody had identified Edges on a solid as a mathematical object • Descartes came close • Counted edges by counting plane angles and dividing by 2

20. # 8 – Venn diagrams • Venn called them Eulerian Circles • Letters to a German Princess • Aid to logic • See “How Euler Did It” – January, 2004

21. # 7 – Algebra = staticsCalculus = dynamics • Calculus is the way to study the world • Every problem is an optimization problem

22. # 6 - • Mixed partial derivatives are equal • Euler knew of no counterexamples, so he did not give continuity conditions

23. # 5 - Precalculus • Introductio in analysin infinitorum • All the prerequisites to calculus

24. # 4 – Transit of Venus • 1761 and 1769 • Astronomical unit (distance to sun) • Longitude • International scientific cooperation • Eli Maor, Thomas Pynchon

25. # 3 - Coauthorship • Co-published with Johann Albrecht and with Charles on Paris Prize • No earlier important work was coauthored • Erdos couldn’t have functioned without coauthorship

26. # 2 - • Modern calculus curriculum • First example of chain rule for a transcendental function =

27. # 1 - Function • Function became a mathematical object • Function became an acceptable answer to a problem

28. And that’s not all • 3-d coordinate systems • Best shape for teeth on gears • Telescopes and microscopes • Logarithms in theory of music • …