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### Twenty Other Ideas

Countdown of two dozen of Euler’s big ideas that don’t have his name on them

# 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

#25 – Fourier series

- 1770s
- Odd functions only
- Elliptical orbits
- Also an early use of subscript-like notation
- [0], [4], [8], etc.

#24 - Paddle wheel, Screw propeller

- Described for 1753 Paris Prize
- Propulsion of ships without wind
- 2nd place
- Actually built about 80 years later

# 23 - Centrifugal pump

- Invented at the command of Frederick the Great
- Developed about a hundred years later
- New patents, often for nautical applications

# 22 – Differential equationsof fluid dynamics

- Conservation of mass in a stream line
- Equation of continuity

# 21 – Knight’s tour

- “… and sufficient” part of Koenigsburg Bridge Problem

# 20 - Statistics of observational data

- Best fit equations for observation of a comet
- Used absolute value, not least squares

# 19 – Partition numbers

- Naude’s problem
- How many ways can you write n as a sum?
- Ramanujan

# 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

# 17 – Zeta function

- Sum of reciprocals of nth powers
- Riemann extended it from positive reals to complex plane
- Sum-Product formula -

# 16 – Gamma function

- First letter to Goldbach
- Generalized n!
- Suggested fractional derivatives

# 15 – FLT n = 4

- First published proof
- Fermat probably did it
- Also had a false general proof, never published

# 14 – Density of primes

- Showed
diverges

# 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.

# 12 – elliptic integrals

- Summation formula for elliptic integrals
- Generalizes trigonometric functions
- Also series for arc length of an ellipse

# 11 - Derangements

- Permutations that move every element
- Showed probability approaches 1/e
- Genoese lottery
- Command of Frederick II

# 10 – integrating factor

- Reduces order of a differential equation
- Often attributed to Clairaut
- Euler was 2 years earlier

# 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

# 8 – Venn diagrams

- Venn called them Eulerian Circles
- Letters to a German Princess
- Aid to logic
- See “How Euler Did It” – January, 2004

# 7 – Algebra = staticsCalculus = dynamics

- Calculus is the way to study the world
- Every problem is an optimization problem

# 6 -

- Mixed partial derivatives are equal
- Euler knew of no counterexamples, so he did not give continuity conditions

# 5 - Precalculus

- Introductio in analysin infinitorum
- All the prerequisites to calculus

# 4 – Transit of Venus

- 1761 and 1769
- Astronomical unit (distance to sun)
- Longitude
- International scientific cooperation
- Eli Maor, Thomas Pynchon

# 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

# 1 - Function

- Function became a mathematical object
- Function became an acceptable answer to a problem

And that’s not all

- 3-d coordinate systems
- Best shape for teeth on gears
- Telescopes and microscopes
- Logarithms in theory of music
- …

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