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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

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Twenty other ideas

Twenty Other Ideas

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


26 laplace transform
# 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
#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
#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
# 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 equations of fluid dynamics
# 22 – Differential equationsof fluid dynamics

  • Conservation of mass in a stream line

  • Equation of continuity


21 knight s tour
# 21 – Knight’s tour

  • “… and sufficient” part of Koenigsburg Bridge Problem


20 statistics of observational data
# 20 - Statistics of observational data

  • Best fit equations for observation of a comet

  • Used absolute value, not least squares


19 partition numbers
# 19 – Partition numbers

  • Naude’s problem

  • How many ways can you write n as a sum?

  • Ramanujan


18 generating functions
# 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
# 17 – Zeta function

  • Sum of reciprocals of nth powers

  • Riemann extended it from positive reals to complex plane

  • Sum-Product formula -


16 gamma function
# 16 – Gamma function

  • First letter to Goldbach

  • Generalized n!

  • Suggested fractional derivatives


15 flt n 4
# 15 – FLT n = 4

  • First published proof

  • Fermat probably did it

  • Also had a false general proof, never published



13 continued fractions
# 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
# 12 – elliptic integrals

  • Summation formula for elliptic integrals

  • Generalizes trigonometric functions

  • Also series for arc length of an ellipse


11 derangements
# 11 - Derangements

  • Permutations that move every element

  • Showed probability approaches 1/e

  • Genoese lottery

  • Command of Frederick II


10 integrating factor
# 10 – integrating factor

  • Reduces order of a differential equation

  • Often attributed to Clairaut

  • Euler was 2 years earlier


9 e edges
# 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
# 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 statics calculus dynamics
# 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
# 5 - Precalculus

  • Introductio in analysin infinitorum

  • All the prerequisites to calculus


4 transit of venus
# 4 – Transit of Venus

  • 1761 and 1769

  • Astronomical unit (distance to sun)

  • Longitude

  • International scientific cooperation

  • Eli Maor, Thomas Pynchon


3 coauthorship
# 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


# 2 -

  • Modern calculus curriculum

  • First example of chain rule for a transcendental function

=


1 function
# 1 - Function

  • Function became a mathematical object

  • Function became an acceptable answer to a problem


And that s not all
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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