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 2 nd 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 equations of 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 n th 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 = statics Calculus = 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

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

# 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 …