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 integralDidn’t follow through, like Laplace didDid Laplace really say “Read Euler. Read Euler. He is the Master of us all!”No

#25 – Fourier series

1770sOdd functions onlyElliptical orbitsAlso an early use of subscript-like notation[0], [4], [8], etc.

#24 - Paddle wheel, Screw propeller

Described for 1753 Paris PrizePropulsion of ships without wind2nd placeActually built about 80 years later

# 23 - Centrifugal pump

Invented at the command of Frederick the GreatDeveloped about a hundred years laterNew patents, often for nautical applications

# 22 – Differential equationsof fluid dynamics

Conservation of mass in a stream lineEquation of continuity

m dxdy dz0yx z

vv vd

dt x y z

# 21 – Knight’s tour

“… and sufficient” part of Koenigsburg Bridge Problem

# 20 - Statistics of observational data

Best fit equations for observation of a cometUsed absolute value, not least squares

# 19 – Partition numbers

Naude’s problemHow many ways can you write n as a sum?Ramanujan

# 18 – Generating functions

Invented them to solve the partition problem in 1741Using the coefficients of a power series to count somethingRelations with recursive calculations

# 17 – Zeta function

Sum of reciprocals of nth powersRiemann extended it from positive reals to complex planeSum-Product formula - 1

1 1

1 ss

p prime pn

# 16 – Gamma function

First letter to GoldbachGeneralized n!Suggested fractional derivatives

# 15 – FLT n = 4

First published proofFermat probably did itAlso had a false general proof, never published

# 14 – Density of primes

Showed diverges

1

p prime p

# 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 integralsGeneralizes trigonometric functionsAlso series for arc length of an ellipse

# 11 - Derangements

Permutations that move every elementShowed probability approaches 1/eGenoese lotteryCommand of Frederick II

# 10 – integrating factor

Reduces order of a differential equationOften attributed to ClairautEuler was 2 years earlier

# 9 – E = edges

Before Euler, nobody had identified Edges on a solid as a mathematical objectDescartes came closeCounted edges by counting plane angles and dividing by 2

# 8 – Venn diagrams

Venn called them Eulerian CirclesLetters to a German PrincessAid to logicSee “How Euler Did It” – January, 2004

# 7 – Algebra = staticsCalculus = dynamics

Calculus is the way to study the worldEvery problem is an optimization problem

# 6 -

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

2 2f f

y x x y

# 5 - Precalculus

Introductio in analysin infinitorumAll the prerequisites to calculus

# 4 – Transit of Venus

1761 and 1769Astronomical unit (distance to sun)LongitudeInternational scientific cooperationEli Maor, Thomas Pynchon

# 3 - Coauthorship

Co-published with Johann Albrecht and with Charles on Paris PrizeNo earlier important work was coauthoredErdos couldn’t have functioned without coauthorship

# 2 -

Modern calculus curriculumFirst example of chain rule for a transcendental function=

xeeede

xe xe e xe e e xe e e e dx

# 1 - Function

Function became a mathematical objectFunction became an acceptable answer to a problem

And that’s not all

3-d coordinate systemsBest shape for teeth on gearsTelescopes and microscopesLogarithms in theory of music…