Leonhard Euler (born 15th April 1707)

“An adventurer…..through a

wonderful mathematical

landscape”

Why the history of mathematics matters

• Story of mathematicians

• Story of mathematics

• Story of learners

Carl Friedrich Gauss

“The study of Euler’s work will remain the

best school for the different fields of

mathematics and nothing else can replace

it.”

Pierre-Simon de Laplace

“Read Euler, read Euler he is the master of

us all”

The father

• Paulus Euler (1670-1745)

• Studied Maths at Basel under

Jakob Bernoulli

• Pastor – Riehen Village (from

1708)

• Euler’s first maths teacher

• Wanted son to study theology

and become a pastor

The high quality textbook

Die Coss - Christoffs Rodolff; mit schönen Exempeln der Coss

durch Michael Stifel gebessert und sehr gemehrt (1553)

Euler’s phi (ɸ) function (totient function) (1760)

ɸ (m) = number of integers (<m) co-prime to m

ɸ(10) = 4 (1,3,7,9)

Explore the landscape:

a) Try the function for yourself

b) What about prime numbers: ɸ(p)?

c) What about powers of prime numbers: ɸ(pα)?

d) What about a product of primes: ɸ(pq)?

e) What about any integer: ɸ(n)?

The mentor

• Johann Bernoulli (1670-1745)

• Professor of Maths at Basel

University (~100 students)

• Euler started at university

October 1720 (age 13)

• Euler’s mentor

• Persuaded Euler’s father that

his son should pursue maths

Personal tuition “I soon found an opportunity to be introduced to a famous professor

named John Bernoulli . . . True, he was very busy and so refused

outright to give me private lessons.”

“I was given permission to visit him freely every Saturday

afternoon and he kindly explained to me everything I could not

understand . . . And this, undoubtedly, is the best method to

succeed in mathematical subjects.”

However, Bernoulli supervised Euler’s study by posing

problems to Euler and by recommending mathematical reading.

This was done on Saturday and Euler spent the rest of the week

solving the problems and trying to trouble his teacher with as

few questions as possible.

The friend

• Daniel Bernoulli (1670-1745)

• Invited Euler to Academy of

Sciences in St Petersburg

• Euler arrived in 1727

• They worked and lodged together

• Daniel back to Basel in 1733

• Euler took over as professor of

mathematics (left in 1741)

The Basel problem

An unexpected solution

“Now, however, against all expectations I have

found an elegant expression for the sum of the series

1+ 1/4 + 1/9 + 1/16 + etc, which depends on the

quadrature of the circle…..I have found that six

times the sum of this series is equal to the square of

the circumference of a circle whose diameter is 1”

Euler, 1735

The sponsors

Frederick the Great (1712-1786)

• Invited Euler to the Berlin Academy

• Euler worked there from 1741-1766

• Productive and peaceful period

• Relationships within the court deteriorated

Catherine the Great (1729-1796)

• Welcomed Euler back to St Petersburg

Academy in 1766

• He remained there until his death

• Most prolific period

Key places

A new branch of geometry “In addition to that branch of geometry which is concerned with

magnitudes, and which has always received the greatest attention,

there is another branch, previously almost unknown, which Leibniz

first mentioned, calling it the geometry of position. This branch is

concerned only with the determination of position and its properties;

it does not involve measurements, nor calculations made with them.

It has not yet been satisfactorily determined what kind of problems

are relevant to this geometry of position, or what methods should be

used in solving them”

The Königsberg bridges problem

Euler’s schematic figure

Lots of practice Years publications percentage in Euler’s lifetime (approx.)

1725-34 35 5%

1735-44 50 7%

1745-54 150 20%

1755-64 110 14%

1765-74 145 19%

1775-83 270 35%

Opera omnia (Series I-III)

algebra, number theory, analysis 40%

mechanics, physics 28%

geometry, including trigonometry 18%

astronomy 11%

architecture, ballistics, philosophy,

theory of music, theology, etc. 3%

Euler’s prime number quadratic (1772)

P(n) = n2 + n + 41

Reflections on learning mathematics 1

• Supportive parents

• Go to an academy with a good sponsor

• Learn from well established textbooks

• Arrange to have private tuition

• Do lots and lots of practice

Reflections on learning mathematics 2

• Have some good maths friends

• Try and do lots in your head

• Read some good mathematics

• Write down your thoughts and share them with others

• Be resilient

• Stay curious

• Just enjoy the exploration

Reflections on developing mathematics

• Build on the work of others

• Lay the foundations for those that follow

• Develop the language and tools needed

• Go with your intuition

• Rigorously deduce new results

• Refine and improve your work

• Be prepared to invent new branches of the subject

Leonhard Euler (died 18th September 1783)

“All celebrated mathematicians

now alive are his disciples:

there is no one who…. is not

guided and sustained by his

genius”

Marquis de Condorcet