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.”
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.
“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.”
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
mechanics, physics
geometry, including trigonometry
astronomy
architecture, ballistics, philosophy,
theory of music, theology, etc.
40%
28%
18%
11%
3%
Euler’s prime number quadratic (1772)
P(n) =
2
n
+ n + 41
Reflections on learning mathematics 1
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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
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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
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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