Euler

Tom Rike
Berkeley Math Circle March 14, 2004

1 Background
Leonhard Euler was born in 1707 in Basel, Switzerland and died in 1783 in St. Petersburg,
Russia. He influenced all areas of mathematics in the Eighteenth Century, consolidating all
that was known in any given branch, extending these branches, and expanding the frontiers
by inventing new branches. He was a genius with a phenomenal memory, amazing insight and
a selfless pursuit for the truths in mathematics. In several instances, he withheld publication
of his results when he knew of other younger or less well known mathematicians, who had
also arrived at the sane result. He wrote textbooks that are still models of our textbooks in
use today. He did not try to hide his methods, but rather shared the excitement of discovery
that he experienced. The great mathematician Laplace said “ Read Euler, read Euler.
He is the master of us all.” Sadly, until recently one could not do this in English.
Although the collected works of Euler take up over eighty volumes, none were available in
English. A modern proponent of reading Euler was George Polya in his great book on how to
do mathematics, Mathematics and Plausible Reasoning [1]. Springer-Verlag has reprinted a
few of his most famous texts in English. They are Elements of Algebra [2] and Introduction to
Analysis of the Infinite[3],[4]. Two recent publications from the Mathematical Association of
America were invaluable in my preparations. They are Journey Through Genius and Euler,
The Master of Us All, by William Dunham. [5],[6]. I recommend them to anyone who wants
to know more about the great moments in mathematics. A major part of this talk is based
on the material in these books. Dunham opens his book with an inscription for the great
architect Christopher Wren who is buried in St. Paul’s Cathedral. “Lector, si monumentum
requiris, circumspice”. Today, I say to those of you looking for the monument in honor of
Euler. “ Auditor, si monumentum requiris, in mathematica circumspice”.

2 Geometry
One of the oldest inequalities about triangles is that relating the radii of the circumcircle
and incircle. It was proved by Euler and is contained in the following theorem and corollary.
Proofs are given in Geometry Revisited by Coxeter and Greitzer [7]. It is published by the
Mathematical Association of America and should be on the bookshelf of everyone interested
in geometry.

Theorem [Euler 1765] Let O and I be the circumcenter and incenter, respectively,
of a triangle with circumradius R and inradius r; let d be the distance OI. Then
d2 = R2 − 2Rr

1
Corollary In a triangle with circumradius R and inradius r, R ≥ 2r.

Theorem [Euler 1767] In a triangle, let O be the circumcenter, G be the centroid, and
H be the orthocenter. The points O, G and H are collinear in that order and GH = 2GO.
The line is known as the Euler line.

3 Number Theory
Euler completed Euclid’s theorem about even perfect numbers showing that there are no
others than the ones Euclid found.

Theorem [Euclid Book IX.36] If as many numbers as we please beginning from a unit
be set out continuously in double proportion, until the sum of all becomes prime, and if the
sum multiplied into the last make some number, the product will be perfect.

Euler used the continuous to find insights into the discrete. He found a fantastic relation
between an infinite series of reciprocals of powers of integers and an infinite product of ex-
pressions indexed by primes. This led Riemann to the most famous unsolved problem in
mathematics, the Riemann Hypothesis. See the wonderful account of all of this in the re-
cently published Prime Obsession [10].
∞


1 1
Theorem [Euler 1737] = 1 , where p is prime.
p 1 − ps
s
k=1 k
n
Fermat went to his grave thinking that his conjecture that 22 + 1 generated prime numbers
5
for n ≥ 0 was true. Euler in 1730 showed that 22 + 1 = 4, 294, 967, 297 is not prime. Al-
though Fermat writes that he has proved his “little” theorem, the proof was never published
and Euler remedied this by publishing a proof in 1731. During his lifetime he came up with
at least three different proofs, including a generalization of the theorem using a function he
invented, the Euler φ function, although it was Gauss who first used the Greek letter φ
to denote the function in his great masterpiece, Disquisitiones Arithmeticae. The function
φ(n) gives the number of positive integers less than or equal to n that are relatively prime
to it.

Theorem [Fermat 1640] If p is a prime then p divides ap − a.

Theorem [Euler 1750] aφ(n) ≡ 1 mod n.

Fermat also claimed to have a proof that every prime of the form 4k + 1 has a unique
representation as a sum of two squares. Euler in 1749, published a proof. He was also able
to show in 1751 that every number could be represented as the sum of four rational squares,
but was unable to make the next step to integer squares. His young protege, Joseph Louis
Lagrange, building on Euler’s work, completed the proof in 1772. This must have spurred
Euler on, since later in the year he published a paper where he “...begins by congratulating
Lagrange on his achievement, then, rightly describing Lagrange’s proof as ‘far-fetched and
laborious’, he proceeds to give a new and elegant variant of the proof for the sum of two
squares showing finally that applies equally well ... to X 2 + Y 2 + Z 2 + T 2 .” [11]
4 Infinite Series
One of the highlights of the year in calculus classes that I teach (at least for me) , comes after
the Advanced Placement exam in May. We see how the 28 year old Euler in 1735 solved a
problem known as the Basel Problem. This problem had been proposed by Jakob Bernoulli
in 1689 when he collected all of the work on infinite series of the 17th century in a volume
entitled Tractatus De Seriebus Infinitis. Bernoulli’s comment in this volume was that the
evaluation “is more difficult than one would expect”. Little did he know how difficult it
really was and that it would take almost 50 years for the problem to be solved. The problem
is to find the sum of the reciprocals of the squares of all of the positive integers. It was
known that the sum was less than two. After seeing Euler’s brilliant insight and proof, we
then see how the problem can be solved to the satisfaction of mathematicians today, using
only knowledge gained in first year calculus. First Euler found a way to sum the series to six
decimal places. ( To do this term by term would require over 1,000,000 terms.) Of course,
Euler found a clever way around this. If you know some calculus, see How Euler Did It, Dec
2003, by Ed Sandiver at MAA Online [9]. Then he found an even better way to make the
series converge faster, now known as the Euler-MacLaurin Summation Formula, finding
the sum to 20 decimal places. Euler then somehow recognized π in the answer and started
experimenting with the infinite polynomial expansion for sin x. Using analogous results from
the theory of nth degree polynomials he was able to find the exact sum.
∞

1 π2
Theorem [Euler 1735] 2
= .
k=1 k 6

5 Areas for Further Research

We have only scratched the surface. In November of 1983, the Mathematical Association of
America journal, Mathematics Magazine devoted an entire issue to commemorate the two
hundredth anniversary of the death of Euler [8]. A glossary at the end of the magazine col-
lected over forty theorems, terms, and formulae that contain the name Euler. In geometry
there is Euler’s (nine-point) Circle. From the classification of polyhedra comes the Euler
Characteristic and its generalization leading to topology. The Königsburg Bridge problem
led to the Euler Circuit which opened up another new branch of mathematics, graph theory.
To represent logical relations the Euler (Venn) Diagrams were born. In combinatorics we
have the Eulerian Numbers and Euler’s Theorem on Partitions. Euler’s Theorem for pentag-
onal numbers and its connection to partitions and the pattern in the sequence of numbers
generated by the sum of the divisors of n is astounding. See [1] pp 90-101. In attempting to
answer to the question about what value should be assigned 12 !, the gamma function (Euler’s
Second Integral) was invented. Euler’s formula, eiθ = cos θ + i sin θ that provides the link be-
tween exponential functions and trigonometric
 functions is another jewel in the crown. Then
1 1 1
there is Euler’s constant, γ = n→∞lim 1 + + + · · · + − ln n ≈ .57721566490153286,
2 3 n
which appears in many places in numerical analysis. Euler calculated its value to 16 decimal
places. It is still not known whether γ is rational or irrational.
6 Some Problems
1. What is the answer to the following clue from a cryptic crossword in The Guardian?
Ideal anesthetic, such as 28. (7,6) [ The ordered pair (7,6) means the answer
consists of a seven letter word followed by a six letter word. ]
∞

1
2. Find 2
.
n=1 (2n)

∞

1
3. Find .
n=1 (2n − 1)
2

∞

1
4. Find 4
.
n=1 n

7 References
1. George Polya, Mathematics and Plausible Reasoning, Princeton University Press, 1954.

2. Euler, Elements of Algebra, Springer-Verlag, Reprint of London: Longman, Ormew,

and Co. 1840 edition.

3. Euler, Introduction to Analysis of the Infinite, Book I, Springer-Verlag N.Y., 1988.

4. Euler, Introduction to Analysis of the Infinite, Book II, Springer-Verlag N.Y., 1990.

5. William Dunham, Journey Through Genius, John Wiley and Son, 1990.

6. William Dunham, Euler, The Master of Us All, Mathematical Association of America,

1999.

7. H.S.M. Coxeter, S.L. Greitzer. Geometry Revisited. Mathematical Association of

America, 1967.

8. A Tribute to Leonhard Euler, Mathematics Magazine, Vol. 59 No. 5, November 1983.

9. Ed Sandifer, How Euler Did It, MAA Online, Nov. 2003, Dec. 2003, Jan. 2004,
http://www.maa.org/news/columns.html

10. John Derbyshire, Prime Obsession, Joseph Henry Press, 2003.

11. André Weil, Number Theory: An Approach Through History, Birkhäuser Boston Inc.,
1984.

12. Euclid, The Bones: A Handy, Where-to-find-it Pocket Reference Companion to Eu-
clid’s Elements, Green Lion Press, 2002.

If you have comments, questions or find glaring errors, please contact me by e-mail at the
following address: tricycle222@earthlink.net