A few of Euler's masterpieces

12 April 2018

Leonard Euler wrote more mathematics than anyone in history. It is said that he was responsible for around a third of all the mathematics, physics and mechanical engineering research published in all of Europe between the years 1726-1800. Much of our modern notation is due to him. He left his mark on every subject he touched. In fact, there is a whole Wikipedia page dedicated to simply listing all the things named after him. Almost everything on the list has its own Wikipedia page. Instead of attempting the impossible by trying to summarise of all of his work, we will present a few personal favourites from the world of pure maths and hope that it encourages others to read further and find their own personal favourites.

Geometry

There’s quite a lot going on in this picture but let’s just focus on the miraculous red line in the middle known as the Euler line.

What’s so miraculous about it? After reading what it is, perhaps you’ll agree that it’s very existence is a miracle. Start with three arbitrary points $A, B, C$ and draw the triangle $ABC$. Next, construct the three perpendicular bisectors of the edges of $ABC$. These are the green lines and they all meet at a single point which we label $O$. The gold lines are the medians of the triangle. They are the lines through the vertices and opposing midpoints and they also meet in a single point, which we label $G$. The blue lines are constructed by dropping a perpendicular from each vertex to its opposite edge of the triangle. Again, they meet in a single point which we call $H$. It turns out, and this is what Euler proved, that no mater how the original points $A,B,C$ are arranged, the points $O,G,H$ always line up in on straight line. What’s more, the distance $GH$ is always exactly twice that of $OG$.

Analysis

Logarithms are introduced in school nowadays as being related to exponentiation by the formulas
$$y= a^x \text{ if and only if } x = \log_a(y).$$
It was Euler who first clearly perceived logarithms in this way. Before Euler, logarithms were used by scientists and engineers to simplify calculations by converting multiplication (which was hard) into addition (which was easier). Euler recognised the significance of logarithms as mathematically interesting functions in their own right, independently of their use in calculations. He observed that $a^\delta$ is very slightly larger than 1 when $\delta$ is very slightly larger than 0. In fact, for $\delta$ positive but very small, $a^\delta \approx 1 + k \delta$ for some proportionality constant $k$ which depends on $a$. He gives the numerical examples, $a=10$, $\delta = 0.000001$ for which $k = 2.3026$ and $a=5$, $\delta = 0.000001$ for which $k = 1.60944$, and found that the number $e = \sum_{n=1}^{\infty} \frac{1}{n!} = 2.7182818284\ldots$ is exactly that number with proportionality constant $k=1$. This number $e$ is appropriately called Euler’s number. The power series
$$e^x = 1+x+\frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots.$$
is also due to him. Many expressions become simpler because $k=1$ and this is why it is natural to take logarithms to the base $e$. He also discovered the power series
$$\log_e(1+x) = x-\frac{x^2}{2} + \frac{x^3}{3} – \frac{x^4}{4} + \frac{x^5}{5} – \cdots.$$

Number theory

“These works are recorded to have been completed in six days $\ldots$ because six is a perfect number – not because God required a protracted time, as if He could not at once create all things,$\ldots$ but because the perfection of the works was signified by the number six. For the number six is the first which is made up of its own parts, i.e., of its sixth, third and half, which are respectively one, two and three, and which make a total is six.”

This is an excerpt from St Augustine’s City of God (Part XI Chapter 30) explaining that God created the world in six days because six is the first perfect number. A perfect number is a positive integer which is equal to the sum of all its proper divisors, so excluding the number itself. For example, 6 is perfect because its proper divisors are 1, 2 and 3, and 1+2+3 = 6. The next smallest perfect numbers are 28, 496 and 8128. Perfect numbers have been entertaining the imaginations of mathematicians and non-mathematicians alike for literally thousands of years. In fact, as far back as c. 300 BC, Euclid proved in book IX of his Elements that if $n = 2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers then $n$ is a perfect number. A prime number of the form $2^p-1$ is known as a Mersenne prime. Although it had been conjectured previously, it wasn’t until Euler worked on the problem (around 2000 years later!) that someone finally succeeded in proving that if $n$ is an even perfect number then $n=2^p(2^p-1)$ where $p$ and $2^p-1$ are both prime numbers. This result, now called the “Euclid-Euler Theorem”, establishes a strikingly curious one-to-one correspondence between even perfect numbers and Mersenne primes.

Analytic number theory

“The remarks I have decided to present here refer generally to that kind of series
which are absolutely different from the ones usually considered till now.”

Euler wrote a lot of his work in Latin. The quotation above is the first sentence from a paper he wrote whose title translates into English as “Several Remarks on Infinite Series”. Theorem 7 of that paper is the following enigmatic identity
$$1 + \frac{1}{2} + \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6} + \cdots = \frac{2\cdot3\cdot5\cdot7\cdot11\cdot13\cdot17\cdot19\cdots}{1\cdot2\cdot4\cdot6\cdot10\cdot12\cdot16\cdot18\cdots} $$
where the numerator of the right hand side is the product of all the prime numbers and the denominator is the product of all the numbers 1 less than a prime. It is well known that the left hand side of this identity diverges to infinity. In fact, in a different work, Euler proved the stronger statement
$$\lim_{n \rightarrow \infty}\left(-\log n + \sum_{k=1}^{n}\frac{1}{k}\right) = \gamma$$
where $\gamma = 0.57721\ldots$ is the Euler–Mascheroni constant. It follows straight away from the fact that the left hand side diverges that there are infinitely many primes – because the right hand side cannot be a finite product. Euler didn’t stop there though. He used his product formula to prove the much more impressive result that
$$\frac{1}{2} + \frac{1}{3} +\frac{1}{5} + \frac{1}{7} +\frac{1}{11} +\frac{1}{13} + \frac{1}{17} + \cdots = \infty.$$

Combinatorics

This next one is truly astonishing – both the statement and Euler’s proof. It concerns the number of different ways of expressing a positive integer as a sum of other positive integers. For example, there are 15 ways of expressing 7 like this and they are

\begin{align*}
&1+1+1+1+1+1+1+1, \:\: 1+1+1+1+1+2, \:\: 1+1+1+1+3, \\
&1+1+1+2+2, \:\: 1+1+1+4, \:\: 1+1+2+3, \:\: 1+2+2+2, \:\: 1+1+5, \\
&1+2+4, \:\: 1+3+3, \:\: 2+2+3, \:\: 1+6, \:\: 2+5, \:\: 3+4, \: \text{ and } \:7.
\end{align*}

The thing to notice is that there are exactly 5 ways where all the numbers are odd and also exactly 5 ways in which there are no repeats. This is no accident. In fact Euler proved that this always happens.

The number of ways of expressing a given number as a sum of distinct positive integers is the same as the number of ways of expressing it a sum of odd positive integers.

It’s hard to believe this at first since it seems like it has no right to be true, but it is. In order to better appreciate Euler’s ingenious proof, it is worth trying to imagine how helpless you would feel if you were asked to show this in an exam. Euler’s proof is as shocking as the statement.

He starts by noticing that the number of ways of writing $n$ as a sum of distinct positive integers is precisely the coefficient of $x^n$ in the expression $(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots.$ Next, manipulate this infinite product to get
\begin{align*}
(1+x)(1+x^2)(1+x^3)(1+x^4)\cdots &= \frac{(1-x^2)(1-x^4)(1-x^6)(1-x^8)\cdots}{(1-x)(1-x^2)(1-x^3)(1-x^4)\cdots} \\
&=\frac{1}{(1-x)(1-x^3)(1-x^5)(1-x^7)\cdots}
\end{align*}
and expand using the formula for a geometric series to get that this is equal to
$$(1+x+x^2+\cdots )(1+x^3+x^6 + \cdots )(1+x^5+x^{10}+\cdots )(1+x^7+x^{14}+\cdots )\cdots.$$
Now finish by recognising the coefficient of $x^n$ in this last expression as being exactly the number of ways of writing $n$ as a sum of positive odd integers, where now we allow repeats.

Infinite series

Jakob Bernoulli’s 1689 Tractatus de seriebus infinitis was a state-of-the-art account of infinite series, as they were understood at the time. It included results like the fact that the harmonic series $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges and explicitly evaluated a number of convergent series. For example, the geometric series $\sum_{n=1}^{\infty}a^n = \frac{1}{1-a}$ for $|a|<1$, the sum of the reciprocals of the triangular numbers,
$$1 + \frac{1}{3} + \frac{1}{6} +\frac{1}{10} + \frac{1}{15} + \cdots = \sum_{n=1}^{\infty}\frac{1}{n(n+1)} = 2,$$
and others like $\sum_{n=1}^{\infty}\frac{n^2}{2^n} = 6$ and $\sum_{n=1}^{\infty}\frac{n^3}{2^n} = 26$ were all known at the time. At some point Jakob decided to think about $\sum_{n=1}^{\infty}\frac{1}{n^2}.$ He knew that it converged but tried and failed, as did a number of others, to evaluate it explicitly. Concerning this sum, the Tractatus included the line

“If anyone finds and communicates to us that which thus far has eluded our efforts, great will be our gratitude.”

Euler rose to the challenge in spectacular fashion by showing that
$$1+\frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + \frac{1}{6^2} + \cdots = \frac{\pi^2}{6}.$$
His original argument, although not entirely justified at the time, is undoubtedly the work of a genius. He reasoned as follows. Just like polynomials can be factored according to their roots, Euler factorised $\frac{\sin x}{x}$ according to it’s (infinitely many!) roots, which are $\pm n \pi$ for $n = 1, 2, 3, \ldots$, as
$$\frac{\sin x}{x} = \prod_{n=1}^{\infty}\left(1-\frac{x}{n\pi}\right)\left(1+\frac{x}{n\pi}\right) = \prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2\pi^2}\right).$$
The power series expansion
$$\frac{\sin x}{x} = 1 – \frac{x^2}{3!} + \frac{x^4}{5!} – \frac{x^6}{7!} – \cdots$$
was well known to Euler. To evaluate the sum in question, it just remains to expand the infinite product and compare coefficients of $x^2$ in these two representations of $\frac{\sin x}{x}.$