Euler

Leonhard Euler (1707-1783 AD) can be considered one of the greatest mathematicians in the history. He contributed significantly to all branches of mathematics including geometry, trigonometry, algebra, and calculus. Besides mathematics, he also made his impression in areas of physics such mechanics, astronomy, and optics.

Leonhard Euler

Leonhard Euler

Born in Basel, Switzerland, Leonhard Euler was a student of the famous mathematician Johann Bernoulli at Basel University. As a child, he was an avid reader and learner who loved science and music. His father who was a pastor of a village church hoped that young Leonhard would follow in his footsteps and emphasized that he study religion. Young Leonard studied religion, and was not particularly antagonistic to his father’s beliefs. However, he found out that he had a penchant for mathematics and decided to dedicate his life to developing mathematics. He spent his academic life in Russia and Germany. He spoke German, French, and Russian fluently and wrote usually in Latin.
When he was at St. Petersburg Academy in Russia, Euler was asked to solve a complex astronomy problem. Leonhard Euler provided an elegant solution to the problem within three days, surprising the top scientists and mathematicians of Russia who expected that such work to take months. Euler would concentrate hours and hours on a problem, forgetting thirst, hunger, and sleep. This habit eventually took a toll on his health, and he fell ill. As a result of this illness, he lost the sight in his right eye. Later, the sight in his left eye started deteriorating as a result of a cataract, eventually he was completely blind. Anticipating a major handicap, he quickly trained his sons and assistants to take dictations and edit his dictated notes. He used a large slate mounted on a table. Whenever he had to explain a concept that need an illustration, he would take a chalk and draw a few crude strokes on the slate. His assistants had become experts in interpreting his illustrations. Euler had accepted the fate and taken steps to combat the adversity instead of despairing when it became evident that he would lose his most important sense. He is believed to have remarked to his friends, “Now I will have few distractions to divert me from my work.” This was in fact true because he produced a lot more scientific and mathematical papers after he became blind.
Euler is considered to be the most productive writer of mathematics. He wrote textbooks for children in Russian elementary schools. Everyone liked Euler who was always cheerful and who especially loved children. He had thirteen children of his own but only five survived to be adults. He wrote such a vast number of scholarly articles that it took about 50 pages just to list the titles of his works. His yearly works generated about 800 printed pages, not counting thousands of letters he wrote.
The Seven Bridges of Königsberg is a well-known problem in mathematics. The city of Königsberg in Prussia included two large islands which were connected to each other and the mainland by seven bridges. The problem was to plot a path through the city that would cross each of the bridges once and only once, with the condition that the islands could only be reached by the bridges and not by a boat. The starting and ending points of the path need not be the same. In 1736, Euler proved that such a path was impossible. The proof laid the foundations of a new branches of mathematics: graph theory and topology.
Euler’s formula in geometry is also very famous. He discovered that for any polyhedron that doesn’t intersect itself, the number of faces (F) plus the number of vertices (V) minus the number of edges (E) always equals 2. This can be written: F + V − E = 2.
Much of the notation used by mathematicians today – including e, i, f(x), ∑, and the use of small letters a, b and c to denote the sides of a triangle with corresponding capital letters A, B, and C to denote angles opposite to those sides- was created by Euler. His efforts to standardize these and other symbols including Ï€ and the trigonometric functions helped internationalize mathematics while encouraging collaboration on problems. He even managed to combine several of these together in an amazing way to create one of the most beautiful of all mathematical equations known as Euler’s Identity: \[e^{i\pi} = -1\].
In 1734, Euler solved the Basel problem which asked for the precise summation of the reciprocals of the squares of the natural numbers:

\[\sum_{n=1}^\infty \frac{1}{n^2} = \lim_{n\to\infty}(\frac{1}{1^2}+\frac{1}{2^2}+\cdots + \frac{1}{n^2})\]

 

The Bernoulli family had unsuccessfully tried to solve this problem prior to Euler but Euler proved that the exact sum of the infinite series was \[\frac{\pi^2}{6}\].
Euler never lost the thrill of new discoveries until he breathed his last. Even on the day he died, he spent hours tracing the orbit of Uranus, a newly discovered planet at that time. With his passing, the world lost a very productive mathematician but his influence on modern mathematics still continues to this day.