Leonhard Euler

1. Biography.

Leonhard Euler was born in the year 1707 in Basel, Switzerland with his father being a pastor and his mother a pastor’s daughter.

A bit later he and his family moved to Riehen where he would meet the Bernoulli family and excel in his studies of mathematics, more especially under the guidance of Jacob Bernoulli which allowed him to earn his master’s degree at the University of Basel in his teens.

In 1727, Leonhard moved to St. Petersburg, Russia where he would not only become an associate and professor of physics at the St. Petersburg Academy of Sciences but serve as a medic in the Russian Navy.

He would later succeed Daniel Bernoulli as the chair of mathematics in 1733.

A few years later he would lose complete vision in both eyes, but would continue working and publish articles in not only mathematics but in the fields of astronomy/ lunar motion, acoustics, mechanics and music. In September of 1783, Leonhard passed away due to a brain hemorrhage.

Six years later in 1741, Leonhard moved and became the mathematics director at the Berlin Academy where he would stay for twenty five years producing hundreds of papers. During this time he created the concept of a function with the inclusion of variables, complex numbers, formulas for differentiation and integration, and developments in the theories of linear differential equations which are helpful in physics.

In 1766 Leonhard moved back to Russia with the invitation from Catherine II and became head of the St. Petersburg Academy. A few years later he would lose complete vision in both eyes, but would continue working and publish articles in not only mathematics but in the fields of astronomy/ lunar motion, acoustics, mechanics and music. In September of 1783, Leonhard passed away due to a brain hemorrhage.

2. Euler's Contributions to Discrete Mathematics.

A. The 7 bridges of Königsberg

In 1735 began the study of graph theory when he presented the solution to the problem known as the Seven Bridges of Königsberg. Königsberg was a merchant city in the Pregal River now known as Kaliningrad, Russia. The river segregated the city into four regions which were connected by seven bridges; five of the bridges connected the island of Kneiphof to the mainland and the remaining two crossed the two branches of the river. The question at the time was: Can one walk across all seven bridges without crossing the same one twice? But in 1735 Euler presented the solution that no path exists; he was able to do this with an abstraction of the problem as a graph where each of the land areas were denoted with letter A, B, C, ad D, the nodes were pieces of land, and the links were bridges. The nodes refer to the vertices or the individual points on the graph, and the edges refer to the lines that connect these nodes together, and in this case would denote the bridges. With the illustration and graphing of this city, Euler came to the conclusion that for a successful route each node must have an even number of edges; Königsberg has four nodes with odd edges leading to no path that could satisfy the problem. Euler’s graph theory has been applied to many domains in life such as transportation, social networks, urban planning, economics, and biological systems.

- [HAN] Königsberg graph and Euler's graphical representation

B. Euler's Graph Formula

Euler is also known to be the mathmatician to have esatiblished the graph theory formula : V – E + F = 2

relating the number of vertices, edges, and faces of a convex polyhedron, and hence of a planar graph. The constant 2 in this formula is now known as the Euler characteristic for the graph.

C. Knight's Tour

Leonhard Euler was the first mathmatician to study the Knight's Tour, in 1759. The Knight's Tour consists of the sequence of moves the knight makes on the chessboard to have at least went through each square exactly once. He will later publish his results and ideas about that problem in the "Solution d'une question curieuse qui ne paraît soumise à aucune analyse" .

3. Reflections on the constructed nature of the digital world.

Many sources where available on Euler's contribution to discrete mathematics. However, some of them were not complete and trustworthy. Overall, I managed to find sources which seem reliable, inasmuch as they agree on the facts on Euler, and on the important contributions he made in his life.

In addition to that, I found more reliable the sources I found in french. As an example the french wikipedia page on Euler has entire other sections and a small line with a link leading to his contribution to another problem in discrete math , known as The Knight's Tour.

I imagine that if I had been researching a different person - or perhaps different type of person (not a mathematician), the constructed nature of the digital world would be even more apparent.

References.

[WikEuler] The Wikipedia page Euler, accessed February 2024.

[HPP] Portrait of Euler Its use is licensed under the Creative Commons Attribution 2.5 Generic license.

[Abu] "The Birth of Graph Theory: Leonhard Euler and the Königsberg Bridge Problem .", Science and Its Times: Understanding the Social Significance of Scientific Discovery. Encyclopedia.com , accesed Feb. 2024

[Singh] Carlson, Stephan C., "Graph Theory", Britannica, accesed February 2024.

[HAN] Luisnatera page Leonhard Euler and Graph Theory, accessed February 2024.

[WFS] The Wikipedia page Knight's Tour, accessed February 2024.