The Life and Discoveries of Leonhard Euler

Abstract

This paper is a brief biography of the Swiss mathematician Leonhard Euler, and an overview of some major contributions he made to various fields of mathematics.

"Read Euler, read Euler, he is the master of us all." - Pierre-Simon Laplace - Journal des Savants, p. 51, Jan. 1846

1. Introduction

Leonhard Euler is quite possibly the most prolific mathematician of all time. Truly, it is impossible to overstate the sheer magnitude of his influence on mathematics. It has been said that discoveries in math must be named after the second person to come across it, since Euler always seems to be first. This 18th century intellectual titan demonstrated the pinnacle of human achievement by not only standing on the shoulders of giants such as Leibniz and Newton, but also laying the foundation for new discoveries to be made in the future. While it would be impractical to enumerate Euler's vast achievements, we will cover his life, his discovery of graph theory and topology, his explorations in number theory, and the impact of his work in algebra and general mathematics.

2. The Life of Euler

Born in 1707 in Basel, Switzerland, Euler came from a middle class family. His father was a Protestant minister who also happened to be a mathematics journeyman, and Euler's father even studied under Jakob Bernoulli during his theological university endeavors. Euler's father began to teach him mathematics from a young age and hired him a private tutor to ensure he would further his studies.

In 1720, Euler enrolled in University to study philosophy, but despite his prescribed course of study became enraptured with math teachings from the likes of Johann Bernoulli (the brother of Jakob Bernoulli, who taught Euler's father). Euler would even be given private instruction by Johann Bernoulli due to Euler's enthusiasm for the subject. After Euler and his father came to terms with Euler's true passion, he dedicated himself wholly to studying mathematics in university and made rapid progress. After displaying impressive aptitude in a variety of math contests, Euler accepted a call to the Academy of Sciences in St. Petersburg, Russia. It was here that he began to meet other prominent mathematicians of his era, such as Christian Goldbach of number theory fame. It was during his time at the Academy that he came to international fame due to his impactful publications, such as a paper solving the Seven Bridges of Königsburg or his solution to the Basel problem.

During his tenure there in 1734, he married Katharina Gsell, and they began a family life, having a total of 13 children of which only 5 survived to adulthood.

During this period of his life, in 1738 he fell seriously ill and lost all vision in his right eye, which would precede his later total blindness. Shortly thereafter, he and his family packed up and left for Prussia upon an invitation from Frederick II (Frederick the Great) to help establish a Prussian Academy of Sciences.

Euler swiftly rose in the ranks, moving from being the director of the Mathematics Class to being the President of the Prussian Academy in a under a decade; however, this would not be a position he held for long as Frederick II declared himself the President of the Academy.

Euler would take himself and his family back to Russia in defiance of the perceived disrespect he experienced from the ruler.

Catherine the Great, the reigning empress of Russia when Euler returned in 1766, welcomed him with open arms, and bestowed upon him the prestige he was owed as he rejoined the Academy of Sciences in St. Petersburg. It was at this time that Euler's personal life took a turn for the worse: he went totally blind in 1771, lost his house in the great fire of St. Petersburg later in the year, and then lost his wife who died in 1773.

Despite all these setbacks, Euler continued to churn out an incredible volume of works. Upon becoming totally blind, he is quoted to have said "Now I will have less distraction."^[1] This was the time when he published two of his most famous works: "Letters to a German Princess" and "Algebra", both of which were aimed at an audience of lower math sophistication compared to most of his works. In contrast to the higher level works that he was generating for his peers, "Letters" and "Algebra" were intended to be teaching material. Euler's works at this time were done in his head and transcribed by his close friends and family.

He continued to write and publish until his death from a stroke on September 18th, 1783. His family continued to publish the works Euler had either yet to send out or finish, and articles of Euler continued to be published for decades after his death due to their extreme volume.

3. Graph Theory and Topology

One of Euler's earliest and most well known works is his solution to the Seven Bridges of Königsburg problem. The problem statement as written by Euler is as follows:

"...in Königsburg in Prussia, there is an island A, called the Kneiphof ; the river which surrounds it is divided into two branches... and these branches are crossed by seven bridges... it was asked whether anyone could arrange a route in such a way that he would cross each bridge once and only once... From this, I have formulated the general problem: whatever be the arrangement and division of the river into branches, and however many bridges there be, can one find out whether or not it is possible to cross each bridge exactly once?"^[2]

Euler solved the problem by contriving a clever method of expressing what it means to "travel" from one location to another across a "bridge". He expressed each land area (or vertex) as a capital letter, and each bridge (or edge) as a lowercase letter. To define a sequence of bridge crossings (a walk along a sequence of edges), Euler writes ABCD to express starting at A, going then to B, then from B to C, then from C to D.

From this, he found the corollary that the number of bridges crossed in a trail (a walk with no duplicate edges) is one less than the number of letters in a walk, and that therefore the number of letters required to cross the seven bridges must be eight. He also concluded that any vertex with an odd number n of connected edges must appear 2n - 1 times.

From this and the problem statement of the Seven Bridges, he concluded that it is impossible to cross all seven bridges once and only once, as it would require a trail of 9 vertices, which contrasts the finding that the number of letters required to cross the seven bridges must be eight.

He finally concluded that for a trail to exist on a graph (a set of vertices and edges), there can only be either zero or two vertices with an odd degree (number of edges connected to a vertex) due to the possibility of starting and ending on one of the vertices of odd degree. Such a walk is named a Eulerian Path.

Euler also solved a similar problem that adds the additional constraint that a path start and end at the same vertex, and a walk that satisfies this condition is an Eulerian Circuit. The conditions for an Eulerian Circuit to exist are that a graph be connected (there exists a walk from any given node in a graph to any other given node) and that there be no nodes of odd degree. Since this is merely a stricter set of conditions to an Eulerian Path, it is true that all Eulerian Circuits are also Eulerian Paths, though the converse is not true.

While all of these discoveries are elementary, they represent a more fundamental concept which is geometry independent of distance: since it does not matter how long an edge is, only what vertices it connects, it provided a framework to analyze geometric truths without needing to consider size. In particular, the realization that the layout of a graph did not affect its properties, which were immutable, led to the development of topology.^[3] Topology is the study of geometric phenomenon which are independent of specific shape and magnitude but rather focus on certain properties which are more generally applicable to objects which can be continuously deformed into one another. Euler carried his discoveries in graph theory forward into topology with his statement of the Euler Characteristic of polyhedra. Simply put, for any^[4]polyhedron, the object's number of vertices plus its number of faces minus its number of edges is always equal to 2. This property is called a topological invariant and the Euler Characteristic was one of the first topological invariants to be discovered.^[5]Since Euler created topology as a field of study, it has grown tremendously in scope and has been applied to solve problems from many different disciplines, such as computational protein folding in biology or topological mechanics in physics.

4. Algebra and General Advancements

The Basel problem, named after Euler's hometown, concerns the infinite sum of reciprocals of squares^[6]:

(4.1)

Euler demonstrated himself early on to be a master with infinite sums by showing the sum to be equal to . He did this by using a polynomial of infinite degree^[7], as discovered by

Newton, and the series expansion of the sine function. Applications of the calculus such as this were Euler's specialty, and he did it more deftly than any who came before him. His efficiency with the calculus was due in large part to his standardization of notation by combining elements of Newton's method of Fluxions and Leibniz's differential notation into a single unified format and the majority of his combined notation for calculus is used to this day. He also standardized many other parts of mathematics' notation, such as function notation and the symbols for many mathematical constants.^[8]His skill in calculus and algebra led him to develop what is accepted by many to be the most beautiful equation in mathematics; simply known as Euler's Identity:

(4.2) e^iπ + 1 = 0

Euler's Identity, besides combining five of the most fundamental constants in mathematics (the circle constant, the exponential growth constant, the imaginary unit or i, one and zero), also describes geometry in the complex plane as the angle varies in the more general form of the equation known as Euler's Formula:

(4.3) e^iθ = cos(θ) + isin(θ)

Put simply, rotating a point in the complex plane by _π radians is equivalent to multiplying it by the real number negative one. The exponential growth constant, e, which is the infinite sum of reciprocals of the factorial function, is often called Euler's number due to both his standardization of its symbol and his discoveries of the number's properties and applications. Euler found that the number was irrational and introduced it as the base for natural logarithms which could be used in the solution to various physical problems as well as many problems in computational calculus. He also developed a way to numerically approximate certain differential equations, using Euler's Method. Generally, this method approximates a curve as being polygonal at certain points and estimates that the vertices of the curve do not vary significantly from the original curve over small step sizes and a small interval of computation.

5. Conclusion

Leonhard Euler did more work across the discipline of mathematics in a single year than most mathematicians can hope to do their entire life. His vast body of work speaks for himself as to how essential his contributions were to propelling modern mathematics forward to where we are today. His discovery of graph theory and topology are only two examples of the many fields he effectively started to solve problems that persisted through generations. Furthermore, his standardization of notation and work with computational calculus and algebraic intuition helped to bring academia up to the same level of work in mathematics and ensured that there would be consistency between mathematicians, aiding in the comprehension of new works and the speed with which new works could propagate. With that many contributions, it is no surprise that everything in mathematics has to be named after the second person to discover it lest it all be named after Euler.

References

[1] Walter Gautschi. "Leonhard Euler: His Life, the Man, and His Works". In: Proceedings of the International Congress of Industrial and Applied Mathematics (2008). doi: 10.1137/070702710. url: https://www.cs.purdue.edu/homes/wxg/EulerLect.pdf.

[2] Leonhard Euler. Solutio Problematis Ad Geometriam Situs Pertinetis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1736. isbn: 0 19 853916 9.

[3] Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson. Graph Theory 1736-1936. 1998.

[4] Edward Early. "On the Euler Characteristic". In: MIT Undergraduate Journal of Mathematics (1999).

[5] Florian Cajori. "The History of Notations of the Calculus". In: Annals of Mathematics, Second Series Vol. 25.No. 1 (Sep., 1923) (1923), pp. 1–46. url: https://www.jstor. org/stable/1967725.

[6] Web Page. 2018. url: http://www-history.mcs.st-and.ac.uk/HistTopics/Perfect_numbers.html.

[7] Web Page. 1998. url: https://math.dartmouth.edu/~jvoight/notes/perfelem.pdf.

[8] Web Page. url: http://mathworld.wolfram.com/QuadraticReciprocityTheorem.html.

[1] Walter Gautschi. "Leonhard Euler: His Life, the Man, and His Works". In: Proceedings of the International Congress of Industrial and Applied Mathematics (2008). doi: 10.1137/070702710. url: https://www.cs.purdue.edu/homes/wxg/EulerLect.pdf.

[2] Leonhard Euler. Solutio Problematis Ad Geometriam Situs Pertinetis. Commentarii Academiae Scientiarum Imperialis Petropolitanae, 1736. isbn: 0 19 853916 9.

[3] Norman L. Biggs, E. Keith Lloyd, and Robin J. Wilson. Graph Theory 1736-1936. 1998.

[4] While Euler believed this to be true for all polyhedra, in reality it only holds for convex polyhedra. He was correct that the Euler Characteristic is a topological invariant however; this was not proven by Euler.

[5] Edward Early. "On the Euler Characteristic". In: MIT Undergraduate Journal of Mathematics (1999).

[6] This is equivalently stated as being the value of the Riemann-Zeta function for the real input of two, though the significance of this would not be realized in Euler's time.

[7] Notably, he did this using unproven assumptions about the behavior of infinite degree polynomials being similar to polynomials of finite degree; though his result was correct.

[8] Florian Cajori. "The History of Notations of the Calculus". In: Annals of Mathematics, Second Series Vol. 25.No. 1 (Sep., 1923) (1923), pp. 1–46. url: https://www.jstor.org/stable/1967725.