本文是数学专业的留学生作业范例，题目是“The Life and Discoveries of Leonhard Euler(莱昂哈德·欧拉的生平和发现)”，本文简要介绍了瑞士数学家莱昂哈德·欧拉的生平，并概述了他在数学各个领域的主要贡献。“读欧拉，读欧拉，他是我们所有人的主人。”——皮埃尔-西蒙-拉普拉斯——《学者杂志》，第51页1846年1月

** Abstract 摘要**

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引言**

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.

莱昂哈德·欧拉很可能是有史以来最多产的数学家。的确，他对数学的影响之大是无法夸大的。有人说，数学上的发现必须以第二个人的名字命名，因为欧拉似乎总是第一个。这位18世纪的知识巨人不仅站在像莱布尼茨和牛顿这样的巨人的肩膀上，而且还为未来的新发现奠定了基础，从而证明了人类成就的顶峰。虽然要列举欧拉的巨大成就是不现实的，但我们将涵盖他的一生，他对图论和拓扑的发现，他对数论的探索，以及他在代数和一般数学方面的工作的影响。

** 2. The Life of Euler 欧拉的一生**

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.

欧拉于1707年出生于瑞士巴塞尔的一个中产阶级家庭。他的父亲是一名新教牧师，碰巧也是一名数学学徒，欧拉的父亲甚至在他就读神学院期间师从雅各布·伯努利。欧拉的父亲在他很小的时候就开始教他数学，并为他聘请了一位私人教师，以确保他能够继续深造。

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.

尽管遇到了这些挫折，欧拉还是继续创作了大量的作品。在完全失明后，他曾说:“现在我不会再分心了。”正是在这个时候，他出版了他最著名的两部作品:《给一位德国公主的信》和《代数》，这两部作品的目标读者是与他的大部分作品相比，数学水平较低的人群。与他为同龄人创作的更高层次的作品相比，《字母》和《代数》意在作为教学材料。欧拉在这个时期的作品都是在他的脑海中完成的，并由他的密友和家人记录下来。

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图论与拓扑学**

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]

欧拉最早和最著名的作品之一是他对Königsburg问题的七座桥的解决方案。欧拉所写的问题表述如下:

"...在普鲁士Königsburg，有一个岛屿A，叫做克尼弗夫;环绕它的河流分成两条支流。这些树枝上有七座桥…有人问，是否有人能安排一条路线，让他每座桥都过一次，而且只过一次……因此，我提出了一个普遍的问题:无论河道怎样布置，怎样划分，无论有多少座桥，能不能每座桥恰好过一次呢?”[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 代数与普通高等**

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:

和正弦函数的级数展开。诸如此类的微积分应用是欧拉的专长，他比前人做得更熟练。他在微积分上的效率很大程度上是由于他的符号标准化，他将牛顿的流通量法和莱布尼茨的微分符号结合成一种统一的格式，他的大多数组合符号直到今天还在使用。他还标准化了数学符号的许多其他部分，如函数符号和许多数学常数的符号。他在微积分和代数方面的技能使他发展出被许多人接受，被称为数学中最美丽的方程;简称欧拉恒等式:

(4.2) eiπ + 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) eiθ = 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.

简单地说，将复平面上的一个点旋转π弧度等于将其乘以实数- 1.指数增长常数e，它是阶乘函数的无穷倒数和，由于欧拉符号的标准化和他对数字的性质和应用的发现，经常被称为欧拉数。欧拉发现这个数是无理数，并将它作为自然对数的基础引入到各种物理问题的解决中，也可以用于计算微积分中的许多问题。他还开发了一种方法，利用欧拉法对某些微分方程进行数值逼近。通常，该方法将曲线在某些点近似为多边形，并估计曲线的顶点在较小的步长和较小的计算时间间隔上与原始曲线没有显著变化。

** 5. Conclusion结论**

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.

莱昂哈德·欧拉在一年中所做的研究，比大多数数学家一生所做的还要多。他大量的工作证明了他的贡献对于推动现代数学发展到今天是多么重要。他发现的图论和拓扑学只是他开始有效解决的许多领域中的两个例子，这些问题持续了几代人。此外,他的符号和标准化处理计算微积分和代数的直觉帮助学术界相同级别的数学工作,保证会有数学家之间的一致性,帮助理解的新工作和新作品的速度传播。有了这么多的贡献，数学中所有的东西都以第二个人的名字命名就不足为奇了，以免所有的东西都以欧拉的名字命名。

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