Math Tutorials and More
by George

When I was a mathematics stu­dent in college I often encountered the name Euler — Euler equations, Euler functions, Euler theorems, etc. Un­fortun­ately, I didn't take the time to investigate the man behind the name. It turns out that Leonhard Euler was one of the greatest math­ematicians and scientists of all time. Mathematics historian Morris Kline ranks him at the highest level with Archimedes, Newton, and Gauss [1].

Euler made significant contributions to nearly every branch of mathematics — analysis (calculus, differential equations, calculus of variations), algebra, number theory, complex variables, Euclidean and differential geometry, topology, graph theory, and combinatorics. He also made considerable advances and discoveries in many branches of physics — mechanics, astronomy, electricity and magnetism, light and color, hydraulics, optics, acoustics, and elasticity. Science historian Clifford Truesdell regarded him as the dominate theoretical physicist of the eighteenth century [2]. If that wasn't enough, he also wrote landmark papers concerning the building and navigation of ships, artillery theory, and the foundations of actuarial science. The famous math­ematician Pierre Simon Laplace made the statement

Lisez Euler, lisez Euler, c'est notre maître à tous.
(Read Euler, read Euler, he is our master in everything.)

In addition to his many discoveries, he also clarified many existing areas of mathematical science and established much of the notation we use today. For example, he established the use of

• $e$ for the base of the natural logarithm
• $\pi$ for the ratio of the circumference to the diameter of a circle
• $f(x)$ for the value of a function
• $\sin x$ and $\cos x$ for values of the sine and cosine functions
• $i$ for the imaginary unit
• $\sum$ for summation
• $\Delta$ for finite difference

The following terminology in mathematics and physics is associated with Euler: Euler angles, Euler circuits, the Euler $\phi$-function, the Euler-Lagrange equation, Euler's identity, Eulerian mechanics, the Euler-Maclaurin summation formula, Euler's addition theorem for elliptic integrals, the Euler-Descartes formula, and many more.

Euler was certainly one of the most prolific scientific authors of all time. During his lifetime he had more than 500 books and articles published. An additional 400 of his manuscripts were published after his death. Historian Clifford Truesdell estimated that approximately one third of all publications in the fields of mathematics, theoretical physics, and engineering mechanics between the years 1725 and 1800 were authored by Euler [2].

Today math­ematicians such as Euler are not well known by the public. However, in Euler's day, mathematics was considered the highest form of knowledge and he was better known by the general public then such literary and music greats as Swift and Bach. His collected works, entitled Euleri Opera Omnia, is contained in 80 volumes, many of which exceed 500 pages. A picture of this collection is shown in Figure 1.

The works in this collection are distributed as follows:

40%	Advanced Algebra, Number theory, and Mathematical Analysis
28%	Mechanics and Physics
18%	Geometry
11%	Astronomy
2%	Artillery, Architecture, and Naval science
1%	Various other subjects

Euler was well known for the clarity of his exposition. Clifford Truesdell [2] says of Euler's writings

It was Euler who first in the western world wrote mathematics openly, so as to make it easy to read. He taught his era that the infinitesimal calculus was something any intelligent person could learn, with application, and use. He was justly famous for his clear style and for his honesty to the reader about such difficulties as there were.

In many cases Euler arrived at his results through an inductive process involving the consideration of many special cases. Unlike most authors Euler did not hide this path of discovery from his readers. He described in detail the examples and reasoning that led him to the result. Thus, the reader is exposed to the thought process of a master.

Euler was gifted with an extraordinary memory and an ability to perform complicated calculations in his head. He could recite from memory the epic poem Aeneid in Latin (approximately 400 pages) and could tell you the first and last line on each page of the original text he used. Once when he had insomnia he calculated in his head the first six powers of all the numbers from 1 to 100 and committed this table to memory. He frequently amused his friends by recalling some of these results.

At another time two of his students performed a difficult calculation involving the sum of a power series up to the seventeenth term and their answers differed in the fifth digit. They went to Euler to settle their dispute. Euler performed the calculation in his head and not only got the correct answer, but was able to point out where each of the students had erred. This ability served him well during the last 17 years of his life when he was almost totally blind. Nearly one half of his works were composed during this final period of his life. The French math­ematician and physicist François Arago made this statement

Euler calculated without apparent effort, as men breathe, or as eagles sustain themselves in the wind.

Euler was universally admired, not only for his genius, but also for his character. He seemed to have had no desire to advance his own career at the expense of others. Clifford Truesdell wrote [2]

He was exceptionally generous, never once making a claim of priority and in some cases actually giving away discoveries that were his own. He was the first to cite the work of others in what is now regarded as the just way, that is, so as to acknowledge their worth.

Euler was one of the principal developers of what is now called the Calculus of Variations. After spending much time and effort in developing this area he received a letter from a young French math­ematician Joseph-Louis Lagrange proposing a new analytic approach for solving variational problems. Euler immediately recognized the superiority of Lagrange's approach and set about publicizing Lagrange's discovery, giving him full credit. He even refused to publish his own work on the subject until after Lagrange's work was published. They became lifelong friends. Euler also delayed publishing his treatise on hydrodynamics until his friend Daniel Bernoulli had published his.

Euler fathered 13 children, only five of which reached adolescence. His children provided him with 38 grandchildren. He was devoted to his family and said that much of his best work was accomplished with children playing at his feet and crawling on his lap. Euler was a committed Christian and frequently expressed awe at the works of the Creator. Euler was particularly impressed by the design of the eye. Here is one statement that he made concerning the eye

though we are very far short of a perfect knowledge of the subject, the little we do know of it is more than sufficient to convince us of the power and wisdom of the Creator. … We discover in the structure of the eye perfections which the most exalted genius could never have imagined

Concerning the calculus of variations he wrote

For since the fabric of the universe is most perfect and the work of a most wise creator, nothing at all takes place in the universe in which some rule of the maximum or minimum does not appear. — Leonhard Euler, Methodus Inveniendi Uneas Curvas, 1st addition, art. 1 (1744).

Each night, until he lost his sight, he read to his family from the Bible and discussed with them the meaning of what was read. His faith was often ridiculed by Enlightenment philosophers such as Voltaire. In defense of his faith he wrote the document Defense of the Divine Revelation against the Objections of the Freethinkers [Leonhardi Euleri Opera Omnia, Ser. 3, Vol. 12].

There is a Table of Contents for this paper (labeled Contents) on the top bar. To view or download a copy of this paper in Adobe Acrobat format you need to click on the link Euler (PDF).

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