Leonhard Euler | An Introduction

Born in Basel, Switzerland on 15th April 1707, Leonhard Euler was arguably the brightest mathematician of all time. The Swiss mathematician and physicist is considered a pioneer in many fields of mathematics. He introduced a lot of the mathematical terminology and notation used today and he is considered the father of mathematical analysis where, for instance, he introduced the notation of a mathematical function, f(x). His contributions to the field of mathematics are in analytic and differential geometry, calculus, the calculus of variation, differential equations, series and the theory of numbers. In physics, although really all his contributions to mathematics apply to physics, he introduced both rigid body mechanics and analytical mechanics (Kline, 401-402).

Born to Paul Euler and Margarete Bruckner, Leonhard was the first of six children. He grew up in Riehen but attended school in Basel. Although mathematics was not taught in his school his father had kindled his interest in the subject (Paul Euler had been friends with another great mathematician at the time, Johann Bernoulli) by giving him lessons at home. Euler entered university at the age of 13 at the University of Basel. Although his official courses of study were philosophy and law, Euler met with Johann Bernoulli who advised Euler and gave him help with his mathematical studies on Saturday afternoons (Stillwell, 188).

Euler lived and worked mainly in Russia and Germany. First he joined the faculty at St Petersburg Academy of Sciences were he worked at first in the medical department then he was quickly promoted to a senior position in the department of mathematics through the influence of his friend Daniel Bernoulli. He also helped the Russian government on many projects including serving in the Russian navy as a medical Lieutenant. After the death of Catherine I in1740 and because of the tough conditions that ensued, Euler moved to Germany at the invitation of Prussian King, Frederick II to the Prussian Academy of Science were he stayed for the next 25 years of his life. Euler gave much service to the Academy which compensated him generously. He sent most of his works to be published there, served as a representative as well as advising the Academy on its many scientific activities. It is there that he reached the peak of his career writing about 225 memoirs on almost every topic in physics and mathematics (Varadarajan, 11).

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Euler returned to St. Petersburg in 1766 under the invitation of the then czarina Catherine the Great (Catherine II) to the St Petersburg Academy. During this period he lost almost all his eyesight through a series of illnesses becoming nearly totally blind by 1771. Nevertheless, his remarkable memory saw him writing about 400 memoirs during this time. It is said that he had a large slate board fixed to his desk where he wrote in large letters so that he could view dimly what was being written. He died on the 18th day of September 1783 due to cerebral hemorrhaging. It is also recorded that he was working even to his last breath; calculations of the height of flight of a hot air balloon were found on his board (Varadarajan 13).

Euler’s contribution to Mathematics and Physics was a lot. His ideas in analysis led to many advances in the field. Euler is famously known for the development of function expressions like the addition of terms, proving the power series expansion, the inverse tangent function and the number e:

∑ (xn/n!)= lim ((1/0!) + (x/1!) + (x2/2!) +…+ (xn/n!) ) =ex

The power series equation in fact helped him solve the famous 1735 Basel problem:

∑ (1/n2) = lim ((1/12) + (1/22) + (1/32) + …+(1/n2)) = π2/6

He introduced the exponential function, e, and used it plus logarithms in analytic proofs. He also defined the complex exponential function and a special case now known as the Euler’s Identity:

eiφ = cos φ + isin φ

And

eiπ + 1= 0 (Euler’s Identity)

In fact, De Moivre’s formula for complex functions is derived from Euler’s formula. Similarly, De Moivre is recognized for the development of calculus of variations, formulating the Euler-Lagrange equation. He was also the first to use solve problems of number theory using methods of analysis. Thus, he pioneered the theories of hyperbolic trigonometric functions, hyper geometric series, the analytic theory of continued fractions and the q-series. In fact, his work in this field led to the progress of the prime number theorem (Dunham 81).

The most prominent notation introduced by Euler is f(x) to denote the function, f that maps the variable x. In fact he is the one who introduced the notion of a function to the field of mathematics. He introduced, amongst others, the letter ∑ to mean the sum, π for the proportion regarding the perimeter of a circle up to the span or the diameter, i for the imaginary unit, √(-1) and the e (2.142…) to represent the base of the natural logarithm.

Euler also contributed to Applied Mathematics. Interestingly enough, he developed some Mathematics applications into music by which he hoped to incorporate musical theory in mathematics. This was however, not successful. This notwithstanding, Euler did solve real-world problems by applying analytical techniques. For instance, Euler incorporated the Method of Fluxions which was developed by Newton together with Leibniz’s differential calculus to develop tools that eased the application of calculus in physical problems. He is remembered for improving and furthering the numerical approximation of integrals, even coming up with the Euler approximations. More broadly, he helped to describe many applications of the constants π and e, Euler numbers, Bernoulli numbers and Venn diagrams.

The Euler-Bernoulli beam equation (one of the most fundamental equations in engineering) is just one of the contributions of the mathematician to physics. He used his analytical skills in classical mechanics and used the same methods in solving celestial problems. He determined the orbits of celestial bodies and calculated the parallax of the sun. He differed with Newton (then the authority in physics) on his corpuscular theory of light. He supported the wave theory of light proposed by Hugens.

Euler’s contributions to graph theory are at the heart of the field of topology. He is famously known to have solved the Seven Bridges of Konigsberg problem, the solution of which is considered the first theorem of planar graph theory. He introduced the formula

V-E+F=2

It is a mathematical formula relating vertices, edges and faces of a planar graph or polyhedron. The constant in the above formula in now called the Euler characteristic.

Euler is also recognized for the use of closed curves in the provision of explanation concerning reasoning which is of syllogistic nature. Afterwards the illustration or diagrams were referred to as the Euler diagrams.

The Number Theory is perhaps the most difficult branch of mathematics. Euler used ideas in analysis linking them with the nature of prime numbers to provide evidence that the total of all the reciprocals of prime numbers diverges. He also discovered the link between the primes Riemann zeta function, what is now called the Euler product formula for the Riemann zeta function. Euler made great strides in the Lagrange four-square theorem while proving Fermat’s theorem on the sum of two squares, Fermat’s Identities and Newton’s identities. Number theory consists of several divisions which include the following: Algebraic Number Theory, Combinational Number theory, Analytic Number Theory, Transcendental number theory, Geometric number theory and lastly we have the Computational Number Theory.

For his numerous contributions to academia, Euler won numerous awards. He won the Paris Academy Prize twelve times over the course of his career. He was elected as a foreign member, in 1755, of the Royal Swedish Academy of Sciences while his image has been featured on many Russian, Swiss and German postage stamps. Above all, he was respected greatly amongst his academic peers demonstrated by a statement made by the great French mathematician, Laplace to his students to read Euler since he was the master of them all (Dunham xiii).

Though not all of the proofs of Euler are satisfactory in regard to the current standards or principles used in mathematics, the ideas created by him are of great importance. They have set a path to the current mathematical advancements.

To conclude, we can therefore say that Euler is a very significant person in the development and advancement of Mathematics. His work has contributed a lot to mathematics up to the current period.

References

Dunham, William. Euler: The Master of Us All. Dolciani Mathemathical Expositions Vol. 22. MAAA, 1999.

Kline, Morris. Mathematical Thoughts from Ancient to Modern Times, Vol 2. New York: Oxford University Press, 1972.

Stillwell, John. Mathematics and its History. Undergraduate Texts in Mathematics. Springer, 2002.

Varadarajan, V. S. Euler Through Time: A New Look at Old Themes. AMS Bookstore, 2006.