Leonhard Euler: Life and Contributions
Leonhard Euler was one of the biggest mathematicians of the history, a man whose devotion was mathematics. One of the many geniuses of his time who was the Director of Mathematics at the Academy of Petersburg. Moreover, he made contributions not only focusing on mathematics but in other areas such as physics, mechanics, and astronomy. This paper will go throughout his life, since he was born, young life, periods of enlightenment, and some of the contributions he did in mathematics.
Euler’s Early Life
Leonhard Euler was born on April 15,1707 in Basel (Switzerland), the first-born of Paul Euler and Margaretha Brucker. He was baptized at St. Martin’s Church on April 17 and was named Leonhard after his godparent. There is not a register to where exactly in the city he was born, then people speculate Margaretha gave birth at home which was common those days. Even though Paul Euler studied theology (pastor of St. Jakob church), he was interested in mathematics which led him to take curricula from Jakob Bernoulli who was a famous mathematician. A year and a half after Euler were born, he ended up serving in the village of Riehen just in the suburbs of the big city of Basel where he was given the position of Protestant Minister. In fact, Paul was Leonhard first teacher and taught him theology and mathematics at home. His dad even hired a math tutor in order to provide a better education. By the early age of 8 years old he was sent back to Basel to a Latin School.
By the age of 12 years old, Euler already enrolled at the University of Basel which in that time was not very usual for a boy of that age. He started his career in theology and he only was taking some first-year math classes. As the first math course at the university he decided to take elementary mathematics, a class taught by Johan Bernoulli, brother of Jakob who had already died. Bernoulli put his attention into Euler since he was good at the course. Such attention led him to motivate Leonhard to pursue a mathematics career by recommending extra book readings. As Marquis de Condorcet stated in his article “Eulogy to Mr. Euler”, Euler was able to manage a friendship with Bernoulli who offered private lessons to help him with his new readings. Marquis de Condorcet also explained that Paul made Leonhard to renounce his math career and focus more on theology when he just got his Master of Arts. Fortunately, later he was convinced that his son was meant to become someone important in mathematics. Then, Euler returned to study with Bernoulli.
Out in the Real World.
At the age of nineteen, after graduating and without any experience or any research, he decided to enter a competition created by the Paris Academy of Sciences. In such competition many great minds where participating, yet Euler won the second prize. Who could have imagined such young boy was able to stand out? That was when Euler began earning some respect in the community. At the age of 20, Euler decided to apply for the chair of the physic’s department at the University of Basel, but that was such a big position which he did not have the enough experience to be able to fulfill. Hence, he did not get the job. Right after, the Academy of Sciences in St.
Petersburg offered him a position which he took.
During his period on St. Petersburg, he learned and mastered Russian, it was a period of such enlightenment. More importantly, he found someone, Leonhard Euler married Katharina Gsell in January 1734. They had 14 children but only 5 of them reached adulthood. His first-born was Johann Albrecht, who later in life, he was able to assist his dad in mathematics and became a mathematician himself. Unfortunately, Euler suffered from a serious illness in 1735. According to Ronald Calinger in his book “Mathematical Genius in the Enlightenment”, there was not diagnose of Euler’s illness then it was supposed he had scrofula. Scrofula was a quite common illness in that time, which now it is called tuberculosis adenitis. Then two years later he lost sight in his right eye. Moreover, people say he was aware of the disease but opted to not tell anyone until Bernoulli decided to communicate Euler’s family about it in 1735.
Times of Changes
In 1741 he began another journey with his family, they moved out to Berlin, where Frederick II offered him, at the academy he founded, to be the director of the mathematics’ faculty. After many years of devotions to the Academy, once the presidency became vacant, Frederick II decided to proclaim himself for the position even though Euler asked for it and many other people recommended Euler to be the right one. After this, Euler realized that his faith was no longer in Berlin and decided to move back to St. Petersburg where he was more than welcomed at the Academy of Sciences. By this time, he already had published more than 20 memoirs, around 10 major treatises, and around 200 letters.
Once he came back to the Academy of Sciences, he was active all the time such that he was able to publish around 400 papers about number theory, analysis, physics, mechanics, statistics, geometry, probability theory, and cartography. He was working back and forth in all areas where mathematics was applied. The recollection of 886 papers and articles are in the “opera omnia”, which has around 70 volumes mostly written in Latin. Euler even said that he could not have done such things without help, hence he gave credit to Albrecht, Krafft, Lexell, Fauss, and his son. He was a man of a great mind, as Marquis de Condorcet said:
“..I have personally experienced the impossibility to follow the details and to provide the knowledge of the astonishing amount of discoveries, new methods, ingenious views covering more than thirty works published outside and the more than seven hundred Mémoires of which two hundred were deposited before his death”
Eulogy to Mr. Euler, by the Marquis de Condorcet. Published in 1783.
Leonhard Euler went completely blind in 1771, but that did not stop him at all. He was still doing research in some topics and working on proves or disproves. With the help of his son and daughter, he was still able to contribute his ideas to the math community. Unfortunately, his wife died in 1773, but he got married 3 years later in order to have a wife that will take care of him. He died from a stroke in St. Petersburg on September 18 in 1783. The Academy of Science gave him tribute by still publishing about him and his works for around 50 years after his dead. He was a loved and a appreciated member of the academy, as Marquis de Condorcet wrote that eight out of the sixteen professors were trained by Euler and “…all of are known through their works… and are proud to add the title of Euler’s disciples”.
Euler’s Contributions
Generalizing Mathematics
Euler was the one that by using following notations was able to generalize math.
• The base of the natural logarithm e = 2. 718..
• Pi, the ratio that Archimedes of Syracuse approximated. Euler used it as π = 3.1416…
• Square root of negative one as i, in 1770.
• He was the first one to use sine, cosine, tangent as functions.
• He promoted the notation f(x) for functions.
So far there are most important constants in mathematics are e and π but then in
1740 Leonhard Euler came up with another constant which became the third most important. Euler’s constant has not been proved to be rational or not rational, which is a mystery still. The constant has a value of 0.57721 that is represented by the lowercase Greek letter gamma γ. It has a connection with the assumption that harmonic series diverges which Jakob Bernoulli proved. Moreover, his brother, Johan Bernoulli was working on when Euler was a kid, then later, he gathered that information and ended up with the following:
𝛾
𝑛→∞ 2 3 𝑛
In 1790 Lorenzo Mascheroni worked on this limit as when Euler left his work, and he computed it to 32 decimal places. Hence γ is called the Euler-Mascheroni constant.
The Euler-Fermat Theorem
This is a well know formula used in cryptology. This theorem was published as E271 in 1763. It stated that for any 𝑎 coprime to 𝑛 such that 𝑎 ∈ ℕ we have:
𝑎𝜑(𝑛) ≡ 1(𝑚𝑜𝑑 𝑛)
Fermat is credited to such Euler’s expression since he used the “Little Fermat” theorem
𝜑(𝑝) = 𝑝 − 1
Where p is a prime number and n = p.
Nowadays, the Euler-Fermat theorem is used in cryptology in order to solve RSA encryption. This is how people send coded messages with publics keys and use Fermat’s theorem in order to find n, which is the product of p and q. Then use the EulerFermat theorem to decode the message.
The famous formula
One of the most famous formula in mathematics and science is the so-called Euler’s Formula, which is also called Euler’s Identity. This equation (or formula) has gained an important popularity such that in 1988, a survey done by David Wells for Mathematical Intelligencer won the title of “The most beautiful theorem”.
Which is the following:
ⅇ𝜋ⅈ + 1 = 0
𝑜𝑟 𝑎𝑙𝑠𝑜 𝑤𝑟𝑖𝑡𝑡ⅇ𝑛 𝑎𝑠
ⅇ𝜋ⅈ = −1
An interesting fact is that Euler did not exactly wrote those terms, since he introduced i in 1770 and his work about this formula was formulated around the decade of 1740’s. Euler spend many years writing a precalculus textbook, in Latin called “Introductio in analysin infinitorum” (which in the Euler’s system archive is E101). Such book was published in 1748.
Sandifer in his book “How Euler Did Even More” stated that in chapter 8 of the Euler’s Introductio was the first time he introduced sine, cosine, and tangent as functions. He was working with them as we usually do today. After he started using complex numbers, and the famous trig identity (𝑠𝑖𝑛 𝑧)2 + (𝑐𝑜𝑠 𝑧)2 = 1. There are some speculations in whether he knew DeMoivre’s formula or not in that time since there are not sources of his ideas in the book. Either way, he wrote some similar versions to DeMoivre’s formula in order to deduce the following:
ⅇ𝑙̇𝑣 = 𝑐𝑜𝑠 𝜈 + 𝑖 𝑠𝑖𝑛 𝑣
Which is where Euler stopped, without considering at any time the case where v which
is the angle could be pi. Hence the Euler’s Formula (identity) is only the case where the angle is qual to pi. Thanks to Euler complex analysis made a giant step in history.
References
• Calinger, Ronals S. “Leonhard Euler, Mathematical Genius in the Enlightenment”
(2016). Princeton University Press, pdf copy.
• De Condorcet, Marquis. “Eurology to Mr. Euler” (1783). History of the Royal
Academy of Science, Paris 1783. Pp 37–68. Found at The Euler Archive
• Dunham, William. “Journey through Genius. The Great Theorems of
Mathematics” (1990). Penguin Books, 1st edition, pdf copy.
• Euler, Leonhard. “Introductio in analysin infinitorum, volume 1” (1748). Found at
The Euler Archive E101
• Fuss, Nicolas. “Eulogy of Leonhard Euler” (1783). Translated by John S.D. Glaus in
2005. The Euler Archive.
• Gautschi, Walter. “Leonhard Euler: His Life, the Man, and His Works”, (2008) Vol.
50, №1, pp. 3–33.
• Handmann, Jakob Emanuel. Euler’s pastel painting (1753).
• Sandifer, C. Edward. “How Euler Did Even More” (2014). The Mathematical Association of America, pdf copy.
