One of the greatest mathematicians of the 18th century, Leonhard Euler was responsible for important advancements in a number of different areas, especially number theory. There are Euler theorems, Euler numbers and Euler formulas in a different fields, each distinct because Euler contributed to so many specialties.

In the field of philosophy, Leonhard Euler was the first to systematically employ logic diagrams to map out the structure of logical statements and arguments. Circles to represent classes and diagram syllogisms had earlier been used by Leibniz and others, but Euler was the first to explain in detail how the circles can and should be used. This understanding would later be expanded upon by John Venn and such diagrams are now known as Venn Diagrams.

## Early life

Leonhard Euler was born on 15 April 1707, in Basel, Switzerland, to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, another pastor’s daughter. He had two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich.^{[9]} Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, Switzerland, where Leonhard spent most of his childhood. Paul was a friend of the Bernoulli family; Johann Bernoulli, then regarded as Europe’s foremost mathematician, would eventually be the most important influence on young Leonhard.

Euler’s formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, at age thirteen, he enrolled at the University of Basel. In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil’s incredible talent for mathematics.^{[10]} At that time Euler’s main studies included theology, Greek and Hebrew at his father’s urging to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician.

In 1726, Euler completed a dissertation on the propagation of sound with the title *De Sono*.^{[11]} At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the *Paris Academy Prize Problem* competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as “the father of naval architecture”, won and Euler took second place. Euler later won this annual prize twelve times.^{[12]}

## Career

### Saint Petersburg

Around this time Johann Bernoulli’s two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia.^{[13]}^{[14]} When Daniel assumed his brother’s position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.^{[15]}

Euler arrived in Saint Petersburg on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in Saint Petersburg. He also took on an additional job as a medic in the Russian Navy.^{[16]}

The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy to lessen the faculty’s teaching burden. The academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions.^{[12]}

The Academy’s benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler’s arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility, suspicious of the academy’s foreign scientists, cut funding and caused other difficulties for Euler and his colleagues.

Conditions improved slightly after the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department.^{[17]}

On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium.^{[18]} The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood.^{[19]}

### Berlin

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the *Berlin Academy*, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: the *Introductio in analysin infinitorum*, a text on functions published in 1748, and the *Institutiones calculi differentialis*,^{[20]} published in 1755 on differential calculus.^{[21]} In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.

In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick’s niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled *Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess*.^{[22]} This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works and was published across Europe and in the United States. The popularity of the “Letters” testifies to Euler’s ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist.^{[21]}

Despite Euler’s immense contribution to the Academy’s prestige, he eventually incurred the ire of Frederick and ended up having to leave Berlin. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs, in many ways the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick’s court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire’s wit.^{[21]} Frederick also expressed disappointment with Euler’s practical engineering abilities:

I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry!

^{[23]}

## Personal life

### Eyesight deterioration

Euler’s eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Euler’s vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as “Cyclops”. Euler remarked on his loss of vision, “Now I will have fewer distractions.”^{[24]} He later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, a failed surgical restoration rendered him almost totally blind. He was 59 years old then. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. For example, Euler could repeat the *Aeneid* of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler’s productivity on many areas of study actually increased. He produced, on average, one mathematical paper every week in the year 1775.^{[25]} The Eulers bore a double name, Euler-Schölpi, the latter of which derives from *schelb* and *schief*, signifying squint-eyed, cross-eyed, or crooked. This suggests that the Eulers had a susceptibility to eye problems.^{[26]}

### Return to Russia and death

In 1760, with the Seven Years’ War raging, Euler’s farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler’s estate, with Empress Elizabeth of Russia later adding a further payment of 4000 roubles—an exorbitant amount at the time.^{[27]} The political situation in Russia stabilized after Catherine the Great’s accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. All of these requests were granted. He spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage.

Three years after his wife’s death, Euler married her half-sister, Salome Abigail Gsell (1723–1794).^{[28]} This marriage lasted until his death. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences.^{[29]}

In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with a fellow academician Anders Johan Lexell, when he collapsed from a brain hemorrhage. He died a few hours later.^{[30]} Jacob von Staehlin-Storcksburg wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss, one of Euler’s disciples, wrote a more detailed eulogy,^{[31]} which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet, wrote:

il cessa de calculer et de vivre— … he ceased to calculate and to live.^{[32]}

Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Goloday Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director’s seat and, in 1837, placed a headstone on Euler’s grave. To commemorate the 250th anniversary of Euler’s birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.

## Contributions to mathematics and physics

Euler is the only mathematician to have *two* numbers named after him: the important Euler’s number in calculus, *e*, approximately equal to 2.71828, and the Euler–Mascheroni constant γ (gamma) sometimes referred to as just “Euler’s constant”, approximately equal to 0.57721. It is not known whether γ is rational or irrational.Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes.^{[25]} Euler’s name is associated with a large number of topics.