[7] Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory".

[11] Euler is credited with being the first to develop graph theory (partly as a solution for the problem of the Seven Bridges of Königsberg, which is also considered the first practical application of topology).

Euler formulated the partial differential equations for the motion of inviscid fluid,[11] and laid the mathematical foundations of potential theory.

[19] From a young age, Euler received schooling in mathematics from his father, who had taken courses from Jacob Bernoulli some years earlier at the University of Basel.

[23] In 1726, Euler completed a dissertation on the propagation of sound titled De Sono,[24][25] with which he unsuccessfully attempted to obtain a position at the University of Basel.

[28] Johann Bernoulli's two sons, Daniel and Nicolaus, entered into service at the Imperial Russian Academy of Sciences in Saint Petersburg in 1725, leaving Euler with the assurance they would recommend him to a post when one was available.

[29][30] 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.

[26] 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.

[32] 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.

[28] The academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died before Euler's arrival to Saint Petersburg.

[37] Concerned about the continuing turmoil in Russia, Euler left St. Petersburg in June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia.

[46] 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 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.

[45] 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 rubles—an exorbitant amount at the time.

In addition to his own research, Euler supervised the library, the observatory, the botanical garden, and the publication of calendars and maps from which the academy derived income.

[55] 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.

[37] Early in his life, Euler memorized Virgil's Aeneid, and by old age, he could recite the poem and give the first and last sentence on each page of the edition from which he had learnt it.

Euler blamed the cartography he performed for the St. Petersburg Academy for his condition,[65] but the cause of his blindness remains the subject of speculation.

With the aid of his scribes, Euler's productivity in many areas of study increased;[68] in 1775, he produced, on average, one mathematical paper per week.

[42] 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 Anders Johan Lexell when he collapsed and died of a brain hemorrhage.

[71] The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones.

While some of Euler's proofs are not acceptable by modern standards of mathematical rigour[73] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances.

Euler's use of power series enabled him to solve the Basel problem, finding the sum of the reciprocals of squares of every natural number, in 1735 (he provided a more elaborate argument in 1741).

The concept is regarded as a fundamental theorem within number theory, and his ideas paved the way for the work of Carl Friedrich Gauss, particularly Disquisitiones Arithmeticae.

[91] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges.

[111] Even when dealing with music, Euler's approach is mainly mathematical,[112] for instance, his introduction of binary logarithms as a way of numerically describing the subdivision of octaves into fractional parts.

[125] He insisted that knowledge is founded in part on the basis of precise quantitative laws, something that monadism and Wolffian science were unable to provide.

[129] Mathematician François Arago said, "Euler calculated without any apparent effort, just as men breathe and as eagles sustain themselves in air".

[146] The Euler Archive was started at Dartmouth College[147] before moving to the Mathematical Association of America[148] and, most recently, to University of the Pacific in 2017.

Series IV, which contains the massive amount of Euler's correspondence, unpublished manuscripts, and notes only began compilation in 1967.