According to the source, Leonhard Euler was the son of Paul Euler, a Lutheran pastor who studied theology and mathematics (he had heard Jacob Bernoulli’s lectures and courses, Bernoulli being one of the most famous mathematicians of that period). Moreover, the mathematician Johann Bernoulli, the younger brother of Jacob Bernoulli, decisively influenced Leonhard Euler’s career.
In 1720, at age 13, Leonhard Euler enrolled at the University of Basel, in order to prepare himself for a theological and philosophical career. But during his student times, he realized that his vocation is mathematics, being guided by his teacher, Johann Bernoulli. In 1723, he graduated from Basel Faculty, receiving a master’s degree in philosophy. In his master’s thesis, he compared and contrasted the philosophical ideas of René Descartes and Isaac Newton. Later in 1726, Euler concluded his theological studies, also at the University of Basel.
In 1727, he was awarded with the grand prize in a competition of Paris Academy dedicated to the best arrangement of masts on a ship. In his career, Euler won 12 times the Paris Academy Award.
Also in 1727, at the age of 20, Euler supported his thesis, “On acoustic”, based on which he applied for a Physics professor post at the University of Basel, but his request was denied.
During the same period, the two sons of Johann Bernoulli, Daniel and Nicolas, were active at the Imperial Academy of Sciences in St. Petersburg, and in 1726, after the death of Nicolas, Daniel took over the chair of mathematics and physics, freeing the department of medicine. Euler was recommended for this post and moved to St. Petersburg in 1727. He was promoted to a higher position in the mathematics department, being the leader of Mathematics Commission of the Academy.
At St. Petersburg, Euler published scientific papers and continued mathematical research, working with Daniel Bernoulli.
Due to exhaustion, Euler suffered a stroke and lost the sight in his right eye, at age 28.
After the death of Tsar Peter the Great and his successor, Catherine I, the power was gained by Peter II and Euler was forced to leave Russia; in 1741 accepted the proposal of Frederick the Great of Prussia to go to the Academy of Berlin. Euler lived in Berlin for 25 years, during which he wrote more than 380 articles and 200 scientific letters and published two of his books based on mathematical analysis.
In 1766, Euler left Berlin and returned to St. Petersburg, where he lived for 17 years, characterized by an extraordinary productivity, but also by a complete vision loss. Thanks to his excellent memory, Euler managed to continue his career aided by his sons.
Euler worked in almost all branches of mathematics, including geometry, infinitesimal calculus, trigonometry, algebra and number theory. In his numerous research, Euler introduced and popularized conventions scoring. He introduced the concept of function and was first to note f(x) for the application of f function to element x. He also introduced the modern notation for the trigonometric functions, the letter “e” for the base of the natural logarithm (presently known as “the number of Euler”), the Greek letter Σ (Sigma) for the amount and the letter “i” for the imaginary unit, according Wikipedia.
The development of infinitesimal calculus boosted the mathematical research in the 18th century, and mathematicians Bernoulli, family friends of Euler, were among those responsible for the progress in this area. Thanks to their influence, infinitesimal calculus became the main subject of Euler’s activity. Thus, he remained well known in mathematical analysis for frequent use of the power series – expressing functions using amounts with an infinite number of terms.
Euler introduced the use of exponential and logarithmic functions in analytic calculation. He found new ways to express various logarithmic functions using power series and successfully defined logarithms for complex numbers, thus extending the scope of logarithms.
Euler is also the one who defined the exponential function for complex numbers and made the connection between this and trigonometric functions, by a formula. A particular case of this formula leads to “Euler’s identity”.
Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving polynomial equations of the fourth degree (quartic equations). He also found a way to calculate integrals with complex limits, thus foreshadowing the development of modern complex analysis and invented calculus of variations, including the well-known Euler-Lagrange equation.
Euler was the first mathematician to use analytical methods to solve problems of the theory of numbers. In this regard, he united two different areas of mathematics (number theory and analysis), introducing a new field of study: analytical theory of numbers. In this new field, Euler created the theory of hypergeometric series, the theory of hyperbolic trigonometric functions and the analytical theory of continued fractions. For example, he demonstrated the infinity of prime numbers using the divergence of the harmonic series and analytic methods to gain an understanding of how the prime numbers are distributed. Euler’s work in this area have enabled further development of the prime number theorem.
Euler demonstrated “Newton’s identity”, “Fermat’s Little Theorem”, Fermat’s “two squares theorem” and Lagrange’s “four squares theorem”
Some of Euler’s greatest successes are found in concrete problem solving, from the real-world, using analytical methods. Thus, he created numerous applications using Bernoulli numbers, Fourier series, Venn diagrams, Euler numbers, e and π constants, continued fractions and integrals.
He integrated Leibniz’s differential calculus with Newton’s flow method and developed new methods for an easier application of differential calculus in mechanical problems. He has made great strides in improving the numerical approximation of integrals, creating a method that today is known as Euler approximations.
Euler demonstrated, simultaneously with Scottish mathematician Colin Maclaurin (but independently of this one), the Euler-Maclaurin formula. He also introduced the Euler-Mascheroni constant.
In fluid mechanics, Euler formulated the equation system describing a fluid movement; along with the continuity equation, this system is currently known as “Euler equations for ideal fluids”.
In addition to the successful implementation of his analytical calculation methods to the Newtonian mechanical issues, Euler applied these methods to astronomy problems. His work in this area have been recognized with numerous prizes awarded by the Academy of Sciences in Paris throughout his career. His achievements include determining with great accuracy the orbits of comets and other celestial bodies, as well as understanding the nature of comets.
Euler is the first one to illustrate the syllogistic reasoning, using closed curves, in 1768. These flowcharts are known as Euler diagrams.
In 1773, after 40 years of marriage, his wife, Katharina, passed away. After three years, Euler married Katharina’s sister, Salome Abigail. Euler and Katharina had 13 children, but only 3 of them survived. And one of the three became a mathematician.
Euler passed away on September 18, 1783, in St. Petersburg at age 76.