Leonhard Euler
Life
Leonhard Euler was born April 15th, 1707 in Basel, Switzerland. His father taught him preliminary math until he attended the University of Basel. There he studied under John Bernoulli, eventually obtaining a Master of Arts degree.
It was at the University of Basel that Euler met Daniel and Nicholas Bernoulli, the two sons of John. In 1727 Daniel and Nicholas helped Euler attain a position as an associate of the Academy of Sciences in St. Petersburg. Three years later Euler became the professor of physics and three years after that, the chair of mathematics.
Frederick the Great coerced Euler to move to Berlin to be a member of the Academy of Sciences and professor of mathematics in 1741 until he returned to St. Petersburg in 1766. He lost his sight to a cataract in his left eye, but that didn't stop him from continuing to solve the unsolved problems of his era or from reshaping countless disciplines.
Leonhard Euler died November 18th, 1783 in St. Petersburg, Russia at the age of 76.
Works
Euler's body of work is so prolific just listing the titles would fill a book. And after his death in 1783 papers he wrote continued to be published for the next 50 years. Even today his work contributes to modern discoveries; research papers are littered with countless citations of Euler's publishings.
Euler enjoyed simplifying and improving upon techniques no matter how elementary, no matter how established, and no matter the discipline. He is well known for elegance in his math and the ability to state complex problems in simple terms.
The influence of Introductio in Analysin infinitorum, published in 1748 can still be seen today in modern textbooks about algebra, polynomials, and trigonometry as well as in the syntax used for trigonometric functions and geometric diagrams.
It is in Introductio in Analysin infinitorum that Euler showed $\cos{θ}+\mathrm{i}\sin{θ}=\mathrm{e}^{\mathrm{i}θ}$. Or in other words that there is a relationship between trigonometric and exponential functions.
In 1755 Euler wrote Institutiones Calculi Differentialis, which is considered the first complete textbook on differential calculus. It was followed by its counterpart Institutionum Calculi Integralis, released as three volumes from 1768 to 1770. In it Euler created the beta and gamma functions.
Famous Theorems
Aspiring mathematicians like to rank Euler's theorems and while the list is subjective and the qualities on which to rank are not well-defined, there are a few theorems that inevitably appear at or near the top.
$\cos{θ}+\mathrm{i}\sin{θ}=\mathrm{e}^{\mathrm{i}θ}$ is called Euler's Formula and was introduced in Introductio in Analysin infinitorum. It leads directly to Euler's identity, although the identity doesn't appear in any of Euler's papers.
Euler's identity, $\mathrm{e}^{\mathrm{i}π} + 1 = 0$ was ranked as the most beautiful theorem in mathematics in 1988 by Mathematical Intelligencer and in 2004 as the second greatest equation by Physics World.
Another formula often topping the list of Euler's greatest contributions is the polyhedron formula. Euler discovered the formula in 1750, positing that convex polyhedron obey the rule $V - E + F = 2$, where V is the number of vertices of the solid, E is the number of edges, and F is the face count. He expressed astonishment in a letter to his friend Christian Goldbach that those before him could have missed such a fundamental property, however, he had difficulty with the proof. Euler thought he had finally proved the formula in his paper E231 - Proof of some of the properties of solid bodies enclosed by planes, but it has since been shown to be incomplete.
Leonhard Euler entered the mathematical stage by solving the Basel problem, which gained fame after prominent mathematicians had tried and failed to solve it. In 1734 Euler gave the solution $\frac{π^2}{6}$ and proved it with an infinite-product. In 1741 he solved it again! This time with integrals.
It is easy to forget about some of his lesser known work. Leonhard was a physicist, theologist, physiologist, theorist, practitioner, inventor, husband, and a father. His influence was vast and his work — truly remarkable.