Euler's first teacher was his father and since he had studied under Jacob Bernoulli he was a well trained math teacher and was able to pass what he knew onto his son. Research shows that Euler wasn't sent to the greatest schools while growing up and self taught mathematics by reading text books. Euler was suppose to follow his father into the church but Johann Bernoulli found the great potential that Euler had and started private tutoring him. Even though his passion was in mathematics Euler followed his father's wishes and began studying theology. After realizing that there wasn't a future for him in the church he got his father's blessing to change his studies and focus on math.
Euler by the age of 20 in 1726 had countless papers in print, short articles on isochronous curves in a resisting medium and was submitting papers into the Paris Academy Grand Prize and took second place behind Bouguer. After this achievement he was offered a post in St. Petersburg teaching in applications of mathematics and appointed to the mathematical-physical division. In the coming years he was appointed to the senior chair of mathematics and was able to start a family. In 173 he became sick and almost died, while being sick his eye sight started to diminish till he was completely blind in one eye and going blind in the other.
In 1741 he and his family left St. Petersburg and went to Berlin where he worked at the Academy of Science. Euler spent the next 25 years in Berlin where he wrote 380 articles, books Calculus of variations, planetary orbits, artillery and ballistics, navigation, motion of the moon, and differential calculus. By the year of 1759 Euler assumed leadership of the Berlin Academy. He only stayed there for another 7 years when he decided to move back to St. Petersburg about this time Euler went completely blind, this however didn't stop him from doing his mathematics because he had a remarkable memory and was able to write articles with the help of his students. But in the year 1783 at nearly 77 years old Euler died.
In his 77 years the mathematics that he had a part in grew and took off into what we know today. Here are just some of the topics he had a part in:
- Geometry and Trigonometry
- considered sin and cos
- Differential calculus
- integrating factors
- Number Theory
- if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1
- Lunar theory with Clairaut
- Three body problem
- wave theory of light
- Eulers notations
- i for the square root on -1
- π for pi
- ∑ for summation
- finite differences Δy and Δ2y
- The Basel problem
- find a closed form for the sum of the infinity series ζ(2) = ∑ (1/n2)
- And much much more.
Looking back over Euler's life it is astonishing the things that he was able to accomplish and was able to do even close to the end when he was blind. I can say that I believe that math wouldn't be what we know it to be without Euler and what he gave future mathematics.