Leonhard Euler was a Swiss mathematician. He was born 1707 in Basel, in the German speaking part of Switzerland. At the age of 20 (in 1727) he went to St. Petersburg in the Tsardom of Russia.
In 1741 (he already was blind on his right eye) he went to Berlin in the Kingdom of Prussia, part of the Holy Roman Empire.
In 1766 he went back to St. Petersburg. Five years later (from 1771 on) he was completely blind. He died 1783, at the age of 76 years in St. Petersburg.
Euler never lived in an English-speaking country, and all of his writings (books and letters) were written either in Latin or in German language.
This means that there is absolutely no reason why his name should be pronounced as if it was an English word. The correct pronunciation is German style.
The German word
means "owl", which is a bird. And a person who breeds owls is an
And so the complete pronunciation of the mathematicians name is: