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According to Internet (actually, Wikipedia and Wolfram MathWorld), I have two information: It was Euler who first introduced the symbol $e$ (before people used $b$); the symbol is to honor Euler.
I find this very strange. My theory is that $e$ is for "exponential" not for Euler.
So what is the evidence that Euler was using the symbol to "honor" himself?
On the The number e page, one can find the following:
As far as we know the first time the number e appears in its own right is in 1690. In that year Leibniz wrote a letter to Huygens and in this he used the notation b for what we now call e. [...] Retrospectively, the early developments on the logarithm became part of an understanding of the number e.
Later:
So much of our mathematical notation is due to Euler that it will come as no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from "exponential", but it may have just be the next vowel after "a" and Euler was already using the notation "a" in his work. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731
Some other details are given here:
Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Euler's Mechanica (1736). While in the subsequent years some researchers used the letter c, e was more common and eventually became the standard.
You are right and your Internet source is wrong. First, Euler introduced this notation himself. And clearly this stood for "exponential". Euler was a modest person, and I cannot imagine him naming anything for himself. Second, if someone wanted to name something after Euler, s/he would choose capital E.
It was common among mathematicians of the 17th century to use vowels; thus Fermat's writing on adequality is full of A's and E's. Since A is used much too often the choice of the next vowel, E, was all too natural. I don't think Euler used the symbol to honor himself. Possibly once he chose E, the symbol was retained by later mathematicians so as to honor Euler.
