How did Leonhard Euler and his contemporaries actually pronounce his surname?

by apolotary

This question has been bothering me for a while. Here’s the thing:

Nowadays I have seen the accepted pronunciation being either “oi-ler” or “yu-ler.” For instance, there is a frustratingly short discussion on stackexchange here
However, in Russian his surname is commonly pronounced as “ei-ler.” It is commonly known that he spent half of his academic career in Russia, spoke Russian, and presumably that’s how he wrote his surname in Russian himself. A scan of an almost contemporary book sort of confirms that

So my question is: which is more historically accurate? “Eiler” or “oiler”? I found the only relevant reply here that says that because of the phonetical shift in German, the pronunciation of “eu” has changed from “ei” to “oi,” but since the reply is unsourced, I take it with a grain of salt.

I found the only relevant reply here that says that because of the phonetical shift in German, the pronunciation of “eu” has changed from “ei” to “oi,”

The pronunciation of "eu" has never changed, it's always been /ɔʏ/:

Old High German has a diphtong /iʊ/, denoted in writing as the digraph "iu".
In Middle High German, the diphtong becomes a long vowel, /y:/. It's still written"iu", however.
In New High German, the monophtong becomes a diphtong again, /ɔʏ/. But this time the spelling changes as well; /ɔʏ/ is denoted using "eu", not "iu". Why? Nobody really knows. (The digraph "eü" is also attested for the phoneme but is obsolete in modern German. Also, "äu" sounds the same as "eu" in modern German, but that's actually coincidence.)

but since the reply is unsourced

Martin Aedler, The High Dutch Minerva, London 1680. Probably the first comprehensive grammar/textbook aimed at teaching German to the English. Contains a section on pronunciation that says German "ey", "ew", "eu" are equivalent to English "oi". (page 5) (The former two are spellings that were already archaic at the time.)

Benedictus Beiler, A New German Grammar, London 1736. Contains a section on pronunciation that says German "eu" is equivalent to English "oi". (page 4)