Now for the mysterious "I" ....
The family history (see
http://digipres.cjh.org:1801/delivery/DeliveryManagerServlet?dps_pid=IE1017160) records that Jacob arrived in the US in 1773, where he was granted a licence to trade with the Indians.
Lo and behold, Jacob
Isaiah Cohen was granted such a licence on 12 May 1773
(
http://www.rootschat.com/links/01pkk/). I find it hard to believe that that was not Jacob I. Cohen.
In secular records, German Jews were generally referred to as given name + father's given name + byname/surname (if one existed). The vast majority of Jews in German territories did not adopt hereditary surnames until the early 19th century. In larger communities, this patronymic form served to distinguish between people with the same given name and surname.
My 4th great-grandfather appears in German records as Matthes Jacob Levi (d. 1803); he was Matthes son of Jacob Levi. His grandson (my 2nd ggf) appears as Matthias David Levy; he was Matthias son of David Levy.
The above would suggest that Jacob and Israel were the sons of Isaiah. But then what about Joshua?
Here's my theory, for what it's worth. Joshua is the English form of the Hebrew name
Yehoshua, sometimes shortened to
Yeshua. Isaiah comes from
Yesha'yahu. The names are virtual synonyms; the former means "God saved", the latter "Saved by God".
However, in secular German records of Jews, the name Josua is extremely unusual, whereas Isaias/Jesaias is not uncommon. It is not beyond the bounds of possibility that Jacob's and Israel's father used Isaias/Isaiah as his secular name.
Moving to the "I" used as a middle initial by Israel's children, it is perfectly logical that each child had the name Israel as a patronymic, e.g. David [son of] Israel Cohen.
This was a just a bit of an academic digression ... the result of being jobless in lockdown
Justin