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Old English finn, from Proto-Germanic *finno (cf. Middle Low German vinne, Dutch vin), perhaps from Latin pinna "feather, wing" (see pin (n.)); or, less likely, from Latin spina "thorn, spine" (see spike (n.1)).
U.S. underworld slang sense of "$5 bill" is 1925, from Yiddish finif "five," from German fünf (see five) and thus unrelated. The same word had been used in England 1868 to mean "five pound note" (earlier finnip, 1839).
A five-dollar bill; five dollars: I gave my pal a fin/ It was the fin seen round the world. Where Reagan got the five bucks is a mystery
[1920s+ Underworld; fr Yiddish finif, ''five'']