> https://www.etymonline.com/word/amuse states that this is the "to, at" > prefix from Latin in a sense of "causing to", not the "not, without" one > from Greek.
Funnily [oops, I guess I meant "amusingly"] enough, the two end up meaning kind of the same: in that what happens is that attention is diverted, so when you "amuse", you cause thinking of something which prevents thinking of what you'd have been thinking of if it weren't for that amusement. Stefan