early 15c., "an undertaking," formerly also enterprize, from Old French enterprise "an undertaking," noun use of fem. past participle of entreprendre "UNDERtake, take in hand" (12c.), from entre- "between" (see entre-) + prendre "to take," contraction of prehendere (see prehensile). Abstract sense of "adventurous disposition, readiness to undertake challenges, spirit of daring" is from late 15c.
I heed the Etymological Fallacy, but how should the etymology be interpreted, to connect the prefixes 'entre-' with 'under-'? .
Update on 2015 Dec 4: I might have answered my own question. ELU references
pp 210-214, An Analytic Dictionary of English Etymology, An Introduction (2008) by Anatoly Liberman that explicate 'under-' in 'understand', which is summarised here whose final 2 paragraphs recapitulate Etymonline below (which I failed to scrutinise):
[...] the under is not the usual word meaning "beneath," but from Old English under, from PIE *nter- "between, among"
That is the suggestion in Barnhart, but other sources regard the "among, between, before, in the presence of" sense of Old English prefix and preposition under as other meanings of the same word. "Among" seems to be the sense in many Old English compounds [...]