to spend the night
aulizomai describes remaining overnight at a place, whether outside a city or on a mountain.
The two attestations show remaining overnight at a specified location: outside the city at Bethany and on the mountain called Olivet.
Senses in use
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Remaining overnight at a location · dominant
In Matthew 21:17, the aorist passive form occurs after going out of the city to Bethany; the APB renders the clause "and spent the night there." In Luke 21:37, the imperfect middle form occurs in the contrast between daytime teaching and nighttime departure to the mountain called Olivet; the APB renders "at night he went out and spent the night on the mountain called Olivet."
In the Anselm Project Bible: Across both verses, the APB presents the action with "spent the night," locating it at Bethany in Matthew 21:17 and on the mountain called Olivet in Luke 21:37. The verse is read as a whole, but both clauses present an overnight stay.