The Collaborative International Dictionary
Engaged \En*gaged"\, a.
Occupied; employed; busy.
Pledged; promised; especially, having the affections pledged; promised in marriage; affianced; betrothed.
Greatly interested; of awakened zeal; earnest.
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Involved; esp., involved in a hostile encounter; as, the engaged ships continued the fight.
Engaged column. (Arch.) Same as Attached column. See under Attach, v. t.
Wikipedia
In architecture, an engaged column is a column embedded in a wall and partly projecting from the surface of the wall, sometimes defined as semi or three-quarter detached. Engaged columns are rarely found in classical Greek architecture, and then only in exceptional cases, but in Roman architecture they exist in abundance, most commonly embedded in the cella walls of pseudoperipteral buildings.