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1. with object occupy, attract, or involve (someone's interest or attention):

｟かたく｠ 〈人･物などが〉〈注意･関心〉を引く, 引きつける(→ engaging)

e.g. he plowed on, trying to outline his plans and engage Sutton's attention.

(engage someone in) cause someone to become involved in (a conversation or discussion):

〖engage A in B〗 A〈人〉をB〈会話･議論など〉に引き込む, 加える

e.g. they attempted to engage Anthony in conversation.

2. no object (engage in) participate or become involved in:

｟かたく｠ ; 〖engage in ［on, upon］ A〗 〈人が〉A〈行為など〉を行う, Aに携わる, 関係する, 参加［従事］する

e.g. organizations engage in a variety of activities

e.g. some are actively engaged in crime.

(engage with) establish a meaningful contact or connection with:

｟かたく｠ 〈人が〉(よりよい理解のために) 【事･人々などに】かかわる, ふれあう «with»

e.g. the teams needed to engage with local communities.

3. with object arrange to employ or hire (someone):

｟かたく｠ «…するために/…として» 〈人など〉を雇用する «to do/as» (!専門職を臨時に雇うことを含意; → employ)

e.g. he was engaged as a trainee copywriter.

with infinitive pledge or enter into a contract to do something:

｟やや古｠ ; 〖～ to do/that節〗 〈人が〉…すると［…ということを］誓約［保証, 請け負い］する.

e.g. he engaged to pay them $10,000 against a bond.

dated reserve (accommodations, a place, etc.) in advance:

｟古｠ 〈座席･客室など〉を予約する(reserve)

e.g. he had engaged a small sailboat.

4. (with reference to a part of a machine or engine) move into position so as to come into operation:

«…と» 〈歯車などが〉かみ合う «with»

e.g. no object : the clutch will not engage

【歯車などと】〈歯車など〉をかみ合わせる «with» (↔ disengage)

e.g. with object : he engaged the gears and pulled out into the road.

5. with object (of fencers or swordsmen) bring (weapons) together preparatory to fighting.

〘フェンシング〙 〈剣〉を組み合わせる

enter into conflict or combat with (an enemy):

｟かたく｠ 〈敵〉と交戦する, 〈味方〉を交戦させる.

e.g. tank and infantry units engaged the enemy.

late Middle English (formerly also as ingage): from French engager, ultimately from the base of gage1. The word originally meant ‘to pawn or pledge something’, later ‘pledge oneself (to do something’), hence ‘enter into a contract’ (mid 16th century), ‘involve oneself in an activity’, ‘enter into combat’ (mid 17th century), giving rise to the notion ‘involve someone or something else’.