In a way that is attractive, interesting, or persuasive; done in a manner that makes something seem desirable or worthy of consideration.
From 'appeal' (to make a request or to be attractive) plus '-ing' (present participle) plus '-ly' (forming adverbs). The root 'appeal' comes from Old French 'apeler,' meaning to call to or address.
There's something psychologically interesting about 'appealingly'—it suggests a power of attraction or persuasion. A smile can be appealing, a proposal can be made appealingly, showing how the word bridges both requests and charm.
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