Present participle of address; speaking to, dealing with a problem, or directing attention to a particular matter.
From Old French 'adresser' meaning 'to make straight, direct', from 'a-' (to) + 'dresser' (to arrange, set up). The meaning evolved from physical directing to speaking to someone or tackling issues.
The phrase 'addressing an envelope' preserves the original meaning of 'making straight' or 'directing' - you're literally directing the mail to its destination by writing the address!
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