In a manner that has been digested or thought through carefully; with calm reflection.
From 'digested' (past tense of digest, from Latin digerere meaning 'to carry apart, distribute') plus the adverb suffix '-ly'. The meaning evolved from the physical process of breaking down food to the metaphorical process of mentally processing information.
This word captures an interesting evolution in language: 'digestion' started as a purely physical process but became a metaphor for thinking. When you 'digest' bad news, your brain is doing cognitive work similar to what your stomach does with food!
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