In a way that relates to or involves surgery, or with the precision and care of surgical work.
From 'surgery' (from Greek 'cheirourgos,' hand-work or handiwork), with the adverbial suffix '-ally' added. The adverb form emerged in medical English to describe actions performed with surgical precision.
The Greek root 'cheiro' means 'hand'—surgery was literally 'hand-work' before modern instruments! When we say something is done 'surgically,' we're borrowing the ancient idea that skilled hands can accomplish what needs careful attention.
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