'antimethodically' originates from modern English, composed of the prefix 'anti-' (from Greek 'anti' meaning 'against') plus 'methodical' (from Greek 'methodos' via Latin/French meaning 'pursuit' or 'following a way'), with the adverbial suffix '-ly'.

'method' comes from Greek 'methodos' -> Latin/Old French forms -> Middle English 'method'; 'methodical' developed in English by adding the suffix '-ical' to 'method'; 'anti-' was prefixed to create 'antimethodical', and then '-ly' produced the adverb 'antimethodically'.

Initially the elements meant 'against' + 'a way/pursuit', and over time the combined form came to mean 'in a way that is against or not following a method' (i.e., not systematic).