1540s, "clean, free from dirt," from Anglo-French neit, French net "clear, pure" (12c.), from Latin nitidus "well-favored, elegant, trim," literally "gleaming," from nitere "to shine," from PIE root *nei- "to shine" (source also of Middle Irish niam "gleam, splendor," niamda "shining;" Old Irish noib "holy," niab "strength;" Welsh nwyfiant "gleam, splendor").
From 1540s as "well-shaped, well-proportioned; characterized by nicety of appearance." Meaning "inclined to be tidy" is from 1570s; sense of "in good order" is from 1590s. Of liquor, "straight, undiluted," c. 1800, from meaning "unadulterated" (of wine), which is first attested 1570s. Informal sense of "very good, desirable" is noted by 1934 in American English, but in many earlier senses in English since 17c. neat seems to be simply a vague commendatory word; variant neato is teenager slang, by 1968. Related: Neatly; neatness.
word-forming element making verbs (such as darken, weaken) from adjectives or nouns, from Old English -nian, from Proto-Germanic *-inojan (also source of Old Norse -na), from PIE adjectival suffix *-no-. Most active in Middle English and early modern English, hence most verbs in -en are comparatively recent.
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Definitions of neaten from WordNet
put (things or places) in order;
Synonyms: tidy / tidy up / clean up / straighten / straighten out / square away