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“correction” in Latin

The English word correction itself goes back to Latin regō, through corrēctiō.

Riddle & Arnold A Copious and Critical English–Latin Lexicon, 1864

correction
correctio: emendatio: (the words are found in this connection and order), correctio et emendatio. SYN. in To CORRECT, v.] ☞ Correctio is also the rhetorical figure, ἐπανόρθωσις, when an orator corrects himself, for the purpose of adding something stronger, etc. (☞ correctura = the office of a corrector, a sort of land-agent or steward in the time of the emperors). The labour of correction, *emendandi cura - *molestia corrigendi et emendandi (Krebs.). The most careful correction of the press, *cura, qua nulla eruditior cogitari potest, in plagulis de prelo corrigendis: to employ a person in the correction of the press, *alicui librum ab operarum mendis (or erratis) liberandum tradere: to attend to the correction of the press, *curam tollendis operarum erratis or mendis insumere: not to attend to the corrections that have been made, *non inspicere, quae emendata sunt (e.g., in an exercise).
Under correction, bona hoc tua venia dixerim (when offence is to be deprecated; e.g., “points of which the Roman augurs, I am speaking under correction are now themselves ignorant,” quae quidem nunc a Romanis auguribus ignorantur; bona hoc tua venia dixerim, Cic., Div., 1, 15, 25): *salvo tamen tuo judicio (with deference to your better judgement): nisi tu aliter existimas, sentis, censes: nisi tibi aliter videtur (unless you think differently). But, to speak under correction you do not prove your assertion, sed da mihi nunc; satisne probas? (Cic., Acad., 1, 3, 9.)
Chastisement; Vid:
House of correction, ergastulum: to put anybody into a house of correction, aliquem in ergastulum ducere or dare: to be confined in a house of correction, esse in ergastulo.

Quick answer

What is the Latin word for correction?
The usual Latin words for “correction” are corrēctiō, ēmendātiō. The right choice depends on the sense; see the distinctions below.

Entry from J. E. Riddle & T. K. Arnold, A Copious and Critical English–Latin Lexicon (rev. C. Anthon, New York: Harper, 1864), digitised by Fergus J. P. Walsh, CC BY-SA 4.0; Latin markup adapted by latindictionary.co.