Etymology
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castigate (v.)

"to chastise, punish," c. 1600, from Latin castigatus, past participle of castigare "to correct, set right; purify; chastise, punish," from castus "pure" (see caste) + agere "to do" (from PIE root *ag- "to drive, draw out or forth, move"). The notion behind the word is "make someone pure by correction or reproof." Compare purge (v.), from purus + agere. Related: Castigated; castigating; castigator; castigatory.

If thou didst put this soure cold habit on To castigate thy pride, 'twere well. [Shakespeare, "Timon" IV.iii (1607)]
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castigation (n.)

"punishment, correction, chastisement," late 14c., castigacioun, from Latin castigationem (nominative castigatio) "a correcting, reproof, chastising," noun of action from past-participle stem of castigare "to correct, set right; purify" (see castigate).

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chastise (v.)

"to inflict pain upon to punish and recall to duty, to punish for the purpose of correcting or reclaiming," c. 1300, chastisen, from Old French chastiier "to warn, advise, instruct; chastise, admonish; punish; dominate, tame" (12c., Modern French châtier), from Latin castigare "to set or keep right, to reprove, chasten, to punish," literally "to make pure" (see castigate). Or perhaps from Middle English chastien (see chasten) + -ise, though this would be early for such a native formation. The form of the modern word "is not easily accounted for" [OED]. Related: Chastised; chastising.

He alone may chastise who loves. [Rabindranath Tagore, "The Crescent Moon," 1913]
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*kes- 
Proto-Indo-European root meaning "to cut."

It forms all or part of: caret; cashier (v.) "dismiss;" cassation; caste; castellan; castellated; Castile; castle; castigate; castrate; castration; chaste; chastity; chateau; chatelaine; Chester; forecastle; incest; quash (v.) "make void, annul."

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit sastra- "knife, dagger;" Greek keazein "to split;" Latin carere "to be cut off from," cassus "empty, void;" Old Church Slavonic kosa "scythe."
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*ag- 

Proto-Indo-European root meaning "to drive, draw out or forth, move."

It forms all or part of: act; action; active; actor; actual; actuary; actuate; agency; agenda; agent; agile; agitation; agony; ambagious; ambassador; ambiguous; anagogical; antagonize; apagoge; assay; Auriga; auto-da-fe; axiom; cache; castigate; coagulate; cogent; cogitation; counteract; demagogue; embassy; epact; essay; exact; exacta; examine; exigency; exiguous; fumigation; glucagon; hypnagogic; interact; intransigent; isagoge; litigate; litigation; mitigate; mystagogue; navigate; objurgate; pedagogue; plutogogue; prodigal; protagonist; purge; react; redact; retroactive; squat; strategy; synagogue; transact; transaction; variegate.

It is the hypothetical source of/evidence for its existence is provided by: Greek agein "to lead, guide, drive, carry off," agon "assembly, contest in the games," agōgos "leader," axios "worth, worthy, weighing as much;" Sanskrit ajati "drives," ajirah "moving, active;" Latin actus "a doing; a driving, impulse, a setting in motion; a part in a play;" agere "to set in motion, drive, drive forward," hence "to do, perform," agilis "nimble, quick;" Old Norse aka "to drive;" Middle Irish ag "battle."

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