Etymology
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quittance (n.)

c. 1200, cwitance, quitaunce, "payment, compensation;" c. 1300, "a discharge from a debt or an obligation," from Old French quitance (Modern French quittance), from quiter "clear, establish one's innocence;" also transitive, "release, let go, relinquish, abandon" (12c.), from quite "free, clear, entire, at liberty; discharged; unmarried," from Medieval Latin quitus, quittus, from Latin quietus "free" (in Medieval Latin "free from war, debts, etc."), also "calm, resting" (from PIE root *kweie- "to rest, be quiet"). The Middle English word also is in part from Medieval Latin quittantia, a variant of quietantia.

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*kweie- 
*kweiə-, also *kwyeə-, Proto-Indo-European root meaning "to rest, be quiet."

It forms all or part of: acquiesce; acquit; awhile; coy; quiesce; quiescent; quiet; quietism; quietude; quietus; quit; quitclaim; quite; quit-rent; quittance; requiescat; requiem; requite; while; whilom.

It is the hypothetical source of/evidence for its existence is provided by: Avestan shaitish "joy," shaiti- "well-being," shyata- "happy;" Old Persian šiyatish "joy;" Latin quies "rest, repose, quiet;" Old Church Slavonic po-koji "rest;" Old Norse hvild "rest."
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omittance (n.)

"omission, failure or forbearance to do something," c. 1600, perhaps coined by Shakespeare, who used it in wordplay ("Omittance is no quittance"), from omit + -ance.

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