Etymology
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installment (n.1)

"a partial payment on account of debt due," 1776, earlier "the arrangement of payment" (1732), an alteration of Anglo-French estaler "to fix payments," from Old French estal "fixed position, place; stall of a stable, market, or choir," from a Germanic source akin to Old High German stal "standing place" (see stall (n.1)). General sense of "a part of a whole, furnished or produced in advance of the rest" is from 1823. Installment plan is from 1894. 

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installment (n.2)

also instalment, 1580s, "induction into office, act of installing," from install + -ment.

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*stel- 

Proto-Indo-European root meaning "to put, stand, put in order," with derivatives referring to a standing object or place.

It forms all or part of: apostle; catastaltic; diastole; epistle; forestall; Gestalt; install; installment; pedestal; peristalsis; peristaltic; stale (adj.); stalk (n.); stall (n.1) "place in a stable for animals;" stall (n.2) "pretense to avoid doing something;" stall (v.1) "come to a stop, become stuck;" stallage; stallion; stele; stell; still (adj.); stilt; stole (n.); stolid; stolon; stout; stultify; systaltic; systole.

It is the hypothetical source of/evidence for its existence is provided by: Greek stellein "to put in order, make ready; equip or dress with weapons, clothes, etc.; prepare (for a journey), dispatch; to furl (sails);" Armenian stełc-anem "to prepare, create;" Albanian shtiell "to wind up, reel up, collect;" Old Church Slavonic po-steljo "I spread;" Old Prussian stallit "to stand;" Old English steall "standing place, stable," Old High German stellen "to set, place."

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pension (n.)

late 14c., pensioun, "payment for services," especially "a regular reward or annual payment out of a will or benefice" (early 14c., in Anglo-Latin), from Old French pension "payment, rent" (13c.) and directly from Latin pensionem (nominative pensio) "a payment, installment, rent," from past-participle stem of pendere "to hang, cause to hang; weigh; pay" (from PIE root *(s)pen- "to draw, stretch, spin"). For the financial sense of the Latin verb, see pound (n.1).

Meaning "regular payment to a person in consideration of past service" is from 1520s, hence "periodic payment made to a person retired from service on account of age or disability" (originally especially government pay to soldiers and sailors). Meaning "boarding house, boarding school" is attested from 1640s, from a sense in French based on the meaning "money paid for board," and in English it is usually in reference to places in France or elsewhere on the Continent.

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inauguration (n.)

"ceremonial investiture with office; act of solemnly or formally introducing or setting in motion anything of importance or dignity," 1560s, from French inauguration "installation, consecration," and directly from Late Latin inaugurationem (nominative inauguratio) "consecration," presumably originally "installment under good omens;" noun of action from past-participle stem of inaugurare "take omens from the flight of birds; consecrate or install when omens are favorable," from in- "on, in" (from PIE root *en "in") + augurare "to act as an augur, predict" (see augur (n.)).

INAUGURATIO was in general the ceremony by which the augurs obtained, or endeavoured to obtain, the sanction of the gods to something which had been decreed by man; in particular, however, it was the ceremony by which things or persons were consecrated to the gods .... If the signs observed by the inaugurating priest were thought favourable, the decree of men had the sanction of the gods, and the inauguratio was completed. [William Smith (ed.), "Dictionary of Greek and Roman Antiquities," 1842]
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