Etymology
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other (adj., pron.)

Old English oþer "second, the second of two; additional, further" (adj.), also as a pronoun, "one of the two; a different person or thing from the one in view," from Proto-Germanic *anthera- (source also of Old Saxon athar, Old Frisian other, Old Norse annarr, Middle Dutch and Dutch ander, Old High German andar, German ander, Gothic anþar "second, other").

These are from PIE *an-tero-(source of Lithuanian antras, Old Prussian anters "other, second), which is perhaps a variant of *al-tero- "the other of two" (source of Latin alter), from root *al- "beyond" + adjectival comparative suffix *-tero-.  Or the first element might be the pronoun *eno-, *ono- [Boutkan]. The Old English, Old Saxon, and Old Frisian forms show "a normal loss of n before fricatives" [Barnhart].

The sense of "second" was detached from this word in English (which now uses second (adj.), which is from Latin) and German (zweiter, from zwei "two") to avoid ambiguity. In Scandinavian, however, the second floor is still the "other" floor (Swedish andra, Danish anden). Also compare Old English oþergeara "next year."

As an adverb, "secondly" (late Old English); "otherwise" (c. 1200); "in addition" (mid-14c.).

The other woman "a woman with whom a man begins a love affair while he is already committed" is from 1855. The other day originally (late Old English) was "the next day;" later (c. 1300) "yesterday;" and now, loosely, "a day or two ago" (early 15c.). OED notes that the other place was euphemistic for Hell or "Oxford as regarded in Cambridge (and vice versa)." Phrase other half in reference to either the poor or the rich, is recorded from c. 1600.

La moitié du monde ne sçayt comment l'aultre vit. [Rabelais, "Pantagruel," 1532]
Halfe the world knowes not how the other halfe li[v]es. [George Herbert, "Outlandish Proverbs," 1640]
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each other 
reciprocal pronoun, originally in late Old English a phrase, with each as the subject and other inflected (as it were "each to other," "each from other," etc.).
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otherness (n.)

"state or quality of being other," 1580s, from other + -ness.

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others (n.)

"the remaining ones," Old English, plural of noun use of other (q.v.).

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otherwise (adv.)

contracted from Old English phrase on oðre wisan "in the other manner" (see other + wise (n.)), which in Middle English became oþre wise, and mid-14c. oþerwise. As an adjective from c. 1400. Also in Middle English were otherwhere "elsewhere;" otherwhat "something else" (pron.).

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another (pron., adv.)
"not this, not the same; someone or something else," early 13c., merger of an + other. Old English used simply oþer. Originally "a second of two." Compound reciprocal pronoun one another is recorded from 1520s.
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alter (v.)

late 14c., "to change (something), make different in some way," from Old French alterer "to change, alter," from Medieval Latin alterare "to change," from Latin alter "the other (of the two)," from PIE root *al- (1) "beyond" + comparative suffix -ter (as in other). Intransitive sense "to become otherwise" first recorded 1580s. Related: Altered; altering.

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alteration (n.)

late 14c., alteracioun, "change, transformation, action of altering," from Old French alteracion "change, alteration" (14c.), and directly from Medieval Latin alterationem (nominative alteratio), noun of action from past-participle stem of Late Latin alterare "to change," from Latin alter "the other (of the two)," from PIE root *al- (1) "beyond" + comparative suffix -ter (as in other). Meaning "change in character or appearance" is from 1530s; that of "change in ready-made clothes to suit a customer's specifications" is from 1901. Related: Alterations.

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otherworldly (adj.)

1854, "governed in this life by motives relating to consideration of an afterlife," from other + world + -ly (1). By 1873 as "of or pertaining to a world of imagination."

Otherworldliness is recorded from 1819. Phrase other world is from c. 1200 (oþre weorlde) as "afterlife, spirit-land, world to come;" c. 1300 as "world of idealism or fantasy, a state of existence different from normal," but otherworldliness seems to have been formed from worldliness. Leigh Hunt used it first in print, in "The Examiner" [Dec. 19, 1819], but a reported use of it by Coleridge, printed in Thomas Allsop's selections from Coleridge's letters and conversations (1836), which apparently cover the years 1818-22, was better-known thereafter, and the word is sometimes attributed to Coleridge:

As there is a worldliness or the too much of this life, so there is another-worldliness, or rather other worldliness, equally hateful and selfish with this worldliness.

Hunt, in his "Autobiography" (1850), writes:

I hope I am not giving fresh instance of a weakness which I suppose myself to have outgrown; much less appropriating an invention which does not belong to me; but an accomplished authoress one day (Mrs. Jameson), at the table of my friend Barry Cornwall, quoted the term "otherworldliness" from Coleridge. I said Coleridge was rich enough not to need the transference to him of other men's property; and that I felt so much honoured by the supposition in this instance, that I could not help claiming the word as my own. If Coleridge, indeed, used it before me, I can only say that I was not aware of it, and that my own reflections, very much accustomed to that side of speculation, would have suggested an identical thought.
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