each other

Old English ælc (n., pron., adj.) "any, all, every, each (one)," short for a-gelic "ever alike," from a "ever" (see aye (adv.)) + gelic "alike" (see like (adj.)). From a common West Germanic expression *aina-galīk (source also of Dutch elk, Old Frisian ellik, Old High German iogilih, German jeglich "each, every"). Originally used as we now use every (which is a compound of each) or all; modern use is by influence of Latin quisque. Modern spelling appeared late 1500s. Also see ilk, such, which.

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]