other (adj.)youdaoicibaDictYouDict
Old English oþer "the second" (adj.), also as a pronoun, "one of the two, other," from Proto-Germanic *antharaz (cognates: Old Saxon athar, Old Frisian other, Old Norse annarr, Middle Dutch and Dutch ander, Old High German andar, German ander, Gothic anþar "other").

These are from PIE *an-tero-, variant of *al-tero- "the other of two" (source of Lithuanian antras, Sanskrit antarah "other, foreign," Latin alter), from root *al- (1) "beyond" (see alias (adv.)) + adjectival comparative suffix *-tero-. The Old English, Old Saxon, and Old Frisian forms show "a normal loss of n before fricatives" [Barnhart]. Meaning "different" is mid-13c.

Sense of "second" was detached from this word in English (which uses second, 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."

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 (mid-12c.) was "the next day;" later (c. 1300) "yesterday;" and now, loosely, "a day or two ago" (early 15c.). 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]
speech (n.)youdaoicibaDictYouDict
Old English spæc "act of speaking; power of speaking; manner of speaking; statement, discourse, narrative, formal utterance; language," variant of spræc, from Proto-Germanic *sprek-, *spek- (cognates: Danish sprog, Old Saxon spraca, Old Frisian spreke, Dutch spraak, Old High German sprahha, German Sprache "speech;" see speak (v.))

The spr- forms were extinct in English by 1200. Meaning "address delivered to an audience" first recorded 1580s.
And I honor the man who is willing to sink
Half his present repute for the freedom to think,
And, when he has thought, be his cause strong or weak,
Will risk t' other half for the freedom to speak,
Caring naught for what vengeance the mob has in store,
Let that mob be the upper ten thousand or lower.

[James Russell Lowell, "A Fable for Critics," 1848]