Etymology
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remorse (n.)

"intense and painful self-condemnation and penitence due to consciousness of guilt; the pain of a guilty conscience," late 14c., from Old French remors (Modern French remords) and directly from Medieval Latin remorsum"a biting back or in return," noun use of neuter past participle of Latin remordere "to vex, torment disturb," literally "to bite back, bite again" (but seldom used in the literal sense), from re- "back, again" (see re-) + mordēre "to bite," which is perhaps from an extended form of PIE root *mer- "to rub away, harm."

The sense evolution was via the Medieval Latin phrase remorsus conscientiæ (Chaucer's remors of conscience, also translated into Middle English as ayenbite of inwit). Middle English also had a verb, remord "to strike with remorse, touch with compassion, prick one's conscience" (late 14c.), from Latin remordere. Richard Brome's "Merry Beggars" (1641) delighted that they had 

No bargains or accounts to make,
No land or lease to let or take:
Or if we had, should that remore us
When all the world's our own before us[?]  
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feel (v.)

Old English felan "to touch or have a sensory experience of; perceive, sense (something)," in late Old English "have a mental perception," from Proto-Germanic *foljanan (source also of Old Saxon gifolian, Old Frisian fela, Dutch voelen, Old High German vuolen, German fühlen "to feel," Old Norse falma "to grope"), which is of uncertain origin, possibly from a PIE *pal- "to touch, feel, shake, strike softly" (source also of Greek psallein "to pluck" the harp), or from PIE root *pel- (5) "to thrust, strike, drive."

In Germanic languages, the specific word for "perceive by sense of touch" has tended to evolve to apply to the emotions. The connecting notion might be "perceive through senses which are not referred to any special organ." Sense of "be conscious of a tactile sensation, sense pain, pleasure, illness, etc.; have an emotional experience or reaction," developed by c. 1200, also "have an opinion or conviction;" that of "to react with sympathy or compassion" is from mid-14c. Meaning "to try by touch" is from early 14c. From late 14c. as "know (something) beforehand, to have foreknowledge of." To feel like "want to" attested from 1829.

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mercy (n.)

late 12c., "God's forgiveness of his creatures' offenses," from Old French mercit, merci (9c.) "reward, gift; kindness, grace, pity," from Latin mercedem (nominative merces) "reward, wages, pay, hire" (in Vulgar Latin "favor, pity;" in Medieval Latin "thanks; grace"), from merx (genitive mercis) "wares, merchandise" (see market (n.)). In Church Latin (6c.) it was given a specific application to the heavenly reward earned by those who show kindness to the helpless and those from whom no requital can be expected.

Meaning "disposition to forgive or show compassion" is attested from early 13c. Sense of "an act or exercise of forbearance or good will" is from c. 1300. As an interjection, attested from mid-13c. (short for may God have mercy, have mercy on me, etc.).  Many of the English senses are found earlier in French, but in French the word largely has been superseded by miséricorde except as a word of thanks. Sense of "discretionary action" (as in at (one's) mercy) is from mid-14c. Seat of mercy "golden covering of the Ark of the Covenant" (1530), hence "the throne of God," is Tyndale's loan-translation of Luther's gnadenstuhl, an inexact translation of Latin propitiatorium, ultimately a rendering of Hebrew kapporeth, literally "propitiatory."

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charity (n.)

late Old English, "benevolence for the poor," also "Christian love in its highest manifestation," from Old French charité "(Christian) charity, mercy, compassion; alms; charitable foundation" (12c.), from Latin caritatem (nominative caritas) "costliness; esteem, affection," from carus "dear, valued" (from PIE *karo-, from root *ka- "to like, desire").

In the Vulgate the Latin word often is used as translation of Greek agape "love" — especially Christian love of fellow man — perhaps to avoid the sexual suggestion of Latin amor). The Vulgate also sometimes translated agape by Latin dilectio, noun of action from diligere "to esteem highly, to love" (see diligence).

Wyclif and the Rhemish version regularly rendered the Vulgate dilectio by 'love,' caritas by 'charity.' But the 16th c. Eng. versions from Tindale to 1611, while rendering agape sometimes 'love,' sometimes 'charity,' did not follow the dilectio and caritas of the Vulgate, but used 'love' more often (about 86 times), confining 'charity' to 26 passages in the Pauline and certain of the Catholic Epistles (not in I John), and the Apocalypse .... In the Revised Version 1881, 'love' has been substituted in all these instances, so that it now stands as the uniform rendering of agape. [OED]

The general sense of "affections people ought to feel for one another" is from c. 1300. Also from c. 1300 as "an act of kindness or philanthropy," also "alms, that which is bestowed gratuitously on a person or persons in need." The sense of "charitable foundation or institution" in English is attested by 1690s. The meaning "liberality in judging others or their actions" is from late 15c. A charity-school (1680s) educated (and sometimes housed and fed) poor children and was maintained by voluntary contributions or bequests.

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