Etymology
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sad (adj.)

Old English sæd "sated, full, having had one's fill (of food, drink, fighting, etc.), weary of," from Proto-Germanic *sathaz (source also of Old Norse saðr, Middle Dutch sat, Dutch zad, Old High German sat, German satt, Gothic saþs "satiated, sated, full"), from PIE *seto-, from root *sa- "to satisfy." Related: Sadder; saddest.

In Middle English and into early Modern English the prevailing senses were "firmly established, set; hard, rigid, firm; sober, serious; orderly and regular," but these are obsolete except in dialect. The sense development seems to have been via the notion of "heavy, ponderous" (i.e. "full" mentally or physically), thus "weary, tired of." By c. 1300 the main modern sense of "unhappy, sorrowful, melancholy, mournful" is evident. An alternative course would be through the common Middle English sense of "steadfast, firmly established, fixed" (as in sad-ware "tough pewter vessels") and "serious" to "grave." In the main modern sense, it replaced Old English unrot, negative of rot "cheerful, glad."

By mid-14c. as "expressing or marked by sorrow or melancholy." The meaning "very bad, wicked" is from 1690s, sometimes in jocular use. Slang sense of "inferior, pathetic" is from 1899; sad sack is 1920s, popularized by World War II armed forces (specifically by cartoon character invented by Sgt. George Baker, 1942, and published in U.S. Armed Forces magazine "Yank"), probably a euphemistic shortening of the common military slang phrase sad sack of shit.

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sadder (adj.)

"more sad," Middle English saddere, comparative of sad (adj.).

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sadly (adv.)

c. 1300, "heavily," also "solidly," from sad + -ly (2). The main modern meaning "sorrowfully" begins by mid-14c.

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sadness (n.)

early 14c., sadnesse, "seriousness," from sad + -ness. Meaning "sorrowfulness, dejection of mind" is by c. 1500, perhaps c. 1400, but throughout Middle English the word usually referred to "solidness, firmness, thickness, toughness; permanence, continuance; maturity; sanity."

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sate (v.)

"to satisfy, fill full, surfeit," c. 1600, probably an alteration (by influence of Latin satiare "satiate") of Middle English saden "become weary or indifferent; satiate," from Old English sadian "to satiate, fill; be sated, get wearied" (see sad (adj.)), ultimately from PIE root *sa- "to satisfy." Related: Sated; sating.

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sadden (v.)

"to make sorrowful," 1620s, from sad (adj.) + -en (1); earlier "to make solid or firm" (c. 1600). The earlier verb was simply sad, from Middle English saden "become weary or indifferent," also "make (something) hard or stiff," from Old English sadian, which also could be the source of the modern verb. The intransitive meaning "to become sorrowful" is from 1718. Related: Saddened; saddening.

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*sa- 
*sā-, Proto-Indo-European root meaning "to satisfy."

It forms all or part of: assets; hadron; sad; sate; satiate; satiety; satisfy; satire; saturate; saturation.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit a-sinvan "insatiable;" Greek hadros "thick, bulky;" Latin satis "enough, sufficient;" Old Church Slavonic sytu, Lithuanian sotus "satiated;" Old Irish saith "satiety," sathach "sated;" Old English sæd "sated, full, having had one's fill, weary of."
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trist (adj.)
"sorrowful," early 15c., from French triste "sad, sadness" (10c.), from Latin tristis "sad, mournful, sorrowful, gloomy." Re-borrowed late 18c. (as "dull, uninteresting") as a French word in English and often spelled triste.
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sorrowful (adj.)
Old English sorgful "sad, anxious, careful; distressing, doleful;" see sorrow (n.) + -ful. Related: Sorrowfully; sorowfulness.
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droopy (adj.)

"dejected, sad, gloomy," c.1200, drupie, perhaps from droop, perhaps from Old Norse drupr "drooping spirits, faintness."

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