Etymology
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rigor (n.)

late 14c., rigour, "harshness, severity in dealing with persons; force; cruelty," from Old French rigor "strength, hardness" (13c., Modern French rigueur), from Latin rigorem (nominative rigor) "numbness, stiffness, hardness, firmness; roughness, rudeness," from rigēre "be stiff" (from PIE root *reig- "stretch; be stretched; be stiff").

Also, in medieval medicine, "a sudden chill" (c. 1400). From early 15c. as "exactness, strictness without indulgence" (of discipline, the law, etc.). Compare rigidity. Rigorism "rigidity in principles or practice" (originally religious) is from 1704. Rigidist is by 1716.

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rigor mortis 

"characteristic stiffening of the body caused by contraction of muscles after death," 1837, from Latin rigor "stiffness" (see rigor) + mortis, genitive of mors "death" (see mortal).

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de rigueur 
1849, French, literally "of strictness," thus "according to obligation of convention." See rigor.
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rigour (n.)
chiefly British English spelling of rigor (q.v.); for spelling, see -or.
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rigorous (adj.)

early 15c., of persons, "strict, exacting, harsh, stern;" of laws, actions, etc., "marked by inflexibility, severe, exacting," hence "unmitigated, merciless;" from Old French rigorous (13c., Modern French rigoureux), from Medieval Latin rigorosus, from Latin rigor "stiffness, firmness" (see rigor). The meaning "scrupulously accurate, precise" is from 1650s. Related: Rigorously; rigorousness (c. 1400).

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rigidity (n.)

1620s, "stiffness, inflexibility," especially in mechanics, "resistance to change of form;" 1620s, from Latin rigiditas "stiffness," from rigidus "hard, stiff, rough, severe" (see rigid). By 1650s as "strictness, severity," but rigidity tends to be used of physical stiffness, while rigor is more active or moral. Rigidness (1640s) "perhaps holds a middle position" [Century Dictionary].

Rigidity is directly opposed to flexibility, and only indirectly to malleability and ductility, which depend chiefly on relations between the tenacity, the rigidity, and the limit of elasticity. [Century Dictionary]
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*reig- 

Proto-Indo-European root meaning "stretch; be stretched; be stiff" 

It forms all or part of: reach; rigid; rigidity; rigor.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit rjyati "he stretches himself," riag "torture" (by racking); Greek oregein "to reach, extend;" Latin rigidus "hard, stiff, rough, severe," rigēre "be stiff;" Lithuanian raižytis "to stretch oneself;" Old Irish ringid "torture," rigim "I stretch;" Middle High German ric "band, string;" Old High German reihhon, Old English ræcan, ræcean "to reach, achieve," on notion of "to stretch out." 

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inclemency (n.)

"state or character of being inclement," 1550s, from French inclémence and directly from Latin inclementia "rigor, harshness, roughness," from inclemens "harsh, unmerciful" (see inclement).

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stiff (n.)
"corpse, dead body," 1859, slang, from stiff (adj.) which had been associated with notion of rigor mortis since c. 1200. Meaning "working man" first recorded 1930, from earlier genitive sense of "contemptible person," but sometimes merely "man, fellow" (1882). Slang meaning "something or someone bound to lose" is 1890 (originally of racehorses), from notion of "corpse."
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moderation (n.)

early 15c., moderacioun, "quality of being moderate or temperate; a lessening of rigor or severity," from Old French moderacion (14c.) "alteration, modification; mitigation, alleviation" and directly from Latin moderationem (nominative moderatio) "a controlling, guidance, government, regulation; moderation, temperateness, self-control," noun of action from past-participle stem of moderari "to regulate, mitigate, restrain, temper, set a measure, keep (something) within measure," from PIE root *med- "take appropriate measures." Meaning "act of moderating or restraining" is from 1520s.

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