Etymology
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Words related to ratio

*re- 

*rē-, Proto-Indo-European root meaning "to reason, count;" a variant of PIE root *ar-, also arə-, "to fit together." 

It forms all or part of: Alfred; arraign; arithmetic; Conrad; dread; Eldred; Ethelred; hatred; hundred; kindred; logarithm; Ralph; rate (n.) "estimated value or worth;" rathskeller; ratify; ratio; ration; read; reason; rede; rhyme; riddle (n.1) "word-game;" rite; ritual.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit radh- "to succeed, accomplish;" Greek arithmos "number, amount;" Latin reri "to consider, confirm, ratify," ritus "rite, religious custom;" Old Church Slavonic raditi "to take thought, attend to;" Old Irish im-radim "to deliberate, consider;" Old English rædan "to advise, counsel, persuade; read;" Old English, Old High German rim "number;" Old Irish rim "number," dorimu "I count."

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Logos (n.)

1580s, "the divine Word, second person of the Christian Trinity," from Greek logos "word, speech, statement, discourse," also "a computation, account," also "reason, judgment, understanding," from PIE *log-o-, suffixed form of root *leg- (1) "to collect, gather," with derivatives meaning "to speak," on notion of "to pick out words." The Greek word was used by Neo-Platonists in metaphysical and theological senses involving notions of both "reason" and "word" and subsequently picked up by New Testament writers.

ratiocination (n.)

"process of reasoning, mental process of passing from the cognition of premises to the cognition of the conclusion," 1520s, from Latin ratiocinationem (nominative ratiocinatio) "a reasoning, calm reasoning," noun of action from past-participle stem of ratiocinari "to reckon, compute, calculate; to deliberate, meditate; to reason, argue, infer." This is a compound of ratio "reckoning, calculation," also "judgment, reason" (see ratio) + -cinari, which probably is related to conari "to endeavor, to try," from PIE *kona-, from root *ken- "to hasten, set oneself in motion" (see deacon).

Most writers make ratiocination synonymous with reasoning. J.S. Mill and others hold that the word is usually limited to necessary reasoning. [Century Dictionary]
ration (n.)

1550, "the mental process of reasoning," later, "the relation of one number to another" (1660s), then "fixed allowance of food or other means of subsistence for a fixed period of time" (1702, often rations, from French ration in this sense).

All are from Latin rationem (nominative ratio) "reckoning, numbering, calculation; business affair, procedure," also "reason, reasoning, judgment, understanding," in Medieval Latin "a computed share or allowance of food." This is from rat-, past participle stem of reri "to reckon, calculate," also "think" (from PIE root *re- "to reason, count").

The first sense listed for the English word is obsolete; the second has gone with ratio. The military pronunciation (rhymes with fashion) took over in English from the preferred civilian pronunciation (rhymes with nation) during World War I. That war also gave the word a specific sense of "officially limited allowance for civilians in times of war or dearth" (by 1917).

rational (adj.)

late 14c., racional, "pertaining to or springing from reason;" mid-15c., of persons, "endowed with reason, having the power of reasoning," from Old French racionel and directly from Latin rationalis "of or belonging to reason, reasonable," from ratio (genitive rationis) "reckoning, calculation, reason" (see ratio).

In arithmetic, "expressible in finite terms," 1560s. Meaning "conformable to the precepts of practical reason" is from 1630s. Related: Rationally. It is from the same source as ratio and ration; the sense in rational is aligned with that in related reason (n.), which got deformed in French.