ratio (n.)

*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."

"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]