c. 1400, "act, process, or method of arithmetical calculation," from Latin computationem (nominative computatio), noun of action from past participle stem of computare "to sum up, reckon, compute," from com "with, together" (see com-) + putare "to reckon," originally "to prune," from PIE root *pau- (2) "to cut, strike, stamp." From 1713 as "a result of computation, amount reckoned."
late 14c., "art, manner, or practice of computing by numbers," also "the process of making a horoscope," from Late Latin calculationem (nominative calculatio) "a computation, calculation, reckoning," noun of action from past-participle stem of calculare "to reckon, compute," from Latin calculus "reckoning, account," originally "pebble used in counting," diminutive of calx (genitive calcis) "limestone" (see chalk (n.)). From early 15c. as "the result of reckoning, the solution for a problem."
1630s, in theological writing, "reason, rationale," from Latin ratio "a reckoning, account, a numbering, calculation," hence also "a business affair; course, conduct, procedure," also in a transferred sense, of mental action, "reason, reasoning, judgment, understanding, that faculty of the mind which forms the basis of computation and calculation." This is from rat-, past-participle stem of reri "to reckon, calculate," also "to think, believe" (from PIE root *re- "to think, reason, count").
Latin ratio often was used to represent or translate Greek logos ("computation, account, esteem, reason") in works of philosophy, though the range of senses in the two do not overlap (ratio lacks the key "speech, word, statement" meaning in the Greek word; see Logos).
The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos). The general or extended sense of "corresponding relationship between things not precisely measurable" is by 1808.
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.
"art of computation, the most elementary branch of mathematics," mid-13c., arsmetike, from Old French arsmetique (12c.), from Latin arithmetica, from Greek arithmetike (tekhnē) "(the) counting (art)," fem. of arithmetikos "of or for reckoning, arithmetical," from arithmos "number, counting, amount," from PIE *erei-dhmo-, suffixed variant form of root *re- "to reason, count."
The form arsmetrik was based on folk-etymology derivation from Medieval Latin ars metrica; the spelling was corrected early 16c. in English (though arsmetry is attested from 1590s) and French. The native formation in Old English was tælcræft, literally "tell-craft."