"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."
word-forming element forming nouns meaning "rule or government by," from French -cratie or directly from Medieval Latin -cratia, from Greek -kratia "power, might; rule, sway; power over; a power, authority," from kratos "strength," from PIE *kre-tes- "power, strength," suffixed form of root *kar- "hard." The connective -o- has come to be viewed as part of it. Productive in English from c. 1800.
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Harper, D. (n.d.). Etymology of arithmocracy. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/arithmocracy