"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 of Greek origin meaning "divination by means of," from Old French -mancie, from Late Latin -mantia, from Greek manteia "oracle, divination," from mantis "one who divines, a seer, prophet; one touched by divine madness," from mainesthai "be inspired," which is related to menos "passion, spirit" (from PIE *mnyo-, suffixed form of root *men- (1) "to think," with derivatives referring to qualities and states of mind or thought). Compare mania.
<a href="https://www.etymonline.com/word/arithmancy">Etymology of arithmancy by etymonline</a>
Harper, D. (n.d.). Etymology of arithmancy. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/arithmancy