"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."
late 14c., from Latin geometricus "of geometry" (from geometria; see geometry) + -al. Since 16c. it has been opposed to arithmetical in ratio, proportion, etc., reflecting the fact that problems of multiplication formerly were dealt with by geometry, not arithmetic. Related: Geometrically.
<a href="https://www.etymonline.com/word/arithmetical">Etymology of arithmetical by etymonline</a>
Harper, D. (n.d.). Etymology of arithmetical. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/arithmetical