arithmomania n.

"art of computation, the most elementary branch of mathematics," mid-13c., arsmetike, from Old French arsmetique (12c.), from Latin arithmetica, from Greek arithmetikē (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., "mental derangement characterized by excitement and delusion," from Late Latin mania "insanity, madness," from Greek mania "madness, frenzy; enthusiasm, inspired frenzy; mad passion, fury," related to mainesthai "to rage, go mad," mantis "seer," menos "passion, spirit," all of uncertain origin, perhaps from PIE *mnyo-, suffixed form of root *men- (1) "to think," with derivatives referring to qualities and states of mind or thought.

Mania is manifested by psychic elevation, increased motor activity, rapid speech and the quick flight of ideas. [Scientific American, September 1973]

Sense of "fad, craze, enthusiasm resembling mania, eager or uncontrollable desire" is by 1680s, from French manie in this sense. Sometimes nativized in Middle English as manye. Used since 1500s as the second element in compounds expressing particular types of madness (such as nymphomania, 1775; kleptomania, 1830; megalomania, 1890), originally in Medical Latin, in imitation of Greek, which had a few such compounds, mostly post-classical: gynaikomania (women), hippomania (horses), etc.