"the medical treatment of mental diseases," 1846, from French psychiatrie, from Medieval Latin psychiatria, literally "a healing of the soul," from Latinized form of Greek psykhē "mind" (see psyche) + iatreia "healing, care" (see -iatric).
Middle English -ik, -ick, word-forming element making adjectives, "having to do with, having the nature of, being, made of, caused by, similar to," from French -ique and directly from Latin -icus or from cognate Greek -ikos "in the manner of; pertaining to." From PIE adjective suffix *-(i)ko, which also yielded Slavic -isku, adjectival suffix indicating origin, the source of the -sky (Russian -skii) in many surnames. In chemistry, indicating a higher valence than names in -ous (first in benzoic, 1791).
In Middle English and after often spelled -ick, -ike, -ique. Variant forms in -ick (critick, ethick) were common in early Modern English and survived in English dictionaries into early 19c. This spelling was supported by Johnson but opposed by Webster, who prevailed.
<a href="https://www.etymonline.com/word/psychiatric">Etymology of psychiatric by etymonline</a>
Harper, D. (n.d.). Etymology of psychiatric. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/psychiatric