c. 1600, "introduce to some practice or system," also "begin, set going," from Late Latin initiatus, past participle of initiare "to begin, originate," in classical Latin only in the sense "to instruct in mysteries or sacred knowledge." This is from initium "a beginning; an entrance," also in plural initia "constituent parts; sacred mysteries," a noun use of the neuter past participle of inire "to go into, enter upon, begin," from in- "into, in" (from PIE root *en "in") + ire "to go" (from PIE root *ei- "to go").
In some senses the English word is a back-formation from initiation. Related: Initiated; initiates; initiating; initiator.
-ory
adjective and noun suffix, "having to do with, characterized by, tending to, place for," from Middle English -orie, from Old North French -ory, -orie (Old French -oir, -oire), from Latin -orius, -oria, -orium.
Latin adjectives in -orius, according to "An Etymological Dictionary of the French Language," tended to "indicate a quality proper to the action accomplished by the agent; as oratorius from orator; laudatorius from laudator. The neuter of these adjectives was early employed as a substantive, and usually denoted the place of residence of the agent or the instrument that he uses; as praetorium from praetor; dormitorium from dormitor; auditorium, dolatorium.
"These newer words, already frequent under the Empire, became exceedingly numerous at a later time, especially in ecclesiastical and scholastic Latin; as purgatorium, refectorium, laboratorium, observatorium, &c." [transl. G.W. Kitchin, Oxford, 1878]