early 15c., "the giving of medicine (in a specified amount or at a stated time)," from Old French dose (15c.) or directly from Medieval Latin dosis, from Greek dosis "a portion prescribed," literally "a giving," used by Galen and other Greek physicians to mean an amount of medicine, from stem of didonai "to give" (from PIE root *do- "to give").
Meaning "quantity of medicine given or prescribed" is from c. 1600. Extended sense, in reference to anything given to be "swallowed," literal or figurative, is from c. 1600. Slang meaning "a case of venereal disease" is by 1914.
word-forming element in nouns of act, process, function, condition, from Old French and French -age, from Late Latin -aticum "belonging to, related to," originally neuter adjectival suffix, from PIE *-at- (source of Latin -atus, past participle suffix of verbs of the first conjugation) + *-(i)ko-, secondary suffix forming adjectives (see -ic).
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Harper, D. (n.d.). Etymology of dosage. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/dosage