late 15c. (Caxton), "to plan, to scheme," from Late Latin projectare "to thrust forward," from Latin proiectus, past participle of proicere "stretch out, throw forth; hold in front; fling away; drive out," from pro- "forward" (from PIE root *per- (1) "forward") + combining form of iacere (past participle iactus) "to throw" (from PIE root *ye- "to throw, impel"). The notion is to "cast forward in the mind."
Meaning "to throw out or forward" physically is from 1590s. Intransitive sense of "to stick out, protrude beyond the adjacent parts, extend beyond something" is from 1718 (also an architectural sense in the Latin verb). Meaning "to cast an image on a screen" is recorded from 1865. Psychoanalytical sense, "attribute to another (unconsciously)" is from 1895 (implied in a use of projective), probably a figurative use from the meaning "throw the mind into the objective world" (1850). Meaning "convey to others by one's manner" is recorded by 1955. Related: Projected; projecting.