project (n.)
c.1400, "a plan, draft, scheme," from Latin proiectum "something thrown forth," noun use of neuter of proiectus, past participle of proicere "stretch out, throw forth," from pro- "forward" (see pro-) + combining form of iacere (past participle iactus) "to throw" (see jet (v.)).

Meaning "scheme, proposal, mental plan" is from c.1600. Meaning "group of low-rent apartment buildings" first recorded 1935, American English, short for housing project (1932). Related: Projects. Project manager attested from 1913.
project (v.)
late 15c., "to plan," from Latin proiectus, past participle of proicere (see project (n.)). Sense of "to stick out" is from 1718. 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). Meaning "convey to others by one's manner" is recorded by 1955. Related: Projected; projecting.