Root of "Project" and "Manage"

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" (from PIE root *ye- "to throw, impel").

manage (v.)

1560s, "to handle, train, or direct" (a horse), from the now-obsolete noun manage "the handling or training of a horse; horsemanship" (see manege, which is a modern revival of it), from Old French manège "horsemanship," from Italian maneggio, from maneggiare "to handle, touch," especially "to control a horse," which ultimately from Latin noun manus "hand" (from PIE root *man- (2) "hand").

Extended sense of "control or direct by administrative ability" any sort of business is by 1570s; meaning "to wield (a tool or object) by hand" is from 1580s. Meaning "effect by effort" (hence "succeed in accomplishing") is by 1732. Intransitive sense of "get by, carry on affairs" is suggested by 1650s, in frequent use from mid-19c. Related: Managed; managing. Managed economy was used by 1933.