Jet propulsion only literally and figuratively took off with the invention of the rocket by the Chinese in the 13th century.
Rocket exhaust was initially used in a modest way for fireworks but gradually progressed to propel formidable weaponry; and there the technology stalled for hundreds of years.
Archytas, the founder of mathematical mechanics, as described in the writings of Aulus Gellius five centuries after him, was reputed to have designed and built the first artificial, self-propelled flying device. This device was a bird-shaped model propelled by a jet of what was probably steam, said to have actually flown some 200 meters.
And there you have how jet propulsion started.