It has actually been known that the Earth was round since the time of the ancient Greeks. It was Pythagoras who first proposed that the Earth was round sometime around 500 B.C. he based his idea on the fact that he showed the Moon must be round by observing the shape of the terminator (the line between the part of the Moon in light and the part of the Moon in the dark) as it moved through its orbital cycle. Pythagoras reasoned that if the Moon was round, then the Earth must be round as well. After that, sometime between 500 B.C. and 430 B.C., a fellow called Anaxagoras determined the true cause of solar and lunar eclipses - and then the shape of the Earth's shadow on the Moon during a lunar eclipse was also used as evidence that the Earth was round.
Around 350 BC, the great Aristotle declared that the Earth was a sphere (based on observations he made about which constellations you could see in the sky as you travelled further and further away from the equator) and during the next hundred years or so, Aristarchus and Eratosthenes actually measured the size of the Earth!