The Greek philosopher Aristotle (384-322 BC) argued in his writings that the Earth was spherical, because it casted a circular shadow on the Moon during a lunar eclipse. Another reason was that some stars visible from Egypt are not seen further north The full quotation can be found at http://www.phy6.org/stargaze/Saristot.htm
The Alexandria philosopher Eratosthenes (en.Wikipedia.org/wiki/Eratosthenes) went one step further and actually estimated how large the Earth was. He was told that on midsummer day (June 21) in the town of Syene in southern Egypt (today Aswan, near a huge dam on the river Nile) the noontime Sun was reflected in a deep well, meaning that it was right overhead, at zenith. Eratosthenes himself lived in Alexandria, near the river's mouth, north of Syene, about 5000 stadia north of Syene (the stadium, the size of a sports arena, was a unit of distance used by the Greeks). In Alexandria the Sun on the corresponding date did not quite reach zenith, and vertical objects still threw a short shadow. Eratosthenes established that the direction of the noon Sun differed from the zenith by an angle that was 1/50 of the circle, that is, 7.2 degrees, and from that he estimated the circumference of the Earth to be 250,000 stadia.
Eratosthenes also headed the royal library in Alexandria, the greatest and most famous library in classical antiquity. Officially it was called “temple of the muses" or “museion," from which our modern “museum" is derived.
Other estimates of the size of the Earth followed. Some writers reported that the Greek Posidonius used the greatest height of the bright star Canopus above the horizon, as seen from Egypt and from the island of Rhodes further north (near the southwestern tip of Turkey). He obtained a similar value, a bit smaller. The Arab Khalif El Ma'mun, who ruled in Baghdad from 813 to 833, sent out two teams of surveyors to measure a north-south baseline and from it also obtained the radius of the Earth. Compared to the value known today, those estimates were pretty close to the mark.
The idea of sailing westward to India dates back to the early Romans. According to Dr. Irene Fischer, who studied this subject, the Roman writer Strabo, not long after Erathosthenes and Posidonius, reported their results and noted:
“if of the more recent measurements of the Earth, the one which makes the Earth smallest in circumference be introduced---I mean that of Posidonius who estimates its circumference at about 180,000 stadia, then. . . "
and he continues:
“Posidonius suspects that the length of the inhabited world, about 70,000 stadia, is half the entire circle on which it had been taken, so that if you sail from the west in a straight course, you will reach India within 70,000 stadia. "
Notice that Strabo---for unclear reasons---reduced the 250,000 Stadia of Eratosthenes to 180,000, and then stated that half of that distance came to just 70,000 stadia. Handling his numbers in that loose fashion, he could argue that India was not far to the west.