Around 250 BCE, he noted that the water wells and pillars in the city of Syene (now Aswan in Egypt) didn’t cast a shadow at noon on the summer solstice because the Sun was directly overhead. Yet at the same time and date, the shadows were long and drawn out in the city of Alexandria, some 800 kilometers (500 miles) away. He knew that the Sun was massive and its rays would be relatively parallel when they hit the Earth, so why were the shadows so different? The explanation, he deduced, must be that the Earth is curved and therefore a sphere. In fact, he managed to work out that the angle of the Sun's rays was approximately 7 degrees (or as he put it, one-fiftieth of a circle). From this, he was able to work out a surprisingly accurate estimate of the size of our planet.
Needless to say, rejection of this idea is not new to the era of celebrities and social media. Throughout more or less every time period in human history, from truly brilliant medieval Islamic scholars to 19th-century pseudoscientists, the acceptance of the spherical Earth has been an oddly contentious issue.
If it's any consolation, at least a handful of ancient Greeks have rested easily for the past 2,000 years, quietly thinking "well, I told you so".