It’s amazing how coincidence and a little critical thinking goes a long way. Homer the poet believed the Earth to be a flat disk, so if you walk far enough, you’ll fall off the end. Pythagoras believed it was a round ball, but how big was it? Eratosthenes, a Greek scholar living in Egypt 2,300 years ago had heard that on the longest day of summer the midday sun shone right to the bottom of a well in the town of Syene which was near what is now the Aswan Dam. At the same time, he found out that the sun was not directly overhead to the north in Alexandria, the shadow cast there wasn’t vertical instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12′). He ‘knew’ that: 1) on the day of the summer solstice, the midday sun was directly over the Tropic of Cancer; 2) Syene was on this tropic; 3) Alexandria and Syene lay on a direct north-south line and 4) the sun was a relatively long way away so the rays of the sun didn’t spread out like the beam of a torch but were parallel instead. According to legend, he had someone walk from Alexandria to Syene to measure the distance – no afternoon stroll, for sure – it turned out to be 5000 stadia or, at 185 m per stadion, about 925 km, but I can’t imagine anybody doing a repeat measurement for accuracy since it would have taken over a month to make the trip. How long was a stadion? Various theories exist, but a good approximation might be that one Roman mile is 8 stadia, or ‘stades’ or 5000 ‘Roman feet’. From these observations, he thus concluded that, since the angular deviation of the sun from the vertical at Alexandria was also the angle of the subtended arc, the linear distance between Alexandria and Syene was 1/50 of the circumference of the Earth which thus must be 50×5000 = 250,000 stadia or just over 40,000km. The diagram isn’t precise – the Tropic isn’t this far north, but, you get the idea. The Roman mile is shorter than its present day counterpart, in case anybody noticed. At the Equator, the Earth is 24, 902 modern miles round. More or less.

Just for completeness, here’s a fun little exercise on finding the circumference using data from the space station.

Now we know how big it is, we can use Newton’s Law of Gravitation to find its mass. Imagine a 1kg mass anywhere on the circumference r, found because we now know its circumference. G and g are both known so its mass drops out nicely. Having found its mass we can use density =mass/volume to find the average density of the material of the Earth.