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Ancient Astronomy - Stoneage
There is evidence that humans started stargazing as far back as the Stone Age. Stonehenge, as well as being a popular New Age destination, is probably a good example of an early astronomical site. There are other ancient astronomy sites in the UK, including a tomb built in the Orkney islands, thought by some to have been built by astronomer priests who passed on their knowledge to pilgrims and knew that the Earth was round.
 
bronze head (courtesy of N.S Rajaram)

India and China
In India, the first references to astronomy appeared in the Rig Veda - the text which forms the basis of Vedic philosophy - in around 2000 BC. Astronomical ideas were a central part of Vedic philosophy. In the 5th Century BC, the astronomer Aryabhatta proposed a heliocentric theory of gravitation, thus predating Copernicus by about 1000 years.

Astronomy was also a big feature in ancient Chinese civilisation. The Hun Tian school envisioned an egg-like universe with the Earth as the yoke, and the The Xuan Ye theory suggested an infinite universe with suspended celestial bodies - a revolutionary idea at the time. The first records of the appearance of Halley's comet can be tracked back to maybe as early as 1059 BC, so the Chinese were very early starters when it came to astronomy.

OU courses
Egypt & Mesoptamia 19th Century