[This article](https://www.scientificamerican.com/article/how-ancient-humans-studied-and-predicted-solar-eclipses/) has a really good breakdown for it. Basically, with good enough record-keeping, you can spot patterns and then start predicting the future. As a species, we’re incredibly good at spotting patterns!
This is an example of lunar eclipses, but the [Gold Hats](https://en.m.wikipedia.org/wiki/Golden_hat) are an [incredible display](https://artsandculture.google.com/story/DAVRgpAwHmLsLw?hl=en) of how these calendars can be encoded and used to make calculations. These hats date from Bronze Age Europe, about 3000 years ago! The hats tracked both the solar and lunar calendars.
Cultural belief and religious systems back then (and still in modern religions – look at Islam’s use of the moon and lunar calendars) were inextricably tied with natural cycles, whether that was the seasons or the passage of the sun, moon and stars. Because these were of such importance to them, they would devote a lot of time to recording times and dates – time between new moons, dates of the summer and winter solstices (look at Maes Howe in Scotland for a good example of this) etc..
For instance, with the winter solstices, you wouldn’t actually have to time how long it took between sunrise and sunset. You could just take measurements of how high the sun got in the sky at its peak using a stick and something to mark it with. Once it stopped getting lower and started getting higher again, you’d know the solstice had passed.
With regards to viewing them, you could observe it through a natural or artificial pinhole camera – even the sunlight passing through leaves will show the eclipse on the ground. They may also have had oral traditions about not looking for too long, given the damage it can cause.
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