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The Babylonian astronomical observatories may have been placed on the flat roofs of ziggurats.
Muslim websites falsely claim that Muslims built the first astronomical observatories.
We know that astronomical observatories existed in Babylonia in pre Islamic times because of the Babylonian writings on clay tablets about astronomy.
The ancient Babylonian calculation of the length of the month was off by only nine-tenths of a second per month and was not improved upon until modern times. http://en.wikipedia.org/wiki/Lunar_theory
The article at https://en.wikipedia.org/wiki/History_of_astronomy says
“The last stages in the development of Babylonian astronomy took place during the time of the Seleucid Empire (323–60 BC). In the third century BC, astronomers began to use “goal-year texts” to predict the motions of the planets. These texts compiled records of past observations to find repeating occurrences of ominous phenomena for each planet. About the same time, or shortly afterwards, astronomers created mathematical models that allowed them to predict these phenomena directly, without consulting past records. A notable Babylonian astronomer from this time was Seleucus of Seleucia, who was a supporter of the heliocentric model.”
Image by Hardnfast, via Wikimedia Commons.