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My understanding is that he counted the number of days between every other equinox. The first year he would have counted 365 days, but after 4 years the tally would have reached 1461 days. Dividing this by 4 he would have had an average length of 365.25 days per year. Using records from older Babylonian and Greek astronomers he would have been able to achieve a higher accuracy. Am I on the right track?

Bonus question: how did he measure the lengths of the seasons?

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    Hipparchus had access to Babylonian astronomical data which spanned centuries, as I mentioned here: https://astronomy.stackexchange.com/a/47584/16685 – PM 2Ring Apr 03 '23 at 12:10

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