The Sumerians figured out the length of the year a lot earlier than 45BC. The first observation you might make if you were to do this yourself is that seasons follow a regular pattern and that pattern coincides with celestial observations. In the northern hemisphere, the sun rises and sets farther to the south in the winter than it does in the summer. Also, certain stars and constellations are only visible during certain times of the year.
Something you might try to do is count the number of days between the most southerly sunrises. All you need to do that is three sticks, some way of tallying numbers (a jar and a bunch of pebbles will do), and a whole lot of patience. You don’t know what day it is because calendars don’t exist, but you know it’s autumn because it’s getting colder and the leaves are changing, from that you know the sunrise should be moving south. You find a spot that has a good, unobstructed view, maybe a hill top overlooking the sea and you drive one of your sticks into the ground. The next morning, you get up at dawn and see where the sun rises. You take your second stick and drive it into the ground so that the two sticks and the sun form a straight line. The next day you do the same thing, drive the third stick into the ground and note that it is to the right of the second. Keep doing that every morning, leapfrogging the second and third stick over one another until you get to a morning where the latest observation is to the left of the last. That is your reference for the start if the year, toss a pebble into the jar. Keep doing that every day and you’ll notice that some time during the summer, the sticks will change course again and you’ll have something like 180-200 pebbles in the jar. Keep going and once the sticks get back to where you started counting (you’ll know because they change direction again) you can count count the pebbles and there should be about 365 of them. Do that a couple more times and you can be sure of your results.
Another way you can check your work is by following a particular star, let’s use Sirius because it’s the brightest and close to an easily recognized constellation. Conveniently, the first night you would be able to see Sirius after the sun sets is close to the winter solstice. If you start putting pebbles in your jar each day after that and keep going until it becomes visible the next winter, you should also wind up with about 365 pebbles in your jar
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