Stop, stop, everybody STOP THIS RIGHT NOW!!!
Whew. Sorry. I let the WSH get away with a reply that, while a little fuzzy, was not really wrong, but the level of misinformation has since been growing exponentially and I have to jump in. Okay! First things first:
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When we talk about the Babylonian influence on the number of degrees in a circle, we are not talking about Sumerians (of whose achievements in geometry or astronomy we still know very little) but about their Semitic successors, the Babylonians, and specifically the Babylonians of the Late-Babylonian and Seleucid periods, after about 1000 BCE.
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The calendar of 360 days per year was not in actual use in this period (or anywhere else that I know of; it shows up as an “ideal” calendar in various contexts but not as a practical timekeeping device). Are you thinking of the Egyptian civil calendar with 12 months of 30 days each plus 5 epagomenal or year-end days? The Babylonian calendar used 12 months of either 29 or 30 days, depending on the first visibility of the crescent moon, with intercalary months (I never heard of Babylonian intercalary weeks or days!) thrown in on an ad hoc basis to keep the months aligned with the seasons. In the last half of the first millennium BCE, intercalation was standardized as the 19-year or “Metonic” cycle similar to that in the modern Jewish calendar, with seven intercalary months at assigned intervals in 19 years. (This system, by the way, produces an error of only about 1 day in about 7000 years, so you can quit sneering at the ancients’ “imprecision.” Sheesh.)
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We get weeks not from the Babylonians, who did note the phases of the moon but didn’t implement any seven-day cycle to correspond with them, but from other Semitic calendaric traditions, notably the ancient Hebrews (cf. Genesis).
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Babylonian astronomers divided the day into 12 “beru”, each of which contained 30 “ush” or time-degrees, making 360 time-degrees in a day. The corresponding unit of distance was applied to the rotating sky, denoting the amount of its motion in one time-degree. (Yes, it was handy to have a unit that roughly corresponded to the average daily solar velocity, but don’t kid yourself that at this period anybody actually thought there were 360 days in a year!) This was used in measuring ecliptic longitudes (the twelve zodiacal signs, originally constellations of unequal lengths, were by this time standardized into equal twelfths), giving twelve signs of 30 degrees each. The sexagesimal system was borrowed (and never returned!) probably sometime in the Achaemenid period by the Greeks, who divided the equator (of earth or sky) into 60 parts. (The earliest surviving evidence for this sexagesimal division is in the work of Eratosthenes, around 250 B.C.E.) Over the subsequent years of development of arc and angle measurement among Greek scientists, this division was broken up into various smaller parts (sometimes 360, sometimes 720), and by about the middle of the second century B.C.E. the division into 360 had evolved into the accepted standard. And that’s why there are 360 degrees in a circle; it’s really not all that “ancient” a tradition, with its origins only about 3000 years ago and reaching the level of standard scientific convention nearly 1000 years later.
(I get all this, by the way, from O. Neugebauer, A History of Ancient Mathematical Astronomy, 3 vols., Berlin: Springer-Verlag 1975, and Hermann Hunger and David Pingree, Astral Sciences in Mesopotamia, Brill: Leiden 1999.)
Thank you! I feel much better now. But I refuse to get into the question of how they measure angles in Radia.