The Babylonians used a base 60 number system (not base 6). This strange base system is certainly connected with the notion of degrees, but my guess would be that degree measure came first. It seems clear that degrees were devised by ancient astronomers who noticed that the sun moved one degree each day (about our fixed Earth) past the stars that appeared to be fixed to an external heavenly sphere. It took one month for the sun to move the 30 degrees from one sign of the Zodiac to the next.

Since there are really 365 days in a year (not 360) why isn't a degree defined to be 1/365 of a circle instead of the official 1/360? I have seen several explanations, most not very convincing. My current favorite explanation is that the number 360 is a compromise between the solar year of about 365.25 days and the lunar year (consisting of 12 months of 29.5 days each) of about 354.37 days. Of course the months fit well with the Zodiac, which played an important role in their astronomy. (Another nifty explanation, but which I've never seen confirmed by a reliable source: an ancient calendar showing a year of 360 days has been discovered, so maybe the year at that time was 5 days shorter than our year!)

Here's how the Greek system worked (based on the system they took from the earlier Babylonians and Egyptians). The stars were mapped out on a sphere much as they are today. The moon, the sun, and all the planets moved through the central region known as the Zodiac. If an ancient Greek looked due south at the stars at exactly 12:00 midnight (solar time) on the first day of spring, say, then he would be looking at the stars on the border between the signs Virgo and Libra. That would tell him that the sun was passing (on the opposite side of the Earth) out of Pisces and into Aires. Thus the official rule that spring begins when the sun moves into Aires. (Warning: I'm giving you the ancient Greek rule; the stars aren't fixed and have moved considerably in the past 2500 years; you'll have to check an encyclopedia to find out where the sun is in spring these days.) By looking at the stars at exactly the same time every evening, one could plot the path of the sun as it moved one degree each day.