The advantages of hexagesimal arithmetic

In my speculative phase, the main reason I could think of for choosing 60 as a base is its number of divisors: it is divisible by 1, 2, 3, 4, 5 and 6. I suspect that having more divisors of the base would be of greater importance to ancient mathematicians than to us today, since they needed to do all their calculations by hand (and maybe without the streamlined algorithms of today), and having your choice of base divide cleanly into many numbers makes that much smoother. Furthermore, the choice of 60 is pretty efficient in terms of how many divisors it packs into a relatively small number: the lowest base which has more distinct divisors is 120 - double the size for just a couple more divisors.

We use base 60 today in several different contexts - most notably, our system of time. Having 60 minutes in an hour makes the hour easy to divide into thirds, quarters, and even fifths and sixths. If there were, say 100 minutes in an hour, a third of an hour would involve a repeating decimal quantity of minutes, which is not at all an intuitive concept, and much harder to measure. The degree system for angle measurement is also, in a way, base 60 - there are 360 = 6*60 degrees in a circle. Again, this makes it easy for us to divide common angles into 3, 4, or 5. This system is not used in just math, but in geography, with our longitudinal and latitudinal distances measured in degrees. Degrees are also divided further into minutes and seconds, which are again base 60.

One interesting thing I found in my research was that there were developed finger-counting methods for duodecimal and hexagesimal number systems. The finger-counting method for our decimal system is very intuitive, but more complicated methods have been documented from the Middle East that counted to twelve, or to 60. These methods, according to a book by Samuel Macy, may have contributed to our usage of 12 hours in the day, which in turn is linked to the division of the hour into 60 minutes, and the minute into 60 seconds (see https://books.google.ca/books?id=xlzCWmXguwsC&pg=PA92&lpg=PA92&redir_esc=y#v=onepage&q&f=false). So one of the perceived drawbacks of base 60 (or base 12) - that it's more difficult to count on fingers - may not be true at all!