I used to wonder why we divide a circle into three-hundred and sixty parts, each of which we call "one degree". I thought that was an odd number to use. Why don't we use a nice round number like one-hundred? Just recently, I think I came across the answer.
I said to imagine that our circle is divided up into arbitrary somethings which we may choose to call "degrees" but that the number of degrees could be any number, not just 360. I then further said that we could take our circle and divide it up into arbitrary numbers of equal size sectors. We could have two sectors, three sectors, ten sectors, seventeen sectors and so forth and depending on how many sectors we'd chosen, each sector would contain the total number of arbitrary degrees we'd chosen divided by the arbitrary number of sectors.
That division calculation would yield the number of degrees per sector which number could have a remainder or which number could be an integer. For example, if we let there be 420 arbitrary degrees in our circle and we divide that circle into 17 sectors, each sector would have 420/17 = 24.70588... degrees with a non-zero remainder but if we let there be 285 arbitrary degrees and 19 sectors, each sector would have 285/19 = 15 degrees with a zero remainder.
If we examine assignments of the number of degrees for a complete circle versus the number of sectors into which we divide that circle up, we can examine which division ratios yield non-integral numbers with non-zero remainders and which division ratios yield integral numbers with remainders of zero. That's what was done in the following spreadsheet result.
In each case for which the division remainder is zero, the number result is shown but for where the remainder is non-zero, the spreadsheet cell value is forced to zero. The number of non-zero cells in each column is counted up and that tally is the number of sectors having integral numbers of degrees.
Clearly, this matrix could be extended indefinitely to larger numbers for our "degrees" definitions and to larger numbers of sectors, but having gone this far is quite sufficient to illustrate the intended point.
It turns out that for circles divided up into as many as fifty sectors, the number 360 of arbitrary degrees yields eighteen sector choices for which the resulting sectors contain integral numbers of degrees.
So far as I have been able to determine, this is the largest number of such sectors for arbitrary degree assignments from one to 719. It then happens that 720 yields twenty sectors having integral numbers of degrees, but those degrees are so small as to be of little practical use when using a physically manageable protractor on a notebook sized sheet of paper.
The number 360 is not arbitrary at all. It is the optimum choice for the definition of degrees because it yields the largest practical repertoire of sector division choices for which the numbers of sector angles are integers.
To which genius of the pre-spreadsheet era do we credit this optimization? I have no idea.