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There are 360° in a circle. We learn such things by rote, but seldom marvel at them. 360 can be factored into 2 · 2 · 2 · 3 · 3 · 5, which is 2³ · 3² · 5. While that seems less than remarkable, 360 is divisible by 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360, an unusually large number of divisors. Mathematicians call numbers like this highly composite, defined as a positive integer with more divisors than any smaller positive integer. A circle can be evenly divided into three hundred sixtieths, one hundred eightieths, one hundred twentieths, ninetieths, seventy-seconds, sixtieths, forty-fifths, fortieths, thirty-sixths, thirtieths, twenty-fourths, twentieths, eighteenths, fifteenths, twelfths, tenths, ninths, eighths, sixths, fifths, quarters, thirds, halves, and itself. Spelling these out was not really necessary, but I also marvel at how, given a sequence of numbers, the words follow so easily.

It's hard to know what we would do with many of these. If we divide a circle into eighteenths or thirty-sixths, what do we end up with exactly? But, to divide a circle not into 90° as we do when we cut one into quarters, but into ninetieths (1/90) would be like dividing a clock face into periods of two-thirds seconds each. Something about that made me wonder, and kept me wondering for days. If our clocks were adjusted to have second-and-a-half hands instead of second hands, or if we had forty minutes-and-half to the hour, would our lives be all that disrupted? Our clocks would chime at ten, twenty, thirty and on the hour. We could divide the day into almost any sequence of periods we liked. The day would still be the same span of time. It would be dressed up in different numbers, but we would still feel its inevitable, inexplicable progress.

There are sixty seconds in a minute and it takes the second hand one revolution to reach sixty. There are sixty minutes in an hour and it also takes the minute hand one revolution to reach sixty. There are twelve hours in a day, twelve in a night, and it takes the hour hand one revolution for each of them to reach twelve. It's hard to imagine the world being otherwise. But imagine, nonetheless, that our clocks were adjusted to have ninety seconds per revolution and ninety minutes to the hour. How many hours would we be left with in a day? If the hours in a day were more or less than a single revolution of the hour hand, we would be forced to calculate the time of noon each day, and midnight, and what time dinner was. There's a logic to clocks that normally escapes us.

Clocks start with the hour hand and move backwards. Although we think of the day as an accumulation of hours, minutes, seconds, the day is actually the thing itself, the hours, minutes, seconds merely divisions of it, not the things out of which it is made. If this seems like a meaningless distinction, just alter the duration of the second and follow the consequences. Increase the second by 50% and noon arrives at 8:00 AM. It's not seconds that determine how long the day is, it's the day that determines how long seconds are.

Of course, dividing the day into mechanical parts is entirely artificial. Time marches on whether we account for it or not. Still, there's a logic in clocks, as I said before, that normally escapes us.

If we return to the hands on the clock, the missing hand is the one that moves twelve times slower than the hour hand. In other words, the hand that would measure six days and six nights (12 · 12 = 144, 144 / 24 = 6) We might call it the six day week hand, or maybe just the days hand. Sixty of these would equal a year of 360 days, and sixty of those a great cycle of sixty years, the equivalent of our century. To those who say that ten is the natural way to count years, because we have ten fingers, I would ask, why then do we count the hours by twelve, the minutes and seconds by sixty, rather than ten?

The year, the actual year, divides rather awkwardly into 365¼ days… more or less. Close enough to remind us of 360, but also off enough to make us wonder. The year, of course, is based on earthly revolutions and rotations, not on numbers per se. The year is an enormous circle. It mirrors the circularity of our clocks, or else clocks seek to mirror the circularity of the year. It seems more than reasonable that the number of days in a year and the number of degrees in a circle should be the same. But, how could we ever make due with clocks that were divided into 365¼°… more or less? The number 360 is the nearest highly composite number to the rather imprecise number of days in an actual year. The next nearest are 240 and 720, neither of which brings us any closer.

In the end, it is a desire for perfection that gives us 360. Obviously, in a perfect world, in a world of perfect days and ways, there would be 360 days in an ordinary year. We would have no need for imprecision trailing behind, no need whatsoever for more or less. Everything would be exactly as it was meant to be.
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