Jibaizi returned from a very long trip recently, to find a copy of Jon Bosak's The Old Measure sitting on the night-stand. The book is a erudite inquiry into the history and development of the traditional weights and measures: the pound, the gallon, the foot.
I was interested by the ancient emphasis on 12 as a divisor. It's not an obvious choice, mathematically; why 2 times 2 times 3? Why not 2 times 3? Why did the Babylonians choose 60 as a number base --60 is 3 times 4 times 5. Why not 30, which is 2 times 3 times 5, and the product of the first three primes? Why skip two? Of course, 12 and 60 are still deeply embedded in daily life – they are the basic units of time. Why? It begs the question to say "the ancients measured the heavens, and our concept of time comes from their measurements of the universe as a clock", or to claim that 12 is a rough analogy to the number of full moons in a year.
I have a speculation to offer.
Consider that division and fractions were messy. The Old Measure points out that Egyptians generally calculated with reciprocals rather than true fractions. So any fraction such as 2/5 would have to be written instead as (1/3 + 1/15). Evidently reciprocals were more intuitive.
Most reciprocals can be represented (and visualized) as regular polygons inscribed within a circle. Here are the first five regular (convex) polygons.
Note that there is no two-sided polygon – so we have to start with the triangle, representing division into thirds, followed by the square, representing division of the circle into quarters. Multiplying these two "reciprocals" together give you the dodecagon, with 12 sides:
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equals 12: |
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In addition to being the Egyptian division of day and night into hours, twelve spokes was a ancient division for the wheel -- for example:
Twelve spokes, one wheel, navels three.
Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.
– Dirghatamas , Rigveda 1.164.48 (cited here)
Of course, 12 is deeply embedded in English counting –one through twelve are primitive numbers, twelve inches make a foot, twelve jurors make a jury, and so forth.
Twelve as a division of the whole is a natural geometric outcome of taking the two smallest polygons and combining them (by multiplication) to get a finer fractional division of the circle. This is a much more satisfying explanation than claiming that it represents the number of months in the year – the lunar length of the year has been known since deep antiquity to be closer to 13 months.
I'm sure you already see where this is going. Let's multiply from triangle through pentagon:
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equals 60: |
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Triangle times Square times Pentagon yields the sixty-fold division of the circle beloved of the Babylonians. There aren't as many just-so stories about why 60 was chosen as a significant number, compared to the stories about 12; people say it's because it has lots of factors, but so does 30 (2 times 3 times 5), and 30 is close to the number of days per month – certainly as close as 12 is to the number of months per year.
Next, let's add the hexagon:
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equals 360 |
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Of course, 360 is the number of degrees in a circle. Apparently this was a Hellenic invention; it's alleged to have been chosen because it's close to 365. It's more satisfying to think of 360 as the extension of a series, than to think of it as a not-very-accurate approximation to the number of days in a year.
So a simple geometric progression based on the regular polygons gives us all of the common units seen in traditional measure: 12 (inches, Troy ounces, hours); 60 (minutes, seconds, Babylonian sexagesimal counting.
(Adding a heptagon gives 2520 as the next circular division in sequence. This is curiously similar to the tun that Bosak reports (p 44) as being composed of 252.0 wine gallons, but otherwise seems to have no special meaning. However, such fine division of the circle had little application for the ancients.)
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