There are two main phases of Babylonian mathematics, during the Old Babylonian (roughly 1800-1500 B.C.) and the Seleucid (roughly 300-100 B.C.) periods. Besides astronomical texts, we find two main genres: collections of geometrical and algebraic word problems and tables of multiples, fractions, etc. Our sample text here belongs to the former. An Old Babylonian text, Akkadian with many Sumerian logograms, excerpted from a collection of mathematical school exercises, now kept in the British Museum:

IGI-4-GÁL 10 SHU.SI qa a-na i-si-iq-tim
u-ul i-sa-an-ni-iq qá-qá-rum UGU qá-qá-rum
en-nam SUKUD; ZA-E 1,40 SUKUD GISHDIB.DIB du8-a
36 ta-mar 36 a-na 30 i-shi 18 ta-mar.
18 a-na 2/30 i-shi 45 ta-mar KI UGU KI DIRI
ki-a-am ne-pé-shum

And, the (literal) translation:

A water clock is opened. 1/2 Litre (of water) is in the water clock.
4 parts of 10 fingers (of water), measured with respect to the litre, before the measuring mark,
are missing. Measured surface to surface,
what is the height? You taker the reciprocal of the height of the water clock, 1,40.
You get 0,36. Multiply 0,36 by 0,30. You get 0,18.
You multiply 0,18 by 2,30. You get 0,45, surface to surface.
That's the way you do it.

What the hell does that mean? Is this an old form of new math? What kind of cockamemey nonsense is this?

See, now, this is where it starts getting fun. I should probably mention that Babylonian math is sexagesimal (base 60). The cuneiform system of writing basically uses two different symbols to represent numbers. A single, vertical wedge represents a 1 (simple enough), two represent 2, three wedges represent 3, and so on, up to 9. Of course, we're in sexagesimal, with no place markers, so a single wedge can stand for 1 (600), 60 (601), 3600 (602), and so forth. For 10, cuneiform (at least in the stages that interest us here) uses a...well...'Winkelhacken'. The only people to come up with a single word to describe this are the Germans, but it's basically a squat wedge open to the right with no extensions of the sides. Just trust me on this one. Either way, this winkelhacken represents the multiples of 10, (600x10, 601x10, 602x10, and so forth) as well as the fractions 10/60, 10/3600, etc. How do you decide which value to use? Quite simply, the math only works with one interpretation. The assyriologist Otto Neugebauer invented a system of notation where we seperate the multiples of 60 from the single digits up to 60 with a comma, so that a winkelhacken followed by 4 vertical wedges is transliterated 1,4; fractions are seperated by a semicolon, so that half of a litre (divided into 60 units) is written 0;30, etc. In the end it's just easier that way, and we don't have to decide on a value until we translate.

But back to our water clock. It just so happens that the British Museum also has one of these water clocks. The earliest attested ones date to the Ur III period (roughly 2112-2004 B.C.) in Mesopotamia; ours dates to around 1800 B.C. or so. They show up in Egypt roughly around the same time, leading inevitably to countless masturbatory articles in referenced journals about which came first. I say, who the heck cares? What did they look like, how did they work? The one in the British Museum is simply a large stone cylinder with a hole in the bottom (the word for water clock, DIB.DIB, is onomatopoeic, imitating the dripping water). Beneath this was placed a smaller container with measuring marks on the sides, into which the water from the larger container could flow or drip. Whenever the water had reached a mark, another period of time was up (the length of time differed, but was generally 3 hours or so). But, you'll be saying, that's the dumbest thing I've ever heard! Nobody can measure time accurately like that! But before you point out that the time it takes for the water to drip out of the tub is proportional to the square root of the height of the water, let me interupt. We know this already. So did they (the Babylonian 'they'). But, they also knew, even if they never saw a graph of a square root curve, that the difference is only substantial when the water level approaches 0, and the more water inside, the less difference in rate of flow (i.e., the curve begins to level out). Keep adding more water as the water flows out, and it works. Do you have a better way to invent mathematics, chronometrics, and astronomy simultaneously? I don't think so.

Back to our word problem. We know what the thing looks like, have an amount of water which has flowed out, A (=1/2 litre, 0;30), an amount of water as a standard measure, B (=1,0 litre), the amount of water which has flowed out, i (measured in fingers, 1/4 of 10 fingers =2;30 fingers), and the height of the water clock, h(=1,40 (also fingers). We want to calculate the difference in height between the first and second surfaces, α. With a little magic, we can come up with the formula B:A=i:ah, in other words, the proportion of our standard initial volume B to the water which has flowed out, A, is equal to the proportion of the difference in height of the water when amount B flows out to the difference in height of the water when amount A flows out, represented as a fraction (a) of the total height (h). All right, so A/B=ah/i. Solve for a, and we get a=(1/h)xAxi (if you want to be precise, also multiply by 1/B, but since B=1, we'll just say the units litre of A/B cancel out). Our Babylonian mathematician began, in line 4, by calculating 1/h. Since this was a school problem, he gave his result in the next line, 0,36, then multiplied this by 0,30, and got 0,18. He then multiplied this result by i, 2;30, and got the answer 0,45.

Now, let's see if this works in decimal. The height of our waterclock is 1,40, which in decimal would be 1 + 40/60 or 5/3. A is easy: 1/2 litre. h, 2;30, is 2 + 30/60 or 5/2. In other words, a = (3/5)(1/2)(5/2)=3/4. The reciprocal of 5/3 is 3/5, or 0,36 (0 + 36/60). He then multiplied this by 0,30, or 0 + 30/60, and got 0,18 sexagesimal, or 3/10. He finally multiplied this by i, 2;30, which in decimal would be 5/3, and got sexagesimal 0,45. Sexagesimal 0,45 = 0 + 45/60 = 3/4. What do you know? The right answer.

Joy of all joys, after all this nonsense, we have a fully functioning water clock. And of course, what do we do with this water clock? Why, we measure the heliacal risings and settings of the stars and planets, of course! But we'll leave that for another node.

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