I think that there's evidence that the Babylonians were pretty good at arithmetic, and if you practice it, as I have, you'll amaze yourself at how much you can do in your head using base 60. Here's an example of how they probably had ideas about limiting processes. YBC7289 gives strong evidence that they (some of them, anyway) knew the sometimes-called Heronian algorithm for square roots. If you want the sq root of 2, make a guess, say 1. Divide into 2 and get 2, so 1's too small and 2's too big. Take the average, 3/2. Divide into 2 and get 4/3. Average 17/12. Next stage 577/408. It's clear that the Babylonians got that far, but I reckon that they also noticed that, after the first guess, all the answers were too big, and that they differed from one another by surprisingly smaller amounts, AND that the process could be continued indefinitely. Let's see what happens if you translate 577/408 into Babylonian. Correct to 3 sexagesimal places, it's 1, 24 51 11. But that's not what they inscribed: 1, 24 51 10 --- because they KNEW that the former was too big, and they probably had a good idea of how much too big it was. As I said, if you work in base 60, calculations are easier, but as most of us are used to base 10, we'll have to do the harder comparison: 577/408 = 1.414215686274509803921568627 1+24/60+51/60^2+10/60^3 = 1.414212962962962962962962963 sqrt(2) = 1.414213562373095048801688724 (577^2+2*408^2)/(2*408*577) = 1.414213562374689910626295579 I'm sure the Babylonians, some of them, were good at geometry, and the above, and the famous Plimpton tablet, were obviously inspired by geometry (actually, astronomy), but the Plimpton tablet is a monumental piece of ARITHMETIC. R. On Thu, 3 Mar 2011, Veit Elser wrote:
On Mar 2, 2011, at 9:11 PM, Andy Latto wrote:
But I think that's just wrong. The babylonians knew how to multiply, and did it arithmetically, rather than by use of geometry.
I don't agree, unless you can prove to me the Babylonians were manipulating algebraic expressions. The term "geometric" probably derives from the context in which the progression arises, such as subdividing plane geometry figures into infinite sequences of similar figures. A famous one involves dividing up the golden rectangle into a square and a smaller golden rectangle. Could the Babylonians have dealt with this example arithmetically?
Veit
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