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July 03, 2006
"Do not say a little in many words, but a great deal in a few."
The Babylonians knew about "Pythagorean" triplets and the "Pythagorean" theorem almost 4,000 years ago, according to a tablet known as Plimpton 322:
The numbers in the first column are interesting since each is a perfect square and subtracting one from each leaves a perfect square. Consider, for instance, line 11. The number 1:33:45 represents 1 + 33/60 + 45/3600 = 1 + 9/16 = 25/16 which is the square of 5/4. One less than 25/16 is 9/16, the square of 3/4. The second and third entries in this row represent these fractions: 45 represents 45/60 = 3/4 and 1:15 represents 1 + 15/60 = 5/4. (Or perhaps 45 represents 45 and 1:15 represents 75 in which case these two entries are proportional to the fractions.)
For another example, consider row 5. 1:48:54:01:40 represents 1 + 48/60 + 54/3600 + 1/216000 + 40/12960000 = 1 + 4225/5184 = 9409/5184, which is the square of 97/72. And 4225/5184 is the square of 65/72. The second and third entries in row 5 are 1:05 representing 65, and 1:37 representing 97. Nearly always, the second and third entries aren't equal to the square roots, but just proportional to them.
This table is usually considered in relation to Pythagorean triples. In that interpretation, the second column is a, one side of a right triangle or the width of a rectangle, and the third column is c, the hypotenuse of the right triangle or the diagonal of the rectangle, and the other side of the triangle or rectangle, b, doesn't appear on the table. In this interpretation the first column is then (c/b)2 = 1 + (a/b)2.
The Old Babylonians knew the Pythagorean theorem (better called the rule of the right triangle for them since there's no evidence that they had a proof; Gillings calls the term "the Pythagorean theorem" a true mumpsimus), since there are examples of its use in various problems of the period. Along with the headings of the second and third columns, that justifies believing that this table relates to Pythagorean triples and right triangles.
Some historians have noticed that (1) each first column entry is the square of the cosecant of an angle of a right triangle, and (2) the associated angles are roughly one degree apart. So they have suggested that this is a trigonometric table of squares of cosecants for 45° down to 30°. This would be the earliest instance of cosecants by millennia, and the earliest instance of a trigonometric function by over a thousand years (the first trig function being the chord of an angle). It would also be the earliest instance of degree measurement by over a thousand years. In other words, that's a bold claim. A weaker claim is that the table was constructed to make the first column uniformly decreasing.
....sorry, that's it. I really meant to post more but ran out of time...
Posted by Jon Rubin at July 3, 2006 10:45 PM
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