I’m sure we all remember Pythagoras’s formula for the relationship of sides of a right-angled triangle: the square of the hypotenuse is equal to the square of the other two sides. You probably measured it out with sides of 3, 4 and 5 units (9 plus 16 is 25). These three numbers are the smallest whole-number solution to the formula and have undoubtedly been used in teaching this relationship ever since they were first discovered. Pythagoras is known as the first person to describe himself as a philosopher – a lover of knowledge – but the Mesopotamians clearly pipped him to the post on this, his most famous theorem.
Today’s tablet is known as Plimpton 322, and was supposed to have been discovered at the city of Larsa some time in the early 1920s, at which point it had been in the ground for close to 3800 years. You may have heard about this tablet. It was the subject of some excitement a couple of years ago, when Daniel Mansfield and Norman Wildberger wrote a paper describing this tablet as “the only completely accurate trigonometric table”. It was picked up by the Guardian and also Popular Science and National Geographic magazines who all wrote up the University of New South Wales’ press release.
What makes this tablet interesting is that all the rows are examples of Pythagorean triples (even though they predate Pythagoras over 1000 years). Barring a few errors, possibly of transcription, possibly of calculation, each row represents a whole-number solution to a narrower and narrower triangle. These are no toy examples like 3:4:5; among the smallest is 65:72:97, and the largest is 12709:13500:18541 which would have taken some fairly complex mathematics to calculate. This had been known for some 70 years or so before the recent fuss, so what was it all about? What made it so accurate?
Mansfield and Wildberger’s argument that this is a more accurate trigonometric table than those we have today appears to be based on the idea that whole number ratios are more accurate than irrational numbers. Mansfield particularly has the reputation of preferring whole number ratios – a field he founded called rational trigonometry. In this field of mathematics, a fraction like ½ which in decimal notation is 0.5 and so terminates at one decimal place, is “better” than 1/3 which continues infinitely as 0.333… At least it does in our base 10 counting system. So if we take a step back, the improvement in accuracy, or at least in the number of ratios which can be expressed with a finite number of digits, is just an artifact of counting in base 60. This may actually have been valuable in ancient Mesopotamia, where a whole number of units could be used in setting out the groundworks for a new building or dividing a field. It is also much easier to work with mathematically as we’ll cover on day 75. But nowadays we have computers that can calculate as many places after the decimal as we need to get the required accuracy.
Dare I say it, Mansfield’s aversion to infinite numbers of decimal places is somewhat… irrational.
