1.7 Babylonian mathematical style
Not only should you have learnt through this exercise more about the Babylonian mathematical style, but also, on another level, you should have gained more experience in the endeavour of trying to understand past mathematics. The model that we have been trying out can be characterised thus: use any means, any symbolism or notation that occurs to you, to find your way into the problem, then check rigorously to see how much of your new understanding is more a projection backwards from your own time and techniques. First, try to understand what they might have been doing. Then address the harder questions of how and why. As this process becomes more familiar, you will find it increasingly easy both to respond to past mathematics on its own terms, and to understand and evaluate historical questions and concerns. (Doing this may also produce the added bonus of a better understanding of your own mathematics.)
The Babylonian ‘quadratic’ problem above is fairly characteristic of problem texts; some have two unknowns (length and width) and two conditions connecting them, and a few even have three unknowns and three conditions. Some involve cubes or higher powers of unknown numbers which are to be found. The problems are expressed sometimes fairly abstractly, as we have seen, sometimes in terms of more concrete imagery that seems direct training for practical problems which a scribe might be called upon to solve professionally. (See, for example, the collection of problems in the attached pdf concerning a number of workers digging a volume of earth for so many days at a certain expected productivity.) Yet even apparently practical problems can have flamboyantly unrealistic solutions—on one tablet (see the attched pdf), the problem is to discover what area of field can be irrigated by a particular volume of water, and the answer turns out to be a field some 3½ kilometres square, covered with water to a uniform depth of one finger's breadth, a procedure that would tax even the most diligent of Babylonian farmers!