# Old Babylonian mathematics

Most of the Mesopotamian mathematics we know comes from the Old Babylonian period, principally from the second half (say, roughly 1800-1600). At first, the archaeological evidence was such that Old Babylonian mathematics seemed to appear fully-formed out of nowhere, flourish briefly and then disappear again for a thousand years. Gradually, some of the gaps in our knowledge are being filled in, but it should be remembered that our picture of Mesopotamian mathematics is inevitably skewed by the accidents of discovery.

We possess several hundred Old Babylonian mathematical tablets. They are conventionally divided into table-texts and problem texts. Here, we give a brief overview of the different types of tablets and summarize what we can learn about the characteristics of Old Babylonian mathematics from them.

### Table texts

The first striking fact we learn about Old Babylonian mathematicians is their penchant for making arithmetical tables. Tables, such as multiplication tables have a very simple structure. They are (comparatively) easy to translate, even if you don't know what the symbols all mean, and they played a very important role in our discovery and decipherment of Mesopotamian mathematics. First, it was clear that the Old Babylonian used a simple positional notation system with a base of sixty. They had a sign for 'one'  which is just bundled to represent 'two' , 'three' , and so on, until you get to 'ten', which has a different sign . Continue bundling the 'one's and 'ten's until you reach 'sixty', at which point you move over a column and use the 'one' symbol again. In this way, an Old Babylonian mathematician could represent any number using combinations of just two symbols. Occasionally, there is a problem for large numbers, because there was no symbol to indicate an empty column, although usually they left some extra space there.

There are no addition or subtraction tables. Presumably, scribes learned to add at about the same time they learned to read and write, so that addition tables would not have been much use. There are, however, lots of multiplication tables. These come in two varieties, single and combined tables. A single multiplication table has products for a single number, called the principal number, while a combined table has several single multiplication tables written in order on one tablet. They seem never to have written a square multiplication table of the sort we tend to use.

The next arithmetical operation we learn is division. There are no division tables. Instead, we find reciprocal tables. The reciprocal of a number n is the fraction 1/n. Instead of dividing by n, an Old Babylonian would multiply by its reciprocal. It's hard to remember reciprocals, so they made tables for them. Just as in our system, there are fractions with infinitely long decimal representations, so too in the sexagesimal system the Babylonians used. The only numbers that have finite reciprocals are ones where the only factors are powers of 2, 3 and 5. These are now called regular numbers, and a fair amount of effort in Babylonian school mathematics went into ensuring that a student only had to find reciprocals of regular numbers.

There are hundreds of multiplication tables, many of them written by students as exercises, and many reciprocal tables. Additionally, there are a few examples of tables of squares, square and cube roots (but no cubes), and some powers. There are some tablets used in problems to do with finding the market rate for goods. Finally, there are a series of tablets with what are called 'coefficient lists'. These list conversion factors for problems involving weights and measures, and also some geometrical coefficients, such as the ratio of diagonal to side of a square.

### Problem texts

The other major class of Old Babylonian mathematical tablets is the 'problem texts'. These give a number of problems to be solved. Sometimes all the problems on a tablet are related exercises in one topic, other times there are many different unrelated topics. From the problems set, we can determine what were the major interests of Old Babylonian mathematicians. First, it should be noted that most of these tablets are intended for use in schools. They tend to be quite cryptic and abbreviated, and use a specialized vocabulary. Certainly, a student then would have had the exercises supplemented and explained orally by a teacher, but it is only what was written down that has been preserved. The problem for scholars is rather as if you were to try to reconstruct modern mathematics armed only with the exercises from the text-book and a few worked examples.

Some important characteristics of Old Babylonian mathematics stand out. Almost always, the goal of a problem is the computation of a number. Students were not asked to produce a figure, nor give what we would call a proof. Because of this emphasis on numeric computation, Old Babylonian mathematics has been called 'algebraic' as opposed to Greek 'geometric', but this is to give a somewhat misleading categorization. Old Babylonian mathematics was not based on the manipulation of symbols in formulas (what we think of as algebra), but rather on following procedures to obtain an answer (what we would call algorithms). Twentieth-century mathematics has had a huge bias towards algebra, but the recent rise of computers with their algorithmic programming requirements has led to a more sensitive view of Mesopotamian mathematics.

The Babylonians were strong believers in word problems. Apart from a few procedure texts for finding things like square roots, most Old Babylonian problems are couched in a language of measurement of everyday objects and activities. Students had to find lengths of canals dug, weights of stones, lengths of broken reeds, areas of fields, numbers of bricks used in a construction, and so on. For most of the problems, then, a good knowledge of Mesopotamian weights and measures systems is needed. Modern scholars have had a difficult task unraveling these systems, and much is still being learned, for example about brick coefficients.

Like modern math word problems, Babylonian mathematics problems had little to do with the real world. Some of the classic examples include finding the original length of a broken reed that was used to measure a field, and flooding fields of several square kilometers to a depth of one finger, for irrigation. However, many of the problems do use realistic coefficients and settings, especially those concerned with defense construction and economic activities, although the standardized work-rates used in the problems were no longer used in the outside world.

Since Babylonian mathematical tables cover multiplication, division (as reciprocals), squares and square roots, we would expect to see problems with computations involving these operations. And this is exactly what we do see. Babylonian 'algebra' problems include both linear and quadratic topics in one or more variables. Some problems involve a single 'equation,' others systems of linear or quadratic equations. They often display considerable ingenuity is setting up and solving problems of great complexity.

Some problem tablets give just the statement of an exercise, others the problem and the correct answer, but the ones that are most helpful to us give the statement of the problem and then follow it with the outline of the procedure the student should follow to get the answer. The Babylonians never stated a general procedure, instead they gave worked examples. The student would work through lots of examples until they could do the problem with any set of initial values. That they were working from a general procedure is clear from the examples of exercises we have where at one step a student will have to multiply by 1, because in a different example, that would not be a 1.

Another way we can see the Babylonian's focus on the procedure is in texts which have related groups of problems. Often the answer is always the same for every problem in the group. This makes the computations easy for the teacher, or student, to check, and emphasizes that it is the 'workings' that count.

The Babylonians did use geometrical constructions in their problems, although, as mentioned before, the purpose of a problem is the computation of a number. For geometric objects, this typically means lengths of sides or diagonals, or determination of area or volume. The Old Babylonians had no measurement of angle, which to us is such a basic part of geometry. Angles are a later development. Some tablets do have figures drawn on them, and there are surviving problems to do with all the standard shapes such as squares, rectangles, triangles, trapezoids, circles, and so on. Some of the problems become very sophisticated. Solid geometry is dominated by bricks and ramps, but cylinders and truncated cones and pyramids do make an appearance.

The level of mathematical skill displayed in Old Babylonian mathematics goes considerably beyond that required for day-to-day usage, even as a scribe. There are some examples of virtuoso work, and it is not clear to us how much mathematical training the average scribe had. We do not know if every student would have been expected to understand all the mathematics for which we have evidence, or if there was some degree of specialization.

At the end of the Old Babylonian period, the archaeological record becomes sparse and broken. We do not know if the Old Babylonian period really represented a unique flourishing before a period of decline, or if the same skills were maintained during what is still a fairly 'dark' period. The next big collection of mathematical tablets belongs to the mathematical astronomy of the Neo-Babylonian, Persian and Seleucid periods a thousand years after the end of the Old Babylonian time. Certainly at that time, the scribes could perform at the level of the Old Babylonians, although there is no evidence of any great developments during the intervening gap.

Further reading: The most readable introductions to Old Babylonian mathematics remain O. Neugebauer's 'Exact Sciences in Antiquity' and A. Aaboe's 'Episodes From the Early History of Mathematics,' though these are somewhat dated now. The best single reference starting point is Friberg's terse and difficult, but comprehensive article Mathematik in RLA.

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