### Mathematics and Economics

The Sumerians developed a complex system of metrology c. 4000 BC. This metrology advanced resulting in the creation of arithmetic, geometry, and algebra. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.

The period 2700-2300 BC saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system. The Sumerians were the first to use a place value numeral system. There is also anecdotal evidence the Sumerians may have used a type of slide rule in astronomical calculations. They were the first to find the area of a triangle and the volume of a cube.

By 3000 BC, the Sumerians were drawing images of tokens on clay tablets to keep records of their goods and supplies - what one might call the first bookkeeping system.

Different types of goods were represented by different symbols, and multiple quantities represented by repetition. Three units of grain were denoted by three 'grain-marks', five jars of oil were denoted by five 'oil-marks' and so on.

Every type of good for which they kept records, had its own distinctive sign. The increasing complexity of economic life led to a great proliferation of styles of tokens. Each of these tokens now had to be rendered by their own sign, and, of course, all the signs had to be learned.

Recording a delivery or disbursement of three jars of oil by writing the oil-jar symbol three times is simple and convenient. Recording a delivery or disbursement of several hundred jars of oil the same way is no longer so convenient and is also a system to prone to error.

The large temple complexes that developed in the late fourth millennium, such as the temple of Inanna at Uruk, were considerable economic enterprises, dealing in large quantities of goods and labor.

Gradually, a new system had to be developed. The first great innovation after the act of writing was the separation of the quantity of the good from the symbol for the good. That is, to represent three units of grain by a symbol for 'three' followed by a symbol for 'grain-unit' in the same way that we would write 3 sheep or 3 cows or, more generally, 3 liters or 3 kilometers.

A system of this sort is a metrological numeration system, a system of weights and measures. The 'three' symbol is not completely abstract, but is given value by its context, by having the units appended. The development of this concept over the third millennium is a fascinating and extremely complex story that is as yet only partially understood.

Whereas we use the same number signs, regardless of their metrological meaning (the '3' for sheep is the same sign as the '3' for kilometers or jars of oil), the Sumerians used a wide variety of different symbols.

Nissen, Damerow and Englund have identified around 60 different number signs, which they group into a dozen or so metrological systems.

Any metrological system contains a number of different-sized units with fixed conversion factors between them, so that, for example, there are 12 inches in a foot and three feet in a yard, and so on.

Just as in our old weight and measure systems, Sumerian metrology featured all sorts of conversion factors, although it is notable that they were all simple fractions of 60.

In the basic sexagesimal system used for counting most discrete objects, a single object, a sheep or cow or fish, is denoted by a small cone.

Ten cones equaled one small circle; six small circles equaled one big cone, ten big cones equaled was a big cone with a circle inside it, six of those was a large circle and ten large circles was given by a large circle with a small circle inside. This last unit was then worth 10x6x10x6x10 = 36000 base units.

Note that the circle and "cone-shape" could be easily made by a stylus pressing on the clay, either vertically for the circle or at an angle for the cone.

For discrete ration goods a 'bisexagesimal' system was used with conversion factors 10, 6, 2, 10 and 6, so that the symbol for the Largest quantity, this time a large circle containing two small circles, denoted 6x10x2x6x10=7200 base units.

Yet another system was used for measuring grain capacity. Here the conversion factors were 5, 10, 3, and 10, so that the largest unit, a large cone containing a small circle, was worth 10x3x10x5=1500 of the small units.

Adding to the confusion for modern scholars attempting to unravel these complex metrological systems was the fact that a single sign might be used in several systems, where it could mean different multiples of the base unit.

In particular, the small circle could mean 6, 10 or 18 small cones, depending on context (as well as other multiples of base units denoted by other symbols).

Gradually, over the course of the third millennium, these signs were replaced by cuneiform equivalents so that numbers could be written with the same stylus that was being used for the words in the text.

The final step in this story, occurring probably some time in the Ur III period, right at the end of the third millennium, was the introduction of a sexagesimal place value system.

The number of signs was reduced to just two: a vertical wedge derived from the small cone often used for the base unit, and a corner wedge, derived from the small circle.

The corner wedge had a value of ten vertical wedges. In the sexagesimal counting system described above, the next size unit was the large cone, worth six circles.

In the place value system, this unit was denoted by the same-sized vertical wedge as the base unit, and it was worth six corner wedges. Now the pair of symbols could be repeated in an indefinitely larger alternating series of corner and vertical wedges, always keeping the same conversion factors of 10 and 6.

The price paid was that a vertical wedge could now mean 1, or 60 (6x10), or 3600 (60x60), and so on. Its actual value was determined by its place.

The sexagesimal place-value system greatly facilitated calculations, but, of course, at the end of the day, the final answer had to be translated back into the underlying metrological system of units.

So a problem would be stated in proper units and the solution would be given in proper units, but the intermediate calculations were carried out in the new sexagesimal place value system.