The Mesopotamians were accomplished mathematicians and students of the heavens. Many of the systems that they developed underlie those used throughout the world today: positional notation, the divisions of time and space (seconds, minutes, degrees in a circle), the mapping of the firmament. Even writing began with Mesopotamian accounting devices.
Mathematics
The Sumerians used mathematics initially for accounting and later developed it to solve other practical problems. Fourth-millennium clay tablets bear witness to the emergence of numerical written records. By the third millennium more complex accounts were being kept, mathematics was being employed in civil engineering, construction, and the organization of labor, and theoretical problems in arithmetic and geometry were also being considered. Figuring became a key part of education and by the OB period, from which most of the surviving texts come, complex mathematics were being used.
The Number System. The Mesopotamians used a base-sixty counting system. Evidence indicates that in early times there were separate numbering systems for counting different things: sheep, measures of grain, jars of oil, and so on. With early writing, however, abstract counting came into use. Instead of the sign for a commodity being written the appropriate number of times (tallied), numbers and commodities were represented separately, so that a number of sheep, for example, was written as a number sign and the sign for sheep. Until the late third millennium, however, different systems were used for counting discrete objects, discrete rations, and measures of grain. By the Ur III period a single sexagesimal positional system was used for writing numbers when performing calculations, although the different commodities were still written at the beginning and end of the sum with their own units.
Abstract numbers were counted in powers of sixty: 1, 60, 3,600, 216,000, and even 12,960,000. These were broken down into decimal units—six lots of ten, six lots of six hundred, and so on. Although initially there were separate signs for each of these, by the Ur III period the signs had become greatly simplified, using vertical wedges for units and powers of sixty, and diagonal wedges for tens and tens of powers of sixty, and relying on position rather than shape to indicate the number in question. A vertical wedge stood for a unit and numbers up to nine were written as an arrangement of these wedges in two rows. Ten was written as a slanting wedge and numbers up to fifty-nine were written as a combination of tens and units, the tens arranged to the left of the units. Sixties were again written vertically, to the left of the tens, and six hundreds diagonally to the left of the sixties, and so on, up to the fourth power of sixty. Fractions were similarly written, in the manner of decimals, to the right of units, although there were also separate symbols for the most common fractions: one-half, one-third, two-thirds, and five-sixths. One problem with positional notation is the need to indicate the absence of a value in a particular position, for example to distinguish the Babylonian number 1, 0, 1 (3,601) from 1, 1 (61). When writing figures the Babylonians sometimes left a gap as a placeholder, and from around 700 b. c.e. onward they occasionally made a mark instead. In most cases, however, they relied on the context to make the appropriate value clear.
Weights and Measures. Mesopotamian units of weight, length, area, and volume used various factors and multiples of 60, particularly 3, 6, 10, 60, and 180 (see table 10.1 and photo p. 132). This is clearest in the units of weight; length also introduced a multiple of 2, and in measures of capacity 5 was also a multiple.
These measures were standardized by the Akkadian king Naram-Sin, including the gur, which he fixed at 300 silas = 5 barigas (about 300 liters). This standardization produced a neat interrelationship among the units of length, area, and volume.
Geometry, Arithmetic, and Others Mathematics. Many mathematical exercises survive in texts, mainly from the Old Babylonian period, although some go back to the Ur III period and even to ED Illa. These exercises were probably mainly didactic aids used in school, demonstrated by the teacher and copied and learned by his pupils. The problems often dealt with practical matters: for example, calculating the size of siege ramp that could be built with a given volume of soil and the time required to build it; the volume of water required to irrigate a
Weights and Measures
Certain area of land; the number of laborers needed to construct or clean out a canal and the quantity of food rations they would require; the amount of grain needed to sow a field of a given area and the yield to be expected from it; or prices of commodities and interest rates on loans. Other problems, however, were theoretical and might involve quadratic equations or arithmetic progressions. Some texts give problems and worked solutions, step-by-step, while others just give long lists of related problems. Although early texts were in Sumerian, from the OB period onward Akkadian was generally used.
A large number of tables were available to aid in solving these problems and in undertaking calculations in the real world, such as surveying land (particularly for dealing with sales and inheritance), calculating the prices of commodities, or computing simple or compound interest. These included straightforward multiplication, division, and metrological tables, but also more advanced mathematical information such as squares, square and cube roots, reciprocals, coefficients, and lists of key numbers. A good approximation to root two was calculated, 1.414212963 (the correct value, to ten significant figures, is 1.414213562), but for pi the Babylonians generally used 3, although lists of reciprocals show that they were aware that 3.125 was a closer approximation (the value used today, to four significant figures, is 3.142).
Their calculations required a good grasp of plane and solid geometry, including knowledge of "Pythagoras's Theorem" (some 1,300 years before Pythagoras), and skill with dealing with triangles, circles, trapezoids, and regular polygons as well as squares and rectangles; and prisms, cylinders, and truncated pyramids as well as cubes and cuboids; but they did not use cones, pyramids or spheres, nor did they work with angles. Some calculations were expressed as algebraic problems with one or several unknown values, and much of the methodology of problem solving was algebraic, although the Babylonians did not use symbolic notation.
Calendrics
Divisions of Time. The Babylonians used a calendar based on cycles of both the moon and the sun. Time was divided into solar days, lunar months, and luni-solar years. Days ran from sunset to sunset and were divided into four parts or into twelve "double-hours." Astronomical texts such as Enuma Anu Enlil provided information on how to calculate the length of daylight at different times in the year.
A cycle of the moon lasts a fraction over twenty-nine days. Mesopotamian months were reckoned as the time from the first sighting of the new moon on the western horizon to its first appearance in the following month: They were therefore either twenty-nine or thirty days long. The first, seventh, and fifteenth days of the month were marked by special religious observances. These provided a measure of the passing of time within the month: There was no division of the month into weeks (an innovation of the Roman period).
The year had twelve lunar months, making it only 354 days long. This is shorter than the solar year of 365 1/4 days, and if left uncorrected, the lunar-based calendar would rapidly get out of step with the solar calendar.
Calary month was therefore inserted periodically to correct this discrepancy, usually after the sixth month or at the end of the year; the introduction of this month was made known by royal decree. Although this practice is attested to in Babylonia from the third millennium onward, in Assyria it was not certainly adopted until the twelfth century B. C.E. (see table 10.2).
A cuneiform tablet bearing a copy of the Assyrian King List. (Library of Congress)
The Babylonian year began with the month Nisannu in the spring (March/April), whereas Assyria originally started the year in the autumn, although by the first millennium this had been brought in line with the Babylonian year. The Babylonian New Year (resh shatti), which fell around the spring equinox, was celebrated in Babylon with a major festival that lasted for twelve days.
The Passing of Time. By the time of the Akkadian Empire, records were dated by reference to the reigning king and an important event within that year, a method that continued in use in Babylonia until around 1500 B. C.E. Official lists of these year names were kept from Ur III times onward. A year might
Be named for the founding of a temple, the appointment of a particular individual as chief priest of the city deity, the installation of a divine image, or even the furnishing of the god's statue with a new garment. Other year names marked significant military achievements. Rim-Sin I of Larsa was so satisfied with his defeat of Isin in 1794 that he dated all the remaining years of his reign as so many years after this victory. Public works might also be celebrated, such as the construction of a canal or a city wall. Sometimes more than one event was recorded in the year name.
From the Kassite period onward, dates in Babylonia were reckoned by the king's regnal years. In addition, Babylonian priests kept a daily record of omens and observations of the heavens and the world around them, for the sake of divination, often including references to important contemporary political events and other interesting occurrences; these were compiled into lists that allowed particular years to be identified and the lapse of time recorded. When Ashurbanipal sacked Susa and recovered the statue of Inanna stolen from Uruk, he was able to claim that 1,635 years had elapsed since the Elamites had carried it off.
Other historical records included king lists, which named the kings and gave the lengths of their reigns. The Sumerian King List, known from OB texts but probably first compiled in the Ur III period, began before the Flood and ended when Hammurabi conquered Isin. It included only some of the Sumerian city-states: Lagash, for example, was omitted. The Babylonian King List began with the First Dynasty of Babylon, and the Assyrian King List started with a number of kings "who lived in tents," the tribal ancestors of Shamshi-Adad. The Synchronistic King List gave details of Assyrian kings, their contemporary Babylonian monarchs, and their viziers.
In some periods chronicles were compiled giving a year-by-year list of events. The earliest surviving example, the Eponym Chronicle, comes from eighteenth-century Mari. Royal annals, recorded by Assyrian kings from the thirteenth century onward, gave details of campaigns, dated by the regnal years of each king. In addition, from the nineteenth century b. c.e. the Assyrians used a system of naming each year after an important official (limmu). These names, often accompanied by details of a significant event during the year, were preserved in eponym lists, which provided the main dating framework for Assyrian timekeeping: the relevant eponym year was noted when writing legal, administrative, and business documents. Many fragments of the eponym lists have survived, allowing parts of the list to be reconstructed, but only for the period 910 to 649 b. c.e. is an unbroken sequence known.
The inclusion of material in the Assyrian annals was very selective, only victories being recorded. In contrast, the Babylonian Chronicle was largely impartial and accurate, reporting reverses as well as successes. It covered events in Babylonia, dated by regnal year, but also mentioned some things that took place in neighboring Elam and Assyria. The Babylonian Chronicle was begun in 747 B. C.E. under Nabonassar and continued into the Seleucid period, although there are many gaps in the sequence. Texts were usually labeled as extracts from a more comprehensive record. Reference on one tablet to it being copied from a waxed writing board suggests that day-to-day records were kept on ephemeral media and were subsequently collated and copied onto the more permanent medium of clay tablets.
Astronomy
Determining the will of the gods and their intentions for the future was of key importance; this information was communicated in various ways that could be interpreted by those with the requisite knowledge. Omens and portents were therefore collected and recorded, along with information on the events that followed them. A major source of such portents, and one that became increasingly important in the first millennium, was the movement of heavenly bodies. In observing and recording these phenomena Mesopotamian sages, and particularly those of Babylonia, became extremely knowledgable about the regular courses of the stars, planets, moon, and sun. Some celestial events, particularly eclipses, were seen as boding ill: In a number of recorded instances, a substitute king was enthroned to deflect the danger of the eclipse from the real monarch.
A number of astronomical texts survive, the earliest dating from around 1700 B. C.E. These contained celestial observations and calculations to enable the patterns of the heavens to be predicted. The second-millennium Enuma Anu Enlil ("When Anu and Enlil"), a compilation of astronomical observations and the events that they presaged, was the most popular work, known in numerous Neo-Assyrian copies. It was in seventy tablets, covering various phenomena: phases of the moon, lunar eclipses, and conjunctions of the moon with planets and fixed stars; the sun, including solar eclipses and coronas; meteorological portents, including thunderstorms and clouds; and a compendium of stellar and planetary movements.
Other important texts included "Astrolabe," concerned with the heliacal rising of three fixed stars over the course of the year, and MUL. APLIN "Plough Star," a two-tablet compendium of all the movements of celestial bodies known in the seventh century b. c.e. when it was compiled. These included the paths of three major constellations of fixed stars, associated respectively with Anu, Enlil, and Ea. These texts also gave arithmetic information for calculating when intercalary months should be inserted into the calendar and establishing the variation in daylight hours, month by month, throughout the year.
Cartography and “The Babylonian Map of the World”
Among the clay tablets from the later third millennium onward are a number of maps and plans giving designed layouts for houses or temples, with written measurements. A seated statue shows Gudea with a temple plan and architect's rule on his knees. Others are land survey plans, setting out measured arrangements of fields, with information on the quality of the soil and their current use. One tablet of particular interest shows a scale plan of the city of Nippur, dated around 1500 b. c.e. It is carefully labeled, with measurements, and shows the city's walls and gates, the river Euphrates and canals, and a number of temples including the Ekur, temple of Nippur's patron god Enlil.
The “Babylonian Map of the World,” showing Babylon, Mesopotamia, the surrounding ocean, and the lands beyond, described in the text at the top of the tablet. (Zev Radovan/Land of the Bible Picture Archive)
One large area is labeled "garden in the city." Another map gives details of royal and temple properties around Nippur, and there are area maps on which the Euphrates, canals, roads, and a city are marked, including one of the Akkadian period showing mountains and cultivated land near Nuzi and labeled with compass directions.
The most ambitious map ("the Babylonian Map of the World") schematically represents the world as known to the Babylonians around 700 b. c.e., surviving in a later copy. Two concentric circles surround the mapped area, representing the ocean, and two parallel vertical lines represent the Euphrates, running down to the marshes in the south, which are identified with a label. Another label placed across the river marks the location of Babylon. At the top a semicircle indicates the mountains and around the right side Urartu, Assyria, and Der are named, with Susa at the bottom. Outside the ocean captions identify strange lands inhabited by fabulous creatures. This is the earliest known attempt to map the world.
World History
