Measuring the Gardens of Eden, by Jenia Tevelev 1
A map of the area around Gasur, near Kirkuk in northern Iraq, drawn up some time in the Sargonic period (2200 BCE). The central area, below the Rahium river, is described as irrigated gardens belonging to Arala.
Mesopotamia, Ancient Greek Μεσοποταµία, "between rivers (Euphrates and Tigris) The Sumerian civilization appeared before 3500 BCE - cradle of civilization Cities, irrigation systems, a legal system, administration, and even a postal service. The first writing script developed. Sumerians and Akkadians (including Assyrians and Babylonians) dominated Mesopotamia until the fall of Babylon in 539 BCE, when it was conquered by the Achaemenid (Old Persian) Empire.
First Work of Literature The Epic of Gilgamesh originated with Sumerian poems dating from the Third Dynasty of Ur (around 2100 BCE). The 11th tablet describes the meeting of Gilgamesh with Utnapishtim. Like Noah in the Hebrew Bible, Utnapishtim had been forewarned of a plan by the gods to send a great flood. He built a boat and loaded it with all his possessions, his kith and kin, domesticated and wild animals and skilled craftsmen of every kind. 4
First Code of Laws King Hammurabi established the Old Babylonian state. Mathematical culture flourished in this period. The Code of Hammurabi (around 1750 BCE) contains 282 laws. According to tradition, Hammurabi received the code from Shamash, the patron Sun god of Babylon 5
Geometric patterns 6
Counting aides 7
First astrological/astronomical tables (e.g. famous MUL.APIN) 8
First Bureaucracy The highly centralized Sumerian and Accadian states required large bureaucracies. They recorded aspects of the quantitative measurement of land, livestock, and labour, not only to account for what had already been acquired or produced but also to make shortterm predictions of costs and yields - E. Robson Pedagogical curricula started to develop to train scribes more effectively 9
Cuneiform The cuneiform culture in Mesopotamia was coextensive with the mathematical culture. Cuneiform was incised on virtually indestructible clay tablets. The earliest known system of writing The original Sumerian script (4th millennium BCE) was adapted for writing in Akkadian and other languages just like the Roman alphabet is now used for many languages and not just Latin. Sumerian was the first language in the world to be written down. It became completely extinct around 1 BCE. By the 2nd century CE all knowledge of how to read cuneiform was lost until its decipherment in the 19th century. About half a million cuneiform tablets have been excavated. Only about 1/10 of them have been read in modern time as there are only a few hundred qualified cuneiformists in the world. 10
In 1835, Henry Rawlinson, a British East India Company army officer, visited the Behistun Inscriptions in Persia. Carved in the reign of King Darius of Persia (522 486 BCE), they consisted of identical texts in the three official languages of the empire: Old Persian, Babylonian, and Elamite. The Behistun inscription was to the decipherment of cuneiform what the Rosetta Stone was to the decipherment of Egyptian hieroglyphs. 11
YBC 7302 12
What s on the tablet? YBC 7302 is from the Old Babylonian period (1700BCE) circular form indicates an exercise by a trainee scribe a vertical wedge = 1 and a corner wedge = 10 13
This tablet is an exercise in computing the area of the circle (which is rather faint) 3 is the circumference. We usually visualize the circle as the area generated by a rotating radius. In ancient Mesopotamia, the circle was visualized as the shape contained within a circumference, no radius is drawn. A = c2 4π c2 12 So 9 is just 3 2 (calculation on the side) But why is the answer 45??? And where is division by 12??? Interpretation 14
Sexagesimal System A place-value number system with base 60 was introduced around 2100 BCE Used now to measure angles, geographic coordinates, and time. For example, a GPS-style latitude N 51 28' 38 of Greenwich is sexagesimal for a decimal fraction 51+28/60+38/3600=51.4772222.. Babylonians didn t use a sexagesimal point or zero may mean 45, but also 45/60 (=3/4), 45*60; etc. 15
Babylonian sexagesimal system was the first ever positional (or place-value) system. In these systems the value of the digit depends on its place. For example, in the decimal system, one 2 in 22 means 2 and another means 20. Likewise, in the Babylonian system may mean 45, but also 45/60 (=3/4), 45*60; etc. To complicate matters (for us), Babylonians didn t use a sexagesimal point or zero. The first systems were sign-value systems. For example, familiar Roman numerals is a sign-value system. MMXVII=? Or Egyptian numerals =? What are the advantages of sign-value systems? 16
Sign-value systems were used in Mesopotamia as well. Often, different symbols would be used for different commodities, measures, or in different trades (similar to imperial units). The sexagesimal system was used only for calculations. When a result was to be inserted into a contract or an account, Babylonian scribes would use one of the sign-value systems. Are there analogues in the modern world when a numerical system is used for calculations but not revealed to the end-user? 17
What s so nice about 60? 60 has a lot of factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. For example, one hour can be divided evenly into sections of 30, 20, 15, 12, 10, etc. minutes. Instead of dividing by a number, Babylonians would multiply by its reciprocal (and shift an invisible sexagesimal point) number 3 5 12 8 18 27 reciprocal 20 12 5 7 30 3 20 2 13 20 What would be an analogous calculation in the decimal system? 18
Now we can solve the mystery of YBC 7302 Instead of dividing 9 by 12, Babylonians would multiply by a reciprocal: 9*5=45 45 is in fact not 45 but 45/60=3/4 19