1. Historical Orientation – Mesopotamia Once again: “The past is a foreign country; they do things differently there” L.P. Hartley, The Go-Between Today, we'll again do a bit of orientation (in location and time) for the second of the ancient civilizations whose mathematics was especially important for the development of the subject.

2. Ancient Mesopotamia the “land between the rivers” Tigris and Euphrates – mostly contained in current countries of Iraq, Iran, Syria.

3. Another very long history ~5500 BCE -- First village settlements in the South ~3500 - 2800 BCE -- Sumerian city-state period, first pictographic texts ~3300 - 3100 BCE -- first cuneiform writing created with a reed stylus on a wet clay tablet, then sometimes baked in an oven to set combined with a pretty dry climate, these records are very durable!

4. A tablet with cuneiform writing Note the limited collection of forms you can make with a wedge-shaped stylus:

5. Cuneiform writing Different combinations of up-down and sideways wedges were used to represent syllables Was used to represent many different spoken languages over a long period – 1000 years + We'll see the way numbers were represented in this system shortly

6. Concentrate on southern area ~2800 - 2320 BCE -- Early Dynastic Period, Old Sumerian literature ~2320 - 2180 BCE -- Akkadian (Sumerian) empire, first real centralized government ~2000 BCE -- collapse of remnant of Sumerian empire ~2000 - 1600 BCE -- Ammorite kingdom "Old Babylonian Period" (roughly contemporaneous with Egyptian Middle Kingdom)

7. Cultural landmarks  Hammurabi Code  Mathematics texts that we will study in detail starting next time  Editing of Sumerian Epic of Gilgamesh  Educational system focused on scribal schools training youths for careers in religious and government institutions, as well as record- keeping for private citizens. Mathematically trained scribes were professionals.

8. Later history This part of the world has been fought over and conquered repeatedly – most recently, of course, in the two Iraq wars of the 1990's and 2000's CE – a very complicated story! Also figures in Biblical history (“Babylonian captivity” of Jewish people) 612 - 539 BCE -- “New Babylonian” period (Nebuchadnezzar) height of Babylonian astronomy – interaction with Greek mathematics and science in period immediately following

9. Decipherment of cuneiform  As was the case with Egyptian hieroglyphics, cuneiform inscriptions could not be read until scholarly work in the 19 th century – work of Henry Rawlinson, Edward Hincks, others  The “Rosetta Stone'” – a series of parallel texts in different languages, found in present- day Iran – created by Darius I, the same Persian king who invaded Greece, was defeated at the battle of Marathon, 490 BCE  Old Persian, Elamite, Babylonian forms were deciphered in that order

10. The Behistun inscriptions

11. Babylonian numerals

12. A base b = 60 positional system  These symbols served as the “digits” for a positional base 60 number system  For example, to write a number like 142, the Babylonians would break it up as 142 = 2 x 60 + 22, and write the “digit” for 2, followed by the “digit” for 22  There is a potential ambiguity here – do you see why?

13. Features of Babylonian mathematics  Number system looks clumsy to us, but they used it very effectively for calculation with large numbers  One reason this was possible – they made quite extensive use of tables of information as memory/calculation aids (will see an example next time)  They went (much) farther than the Egyptians did into algebra and “geometric algebra”

14. And after that, …  Baghdad under Muslim caliphate was a world center of learning during “dark ages” in Europe  House of Wisdom – something like a university or research institute  Scholars there collected, studied, and extended the works from the classical Greek and Roman eras  Were transmitted back to Europe during the Renaissance – will discuss this later(!)