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Mathematics in Ancient Egypt and Mesopotamia Mathematics in Ancient Egypt and Mesopotamia Waseda University, SILS, History of Mathematics Mathematics in Ancient Egypt and Mesopotamia Outline Introduction Egyptian mathematics Egyptian numbers Egyptian computation Some example problems Babylonian Mathematics Babylonian numbers Babylonian computation Some example problems Mathematics in Ancient Egypt and Mesopotamia Introduction How do historians divide up history? The large scale periodization used for (Western) history is the following: I Ancient: the distant past to, say, 5th or 6th century ce I Medieval: 6th to, say, 15th or 16th century I Modern: 16th century to the present Mathematics in Ancient Egypt and Mesopotamia Introduction Ancient cultures around the Mediterranean Mathematics in Ancient Egypt and Mesopotamia Introduction How do we study ancient history? I What are our ancient sources? I material objects I images I texts a. found as ancient material objects b. transmitted by tradition I What is the condition of the sources? I Wherever possible, we focus on reading and understanding texts. I When we study objects, without any textual support or evidence, it is very easy to be mislead, or to have very open-ended and unverifiable interpretations. Mathematics in Ancient Egypt and Mesopotamia Introduction How can we interpret these objects without texts?1 1 The pyramids of Giza. Mathematics in Ancient Egypt and Mesopotamia Introduction Or how about these?2 2 Stonehenge in Wiltshire, England. Mathematics in Ancient Egypt and Mesopotamia Introduction But . how do we interpret these texts? Mathematics in Ancient Egypt and Mesopotamia Introduction Things we might want to know about a text I What does it say, what are its contents? I Who wrote it? I What was its purpose? I What can we learn from the text itself about its author? I How much do we need to know about the author to understand the text? I Can we situate it in a broader context? I The “horizontal” context of the society in which it was composed? I The “vertical” context of earlier and later developments? Mathematics in Ancient Egypt and Mesopotamia Introduction Evidence for Egyptian and Babylonian mathematics I Egypt: A handful of Old Egyptian Hieratic papyri, wooden tables and leather rolls, a few handfuls of Middle Egyptian Demotic papyri, less than a hundred Greek papyri, written in Egypt and forming a continuous tradition with the older material. I Mesopotamia: Thousands of clay tablets containing Sumerian and Assyrian, written in cuneiform. I All of this material is scattered around in a number of different library collections, poses many difficulties to scholars and involves many problems of interpretation. I These texts are written in dead languages. Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics The place of mathematics in Egyptian culture I Mediterranean Sea Ancient Egypt was an autocratic society Jerusalem Damietta Gaza Rosetta Dead Sea Rafah Alexandria Buto Sais Tanis Pelusium Naukratis Busiris ruled by a line of Pharaohs, who were Avaris N Wadi Natrun NW NE Bubastis W E Nile Delta SW SE Merimda Great Bitter S Lake Heliopolis 0 (km) 100 Cairo 0 (mi) 60 Giza Memphis Saqqara Helwan Sinai thought to be divine. Dahshur Timna Faiyum Lower Lake MoerisMeydum Lahun Egypt Herakleopolis Serabit al-Khadim I Gulf of Suez Nile river Bahariya Oasis Egyptian, a Semitic language, was Gulf of Aqaba Beni Hasan Hermopolis Amarna written in two forms, Hieroglyphic and Asyut Badari Eastern Desert Qau Western Desert Akhmim Red Sea Thinis Abydos Nile river Dendera Hieratic. Quseir Kharga Oasis Naqada Koptos Wadi Hammamat Thebes (Luxor and Karnak) Dakhla Oasis Tod Upper Hierakonpolis I Egypt A very small group of professional Edfu Kom Ombo Aswan First Cataract Bernike scribes could read and write. Dunqul Oasis Nabta Playa Wadi Allaqi I The only evidence we have for Abu Simbel Buhen Second Cataract Kush mathematics in ancient Egypt comes Wadi Gabgaba Nubian Desert Third Cataract from the scribal tradition. Kerma Nile river Kawa Fourth Cataract I Napata Gebel Barkal The study of mathematics was a key Fifth Cataract component of a scribe’s education. Meroe Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Our evidence for Egyptian mathematics I Ancient Egyptian mathematics is preserved in Hieratic and Demotic on a small number of papyri, wooden tablets and a leather roll. I Middle and late Egyptian mathematics is preserved on a few Demotic and Coptic papyri and many more Greek papyri, pot sherds and tablets. I This must be only a small fraction of what was once produced, so it is possible that our knowledge of Egyptian mathematics is skewed by the lack of evidence. Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics An example: The Rhind Papyrus, complete Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics An example: The Rhind Papyrus, end Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics An example: The Rhind Papyrus, detail Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Egyptian numeral system I A decimal (base-10) system I Not a place-value system. Every mark had an absolute value. I Unordered Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Egyptian numeral system Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Some examples: An inscribed inventory Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Some examples: Accounting3 3 This text uses both Hieroglyphic and Hieratic forms. Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Egyptian fraction system I 1 The Egyptians only used unit fractions (we would write n ). I They wrote a number with a mark above it, and had special 1 1 1 2 symbols for 2 , 3 , 4 and 3 . I This makes working with their fractions different from what we learned in school. I Doubling an “even-numbered” fraction is simple (ex. 1 1 1 10 + 10 = 5 ), however doubling an “odd-numbered” 1 1 1 1 fraction is not straightforward (ex. 5 + 5 = 3 + 15 )... I And, collections of unit fractions are often not unique (ex. 1 1 1 1 1 1 1 5 + 5 = 3 + 15 = 4 + 10 + 20 ). I ¯ 2 ¯ 1 ¯ 1 We often write Egyptian fractions as 3 = 3 , 2 = 2 , 3 = 3 , ¯ 1 1 4 = 4 , etc. Why not just use n ? Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian numbers Egyptian fraction system Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian computation Multiplication I The Egyptians carried out multiplication by a series of successive doublings and then additions, as well as inversions (flipping reciprocals, 2 24 ! 24¯ 2).¯ For example, to multiply 12 by 12, we find, in Rhind P. #32: . 12 2 24 n 4 48 n 8 96 sum 144 I The scribe has written out a list of the successive doubles of 12 then put a check by the ordering numbers (1, 2, 4, 8, ...) that total to 12. I The doubles that correspond to these ordering numbers are then summed (48 + 96 = 144). Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Egyptian computation Division I Division is an analogous process, using halving and unit fractions. For example, to divide 19 by 8 we find, in Rhind P. #24: . 8 n 2 16 2¯ 4 n 4¯ 2 n 8¯ 1 sum 19 I In this case, the numbers on the right side sum to 19 (16 + 2 + 1 = 19), so the ordering numbers must be added together to give the answer (2 + 4¯ + 8).¯ The answer is not stated explicitly. Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Some example problems The method of false position, 1 I , , , , A class of problems, known as h. problems ( h. means “heap” or “quantity”), reveals a method for solving problems of the form x + ax = b. For example, in Rhind P. #26, we have: I “A quantity, its 4¯ [is added] to it so that 15 results. What is x the quantity? [That is, x + 4 = 15] Calculate with 4. [The assumed, “false” value.] You shall callculate its 4¯ as 1. Total 5. Divide 15 by 5. n . 5 n 2 10 3 shall result Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Some example problems The method of false position, 2 I Multiply 3 times 4 . 3 2 6 n 4 12 12 shall result n . 12 n 4¯ 3 Total 15” I We start with an assumed value, 4, and find out what part of the final result, 15, it produces. Then we correct by this part. I Why would the text treat such trivial calculations in this kind of detail? Mathematics in Ancient Egypt and Mesopotamia Egyptian mathematics Some example problems “I met a man with seven wives. ”4 I P. Rhind #79: “There were 7 houses, in each house 7 cats, each cat caught 7 mice, each mice ate 7 bags of emmer, and each bag contained 7 heqat. How many were there altogether?” I Answer (modern notation): 7 + 72 + 73 + 74 + 75 = 19607. I Is this a real problem? What is the point of it? 4 An old English nursery rhyme: “As I was going to St. Ives: I met a man with seven wives: Each wife had seven sacks: Each sack had seven cats: Each cat had seven kits: Kits, cats, sacks, wives: How many were going to St. Ives?” Mathematics in Ancient Egypt and Mesopotamia Babylonian Mathematics The place of mathematics in Mesopotamian culture I Ancient Mesopotamian culture had two primary institutions, the King and the Temple, but wealthy merchants also played an important role in society. I Mathematics was practiced by clans of literate scribes. I They made their living working as priests, scribes and accountants—mathematics was a side product of their primary role. Mathematics in Ancient Egypt and Mesopotamia Babylonian Mathematics The place of mathematics in Mesopotamian culture I Mathematics was used for both practical purposes and to create professional distinctions.
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