History 107 Lecture 7

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History 107Lecture 7

• Middle Kingdom Egypt

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Pyramid Building and Egyptian Math

• How did Egyptians build the Great Pyramid of
  Cheops and other impressive structures?
  
• Essential tools Egyptian mathematics and
  geometry
  
• Egyptians used symbols for numbers (one, ten, one
  hundred, one thousand)
  
• Writing from right to left
• No knowledge of multiplication or division
  (everything was addition and subtraction)
  

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An example of Egyptian Math

• Compute 19 X 6 using Egyptian methods
• Today
• 19 X 6 114
• Ancient Egypt
• Double 19 (to get 38)
• Double 38 (to get 76)
• Add the results (38 76 114)

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An example of Egyptian Math

• The problem would have been organized like this
• 1 19
• \ 2 38
• \ 4 76
• 6 114 (Total)

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An example of Egyptian Math

• And written right to left like this
• 91 1
• 83 2 \
• 67 4 \
• 411 6
• But using Egyptian symbols rather than Arabic
  numerals
  

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Fractions in Egypt

• Egyptians also wrote fractions out long hand,
  since each fraction could only have a numerator
  of 1.
  
• For example, 3/4 was written as 1/2 1/4
• 31/51 would be written as
• 1/2 1/17 1/34 1/51

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Complex Calculations in Egypt

• Despite their limitations, Egyptians could
  compute the volume of a regular and truncated
  pyramid, volume of a cylinder, area of triangles,
  trapezoids, rectangles, and circles.
• Egyptians could not calculate pi as the Greeks
  did 1,000 years later, but had close
  approximations
  

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Volume of a Pyramid

• Formula for the volume of a pyramid is
• vol. 1/3 X length X width X height
• Thus for the Great Pyramid of Cheops
• vol. 1/3 X 756 ft. X 756 ft. X 481 ft.
• vol. 91,636,272 ft. (interior space)
• Page 40 of the text book suggests a volume of
  about 85 million cubic feet which seems to be an
  error

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Small Group Exercise Egyptian Math Story Problem

• Imagine that you work at the granary in ancient
  Egypt and you need to construct holding
  containers for corn that will be used throughout
  the year
• The formula for the area of a circle today is pi
  r2. To compute the volume of a cylinder, you
  simply multiply the height of the cylinder by the
  result, thus
• vol. height pi r2
• But in ancient Egypt (before pi had been
  discovered) Egyptians used a different formula to
  compute volume
  
• vol. height (8/9 diameter)2
• Considering this, about how much grain could you
  store in a cylinder of height 6 and diameter 9?
  
  
• Solve this problem by hand using both Egyptian
  and modern methods

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Small Group Exercise Solutions

• The volume of a cylinder 6 high with a diameter
  of 9 can be
  
• computed as follows
• Ancient Egypt Modern World
• vol. height (8/9 diameter)2 vol. height
  pi radius2
  
• vol. 6 (8/9 9)2 vol. 6 3.1417 4.52
• vol. 6 (8)2 vol. 6 3.1417 20.25
• vol. 6 64 vol. 6 63.619425
• vol. 384 vol. 381.71655

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Math Summary Our distance from ancient Egypt

• Doing a few computations helps to clarify how
  distant we are from ancient Egyptiansnot only in
  how we live, but in how we think.
  
• And we want to know more How was grain
  preserved? Who controlled it? How much did a
  family need to survive? Was it taxed?
  

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Chronology of Middle Kingdom Egypt (c. 2055-1650
B.C.E.)

• During the Middle Kingdom, Memphis (near Cairo)
  was neglected and the capital moved to Thebes in
  Upper Egypt
  
• Egyptian middle class thrives
• Pyramid building replaced by public works
• Valley of Kings in Thebes is religious center
• Popular literature develops, including Egyptian
  Book of the Dead (afterlife not just for the
  pharaohs)
  
• Middle Kingdom comes to an end when western Asian
  invaders called the Hyksos defeat Egypt
  

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Chronology of New Kingdom Egypt (c. 1550-1075
B.C.E.)

• South Egypt revolts against the Hyksos by 1550
  B.C. the Hyksos have been defeated
  
• Ahmose, a great new pharaoh, uses horse and
  chariot technology to expand Egypts territory
  
• After securing Palestine and Syria, Egypt becomes
  a great empire at the height of its power
  
• Queen Hatshepsut (r. 1479 B.C.) is a
  controversial female ruler for her step son
  Thutmosis III
  
• Thutmosis III blots out his mother-in-laws name
  during a power struggle and expands the empire
  (Israel first mentioned as a rival)
  
• New Kingdom Egypt pharaohs give thanks to
• Amon re for victory