Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt 01
Week 07 Lecture 02
Introduction to Egypt
P yramids
Mathematics
This slideshow was last updated 23 January, 2016
1/24/2016
1
Montclair State University Department of Anthropology
Anth 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Week 07 Ancient Egypt
The learning objectives for week 07 are:
– to appreciate the nonwestern origins of writing and
the alphabet
– to understand what scholars know about the ancient
pyramids
– to know a few basic facts about life in ancient Egypt
– to appreciate some of the major medical advances
made in ancient Egypt
– to understand the likely Egyptian influences on the
Judeo-Christian tradition
2
Montclair State University Department of Anthropology
Anth 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Week 07 Ancient Egypt
Terms you should know for week 07 are:
–
–
–
–
–
Hieroglyph
C artouche
Rosetta Stone
Kufu pyramid
Monotheism (origins)
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Montclair State University Department of Anthropology
Anth 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Week 07 Lecture 02 Ancient Egypt and the
P yramids
Sources:
Gillings, Richard J. 1972. Mathematics in the Time of the Pharaohs. New York: Dover Publications, Inc.
Hurry, Jamieson B. 1926. Imhotep: The Vizier and Physician of King Zoser. Oxford: Oxford University Press.
Joseph, George Gheverghese. 1991. The Crest of the Peacock: Non-European Roots of Mathematics. London: I.
B. Tauris and Co. Ltd. Page 238.
Livio, Mario. 2002. The Golden Ratio: The Story of Phi, The World’s Most Astonishing Number.New York:
Broadway Books. Page 56 (the diagram and some of the calculations).
Robbins, Gay and Charles Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. New
York: Dover Books [orig. The British Museum)
Smith, W. Stevenson. 1958. The Art and Architecture of Ancient Egypt. New Haven: Yale University Press.
Trigger, B. G., B. J. Kemp, D. O’Conner and A. B. Lloyd. 1983. Ancient Egypt: A Social History. Cambridge:
Cambridge University Press.
Zaslavsky, Claudia. 1990 [1973]. Africa Counts: Number and Pattern in African Culture. Brooklyn, New York:
4
Lawrence Hill Books. Page 22
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Modern Egypt
1/24/2016
5
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt
• Egypt the oldest – or 2nd oldest after
Sumer – organized political system on
earth
• Egypt and Eastern Mediterranean the
sources of much of the knowledge often
attributed to ancient Greece
•1/24/2016
But Egypt much older
6
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Historical P eriods
P haraonic
3000 BC to 341
Greek (Alexander and
Ptolemies)
Roman Rule
332 BC to 30 BC
Arab Conquest
640 AD to 1517
Turkish (Ottoman) Rule
1517 to 1882
British Occupation
1882 to 1952
Independent Egypt
1952 to present
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BC
30 BC to 638 AD
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt
• Egyptian history so long it’s easy to get
confused
• Longest single historical sequence of any
modern or recent culture
• Our main interest: P haraonic Egypt of Old
and Middle Kingdoms
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
•
•
•
•
Ancient Egypt
Predynastic Egypt: 4,500 to 3,000 BC
Early dynastic: dynasties 1 – 3
Includes “Menes,” possibly mythical first
pharaoh
Dynasties 1 – 3 run from 3,100 to 2,613 BC
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
•
•
•
•
•
Ancient Egypt
Our first main interest – Old Kingdom
Dynasties 4 – 8
2,613 to 2,160 BC
Especially the 4th Dynasty
Great pyramids of Giza
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
•
•
•
•
•
Ancient Egypt
Next major interest: Middle Kingdom
Dynasties 9 – 11
2025 to 1,700 BC
1,700 BC time of historical Abraham of
Bible
Many mathematical and scientific
achievements
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt
• Important to absorb –
• by time of Abraham Egypt had been a
flourishing civilization for 1,300 years or
more
• Egyptian philosophy may have influenced
Old Testament ideas – examples in week
07 lecture 03
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Some Major C ontributions
1. The alphabet
2. Irrigation
3. Calendar and star clock
4. P apyrus paper
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Major C ontributions
5. Mathematics
–
–
–
–
–
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Addition and subtraction
Multiplication and division
Fractions
Rectangular Geometry
Solid Geometry
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Major C ontributions
6. Architecture
P yramids
• Many Egyptian contributions dovetail with
those of Mesopotamia and ancient
C anaan
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Modern Egypt
1/24/2016
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt
“Egypt is the
gift of the Nile”
– Greek
historian
Herodotus
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt
1/24/2016
18
Montclair State University
Department of Anthropology
ANTH 140: Non Western Contributions
to the Western World
Dr. Richard W. Franke
Ancient Egypt:
The P yramids
1/24/2016
19
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
1. About 80 pyramids in ancient Egypt
2. Largest are 3 pyramids of Giza, built
about 2,600 BC
3. Giza Plateau a limestone hill on west
bank of Nile, overlooking suburbs of
C airo
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
4. P yramids were burial sites for pharaohs
5. P yramids may have been connected with
mummification
6. P yramid partly to help passage of soul to
afterlife where Egyptians believed it
needed a body – more on ancient
Egyptian religion later
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
7. P yramid stocked with supplies to sustain
the soul on journey to the afterlife
– Also possessions for the afterlife
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Giza P yramid Facts
1/24/2016
23
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Giza
P yramid Facts
The three Giza
pyramids are Khufu
(Cheops)
Khafra (Chefren) –
also the Sphinx
Menkaure
(Mycerinus)
(Greek versions in
parenths)
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
8. Scholars do not know how the pyramids
were designed or built.
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25
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt:
P yramid
Facts
9. One
common
theory is that
earthen
ramps were
used
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
1/24/2016
Ancient
Egypt:
P yramid
Facts
10. With
Lebanese
cedar
rollers or
sleds to
push the
large
stones into27
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
11. More difficult to explain is how they
calculated and built the great Khufu
pyramid at a consistent angle of inclination
of 51º 50’ 35”.
12. More about the mathematics of this
strange angle later.
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid
Facts
13. P erhaps they used
a plumb line – known
to Egyptian
carpenters – but we
can’t figure out how
they measured the
angle itself.
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: P yramid Facts
14. Equally difficult to understand is how
they set the giant building with the four
sides almost perfectly facing north, south,
east and west – an engineering feat
difficult event today.
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu
P yramid Facts
15. P ossibly they did it
with their knowledge
of star clocks
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31
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Facts
16. The Khufu pyramid is known as the
Great P yramid of Giza
17. Also the “Cheops” pyramid using the
Greek pronunciation
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32
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu
P yramid Facts
18. Strange
mathematical
properties of this
pyramid have
fascinated scholars
for centuries
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33
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Facts
18a. Strange mathematical properties of this
pyramid have fascinated scholars for
centuries
– So much so that critics sneer at “pyram-idiots”
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34
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
1/24/2016
Ancient Egypt:
Khufu P yramid
Facts
19. But there can
be little doubt
the pyramid
displays
knowledge of
important
mathematical
functions
35
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Facts
20. The height (originally) is 481 ft = 146.6
meters
21. The length of each base = 756 ft
(precise to within a couple of inches on
each side)
22. The base area = 13 acres or 7 city
blocks
23. Is the most massive building ever
1/24/2016
erected with 2x the volume and 30x the
36
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt:
Khufu P yramid
Facts
24. Stones weigh
avg of 2.5 tons
each – largest
weigh 70 tons
25. Was world’s
tallest building for
4,000 years
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
26. The perimeter of the base (921.6
meters) divided by 2x the height (921.6 /
[2*146.6 or 293.2] = 3.1432, the most
exact known value of π pi in the ancient
world
1/24/2016
38
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient
Egypt:
Khufu
P yramid
Facts
27. As noted
earlier…
1/24/2016
39
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt:
Khufu P yramid
Mysteries
28.The angle of
inclination of the
face to the base
is 51º 50’ 35”
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40
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Facts
29. This angle corresponds almost exactly
to the requirements of a complex
mathematical problem involving the
construction of nine triangles inside a
circle.
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41
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
30. In ancient Indian mathematics, this
construction is called a “sriyantra,” and
has important mystical as well as
mathematical qualities.
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42
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
31. A sriyanta…
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43
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
32. On the Great P yramid this angle
produces a slope height of 186.4 meters…
1/24/2016
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
33. One-half of any side of the square base
= 11 5.2 meters (= the secant of the angle,
an important function in trigonometry)
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45
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
34. Dividing the slope length by the secant
gives the number 1.61 8 = (½ + √5), known
in ancient math as the “golden ratio,” also
known as ϕ phi
35. This ratio figures in art and architecture,
and is one of the main principles of the
proportions of the P arthenon in Athens
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46
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
36. This function ϕ has strange and
important mathematical properties.
37. Its square is equal to itself plus 1 while
its reciprocal is itself minus one.
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47
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
38. It also displays a feature known today as
the Fibonacci sequence
39. In a string of numbers where each
equals the sum of the previous two, the
longer the sequence goes, the closer to ϕ
the ratio of the last two numbers becomes.
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48
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
40. The Fibonacci sequence has important
applications in genetics and in some
investment strategies.
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49
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid
Mysteries
41. The “Golden Ratio”
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h=481.4 feet (186.4
meters)
2a=755.79 ft.
a=377.9
s=612.01 (see next
slide)
s/a=1.619502;
very close to
Ф=1.61803
C alculations in metric
numbers give an
even closer
50
correspondence.
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid
Mysteries
42. The “Golden Ratio”
1/24/2016
s is found using the
P ythagorean
formula:
c2 = a2 + b2, or, in this
case;
s2=a2 + h2
s2 = (377.9)2 +
(481.4)2 =
142,808.4
+231,745.96=
374,554.36
√374,554.36=612.01
51
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Khufu P yramid Mysteries
43. What reason is there to believe these
values were chosen intentionally rather
than by chance?
44. The creation of the golden ratio requires
that the angle of incline of the face of the
pyramid must be 51.5°. In fact, it is 51° 50’
35”.
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52
Sources: Livio, Mario. 2002. The Golden Ratio: The Story of Phi, The World’s Most Astonishing
Number.New York: Broadway Books. Page 56 (the diagram and some of the calculations);
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
45. The mathematical mysteries in the
Khufu pyramid indicate Egyptian
mathematical knowledge by 2,500 BC , at
the time of the Old Kingdom 4th Dynasty
46. From 2,000 BC – the Middle Kingdom –
documents exist showing additional
developments altho many could have
existed already in the Old Kingdom
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53
Ancient Egypt: Mathematical Achievements
47. A simple and easy to learn set of math
symbols – except for the higher numbers
Montclair State University
Department of Anthropology
ANTH 140: Non Western
Contributions to
the Western World
Dr. Richard W. Franke
1/24/2016
54
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
1/24/2016
55
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
47. See for example, Claudia Zaslavsky. 1990. Africa Counts:
Number and Pattern in African Culture. Brooklyn, New York:
Lawrence Hill Books. P age 22, where they are written right to left.
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56
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
48. Multiplication
– Requires only knowledge of addition and
– Knowledge of the times 2 table
– No other tables need be memorized
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57
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
49. Used doubling or halving (times 2
tables)
– Example: multiply 8 times 7
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58
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
50. The problem
on pages 20
and 22 of
Zaslavsky (see
slides 4 or 56
for full
reference)
1/24/2016
59
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Source: Joseph, George Gheverghese. 1991. The Crest of the Peacock: NonEuropean Roots of Mathematics. London: I. B. Tauris and Co. Ltd. Page
238.
1/24/2016
Ancient Egypt:
Mathematical
Achievement
s
51. A larger
problem with
explanation
by a modern
mathematician
60
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
51a. Ironically, the Egyptian method was
so convenient to learn and to use that it
helped perpetuate the Roman numerals
system far into the European
Renaissance – even though the IndianArabic numbers were far more universal
and flexible.
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61
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
51b. Here is an
example of
multiplication using
Roman numerals
showing how much
easier they are with
the Egyptian method.
1/24/2016
Source: Al-Khalili, Jim. 2010. The House of Wisdom: How Arabic
Science Saved Ancient Knowledge and Gave Us the
Renaissance. New York: Penguin Books. Page 260.
62
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
52. The Ahmes P apyrus is named after the
scribe who composed it in about 1650 BC
from a work 3 centuries older.
53. Sometimes called the “Rhind
Mathematical P apyrus” after the British
collector who donated it to the British
Musuem in 1858 AD.
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63
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
1/24/2016
Ancient Egypt: The
Ahmes P apyrus
54. The Ahmes
P apyrus contains the
solutions to 87
mathematical
problems along with
a fractions
decomposition table
for odd values of n
from 3 to 101 .
64
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
55. Egyptian mathematicians may have
been the first to develop a systematic
understanding of fractions.
56. They used a system known as “unit
fractions” – meaning that all fractions were
expressed as combinations of 1/n.
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65
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
57. Writing Fractions in Ancient Egyptian
Hieroglyphs
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66
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
58. The Ahmes P apyrus decomposition
table contains unit fraction decompositions
for about 28,000 fractions
59. Modern mathematicians have analyzed
the table in detail.
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67
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
60. The table gives the smallest and most
efficient decomposition for each fraction –
equal in every case to the preferred
solution using modern math and computer
algorithms
61. The table does not contain a single
mathematical error
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68
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
62. Modern mathematicians consider the
table a major event in the history of math –
similar to the discovery of logarithms
63. The next slide shows a sample of the
2/n part of the table…
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69
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The
Ahmes P apyrus
64. Some of the 2/n
decompositions
among the 28,000 in
the Ahmes P apyrus
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70
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
65. Why use unit fractions? It could be a
question of pure intellectual fascination
with numbers – something thought earlier
to have begun only with the Greeks 1,500
years later.
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71
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
66. Among the 87 problems and solutions is
the famous “9 loaves” problem.
67. Problem 6: How do you divide 9 loaves
of bread among 10 men?
68. One solution: give 9 men 9/10 of a loaf
each and the 10th man the 1/10
remainders.
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72
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
69. Simple solution on previous slide is
highly unfair to the 10th man: his bread will
get stale more quickly.
70. Egyptian solution: look up n/10 on the
decomposition table for n=9.
71. This gives 9/10 = 2/3 + 1/5 + 1/30
72. Using Egyptian division principles, we
learn that…
1/24/2016
73
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
73. 7 men each receive 3 pieces of bread of
sizes 2/3, 1/5 and 1/30 loaves; 3 men
each receive 4 pieces consisting of 2 1/3
pieces, a single 1/5 piece, and a single
1/30 piece.
74. This solution provides the most just
division of the bread both numerically and
in terms of preserving it.
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74
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Berlin P apyrus
75. Earliest known example of non linear
simultaneous equations –
76. The Berlin P apyrus – not quite as
extensive as the Ahmes P apyrus contains
the following problem…
1/24/2016
75
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Berlin P apyrus
77. It is said to thee that the area of a square
of 100 square cubits is equal to that of two
smaller squares. The side of one is ½ + ¼
of the other. Le me know the sides of the
two unknown squares.
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76
Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: The Ahmes P apyrus
78. Egyptian mathematicians solved this
problem through a procedure known as
false position…you can read the details in:
Source: Joseph, George Gheverghese. 1991. The
Crest of the Peacock: Non-European Roots of
Mathematics. London. I. B. Tauris & Co. Ltd. P age 78.
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Ancient Egypt: Mathematical Achievements
79. To see its significance, we solve the
problem using modern algebra
• Equation 1: x2 + y2 = 100
• Equation 2: x = 3y
4
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
80. Now substitute the value of x in equation
2 in terms of y back into equation one for
x:
• Equation 3: 3y 2 + y2 = 100
4
• Equation 5: 9y2 + y2 = 100
16
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
81. Multiply through by 16 to get
• Equation 6: 9y2 + 16y2 = 1600
• Equation 7:
25y2 = 1600
• Equation 8:
y2 = 1600/25 = 64
• Equation 9:
y2 = 64
• Equation 10:
y = 8
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
82. Now substitute the solved value for y
back into either Equation 1 or Equation 2.
For Equation 2:
• Equation 2 was: x = 3y
4
• Equation 11
x = 3*8 = 24 = 6
4
4
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
83. Test by putting both values into both
equations:
• Equation 12: 62 + 82 = 100
36 + 64 = 100
• Equation 13: 6 = ½ + ¼ of 8
• Equation 14: 6 = 4 + 2 = 6
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Montclair State University Department of Anthropology
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Ancient Egypt: Mathematical Achievements
84. Squaring the Circle and Calculating Pi
(π):
Another Mathematical Achievement of
Ancient Egypt
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
85. Pi is a mathematical constant for which the
Greek letter π was first used by Swiss
mathematician Leonhard Euler in 1727. Pi has
been important in the development of
mathematics. The value of pi was thought by
many to be the key to squaring the circle, one of
mathematics’ enduring quests. It is also essential
for computing the circumference and the area of
a circle and the volumes of cylinders and other
round shapes.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
86. The earliest known attempt to calculate
the value of pi comes from ancient Sumer
(Babylonia) around 2000 BC . They came
up with 3⅛ or 3.125. The logic behind their
calculation is not known.
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Montclair State University Department of Anthropology
ANTH 140: Non Western Contributions to the Western World
Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
87. By the Egyptian Middle Kingdom (2000–
1700 BC), the Ahmes P apyrus, indicates
that Egyptian mathematicians had
developed an approximation of pi at
3.1605. The Egyptian and Sumerian
values compare with the modern value of
3.14159… now available on some web
sites to more than one trillion places.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
88. The first correct calculation of pi to the
second decimal place was achieved by the
Greek scientist and math wizard
Archimedes, in about 250 BC . Thus the
Egyptian value was the most accurate in
the world for about 1,700 years. The Old
Testament (I Kings 7:23) gives the far less
accurate value of 3.0.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
89. The Egyptian value is derived from a
discovery made at least by 2000 BC that
the area of a circle is equal to the square
of 8/9 of the diameter or (8d/9)2 – a
simpler and fairly close approximation of
what one gets using the modern formula.
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Montclair State University Department of Anthropology
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Ancient Egypt: Mathematical
Achievements
90. C onsider a circle with a diameter of 9
units. Using the Egyptian formula:
(8x9)2= 722 = 5,184 = 64
(9)2
81
81
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
91. Using the modern formula of πr2, we
have:
3.1415x4.52 (the radius is ½ the diameter)
= 3.14x20.25 = 63.62.
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Montclair State University Department of Anthropology
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Ancient Egypt: Mathematical Achievements
92. We can derive the Egyptian value of pi
using simple algebra.
Recall that in modern algebra the area of a circle =πr2
and that r, the radius equal ½ the diameter = d
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
92a. We can therefore write in place of “r” d/2 and
in place of r2 we can thus write (d/2)2
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Montclair State University Department of Anthropology
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Ancient Egypt: Mathematical Achievements
92b. The modern formula for the area of a
circle can thus be written as
Area = π (d/2)2
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
92c. We can now set the modern and
ancient Egyptian formulas equal to each
other and solve for pi:
π (d/2)2 = (8d/9)2
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Montclair State University Department of Anthropology
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Ancient Egypt: Mathematical Achievements
92d.
π = (8d/9)2
(d/2)2
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
92e. Now multiply out the components:
π = (8d/9)2 = 64d2 / 81
(d/2)2
d2 /4
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
92f. Invert and cross-multiply:
π = 64d2 X
4
81
d2
The d2s cancel each other out
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
92g.
pi = 256/81 = 3.1605
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
93. How did the Egyptians actually derive
pi? Problem Number 48 of the Ahmes
(Rhind) Mathematical P apyrus contains a
drawing of an octagon inside a square.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
94. This drawing and
associated notes by the
Egyptian scribe indicate
that s/he was aware
that a circle with
circumference of 9 is
very close to a square
with a side of 8, as we
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saw in the example
earlier.
95. The scribe first draws a
square inside which s/he
then draws a circle and
an octagon.
100
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Ancient Egypt:
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Achievements
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96. The scribe notes
from naked eye
observation that the
portions of the
octagon outside the
circle are
approximately equal
to those inside the
circle.
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Ancient Egypt:
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Achievements
97. Thus it becomes
possible to inscribe
an octagon inside a
square as a stand-in
for the circle.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
98. We can see that
each corner area
equals 4 ½ squares (3
full squares plus 3
times ½ squares).
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
99. So, now the
scribe can move
the squares in
such a way as to
allow for a
straightforward
calculation of the
area:
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Dr. Richard W. Franke
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
100. Nine squares
are moved to the
vertical and
horizontal sides –
4 ½ for each of
the corners of the
previous octagon.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt:
Mathematical
Achievements
101 . The nine squares
produced by the
diameter of the circle
thus transform
geometrically into an
8 by 8 square, for
which the scribe can
easily calculate the
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area.
107
Montclair State University Department of Anthropology
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Ancient Egypt:
Mathematical
Achievements
102. The error lies in
the fact that one of
the squares is used
twice. This error is
small, but was not
corrected until
Archimedes 1,700
years later.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
103. Ahmes – the author of the papyrus, but
probably not the actual mathematician
who discovered these relationships –
recorded the first known attempt to square
the circle and the first formal mathematical
presentation of the mechanisms to find the
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value of pi.
Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
104. The Egyptian discovery of the close
relationship between a circle of diameter 9 and a
square of sides 8, and the connection between
this discovery and their understanding of pi, may
have remained unknown until 1972 when
Australian mathematician Richard J. Gillings
reproduced it, based on the Ahmes (Rhind)
papyrus.
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Ancient Egypt
105. Problems 41– 43
of the Ahmes
papyrus show the
use of the Egyptian
understanding of pi
to calculate nearly
precise volumes of
cylinders –
problems 41– 43, 48
and 50
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical
Achievements
106. Problems 41–43 of the Ahmes papyrus
suggest that the Egyptians might have used the
“Pythagorean” formula to come up with the 8/9
equivalent for pi.
But many scholars believe the Egyptians of the
Old Kingdom did not have knowledge of the
Pythagorean formula.
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Montclair State University Department of Anthropology
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Ancient Egypt
107. Interested students may consult
Robins, Gay and Shute, Charles. 1987. The Rhind
Mathematical Papyrus: An Ancient Egyptian Text.
London: The British Museum and New York: Dover
Books. Page 45.
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Ancient Egypt: Mathematical Achievements
Geometric Progressions
108. . Based on their understanding of the
sequence of the powers of 2, on which
their mathematics is essentially based, the
Egyptians were apparently the first culture
to consciously study geometric
progressions.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
Geometric Progressions
109. They experimented with progressions
using 3, 5, and 7. They were especially
interested in the series 7x7; 7x7x7;
7x7x7x7, etc.
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
Geometric Progressions
110. Problem 79 of the Ahmes P apyrus contains a problem
in computing and summing the geometric progression of
75, with real elements:
– Houses
7
– C ats
49
– Mice
343
– Spelt (a unit of grain)
2,401
– Hekat (a unit of weight)
16,807
– Sum ofthe progression……..…19,607
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11 1. This
may be the
origin of
the famous
poeticriddle,
attributed
later to the
small
English
market
town of St.
Ives, near
the city of
Cambridge
:
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
Geometric Progressions
As I was going to St Ives
Imet a man with seven
wives.
Each wife had seven
sacks,
Each sack had seven
cats,
Each cat had seven kits;
Kits, cats, sacks and
wives –
How
many were going to
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St Ives?
118
Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Geometric Progressions
112. Here is the Egyptian solution:
–
–
–
–
–
Wives
7
Sacks
49
C ats
343
Kits
2,401
Sum of the progression……… 2,800
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Montclair State University Department of Anthropology
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Dr. Richard W. Franke
Ancient Egypt: Mathematical Achievements
Geometric Progressions
11 1. There are at least two other solutions to
this puzzle: neither of them requires
knowledge of mathematics.
C an you figure them out?
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Ancient Egypt – Week 07
End of Lecture 02
Introduction
Pyramids
Mathematics
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