Mathematics in Ancient Egypt - Vrije Universiteit...
Transcript of Mathematics in Ancient Egypt - Vrije Universiteit...
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Mathematics in Ancient Egypt
Philippe Cara
Department of Mathematics
http://www.vub.ac.be/DWIS
BEST Summerschool “Pyramids in the Cosmos”
VUB, 7th September 2001
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1
The use of Mathematics in AncientEgypt
• Partitioning of fertile grounds
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1
The use of Mathematics in AncientEgypt
• Partitioning of fertile grounds
• Construction of pyramids
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1
The use of Mathematics in AncientEgypt
• Partitioning of fertile grounds
• Construction of pyramids
• Administration
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1
The use of Mathematics in AncientEgypt
• Partitioning of fertile grounds
• Construction of pyramids
• Administration
• Calendar. . .
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Sources
• Wrote mainly on papyrus
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Sources
• Wrote mainly on papyrus
• Not suited for long conservation
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Sources
• Wrote mainly on papyrus
• Not suited for long conservation
• Hieroglyphics and wall-paintings
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Sources
• Wrote mainly on papyrus
• Not suited for long conservation
• Hieroglyphics and wall-paintings
• Babylonians had clay-tablets
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Sources
• Wrote mainly on papyrus
• Not suited for long conservation
• Hieroglyphics and wall-paintings
• Babylonians had clay-tablets
• Mainly “real life” problems, no general methods!
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Rhind papyrus
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• 1650 BC
• Henry Rhind (1833–1863)
• 87 problems with answer
• Main source of information
• Tables with fractions
• Ahmes
• Leather Roll
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Back of Rhind papyrus
2/5 = 1/3 + 1/15
2/7 = 1/4 + 1/28
2/9 = 1/6 + 1/18
2/11 = 1/6 + 1/66
2/13 = 1/8 + 1/52 + 1/104
2/15 = 1/10 + 1/30...
2/101 = 1/101 + 1/202 + 1/303 + 1/606
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Theorem
Every fraction can be written as a sum of unitfractions.
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Theorem
Every fraction can be written as a sum of unitfractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
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Theorem
Every fraction can be written as a sum of unitfractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
p
q− 1n
=np− qnq
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Theorem
Every fraction can be written as a sum of unitfractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
p
q− 1n
=np− qnq
Suppose np− q > p then np− p > q or (n− 1)p > q
1n<
1n− 1
6p
q
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Remark
There are many ways to write a fraction as sum of
unit fractions.
1n
=1
n+ 1+
1n(n+ 1)
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Multiplication 1
7× 22 =?
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Multiplication 1
7× 22 =?
1 7
✔ 2 14
✔ 4 28
8 56
✔ 16 112
22 154
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Division
154÷ 7 =?
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Division
154÷ 7 =?
1 7
2 14 ✔
4 28 ✔
8 56
16 112 ✔
22 154
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Multiplication 2
5 + 7/8× 12 + 2/3 =?
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Multiplication 2
5 + 7/8× 12 + 2/3 =?
✔ 1 12+2/3
2 25+1/3
✔ 4 50+2/3
✔ 1/2 6+1/3
✔ 1/4 3+1/6
✔ 1/8 1+1/2+1/12
5+7/8 99+1/2+1/4
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Writing of numbers
• Decimal
• But not positional
1 10 100 1000 10000 100000 1000000
� � � � µ � �
������� � �
� � �� � = 275
� µ µ µ µ µ ����� � � = 152023
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Writing of numbers
• Decimal
• But not positional
1 10 100 1000 10000 100000 1000000
� � � � µ � �
������� � �
� � �� � = 275
� µ µ µ µ µ ����� � � = 152023No zero symbol needed!!
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Moscow papyrus
• 1850 BC
• Golenischev
• 5 metres long, 8 cm high
• 25 problems
• Bad handwriting
• Volume of truncated pyramid
• Surface of half a sphere
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Other papyri
• Reisner papyri (1880 BC)
• Kahun papyri (1800 BC)
• Rollin papyrus (1350 BC)
• Harris papyrus (1167 BC)
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Mathematics used for building pyramids=
GEOMETRY
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Weights and measures
• 1 cubit = 52.3 cm
• Long distances: ropes with knots
• Shorter distance: ruler
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Egyptian ruler
The cubit of King Amenhotep I (1559 – 1539 BC)
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Other units for distances
• 1 palm = 1/7 of a cubit
• 1 finger = 1/4 of a palm
• 1 hayt = 1 khet = 100 cubits
• 1 remen = half the length of the diagonal of a
square with side one cubit. That is√
22 cubit.
Useful when measuring land areas.
• 1 double remen =√
2 cubit.
• 1 aura = 1 setat = area of square with side 100
cubits, hence 10000 square cubits.
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Weights or volumes
• 1 hekat = 1/30 of a cubic cubit of grain
• 1 hinu = 1/10 of a hekat
• 1 ro = 1/320 of a hekat
• Horus eye to write fractions of a hekat
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Beer and Pesu
• 1 des = approx. half a liter
• The pesu is a unit for measuring the strength of
beer, bread or cakes.
• If one hekat of grain was used to make 5 des of
beer, it was said to have a pesu of 5.
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Beer and Pesu
• 1 des = approx. half a liter
• The pesu is a unit for measuring the strength of
beer, bread or cakes.
• If one hekat of grain was used to make 5 des of
beer, it was said to have a pesu of 5.
• The less the pesu, the stronger the beer!!!
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Rhind problem 76
If you want to trade 1000 des of beer of pesu10 for beer of pesu 20, how many des do youget?
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Sekhed of a pyramid
is the inclination of any one of the four triangular
faces to the horizontal plane of its base.
a/2
a
α
h h
sekhed =a
2hIn fact the sekhed is the cotangent of the slope α.
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Rhind problem 57
The sekhed of a pyramid is 5 palms and 1finger and the side is 140 cubit. What is theheight?
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Sekheds of some well-known pyramids
Name Sekhed
Chephren, Ouserkaf 3/4 cubit
Neferirka-Re, Teti, Pepi 21 fingers
Cheops, Snofru 11/14 cubit
Neouser-Re 22 fingers
Sesostris 6/7 cubit
Amenemhat III 9/14 cubit
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Pythagoras’ theorem
• The sum of the squares of the right anglesides of a rectangular triangle is the square ofthe remaining side.
a2 + b2 = c2a
bc
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Pythagoras’ theorem
• The sum of the squares of the right anglesides of a rectangular triangle is the square ofthe remaining side.
a2 + b2 = c2a
bc
• If a triangle fulfills the above then it is rect-angular.
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Construction of right angles 1
p
p
p
p
k0
k0k0
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Property of a rectangle
• The diagonals of a rectangle meet eachotherhalfway.
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Property of a rectangle
• The diagonals of a rectangle meet eachotherhalfway.
• If a quadrilateral fulfills the above then it is arectangle.
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Construction of right angles 2
m
m
mm
a
bc
d
a
b
p pa
d
b pa
a d a
b pa
d
R R
R R
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Area of a triangle?
Rhind problem 49: Multiply half a side with the
other side.
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Area of a triangle?
Rhind problem 49: Multiply half a side with the
other side.� � � � �� � � � �� � � � �� � � � �
� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �� � � �� � � �� � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � �� � � �� � � �� � � �� � � �
� � � �� � � �� � � �� � � �� � � �
1/2
b
1/2
c
a
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31
Area of a quadrilateral?
b
c
d
a
a+ c
2· b+ d
2
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32
Rhind problem 48
A = (d− d9)2
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33
The area of a circle 1
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33
The area of a circle 1
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33
The area of a circle 1
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33
The area of a circle 1
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34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
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34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
πd2
4≈ 64
81d2
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34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
πd2
4≈ 64
81d2
π ≈ 46481
=25681
= 3.1604938 . . .
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34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
πd2
4≈ 64
81d2
π ≈ 46481
=25681
= 3.1604938 . . .
• Babylonians: π ≈ 3.125
• Bible: π ≈ 3
![Page 60: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds](https://reader030.fdocuments.net/reader030/viewer/2022041300/5e0eb5f0391b6c10a718b2e9/html5/thumbnails/60.jpg)
35
Area of a basket
Moscow problem 10: Find the area of a basket
with given “mouth” d.
d
They use the formula
A = 26481d2
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35
Area of a basket
Moscow problem 10: Find the area of a basket
with given “mouth” d.
d
They use the formula
A = 26481d2
This is
A = 225681r2
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35
Area of a basket
Moscow problem 10: Find the area of a basket
with given “mouth” d.
d
They use the formula
A = 26481d2
This is
A = 225681r2
and since π ≈ 25681 we find
A = 2πr2
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36
Volume of a truncated pyramid
a
b
h
V =13h(a2 + ab+ b2)
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37
How did they do it?
h’
a a
b
bh
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38
Mysticism. . .
• The ratio of twice the side to the height of the
Great pyramid is π.
• In the dimensions of the Great pyramid are en-
coded: the radius of th earth, the density of earth,
the distance between earth and sun, . . .
• All important dates of human history
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39
![Page 67: Mathematics in Ancient Egypt - Vrije Universiteit Brusselhomepages.vub.ac.be/~pcara/Hobby/egypres.pdf · The use of Mathematics in Ancient 1 Egypt Partitioning of fertile grounds](https://reader030.fdocuments.net/reader030/viewer/2022041300/5e0eb5f0391b6c10a718b2e9/html5/thumbnails/67.jpg)
40
Charles Piazzi Smyth
• 1819–1900
• Astronomer
• Professor at university of Edinburgh
• Fellow of the Royal Society
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41
Verification
William Matthew Flinders Petrie (1853–1942)
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42
Other pyramidologists
• Charles Lagrange
• David Davison
• Georges Barbarin
• Robert Bauval
• . . .