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
The use of Mathematics in Ancient
Egypt
• Partitioning of fertile grounds
1
The use of Mathematics in Ancient
Egypt
• Partitioning of fertile grounds
• Construction of pyramids
1
The use of Mathematics in Ancient
Egypt
• Partitioning of fertile grounds
• Construction of pyramids
• Administration
1
The use of Mathematics in Ancient
Egypt
• Partitioning of fertile grounds
• Construction of pyramids
• Administration
• Calendar. . .
1
Sources
• Wrote mainly on papyrus
2
Sources
• Wrote mainly on papyrus
• Not suited for long conservation
2
Sources
• Wrote mainly on papyrus
• Not suited for long conservation
• Hieroglyphics and wall-paintings
2
Sources
• Wrote mainly on papyrus
• Not suited for long conservation
• Hieroglyphics and wall-paintings
• Babylonians had clay-tablets
2
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!
2
Rhind papyrus
3
4
• 1650 BC
• Henry Rhind (1833–1863)
• 87 problems with answer
• Main source of information
• Tables with fractions
• Ahmes
• Leather Roll
5
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
6
Theorem
Every fraction can be written as a sum of unit
fractions.
7
Theorem
Every fraction can be written as a sum of unit
fractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
7
Theorem
Every fraction can be written as a sum of unit
fractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
p 1 np − q
− =
q n
nq
7
Theorem
Every fraction can be written as a sum of unit
fractions.
Given p/q with p < q, we subtract the largest unit
fraction 1/n. Repeat till a unit fraction is left. . .
p 1 np − q
− =
q n
nq
Suppose np − q > p then np − p > q or (n − 1)p > q
1
1
p
<
6
n n−1 q
7
Remark
There are many ways to write a fraction as sum of
unit fractions.
1
1
1
=
+
n n + 1 n(n + 1)
8
Multiplication 1
7 × 22 =?
9
Multiplication 1
7 × 22 =?
1
7
✔ 2 14
✔ 4 28
8 56
✔ 16 112
22
154
9
Division
154 ÷ 7 =?
10
Division
154 ÷ 7 =?
1
7
2 14 ✔
4 28 ✔
8 56
16 112 ✔
22
154
10
Multiplication 2
5 + 7/8 × 12 + 2/3 =?
11
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
11
12
Writing of numbers
• Decimal
• But not positional
1

10
100
‰

‰ ‰
µµµµµ
ŒŒ
  





1000
10000
100000
1000000
Œ
µ

‹
= 275
= 152023
12
Writing of numbers
• Decimal
• But not positional
1

10
100
‰

‰ ‰
µµµµµ
ŒŒ
  





1000
10000
100000
1000000
Œ
µ

‹
= 275
= 152023
No zero symbol needed!!
Moscow papyrus
• 1850 BC
• Golenischev
• 5 metres long, 8 cm high
• 25 problems
• Bad handwriting
• Volume of truncated pyramid
• Surface of half a sphere
13
Other papyri
• Reisner papyri (1880 BC)
• Kahun papyri (1800 BC)
• Rollin papyrus (1350 BC)
• Harris papyrus (1167 BC)
14
Mathematics used for building pyramids
=
GEOMETRY
15
Weights and measures
• 1 cubit = 52.3 cm
• Long distances: ropes with knots
• Shorter distance: ruler
16
Egyptian ruler
The cubit of King Amenhotep I (1559 – 1539 BC)
17
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.
18
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
19
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.
20
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!!!
20
Rhind problem 76
If you want to trade 1000 des of beer of pesu
10 for beer of pesu 20, how many des do you
get?
21
Sekhed of a pyramid
is the inclination of any one of the four triangular
faces to the horizontal plane of its base.
h
h
α
a/2
a
a
sekhed =
2h
In fact the sekhed is the cotangent of the slope α.
22
Rhind problem 57
The sekhed of a pyramid is 5 palms and 1
finger and the side is 140 cubit. What is the
height?
23
Sekheds of some well-known pyramids
Name
Chephren, Ouserkaf
Neferirka-Re, Teti, Pepi
Cheops, Snofru
Neouser-Re
Sesostris
Amenemhat III
Sekhed
3/4 cubit
21 fingers
11/14 cubit
22 fingers
6/7 cubit
9/14 cubit
24
25
26
Pythagoras’ theorem
• The sum of the squares of the right angle
sides of a rectangular triangle is the square of
the remaining side.
c
b
a2 + b2 = c2
a
26
Pythagoras’ theorem
• The sum of the squares of the right angle
sides of a rectangular triangle is the square of
the remaining side.
c
b
a2 + b2 = c2
a
• If a triangle fulfills the above then it is rectangular.
Construction of right angles 1
k0
p
k0
p
p
k0
p
27
Property of a rectangle
• The diagonals of a rectangle meet eachother
halfway.
28
Property of a rectangle
• The diagonals of a rectangle meet eachother
halfway.
• If a quadrilateral fulfills the above then it is a
rectangle.
28
Construction of right angles 2
a
d
d
a
m
m
b
c
b
p
R
a
R
d
pa
a
d
m
m
R
R
b
pa
b
pa
29
Area of a triangle?
Rhind problem 49: Multiply half a side with the
other side.
30
30
Area of a triangle?
Rhind problem 49: Multiply half a side with the
other side.
1/2
1/2
c
b
a
Area of a quadrilateral?
b
c
a
d
a+c b+d
·
2
2
31
Rhind problem 48
d 2
A = (d − )
9
32
The area of a circle 1
33
The area of a circle 1
33
The area of a circle 1
33
The area of a circle 1
33
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
d2 64 2
π ≈ d
4
81
34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
d2 64 2
π ≈ d
4
81
64 256
π≈4 =
= 3.1604938 . . .
81
81
34
The number π
Definition: The ratio of the area of a circle and the
square of its radius is denoted by π.
d2 64 2
π ≈ d
4
81
64 256
π≈4 =
= 3.1604938 . . .
81
81
• Babylonians: π ≈ 3.125
• Bible: π ≈ 3
34
Area of a basket
Moscow problem 10: Find the area of a basket
d
with given “mouth” d.
They use the formula
64 2
A=2 d
81
35
Area of a basket
Moscow problem 10: Find the area of a basket
d
with given “mouth” d.
They use the formula
64 2
A=2 d
81
This is
256 2
A=2
r
81
35
Area of a basket
Moscow problem 10: Find the area of a basket
d
with given “mouth” d.
They use the formula
64 2
A=2 d
81
This is
and since π ≈
256
81
256 2
A=2
r
81
we find
A = 2πr2
35
Volume of a truncated pyramid
b
h
a
1
V = h(a2 + ab + b2)
3
36
37
How did they do it?
h’
b
b
h
a
a
Mysticism. . .
• The ratio of twice the side to the height of the
Great pyramid is π.
• In the dimensions of the Great pyramid are encoded: the radius of th earth, the density of earth,
the distance between earth and sun, . . .
• All important dates of human history
38
39
Charles Piazzi Smyth
• 1819–1900
• Astronomer
• Professor at university of Edinburgh
• Fellow of the Royal Society
40
Verification
William Matthew Flinders Petrie (1853–1942)
41
Other pyramidologists
• Charles Lagrange
• David Davison
• Georges Barbarin
• Robert Bauval
•...
42