Download Egyptian Geometry

Activity
Egyptian Geometry
Purpose:
To explore some ancient geometry and mathematical concepts.
Required Equipment and Supplies:
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●
This activity sheet
Scratch paper
Discussion:
The general accepted account of the origin of geometry is that it came into being in ancient Egypt,
where the yearly inundations of the Nile demanded that the size of the landed property be resurveyed
for tax purposes. Indeed, the named geometry, a compound of two Greek words meaning earth and
measure, seems to indicate that the subject arose from necessity of land surveying.
Most of our knowledge of the order of mathematics in Egypt is derived from two sizable papyri, each
named after its former owner – The Rhind Papyrus and the Golenischev Papysrus. The Rhind Papyrus
starts with a bold premise. Its content has to do with “a thorough study of all things, insights into all
that exists, knowledge of all obscure secrets.” It soon becomes apparent that we are dealing with a
practical handbook of mathematical exercises and the only secrets are how to multiply and divide
(which is not the easiest do in the Egyptian number system). Nonetheless, the 85 problems contained
therein give us a pretty clear idea of the character of the Egyptian mathematics.
Procedure:
Step 1: Multiplication of two numbers was accomplished by successively doubling one of the numbers
and then adding the appropriate duplication to form the product.
Ex: 19 ⋅ 71
x
1
71
x
2
142
4
284
8
568
16
1136
19
1349
x
Total
The Egyptian mathematician would not double further than 16 as the next one would yield a number
larger than 19. Because 19 = 1 + 2 + 16, we would mark these rows to indicate that these should be
added together. The answer to our problem would then be
1349 = 71 + 142 + 1136 = (1 + 2 + 16) 71 = 19 ⋅ 71
Activity
Task: Find you own solution to the following problems by using the Egyptian method of doubling.
a) 18 ⋅ 25
b) 85 ⋅ 21
c) 26 ⋅ 33
The Egyptian division could be described as multiplication in reverse – where the divisor is repeatedly
doubled to give the dividend.
Ex: 91/7
x
1
7
2
14
x
4
28
x
8
56
13= 8+4+1
91
Total
But what if the dividend was not that cooperative? Then fractions had to be introduced and this is the
cumbersome drawback of the Egyptian system.
Ex. 35/8
1
8
2
16
4
32
1/2
4
x
1/4
2
x
1/8
1
1 1
+
4 8
35
x
Total
4+
Task: Find, in the Egyptian fashion, the following quotients.
a) 184÷8
b) 19÷8
c) 61÷8
Discussion:
What influence do you think the way of multiplying and dividing might have had on the evolution of
mathematics in Egypt?
Activity
Step 2: In the great dedicatory inscription, of about 100 B.C., in the Temple of Horus at Edfu, there
are references to numerous four-sided fields that were gifts to the temple. For each of these, the areas
were obtained by taking the product of the averages of the two pairs of the opposite sides, that is, they
used the following formula
1
A = (a + c)(b + d )
4
where a,b,c and d are the length of the consecutive sides. The formula is obviously incorrect, it does
give a pretty good answer if the field is approximately rectangular.
Task: Create three different quadrilaterals. Use the incorrect formula and use the Pythagorean
Theorem to calculate the area of the quadrilaterals. Finally calculate the relative error.
Step 3: The geometrical problems of the Rhind Papyrus are those numbered 41-60, and are largely
concerned with the amounts of grain stored in rectangular and cylindrical granaries. Perhaps the best
achievement of the Egyptians in two-dimensional geometry was their method for finding the area of a
circle, which appears in Problem 50:
1
of the diameter,
9
namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of
land.
Example: A round field of diameter 9 khet. What is its area? Take away
The scribes' process for finding the area of a circle can be stated as follows: Subtract from the diameter
1
its
part and square the remainder. In modern symbols we would write it as
9
2
d

A =  d −  where d denotes the diameter of the circle.
9

Task: If you compare this with the actual formula for the area of the circle you will be able to get an
approximation for the ratio between the circles circumference and its diameter.
1
as good enough value for all practical purposes for this ratio.
7
How close is the Egyptian approximation to this value?
•
Many students today use 3
•
Archimedes (287 – 212 BC) claims in his book Measurement of a Circle that the area of a circle
is to the square of on diameter as 11 to 14. Show that this geometric rule leads to a value of
22
for π. How accurate is this value compared to what the Egyptians used?
7
•
The sixth-century Hindu mathematician Aryabhata used the following procedure for finding the
area of a circle: Half the circumference multiplied by half the diameter is the area of the circle.
How accurate is this rule?
Activity
Bibliography:
Burton, David M. The History of Mathematics 5th Ed. Boston MA, Mc GrawHill (2003)