mathematics history

1. Egyptian Geometry
Prepared by:
Antipona, Mary Grace

2. Approximating the Area of a
Circle

3. What is Geometry?
- the name "geometry" is a
compound two greek words
meaning "earth" (geo) and
"measure" (metry) seems to
indicate that the subject arose
from the necessity of land
surveying.

4. Origin of Geometry
- The Greek historian Herodotus, who visited the Nile about
460–455 B.C., described how the first systematic geometric
observations were made.They said also that this king
[Sesostris] divided the land among all Egyptians so as to give
each one a quadrangle of equal size and to draw from each
his revenues, by imposing a tax to be levied yearly. But every
one from whose part the river tore away anything, had to go
to him and notify what had happened. He then sent the
overseers, who had to measure out by how much the land
had become smaller, in order that the owner might pay on
what was left, in proportion to the entire tax imposed. In this
way, it appears to me, geometry originated.

5. What is Egyptian Geometry?
Egyptian geometry refers to geometry as it
was developed and used in Ancient Egypt.
Ancient Egyptian mathematics as discussed
here spans a time period ranging from ca. 3000
BC to ca 300 BC.

6. In the great dedicatory inscription, of about 100 B.C., in
the Temple of Horus at Edfu,there are references to
numerous four-sided elds 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 opposite sides,
that is, by using the formula:
A = ¼(a+b)(c+d)
where a, b, c, and d are the lengths of the consecutive
sides. The formula is obviously incorrect. It gives a fairly
accurate answer only when the eld is approximately
rectangular. What is interesting is that this same
erroneous formula for the area of a quadrilateral had
appeared 3000 years earlier in ancient Babylonia

7. 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:
Example of a round eld of a diameter 9 khet.
What is its area? Take away ⅑ of the diameter,
namely 1; the remainder is 8. Multiply 8 times 8; it
makes 64. Therefore it contains 64 setat of land.

8. The scribes’ process for finding the area of a circle
can thus be simply stated: Subtract from the diameter
its ⅑ part and square the remainder. In modern
symbols, this amounts to the formula:
where d denotes the length of the diameter of the
circle. If we compare this with the actual formula for the
area of the circle, namely πd²/4, then

9. so that we get
for the Egyptian value of the ratio of the circle’s
circumference to its diameter. This is a cose
approximation to 317 ; which many students find good
enough for practical purposes.

10. The Volume of a Truncated
Pyramid

11. There are only 25 problems in the Moscow Papyrus,
but one of them contains the masterpiece of ancient
geometry. Problem 14 shows that the Egyptians of
about 1850 B.C. were familiar with the correct formula
for the volume of a truncated square pyramid (or
frustum). In our notation, this is
where h is the altitude and a and b are the lengths of
the sides of the square base and square top,
respectively,

13. It is generally accepted that the Egyptians were
acquainted with a formula for the volume of the
complete square pyramid, and that it probably was the
correct one,
In analogy with the formula A=½bh for the area of a
triangle, the Egyptians may have guessed that the
volume of a pyramid was a constant times ha2. We
may suppose even that they guessed the constant to
be 1=3. But the formula

14. could not very well be a guess. It could have been
obtained only by some sort of geometric analysis or
by algebra from V D (h=3)a2. It is not, however, an
easy task to reconstruct a method by which they
could have deduced the formula for the truncated
pyramid with the materials available to them.

15. Speculations About the
Great Pyramid

16. Egyptian Geometry is used in building the Great Pyramid.
According to Herodotus, 400,000 workmen labored
annually on the Great Pyramid for 30 years—four
separate groups of 100,000, each group employed for
three months. (Calculations indicate that no more than
36,000 men could have worked on the pyramid at one
time without hampering one another’s movements.) Ten
years were spent constructing a road to a limestone
quarry some miles distant, and over this road were
dragged 2,300,000 blocks of stone averaging 212 tons
and measuring 3 feet in each direction. These blocks
were tted together so perfectly that a knife blade cannot
be inserted in the joints.

17. The Great Pyramid has down to the present red
adventurous minds to the wildest speculations. These
pyramid mystics (or as they are sometimes uncharitably
called, pyramidiots) have ascribed to the ancient builders
all sorts of metaphysical intentions and esoteric
knowledge. Among the extraordinary things claimed is that
the pyramid was built so that half the perimeter of the
base divided by the height should be exactly equal to π.
While the difference between the two values
is only 0.00036......their closeness is merely accidental and
has no basis in any mathematical law.

18. The Egyptian priests, according to action that has crept
into the recent literature, told Herodotus that the
dimensions of the Great Pyramid were so chosen that the
area of each face would be the same as the area of a square
with sides equal to the Pyramid's height. Writing 2b for the
length of a side of the base, a for the altitude of a face
triangle,and h for the height of the pyramid, we find that
Herodotus’s relation is expressed by the equation

19. The Pythagorean theorem tells us that because a is
the hypotenuse of a right triangle with legs b and h, then
h² + b² = a², or h² = a² -b². Equating the two expressions
for h², we get
When both sides are divided by a², this last equation
becomes

20. Reviewing everything, we are forced to
conclude that Egyptian geometry never
advanced beyond an intuitive stage, in which
the measurement of tangible objects was the
chief consideration. The geometry of that
period lacked deductive structure—there were
no theoretical results, nor any general rules of
procedure. It supplied only calculations, and
these sometimes approximate, for problems
that had a practical bearing in construction
and surveying.

21. THANK YOU!!