- EGYPT 237
+\/80»9FIGURE 1. Conjectured explanations of the Egyptian squaring of
the circle.Ahmose takes the area of a circle to be the area of the square whose side is
obtained by removing the ninth part of the diameter. In our language the area is the
square on eight-ninths of the diameter, that is, it is the square on ø of the radius.
In our language, not that of Egypt, this gives a value of ð for area problems equal to
^jj*. Please remember, however, that the Egyptians had no concept of the number
ð. The constant of proportionality that they always worked with represents what
we would call ð/4. There have been various conjectures as to how the Egyptians
might have arrived at this result. One such conjecture given by Robins and Shute
(1987, p. 45) involves a square of side 8. If a circle is drawn through the points
2 units from each corner, it is visually clear that the four fillets at the corners, at
which the square is outside the circle, are nearly the same size as the four segments
of the circle outside the square; hence this circle and this square may be considered
equal in area. Now the diameter of this circle can be obtained by connecting one of
the points of intersection to the opposite point, as shown on the left-hand diagram
in Fig. 1, and measurement will show that this line is very nearly 9 units in length
(it is actually y/80 in length). A second theory due to K. Vogel (see Gillings,
1972, pp. 143-144) is based on the fact that the circle inscribed in a square of side
nine is roughly equal to the unshaded region in the right-hand diagram in Fig. 1.
This area is g of 81, that is, 63. A square of equal size would therefore have side
\/63 ~ 7.937 « 8. In favor of Vogel's conjecture is the fact that a figure very similar
to this diagram accompanies Problem 48 of the papyrus. A detailed discussion of
various conjectures, giving connections with traditional African crafts, was given
by Gerdes (1985).The Pythagorean theorem. Inevitably in the discussion of ancient cultures, the ques-
tion of the role played by the Pythagorean theorem is of interest. Did the ancient
Egyptians know this theorem? It has been reported in numerous textbooks, pop-
ular articles, and educational videos that the Egyptians laid out right angles by
stretching a rope with 12 equal intervals knotted on it so as to form a 3-4-5 right
triangle. What is the evidence for this assertion? First, the Egyptians did lay
out very accurate right angles. Also, as mentioned above, it is known that their
surveyors used ropes as measuring instruments and were referred to as rope-fixers
(see Plate 7). That is the evidence that was cited by the person who originally
made the conjecture, the historian Moritz Cantor (1829-1920) in the first volume
of his history of mathematics, published in 1882. The case can be made stronger,
however. In his essay Isis and Osiris Plutarch says the following.