236 9. MEASUREMENThorizontal and vertical displacements, it makes sense to use a larger unit of length
for vertical distances than is used for horizontal distances, even at the expense of
introducing an extra factor into computations of slope. In our terms the seked is
seven times the tangent of the angle that the sloping side makes with the vertical.
In some of the problems the seked is given in such a way that the factor of 7 drops
out. Notice that if you were ordering a stone from the quarry, the seked would
tell the stonecutter immediately where to cut. One would mark a point one cubit
(distance from fingertip to elbow) from the corner in one direction and a point at a
number of palms equal to the seked in the perpendicular direction, and then simply
cut between the two points marked.
In Problem 57 a pyramid with a seked of 5 4 and a base of 140 cubits is given.
The problem is to find its height. The seked given here (f of 7) is exactly that
of one of the actual pyramids, the pyramid of Khafre, who reigned from 2558 to
2532 BCE. It appears that stones were mass-produced in several standard shapes
with a seked that could be increased in intervals of one-fourth. Pyramid builders
and designers could thereby refer to a standard brick shape, just as architects and
contractors since the time of ancient Rome have been able to specify a standard
diameter for a water pipe. Problem 58 gives the dimensions of the same pyramid
and asks for its seked, apparently just to reinforce the reader's grasp of the relation
between seked and dimension.
The circle. Five of the problems in the Ahmose Papyrus (41-43, 48, and 50) involve
calculating the area of a circle. The answers given are approximations, but would
be precise if the value 64/81 used in the papyrus where we would use ð/4 were
exact. The author makes no distinction between the two. When physical objects
such as grain silos are built, the parts used to build them have to be measured.
In addition, the structures and their contents have a commercial, monetary value.
Some number has to be used to express that value. It would therefore not be
absurd -although it would probably be unnecessary—for a legislature to pass a bill
prescribing a numerical value to be used for ð.^3 Similarly, the claim often made
that the "biblical" value of ð is 3, based on the description of a vat 10 cubits from
brim to brim girdled by a line of 30 cubits (1 Kings 7:23) is pure pedantry. It
assumes more precision than is necessary in the context. The author may have
been giving measurements only to the nearest 10 cubits, not an unreasonable thing
to do in a literary description.^4
(^3) However, in the most notorious case where such a bill was nearly passed—House Bill 246 of
the 1897 Indiana legislature—it was absurd. The bill was written by a physician and amateur
mathematician named Edwin J. Goodwin. Goodwin had copyrighted what he thought was a
quadrature of the circle. He offered to allow textbooks sold in Indiana to use his proof royalty-free
provided that the Indiana House would pass this bill, whose text mostly glorified his own genius.
Some of the mathematical statements the legislature was requested to enact were pure gibberish.
For example, "a circular area is to the square on a line equal to the quadrant of the circumference,
as the area of an equilateral rectangle is to the square on one side." The one clear statement is
that "the ratio of the chord and arc of ninety degrees... is as seven to eight." That statement
implies that ð = 16\/2/7 w 3.232488 The square root in this expression did not trouble Dr.
Goodwin, who declared that <J2 = 10/7. At this point, one might have taken his value of ôô to be
160/49 = 3.265306122.... But, in a rare and uncalled-for manifestation of consistency, since he
"knew" that 100/49 = (10/7)^2 = 2, Goodwin declared this fraction equal to 16/5 = 3.2. The bill
was stopped at the last minute by lobbying from a member of the Indiana Academy of Sciences
and was tabled without action.
(^4) However, like everything in the Bible, this passage has been subject to exhaustive and repeated
analysis. For a summary of the conclusions reached in the Talmud, see Tsaban and Garber (1998).