162 7. ANCIENT NUMBER THEORYand "diagonal." Those labels tell us that we are dealing with dimensions of a
rectangle here, and that we should be looking for its length. By the Pythagorean
theorem, that length is \J(p^2 + q^2 )^2 - (p^2 - q^2 )^2 = \JAp^2 q^2 = 2pq. Even with this
auxiliary number, however, it requires some ingenuity to find a formula involving
ñ and q that fits the entries in the first column that can be read. If the numbers
in the first column are interpreted as the sexagesimal representations of numbers
between 0 and 1, those in rows 5 through 14—the rows that can be read—all fit
the formula^1
(p/q - Q/P
\ 2
Assuming this interpretation, since it works for the 10 entries we can read, we
can fill in the missing digits in the first four and last rows. This involves adding
two digits to the beginning of the first four rows, and it appears that there is just
the right amount of room in the chipped-off place to allow this to happen. The
digits that occur in the bottom row are 23,13,46,40, and they are consistent with
the parts that can be read from the tablet itself.
The purpose of Plimpton 322: some conjectures. The structure of the tablet is no
longer a mystery, unless one counts the tiny mystery of explaining the misprint in
row 2, column 3. Its purpose, however, is not clear. What information was the
table intended to convey? Was it intended to be used as people once used tables
of products, square roots, and logarithms, that is, to look up a number or pair of
numbers? If so, which columns contained the input and which the output? One
geometric problem that can be solved by use of this tablet is that of multiplying
a square by a given number, that is, given a square of side a, it is possible to find
the side 6 of a square whose ratio to the first square is given in the first column.
To do so, take a rope whose length equals the side a and divide it into the number
of equal parts given in the second column, then take a second rope with the same
unit of length and total length equal to the number of units in the third column
and use these two lengths to form a leg and the hypotenuse of a right triangle. The
other leg will then be the side of a square having the given ratio to the given square.
The problem of shrinking or enlarging squares was considered in other cultures, but
such an interpretation of Plimpton 322 has only the merit that there is no way of
proving the tablet wasn't used in this way. There is no proof that the tablet was
ever put to this use.
Friberg (1981) suggested that the purpose of the tablet was trigonometrical,
that it was a table of squares of tangents. Columns 2 and 3 give one leg and the
hypotenuse of 15 triangles with angles intermediate between those of the standard
45-45-90 and 30-60-90 triangles. What is very intriguing is that the table contains
all possible triangles whose shapes are between these two and whose legs have
lengths that are multiples of a standard unit by numbers having only 2, 3, and 5
(^1) In some discussions of Plimpton 322 the claim is made that a sexagesimal 1 should be placed
before each of the numbers in the first column. Although the tablet is clearly broken off on the
left, it does not appear from pictures of the tablet—the author has never seen it "live"—that
there were any such digits there before. Neugebauer (1952, p. 37) claims that parts of the initial
1 remain from line 4 on "as is clearly seen from the photograph" and that the initial 1 in line
14 is completely preserved. When that assumption is made, however, the only change in the
interpretation is a trivial one: The negative sign in the formula must be changed to a positive
sign, and what we are interpreting as a column of squares of tangents becomes a column of squares
of secants, since tan^2 è + 1 = sec^2 È.