- PLIMPTON 322 163
as factors. Of all right triangles, the 45-45-90 and the 30-60-90 are the two that
play the most important role in all kinds of geometric applications; plastic models
of them were once used as templates in mechanical drawing, and such models are
still sold. It is easy to imagine that a larger selection of triangle shapes might have
been useful in the past, before modern drafting instruments and computer-aided
design. Using this table, one could build 15 model triangles with angles varying
in increments of approximately 1°. One can imagine such models being built and
the engineer of 4000 years ago reaching for a "number 7 triangle" when a slope of
574/675 = .8504 was needed. However, this scenario still lacks plausibility. Even
if we assume that the engineer kept the tablet around as a reference when it was
necessary to know the slope, the tablet stores the square of the slope in column 1.
It is difficult to imagine any engineering application for that number.
Having failed to find a geometric explanation of the tablet, we now explore
possible associations of the tablet with Diophantine equations, that is, equations
whose solutions are to be rational numbers, in this case numbers whose numerators
and denominators are products of only the first three prime numbers. The left-hand
column contains numbers that are perfect squares and remain perfect squares when
1 is added to them. In other words, it gives u^2 for solutions to the Diophantine
equation u^2 +1 = v^2. This equation was much studied in other cultures, as we shall
see below. If the purpose of the table were to generate solutions of this equation,
there would of course be no reason to give v^2 , since it could be obtained by placing
a 1 before the entry in the first column. The use of the table would then be as
follows: Square the entry in column 3, square the entry in column 2, then divide
each by the difference of these squares. The results of these two divisions are v^2
and u^2 respectively. In particular, u^2 is in column 1. The numbers ñ and q that
generate the two columns can be arbitrary, but in order to get a sexagesimally
terminating entry in the first column, the difference (p^2 + q^2 )^2 - (p^2 - q^2 )^2 — Ap^2 q^2
should have only 2, 3, and 5 as prime factors, and hence ñ and q also should have
only these factors. Against this interpretation there lies the objection that ñ and
q are concealed from the casual reader of the tablet. If the purpose of the tablet
was to show how to generate u and õ or u^2 and v^2 = u^2 + 1, some explanation
should have been given as to how columns 2 and 3 were generated. But of course,
the possibility exists that such an explanation was present originally. After all, it
is apparent that the tablet is broken on the left-hand side. Perhaps it originally
contained more columns of figures that might shed light on the entire tablet if we
only had them. Here we enter upon immense possibilities, since the "vanished"
portion of the tablet could have contained a huge variety of entries. To bring this
open-ended discussion to a close, we look at what some experts in the area have to
say.
In work that was apparently never published (see Buck, 1980, p. 344), D.L.
Voils pointed out that tablets amounting to "teacher's manuals" have been found
in which the following problem is set: Find a number that yields a given number
when its reciprocal is subtracted. In modern terms this problem requires solving the
equation
χ = d,
χ
where d is the given number. Obviously, if you were a teacher setting such a
problem for a student, you would want the solution χ to be such that both χ and
1/x have terminating sexagesimal digits. So, if the solution is to be χ = p/q, we