- How to read it
- What it means
- How the table was produced
- Accounting for the errors
- Further references

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The number 10 was written in a single character and the numbers 20 to 50 were written with multiples of this character:

The numbers up to 59 were obtained by combining these much as we
do. For example, 11 was written as .
But the system was not actually based on 10, but rather on 60 - *
and it was a true floating point system * so that 1 and 60
(as well as, say, 3600) were indistiguishable.
All numbers could be obtained by combining these in order,
writing from left to right as we do. Thus seventy was written
as .
Sometime before 300 BCE, but after Plimpton 322 was written,
a special symbol was devised as a zero, but in Plimpton 322
there is potential confusion because of this problem.
The conventional way to write floating point sexagesimal numbers is
by using comma separators, so that 1,29 is 60+29 = 89 in decimal notation,
and 1,1,1 is 3661.

**The last two columns**.
At any rate, we now know almost enough to read the right hand
columns of the tablet. The last column
(with a few natural interpolations to take into account
missing symbols for *5*, *6*, and *15*, ) simply
numbers the line of numerical data. In the fourth column
is written the word pronounced *ki* which can be translated loosely
here as *number*, so that the fourth
and fifth column together (**number 1**, **number 2**, etc.) just
enumerate the different lines of data.
Interpolations are in green.

For the rest of the columns, we show in order the original column, the sexagesimal numbers written in conventional notation, and then the decimal equivalents. Error corrections are in red. Both interpolations and corrections are explained later on. All the zeroes are interpolated, also, since the Babylonians at this time did not have them.

**Column two**.
The heading of the second column includes the word for *width*.

**Column three**.
The heading of the third column includes the word for *diagonal*.

**Column one**.
The heading of the first column has not been translated.
We shall see in a moment the reasons for placing the `decimal point'.

which form the sides of right triangles. Here is the resulting table of calculations, in modern notation (with discrepancies in square brackets):

A bit later, we shall confirm this theory by explaining the errors.

As for the first column, it contains the values of
*
d ^{2}
/
l^{2}
*. For example, in the first row
$d = 169@, $l = 120@, and
$d

- $p@ and $q@ are both positive;
- $p@ is greater than $q@;
- One of them is odd, the other even;
- $p@ and $q@ are relatively prime.

Here are the values of $p@, $q@, and $m@ for the triples in the table:

The ratio $c/b@ is equal to $(p^{2} + q^{2})/2pq = (1/2)(p/q + q/p)@.
Therefore this ratio, the square of
which appears in the first column of the tablet,
will have a finite expression in
base 60 if $1/p@ and $1/q@ do.
The Babylonians almost certainly understood
the difference between finite sexagesimal
expansions and repeating ones, and in particular
we have found tables of reciprocals $1/p@ for
many values of $p@ where the expansion is finite.
Such numbers $p@ are called **regular** by Neugebauer.
It is not likely to be a coincidence that
the values of both $p@ and $q@ associated to
the rows of the tablet are regular, and in fact
that in all but one case the expansions of $1/p@ and
$1/q@ appear in the tables of reciprocals that have been found.
*It seems plausible, therefore, that
the Babylonians knew how to generate primitive Pythagorean
triples.*

We know something, therefore, about how the tablet was constructed, but we do not know exactly why it was constructed. The ordering of the rows according to the size of the first column suggests that it might have been used in an early form of trigonometry. Perhaps it was constructed from Pythagorean triples just to make arithmetic easier.

*However incomplete
our present knowledge of Babylonian mathematics may be, so much is establshed
beyond any doubt:
we are dealing with a level of mathematical development
which can in many aspects be compared with the mathematics, say, of
the early Renaissance*. (O. Neugebauer in *The exact sciences
in antiquity*)

As confirmation of both the interpretation of the table and this conjecture regarding $p@ and $q@, the four apparent errors can be reasonably explained:

- The number
**[9, 1]**in row 9 should be**[8, 1]**- a simple copying error. **[7,12,1]**in row 13 is the square of**[2,41]**, which would be the correct value - a mistake particularly easy to make since the squares also appear in the conjectured calculation.- The correct value to replace
**[53]**in row 15 is**[1,46]**, which is twice the erroneous value. - As for the fourth error in row 2, where
**[3,12,1]**occurs instead of**[1,20,25]**, there have been a couple of solutions proposed. None are entirely convincing. The possibility proposed by Gillings suggests strongly that those who made up the table had values of $p@ and $q@ at hand.

- O. Neugebauer and A. Sachs,
**Mathematical cuneiform texts**, American Oriental Society, 1945. This, as far as I know, is where the tablet was first analyzed. - O. Neugebauer,
**The exact sciences in antiquity**, Dover, 1969. This is a reprint of the 1957 second edition. Chapter 2 discusses the tablet in detail. - R. J. Gillings, Australian Journal of Science 16 (1953), pages 54-56. Mentioned in Neugebauer's book.
- R. Creighton Buck,
*Sherlock Holmes in Babylon*, the Mathematical Monthly, 1964. - The Akkadian home page. The Internet has a large collection of material on cuneiform and Bablyonian tablets, although not much concerned with mathematics, and no recent photographs of the tablet Plimpton 322.

Written by Bill Casselman.