Plimpton 322

Commentary on the text

Cuneiform writing is clearly recognized on our photo of Plimpton 322. Although the tablet is damaged in spots (especially in the upper left corner where much of the text is lost, and on the right side where a good-sized chip has fallen away), we can still identify five columns of text, organized in fifteen rows, with a row of script along the top edge in symbols of a markedly different style than those in the body of the tablet.

A study of the main body of text reveals that all the writing here is made up of just two kinds of symbol, a thin vertical wedge (), and a wide horizontal wedge ().  These symbols were likely made by pressing a reed stylus into the tablet before the clay had completely dried.

The fourth of the columns contains a repeated pattern of symbols and is uninteresting, but the rightmost fifth column can be seen to contain the "words" , , , , in order down the column.  The next two rows are lost, but in row seven is the combination , followed by the groupings  and . Curiously, the tenth row contains a single horizontal wedge , under which we find the groupings , , , with the rest of the column lost.

Clearly, we are seeing Babylonian numeration, the symbols for the numbers 1, 2, 3, etc. That is, the fifth column of the table is enumerating the rows. Vertical wedges are used to represent units of one (clustered in groups of threes), and horizontal wedges denote units of ten. So the numeration appears to be decimal in form.

Having noted this, we observe that the other columns contain lists of numbers as well. For instance, consider column 3: in the first row, we make out the sequence . This denotes the pair of numbers 2 and 49. The next entry in this column contains three vertical wedges, then one horizontal wedge, two vertical wedges, a short gap, and one vertical wedge; we interpret this as the numbers [3, 12, 1]. The following entry represents the sequence [1, 50, 49].

In this way, we translate the entire tablet, save some strange text at the very top for which we consult Akkadian linguists, to obtain this translation. Notice that we presented three versions here: a transliteration into standard notation, a version which reconstructs the missing text from the damaged portions of the tablet (how this is done will be explained shortly), and a final version that corrects the computational errors made by the Scribe.

Some interesting observations can already be made after a cursory study of the translated tablet:
 

• sequences of numbers in the first column are much longer on average than those in the other columns;
• the first column is ordered by the size of the first number in its sequence;
• there is no symbol for 0 in the original tablet, but its presence is denoted by a gap between other numbers (see especially the entry in row 13 of the first column);
• we never see groups of tens higher than fifty, that is, there is no number in the table higher than 59.

This last observation is critical to understanding how to interpret the tablet, for it indicates that Babylonian numeration is not truly a decimal numeration at all, but rather a sexagesimal numeration, one based on units of 60.

In our familiar decimal numeration, numbers are represented by means of just ten symbols, the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. It is their relative position in a numeral that determines its value. For instance, we distinguish between 247 and 724 because

This is why we call this numeration decimal positional notation: the position of the digits represent successive powers of 10, with the 10⁰, or ones, place at the rightmost position.

This is extended to encompass noninteger values by employing the decimal point after the ones digit and placing additional digits to the right of the point to represent negative powers of 10. So

Sexagesimal numeration behaves similarly. Consider the sequences of numbers at the top of the second and third columns on Plimpton 322. These sequences, [1, 59] and [2, 49] are really two-digit sexagesimal numbers:

These computations illustrate how to convert numbers from sexagesimal into decimal notation. As another example, consider the numbers in row 10 of the second and third columns:

By the way, conversion from decimal to sexagesimal is equally straightforward. To reverse the last computation above (to find the sexagesimal equivalent of 8161), we divide the number by 60, marking the quotient and remainder: 8161 ÷ 60 = 136 r 1. Thus the remainder is our first rightmost (ones) digit. The quotient determines the remaining digits through a repetition of this process: since 136 ÷ 60 = 2 r 16, the second (sixties) digit is 16, and the third (60²) digit is 2. Therefore, 8161 = [2, 16, 1].

This still leaves the question of how we are to understand the numbers in the first column. For if we treat them in the same way as the others, we are then forced to deal with some truly immense values. The number in row 10 of the first column has 8 sexagesimal digits, so its decimal equivalent would be on the order of 100,000,000,000,000! Clearly, something else is afoot.

Recall that the numbers in the first column are ordered not by the number of digits they contain, but by the size of the leftmost digit. This suggests that they are not whole numbers, but that a "sexagesimal point", or more properly, a radix point, should precede each sequence of digits. Consequently, all these numbers should be considered less than 1, decreasing in size from the top down. We use a semicolon for the radix point, so that the entries of the column in rows 1, 5, and 10 should be interpreted as

[; 59, 0, 15] = 59·60^-1 + 0·60^-2 + 15·60^-3 = 0.9834...
[; 48, 54, 1, 40] = 48·60^-1 + 54·60^-2 + 1·60^-3 + 40·60^-4 = 0.8150...
[; 35, 10, 2, 28, 27, 24, 26, 40] = 35·60^-1 + 10·60^-2 + 2·60^-3 + 28·60^-4 + 27·60^-5 + 24·60^-6 + 26·60^-7 + 40·60^-8 = 0.5861...

These decimal equivalents are only 4-place approximations, however. We can always obtain exact values as fractions (which we write in lowest terms):

[; 59, 0, 15] = 59·60^-1 + 0·60^-2 + 15·60^-3 = 60^-3(59·60² + 0·60¹ + 15·60⁰) = 212416/216000 = 14161/14400

Of course, these calculations become more complicated as the number of digits increases; they may require deft use of a calculator or mathematical software. Needless to say, the Babylonian Scribe did not have such tools at his disposal, but then he would not have even considered the problem of converting numbers from sexagesimal to decimal, or back again. For him, sexagesimal was the only numeration he knew.

As a brief aside, we should remark that arithmetic can be performed in sexagesimal notation by the same methods as in decimal. While this is not to say that Babylonian calculators employed precisely these procedures, they point out the similarities in the underlying concepts of arithmetic in both systems of numeration. For instance, addition and subtraction can be performed just as we learned in grade school, provided we remember that carrying and borrowing involve units of 60 instead of 10. Study these examples, showing the arithmetic for  [3, 23, 50] + [1, 45, 25]  and  [2, 0, 15] - [25, 40]:
 

Multiplication and division can also be carried with familiar methods, but tend to be somewhat more tedious. Babylonians employed many shortcuts and clever devices to make these computations easier and more efficient, but an analysis of these methods would take us away from our main focus.

Our discussion so far has uncovered how to interpret the writing on the tablet, but what about its meaning? It was originally thought that Plimpton 322 was one of dozens of tablets that simply recorded inventories of food and merchandise. But in 1946, historians of mathematics Otto Neugebauer and A. J. Sachs discovered that this tablet was evidence that the numbers in the table were computations and not simply records of quantity; the Babylonian Scribe was doing some rather sophisticated mathematics. A clue to support this claim lies in the translation of the text at the top of the tablet, which show that the second, third, and fifth columns are described with the words "width", "diagonal", and "name", respectively. That the fifth column is enumerating the rows of the table has already been ascertained, but the other labels indicate that the numbers represent the dimensions of a right triangle as in the figure below.
 
 

Neugebauer and Sachs confirmed this conjecture by checking that the numbers in columns 2 and 3, when assigned as lengths of a side (width) and hypotenuse (diagonal) of a right triangle, always gave rise, by means of the Pythagorean theorem a² + b² = c² (a = width,b = height, c = diagonal) to a whole number value for the length of the missing side (height). There were some exceptions (namely in rows 2, 9, 13, and 15), but these were explained as errors made by the Scribe, and can be identified as the digits in color in the decoded text. The errors have been corrected in the final form of the table in decimal numeration that we present at this link.
 

This final version of the text shows what Neugebauer and Sachs found after studying the table: the numbers in the first column of the tablet corresponded to values of the quantity (a/b)².  In particular, this means that Scribe knew how to compute all three terms of the triple of numbers a, b, c.  Since the ability to find whole number triplets a, b, c that solve the Pythagorean relation is not a trivial sort of problem, this indicates that the Scribe was possessing of some sophistication in mathematical technique. In other words, the tablet shows that the Scribe knew the Pythagorean Theorem (or something that is its equivalent) more than 1000 years before Pythagoras was born!

Return to calendar