Mesopotamian mathematics is fundamentally about lists (Proust, 2015, p. 215). Numerical calculations in administration, engineering and surveying were performed by trained professionals, called scribes, whose understanding of mathematics was founded in lists. In this section, we redefine several familiar mathematical concepts based on the author’s understanding of source materials and secondary literature. The purpose of these definitions is to highlight the unique aspects of the Mesopotamian list-based approach to mathematics. We begin with a somewhat novel definition of number.
Numbers
Scribes used a sexagesimal (base 60) number system. Their digits were written by accumulating symbols that represent 10 and 1, in that order. For instance, 16 was written as one 10 followed by six 1s. Similarly, 5 was written as no 10s followed by five 1s. We shall write these digits as \(00, 01, 02, 03, \ldots , 58\), and 59. The null digit 00 warrants special attention because it was written as no 10s and no 1s. In other words, the null digit appears as a blank space.Footnote 1
A number is a finite sequence of digits. Because scribes wrote the null digit as blank space, intermediate null digits appear as gaps within the number. More importantly, any leading or trailing null digits are lost in the vacant space surrounding the number.
Definition 1
A number, often called a Sexagesimal Place-Value Notation (SPVN) number, is a finite sequence of sexagesimal digits \(00, 01, \ldots , 58\), and 59 where both the first and last digits are non-null. We separate the digits of a number using a colon. The digit to the left of a colon represents units that are sixty times greater than the digit to the right.Footnote 2
The loss of surrounding null digits means that SPVN numbers are only determined up to a multiple of 60. For example, the number 16 : 00 : 05 itself is ambiguous and can mean any rational number of the form
$$\begin{aligned} 16\times 60^n + 0 \times 60^{n-1} + 5\times 60^{n-2},\, n\in {\mathbb {Z}}. \end{aligned}$$
This ambiguity is sometimes resolved by context.
Decimal numbers are fundamentally different from SPVN numbers since the former can have leading zeros, trailing zeros, and a radix point while the latter can not. Several modern mathematical concepts must be adjusted to accommodate the different and more ancient SPVN number system. For example, in SPVN arithmetic \(42 \times 10 = 7\) which means that 7 is a multiple of 10 with the usual understanding of the term “multiple”. This is clearly unsatisfactory, and the problem is that our usual understanding of the term “multiple” is too modern.Footnote 3Footnote 4
Regular Numbers
Before we can address the concept of multiple we must first recognize that many SPVN numbers are actually units, called regular numbers.
Definition 2
The SPVN number a is regular if there exists some SPVN number \({\overline{a}}\) such that \(a \times {\overline{a}} = 1\), the number \({\overline{a}}\) is the reciprocal of a, and any number which is not regular is irregular.Footnote 5
The standard table of reciprocals (Table 1) is a canonical piece of scribal equipment found throughout the OB period. It lists every one and two-digit regular number from 2 to 1 : 21 together with their corresponding reciprocals, and was memorized by scribes during their training. This meant that scribes could instantly recognize all regular and irregular numbers within this range. These numbers play a fundamental role in Mesopotamian mathematics and, following (Proust, 2012, p. 391), we reflect their importance with a further definition.
Definition 3
An elementary regular number is any number a or \({\overline{a}}\) from the standard table of reciprocals (Table 1). An elementary irregular number is any one or two-digit number between 2 and 1 : 21 that is not regular.
In other words, we regard the elementary regular numbers as obviously regular and the elementary irregular numbers as obviously irregular.
Table 1 The standard table of reciprocals which lists pairs of numbers a and \({\overline{a}}\) such that \(a \times {\overline{a}} = 1\). Some texts omit duplicate entries marked with \({}^*\) that appear elsewhere on the table Scribes knew that certain numbers became smaller when multiplied by certain reciprocals. In fact, they knew precisely which numbers could be reduced and how to reduce them. For example, they knew 5 : 55 : 57 : 25 : 18 : 45 could be reduced with multiplication by the reciprocal of 3 : 45, and that numbers like 7 or 5 : 09 : 01 could not be reduced at all. Their ability to reduce numbers was based on an understanding of “factors” and “multiples” that is slightly different from the usual understanding of these terms.
Factors and Multiples
We return to decimal for a moment. For decimal numbers, it is well known that any integer with final digit 2, 4, 6, 8, or 0 is a multiple of two, and any integer with final digit 5 or 0 is a multiple of five. Other multiples are possible if we look at more digits. For instance, any decimal integer whose final three digits are
$$\begin{aligned} 125, 250, 375, 500, 625, 750, 875 \text { or } 000 \end{aligned}$$
is a multiple of \(5^3\). Generally, if a is a proper factor of \(10^k, k \in {\mathbb {Z}}^+\) then a decimal integer is a multiple of a exactly when its final k digits match one of the prescribed endings
$$\begin{aligned} 1 \times a, 2\times a, \cdots , \left( \frac{10^k}{a} - 1\right) \times a \end{aligned}$$
or are all zero. Our definition of multiple for SPVN numbers is similar, except that we limit the number of final digits to at most two, omit trailing null digits, and have many proper factors to consider because 60 is a superior highly composite number. This definition is one of the novel aspects of the paper.
Let \(A_1\) be the set of proper factors of \(60^1\) in SPVN form
$$\begin{aligned} A_1 = \left\{ 30, 20, 15, 12, 10, 6, 5, 4, 3 ,2\right\} . \end{aligned}$$
Similarly let \(A_2\) be the set of proper factors of \(60^2\) that are not factors of \(60^1\), in SPVN form
$$\begin{aligned} A_2 =&\left\{ 50, 48, 45, 40, 36, 25, 24, 18, 16, 9, 8, \right. \\&\left. 7:30, 6:40, 3:45, 3:20, 2:30, 2:24, 1:40, 1:30, 1:20, 1:15, 1:12 \right\} . \end{aligned}$$
Definition 4
The SPVN number n is a multiple of \(a \in A_k, k \in \left\{ 1,2\right\}\) exactly when the final k digits of n match one of the prescribed endings
$$\begin{aligned} 1 \times a, 2\times a, \cdots , ({\overline{a}}-1) \times a. \end{aligned}$$
Equivalently we say that a is a factor of n.
For instance, the number 5 : 55 : 57 : 25 : 18 : 45 is a multiple of 3 : 45 because it ends with \(18:45 = 5 \times 3:45\). On the other hand, the possible endings for a multiple of 10 are 10, 20, 30, 40, and 50. Since 7 does not match one of these endings we conclude that 7 is not a multiple of 10, despite the fact that \(7 = 42 \times 10\) in SPVN arithmetic. Defining the concept of multiple in terms of the final digits is not an entirely new idea, indeed Neugebauer’s triaxial index grid is just a special case of this definition (Neugebauer, 1934, p. 9–15). See also (Friberg & Al-Rawi, 2016, p. 63–73) and (Proust, 2007, p. 170 footnote 313).Footnote 6
Scribes would only consider factors that were obvious from the final one or two digits of a number. Because they were not willing to look beyond these digits, our definition explicitly excludes the elementary regular numbers 27, 32, 54, 1 : 04, 1 : 21 and their reciprocals from consideration.
Scribes were very familiar with the multiples of \(a \in A_1 \cup A_2\) and learned them by rote in scribal school (Robson, 2008, p. 97–106). To be precise, scribes would memorize lists with the following standard format
$$\begin{aligned} 1 \times a,2\times a, \cdots ,19 \times a,20 \times a,30\times a,40\times a,50\times a. \end{aligned}$$
Such lists are called a-multiplication tables and were memorized to facilitate multiplication. Although it seems that some tables served an additional purpose related to the determination of factors. For example, CBS 6095 (Neugebauer & Sachs, 1945, p.23) (Table 2) is a 3 : 45-multiplication table. Memorization of this table enabled scribes to easily multiply by 3 : 45, but it also allowed them to immediately recognize any multiple of 3 : 45 by matching the final two digits with this table.
Table 2 The 3 : 45-multiplication table How does this definition of factor fit with the Mesopotamian list-based approach to mathematics? The combined table is another standard piece of scribal equipment. It consists of the standard table of reciprocals followed by a homogeneous selection of about forty a-multiplication tables. Instances of the combined table are found throughout OB times across multiple archaeological sites and, aside from isolated variations, are basically uniform in structure (Sachs, 1947, p. 221). While the combined table was almost certainly used for multiplication, the inclusion of the standard table of reciprocals and the curious selection of multiplication tables suggests more is true.
Friberg argued that the combined table could be used to facilitate division (Friberg, 2007, p. 92). Perhaps it is more accurate to say that the combined table served as a reference table for all things related to multiplication, including the identification and removal of factors. The selection of a-multiplication tables includes every \(a \in A_1 \cup A_2\), and this would explain how scribes were able to identify factors based on the final one or two digits alone. The standard table of reciprocals was also included because it showed how to remove these factors. In other words, the combined table relates to the identification and removal of factors in the following sense: the multiplication tables showed which numbers have factors, and the standard table of reciprocals showed how to remove those factors.Footnote 7,Footnote 8,Footnote 9
Factorization
Scribes used their understanding of factors to reduce large numerical problems to smaller problems that could be solved by reference to standard tables. For example, a scribe could reduce the problem \(6:14:24 \times x = 2:36\) by removing a factor of 12 from 6 : 14 : 24. This simplification could be repeated, eventually reducing the problem to \(13 \times x = 5:25\) which is easily solved by reference to the combined table.
Here we define factorization as the process of simplifying a large problem through repeated identification and removal of factors, which halts once the problem is simple enough to be solved directly. The process was first recognized by Sachs in 1947, who called it “the Technique” for finding reciprocals (Sachs, 1947, p. 223). But this is just one application, it has since become apparent that scribes also used factorization to solve linear equations and find square roots. Examples are discussed below.
When does factorization stop? Most scholars agree that a scribe would have “pushed the simplification of the numbers as far as possible by dividing out all regular factors” (Bruins, 1957, p. 28) or until “no regular factors remained” (Britton et al., 2011, p. 534). Here we emphasize that factorization continues at the pleasure of the scribe. It is a tool that helps them solve large problems, and ceases once the problem is small enough to be solved directly. This usually means that all factors are removed, but not always. This subtle distinction is important for our later analysis.
This discussion focuses on factorization in general. It is different to the specific analysis of factorization for regular numbers, such as Sachs (1947); Proust (2012). It is much easier to determine the presence of factors for regular numbers because regular numbers have fewer possible one and two-digit endings. For example, any regular number ending in 2 : 40 must be a multiple of 2 : 40 (Proust 2007, p. 174–175). This level of simplicity does not extend to numbers in general. Indeed, the irregular number 12 : 40 ends in 2 : 40 but it is not a multiple of 2 : 40.
Our first example (Table 3) shows how a scribe computes the reciprocal of 5 : 55 : 57 : 25 : 18 : 45 and is taken from lines 9 to 17 of CBS 1215 #21 (Friberg & Al-Rawi, 2016, p. 74–75). The “computation of (reciprocal pairs of) regular many-place sexagesimal numbers that are not listed in the Standard table of reciprocals” is often referred to as the trailing-part algorithm (Friberg, 1990, p. 550). However, we regard it more generally as a specific application of factorization.
Here and throughout we used bold-font to emphasize those digits used to determine factors during factorization.
Table 3 CBS 1215 #21: computation of a many-place reciprocal Table 3 shows that the scribe computed the reciprocal of 5 : 55 : 57 : 25 : 18 : 45 by iteratively breaking off elementary regular factors. The reciprocal of each factor was recorded on the right, and then the reciprocal of the whole computed from these parts
$$\begin{aligned} \overline{5:55:57:25:18:45} = 6:40 \times 1:20 \times 16^3 = 10:06:48:53:20. \end{aligned}$$
To be precise, the task is to compute the reciprocal of 5 : 55 : 57 : 25 : 18 : 45. The answer is not obvious and so the scribe seeks to reduce the problem by removing an elementary regular factor. The final digits 18 : 45 show this number contains a factor of 3 : 45, which can be removed with multiplication by the reciprocal \(\overline{3:45}=16\). This reciprocal is recorded on the right and the problem is reduced to finding find the reciprocal of
$$\begin{aligned} 5:55:57:25:18:45 \times 16 = 1:34:55:18:45. \end{aligned}$$
Line ten begins with 1 : 34 : 55 : 18 : 45, from which another factor of 3 : 45 is found. The reciprocal 16 is recorded on the right, and the problem is reduced further
$$\begin{aligned} 1:34:55:18:45 \times 16 = 25:18:45 \end{aligned}$$
Line eleven begins with 25 : 18 : 45, from which still another factor of 3 : 45 is found. The reciprocal 16 recorded, and the problem is reduced again
$$\begin{aligned} 25:18:45 \times 16 = 6:45. \end{aligned}$$
Line twelve begins with 6 : 45. The reciprocal of this number is still not obvious, so the scribe reduces the problem by removing an elementary regular factor of 45 (the final two digits \(6:45 = 9 \times 45\) show the factor of 45 is present). The reciprocal \({\overline{45}} = 1:20\) is recorded on the right, and the problem is reduced to finding the reciprocal of
$$\begin{aligned} 6:45 \times 1:20 =9. \end{aligned}$$
Line thirteen begins with 9, the reciprocal \({\overline{9}} = 6:40\) appears on the right and factorization stops because there is nothing left to remove. The product of these reciprocals is calculated in lines fourteen to seventeen:
$$\begin{aligned} \overline{5:55:57:25:18:45}&= 6:40 \times 1:20 \times 16^3 \\&= (8:53:20) \times 16^3 \\&= (2:22:13:20) \times 16^2 \\&= (37:55:33:20) \times 16\\&= 10:06:48:53:20. \end{aligned}$$
In this example, the scribe repeated factorization until no factors remained. This is not the case in 3N-T 362+366 (Robson, 2000, p. 22), which shows the calculation of another many-place reciprocal \(\overline{17:46:40} = 3:22:30\). Lines 4 to 6 are given in Table 4.
Table 4 3N-T 362+366: computation of a many-place reciprocal Line four follows directly from factorization: the scribe does not know the reciprocal of 17 : 46 : 40 and seeks to simplify the problem by breaking off an elementary regular factor. The scribe can immediately see that 6 : 40 is a factor. This is because, having memorized the 6 : 40-multiplication table, a scribe would have recognized the final digits as \(46:40 = 7 \times 6:40\). The scribe records the reciprocal of this factor \(\overline{6:40} = 9\) on the right, and then removes it with \(17:46:40 \times 9 = 2:40\).
Line five begins with 2 : 40. This number could have been broken down into the elementary regular factors \(2:40 = 2 \times 1:20\) but instead factorization halts. This is important because factorization has stopped before all the elementary regular factors were removed. The scribe may have noticed that \(2:40 = 2 \times 1:20\) and computed the reciprocal directly as half of \(\overline{1:20} = 45\)
$$\begin{aligned} \overline{2:40} = {\overline{2}} \times \overline{1:20} = {\overline{2}} \times 45 = 22:30. \end{aligned}$$
This is perhaps why the superfluous 2 appears in line five of the text.
As before, the final line contains the reciprocal \(\overline{17:46:40}\) which was calculated as the product of the individual reciprocals
$$\begin{aligned} \overline{17:46:40} = 9 \times 22:30 = 3:22:30. \end{aligned}$$
This example is important because it emphasizes the idiosyncratic nature of Mesopotamian factorization. Scribes do not necessarily seek to remove each and every factor as with modern factorization. Instead, an individual scribe will cease removing factors once they can solve the problem directly.Footnote 10
Our next example comes from VAT 7532 (Friberg, 2007, p. 23) where, in modern terms, a scribe solves the linear equation
$$\begin{aligned} 6:14:24 \times x = 2:36. \end{aligned}$$
The scribe simply states that \(x = 25\) so we can do no more than guess how they arrived at this answer. Factorization can reduce this problem to
$$\begin{aligned} 13 \times x = 5:25. \end{aligned}$$
This cannot be reduced further, however a scribe would have already recognized the answer because \(13\times 25 = 5:25\) is familiar from the 25-multiplication table. This is typical, almost all known exercises can be solved with factorization and the combined table alone (Sachs, 1952, p. 151).
Our final example shows how factorization was used to calculate square roots such as \(\sqrt{1:07:44:03:45}\) and is taken from the reverse side of UET 6/2 222 (Friberg, 2000, p. 108) (Table 5). The text begins by computing \(1:03:45^2 = 1:07:44:03:45\) in lines one to three. This is followed by a variation on factorization where only square elementary regular factors are removed. As usual the reciprocal of each factor is recorded on the right, but additionally the square root of each factor is recorded on the left.
Table 5 UET 6/2 222: the square root of a many-place number The product of these individual square roots yields the expected result 1 : 03 : 45 in lines six and seven. In modern language, the scribe made two symmetrical calculations
$$\begin{aligned} 1:03:45^{2} & = 1:07:44:03:45 \\ \sqrt {1:07:44:03:45} & = 15 \times \sqrt {18:03:45} \\ & = 15^{2} \times \sqrt {4:49} \\ & = 15^{2} \times 17 = 1:03:45. \\ \end{aligned}$$
The scribe computes the square root of this many-place number by successively removing square factors until the problem is small enough to be solved directly. In line three, a square factor of 3 : 45 (i.e. \(3:45=15^2\)) is apparent from the final two digits. The scribe records the reciprocal \(\overline{3:45}=16\) on the right and the square root \(\sqrt{3:45} = 15\) on the left. Then the factor is removed
$$\begin{aligned} 1:07:44:03:45 \times 16 = 18:03:45. \end{aligned}$$
In line four the procedure repeats. Another square factor of 3 : 45 is identified, its reciprocal and square root recorded, and then the factor is removed
$$\begin{aligned} 18:03:45 \times 16 = 4:49. \end{aligned}$$
The problem has now been reduced to finding the square root of 4 : 49. Factorization halts because the answer, 17, can be found directly from a table of square roots (Friberg, 2007, p. 50).
In summary, scribal mathematics is fundamentally about lists and procedures. Large problems are reduced to small problems by factorization, and small problems are solved directly from standard tables.
Diagonal Triples
A Pythagorean triple is a right triangle whose three sides are all integers where the square of the hypotenuse equals the sum of the squares of the other two sides. The equivalent Mesopotamian understanding of this fundamental object is slightly different.
Definition 5
A diagonal triple is a triple of SPVN numbers \((b,\ell , d)\) corresponding to the sides and diagonal of a rectangle where \(b^2 + \ell ^2 = d^2\) and with \(b < \ell\). If \(\ell = 1\) then the triple is said to be normalized and the special notation \((\beta , 1, \delta )\) is used.
A diagonal triple is a rectangle whose sides and diagonal are SPVN numbers, as distinct from a Pythagorean triple which is a right triangle over the positive integers. We use the term diagonal triple instead of Pythagorean triple from now on, and refer to its measurements as the short side b, the long side \(\ell\) and diagonal d.Footnote 11
For a period spanning 1500 years, from OB times to Seleucid times, it was known that a regular number x generates the normalized diagonal triple \((\beta , 1, \delta )\) where
$$\begin{aligned} \delta ={\overline{2}}(x + {\overline{x}}) \text { and } \beta =\sqrt{\delta ^2 - 1}={\overline{2}}(x - {\overline{x}}). \end{aligned}$$
For example, MS 3971 §3 ( (Friberg, 2007, p. 252–253), also see below) is an exercise where five different regular numbers are used to generate five normalized diagonal triples. The fourth in this series is given in Table 6.
Table 6 MS 3971 §3d: generation of a diagonal triple from the regular number 1 : 20 The procedure begins with the elementary regular number \(x=1:20\). In lines one and two this is used to compute the diagonal as the average of x and its reciprocal \(\delta = {\overline{2}}(x + {\overline{x}}) = 1:02:30\). In line three the scribe computes the square of the diagonal \(\delta ^2 = 1:05:06:15\). In line four the scribe computes the square of the short side according to the “Pythagorean” relation \(\beta ^2 = \delta ^2 - 1 = 5:06:15\). Factorization was almost certainly used during the square root calculation \(\beta = \sqrt{5:06:15} = 17:30\) in line five but all details of this calculation are omitted. In any case, this example demonstrates a key characteristic of Mesopotamian mathematics: the questions were designed so they could be solved using only standard procedures and tables. It is because of this tradition, not lucky coincidence, that this square root can be found using only the techniques we have discussed thus farFootnote 12
$$\begin{aligned} \sqrt{5:06:15} = 30 \times \sqrt{20:25} = 30 \times 5 \times \sqrt{49} = 30 \times 5 \times 7 = 17:30. \end{aligned}$$
MS 3971 §3 and §4, a Small Plimpton 322
It is instructive to prefix our main discussion of Plimpton 322 with a discussion of MS 3971, a sequence of OB mathematical exercises published by Friberg in 2007. The similarities between Plimpton 322 and MS 3971 part §3 are “striking”, as discussed by (Britton et al., 2011, p. 588), (Proust, 2011, p. 664) and (Friberg, 2007, p. 252–254), although the connection between Plimpton 322 and MS 3971 parts §3 and §4 together has not been previously considered.
Part §3 begins with instructions that we should inspect the diagonals of five rectangles aššum 5 ṣilpatum amari-ka, literally “when/because your seeing the 5 diagonals”. A less literal translation would be “Carefully examine the 5 diagonals”. This is followed by the generation of five diagonal triples using the standard method shown above in Table 6. Of interest are the initial generation parameters x, diagonals \(\delta\), intermediate values \(\delta ^2\), and short sides \(\beta\) which have been summarized in Table 7.
Table 7 Generation parameters, intermediate values, and results corresponding to the OB school exercise MS 3971 §3. The text specifies that we are to examine the diagonals, and then in part §4 we are told to find a rectangle with diagonal 7. Of the five given rectangles, only §3d can be rescaled to have diagonal 7 What can be said about these diagonals? The numbers \(1:01,\, 1:05\), and 1 : 08 can be immediately recognized as elementary irregular numbers (recall that the student knows all the one and two-digit regular and irregular numbers between 2 and 1 : 21 by rote). A small amount of simplification is required to deduce that 1 : 00 : 07 : 30 is irregular. This is because once a single factor is removed, what remains is a number without any elementary regular factors
$$\begin{aligned} 1:00:07:30 \times \overline{7:30} = 8:01. \end{aligned}$$
Only 1 : 02 : 30 is regular, which is apparent once a single factor is removed
$$\begin{aligned} 1:02:30 \times \overline{2:30} = 25. \end{aligned}$$
These calculations present no difficulty for a student familiar with the combined table and versed in factorization. Part §3 concludes by repeating “5 diagonals”, so evidently we were supposed to realize something important about these numbers.
Part §3 is immediately followed by part §4 where the student is asked to find a rectangle with diagonal 7 or, in other words, to find a rectangle \((b, \ell ,d)\) which can be rescaled by a factor of \(7{\overline{d}}\) into
$$\begin{aligned} 7 {\overline{d}} (b,\ell ,d) = (7b{\overline{d}}, 7\ell {\overline{d}}, 7). \end{aligned}$$
This might seem trivial, but it is not. The question requires the student to use a rectangle with a regular diagonal (i.e. a rectangle where \({\overline{d}}\) exists). There are infinitely many diagonal triples, but only two have this property: the rectangle from §3d whose diagonal we were told to examine, and the simple (3, 4, 5) rectangle. It is not surprising that the student elects to answer this question with the numerically simple rectangle
$$\begin{aligned} 7\times {\overline{5}} \times (3,\, 4,\, 5) = (4:12,\, 5:36,\, 7). \end{aligned}$$
But it cannot be a coincidence that the other possible rectangle appeared in section §3, especially considering the explicit and repeated emphasis on its diagonal. The alternative answer, using the rectangle from §3d, would have been
$$\begin{aligned} 7\times \overline{1:02:30} \times (17:30,\, 1,\, 1:02:30) = (1:57:36,\, 6:43:12,\, 7). \end{aligned}$$
The lesson from MS 3971 §3 and §4 is that rectangles with regular sides are important because only the regular sides can be rescaled to arbitrary lengths such as 7. Naturally, (3, 4, 5) is the preferred triple on account of its three regular sides. Although diagonal triples with two regular sides are also useful in certain circumstances, such as \((17:30,\, 1,\, 1:02:30)\) in this example.Footnote 13