1.3: A New Reading

As we have just seen, the arithmetical interpretation is unable to account for the words which the Babylonians used to describe their procedures. Firstly, it conflates operations that the Babylonians treated as distinct; secondly, it is based on operations whose order does not always correspond to that of the Babylonian calculations. Strictly speaking, rather than an interpretation it thus represents a control of the correctness of the Babylonian methods based on modern techniques.

A genuine interpretation —a reading of what the Old Babylonian calculators thought and did—must take two things into account: on one hand, the results obtained by the scholars of the 1930s in their “first approximation”; on the other, the levels of the texts which these scholars had to neglect in order to create this first approximation.

In the following chapters we are going to analyze a number of problems in a translation that corresponds to such an interpretation. First some general information will be adequate.

Representation and “variables”

In our algebra we use \(x\) and \(y\) as substitutes or names for unknown numbers. We use this algebra as a tool for solving problems that concern other kinds of magnitudes, such as prices, distances, energy densities, etc.; but in all such cases we consider these other quantities as represented by numbers. For us, numbers constitute the fundamental representation.

With the Babylonians, the fundamental representation was geometric. Most of their “algebraic” problems concern rectangles with length, width and area⁴, or squares with side and area. We shall certainly encounter a problem below (YBC 6967, page 46) that asks about two unknown numbers, but since their product is spoken of as a “surface” it is evident that these numbers are represented by the sides of a rectangle.

An important characteristic of Babylonian geometry allows it to serve as an “algebraic” representation: it always deals with measured quantities. The measure of its segments and areas may be treated as unknown—but even then it exists as a numerical measure, and the problem consists in finding its value.

Units

Every measuring operation presupposes a metrology, a system of measuring units; the numbers that result from it are concrete numbers. That cannot be seen directly in the problem that was quoted above on page 9; mostly, the mathematical texts do not show it since they make use of the place-value system (except, occasionally, when given magnitudes or final results are stated). In this system, all quantities of the same kind were measured in a “standard unit” which, with very few exceptions, was not stated but tacitly understood.

The standard unit for horizontal distance was the nindan, a “rod” of c. 6 m.⁵ In our problem, the side of the square is thus \(\frac{1}{2} \mathrm{NINDAN}\), that is , c. 3 m. For vertical distances (heights and depths), the basic unit was the kùš, a "cubit" of \(\frac{1}{12} \mathrm{NINDAN}\) (that is, c. 50 cm).

The standard unit for areas was the sar, equal to 1 \(\mathrm{NNDAN}^{2}\). The standard unit for volumes had the same name: the underlying idea was that a base of 1 \(\mathrm{NINDAN}^{2}\) was provided with a standard thickness of 1 kùš. In agricultural administration, a better suited area unit was used, the bùr. equal to sar, c. \(6 \frac{1}{2}\) ha.

The standard unit for hollow measures (used for products conserved in vases and jars, such as grain and oil) was the sìla, slightly less than one litre. In practical life, larger units were often used: 1 bán \(=\) 10 sìla, 1 pi = 1‵ sìla, and 1 gur, a "tun" of 5‵ sìla.

Finally, the standard unit for weights was the shekel, c. 8 gram. Larger units were the mina , equal to 1‵ shekel (thus close to a pound)⁶ and the gú, “a load” equal to shekel, c. 30 kilogram. This last unit is equal to the talent of the Bible (where a talent of silver is to be understood).

Additive Operations

There are two additive operations. One (kamārum/ul.gar/gar.gar), as we have already seen, can be translated “to heap \(a\) and \(b\)," the other (waṣābum/da[+.1ex]h) “to join \(j\) to \(S\)." "Joining" is a concrete operation which conserves the identity of \(S\). In order to understand what that means we may think of "my" bank deposit \(S\); adding the interest \(j\) (in Babylonian called precisely ṣibtum, “the joined,” a noun derived from the verb waṣābum) does not change its identity as my deposit. If a geometric operation “joins” \(j\) to \(S\), \(S\) invariably remains in place, whereas, if necessary, \(j\) is moved around.

“Heaping,” to the contrary, may designate the addition of abstract numbers. Nothing therefore prevents from “heaping” (the number measuring) an area and (the number measuring) a length. However, even “heaping” often concerns entities allowing a concrete operation.

The sum resulting from a “joining” operation has no particular name; indeed, the operation creates nothing new. In a heaping process, on the other hand, where the two addends are absorbed into the sum, this sum has a name (nakmartum, derived from kamārum, “to heap”) which we may translate “the heap”; in a text where the two constituents remain distinct, a plural is used (kimrātum, equally derived from kamārum); we may translate it “the things heaped” (AO 8862 #2, translated in Chapter 4, page 60).

Subtractive Operations

There are also two subtractive operations. One (nasāḫum/zi), “from \(B\) to tear out \(a\)" is the inverse of "joining"; it is a concrete operation which presupposes \(a\) to be a constituent part of \(B\). The other is a comparison, which can be expressed "\(A\) over \(B\), \(d\) goes beyond" (a clumsy phrase, which however maps the structure of the Babylonian locution precisely). Even this is a concrete operation, used to compare magnitudes of which the smaller is not part of the larger. At times, stylistic and similar reasons call for the comparison being made the other way around, as a observation of \(B\) falling short of \(A\) (note 4, page 48 discusses an example).

The difference in the first subtraction is called “the remainder” (šapiltum, more literally “the diminished”). In the second, the excess is referred to as the “going-beyond” (watartum/dirig).

There are several synonyms or near-synonyms for “tearing out.” We shall encounter “cutting off” (ḫarāṣum) (AO 8862 #2, page 60) and “make go away” (šutbûm) (VAT 7532, page 65).

“Multiplications”

Four distinct operations have traditionally been interpreted as multiplication.

First, there is the one which appears in the Old Babylonian version of the multiplication table. The Sumerian term (a.rá, derived from the Sumerian verb rá, “to go”) can be translated “steps of.” For example, the table of the multiples of 6 runs:

1 step of 6 is 6

2 steps of 6 are 12

3 steps of 6 are 18

…

Three of the texts we are to encounter below (TMS VII #2, page 34, TMS IX #3, page 57, and TMS VIII #1, page 78) also use the Akkadian verb for “going” (alākum) to designate the repetition of an operation: the former two repeat a magnitude \(s\) \(n\) times, with outcome \(n \cdot s\) (TMS VII #2, line 18; TMS IX #3, line 21); TMS VIII #1 line 1 joins a magnitude \(s\) \(n\) times to another magnitude \(A\), with outcome \(A+n \cdot s\).

The second “multiplication” is defined by the verb “to raise” (našûm/íl/nim). The term appears to have been used first for the calculation of volumes: in order to determine the volume of a prism with a base of \(G\) sar and a height of \(h\) kùš, one “raises” the base with its standard thickness of 1 kùš to the real height \(h\). Later, the term was adopted by analogy for all determinations of a concrete magnitude by multiplication. "Steps of" instead designates the multiplication of an abstract number by another abstract number.

The third “multiplication” (šutakūlum/gu₇.gu₇), “to make \(p\) and \(q\) hold each other"—or simply, because that is almost certainly what the Babylonians thought of, "make \(p\) and \(q\) hold (namely, hold a rectangle)”⁷—is no real multiplication. It always concerns two line segments \(p\) and \(q\). Since \(p\) and \(q\) as well as the area \(A\) of the rectangle are all measurable, almost all texts give the numerical value of \(A\) immediately after prescribing the operation—"make 5 and 5 hold: 25"—without mentioning the numerical multiplication of 5 by 5 explicitly. But there are texts that speak separately about hte numerical multiplication, as "\(p\) steps of \(q\)," after prescribing the construction, or which indicate that the process of "making hold" creates "a surface"; both possibilities are exemplified in AO 8862 #2 (page 60). If a rectangle exists already, its area is determined by “raising,” just as the area of a triangle or a trapezium. Henceforth we shall designate the rectangle which is “held” by the segments \(p\) and \(q\) by the symbol (\(p\),\(q\)), while (\(a\)) will stand for the square which a segment \(a\) "holds together with itself" (in both cases, the symbol designates the configuration as well the area it contains, in agreement with the ambiguity inherent in the concept of "surface"). The corresponding numerical multiplications will be written symbolically as \(p\) \(q\) and \(a\) \(a\).

The last “multiplication” (eṣēpum) is also no proper numerical multiplication. “To repeat” or “to repeat until \(n\)" (where \(n\) is an integer small enough to be easily imagined, at most 9) stands for a "physical" doubling or \(n\)-doubling—for example that doubling of a right triangle with sides (containing the right angle) \(a\) and \(b\) which produces a rectangle (\(a\),\(b\)).

Division

The problem “what should I raise to \(d\) in order to get \(P\)?" is a division problem, with answer \(P \div d\). Obviously, the Old Babylonian calculators knew such problems perfectly well. They encountered them in their "algebra" (we shall see many examples below) but also in practical planning: a worker can dig \(N\) \(\mathrm{NINDAN}\) irrigation canal in a day; how many workers will be needed for the digging of 30 \(\mathrm{NINDAN}\) in 4 days? In this example the problem even occurs twice, the answer being \((30 \div 4) \div N\). But division was no separate operation for them, only a problem type.

In order to divide 30 by 4, they first used a table (Figure 1.2), in which they could read (but they had probably learned it by heart in school⁸) that igi \(4\) is \(15^{\prime}\); afterwards they "raised" \(15^{\prime}\) to \(30\) (even for that tables existed, learned by heart at school), finding \(7^{\circ} 30^{\prime}\).⁹

Figure \(1.2\): Translation of the Old Babylonian table of reciprocals (igi).

Primarily , igi \(n\) stands for the reciprocal of n as listed in the table or at least as easily found from it, not the number \(\frac{1}{n}\) abstractly. In this way, the Babylonians solved the problem \(P \div d\) via a multiplication \(P \cdot \frac{1}{d}\) to the extent that this was possible.

However, this was only possible if \(n\) appeared in the igi table. Firstly, that required that \(n\) was a “regular number," that is, that \(\frac{1}{n}\) could be written as a finite "sexagesimal fraction.”¹⁰ However, of the infinitely many such numbers only a small selection found place in the table—around 30 in total (often, \(1\) \(12\), \(1\) \(15\) and \(1\) \(20\) are omitted "to the left" since they are already present "to the right").

In practical computation, that was generally enough. It was indeed presupposed that all technical constants—for example, the quantity of dirt a worker could dig out in a day—were simple regular numbers. The solution of “algebraic” problems, on the other hand, often leads to divisions by a non-regular divisor . In such cases, the texts write “what shall I posit to \(d\) which gives me \(A\)?", giving immediately the answer "posit \(Q\), \(A\) will it give you.”¹¹ That has a very natural explanation: these problems were constructed backwards, from known results. Divisors would therefore always divide, and the teacher who constructed a problem already knew the answer as well as the outcome of divisions leading to it.

Halves

\(\frac{1}{2}\) may be a fraction like any other: \(\frac{2}{3}\), \(\frac{1}{3}\), \(\frac{1}{4}\), etc. This kind of half, if it is the half of something, is found by raising that thing to \(30^{\prime}\). Similarly, its \(\frac{1}{3}\) is found by raising to \(20^{\prime}\), etc. This kind of half we shall meet in AO 8862 #2 (page 60).

But \(\frac{1}{2}\) (in this case necessarily the half of something may also be a "natural" or "necessary" half, that is, a half that could be nothing else. The radius of a circle is thus the "natural" half of the diameter: no other part could have the same role. Similarly, it is by necessity that exact half of the base that must be raised to the height of a triangle in order to give the area—as can be seen on the figure used to prove the formula (Figure 1.3).

Figure \(1.3\)

This “natural” half had a particular name (bāmtum), which we may translate “moiety.” The operation that produced it was expressed by the verb “to break” (ḫepûm/gaz)—that is, to bisect, to break in two equal parts. This meaning of the word belongs specifically to the mathematical vocabulary; in general usage the word means to crush or break in any way (etc.).

Square and “square root”

The product \(a \cdot a\) played no particular role, neither when resulting from a "raising" nor from an operation of "steps of." A square, in order to be something special, had to be a geometric square.

But the geometric square did have a particular status. One might certainly “make \(a\) and \(a\) hold" or "make a together with itself hold"; but one might also "make \(a\) confront itself" (šutamḫurum, from maḫārum “to accept/receive/approach/welcome”). The square seen as a geometric configuration was a “confrontation” (mitḫartum, from the same verb).¹² Numerically, its value was identified with the length of the side. A Babylonian “confrontation” thus is its side while it has an area; inversely, our square (identified with what is contained and not with the frame) is an area and has a side. When the value of a “confrontation” (understood thus as its side) is found, another side which it meets in a corner may be spoken of as its “counterpart”—meḫrum (similarly from maḫārum), used also for instance about the exact copy of a tablet.

In order to say that \(s\) is the side of a square area \(Q\), a Sumerian phrase (used already in tables of inverse squares probably going back to Ur III, see imminently) was used: "by \(Q\), \(s\) is equal" —the Sumerian verb being íb.si₈. Sometimes, the word íb.si₈ is used as a noun, in which case it will be translated “the equal” in the following. In the arithmetical interpretation, “the equal” becomes the square root.

Just as there were tables of multiplication and of reciprocals, there were also tables of squares and of “equals.” They used the phrases “\(n\) steps of \(n\), \(n^{2}\)" and "by \(n^{2}\), \(n\) is equal" (1\(n\) 60). The resolution of “algebraic” problems, however, often involves finding the “equals” of numbers which are not listed in the tables . The Babylonians did possess a technique for finding approximate square roots of non-square numbers—but these were approximate. The texts instead give the exact value, and once again they can do so because the authors had constructed the problem backward and therefore knew the solution. Several texts, indeed, commit calculational errors, but in the end they give the square root of the number that should have been calculated, not of the number actually resulting! An example of this is mentioned in footnote 8, page 73.