1.2: The First Algebra and the First Interpretation

Before speaking about algebra, one should in principle know what is meant by that word. For the moment, however, we shall leave aside this question; we shall return to it in the end of the book; all we need to know for the moment is that algebra has to do with equations.

Figure \(1.1\): The cuneiform version of the problem BM 13901 #1.

Indeed, when historians of mathematics discovered in the late 1920s that certain cuneiform texts (see the box “Cuneiform Writing,” page 10) contain “algebraic” problems, they believed everybody knew the meaning of the word.

Let us accept it in order to enter their thinking, and let us look at a very simple example extracted from a text written during the eighteenth century bce in the transliteration normally used by Assyriologists—as to the function of italics and small caps, see page 23 and the box “Cuneiform Writing,” page 10 (Figure 1.1 shows the cuneiform version of the text):

1 a.šà^l[am] ù mi-it-ḫar-ti ak-m[ur-m]a 45-e 1 wa-ṣi-tam

2 ta-ša-ka-an ba-ma-at 1 te-ḫe-pe [3]0 ù 30 tu-uš-ta-kal

3 15 a-na 45 tu-ṣa-ab-ma 1-[e] 1 íb.si₈ 30 ša tu-uš-ta-ki-lu

4 lìb-ba 1 ta-na-sà-aḫ-ma 30 mi-it-ḫar-tum

The unprepared reader, finding this complicated, should know that for the pioneers it was almost as complicated. Eighty years later we understand the technical terminology of Old Babylonian mathematical texts; but in 1928 it had not yet been deciphered, and the numbers contained in the texts had to provide the starting point.¹ https://mprl-series.mpg.de/textbooks/2/2/index.html#22 " data-title="Permanent Link" rel="popover" title="title">

Cuneiform Writing

From its first beginning, Mesopotamian writing was made on a flattened piece of clay, which was then dried in the air after the inscription (a “tablet”). In the fourth millennium, the signs were drawings made by means of a pointed stylus, mostly drawings of recognizable objects representing simple concepts. Complex concepts could be expressed through combination of the signs; a head and a bowl containing the daily ration of a worker meant “allocation of grain” (and later “to eat”).

The signs for numbers and measures, however, were made by vertical or oblique impression of a cylindrical stylus.

With time, the character of the script changed in two ways. Firstly, instead of tracing signs consisting of curved lines one impressed them with a stylus with sharp edges, dissolving the curved lines into a sequence of straight segments. In this way, the signs seem to be composed of small wedges (whence the name “cuneiform”).

In the second half of the third millennium, numerical and metrological signs came to be written in the same way. The signs became increasingly stylized, loosing their pictographic quality; it is then not possible to guess the underlying drawing unless one knows the historical development behind the sign. Until around 2000 bce, however, the variations of characters from one scribe to another show that the scribes knew the original drawings.

Let us for instance look at the character which initially depicted a vase with a spout (left).

In the middle we see three third-millennium variants of the same character (because the script was rotated 90 degrees to the left in the second millennium, it is habitual to show the third-millennium script in the same way). If you know the origin, it is still easy to recognize the underlying picture. To the right we see two Old Babylonian variants; here the picture is no longer suggested.

The other change concerns the use of the way the signs were used (which implies that we should better speak of them as “characters”). The Sumerian word for the vase is dug. As various literary genres developed alongside accounting (for instance, royal inscriptions, contracts and proverb collections), the scribes needed ways to write syllables that serve to indicate grammatical declinations or proper nouns. This syllabic system served also in the writing of Akkadian. For this purpose, signs were used according to their approximate phonetic value; the “vase” may thus stand for the syllables dug, duk, tug and tuk. In Babylonian writing, the Sumerian sign might also serve as a “logogram” or “word sign” for a word meaning the same as dug—namely karpatum

Words to be read as logograms or in Sumerian are transliterated in small caps; specialists (cf. Appendix B) often distinguish Sumerian words whose phonetic value is supposed to be known, which are then written in s p a c e d w r i t i n g, from those rendered by their “sign name” (corresponding to a possible reading), which are written as small caps. Phonetic Akkadian writing is transcribed as italics .

Assyriologists distinguish “transcriptions” from “transliterations.” A “transcription” is an intended translation into Akkadian written in Latin alphabet. In a “transliteration” each cuneiform character is rendered separately according to its presumed phonetic or logographic value.

It was already known that these numbers were written in a place-value system with base 60 but without indication of absolute order of magnitude (see the box “The Sexagesimal System,” page 14). We must suppose that the numbers appearing in the text are connected, and that they are of at least approximately the same order of magnitude (we remember that “1” may mean one as well as 60 or ). Let us therefore try to interpret these numbers in the following order:

\(45^{\prime}\left(=\frac{3}{4}\right)-1^{\circ}-1^{\circ}-30^{\prime}\left(=\frac{1}{2}\right)-30^{\prime}-15^{\prime}\left(=\frac{1}{4}\right)-45^{\prime}-1^{\circ}-1^{\circ}-30^{\prime}-1^{\circ}-30^{\prime}\).

In order to make the next step one needs some fantasy. Noticing that \(30^{\prime}\) is \(\frac{1}{2} \cdot 1\) and \(15^{\prime}=\left(30^{\prime}\right)^{2}\) we may think of the equation

\(x^{2}+1 \cdot x=\frac{3}{4}\).

Today we solve it in these steps (neglecting negative numbers, a modern invention):

\begin{aligned}
x^{2}+1 \cdot x=\frac{3}{4} & \Leftrightarrow x^{2}+1 \cdot x+\left(\frac{1}{2}\right)^{2}=\frac{3}{4}+\left(\frac{1}{2}\right)^{2} \\
\Leftrightarrow & x^{2}+1 \cdot x+\left(\frac{1}{2}\right)^{2}=\frac{3}{4}+\frac{1}{4}=1 \\
& \Leftrightarrow\left(x+\frac{1}{2}\right)^{2}=1 \\
\Leftrightarrow & x+\frac{1}{2}=\sqrt{1}=1 \\
\Leftrightarrow & x=1-\frac{1}{2}=\frac{1}{2}
\end{aligned}.

As we see, the method is based on addition, to both sides of the equation, of the square on half the coefficient of the first-degree term —here . That allows us to rewrite the left-hand side as the square on a binomial:

\(x^{2}+1 \cdot x+\left(\frac{1}{2}\right)^{2}=x^{2}+2 \cdot \frac{1}{2} \cdot x+\left(\frac{1}{2}\right)^{2}=\left(x+\frac{1}{2}\right)^{2}\).

This small trick is called a “quadratic completion.”

Comparing the ancient text and the modern solution we notice that the same numbers occur in almost the same order. The same holds for many other texts. In the early 1930s historians of mathematics thus became convinced that between 1800 and 1600 bce the Babylonian scribes knew something very similar to our equation algebra. This period constitutes the second half of what is known as the “Old Babylonian” epoch (see the box “Rudiments of General History,” page 7).

The next step was to interpret the texts precisely. To some extent, the general, non-technical meaning of the vocabulary could assist. In line 1 of the problem on page 9, ak-mur may be translated “I have heaped.” An understanding of the “heaping” of two numbers as an addition seems natural and agrees with the observation that the “heaping” of and (that is, of and ) produces 1. When other texts “raise” (našûm) one magnitude to another one, it becomes more difficult. However, one may observe that the “raising” of 3 to 4 produces 12, while 5 “raised” to 6 yields 30, and thereby guess that “raising” is a multiplication.

In this way, the scholars of the 1930s came to choose a purely arithmetical interpretation of the operations—that is, as additions, subtractions, multiplications and divisions of numbers. This translation offers an example :²

1 I have added the surface and (the side of) my square: \(45^{\prime}\).

2 You posit \(1^{\circ}\), the unit. You break into two \(1^{\circ}\): \(30^{\prime}\). You multiply (with each other) \(\left[30^{\prime}\right] and 30^{r}\):

3 \(15^{\prime}\). You join \(15^{\prime}\) to \(45^{\prime}\): \(1^{\circ}\). \(1^{\circ}\) is the square of \(1^{\circ}\). \(30^{\prime}\), which you have multiplied (by itself),

4 from \(1^{\circ}\) you subtract: \(30^{\prime}\) is the (side of the) square.

Such translations are still found today in general histories of mathematics. They explain the numbers that occur in the texts, and they give an almost modern impression of the Old Babylonian methods. There is no fundamental difference between the above translation and the solution by means of equations. If the side of the square is \(x\), then its area is \(x^{2}\). Therefore, the first line of the text—the problem to be solved—corresponds to the equation \(x^{2}+1 \cdot x=\frac{3}{4}\). Continuing the reading of the translation we see that it follows the symbolic transformations on page 12 step by step.

However, even though the present translation as well as others made according to the same principles explain the numbers of the texts, they agree less well with their words, and sometimes not with the order of operations. Firstly, these translations do not take the geometrical character of the terminology into account, supposing that words and expressions like “(the side of) my square,” “length,” “width” and “area” of a rectangle denote nothing but unknown numbers and their products. It must be recognized that in the 1930s that did not seem impossible a priori—we too speak of \(3^{2}\) as the “square of 3” without thinking of a quadrangle.

But there are other problems. The most severe is probably that the number of operations is too large. For example, there are two operations that in the traditional interpretation are understood as addition: “to join to” (waṣābum/da[+.1ex]h, the infinitive corresponding to the tu-ṣa-ab of our text) and “to heap” (kamārum/gar.gar, from which the ak-mur of the text). Both operations are thus found in our brief text, “heaping” in line 1 (where it appears as “add”) and “joining” in line 3.

The Sexagesimal Place-Value System

The Old Babylonian mathematical texts make use of a place-value number system with base \(60\) with no indication of a "sexagesimal point." In out notation, which also employs place value, the digit "\(1\)" may certainly represent the number \(1\), but also the numbers \(10\), \(100\), \(...\), as well as \(0.1\), \(0.01\), \(...\). Its value is determined by its distance from the decimal point.

Similarly, "\(45\)" written by a Babylonian scribe may mean \(45\); but it may also stand for \(\frac{45}{60}\) (thus \(\frac{3}{4}\)); for \(45 \cdot 60\); etc. No decimal point determines its "true" value. The system corresponds to the slide rule of which engineers made use before the arrival of the electronic pocket calculator. This device also had no decimal point, and thus did not indicate the absolute order of magnitude. In order to know whether a specific construction would ask for \(3.5 m^{3}\), \(35 m^{3}\) or \(350 m^{3}\) of concrete, the engineer had recourse to mental calculation.

For writing numbers between 1 and 59, the Babylonians made use of a vertical wedge () repeated until 9 times in fixed patterns for hte numbers 1 to 9, and of a Winkelhaken (a German loanword originally meaning "angular hook") () repeated until 5 times for the numbers \(10\), \(20\), \(...\), \(50\).

A modern reader is not accustomed to reading numbers with undetermined order of magnitude. In translations of Babylonian mathematical texts it is therefore customary to indicate the order of magnitude that has to be attributed to numbers. Several methods to do that are in use. In the present work we shall employ a generalization of the degree-minute-second notation. If means \(\frac{15}{60}\), we shall transcribe it \(15^{\prime}\), if it corresponds to \(\frac{15}{60 \cdot 60}\), we shall write \(15^{\prime \prime}\). If it represents \(15 \cdot 60\), we write , etc. If it stands for \(15\), we write \(15\) or, if that is needed in order to avoid misunderstandings, \(15^{\circ}\). understood as \(10+5 \cdot 60^{-1}\) will thus be transcribed \(10^{\circ} 5^{\prime}\)

understood as \(30^{\prime}\) thus means \(\frac{1}{2}\).

understood as \(45^{\prime}\) means \(\frac{3}{4}\).

understood as \(12^{\prime}\) means \(\frac{1}{5}\); understood as it means \(720\).

understood as \(10^{\prime}\) means \(\frac{1}{6}\).

may mean or \(16^{\circ} 40^{\prime}=16 \frac{2}{3}\), etc.

may mean , \(1^{\circ} 40^{\prime}=1 \frac{2}{3}, 1^{\prime} 40^{\prime \prime}=\frac{1}{36}\), etc.

Outside school, the Babylonians employed the place-value system exclusively for intermediate calculations (exactly as an engineer used the slide rule fifty years ago). When a result was to be inserted into a contract or an account, they could obviously not allow themselves to be ambiguous; other notations allowed them to express the precisse number they

intended.

Certainly, we too know about synonyms even within mathematics—for instance, “and,” “added to” and “plus”; the choice of one word or the other depends on style, on personal habits, on our expectations concerning the interlocutor, and so forth. Thureau-Dangin , as we see, makes use of them, following the distinctions of the text by speaking first of “addition” and second of “joining”; but he argues that there is no conceptual difference, and that nothing but synonyms are involved—“there is only one multiplication,” as he explains without noticing that the argument is circular.

Synonyms, it is true, can also be found in Old Babylonian mathematics. Thus, the verbs “to tear out” (nasāḫum/zi) and “to cut off” (ḫarāṣum/kud) are names for the same subtractive operation: they can be used in strictly analogous situations. The difference between “joining” and “heaping,” however, is of a different kind. No text exists which refers to a quadratic completion (above, page 12) as a “heaping.” “Heaping,” on the other hand, is the operation to be used when an area and a linear extension are added. These are thus distinct operations, not two different names for the same operation. In the same way, there are two distinct “subtractions,” four “multiplications” and even two different “halves.” We shall come back to this.

A translation which mixes up operations which the Babylonians treated as distinct may explain why the Babylonian calculations lead to correct results; but it cannot penetrate their mathematical thought.

Further, the traditional translations had to skip certain words which seemed to make no sense. For instance, a more literal translation of the last line of our small problem would begin “from the inside of \(1^{\circ}\)" (or even "from the heart" or "from the bowels"). Not seeing how a number 1 could possess an "inside" or "bowels," the translators tacitly left out the word.

Other words were translated in a way that differs so strongly from their normal meaning that it must arouse suspicion. Normally, the word translated “unity ” by Thureau-Dangin and “coefficient” by Neugebauer (waṣītum, from waṣûm, “to go out”) refers to something that sticks out, as that part of a building which architects speak about as a “projection.” That must have appeared absurd—how can a number 1 “stick out”? Therefore the translators preferred to make the word correspond to something known in the mathematics of their own days.

Finally, the order in which operations are performed is sometimes different from what seems natural in the arithmetical reading.

In spite of these objections, the interpretation that resulted in the 1930s was an impressive accomplishment, and it remains an excellent “first approximation.” The scholars who produced it pretended nothing more. Others however, not least historians of mathematics and historically interested mathematicians, took it to be the unique and final decipherment of “Babylonian algebra”—so impressive were the results that were obtained, and so scary the perspective of being forced to read the texts in their original language. Until the 1980s, nobody noticed that certain apparent synonyms represent distinct operations.³