1.4: Concerning the Texts and the Translations

The texts that are presented and explained in the following are written in Babylonian, the language that was spoken in Babylonia during the Old Babylonian epoch. Basically they are formulated in syllabic (thus phonetic) writing—that which appears as italics on page 11. All also make use of logograms that represent a whole word but indicate neither the grammatical form nor the pronunciation (although grammatical complements are sometimes added to them); these logograms are transcribed in small caps (see the box “Cuneiform Writing,” page 10). With rare exceptions, these logograms are borrowed from Sumerian, once the main language of the region and conserved as a scholars’ language until the first century ce (as Latin in Europe until recently). Some of these logograms correspond to technical expressions already used as such by the Sumerian scribes; igi is an example. Others serve as abbreviations for Babylonian words, more or less as viz in English, which represents the shorthand for videlicet in medieval Latin manuscripts but is pronounced namely.

As already indicated, our texts come from the second half of the Old Babylonian epoch, as can be seen from the handwriting and the language. Unfortunately it is often impossible to say more, since almost all of them come from illegal diggings and have been bought by museums on the antiquity market in Baghdad or Europe.

We have no direct information about the authors of the texts. They never present themselves, and no other source speaks of them. Since they knew how to write (and more than the rudimentary syllabic of certain laymen), they must have belonged to the broad category of scribes; since they knew how to calculate, we may speak about them as “calculators”; and since the format of the texts refers to a didactical situation, we may reasonably assume that they were school teachers.¹³

All this, however, results from indirect arguments. Plausibly, the majority of scribes never produced mathematics on their own beyond simple computation; few were probably ever trained at the high mathematical level presented by our texts. It is even likely that only a minority of school teachers taught such matters. In consequence, and because several voices speak through the texts (see page 33), it is often preferable to pretend that it is the text itself which “gives,” “finds,” “calculates,” etc.

The English translations that follow—all due to the author of the book—do not distinguish between syllabically and logographically written words (readers who want to know must consult the transliterations in Appendix B). Apart from that, they are “conformal”—that is, they are faithful to the original, in the structure of phrases¹⁴ as well as by using always distinct translations for words that are different in the original and the same translation for the same word every time it occurs unless it is used in clearly distinct functions (see the list of “standard translations” on page 129). In as far as possible the translations respect the non-technical meanings of the Babylonian words (for instance “breaking” instead of “bisecting”) and the relation between terms (thus “confront itself” and “confrontation”—while “counterpart” had to be chosen unrelated of the verbal root in order to respect the use of the same word for the copy of a tablet).

This is not to say that the Babylonians did not have a technical terminology but only their everyday language; but it is important that the technical meaning of a word be learned from its uses within the Old Babylonian texts and not borrowed (with the risk of being badly borrowed, as has often happened) from our modern terminology.

The Babylonian language structure is rather different from that of English, for which reason the conformal translations are far from elegant. But the principle of conformality has the added advantage that readers who want to can follow the original line for line in Appendix B (the bibliographic note on page 149 indicates where the few texts not rendered in the appendix were published).

In order to avoid completely illegible translations, the principle is not followed to extremes. In English one has to choose whether a noun is preceded by a definite or an indefinite article; in Babylonian, as in Latin and Russian, that is not the case. Similarly, there is no punctuation in the Old Babylonian texts (except line breaks and a particle that will be rendered “:”), and the absolute order of magnitude of place-value numbers is not indicated; minimal punctuation as well as indications of order of magnitude (′,‵ and °) have been added. Numbers that are written in the original by means of numerals have been translated as Arabic numerals, while numbers written by words (including logograms) have been translated as words; mixed writings appear mixed (for instance, “the 17th” and even “the 3rd” for the third).

Inscribed clay survives better than paper—particularly well when the city burns together with its libraries and archives, but also when discarded as garbage. None the less, almost all the tablets used for what follows are damaged. On the other hand, the language of the mathematical texts is extremely uniform and repetitive, and therefore it is often possible to reconstruct damaged passages from parallel passages on the same tablet. In order to facilitate reading the reconstructions are only indicated in the translations (as ^¿…^?) if their exact words are not completely certain. Sometimes a scribe has left out a sign, a word or a passage when writing a tablet which however can be restored from parallel passages on the same or closely kindred tablets. In such cases the restitution appears as〈...〉, while {...} stands for repetitions and other signs written by error (the original editions of the texts give the complete information about destroyed and illegible passages and scribal mistakes). Explanatory words inserted into the texts appear within rounded brackets (...).

Clay tablets have names, most often museum numbers. The small problem quoted above is the first one on the tablet BM 13901—that is, tablet #13901 in the British Museum tablet collection. Other names begin AO (Ancient Orient, Louvre, Paris), VAT (Vorderasiatische Texte, Berlin) or YBC (Yale Babylonian texts). TMS refers to the edition Textes mathématiques de Suse of a Louvre collection of tablets from Susa, an Iranian site in the eastern neighborhood of Babylon.

The tablets are mostly inscribed on both surfaces (“obverse” and “reverse”), sometimes in several columns, sometimes also on the edge; the texts are divided in lines read from left to right. Following the original editions, the translations indicate line numbers and, if actual, obverse/reverse and column.