What we now know about Old Babylonian algebra—its flexibility, its operational power in the solution

## The Scribe School

Old Babylonian mathematics was not the high-status diversion of wealthy and highly intelligent amateurs, as Greek mathematicians were or aspired to be. According to the format of its texts it was taught in the scribe

The word “scribe” might mislead. The scribe certainly knew to write. But the ability to calculate was just as important—originally, writing had been invented as subservient to accounting, and this subordinated function with respect to calculation remained very important. The modern colleagues of the scribe are engineers, accountants and notaries.

Therefore, it is preferable not to speak naively of “Babylonian mathematicians.” Strictly speaking, what was taught number- and quantity-wise in the scribe school should not be understood primarily as “mathematics” but rather as *calculation*. The scribe should be able to *find the correct number*, be it in his engineering function, be it as an accountant. Even problems that do not consider true practice always concern measurable magnitudes

That is one of the reasons that many of the problems

If we really want to find Old Babylonian “mathematicians”*created* the techniques and discovered how to *construct* problems that were difficult but could still be solved. For example we may think of the problem TMS XIX #2 (not included in the present book): to find the sides
and
of a rectangle from its area and from the area of another rectangle

## The First Purpose: Training Numerical Calculation

When following the progression of one of the algebraic texts—in particular one of the more complicated specimens—one is tempted to trust*should* be correct). The reader who has been more suspicious will, on the other hand, have received a good training in sexagesimal arithmetic.

That illustrates one of the functions of algebra in the curriculum: it provided a pretext for training the manipulation of difficult numbers. As the aim of the school was the training of professional routine, the intensive cultivation of sexagesimal arithmetic was obviously welcome.

This observation can be transferred to our own epoch and its teaching of second-degree equations. Its aim was never to assist the copying of gramophone records or CDs to a cassette tape. But the reduction of complicated equations and the ensuing solution of second-degree equations is not the worst pretext for familiarizing students with the manipulation of symbolic algebraic expressions and the insertion of numerical values in a formula; it seems to have been difficult to find alternatives of more convincing direct practical relevance—and the general understanding and flexible manipulation of algebraic formulas and the insertion of numerical values in formulas *are* routines which are necessary in many jobs.

## The Second Purpose: Professional Pride

The acquisition of professional dexterity is certainly a valid aim, even if it is reached by indirect means. Yet that was not the only purpose of the teaching of apparently useless mathematics. Cultural

Quite a few such texts are known. They speak little of everyday routines—the ability to handle these was too elementary, in order to be justified the pride of a scribe

To find the area of a rectangular field from its length and width was also not suited to induce much self-respect—any bungler in the trade could do that. Even the determination of the area of a trapezium was too easy. But to find a length and a width from their sum and the area they would “hold” was already more substantial; to find them from data such as those of AO 8862 #2,*real* scribe, as somebody who could command the respect of the non-initiates.

We have no information about Sumerian and mathematics being used for social screening of apprentice-scribes—one of the functions of such matters in the school of today: Since the scribe school*beyond* what was necessary) were essential components of the professional identity of engineers, architects and officers.^{1}

Even analysis of the cultural function of “advanced” Old Babylonian mathematics may thus teach us something about our own epoch.

## Footnotes

In the nineteenth century, precisely these three groups provided the bulk of subscribers to the *Journal des mathématiques élémentaires* and similar periodicals. The *Ladies’ Diary*, published from 1704 until 1841 and rich in mathematical contents, could also aim at a social group that was largely excluded from Oxford-Cambridge and public-school Latinity and Grecity, to which even genteel women had no access.