8.1: The Origin- Surveyors’ Riddles

On the contrary, the algebra of the Old Babylonian scribe school is no continuation of century- (or millennium-)old school traditions—nothing similar had existed during the third millennium. It is one expresssion among others of the new scribal culture of the epoch. In principle, the algebra might have been invented within the school enviornment—the work on bilinual texts and the study of Sumerian grammar from an Akkadian point of view certainly were. Such an origin would fit the fact that the central vocabulary for surveying and part of that used in practical calculation is in Sumerian or at least written with Sumerian logograms ("length," "width," igi, "be equal by"), while the terms that characterize the algebraic genres as well as that which serves to express problems is in Akkadian.

However, an invention wihtin the scribe school agrees very badly with other sources. In particular it is in conflict with the way problems and techniques belonging to the same family turn up in Greek and medieval sources. A precise analysis of all paralle material reveals a very different story—the material is much too vast to allow a complete presentation of the argument here but part of it is woven into the following discussion.

The surveyors of central Iraq (perhaps a wider region, but that remains a hypothesis in as far as this early epoch is concerned) had a tradition of geometrical riddles. Such professional riddles are familiar from other pre-modern environments of mathematical practitioners (specialists of commercial computation, accounting, master builders, and of course surveying) whose formation was based on apprenticeship and not taken care of by a more or less learned school. As an example we may cite the problem of the "hundred fowls" which one finds in numerous Chinese, Indian, Arabic and European problem collections from the Middle Ages:

Somebody goes to the market and buys 100 fowls for 100 dinars. A goose costs him 3 dinars, a hen 2 dinars, and of sparrows he gets 3 for each dinar. Tell me, if you are an expert calculator, what he bought!¹

There are many solutions. 5 geese, 32 hens, and 63 sparrows; 10 geese, 24 hens and 66 sparrows; etc. However when answering a riddle, even a mathematical riddle, one needs not give an exhaustive solution, nor give a proof (except the numerical proof that the answer fulfills the conditions)² Who is able to give one good answer shows himself to be a competent calculator “to the stupefaction of the ignorant” (as says a manual of practical arithmetic from 1540).

Often the solution of a similar riddle asks for the application of a particular trick. Here, for instance, one may notice that one must buy 3 sparrows each time one buys a goose—that gives 4 fowls for 4 dinars—and 3 sparrows for each two hens—5 fowls for 5 dinars.

Such "recreational problems" (as they came to be called after having been adopted into a mathematical culture rooted in school, where their role was to procure mathematical fun) had a double function in the milieu where they originated. On one hand, they served training—even in today's school, a lion that eats three math teachers an hour may be a welcome variation on kids receiving 3 sweets a day. On the other, and in particular (since the central tricks rarely served in practical computation), they allowed the members of the profession to feel like "truly expert calculators"—a parallel to what was said above on the role of Sumerian and "too advanced" mathematics for the Old Babylonian scribes.

At some moment between 2200 and 1800 bce, the Akkadian surveyors invented the trick that was later called "the Akkadian method," that is, the quadratic completion; around 1800, a small number of geometrical riddles about squares, rectangles and circles circulated whose solution was based on this trick. A shared characteristic of these riddles was to consider solely elements that are directly present in the figures—for instance the side or all four sides of a square, never "3 times the area" or "\(\frac{1}{3}\) of the area." We may say that the problems are defined without coefficients, of, alternatively, with "natural" coefficients.

If \({ }_{4} c\) stands for "the 4 sides" and \(\square(c)\) for the area of a square, \(d\) for the diagonal and \((\ell, w)\) for the area of a rectangle, the list of riddles seems to have encompassed the following problems:

\(\begin{aligned}
c+\square(c) &=110 \\
4^{c+} \square(c) &=140 \\
\square(c)-c &=90 \\
\square(c)-{ }_{4} c &=60(?)
\end{aligned}\)

\(\ell+w=\alpha\), \((\ell, w)=\beta\)

\(\ell-w=\alpha\), \((\ell, w)=\beta\)

\(\ell+w=\alpha\), \((\ell-w)+\) \((\ell, w)=\beta\)

\(\ell-w=\alpha\), \((\ell+w)+\) \((\ell, w)=\beta\);

\(d=\alpha\), \((\ell, w)=\beta\).

Beyond that, there were problems about two squares (sum of or difference between the sides given together with the sum of or difference between the areas); a problem in which the sum of the perimeter, the diameter and the area of a circle is given, and possible the problem \(d-c=4\) concerning a square, with the pseudo-solution \(c=10\), \(d=14\); two problems about a rectangle, already known before 2200 bce, have as their data, one the area and the width, the other the area and the length. That seems to be all.³

These riddles appear to have been adopted into the Old Babylonian scribe school, where they became the starting point for the development of the algebra as a genuine discipline. Yet the school did not take over the riddle tradition as it was. A riddle, in order to provoke interest, must speak of conspicuous entities (the side, all four sides, etc.); a school institution, on the other hand, tends to engage in systematic variation of coefficients—in particular a school which, like that of the Mesopotamian scribes since the invention of writing in the fourth millennium, had always relied on very systematic variation.⁴ In a riddle it is also normal to begin with what is most naturally there (for instance the four sides of a square) and to come afterwards to derived entities (here the area). In school, on the contrary, it seems natural to privilege the procedure, and therefore to speak first of that surface which eventually is to be provided with a "projection" or a "base."

Such considerations explain why a problem collection about squres like BM 13901 moves from a single to two and then three squares, and why all problems except the archaizing #23, "the four sides and the area," invariably speak of areas before mentioning the sides. But the transformation does not stop there. Firstly, the introduction of coefficients asked for the introduction of a new technique, the change of scale in one direction (ant then different changes in the two directions, as in TMS IX #3); the bold variation consisting in the addition of a volume and an area gave rise to a more radical innovation: the use of factorization. The invention of these new techniques made possible the solution of even more complicated problems.

On the other hand, as a consequence of the drill of systematic variation, the solution of the fundamental problems became a banality on which professional self-esteem could not be built: thereby work on complicated problems became not only a possiblility but also a cultural necessity.

One may assume that the orientation of the scribal profession toward a wide range of practices invited the invention of problems outside abstract surveying geometry where the algebraic methods could be deployed—and therefore, even though "research" was no aim of the scribal school, to explore the possibilities of representation. It is thus, according to this reconstruction, the transfer to the school that gave to the cut-and-paste technique the possibility of becoming the heart of a true algebra.

Other changes were less momentous though still conspicuous. In the riddles, 10 was the preferred value for the side of the square, remaining so until the sixteenth century ce; the favorite value in school was \(30^{\prime}\), and when an archaizing problem retained 10 it was interpreted as \(10^{\prime}\).⁵ Finally, as explained above (page 34), the hypothetical “somebody” asking a question was replaced by a professorial “I.”

BM 13901 #23 (page 75), retaining "the four widths and the surface" (in that order) and the side 10 while changing its order of magnitude, is thus a characteristic fossil pointing to the riddle tradition. Even its language is archaizing, suggesting the ways of surveyors not educated in the scribe school. Taking into account its position toward the end of the text (#23 of 24 problems, #24 being the most intricate of all), we may see it as something like "last problem before Christmas."

It appears that the first development of the algebraic discipline took place in the Eshnunna region, north of Babylon, during the early decades of the eighteenth century;⁶ from this area and period we have a number of mathematical texts that for once have been regularly excavated and which can therefore be dated. By then, Eshnunna was a cultural centre of the whole north-central part of Iraq; Eshnunna also produced the first law-code outside the Sumerian south. The text Db₂–146 (below, page 126) comes from a site belonging to the Eshnunna kingdom.

In c. 1761 Eshnunna was conquered by Hammurabi and destroyed. We know that Hammurabi borrowed the idea of a law-code, and can assume that he brought enslaved scholars back. Wheter he also brought scholars engaged in the production or teaching of mathematics is nothing but a guess (the second-millennium strata of Babylon are deeply buried below the remains of the first-millennium world city), but in any case the former Sumerian south took up the new mathematical discipline around 1750—AO 8862 (above, page 60), with its still unsettled terminology and format, seems to represent an early specimen from this phase.

Problems from various sites in the Eshnunna region deal with many of the topics also known from later—the early rectangle variant of the "broken-reed" problem mentioned on page 70 is from one of them. Strikingly, however, there is not a single example of representation. AO 8862, on the other hand, already contains an example, in which a number of workers, their working days and the bricks they have produced are "heaped." It does not indicate the procedure, but clearly the three magnitudes have to be represented by the sides of a rectangle and its area multiplied by a coefficient. A large part of the Eshnunna texts start "If somebody asks you thus [...]," found neither in AO 8862 nor in any later text (except as a rudiment in the archaizing BM 13901 #23).

Not much later, we have a number of texts which (to judge from their orthography) were written in the south. Several text groups obey very well-defined canons for format and terminology (not the same in all groups), demonstrating a conscious striving for regularity (the VAT- and Str-texts all belong here). However, around 1720 the whole south seceded, after which scribal culture there was reduced to a minimum; mathematics seems not to have survived. From the late seventeenth century, we have a fair number of texts from Sippar, somewhat to the north of Babylon (BM 85200 + VAT 6599 is one of them), and another batch from Susa in western Iran (the TMS-texts), which according to their terminology descend from the northern type first developed in Eshnunna. And then, nothing more ... .