8.2: The Heritage

Indeed, in 1595 a Hittite raid put an end to the already weak Old Babylonian state and social system. After the raid, power was grasped by the Kassites, a tribal group that had been present in Babylonia as migrant workers and marauders since Hammurabi's times. This caused an abrupt end to the Old Babylonian epoch and its particular culture.

The scribe school disappeared. For centuries, the use of writing was strongly reduced, and even afterwards scholar-scribes were taught as apprentices within "scribal families" (apparently bloodline families, not apprenticeship formalized as adoption).

Even sophisticated mathematics disappeared. The social need for practical calculation, though reduced, did not vanish; but the professional pride of scholar-scribes now built on the appurtenance to a venerated tradition. The scribe now understood himself as somebody who knew to write, even literature, and not as a calculator; much of the socially necessary calculation may already now have rested upon specialists whose scanty literary training did not quality them as "scribes" (in the first millennium, such a split is fairly certain).

The 1200 years that follow the collapse of the Old Babylonian cultural complex have not left a single algebra text. In itself that does not say much, since only a very small number of mathematical texts even in the vaguest sense have survived (a few accounting texts, traces of surveying, some tables of reciprocals and squares). But when a minimum of mathematical texts proper written by scholar-scribes emerges again after 400 bce, the terminology allows us to distinguish that which had been transmitted within their own environment from that which was borrowed once agin from a "lay" environment. To the latter category belongs a small handful of problems about squares and rectangles. they contain no representation, no variation of coefficients, nothing sophisticated like the "broken reed" or the oil trade, only problems close to the original riddles; it would hardly be justified to speak of them as representatives of an "algebra."

These late texts obviously do not inform us, neither directly nor indirectly, about the environment where the riddles had been transmitted, even though a continuation of the surveyors' tradition is the most verisimilar hypothesis. Sources from classical antiquity as well as the Islamic Middle Ages at least make it clear that the tradition that had once inspired Old Babylonian algebra had survived despite the disappearance of its high-level offspring.

The best evidence is offered by an Arabic manual of practical geometry, written perhaps around 800 ce (perhaps later but with a terminology and in a tradition that points to this date), and known from a Latin twelfth-century translation.⁷ It contains all the problems ascribed above to the riddle tradition except those about two squares and the circle problem—in particular the problem about “the four sides and the area,” in the same order as BM 13901 #23, and still with solution 10 (not \(10^{\prime}\). It also conserves the complex alternation between grammatical persons, the hypothetical "somebody" who asks the question in many of the earliest school texts, the exhortation to keep something in memory, and even the occasional justification of a step in the procedure by means of the quotation of words from the statement as something which "he" has said. Problems of the same kind turn up time and again in the following centuries—"the four sides and the area" (apparently for the last time) in Luca Pacioli's Summa de Arithmetica from 1494, "the side and the area" of a square in Pedro Nuñez’s Libro de algebra en arithmetica y geometria from 1567 (in both cases in traditional riddle order, and in the Summa with solution 10).

as “four feet.”

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Figure \(8.1\): "The area joined to the perimeter" of Geometrica.

Since the dsicovery of Babylonian algebra, it has often been claimed that one component of Greek theoretical geometry (namely, Euclid's Elements II. 1-10) should be a translation of the results of Babylonian algebra into geometric language. This idea is not unproblematic; Euclid, for example, does not solve problems but proves constructions and theorems. The geometric interpretaion of the Old Babylonian technique, on the other hand, would seem to speak in favor of the hypothesis.

However, if we align the ten theorems Elements II. 1-10 with the list of original riddles we make an unexpected discovery: all ten theorems can be connected directly to the list—they are indeed demonstrations that the naive methods of the riddle tradition can be justified according to the best theoretical standards of Euclid's days. In contrast, there is nothing in Euclid that can be connected to the innovations of the Old Babylonian school. Its algebra turns out to have been a blind alley—not in spite of its high level but rather because of this level, which allowed it to survive only in the very particular Old Babylonian school environment.

The extraordinary importance of the Elements in the history of mathematics is beyond doubt. None the less, the most important influence of the surveyors' tradition in modern mathematics is due to its interaction with medieval Arabic algebra.

Even Arabic algebra seems to have origianlly drawn on a riddle tradition. As mentioned above (page 92), its fundamental equations deal with an amount of money (a "possession") and its square root. They were solved according to rules without proof, like this one for the case "a possession and ten of its roots are made equal to 39 dinars":

you halve the roots, which in this question are 5. You then multiply them with themselves, from which arises 25; add them to 39, and they will be 64. You should take the root of this, which is 8. Next remove form it the half of the roots, which is 5. Then 3 remains, which is the root of the possession. And the possession is 9.

Already the first author of a treatise on algebra which we known (which is probably the first treatise about the topic⁸)—al-Khwārizmī, from the earlier ninth century ce—was not satisfied with rules that are not based on reasoning or proof. He therefore adopted the geometric proofs of the surveyors’ tradition corresponding to Figures 3.1, 3.3, 4.1 and, first of all, the characteristic configuration of Figure 4.12. Later, mathematicians like Fibonacci, Luca Pacioli and Cardano saw these proofs as the very essence of algebra, not knowing about the polynomial algebra created by al-Karajī, as-Samaw’al and their successors (another magnificent blind alley). In this way the old surveyors’ tradition conquered the discipline from within; the word census, the Latin translation of “possession,” came to be understood as another word for “square.” All of this happened in interaction with Elements II—equally in debt to the surveyors’ tradition, as we have just seen.

Thus, even though the algebra of the cuneiform tablets was a blind alley—glorious but blind all the same—the principles that it had borrowed from practitioners without erudition was not. Without this inspiration it is difficult to see how modern mathematics could have arisen. As has been said about God: "If he did not exist, one would have had to invent him."