Changing Trends in the Historiography of Mesopotamian Mathematics: An Insider's View

Snow C. P. , The two cultures: And a second look. An expanded version of The two cultures and the Scientific Revolution (London, 1964).

To some extent, of course, all history of science is ridden by the same dichotomy: Is history of science to be done and judged as history, or does it belong within the realm of the sciences? Logically, one should opt for the former answer; according to the down-to-earth sociology of the pay-roll and the institutional affiliation of most historians of science, however, most of them are scientists. Yet, even if the problem is shared by all history of science, it becomes more outspoken when the philology and history of the period involved gives outsiders the impression of an occult science, as is the case of Assyriology.

The article was first presented at the conference on “Contemporary trends in the historiography of science” (Corfu, June 1991), arranged by the Greek Society for the History of Science and Technology, and was accepted for publication by the editors of the proceedings. After several years' irresolution, however, Kluwer Publishers decided that it was too long and omitted it from the volume.

The rather few mathematical texts which we know from the late period, it is true, were written by and for members of the astronomical environment, a context that seems to have influenced the mathematical mode of thought.

A critical bibliography of works from the period 1854 to 1929 with relevance for the understanding of Mesopotamian mathematics will be found on pp. 1–36 in Friberg Jöran , “A survey of publications on Sumero-Akkadian mathematics, metrology and related matters (1854–1982)”, DMC, no. 1982–17.

Neugebauer O. , “Zur Geschichte der babylonischen Mathematik”, QS, i (1929–31), 67–80; and Frank Carl , Straßburger Keilschrifttexte in sumerischer und babylonischer Sprache (Schriften der Straßburger Wissenschaftlichen Gesellschaft in Heidelberg, Neue Folge, Heft 9; Berlin and Leipzig, 1928). The translation of the following passage is mine, as are other translations from original publications (unless otherwise stated).

According to what I was told in 1985 by Kurt Vogel, Schuster was in fact the first to discover the Babylonian solution of second-degree problems.

Neugebauer O. , “Beiträge zur Geschichte der Babylonischen Arithmetik”, QS, i (1929–31), 120–30; and Schuster H. S. , “Quadratische Gleichungen der Seleukidenzeit aus Uruk”, QS, i (1929–31), 194–200. On the periods, see Box I.

Neugebauer O. Struve W. W. , “Über die Geometrie des Kreises in Babylonien”, QS, i (1929–1931), 81–92.

Schuster , “Quadratische Gleichungen …”, 194.

Further explanation and exemplification in Høyrup Jens , “Algebra and naive geometry: An investigation of some basic aspects of Old Babylonian mathematical thought”, AoF, xvii (1990), 27–69, 262–354, pp. 43–54.

It may be accurate that the two “hated each other”, as I was told by Olaf Schmidt, who was close to Neugebauer in the later 1930s — While Bruins's statement that Thureau-Dangin considered the MKT “a flood of errors” probably misrepresents Thureau-Dangin as much as his following remarks on the latter's intention in publishing the Textes mathématiques babyloniens (compare Bruins E. M. , “Requisites for the interpretation of ancient mathematics” ( Janus , lxxi (1984), 107–34, p. 107), with TMB, p. xl, the final paragraph). But even if their mutual feelings may have been acrimonious, the fruitful outcome of the process shows the function of scholarly mores at their best. Competition never prevented either of them from giving advice or learning from criticism, nor from emphasizing the merits of the other's publications. It would be difficult to find more indisputable corroboration of Robert Merton's theses concerning the function of the “institutional imperatives of science” in the “Note on science and democracy”, Journal of legal and political sociology, i (1942), 115–26. One possible exception to this optimistic verdict should perhaps be mentioned, even though I know about it only from rumours and have not been able to verify it: Thureau-Dangin is claimed to have taken care that Neugebauer should not get access to the extremely important mathematical texts from Susa, which had been found already in 1933, and which were only published in problematic form in 1961 as TMS (cf. below). Against the rumour speaks “the generosity with which the text [AO 8862] was made available to me and with which I was given the permission to publish it” by Thureau-Dangin —Neugebauer, “Studien zur Geschichte der antiken Algebra I”, QS, ii (1932–33), 1–27, p. 3. Such generosity is no matter of course among Assyriologists. Cf. also ref. 46.

Thus according to Wolfram von Soden's review (Zeitschrift der Deutschen Morgenländischen Gesellschaft, xciii (1939), 143–52, p. 144).

In MKT, i, a number of presumed Ur III (21st century b.c.) tables of reciprocals had been listed. Still, the mathematical substance of these was evidently soon exhausted, and as long as mathematical procedures and techniques were asked for, only the late, multi-place tables were subjected to further investigation. The above statement does not mean that nobody looked at older mathematical techniques. For one, F.-M. Allotte de la Fuÿe, who had produced important publications on such subjects for decades, continued to do so. But his text material and his results were not understood as belonging to the history of (Babylonian) mathematics.

Schuster , “Quadratische Gleichungen …” (ref. 8), 194. Neugebauer (“Studien … I”, 6) had been even more cautious; he presented the existence of a Sumerian prehistory to Old Babylonian advanced algebra as an hypothesis which was close at hand but unsupported by positive evidence. An important part of the same article is also dedicated to terminological differences and changes, and the statement that “the level of content has not changed much [from c. 1700 to c. 300 b.c.]” is characterized as “evidently only an assertion ‘in first approximation’”.

Neugebauer , “Zur Transkription mathematischer und astronomischer Keilschrifttexte”, Archiv für Orientforschung, viii (1932–33), 221–3, p. 222.

A distinction between “unreflecting” and “critical” reading through the categories of more familiar mathematics is important. Explanation always has to represent the categories that are to be explained by others which can be supposed to be known, and which are necessarily different. “Unreflecting” translation of categories is ‘one-to-one’, while “critical” translation will be ‘network-to-network’. In itself there is nothing wrong in describing a problem “I have added the measuring number of the side and the area of a square, and the result was 110” as “an equation”; this is in fact the closest we can get in terms of familiar notions. But an explanation which stops at this point, instead of discussing the particular character of the “equation”, the way it differs from and the way it is similar to a modern equation in x and y, is no explanation but a replacement of an ancient by a modern conceptual structure — A cover-up, indeed.

A few cases can be found where Neugebauer is misled himself and takes the justification to be the only possible interpretation, for example a commentary to problem no. 3 of the tablet AO 8862 in his “Studien … I”, 21f. In the discussion of the same problem in MKT (i, 120), however, the mistake is eliminated.

Neugebauer Thus , “Studien … I”, 24, and MKT, iii, 79.

Cf. ref. 81.

Neugebauer O. , Vorlesungen über Geschichte der antiken mathematischen Wissenschaften, I: Vorgriechische Mathematik (Berlin, 1934). Another exception is idem, “Zur geometrischen Algebra (Studien zur Geschichte der antiken Algebra III)”, QS, iii (1934–36), 245–59 — An article to which I return below.

Neugebauer , “Zur Geschichte der babylonischen Mathematik” (ref. 6).

Thureau-Dangin F. , “L'Équation du deuxième degré dans la mathématique babylonienne d'après une tablette inédite du British Museum”, RA, xxxiii (1936), 27–48, p. 28.

Thureau-Dangin F. , “L'Origine de l'algèbre”. Académie des Belles-Lettres, Comptes rendus, 1940, 292–319, p. 301.

Thureau-Dangin , “L'Origine de l'algèbre”, 302.

Thureau-Dangin F. , “La Méthode de fausse position et l'origine de l'algèbre”, RA, xxxv (1938), 71–77; idem, “L'Origine de l'algèbre”, 316f.

Gandz S. , “The origin and development of the quadratic equations in Babylonian, Greek, and early Arabic algebra”, Osiris, iii (1937), 405–557.

The problematic nature of this belief can be illustrated on Euclidean material. Even if we accept the theses that, e.g., Elements II, prop. 5 should be read, firstly, as algebra, and, secondly, as an equation and not as an algebraic identity, how do we know that the statement “if a straight line be cut in equal and unequal segments, the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section is equal to the square on the half” ( The thirteen books of Euclid's Elements , ed. and trans. by Heath T. L. , i (Cambridge and New York, 1926), 382) is to be translated into x + y = a, xy = b, and not into ax – x² = b? Indeed Heath, in his commentary, gives the latter equation as his main interpretation and the former only in passing. In certain Babylonian problems, the situation is definitely no better.

And also, it should be remembered, by the lack of obvious connections between the sophisticated mathematical texts and what else was known about Babylonian culture: “One should … not forget that we still know practically nothing about the whole setting of Babylonian mathematics within the framework of the culture as a whole” ( Neugebauer , Vorgriechische Mathematik (ref. 21), 204). It was understood that the texts we possess are training problems, constructed backwards from the solution, and thus school exercises. But texts elucidating the structure, curriculum and ideology of the Babylonian school have been published only since the late 1940s. In 1934 Neugebauer was fully right in maintaining that only a negative conclusion could be attained: Babylonian mathematics was not a child of astronomy and astrology, and not born from religious concerns. Even the relation between ‘practical’ mathematical problems and real computational practice was difficult to specify at a time when tables of technical (‘igi-gub’) constants were unknown (the first were to be published in MCT in 1945).

Zeuthen H. G. , Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886), 5ff. According to al-Nayrīzī's commentary to the Elements, already Hero had begun proving the theorems of Book II “by means of analysis”, which is at the very least a step in the direction toward an algebraic interpretation — Depending, of course, on our definition of that term, but in agreement with Viète's understanding of his own accomplishment as a redemption of analysis ( Codex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii , ed. and trans. by Besthorn R. O. Heiberg J. L. (3 vols, Copenhagen, 1893–1932), ii/l, 27). In the thirteenth century, Jordanus de Nemore modelled his whole reconstruction of Arabic algebra after Elements II and the corresponding propositions of the Data (cf. Høyrup J. , “Jordanus de Nemore, 13^th century mathematical innovator: An essay on intellectual context, achievement, and failure”, AHES, xxxviii (1988), 307–63, 332–6). In his case, the idea that Elements II was a metatheoretically more satisfactory version of al-jabr is thus indubitable.

Neugebauer , “Zur Geschichte der babylonischen Mathematik” (ref. 6), 80.

Neugebauer , “Zur geometrischen Algebra” (ref. 21), 250.

Gandz , “The origin and development of the quadratic equations” (ref. 27).

In early years not least Neugebauer's Vorgriechische Mathematik of 1934 and Gandz's “Origin and development of the quadratic equations” of 1937; later also Neugebauer's Exact sciences in Antiquity (Copenhagen, 1952; the more influential second edition, Providence, R.I., 1957) and van der Waerden's B. L. Science awakening (1st Dutch edn 1950, English transl. 1954; the influential second edition, Groningen, 1962). Undeservedly, Kurt Vogel's Vorgriechische Mathematik, ii: Die Mathematik der Babylonier (Hanover and Paderborn, 1959) and Vajman's A. A. Šumero-vavilonskaja matematika. III-I Tysjačeletija do n. e. (Moscow, 1961) have been much less influential, in Vajman's case because of the language in which the book was written, in Vogel's perhaps because its appearance in a series of high-school textbooks masked its qualities.

Neugebauer had explained his choice of what he considered as “technically adequate” instead of literal translations by the sarcastic observation that “who intends to study the history of terminology by means of a translation, he is anyway not to be saved” (MKT, iii, 5 n20). If this was read at all, then only as a statement that the study of “the history of terminology” was irrelevant to the study of the history of mathematics, and that translations could thus safely be relied upon.

Even though a few writers have maintained, basing their understanding upon one or two simple examples borrowed from the secondary literature, that Babylonian mathematics contained nothing but empirically established numerical schemes. Familiarity with only a modestly broader sample of translations taken from MKT or TMB would have prevented the mistake.

See ref. 34.

The same strengthening of the tendency can be noticed in an extensive article on “Die Algebra der Babylonier” ( Goetsch H. , AHES, v (1968–69), 79–153), which builds exclusively on translations and, even more, on the mathematical commentaries of original editions (see, e.g., p. 118), and whose only reservation concerning symbols arises when the author does not understand that Neugebauer's justifications should not automatically be understood as interpretations (p. 103). The form of the article is illustrative of the general expectation as to how the history of Babylonian mathematics was to be dealt with.

Hofmann J. E. , Geschichte der Mathematik (3 vols, Berlin, 1953–57).

It is immaterial for the present purpose that Hofmann's presentation is also riddled with actual mistakes.

Boyer C. , History of mathematics (New York, 1968).

Eves Howard , An introduction to the history of mathematics (New York, 1964; 3rd edn, 1969), 31.

Kline Morris , Mathematical thought from ancient to modern times (New York, 1972), 8f.

I shall restrict myself to a single reference: The unreserved use of symbolic algebra in Gundlach Karl-Bernhard von Soden Wolfram , “Einige altbabylonische Texte zur Lösung ‘quadratischer Gleichungen’”, Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, xxvi (1963), 248–63. The reason for picking out this particular, thoughtful publication is that von Soden was almost the only scholar at the time to point out the dangers inherent in unreflective modernization — Thus in a slightly later publication on “language, thought and concept formation in the Ancient Orient”, he wrote: “In my opinion, the historians of mathematics translate the Babylonian computations too rapidly into the kind of equations with which we are familiar, often moreover with general numbers, thereby betraying the dissimilitude of the mathematical thought of the ancient Orient” (Sprache, Denken und Begriffsbildung im Alten Orient (Mainz and Wiesbaden, 1974), 28). In spite of the authors' own doubts concerning the procedure, there was no other way to present Babylonian mathematics at the time. Apart from the Susa texts (on which below) and a smaller group of tablets from central Iraq published by Taha Baqir and Albrecht Goetze in Sumer in 1950 and 1951 (see Box IV), only a very few new texts were published between 1945 and 1970. That they were treated according to the canon that had been established by Neugebauer, Sachs and Thureau-Dangin goes without saying, and calls for no supplementary commentary in the present context.

As I discussed the process with my colleague Michel Olsen, he commented that this was exactly what likewise happened within the field of structuralist text analysis.

According to Jim Ritter (personal communication) the hand copies were made in great haste before the tablets were stored away, while the rest of the work was done after the War.

“Apparently”, since the preface only states that the translation (which seems to encompass everything between copying and mathematical commentary) was made in cooperation (p. xi). It is obvious, however, that much in the translation into French and even in the transcription into Akkadian has been derived backwards from the mathematical commentary; the transliteration, on the other hand, is relatively free of this backward influence.

For documentation, I shall refer only to pp. 299–302 and 320–7 of my “Algebra and naive geometry” (ref. 11). The -šu / š u mistake, not mentioned there, is discussed on p. 246 of my “Mathematical Susa Texts VII and VIII: A reinterpretation”, AoF, xx (1993), 245–60.

von Soden Wolfram , Bibliotheca orientalis, xxi (1964), 44–50. The claim that addition and multiplication be interchangeable operations, for instance, is characterized merely as “arbitrary” though with an exclamation mark.

Gundlach von Soden , “Einige altbabylonische Texte zur Lösung ‘quadratischer Gleichungen’” (ref. 44).

Bruins could never make up his mind whether it was Neugebauer or Derek Price who should have been caught in the Plimpton collection trying to break off a piece from Plimpton 322 in order to make the counter-evidence to his theory disappear. He told the story regularly but with changing protagonist. A third variant — Less conspicuously absurd — Can be found on p. 118 in Bruins , “Requisites …” (ref. 12).

Once Bruins discovered that he had made a mistake, he would cite himself in future publications for the correct opinion and make somebody else responsible for the erroneous point of view — Preferably the person who had pointed out his mistake. This can be exemplified by the following sequence: In his “Antecedents of Old Babylonian place notation and the early history of Babylonian mathematics” ( HM , iii (1976), 417–39, p. 432), Marvin A. Powell had pointed out that two mid-third-millennium texts solve the same mathematical problem, one correctly and another wrongly, and based his interpretation on an analysis of the error; in a critical abstract of this paper (Zentralblatt für Mathematik und ihre Grenzgebiete, ccclvii (1978), 7–8), Bruins rejected Powell's interpretation of the first tablet without noticing that his own interpretation was contradicted by the second; in my “Investigations of an early Sumerian division problem, c. 2500 B.C.” ( HM , ix (1982), 19–36, p. 32), I permitted myself to mention this neglect in a footnote; in his “Requisites …”, 134 n5 (see ref. 12), Bruins accuses Powell of having overlooked the existence of the two parallel texts (and identifies them wrongly). David Fowler commented upon this example with the words “I could put together a similar sequence over the Rhind papyrus 2/n table”.

Mahoney Michael , “Babylonian algebra: Form vs. content”, Studies in history and philosophy of science, i (1970–71), 369–80, quotation on p. 375.

Szabó Arpád , Anfänge der griechischen Mathematik (Munich and Budapest, 1969), 455ff.

Unguru Sabetai , “On the need to rewrite the history of Greek mathematics”, AHES, xv (1975), 67–114, p. 77.

Freudenthal Hans , “What is algebra and what has it been in history?” AHES, xvi (1977), 189–200.

Weil André , “Who betrayed Euclid?”, AHES, xix(1978), 91–93.

van der Waerden B. L. , “Defence of a ‘shocking’ point of view”, AHES, xv (1976), 199–210.

Böhme Gemot , “Die Finalisierung der Wissenschaft”, Zeitschrift für Soziologie, ii (1973), 128–44.

Powell Marvin , “Sumerian area measures and the alleged decimal substratum”, ZA, lxii (1972–73), 165–221.

See ref. 52.

Schmandt-Besserat Denise , “An archaic recording system and the origin of writing”, Syro-Mesopotamian studies , i/2 (1977); soon followed by idem, “The invention of writing” ( Discovery: Research and scholarship at the University of Texas at Austin , i/4 (1977), 4–7), and by idem, “The earliest precursor of writing”, Scientific American, ccxxxvii/6 (1978), 38–47 (European pagination).

Friberg Jöran , “The third millennium roots of Babylonian mathematics. I. A method for the decipherment, through mathematical and metrological analysis, of proto-Sumerian and proto-Elamite semi-pictographic inscriptions”, DMC, no. 1978–9; “The early roots of Babylonian mathematics. II: Metrological relations in a group of semi-pictographic tablets of the Jemdet Nasr type, probably from Uruk-Warka”, DMC, no. 1979–15.

Once again, Aisik Vajman should have been mentioned, if only his earlier works on the same matters had not been even more badly published, and not backed by personal contacts. With Marvin Powell as the sole exception, nobody outside the Soviet Union (and few scholars there) seems to have taken serious note of them before Friberg.

“The proto-literate period” in Mesopotamia is dated approximately 3400 b.c. to 3000 b.c. (according to a compromise between not too firmly established calibrated radiocarbon dates and stratigraphic evidence), see Box I. Proto-Elamite writing was used in the Iranian region during the second half of this period. It appears to have been inspired by the invention of writing in Mesopotamia, but makes use of a different inventory of signs; the metrologies, however, are largely but not fully identical.

That is, for instance: The area 3 i k u was denoted by threefold repetition of the sign i k u; in our metrology, by contrast, three hectares are written ‘3 ha’, with separation of quantity (‘3’) from quality (‘ha’).

Computer analysis of the complete material has shown since then that the proto-cuneiform accounting tablets make use of two different counting systems used for counting objects belonging to different categories: One, sexagesimal, with the steps 1, 10, 60, 600, 3600, and 36000; another, ‘bisexagesimal’, containing the steps 1, 10, 60, 120, 1200, and 7200 ( Damerow Peter Englund Robert K. , “Die Zahlzeichensysteme der Archaischen Texte aus Uruk”, chap. 3 (pp. 117–66), in Green M. W. Nissen Hans J. (eds), Zeichenliste der Archaischen Texte aus Uruk, ii (ATU 2; Berlin, 1987), 126f, 133f, 165). Friberg's sequence merges the two.

Friberg , “The early roots of Babylonian mathematics. II” (ref. 63), 33–43.

Denis Soubeyran has published and discussed a collection of mostly mathematical texts from Mari (“Textes mathématiques de Mari”, RA, lxxviii (1984), 19–48); a new group of problems from Central Iraq have been published and discussed by Farouk al-Rawi and Michael Roaf (“Ten Old Babylonian mathematical problem texts from Tell Haddad, Himrin”, Sumer, xliii (1984), 195–218). The discovery of Ebla has brought three texts with mathematical contents (analysis and previous publication history in Friberg Jöran , “The early roots of Babylonian mathematics. III: Three remarkable texts from Ancient Ebla”, Vicino oriente, vi (1986), 3–25). Several new texts have been located by Friberg , cf. below.

So, critical reflection on Schmandt-Besserat's thesis led Stephen Lieberman (“Of clay pebbles, hollow clay balls, and writing: A Sumerian view”, American journal of archaeology, lxxxiv (1980), 339–58) to investigate the Sumerian use of two different ways to write numbers (‘curviform’ and ‘cuneiform’) throughout the third millennium and connect it to a conjectural use of tokens as a computation device (Lieberman did not know about Vajman's and Friberg's work, and therefore accepted the identification of tokens with sexagesimal numbers). Robert M. Whiting analysed Powell's evidence and some supplementary texts in an attempt to push backward the ante quem of the place value system, but neglected to observe Powell's distinction between the prerequisite idea of sexagesimal regularization and extension and the establishment of a place value system stricto sensu (“More evidence for sexagesimal calculations in the third millennium B.C.”, ZA, lxxiv (1984), 59–66).

I persist in disregarding astronomy — For which I apologize to Hermann Hunger, who participated in several workshops. I also omit what a number of regularly participating ‘general discussants’ have contributed from their general competence as historians of science or as Assyriologists: Kilian Butz, Jean-Pierre Grégoire, Wolfgang Lefèvre, Johannes Renger, Jim Ritter, Arpád Szabó, Sabetai Unguru, Kurt Vogel, as well as everybody who only participated once.

Damerow P. , “Die Entstehung des arithmetischen Denkens”, in Rechenstein, Experiment, Sprache , ed. by Damerow P. Lefèvre W. (Stuttgart, 1981), 11–113. A more refined analysis along the same lines is Damerow P. , “Individual development and cultural evolution of arithmetical thinking”, in Ontogeny, phylogeny, and historical development, ed. by Strauss S. (Human Development, ii; Norwood, N.J., 1988), 125–52.

Described in Peter Damerow, Englund Robert Nissen Hans , “Zur rechnergestützten Bearbeitung der archaischen Texte aus Mesopotamien (ca. 3200–3000 v. Chr.)”, Mitteilungen der Deutschen Orient-Gesellschaft, cxxi (1989), 139–52.

Damerow Englund , “Die Zahlzeichensysteme der Archaischen Texte aus Uruk” (ref. 67).

Idem, The proto-Elamite texts from Tepe Yahya (The American School of Prehistoric Research, Bulletin 39; Cambridge, Mass., 1989).

Damerow Peter Englund Robert Nissen Hans , “Die Entstehung der Schrift”, Spektrum der Wissenschaften , February 1988, 74–85; idem, “Die ersten Zahldarstellungen und die Entwicklung des Zahlbegriffs”, Spektrum der Wissenschaften , March 1988, 46–55; Nissen Hans J. Damerow Peter Englund Robert , Frühe Schrift und Techniken der Wirtschaftsverwaltung im alten Vorderen Orient: Informationsspeicherung und -verarbeitung vor 5000 Jahren (Bad Salzdetfurth, 1990). A revised version of the latter work has appeared as Archaic bookkeeping: Writing and techniques of economic administration in the ancient Near East (Chicago and London, 1993).

Englund Robert , “Administrative timekeeping in ancient Mesopotamia”, Journal of the economic and social history of the Orient, xxxi (1988), 121–85.

Høyrup Jens , “Influences of institutionalized mathematics teaching on the development and organization of mathematical thought in the pre-modern period”, Materialien und Studien. Institut für Didaktik der Mathematik der Universität Bielefeld, xx (1980), 7–137.

Friberg , “A survey of publications …” (ref. 5), 137. Strictly speaking, Friberg does not report my original publication but a slightly later Danish essay.

So I later found out - but my real inspiration for the method was vaguely structuralist text analysis.

The texts distinguish sharply between two different operations both traditionally translated as ‘addition’; similarly two different ‘subtractive operations’; no fewer than four ‘multiplications’; and two different ‘halves’.

Høyrup Jens , Babylonian algebra from the view-point of geometrical heuristics: An investigation of terminology, methods, and patterns of thought (Roskilde University Centre, Institute of Educational Research, Media Studies and Theory of Science, 1984). A more readable exposition is Høyrup , “Algebra and naive geometry” (ref. 11).

See ref. 38.

See ref. 5.

Unfortunately but for reasons of space, his article “Mathematik” in Reallexikon der Assyriologie, vii (Berlin and New York, 1990), 531–85 was not allowed to cover the subject-matter as broadly.

Friberg Jöran , “Methods and traditions of Babylonian mathematics. II: An Old Babylonian catalogue text with equations for squares and circles”, Journal of cuneiform studies, xxxiii (1981), 57–64; Friberg Jöran Hunger Hermann al-Rawi Farouk , “‘Seeds and reeds’: A metro-mathematical topic text from Late Babylonian Uruk”, Baghdader Mitteilungen, xxi (1990), 483–557, Tafel 46–48.

Cf. ref. 15.

Nemet-Nejat Karen , Cuneiform mathematical texts as a reflection of everyday life in Mesopotamia (American Oriental Series, 75; New Haven, Conn., 1993). Drafts of the work were presented at the 1988 Berlin Workshop, but the author had taken up the subject independently of the Berlin collaboration.

Gelb Ignace , “Approaches to the study of ancient society”, Journal of the American Oriental Society, Ixxxvii (1967), 1–8, p. 8.

In private conversation, and therefore not quoted verbatim.

In this connection I disregard B. L. van der Waerden's work on Geometry and algebra in ancient civilizations (Berlin etc., 1983), since the fundamental perspective, though certainly a recast, is not that of Babylonian mathematics.

This does not prevent categories of modern mathematics being used when needed as analytical tools (cf. ref. 17) — Even if we stop asking as our fundamental question, “how the equations of the Babylonians looked”, we may still take notice that a problem “I have added [the measuring numbers of] the side and the area of a square, and the result was 110” shares essential features with modern equations, and can be expressed no more adequately by a single word. We still need an Archimedean point from which to describe the world, and purists who refuse to speak about “algebra” and insist, e.g., on “numerical mathematics” are mistaken — “numbers” are no more transhistorically immutable than “algebra”, as shown by Vajman, Friberg and Peter Damerow.

Mahoney , “Babylonian algebra” (ref. 53), 378.

As quoted in ref. 29.