Abstract
Those who nowadays work on the history of advanced-level Babylonian mathematics do so as if everything had begun with the publication of Neugebauer’s Mathematische Keilschrift-Texte from 1935 to 1937 and Thureau-Dangin’s Textes mathématiques babyloniens from 1938, or at most with the articles published by Neugebauer and Thureau-Dangin during the few preceding years. Of course they/we know better, but often that is only in principle. The present paper is a sketch of how knowledge of Babylonian mathematics developed from the beginnings of Assyriology until the 1930s, and raises the question why an outsider was able to create a breakthrough where Assyriologists, in spite of their best will, had been blocked. One may see it as the anatomy of a particular “Kuhnian revolution”.
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Notes
- 1.
My translation, as everywhere in what follows when no translator is indicated.
- 2.
- 3.
Archibald Henry Sayce, when returning to the text (now identified as K 490) in (1887: 337–340), reinterprets the topic as a table of lunar longitudes.
- 4.
That Rawlinson is anyhow also interested in the mathematics per se and not only as a means for chronology (after all, he was interested in everything Assyro-Babylonian) is however revealed by what comes next in the note, namely that “while I am now discussing the notation of the Babylonians, I may as well give the phonetic reading of the numbers, as they are found in the Assyrian vocabularies”.
- 5.
Since Mesopotamian metrology varies much more over the epochs than Oppert had imagined, it is obvious that the comparative method led him astray as often as to the goal. The task may be claimed only to have been brought to a really satisfactory end by Marvin Powell (1990).
- 6.
This is well illustrated by the chapter “History and Chronology [of Chaldaea]” in the second edition of George Rawlinson’s Five Great Monarchies of the Ancient Eastern World from (1871: I, 149–179). The author can still do no better than his brother Henry had done in (1855) – all we find is a critically reflective combination of Berossos and Genesis, with a few ruler names from various cities inserted as if they were part of one single dynasty.
This was soon to change. In (1885: 317–790), Fritz Hommel was able to locate everything from Gudea onward in correct order; absolute chronologies before Hammurapi were still constructed from late Babylonian fancies (Hommel locates Sargon around 3800 bce and Ur-Nammu around 3500 and lets the Ur, Larsa and Isin dynasties (whose actual total duration was c. 350 years) last from c. 3500 until c. 2000 bce – pp. 167f).
- 7.
See (Friberg 1982: 3–27).
- 8.
Actually, Scheil’s understanding was not broadly accepted: Meissner (1920: II, 387) from 1925 does not know about sexagesimal fractions. Meissner also mixes up the place-value and the absolute system.
- 9.
Esoteric numerology certainly left many traces in Mesopotamian sources – but not in sources normally counted as “mathematical”; the only exception is a late Babylonian metrological table starting with the sacred numbers of the gods (W 23273, see (Friberg 1993: 400)). Apart from that, even the text corpus produced by the Late Babylonian and Seleucid priestly environment kept the two interests strictly separate.
- 10.
I disregard the “metrological tables”, which were not yet understood as mediators between the various metrologies and the place value system. I also disregard mathematical astronomy, where the extension of the place value system to fractions had been understood better (Epping 1889: 9f), (Kugler 1900: 12, 14), without this understanding being generalized, cf. (Scheil 1915a: 196).
- 11.
Weidner mentions as the only exception “an occasional notice” by Hommel in a Beilage to the Münchener Neuesten Nachrichten 1908, Aug. 27, Nr. 49, p. 459, which I have not seen. He says nothing about its substance being in any way important, only that it interprets the final clause ne-pé-šum of problems as “quod erat demonstrandum”.
- 12.
This is certainly “whiggish” historiography – and it has to be, if our aim is to locate Neugebauer’s achievement in its historical context.
- 13.
Here and in what follows, when quoting transcriptions and transliterations (also of single words and signs), I follow the conventions of the respective originals. When speaking “from the outside”, on my own, I follow modern conventions. Since the delimitation is not always clear, some inconsistencies may well have resulted.
- 14.
Neugebauer tries to make sense of this impossible formula by interpreting it as an approximationto \( a+\frac{2a\cdot {b}^2}{2{a}^2+{b}^2} \); (Neugebauer 1931a: 95–99) explains the origin of the guess, which he finds inthe music theory of Nicomachos and Iamblichos – classical Antiquity remained a resource when other arguments were not available.
Difficulties in the handling of the sexagesimal system may be the reason that Weidner did not discover that the formula – adding a length and a volume – is impossible because a change of measuring unit would not change the two addends by the same factor (this is the gist of “dimension analysis”).
- 15.
In detail: Weidner supposes the dimensions of the rectangle to be 10 and 40, even though the initial “2 cubits” should make him understand that 10 stands for 10´, and 40 in consequence (if the calculations are to be meaningful) for 40´ – both corresponding to the unit nindan (1 nindan = 12 cubits); instead he wonders (col. 259) what these 2 cubits may be. Weidner therefore supposes the product to be 4000, about which he says that “it is written in cuneiform as 1 6 40, i.e., 1 (3600) + 6 (⋅60) + 40. But this number can also be understood as 1 + 6(⋅\( \frac{1}{60} \)) + 40(⋅\( \frac{1}{3600} \)) = \( \frac{4000}{3600} \) = 1,11”.
A small remark on notations: the ´–´´ notation was used (and possibly introduced) by Louis Delaporte in (1911:132) (´ and ´´ only); Scheil (1916: 139), immediately followed by Ungnad (1916: 366), uses ´, ´´ and °, as does later Thureau-Dangin. Strangely, Neugebauer believed in (1932a: 221) that the °-´-´´ notation had been created “recently” by Thureau-Dangin (similarly MKT I, p. vii n. 5); I have not noticed references from his hand to (Scheil 1915a), but he had referred to Ungnad (1916) on several occasions – e.g., (Neugebauer 1928: 45 n.3). Neugebauer’s own notation goes back at least to (1929: 68, 71).
- 16.
See, e.g., (Thureau-Dangin 1921: 133).
- 17.
For such similarity I rely on (Labat 1963).
- 18.
This reading goes back to (Zimmern 1916: 323).
- 19.
Now nagbu, interpreted “spring, fountain, underground water” ((CAD XI, 108), cf. (AHw 710)). The error was pointed out by Ungnad (1916: 363), who also proposed the reading sukud, “height”.
- 20.
In (1929: 88), Neugebauer still accepted Weidner’s interpretation. Arguments that a verbal imperative was most unlikely and an alternative orthography for mīnûm unsupported by other evidence were first given by Thureau-Dangin (1931: 195f); the idea that it is a (pseudo-)Sumerogram for that word was first hinted at by Albert Schott, see (Neugebauer 1932b: 8 n. 18).
- 21.
No longer needed, since other texts have the Akkadian atta.
- 22.
I wonder whether Weidner was led to this conclusion by numerical necessity alone (1 6 40 being indeed the product of 40 and 1 40) or by the parallel use of έπί in Greek mathematics.
- 23.
Unfortunately, Weidner’s commentary equates this Sumerian word with eşēpu, building on a hint in Delitzsch (1914: 134); Delitzsch’s supportive examples are conjugated forms of waşābum, also the actual equivalent in Old Babylonian mathematical texts.
- 24.
A possible interpretation is offered in (Høyrup 2002: 271f).
- 25.
He does not use the term “reciprocal” but speaks about the operation of dividing 1 by the number in question.
- 26.
To this he links Akkadian ruddûm instead of waşābum – a mistake in the context of the mathematical texts, as it was to turn out when more of these became known.
- 27.
Gadd says kibbatum, but that orthography has already disappeared in (Bezold 1926: 147).
- 28.
According to what is written on p. 6, the hand copies were made in 1914, after which Frank had no more access to the tablets; he only received his old copies and notes in 1925, after which he could resume working on them. Actually, what Frank received through the mediation of a friend were only draft hand copies; what he had originally prepared for an edition arrived too late (Waschow 1932a: 211), cf. (Thureau-Dangin 1934).
- 29.
Full title Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. Abteilung B: Studien.
- 30.
In MKT I, p. 263, Neugebauer characterizes it as umwegig, “roundabout”. A possible understanding of the underlying idea, based on a proposal by Jöran Friberg, is in (Høyrup 2002: 241–244). The procedure itself was perfectly understood by Neugebauer.
- 31.
The interpretation of RI as “Trennungslinie”, the parallel transversal dividing the trapezoidal quadrangle into two strips, is probably an exception to this rule; according to p. 70, n. 14 it was due to V. V. Struve.
- 32.
As I have experienced several times, this does not mean that today’s Assyriologists are generally familiar with the place value system. Indeed, unless they work on astronomical texts or mathematical school texts (very few do), they never see it in use.
- 33.
dù, we remember, had become du8 in Thureau-Dangin’s reform.
- 34.
Footnote 5a in (Neugebauer 1930a: 122) reveals that “subtrahiert” was chosen originally because Neugebauer had mistakenly believed to improve Frank’s reading a-šà(g) dù by changing it into uṣuḫ. The same note shows that Neugebauer is now perfectly aware that the correct literal translation would be “abgespalten”; we may perhaps presume the deviating translation in MKT to be nothing but a slip.
- 35.
“Die Übersetzung ist selbstverständlich im Prinzip eine wörtliche”.
- 36.
This is why Schuster’s discovery should probably be dated in early 1929.
- 37.
Actually, verbal forms of warādum (“to descend”) are involved, but for the immediate technical purpose this was not decisive, as observed by Thureau-Dangin (1932b: 80) in the note where he made the grammatical analysis of the term.
- 38.
In the sense of “standard schemes” – no symbolic writing was of course intended as far as the Babylonians were concerned.
- 39.
- 40.
“[…] les études d’O. Neugebauer, qui ont pour objet plutôt le fond que la forme des textes, apportent au philologue d’utiles données”.
- 41.
This limitation was emphatically pointed out by Neugebauer in (1932b: 24). In (1934a: 204), he was perhaps even more emphatic when pointing out that “we still know practically nothing about how Babylonian mathematics was situated within the overall cultural framework”.
We may take it as an expression of the same explicitly agnostic attitude that Neugebauer never spoke of Babylonian “mathematicians”. We may recognize mathematics in the texts, but nothing was known about the social role of their authors, in particular, whether any social role or identity (even a part-time role or an aspect of identity) would allow this characterization.
- 42.
- 43.
- 44.
In (1932a: 52), Thureau-Dangin still speaks of igû and šibû.
- 45.
This observation had already been made by Hilprecht (1906: 21), but did not make much sense in his context of “Plato’s number”.
- 46.
This idea could possibly explain his otherwise not obvious translation of igi/igi.bi as “Nenner”/“Zähler”.
- 47.
The absolute sexagesimal system is described in (Thureau-Dangin 1898: 81f). That it goes back to the fourth millennium bce was not known in 1898, nor in 1930, but in any case it precedes every hint of use of the place value notation by many centuries; besides, the original curviform character of its signs shows them to belong with the earliest phase of writing.
Neugebauer does discuss the absolute system in (1927: 8–13), but mixing it up with speculations that thwart his understanding.
- 48.
Since they are peripheral to my topic, I shall not document these claims, just refer to (Powell 1976) as a seminal publication.
- 49.
The main theme of this article is the link between, on one hand, tables of cubes and cube roots (known since Rawlinson) and a recently discovered tabulation of n 3 + n 2, on the other the third-degree problems of a text now known as BM 8200 + VAT6599.
- 50.
Among these, I shall mention in particular (Neugebauer 1934b), the first description of the mathematical series texts.
- 51.
It can hence be considered a paradox that Assyriologists, after the appearance of MKT, tended to put aside any tablet containing too many numbers in place-value notation as “at matter for Neugebauer” (as formulated to me with regret by Hans Nissen at one of the Berlin workshops on “Concept Formation in Mesopotamian Mathematics” in the 1980s). As we have seen, the fathers and giants of Assyriology, from Hicks, Rawlinson and Oppert to Thureau-Dangin, considered anything mathematical as very important. Even after the revival of active work on Mesopotamian mathematics during the last three decades and many new insights, an Encyclopedia of Ancient History planned by Blackwell and Wiley in 2009 suggested 500 words for “Mathematics, Mesopotamian” – the same as was dedicated to Mesopotamian hairstyles (I succeeded in raising the limit to 700 words).
- 52.
His characterization of AO 6484 as “arithmetical operations” might suggest exactly such low expectations.
- 53.
In 1930 he published an article on protoliterate (Jemdet Nasr) metrology and mensuration, in 1932 another one on AO 6456, the Seleucid table of reciprocals. On his mathematical interest and competence, see (de Genouillac 1939).
- 54.
I disregard publications where general a priori ideas about the nature of Mesopotamian mathematics enter as part of a broader argument, such as (von Soden 1936).
- 55.
It may perhaps be adequate to recapitulate some elements of what these four Assyriologists did later in connection with Mesopotamian mathematics.
Schuster published oft-cited works on Sumero-Babylonian bilingual texts in 1938 and on Hatto-Hittite bilinguals in 1974 and 2002; he lived until 2002, but seems not to have worked on mathematical texts after 1930.
Struve, as curator of the cuneiform collection of the Ermitage in Leningrad, analyzed its corpus of Ur III accounts, which induced him to draw a very grim picture of the social system that implemented the place value system in its social engineering (Struve 1934) – a picture that has now been amply confirmed by Robert Englund (1990). He lived until 1965 but seems never to have published more on “mathematics proper”.
Waschow prepared an edition of the important Seleucid problem text BM 34568, published in MKT III (pp. 14–22). In his dissertation from (1936), an edition of letters from the Kassite period, he states in the (unpaginated) CV that he had entered active army service in 1934 and was at the moment serving as a non-commissioned anti-aircraft officer while intending to continue scholarly activity in parallel. In 1938 he published 4000 Jahre Kampf um die Mauer, about siege techniques since Old Babylonian times. I can find no later traces of him and assume that he is one of those collaborators of Neugebauer who according to Vogel (private communication) fell in the war.
According to Neugebauer (MKT I, p. ix), Schott contributed intensely to MKT. He was also one of Neugebauer’s intended collaborators in the publication of the corpus of Babylonian astronomical texts, planned around 1935 (see the description in (Neugebauer 1937)) – not realized immediately because of the war. Schott died at the end of the war in 1945 (Thompson 2010). He had also collaborated with the astronomer Paul Neugebauer on other aspects of Mesopotamian astronomy, and he translated the Gilgameš-epic in 1934 (eventually published with revisions by von Soden in 1958).
As we see, “mathematics proper” did not stay central to those three who had the possibility to make Assyriological work after 1936. Though also engagin in other matters, Thureau-Dangin was actually more tenacious as regards mathematics, as expressed in his (1940a, b).
- 56.
As Neugebauer tells with gratitude in (1927: 5), he has also been well counselled and trained by Anton Deimel during a fairly long stay at the Pontificium Institutum Biblicum in Rome, as his initial interest in Mesopotamian mathematics (as a parallel elucidating the foundations of Egyptian mathematics) had first been stimulated by works of Thureau-Dangin (1898, 1921) and Deimel (1922).
- 57.
“They hated each other”, I was told by Olaf Schmidt, Neugebauer’s assistant during his stay in Copenhagen. Schmidt, too gentle to hate anybody as far as I can judge, may have mistaken animosity for genuine hatred.
- 58.
Given the general unreliability of Evert Bruins, I permit myself to disregard what he claimed in a letter to me: that Thureau-Dangin took care that Neugebauer should not get access to the mathematical texts from Susa, which had been found in 1933.
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Høyrup, J. (2016). As the Outsider Walked in the Historiography of Mesopotamian Mathematics Until Neugebauer. In: Jones, A., Proust, C., Steele, J. (eds) A Mathematician's Journeys. Archimedes, vol 45. Springer, Cham. https://doi.org/10.1007/978-3-319-25865-2_5
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