Quasi-algebraic Geometry

  • Jens Høyrup
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)


The texts that were presented in Chapters III and V should give a comprehensive view of that part of Old Babylonian mathematics that can somehow be regarded as “algebraic”. However, if the underlying understanding, the “standard representation” and most of the techniques were geometric; if, furthermore, the authors of the theme texts show by their organization of the material to consider geometric configuration primary with respect to type and technique: then inspection of texts that are concerned with geometry but whose methods cannot be characterized as algebraic in the strict sense is likely to throw new light even on the more properly algebraic techniques and concepts.


Mathematical Text Present Text Lower Width Original Triangle Regular Octagon 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 251.
    Rectilinear figures are said to be given in shape if the angles are given one by one and the ratios of the sides to each other are given“ [ed. Menge 1896: 1].Google Scholar
  2. 252.
    In Babylonian astronomical texts from the first millennium we do find something which modern mathematical thinking might see as quantified angles (being blind to its own conflation of concepts which only merge when a particular theoretical substructure is presupposed). However, by measuring distances in heaven in length units in discordant ways, these texts demonstrate that such distances are not seen in terms of the angle between sighting lines but in the likeness of linear distances or corresponding travel times; see [Powell 1990: §§1.2e, 1.2k-l].Google Scholar
  3. 253.
    Detailed documentation — including both the tables of coefficients and their use in procedure texts — and discussion will be found in [Robson 1999: 34–56].Google Scholar
  4. 255.
    See, e.g., [Allotte de la Fuye 1915: 141].Google Scholar
  5. 256.
    In a single example from the 21st century BCE, in which the calculated area is at least 51% above what could be the true area if the indicated measures are what they pretend to be, the scribe is likely to have created his numbers backwards after the actual division of a terrain which was confined by a curved stream — for instance, by using a measured height as his length. Alternatively, he used his monopoly on mathematical knowledge to pull the leg of his corrupt superior — cf. [Hoyrup 1995].Google Scholar
  6. 257.
    Based on the transliteration, photograph, and hand copy in [Baqir 1950].Google Scholar
  7. 258.
    Since it is the ratio that is multiplied and not the area, “raising” is used instead of “repetition”. The latter operation could only come in play if the surface were multiplied first by 45’, thus producing an isosceles triangle that could be combined with its own mirror image as a square.Google Scholar
  8. 260.
    Of the 222 fields inventoried in the “round tablets” from the province of Lagash that are analyzed by Mario Liverani [1990: 160–166 (= Figures 8–14)], 172 are defined by length (mir, “north”) and width (kur, “east”) alone and thus ideally to be thought of as rectangles, while all of the remaining 50 appear to have beenGoogle Scholar
  9. 261.
    As we shall see when analyzing the text YBC 4675 (p. 247), the bisection problem is indeed solved by means of scaling and by taking the difference between squares.Google Scholar
  10. 263.
    It is often supposed that dal is a logogram for pirkum in the mathematical texts, but inspection of the supposed evidence shows that the conclusion does not follow. Instead, the text YBC 4675 (below, p. 244) uses dal and tallum about the same entity. Moreover, as Joran Friberg reminds me (personal communication), dal and pirkum refer to different entities in the igi.gub table TMS III.Google Scholar
  11. 265.
    Neugebauer suggests that the sign MA may be a writing error for the missing sum, which leaves a NU which is rather hard to explain (neither the logogram for salmum, “statue, picture”, nor a nominalization of the Sumerian prefix nu., “person (taking care) of’, seems very adequate). Thureau-Dangin reads MA NU as Akkadian ma-nu, which he finds inexplicable, but which should come from manum, ”to count“, and thus mean ”the counting“, ”the counted“ or ”the reckoning“.Google Scholar
  12. 268.
    It is also possible that the repetition in rev. 8 replaces a statement of the total length; it is indeed striking that the number T12 is identified in rev. 11 as the length while being unidentified at its first appearance.Google Scholar
  13. 269.
    As such, they suffer of course from the same weakness as other crucial discriminators, experimental or otherwise: they are able to discriminate between the possibilities that we already have in mind — but they cannot exclude explanations of the facts which we never thought of. This is where van der Waerden and Neugebauer are misled by the seemingly irrefutable crucial argument for the numerical and against the geometrical interpretation (the inhomogeneous additions of lines and surfaces, and of volumes and areas): after Pedro Nunez’s Libro de algebra from 1567, no mathematician seems to have imagined the possibility to think in terms of “broad lines”.Google Scholar
  14. 270.
    I use the occasion to make a personal note: as always when I work on this material, I remain deeply impressed by what was done in the thirties by Neugebauer, Thureau-Dangin, Gandz, Vogel, and von Soden — but not least by the profundity of Neugebauer’s insights on points where he followed Newton’s maxim and “made no hypotheses” that were not needed for his actual project.Google Scholar
  15. 271.
    In principle, even a product may of course do, if only it is not conflated with its resulting number but its factors are kept separate; but this requires that the product have a representation that allows us to see it thus — that is, either as a rectangle or some other concrete representation, or by means of symbolic algebra. But symbolic algebra was not at hand, and the only concrete representation of a product in the present context is exactly as a rectangle (e.g., commercial rates and amounts of silver as in TMS XIII would be meaningless here); therefore the surface with scalings appears to be the only possibility.Google Scholar
  16. 272.
    interpret the final.e as the representative of the suffix /-ed/, whence the translation. Normally it would not be written after a vowel [SLa, §243]; but it does happen, and the repeated uncontracted writing u.ub instead of the normal u. in the following shows a tendency to make grammatical elements more explicitly visible than normally; cf. note 277.Google Scholar
  17. 275.
    Based on the transliteration in [Kilmer 1964], with a correction to line 1 proposed by Kazuo Muroi [1998] (in part a reversal to the first transliteration in [Vajman 1961: 257], to which I had no access when preparing the basic version of the manuscript).Google Scholar
  18. 283.
    Unfortunately, no precise dating can be given because this and other mathematical texts from Ur were seemingly brought to the house from which they were excavated as fill [Friberg 2000: 44/]. In the same place a few tablets from Ur III were found, but most datable texts from the location are from the interval 1890 to 1810 BCE. What can be stated with high certainty from stylistic considerations is that the four mathematical problem texts that were found in the house belong together — cf. p. 352.Google Scholar
  19. 284.
    However, a couple of newly published texts from Nippur (CBS 43, CBS 154–921 [ed. Robson 2000: 39/]) reveal the same underlying idea by asking for the side of a square by the phrase kiya imtahhar, referring to the notion of “standing against itself” which we have encountered in BM 13901 no 23 and using the interrogative phrase kiya (“how much each”) which always asks for the value of several magnitudes at a time.Google Scholar
  20. 296.
    The reading of BAR.TA as bar.da, literally “cross-bar”, was proposed by Muroi [1992: 47].Google Scholar
  21. 297.
    Known from Theon of Smyrna [ed., trans. Dupuis 1892: 70–75], and from Proclos’s commentary to Plato’s Republic [Kroll 1899: II, 393–400]; cf. [Hultsch 1900 ].Google Scholar
  22. 298.
    Division by an irregular number was certainly not outside the capabilities of the Old Babylonian scribes. Several school exercises from the mid-third millennium train the division of very large numbers by the irregular divisors 7 and 33; the method was stepwise division and conversion of the remainder into smaller units, structurally similar to our long division. See [Hoyrup 1982] and [Friberg 1986 ]. A few tabulations of the reciprocals of irregular numbers are also known (cf. note 50), and even though we do not know about a case where they were used we may be certain that they were computed.Google Scholar
  23. 300.
    Ed. Cantor 1875: 212, Fig. 40]. The text is also in [Bubnov 1899: 539], but the diagram is omitted.Google Scholar
  24. 309.
    After the transliteration in [MKT I, 279/], cf. [TMB, 130]. Christopher Walker has recognized in the fragment BM 96957 a missing part of the tablet, but since both his hand copy and Eleanor Robson’s transliteration and analysis of the relevant part were only published in preliminary form in [Robson 1995: 269–280] when the manuscript was prepared, I restricted myself to reproducing the two problems that were indubitably in the public domain at that moment. In Robson’s counting, they are #18 and #21.Google Scholar
  25. 310.
    At the moment when I was making the ultimate corrections the complete tablet had been published in [Robson 1999: 231–244]. [Robson 1996] is a proper edition of the brick-problem part of the new fragment, though it is now superseded by the full edition [1999].Google Scholar

Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Jens Høyrup
    • 1
  1. 1.Section for Philosophy and Science StudiesUniversity of RoskildeRoskildeDenmark

Personalised recommendations