Lengths, Widths, Surfaces pp 227-277 | Cite as

# Quasi-algebraic Geometry

## Abstract

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.

## Keywords

Mathematical Text Present Text Lower Width Original Triangle Regular Octagon## Preview

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## References

- 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
- 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 - 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
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- 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
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- 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
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- 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
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*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 - 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 - 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
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- 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 - 296.The reading of BAR.TA as bar.da, literally “cross-bar”, was proposed by Muroi [1992: 47].Google Scholar
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