8. In the reading of numbers expressed in the Babylonian sexagesimal system, uncertainty arises from the fact that the early Babylonians had no symbol for zero. In the foregoing tablets, how do we know, for example, that the last number in the first line is 720 and not 12? Nothing in the symbolism indicates that the 12 is in the place where the local value is “sixties” and not “units.” Only from the study of the entire tablet has it been inferred that the number intended is 12×60 rather than 12 itself. Sometimes a horizontal line was drawn following a number, apparently to indicate the absence of units of lower denomination. But this procedure was not regular, nor carried on in a manner that indicates the number of vacant places.
9. To avoid confusion some Babylonian documents even in early times contained symbols for 1, 60, 3,600, 216,000, also for 10, 600, 36,000.[1] Thus · was 10, ● was 3,600, ⦾ was 36,000.
in view of other variants occurring in the mathematical tablets from Nippur, notably the numerous variants of “19,”¹ some of which may be merely scribal errors:
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They evidently all go back to the form 𒎙𒇲𒁹 or 𒎙𒇳𒁹 (20−1=19).
Fig. 2.—Showing application of the principle of subtraction
10. Besides the principles of addition and multiplication, Babylonian tablets reveal also the use of the principle of subtraction, which is familiar to us in the Roman notation XIX (20−1) for the number 19. Hilprecht has collected ideograms from the Babylonian tablets which he has studied, which represent the number 19. We reproduce his symbols in Figure 2. In each of these twelve ideograms (Fig. 2), the two symbols to the left signify together 20. Of the symbols immediately to the right of the 20, one vertical wedge stands for “one” and the remaining symbols, for instance 𒇲, for LAL or “minus”; the entire ideogram represents in each of the twelve cases the number 20−1 or 19.
One finds the principle of subtraction used also with curved signs;[2] ᗞ●●𒇲ᗞ meant 60+20−1, or 79.
- ↑ See François Thureau-Dangin, Recherches sur l’origine de l’écriture cunéiforme (Paris, 1898), Nos. 485–91, 509–13. See also G. A. Barton, Haverford College Library Collection of Cuneiform Tablets, Part I (Philadelphia, 1905), where the forms are somewhat different; also the Hilprecht Anniversary Volume (Chicago, 1909), p. 128 ff.
- ↑ G. A. Barton, op. cit., Plate 3, obverse.
