Page:A History Of Mathematical Notations Vol I (1928).djvu/29

Rh 12. J. Oppert pointed out the Babylonian use of a designation for the sixths, viz., ⅙, ⅓, ½, ⅔, ⅚. These are unit fractions or fractions whose numerators are one less than the denominators. He also advanced evidence pointing to the Babylonian use of sexagesimal fractions and the use of the sexagesimal system in weights and measures. The occurrence of sexagesimal fractions is shown in tablets recently examined. We reproduce in Figure 4 two out of twelve columns found on a tablet described by H. F. Lutz. According to Lutz, the tablet “cannot be placed later than the Cassite period, but it seems more probable that it goes back even to the First Dynasty period, ca. 2000 B.C.”

13. To mathematicians the tablet is of interest because it reveals operations with sexagesimal fractions resembling modern operations with decimal fractions. For example, 60 is divided by 81 and the quotient expressed sexagesimally. Again, a sexagesimal number with two fractional places, 44(26)(40), is multiplied by itself, yielding a product in four fractional places, namely, [32]55(18)(31)(6)(40). In this notation the [32] stands for 32×60 units, and to the (18), (31), (6), (40) must be assigned, respectively, the denominators 60, 60², 60³, 60⁴.

The tablet contains twelve columns of figures. The first column (Fig. 4) gives the results of dividing 60 in succession by twenty-nine different divisors from 2 to 81. The eleven other columns contain tables of multiplication; each of the numbers 50, 48, 45, 44(26)(40), 40, 36, 30, 25, 24, 22(30), 20 is multiplied by integers up to 20, then by the numbers 30, 40, 50, and finally by itself. Using our modern numerals, we interpret on page 10 the first and the fifth columns. They exhibit a larger number of fractions than do the other columns. The Babylonians had no mark separating the fractional from the integral parts of a number. Hence a number like 44(26)(40) might be interpreted in different ways; among the possible meanings are 44×60²+26×60+40, 44×6O+26+40×60⁻¹, and 44+26×60⁻¹+40×60⁻². Which interpretation is the correct one can be judged only by the context, if at all.

The exact meaning of the first two lines in the first column is uncertain. In this column 60 is divided by each of the integers written on the left. The respective quotients are placed on the right.