Page:A History of Mathematics (1893).djvu/25

 10 times 100, or 1000. But this symbol for 1000 was itself taken for a new unit, which could take smaller coefficients to its left. Thus, denoted, not 20 times 100, but 10 times 1000. Of the largest numbers written in cuneiform symbols, which have hitherto been found, none go as high as a million.[3]

If, as is believed by most specialists, the early Sumerians were the inventors of the cuneiform writing, then they were, in all probability, also familiar with the notation of numbers. Most surprising, in this connection, is the fact that Sumerian inscriptions disclose the use, not only of the above decimal system, but also of a sexagesimal one. The latter was used chiefly in constructing tables for weights and measures. It is full of historical interest. Its consequential development, both for integers and fractions, reveals a high degree of mathematical insight. We possess two Babylonian tablets which exhibit its use. One of them, probably written between 2300 and 1600 B.C., contains a table of square numbers up to $$\scriptstyle{60^2}$$. The numbers 1, 4, 9, 16, 25, 36, 49, are given as the squares of the first seven integers respectively. We have next $$\scriptstyle{1.4=8^2}$$, $$\scriptstyle{1.21=9^2}$$, $$\scriptstyle{1.40=10^2}$$, $$\scriptstyle{2.1=11^2}$$, etc. This remains unintelligible, unless we assume the sexagesimal scale, which makes $$\scriptstyle{1.4=60+4}$$, $$\scriptstyle{1.21=60+21}$$, $$\scriptstyle{2.1=2.60+1}$$. The second tablet records the magnitude of the illuminated portion of the moon's disc for every day from new to full moon, the whole disc being assumed to consist of 240 parts. The illuminated parts during the first five days are the series 5, 10, 20, 40, 1.20 ($$\scriptstyle{=80}$$), which is a geometrical progression. From here on the series becomes an arithmetical progression, the numbers from the fifth to the fifteenth day being respectively 1.20, 1.36, 1.52, 2.8, 2.24, 2.40, 2.56, 3.12, 3.28, 3.44, 4. This table not only exhibits the use of the sexagesimal system, but also indicates the acquaintance of the Babylonians with progressions.