and 49 are given as the squares of the first seven integers, respectively. We have next 1.4=82, 1.21=92, 1.40=102, etc. This clearly indicates the use of the sexagesimal scale which makes 1.4=60+4, 1.21=60+21, 1.40=60+40, etc. This sexagesimal system marks the earliest appearance of the all-important "principle of position" in writing numbers. In its general and systematic application, this principle requires a symbol for zero. But no such symbol has been found on early Babylonian tablets; records of about 200 B.C. give a symbol for zero as we shall see later, but it was not used in calculation. The earliest thorough and systematic application of a symbol for zero and the principle of position was made by the Maya of Central America, about the beginning of the Christian Era.
6. An extension of our knowledge of Babylonian mathematics was made by H. V. Hilprecht who made excavations at Nuffar (the ancient Nippur). We reproduce one of his tablets[1] in Figure 1.
Hilprecht's transliteration, as given on page 28 of his text is as follows:
| Line 1. | 125 | 720 |
| Line 2. | IGI-GAL-BI | 103,680 |
| Line 3. | 250 | 360 |
| Line 4. | IGI-GAL-BI | 51,840 |
| Line 5. | 500 | 180 |
| Line 6. | IGI-GAL-BI | 25,920 |
| Line 7. | 1,000 | 90 |
| Line 8. | IGI-GAL-BI | 12,960 |
| Line 9. | 2,000 | 18 |
| Line 10. | IGI-GAL-BI | 6,480 |
| Line 11. | 4,000 | 9 |
| Line 12. | IGI-GAL-BI | 3,240 |
| Line 13. | 8,000 | 18 |
| Line 14. | IGI-GAL-BI | 1,620 |
| Line 15. | 16,000 | 9 |
| Line 16. | IGI-GAL-BI | 810 |
7. In further explanation, observe that in
| Line 1. | 125=2×60+5, | 720=12×60+0 |
| Line 2. | Its denominator, | 103,680=[28×60+48(?)]×60+0 |
| Line 3. | 250=4×60+10, | 360=6×60+0 |
| Line 4. | Its denominator, | 51,840=[14×60+24]×60+0 |
| Line 5. | 500=8×60+20, | 180=3×60+0 |
| Line 6. | Its denominator, | 25,920=[7×60+12]×60+0 |
| Line 7. | 1,000=16×60+40, | 90=1×60+30 |
| Line 8. | Its denominator, | 12,960=[3×60+36]×60+0 |
- ↑ The Babylonian Expedition of the University of Pennsylvania. Series A: "Cuneiform Texts," Vol. XX, Part 1. Mathematical, Metrological and Chronological Tablets from the Temple Library of Nippur (Philadelphia, 1906), Plate 15, No. 25.
