40+2×40×10²÷60², yielding 42(13)(20), and also by the approximation 40+10²÷{2×40} yielding 41(15). Translated into the decimal scale, the first answer is 42.22+, the second is 41.25, the true value being 41.23+. These computations are difficult to explain, except on the assumption that they involve sexagesimal fractions.
15. From what has been said it appears that the Babylonians had ideograms which, transliterated, are Igi-Gal for “denominator” or “division,” and Lal for “minus.” They had also ideograms which, transliterated, are Igi-Dua for “division,” and A-Du and Ara for “times,” as in Ara–1 18, for “1×18=18,” Ara–2 36 for “2×18=36”; the Ara was used also in “squaring,” as in 3 Ara 3 9 for “3×3=9.” They had the ideogram Ba–Di–E for “cubing,” as in 27-E 3 Ba-Di-E for “3³=27”; also Ib-Di for “square,” as in 9-E 3 Ib-Di for “3²=9.” The sign A–An rendered numbers “distributive.”[1]
EGYPTIANS
16. The Egyptian number system is based on the scale of 10, although traces of other systems, based on the scales of 5, 12, 20, and 60, are believed to have been discovered.[2] There are three forms of Egyptian numerals: the hieroglyphic, hieratic, and demotic. Of these the hieroglyphic has been traced back: to about 3300 B.C.;[3] it is found mainly on monuments of stone, wood, or metal. Out of the hieroglyphic sprang a more cursive writing known to us as hieratic. In the beginning the hieratic was simply the hieroglyphic in the rounded forms resulting from the rapid manipulation of a reed-pen as contrasted with the angular and precise shapes arising from the use of the chisel. About the eighth century B.C. the demotic evolved as a more abbreviated form of cursive writing. It was used since that time down to the beginning of the Christian Era. The important mathematical documents of ancient Egypt were written on papyrus and made use of the hieratic numerals.[4]
- ↑ Hilprecht, op. cit., p. 23; Arno Poebel, Grundzüge der sumerischen Grammatik (Rostock, 1923), p. 115; B. Meissner, op. cit., p. 387–89.
- ↑ Kurt Sethe, Von Zahlen und Zahlworten bei den alten Ägyptern (Strassburg, 1916), p. 24–29.
- ↑ J. E. Quibell and F. W. Green, Hierakonopolis (London, 1900–1902), Part I, Plate 26B, who describe the victory monument of King Ncr-mr; the number of prisoners taken is given as 120,000, while 400,000 head of cattle and 1,422,000 goats were captured.
- ↑ The evolution of the hieratic writing from the hieroglyphic is explained in G. Möller, Hieratische Paläographie, Vol. I, Nos 614 ff. The demotic writing
