20. In the older hieroglyphs 2,000 or 3,000 was represented by two or three lotus plants grown in one bush, For example, 2,000 was (
Hieroglyphic characters); correspondingly, 7,000 was designated by (Hieroglyphic characters). The later hieroglyphs simply place two lotus plants together, to represent 2,000, without the appearance of springing from one and the same bush.
21. The multiplicative principle is not so old as the additive; it came into use about 1600–2000 b.c. In the oldest example hitherto known,[1] the symbols for 120, placed before a lotus plant, signify 120,000. A smaller number written before or below or above a symbol representing a larger unit designated multiplication of the larger by the smaller. Müller cites a case where 2,800,000 is represented by one burbot, with characters placed beneath it which stand for 28.
22. In hieroglyphic writing, unit fractions were indicated by placing the symbol 𓂋 over the number representing the denominator. Exceptions to this are the modes of writing the fractions ½, and ⅔; the old hieroglyph for ½ was 𓐞, the later was 𓐛; of the slightly varying hieroglyphic forms for ⅔, (Hieroglyphic characters) was quite common.[2]
23. We reproduce an algebraic example in hieratic symbols, as it occurs in the most important mathematical document of antiquity known at the present time—the Rhind papyrus. The scribe, Ahmes, who copied this papyrus from an older document, used black and red ink, the red in the titles of the individual problems and in writing auxiliary numbers appearing in the computations. The example which, in the Eisenlohr edition of this papyrus, is numbered 34, is hereby shown.[3] Hieratic writing was from right to left. To facilitate the study of the problem, we write our translation from right to left and in the same relative positions of its parts as in the papyrus, except that numbers are written in the order familiar to us; i.e., 37 is written in our translation 37, and not 73 as in the papyrus. Ahmes writes unit fractions by placing a dot over the denominator, except in case of
- ↑ Ibid., p. 8.
- ↑ Ibid., p. 92–97, gives detailed information on the forms representing ⅔. The Egyptian procedure for decomposing a quotient into unit fractions is explained by V. V. Bobynin in Abh. Gesch. Math., Vol. IX (1899), p 3.
- ↑ Ein mathematisches Handbuch der alten Ägypter (Papyrus Rhind des British Museum) übersetzt und erklärt (Leipzig, 1877; 2d ed., 1891). The explanation of Problem 34 is given on p. 55, the translation on p. 213, the facsimile reproduction on Plate XIII of the first edition. The second edition was brought out without the plates. A more recent edition of the Ahmes papyrus is due to T. Eric Peet and appears under the title The Rhind Mathematical Papyrus, British Museum, Nos. 10057 and 10058, Introduction, Transcription, and Commentary (London, 1923).
