Page:A History Of Mathematical Notations Vol I (1928).djvu/47

modern exponent; thus^[1] Archimedes and Eutocius use the notation (symbol characters) or (symbol characters) for 17/21, and Diophantus (§§ 101–6), in expressing large numbers, writes (Arithmetica, Vol. IV, p. 17), (symbol characters) for 36,621/2,704. Here the sign (symbol characters) takes the place of the accent. Greek writers, even as late as the Middle Ages, display a preference for unit fractions, which played a dominating rôle in old Egyptian arithmetic.^[2] In expressing such fractions, the Greeks omitted the αʹ for the numerator and wrote the denominator only once. Thus μδʺ=¹⁄₄₄. Unit fractions in juxtaposition were added,^[3] as in ζʺ κηʺ ριβʺ σκδʺ=⅐+¹⁄₂₈+¹⁄₁₁₂+¹⁄₂₂₄. One finds also a single accent,^[4] as in δʹ=¼. Frequent use of unit fractions is found in Geminus (first century B.C.), Diophantus (third century A.D.), Eutocius and Proclus (fifth century A.D.). The fraction ½ had a mark of its own,^[5] namely, 𐅵 or 𐅵ʹ, but this designation was no more adopted generally among the Greeks than were the other notations of fractions. Ptolemy^[6] wrote 38°50′ (i.e., 38°½ ⅓) thus, ληʹ 𐅵ʺγʺ. Hultsch has found in manuscripts other symbols for ½, namely, the semicircles (symbol characters), 𐅁, and the sign (symbol characters); the origin of the latter is uncertain. He found also a symbol for ⅔, resembling somewhat the small omega (𐅷).^[7] Whether these symbols represent late practice, but not early usage, it is difficult to determine with certainty.

42. A table for reducing certain ordinary fractions to the sum of unit fractions is found in a Greek papyrus from Egypt, described by

1. G. H. F. Nesselmann, Algebra der Griechen (Berlin, 1842), p. 114.
2. J. Baillet describes a papyrus, “Le papyrus mathématique d’Akhmîm,” in Mémoire publiés par les membres de la Mission archéologique française au Caire (Paris, 1892), Vol. IX, p. 1–89 (8 plates). This papyrus, found at Akhmîm, in Egypt, is written in Greek, and is supposed to belong to the period between 500 and 800 A.D. It contains a table for the conversion of ordinary fractions into unit fractions.
3. Fr. Hultsch, Metrologicorum scriptorum reliquiae (1864—66), p. 173–75; M. Cantor, Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 129.
4. Nesselmann, op. cit., p. 112.
5. Ibid., James Gow, Short History of Greek Mathematics (Cambridge, 1884), p. 48, 50.
6. Geographia (ed. Carolus Müllerus; Paris, 1883), Vol. I, Part I, p. 151.
7. Metrologicorum scriptorum reliquiae (Leipzig, 1864), Vol. I. p. 173, 174. On p. 175 and 176 Hultsch collects the numeral symbols found in three Parisian manuscripts, written in Greek, which exhibit minute variations in the symbolism. For instance, 700 is found to be ψ, ψ^π, ψʹ.