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L. C. Karpinski,[1] and supposed to be intermediate between the Ahmes papyrus and the Akhmim papyrus. Karpinski (p. 22) says: “In the table no distinction is made between integers and the corresponding unit fractions; thus γʹ may represent either 3 or ⅓, and actually γʹγʹ in the table represents 3⅓. Commonly the letters used as numerals were distinguished in early Greek manuscripts by a bar placed above the letters but not in this manuscript nor in the Akhmim papyrus.” In a third document dealing with unit fractions, a Byzantine table of fractions, described by Herbert Thompson,[2] ⅔ is written (Symbol missingsymbol characters); ½, (Symbol missingsymbol characters); ⅓, (Symbol missingsymbol characters) (from (Symbol missingsymbol characters)); ¼, (Symbol missingsymbol characters) (from Δʹ); ⅕, (Symbol missingsymbol characters) (from (Symbol missingsymbol characters)); ⅛, (Symbol missingsymbol characters) (from Ηʹ). As late as the fourteenth century, Nicolas Rhabdas of Smyrna wrote two letters in the Greek language, on arithmetic, containing tables for unit fractions.[3] Here letters of the Greek alphabet used as integral numbers have bars placed above them.

43. About the second century before Christ the Babylonian sexagesimal numbers were in use in Greek astronomy; the letter omicron, which closely resembles in form our modern zero, was used to designate a vacant space in the writing of numbers. The Byzantines wrote it usually ο̄, the bar indicating a numeral significance as it has when placed over the ordinary Greek letters used as numerals.[4]

44. The division of the circle into 360 equal parts is found in Hypsicles.[5] Hipparchus employed sexagesimal fractions regularly, as did also C. Ptolemy[6] who, in his Almagest, took the approximate value of π to be 3+8/60+30/60×60. In the Heiberg edition this value is written γ̄ η̄ λ̄, purely a notation of position. In the tables, as printed by Heiberg, the dash over the letters expressing numbers is omitted. In the edition of N. Halma[7] is given the notation γ̄ ηʹ λʹʹ, which is

  1. “The Michigan Mathematical Papyrus No. 621,” Isis, Vol. V (1922), p. 20—25.
  2. “A Byzantine Table of Fractions,” Ancient Egypt, Vol. I (1914), p. 52–54.
  3. The letters were edited by Paul Tannery in Notices et extraits des manuscrits de la Bibliothèque Nationale, Vol. XXXII, Part I (1886), p. 121–252.
  4. C. Ptolemy, Almagest (ed. N. Halma; Paris, 1813), Book I, chap. ix, p. 38 and later; J. L. Heiberg, in his edition of the Almagest (Syntaxis mathematica) (Leipzig, 1898; 2d ed., Leipzig, 1903), Book I, does not write the bar over the ο but places it over all the significant Greek numerals. This procedure has the advantage of distinguishing between the ο which stands for 70 and the ο which stands for zero. See Encyc. des scien. math., Tome I, Vol. I (1904), p. 17, n. 89.
  5. Αναφορικός (ed. K. Manitius), p. xxvi.
  6. Syntaxis mathematica (ed. Heiberg), Vol. I, Part I, p. 513.
  7. Composition math. de Ptolémée (Paris, 1813), Vol. I, p. 421; see also Encyc. des scien. math., Tome I, Vol. I (1904), p. 53, n. 181.