Page:A History Of Mathematical Notations Vol I (1928).djvu/48

28 L. C. Karpinski, 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 may represent either 3 or ⅓, and actually  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, ⅔ is written ; ½, ; ⅓, (from ); ¼,  (from ); ⅕,  (from ); ⅛,  (from ). As late as the fourteenth century, Nicolas Rhabdas of Smyrna wrote two letters in the Greek language, on arithmetic, containing tables for unit fractions. 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, the bar indicating a numeral significance as it has when placed over the ordinary Greek letters used as numerals.

44. The division of the circle into 360 equal parts is found in Hypsicles. Hipparchus employed sexagesimal fractions regularly, as did also C. Ptolemy who, in his Almagest, took the approximate value of to be 3+$8⁄60$+$30⁄60×60$. In the Heiberg edition this value is written, 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 is given the notation, which is