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

Rh modern exponent; thus Archimedes and Eutocius use the notation or  for $17⁄21$, and Diophantus (§§ 101–6), in expressing large numbers, writes (Arithmetica, Vol. IV, p. 17),  for $36,621⁄2,704$. Here the sign 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. In expressing such fractions, the Greeks omitted the for the numerator and wrote the denominator only once. Thus =$1/undefined$. Unit fractions in juxtaposition were added, as in =⅐+$1/undefined$+$1/undefined$+$1/undefined$. One finds also a single accent, as in =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, namely, or, but this designation was no more adopted generally among the Greeks than were the other notations of fractions. Ptolemy wrote 38°50′ (i.e., 38°½ ⅓) thus,. Hultsch has found in manuscripts other symbols for ½, namely, the semicircles, , and the sign ; the origin of the latter is uncertain. He found also a symbol for ⅔, resembling somewhat the small omega. 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