Page:A History of Mathematics (1893).djvu/33

methods of operation were, of course, radically different from ours. Fractions were a subject of very great difficulty with the ancients. Simultaneous changes in both numerator and denominator were usually avoided. In manipulating fractions the Babylonians kept the denominators (60) constant. The Romans likewise kept them constant, but equal to 12. The Egyptians and Greeks, on the other hand, kept the numerators constant, and dealt with variable denominators. Ahmes used the term "fraction" in a restricted sense, for he applied it only to unit-fractions, or fractions having unity for the numerator. It was designated by writing the denominator and then placing over it a dot. Fractional values which could not be expressed by any one unit-fraction were expressed as the sum of two or more of them. Thus, he wrote $\scriptstyle {1 \over 3}{1 \over 15}$ in place of $\scriptstyle {2 \over 5}$. The first important problem naturally arising was, how to represent any fractional value as the sum of unit-fractions. This was solved by aid of a table, given in the papyrus, in which all fractions of the form $\scriptstyle {2 \over 2n+1}$ (where n designates successively all the numbers up to 49) are reduced to the sum of unit-fractions. Thus, $\scriptstyle {2 \over 7}={1 \over 4}{1 \over 28}$; $\scriptstyle {2 \over 99}={1 \over 66}{1 \over 198}$. When, by whom, and how this table was calculated, we do not know. Probably it was compiled empirically at different times, by different persons. It will be seen that by repeated application of this table, a fraction whose numerator exceeds two can be expressed in the desired form, provided that there is a fraction in the table having the same denominator that it has. Take, for example, the problem, to divide 5 by 21. In the first place, $\scriptstyle {5=1+2+2}$. From the table we get $\scriptstyle {2 \over 21}={1 \over 14}{1 \over 42}$. Then $\scriptstyle {5 \over 21}={1 \over 21}+({1 \over 14}{1 \over 42})+({1 \over 14}{1 \over 42})={1 \over 21}+({2 \over 14}{2 \over 42})={1 \over 21}{1 \over 7}{1 \over 21}={1 \over 7}{2 \over 21}={1 \over 7}{1 \over 14}{1 \over 42}$. The papyrus contains problems in which it is required that fractions be raised by addition or multiplication to given whole numbers or to other fractions.