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GROWTH OF MATHEMATICAL SCIENCE 15 represent the accumulated experiences of previous writers. In one solitary case, however, he has indicated his method, for, after having asserted that is the sum of \ and , he added that therefore two-thirds of one-fifth is equal to the sum of a half of a fifth and a sixth of a fifth, that is, to -^ + $ That so much attention should have been paid to fractions may be explained by the fact that in early times their treat- ment presented considerable difficulty. The Egyptians and Greeks simplified the problem by reducing a fraction to the sum of several fractions, in each of which the numerator was unity, so that they had to consider only the various denominators : the sole exception to this rule being the fraction f. This remained the Greek practice until the sixth century of our era. The Romans, on the other hand, generally kept the denominator equal to twelve, expressing the fraction (approximately) as so many twelfths. In Ahmes' treatment of multiplication, he seems to have relied on repeated additions. Thus, to multiply a certain number, which we will denote by the letter a, by 13, he first multiplied by 2 and got 2a, then he doubled the results and got 4a, then he again doubled the result and got 8a, and lastly he added together a, 4a, and 8a. Now we have used the sign " a " to stand for any number : not a particular number like 3, but any one. This is what Ahmes did, and what we learn to do in what we call " algebra." When Ahmes wished to find a number such that it, added to its seventh, makes 19, he symbolised the number by the sign we translate " heap." He had also signs for our " + ," " ," and " = ." l Nowadays we can write Ahmes' problem as : Find the number x such that x +-^- = 19. Ahmes gave the answer in the form 16 + \ + |-. 1 In this book, I shall take great care in distinguishing signs for what they signify. Thus : 2 is to be distinguished from " 2 " : by