Page:A History of Mathematics (1893).djvu/98

 Rome, but also in Greece and throughout the East, certainly as early as the beginning of the Christian era, and continued to be used in Europe during the Middle Ages. We possess no knowledge as to where or when it was invented. The second mode of calculation, by the abacus, was a subject of elementary instruction in Rome. Passages in Roman writers indicate that the kind of abacus most commonly used was covered with dust and then divided into columns by drawing straight lines. Each column was supplied with pebbles (calculi, whence 'calculare' and 'calculate') which served for calculation. Additions and subtractions could be performed on the abacus quite easily, but in multiplication the abacus could be used only for adding the particular products, and in division for performing the subtractions occurring in the process. Doubtless at this point recourse was made to mental operations and to the multiplication table. Possibly finger-multiplication may also have been used. But the multiplication of large numbers must, by either method, have been beyond the power of the ordinary arithmetician. To obviate this difficulty, the arithmetical tables mentioned above were used, from which the desired products could be copied at once. Tables of this kind were prepared by Victorius of Aquitania. His tables contain a peculiar notation for fractions, which continued in use throughout the Middle Ages. Victorius is best known for his canon paschalis, a rule for finding the correct date for Easter, which he published in 457 A.D.

Payments of interest and problems in interest were very old among the Romans. The Roman laws of inheritance gave rise to numerous arithmetical examples. Especially unique is the following: A dying man wills that, if his wife, being with child, gives birth to a son, the son shall receive $$\scriptstyle{\frac{2}{3}}$$ and she $$\scriptstyle{\frac{1}{3}}$$ of his estates; but if a daughter is born, she shall receive $$\scriptstyle{\frac{1}{3}}$$ and his wife $$\scriptstyle{\frac{2}{3}}$$. It happens that twins are born, a boy and a girl.