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 to be divided, are counted in wholly, even though parts of them project beyond it.

Gerbert made a careful study of the arithmetical works of Boethius. He himself published two works,—Rule of Computation on the Abacus, and A Small Book on the Division of Numbers. They give an insight into the methods of calculation practised in Europe before the introduction of the Hindoo numerals. Gerbert used the abacus, which was probably unknown to Alcuin. Bernelinus, a pupil of Gerbert, describes it as consisting of a smooth board upon which geometricians were accustomed to strew blue sand, and then to draw their diagrams. For arithmetical purposes the board was divided into 30 columns, of which 3 were reserved for fractions, while the remaining 27 were divided into groups with 3 columns in each. In every group the columns were marked respectively by the letters C (centum), D (decem), and S (singularis) or M (monas). Bernelinus gives the nine numerals used, which are the apices of Boethius, and then remarks that the Greek letters may be used in their place.[3] By the use of these columns any number can be written without introducing a zero, and all operations in arithmetic can be performed in the same way as we execute ours without the columns, but with the symbol for zero. Indeed, the methods of adding, subtracting, and multiplying in vogue among the abacists agree substantially with those of to-day. But in a division there is very great difference. The early rules for division appear to have been framed to satisfy the following three conditions: (1) The use of the multiplication table shall be restricted as far as possible; at least, it shall never be required to multiply mentally a figure of two digits by another of one digit. (2) Subtractions shall be avoided as much as possible and replaced by additions. (3) The operation shall proceed in a purely mechanical way, without requiring trials.[7]