By counting rods or by abacus? The practices of Chinese mathematicians’ computations of multiplication and division in the 13th-15th century

The counting rods were applied to computations in the ancient Chinese mathematics and a set of computation rules, procedures and expressions were formulated. In mathematical writings,however, the procedures were usually described by words, instead of using the expressions of the counting rods. Until the 13th century, the expressions of the counting rods became popular in mathematical writings, meanwhile, the computation using abacus developed rapidly. The abacus computation was impacted greatly by those of the rods counting. Mathematicians represented by Yang Hui of the 13th century designed various new calculation procedures, treated computation in new methods, or compiled calculation pithy formula, to improve the computing speed or simplify computing procedure. All these treatments of computations were helpful to the development of abacus techniques. There was a period of parallel calculation using both the computation tools. During the period, whether the computation was carried on by counting rods or by abacus, the process was still described in the terms and expressions of counting rods in writing mathematical works. It made it some difficult for historians to distinguish computations between the rods and the abacus.

Analyzing the computation procedures from the background of mathematical practice is helpful to solve the problem. Whether it is a rod calculation or an abacus calculation, when calculating multiplication and division, especially,applying the so-called ‘addition’ for multiplication and ‘subtraction’ for division,there will be cases that a number more than 10 appears in one digit. Analyzing how a mathematician treating those cases in his practice is important for us to understanding his computation tool. In the present presentation, we will analyze the procedures and expressions of multiplications and divisions in several mathematical works arranged from the 13th to 15th centuries, so as to distinguish the computations between counting rods and abacus.