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How did Seki Takakazu replace counting-rod notation with a tabular system?

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East Asian mathematics Counting boards and markers, or counting rods, were used in China to solve systems of linear equations. This is an example from the 1st century ce.

Seki Takakazu, a central figure in the Japanese wasan tradition, revolutionized mathematical notation by replacing the cumbersome Chinese method of counting rods with a tabular notational system.1

This transition was part of his broader effort to develop a "literal" written algebra. By utilizing this tabular system, Seki simplified the handling of equations involving more than one unknown. He maintained a positional notation for equations with respect to a single unknown, while expressing coefficients through numbers, parameters, or other variables.21

This innovation effectively liberated mathematicians from the physical limitations of counting rods, allowing for more sophisticated algebraic manipulation. In his work Kaifukudai no hō (1683; "Method for Solving Concealed Problems"), Seki detailed important properties related to these computations, which ultimately enabled him and his disciples to advance beyond traditional methods, including the development of procedures equivalent to the computation of determinants to eliminate unknowns in simultaneous equations.21