Ch'in Chiu-Shao

Ch'in Chiu-Shao is a thirteenth century Chinese sage who around 1247 AD composed the nine sections of mathematics. He also developed a scheme for the solution of numerical equations.

The solution involves a first approximation to a root which is guessed. The first coefficient is multiplied by the approximate root and added to the second coefficient, making the first partial sum of the first set. The first coefficient is multiplied by the approximate root and added to the first partial sum, making the second partial sum of the first set. This system is kept up until a first set of n partial sums is made. Then the method starts again. The first partial sum is multiplied by the approximate root and the product is added to the third coefficient, making the first partial sum of the second set. The method keeps on in this way, multiplying the partial sums of the first set by the approximate root and adding to the corresponding partial sums of the second set. This is n-1 times, making the second set of partial sums and so on. This is evidently Horner's reduction of all roots by the amount of approximation. The difference between Ch'in Chiu-Shao and Horner's is that Ch'in Chiu-Shao uses Horner's method of synthetic division in reverse order.

In 1913, in his book titled 'The Development in Mathematics in China and Japan', Yoshio Mikam published a simplification of Ch'in Chiu-Shao process. The process is summed up, below in table, using the equation:-

If we solve this as a quadratic, we find four real approximate roots:-

x = + or - 76, x = + or - 265

For some reason Ch'in Chiu-Shao takes his first approximation 800 to solve this equation (as seen in the table below). This solution in the table needs to be read up from the bottom.

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