Systematic Treatise on Arithmetic, Volume 3

Page 4

Square Fields

The method states: Set the length at 50 and multiply it by the width, which is also 50. The resulting product shi: the dividend or result of multiplication is 2,500 square steps. Divide this by the standard mu conversion factor mufa: the rule that 240 square steps equal 1 mu.

The Method for Determining Position
original: 定位法 (dingweifa). This refers to the technique used by abacus practitioners to determine the decimal place or "order of magnitude" of a result.
First, count from the leading digit of the original product (the dividend), starting from the tens place and moving downwards to the final digit. Place the leading digit of the divisor fa: the number one divides by one position below. Count backwards and upwards to the leading digit of the dividend. If you have 2,000, then the next digit is in the 500s place. All other calculations follow this pattern.

Square Fields Measured Diagonally

Suppose there is a square field measured diagonally from the southeast corner to the northwest corner, and from the southwest corner to the northeast corner. Each diagonal is 70 steps. What is the total area?

Answer: The area is 2,450 square steps. Tax: 10 mu, 0 fen, 2 li, 0 hao, 8 si original: 稅一十畝○二厘○八. This is the land tax expressed in mu and its decimal sub-units, calculated as 10.208 mu.

The method states: Take the diagonal xiexian: literally "slanting string," the hypotenuse or diagonal of 70 and multiply it by itself to get 4,900. Halve this The area of a square can be found by (diagonal² / 2) to get 2,450 square steps as the dividend. Divide this by the standard mu conversion factor of 240 to arrive at the answer. The positioning of the digits is the same as described above.