Systematic Treatise on Arithmetic

The Positioning Method Positioning Method (Dingwei fa); the rule for determining the decimal place or unit value of a result on the abacus simply identifies the position one place above the dan original: shi; a unit of volume for grain this is the tens place to set the unit for taels original: liang; a unit of silver weight. Moving backwards and upwards, the next position the hundreds place sets the tens of taels. Ascending one more position sets the hundreds of taels. In this way, the values are aligned.

This is what is meant by "applying the result to the upper position" in Return Division Return Division (Guichu); a method of abacus division where the quotient is placed on the rods to the left of the dividend. First, count the original dividend (shi) from the hundreds place down to the dan (units). When the leading digit of the divisor (fa) is "2 dan per tael," stop there and move one position forward to designate it as the units of taels. Then, count backwards and upwards through each place to align the result.

The layout for setting the calculation is presented below:

The method states: Perform the Returns original: Gui; refers to the "Nine Returns" division tables.

A vertical calculation diagram representing an abacus or counting board. Circular markers indicate the digits of the result. From top to bottom, they show the digits 2 (貳), 4 (肆), 3 (參), followed by the unit markers for "dan" (石) and "tens" (十). This visually demonstrates the result 243.1.

Observing the Positions

For every two, advance ten remove the value from the current position and advance one digit to the left.
For every six, advance thirty remove the value from the current position and advance three digits to the left.
For every eight, advance forty remove the value from the current position and advance four digits to the left.
These are mnemonic rules for dividing by 2: 2÷2=1, 6÷2=3, 8÷2=4.

A secondary diagram illustrating the physical placement of the "dan" (石) and "tens" (十) units on the calculation board to ensure proper alignment of the silver units.

Positioning

According to the method, since the rate is 2 dan per tael, when you encounter the dan (units) position, the place immediately preceding it is where you find the current taels. By advancing forward through the count and elevating each position, the values will align correctly.