Preface to the Diagrams of Double Differences

The original diagrams for the Chongcha Double Difference (specifically the "Looking toward the Sea Island" problem) only included one diagram. I have now supplemented this with the remaining eight. Regarding the method of similar triangles: when comparing two sets of proportions, the product of the second and third terms is equal to the product of the first and fourth terms. To illustrate this with a diagram, take the first and third terms and join them to form one side, then take the second and fourth terms and join them to form another side. When constructing a rectangular diagram representing the product, it naturally divides into four areas (powers). By using a diagonal chord to bound these, two sets of similar right-angled triangles are formed, making the relationships clear.

In the older diagrams, one had to draw separate squares of equal area outside the main figures. When the two figures were far apart, it was necessary to connect them with winding lines; this is no longer required. The diagrams in this text discuss quadrilaterals and pentagons, which may appear unlike standard right-angled triangles. However, by supplementing the original figures with additional right-angled triangles, they are effectively transformed into right-angled triangles as well. The rule of four-term proportion is mentioned in the "Millet and Rice" section of the Nine Chapters. It is called the "Rule of Three" original: "今有一" lit. "Now there is one", where the first term is the "given rate," the second is the "target rate," and the third is the "required..." the text breaks off here as the final term is missing

Kang