The Original Diagram for Finding Side-Rates in Root Extraction

Although this diagram is included in Master Wu’s Nine Chapters original: 吳氏九章, Wushi Jiuzhang. This refers to Wu Jing's 1450 work, Detailed Analysis of the Mathematical Canon of the Nine Chapters, which preceded this text. covering everything from the square root to the fifth power, it does not explain how to apply it, and the commentary is unclear. Now, according to the diagram, we proceed from the top:

The left side is the accumulated number The dividend or total product.

The right side is the corner-counter The constant unit used in the calculation.

The Supreme Ultimate

original: 太極, Taiji. In Chinese philosophy and mathematics, this represents the singular source from which all numbers and the universe arise.

(1)

(1) (1)

Square Root Method

(1) (2) (1)

Side-Rates

(1) (3) (3) (1)

Third-Power Equivalent to the fourth power in modern notation ($x^4$); in this system, "three-multiplied-power" indicates the number of multiplications beyond the first power.

(1) (4) (6) (4) (1)

Fourth-Power

(1) (5) (10) (10) (5) (1)

Fifth-Power

(1) (6) (15) (20) (15) (6) (1)

Sixth-Power

The left slant is the accumulated number; the right slant is the corner-counter. The numbers arranged in the middle are all "side-rates." Use the side-rate to multiply the estimated quotient, then assign it to the dividend and divide it.

The 6 and 4 serve as the rates for the third-power extraction. Seeking downward from these, one can find the rates for over thirty higher powers, all of which arise naturally from this pattern.


Combine 1, 3, 3, 1 to get 4, 6, 4 yielding the fourth power	This becomes the Cubic rate, and again	Combining? 1, 2, 1 to get 3, 3	For the Square rate, also combine	Combining? 1 to get 2, which is	Now following the diagram from the top

The table describes the additive property of the triangle: by adding adjacent numbers in one row, you produce the coefficients for the next higher power.