General Source of Computational Methods, Volume 12

Finding the Diameter from the Chord and Depth

The method says: Take the saw path (the width of the cut) of 1 foot and halve it to get 5 inches, which acts as the 勾 (Gou): the horizontal base of a right triangle. Square this to get 25 inches as the dividend: the number to be divided. Use the 濇 (Se): the depth of the saw cut (sagitta) of 1 inch as the 股較 (Gu Jiao): the difference between the vertical side and the hypotenuse to divide it as before. This gives 25 inches as the 股 (Gu): the vertical side Specifically, this represents the remaining portion of the diameter minus the depth of the cut. Add the depth of the cut of 1 inch to this, for a total wood diameter of 2 feet 6 inches. This matches the question.

This is the same method used to find the original diameter when a segment is cut from a circular field. Set the saw path of 1 foot as the chord of the segment. Halve it to get 5 inches, and square it to get 25 inches as the dividend. Use the depth of 1 inch as the 矢 (Shi): the sagitta or height of the arc as the divisor. Dividing gives 25 inches. Adding the sagitta depth of 1 inch results in a total of 2 feet 6 inches, which is the original diameter of the round log.

Suppose there is a round log with a diameter of 2 feet 6 inches. If a saw is used to cut into the wood to a depth of 8 inches, what is the length of the saw path?

This question is identical to the diagram on the right. The numbers are now noted within the diagram for easy viewing of the diameter and components.

The answer says: The saw path is 2 feet 4 inches long.

The method says: Take the diameter of 2 feet 6 inches and subtract the depth of 8 inches, leaving 1 foot 8 inches. Multiply this by the depth of 8 inches to get 144 inches?. The mathematical logic continues by taking the square root of 144 to get 12 inches, then doubling it to find the full saw path of 24 inches, or 2 feet 4 inches.