Finding the Hypotenuse and Vertical Side from the Hypotenuse Difference

The answer says: The wood is 5 zhang and 5 inches long. The distance from the wall is 4 zhang, 9 feet, and 5 inches.
In this period, 1 zhang equals 10 feet, and 1 foot equals 10 inches.

The method says: According to the rule for finding the 弦 (Xian): the hypotenuse or the longest side of a right triangle from the difference between sides. Take the wall height of 1 zhang 10 feet as the 勾 (Gou): the horizontal base of a right triangle. Multiply the base by itself to get 100. Take the 1 foot by which the leaning wood exceeds the base as the 弦較 (Xian Jiao): the difference between the hypotenuse and the other side. Divide the squared base by this difference, which remains 100. Add the difference of 1 foot to this to get a total of 101 feet. Halve this to get 50.5 feet, or 5 zhang and 5 inches, which is the length of the wood acting as the hypotenuse. Subtract the excess of 1 foot from the hypotenuse to get 49.5 feet, or 4 zhang, 9 feet, and 5 inches, which acts as the 股 (Gu): the vertical side of a right triangle. This is the distance from the wall. These results match the question.

Now, suppose there is a round log embedded in a wall, and its diameter is unknown. If you saw into it to a depth of 1 inch, and the 鋸道 (Ju Dao): the "saw path" or the length of the chord cut by the saw is 1 foot 10 inches long, what is the diameter of the log?

The answer says: The diameter of the log is 2 feet and 6 inches.

[Diagram: A geometric illustration of the Gougu theorem applied to a leaning object. A vertical line represents a wall, labeled "Wall height of 1 zhang is the base." A horizontal line represents the ground, labeled "From the base of the wood to the wall is the vertical side." A diagonal line represents a piece of wood, labeled "The length of the wood is the hypotenuse." A decorative floral element sits in the upper right. Labels include "Hypotenuse difference of 1 foot" and "Base of the wood."]

This problem uses the property of a circle where the diameter can be found if the depth of a segment (the sagitta) and the length of the chord are known. The calculation follows the formula: Diameter = (Half-Chord squared / Depth) + Depth. Here: (5² / 1) + 1 = 26 inches, which is 2 feet 6 inches.