Illustrated Commentary on Nine Chapters, Vol. 1

...gnomons to observe distance, or the method of using trees to view mountains. All are observed from the side and from the front; they are not something that can be transcended. Matters of this kind, however, are such that the methods established by Cang and his colleagues were still insufficient to cover all numbers. I, Hui, investigated that the Nine Numbers contain the name "Double Difference" chongcha double observation/difference. The original meaning is that it is applied here. Whenever one observes extreme heights, measures great depths, or simultaneously knows their distances, one must use the Double Difference. For the gougu right-angled triangle, one must use the Double Difference as a ratio. Therefore, it is called the Double Difference. Erect two gnomons in the city of Luoyang, each eight feet high, at an equal distance to the north and south on level ground. Measure their shadows at the exact same moment at midday. Use the difference in shadows as the divisor. Multiply the gnomon height by the distance between the gnomons as the dividend. Divide the dividend by the divisor, and add the gnomon height; this is the distance of the sun from the ground. Multiply the southern gnomon's shadow by the distance between the gnomons as the dividend. Divide the dividend by the divisor; this is the distance from the southern gnomon to the point directly below the sun. Use the distance below the sun and the distance of the sun from the ground as the base and altitude of a right-angled triangle. Solve for the hypotenuse, and that is the distance of the sun from the observer. By using a tube with a one-inch diameter to look south at the sun, if the sun fills the tube...