Illustrated Commentary on Nine Chapters, Vol. 1

Then place 100 in the middle position. Multiply the upper [position] once, call out four, one, four, like four. Then place 40 in the middle position, move down the lower position one rank, collect the ten or more from the upper position. In the upper position, two, call out the lower position, one, two, two, like two. Then place 20 in the middle position. Multiply the upper position by four, two, four, like eight. Then place 8 in the middle position. Collect the upper and lower positions, and the middle position yields 168 as the square steps. The answer matches.

The commentary states: Ancient calculations used counting rods. The Han Shu Book of Han states: "The method of calculation uses bamboo, with a diameter of one fen and a length of six inches, 271 pieces, forming one bundle of six-sided rods." The Sunzi Suanjing Master Sun's Mathematical Manual states: "The method of calculation requires first knowing the positions: one is vertical, ten is horizontal, one hundred is standing, one thousand is lying down. The thousand and ten face each other, the ten thousand and hundred correspond to each other." It also says: "Six does not accumulate, five does not stand alone," referring to the counting rods. Its multiplication formula: place the dividend in the upper position, place the divisor in the lower position, and place the multiplier...