# Q.E.D.

14 15 a FrustratIng Frustum horses and moat walls Many ancient manuscripts contain algorithms for calculating the areas or volumes of geometric figures, but not all formul used by the ancients were correct. According to a Babylonian source, the volume of a frustum, or truncated square pyramid, was 1 2 a b2h, whereas the Egyptian Rhind papyrus ca. 1800 B.C.E., indicates the descendants of the pyramid builders were using the correct version 1 3 a 2 ab b 2h. One of the oldest surviving Chinese mathematical treatises Jiuzhang Suanshu Arithmetic in Nine Chapters, ca. 50 B.C.E. also mentions this version of the formula, and Liu Hui ca. 263 A.D., in his commentary, gives a beautiful proof for it. He dissects the frustum or fangting square pavilion into nine pieces four identical pyramids or yangma male horses, four prisms or qiandu moat walls, and a rectangular box, all of which combine into a box and a pyramid. Adding the volumes of these together produces the volume of the frustum opposite, top. This proof assumes that we already know the formula for the volume of a square pyramid see previous page, but we can nevertheless use our frustum dissection to rediscover this in an elegant snakebitesitsown tail way lower, opposite. Other solids volumes from Liu Huis commentary are shown below.