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Before discussing Liu Hui's treatment of the *yangma*, it will be
useful to look at a simpler example of Liu Hui's treatment of the volumes of
rectilinear solids. The example I have chosen here is his derivation of the
volume of a *fangting* : this derivation is mathematically simple and
presents no philological problems.(Liu Hui's proof is translated in full in
Wagner [1975, 15-19].)

Liu Hui's general method in treating a rectilinear solid is to cut it up into parts whose volumes are known. From the formulas for these volumes he then derives the formula for the volume of the solid under discussion. Here he runs into two areas of difficulty: (1) expressing how the solid is to be cut up, and (2) manipulating the formulas for the parts to arrive at the formula for the whole.

Liu Hui is less successful in solving the second difficulty. His technical terminology is apparently not well-developed enough for him to refer to the general case; therefore, he derives the formula by reference to a case of specific dimensions, but in such a way that the proof can easily be generalised to the case of arbitrary dimensions. He gives only the broad outlines of the manipulations, leaving the details to the reader.

The *Jiuzhang suanshu* [1963, 164] gives the calculation of the volume of
a *fangting* as

where_{ }

In his derivation of this formula, Liu Hui refers throughout to the specific
case with a = 1 *chi*, b = 3 *chi*, and h = 1 *chi*. He
describes how this solid can be built up with the standard blocks. See Figure
1. The blocks are:

1 cube, ABDCQRML;He then writes:

4qiandu, CDRQTU, BDRMNS, ABMLIJ, and ACQLKP;

4yangma, CQTGP, DRUHS, BMNFJ, and ALIEK.

The product of [the sides of] the upper and lower squares is [That is to say,ab=] 3 [square]chi, and multiplying by the height gives [abh=] 3 [cubic]chi. This means there is obtained [the volume of] one central cube and one each of theqianduat the four sides.

[Jiuzhang, 164]

and though the text refers to specific dimensions, this statement is also true in the case of arbitrary dimensions. The statement can be verified geometrically: the five pieces can be fitted together to form a box with dimensionsabh= ABDCQRML + CDRQTU + BDRMNS + ABMLIJ + ACQLKP,

andb^{2}h= ABDCQRML + 2(CDRQTU + BDRMNS + ABMLIJ + ACQLKP) + 3(CQTGP + DRUHS + BMNFJ + ALIEK),

Again these statements are given in terms of specific dimensions, but both are valid in the case of arbitrary dimensions. Both can be verified geometrically. Liu Hui concludes thata^{2}h= ABDCQRML.

Thus, implicitly using the distributive law,abh+b^{2}h+a^{2}h= 3 ABDCQRML

+ 3 (CDRQTU + BDRMNS + ABMLIJ + ACQLKP)

+ 3 (CQTGP + DRUHS + BMNFJ + ALIEK).

which was to be proved.

Liu Hui goes on to derive another, equivalent, formula:

Another method: multiply by itself the difference of [the sides of] the squares, multiply by the height, and divide by 3. this gives [the volume of] the four[yangma. then multiply together [the sides of] the upper and lower squares and multiply by the height. This gives [the volume of] the central cube and theqianduat the four sides. Add [these two results] to obtain the volume of thefangting.

which may be verified either algebraically from the first formula or by Liu Hui's geometric method._{ }

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