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Chapter 5 of the *Jiuzhang suanshu* is ostensibly concerned with
earthworks and the amount of labour needed to build them. The only parts of the
chapter which are especially interesting mathematically, however, are those
which give algorithms for calculating the volumes of solids. For each of these
algorithms, Liu Hui gives a derivation. The order of his derivations is
dictated by the order in which the solids are discussed in the *Jiuzhang
suanshu*, but the derivations can be placed in a logical order so that each
depends only-upon those which precede it; Liu Hui is never guilty of circular
reasoning.

In the present article I summarise one of Liu Hui's derivations (Section 4 below) and translate one "Section 6). Below I describe briefly each of the rectilinear solids treated by Liu Hui.[4]

The volume of a box, or rectangular parallelopiped, the product of its three dimensions, is implicitly assumed, and Liu Hui makes no attempt to explain it.

Another feature which will be apparent in the above discussion is that most of the terms for these solids refer to practical objects. This should not lead to the immediate conclusion that ancient Chinese mathematics was only concerned with practical matters: certainly it was more practically oriented than Greek mathematics, but the use of terms like "fodder loft" to denote geometrical figures is not really different from our use of terms like "pyramid."

[4]In an unpublished thesis [Wagner 1975] I have translated Liu Hui's derivations of the volumes of the

[5]The derivation would begin with the fact that
the formula holds for any pyramid with triangular base and with one lateral
edge perpendicular to the base. (Such a pyramid is either the sum or the
difference of two *bienao.) *Next the formula could be derived for the
case of a pyramid whose base is a convex *n*-sided polygon and whose
altitude intersects the base. (Such a pyramid can be divided into n pyramids of
the type considered above.) The general case would be considerably more
complex, but not absolutely beyond Liu Hui's methods.

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