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References

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1. Introduction

Early Chinese attempts at proofs of mathematical propositions are not
well known in the West. Among the mathematicians who attempted to go beyond
practical calculation to a more abstract and reasoned mathematics can be named:
Liu Hui (third century A.D.), Zhao Shuang (third century A.D.), Zu Chongzhi
(429-500), Zu Gengzhi (son of Zu Chongzhi), and Wang Xiaotong (seventh century
A.D.).General information on these persons can be found in the usual histories
of Chinese mathematics [1].

I have elsewhere translated Zu Gengzhi's derivation of the volume of a sphere
[Wagner 1978a]. Here I translate Liu Hui's treatment of the volume of a solid
called a *yangma* : it is a pyramid with rectangular base and with one
lateral edge perpendicular to the base.

As is well known, it is a consequence of a theorem proved by Max Dehn in 1900
that any proof of the volume of a pyramid must use infinitesimal considerations
in one form or another.[2] Liu Hui does in fact
use a limit process, though he has considerable conceptual difficulties with
it.

[1]Of works in Western languages the following
are [in 1979] best: Libbrecht [1973]; Juschkewitsch [1964]; Needham [1959];
Mikami [1913]. By far the best history of Chinese mathematics is Qian [1964].
[2]Gauss asked in 1834 whether a proof of the
volume of a pyramid were not possible without the use of infinitesimal
considerations. This question was the basis of the third of Hilbert's famous 23
problems for mathematicians of the twentieth century. Dehn solved the problem
when he proved that a regular tetrahedron and a prism can in no way be divided
into respectively congruent parts. See Gauss [1900, Bd. 8, 244], Hilbert [1900,
301-302], Dehn [1900; 1902], Jessen [1939].