The History of Mathematical Proof in Ancient Traditions

Reading proofs in Chinese commentaries 443

Figure 13.5 Th e basic structure of algorithms 1 and 2, for the truncated pyramid
with square base.

Dividing Multiplying by one another

(Procedure for the fi eld with the greatest generality)

Cs

3

as

bs

3

Cs

33

Cs. Ci Cs. Ci

9

Dividing

Cs

3

as

bs

3

Cs

3

Cs

3

Cs

3

Figure 13.6 Th e basic structure of algorithm 2′, which begins the computation of the
volume sought for.

Multiplying by one another Dividing

Cs

3

Cs. Ci Cs. Ci

9

Dividing

Ci Ci

3

Ci

3

Th e beginning of algorithm 2′ would instead yield Figure 13.6.

Postfi xing operations to an algorithm within the context of the proof

Let us now return to Liu Hui’s commentary on the algorithm determining
the volume of the truncated pyramid with circular base and read its fol-
lowing section. Th e commentator’s interpretation of the result of the fi rst
section of the algorithm as ‘parts of the product ( jifen ) of 3 truncated pyra-
mids with square base’ produces a foundation upon which his reasoning
can be built. He writes (transformation 4):

Here, one must multiply the denominators, 3, by one another – hence one obtains
9 – to make the divisor, and divide by this. If, in addition to this, one divides by 3,
one obtains the volume of the truncated pyramid with square base.