The History of Mathematical Proof in Ancient Traditions

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Using the same notations for algorithms as above, it can be represented

as follows:

14 Incidentally, this proposition is stated in the Book of Mathematical Procedures (slips 194–5,

Peng Hao 2001 : 111).

Figure 13.2 Th e truncated pyramid with square base.

Ds

Di

Multiplications Division

sum by 3

Multiplication by h

D (^) i > ( D (^) i D (^) s + D (^) i 2 + D (^) s 2 ) h > ( D (^) i D (^) s + D (^) i 2 + D (^) s 2 ) h /3
D (^) s
On this basis, Liu Hui states a fi rst algorithm (algorithm 1) which deter-
mines the volume of the truncated pyramid with square base circumscribed
to the truncated pyramid with circular base which is considered. Quoting
the algorithm of the Classic verbatim – a fact that I indicate by using quota-
tion marks in the translation – his commentary reads:
Th is procedure presupposes ( yi′ ) that the circumference is 3 when the diameter is

1. One must hence divide by 3 the circumferences of the upper and lower circles
   to make the upper and lower diameters respectively. ‘Multiplying them by one
   another, then multiplying each of them by itself ’, adding, ‘multiplying this by the
   height and dividing by 3’ makes the volume of the truncated pyramid with square
   base.
   Th e only transformation (transformation 1) needed to make use of the
   algorithm quoted in this new context is to prefi x its text with two div-
   isions by 3. Th ese operations change the given circumferences into the
   corresponding diameters, the lengths of which are respectively equal to
   the lengths of the sides of the upper and lower circumscribed squares. 14
   Algorithm 1 can be represented as follows: