Reading proofs in Chinese commentaries 439
with square base circumscribed to one with a circular base in case quanti-
ties with fractions occurred, can hence be transformed into algorithm 2′,
without altering the result:
Multiplications
SumsC (^) i > C (^) i 2 + C (^) i · C (^) s + C (^) s 2
C (^) s
Multiplying the denominators, dividing by the result 9, multiplying
the result by h , dividing by 3
Th e essential point now is that algorithm 2′ shares the same initial list of
operations with the algorithm for the truncated pyramid with circular base
as described in Th e Nine Chapters. Th e reason why this fact is important is
that the arguments outlined above allow the interpretation of the ‘meaning’,
namely, the ‘intention’ ( yi ) of the fi rst part of the algorithm, the correctness
of which is to be established. Liu Hui writes (my emphasis):
If one multiplies by one another the upper and lower diameters, then multiplies
each by itself respectively, then adds these and multiplies by the height, this gives
the parts of the product ( jifen ) of 3 truncated pyramids with square base.
Again, this statement is worth analysing in detail. Note, fi rst, that Liu
Hui refers to C i and C s as ‘diameters’. Th is is the meaning of the initial values
entered in the algorithm that was established by bringing to light the pair
of deleted, opposed operations. Th ese values are diameters, with respect to
the denominators. Such an analysis corresponds to the fact that the result
of the fi rst section of the algorithm is interpreted as ‘parts of the product’ in
reference to the ‘procedure for the fi eld with the greatest generality’. More
generally, it is by reference to algorithm 2′, itself obtained from a combina-
tion of three algorithms, that the interpretation of the result of the fi rst part
of the algorithm is made explicit. Algorithm 2′ has been shown to yield the
volume of the circumscribed truncated pyramid. To state the meaning of
the result of its fi rst part as the ‘parts of the product ( jifen ) of 3 truncated
pyramids with square base’, two of its fi nal computations had to be dropped
(dividing the result by 9 and dividing by 3). Each computation relates to
a diff erent algorithm among the algorithms that are combined, and the
structure of the statement highlights the diff erent statuses of the factors
which are left out. Th e proof of the correctness of the algorithm for the
truncated pyramid with square base had established that the fi rst part of its
computations yielded the value of 3 pyramids. Th e proof of the correctness