The History of Mathematical Proof in Ancient Traditions

444 karine chemla

Th e fi rst division ends the ‘procedure for the fi eld with the greatest gen-

erality’. Th e reason underlying its correctness is not mentioned here. Th e

second division ends the algorithm for computing the volume of the trun-

cated pyramid with square base. Mentioning the two divisions in succes-

sion allows making sense of the operations step by step, and hence, globally,

of the result. Moreover, this will prove important for the following part of

the reasoning. 26

As a consequence, by successive transformations of algorithm 1, the fol-

lowing algorithm (algorithm 3) is obtained for determining the volume of

the truncated pyramid with square base circumscribed to the desired trun-

cated pyramid with circular base:

Th e appending of two operations to yield algorithm 3 belonged to the

fi rst line of argumentation, as does the next transformation to be eff ected.

Indeed, once he has obtained an algorithm for the truncated pyramid with

square base, Liu Hui turns to considering how to derive the volume of the

truncated pyramid with circular base on the basis of the volume of the cir-

cumscribed pyramid. It is by a fi ft h transformation of the obtained list of

operations that he achieves this goal: operations are to be postfi xed to the

former sequence to get an algorithm yielding the volume of the truncated

pyramid with circular base inscribed in the obtained pyramid with square

base. Liu Hui fi rst makes a geometrical statement (my emphasis):

To look for the volume of the truncated pyramid with circular base, when knowing

the truncated pyramid with square base, is also like to look for the surface of the

circle at the centre of the surface of the square.

Two words deserve some attention here, which is why I emphasized

them. Th e fi rst one is ‘to look for’ ( qiu ). It regularly introduces the task that

26 Below, we shall meet with cases in which Liu Hui combines two divisions that follow each

other. Th e fact that he does in some cases and does not in others relates clearly to the

argument he is making. Th is feature highlights how carefully the relationship between shaping

a procedure and arguing for the correctness of a procedure is handled.

Multiplications Division Division

Sum by 9 by 3

Multiplication by h

C (^) i > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h > ( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9 >[( C (^) i C (^) s + C (^) i 2 + C (^) s 2 ) h /9]/3
C (^) s