The History of Mathematical Proof in Ancient Traditions

(Elle) #1

462 karine chemla


We can now turn to examining in greater detail the relationship between
proving the correctness of procedures dealing with fractions and establish-
ing the validity of transformations ii and iii. To do so, we shall have to
analyse new samples of proof contained in our Chinese sources. Th is will
give us the opportunity to describe further the specifi cities of the practice
of proof to which our documents bear witness.

Proving the correctness of the general algorithm for division
Let us examine the way in which, in his commentary, Liu Hui establishes
that the ‘procedure for directly sharing’ is correct, before considering why
this argument can be interpreted as related to the validity of transforma-
tion iii.^49 Th e Nine Chapters introduces the algorithm for dividing between
quantities with fractions aft er the two following problems:
(1.17) Suppose one has 7 persons sharing 8 units of cash and 1/3 of a unit
of cash. One asks how much a person gets.
Answer: a person gets 1 unit of cash 4/21 of unit of cash.
(1.18) Suppose again one has 3 persons and 1/3 of a person sharing 6 units
of cash, 1/3 and 3/4 of a unit of cash. One asks how much a person gets.
Answer: a person gets 2 units of cash 1/8 of unit of cash.
In the fi rst problem, the quantity that is to become the dividend contains
one fraction, whereas the second problem leads to both the dividend and
the divisor having fractions. Th e fact that the dividend even contains two
fractions is remarkable. Interestingly enough, such quantities, in which
an integer is followed by a sequence of fractions, occur only in problems
related to similar divisions. 50 We shall see that this is linked to the fact that
Liu Hui uses the operations introduced in his commentary on the addition
of fractions for his proof.
Th ese two problems are in fact the fi rst ones in Chapter 1 for which the
data are neither pure integers nor pure fractions. Moreover, they are the
fi rst problems in which the fractions derive from sharing a unit that is not
abstract. Furthermore, problem 1.18 mixes together fractions of diff erent

49 Th e critical edition and the translation of this piece of commentary can be found in CG2004:
166–9.
50 In addition to the situation examined here, this also designates problems linked to the
‘procedure for the small width’, which opens Chapter 4. Th e procedure provides another
way of carrying out the division. For comparison, I refer the reader to the introduction to
Chapter 4 in CG2004. Th e interpretation of the ‘procedure for directly sharing’ requires an
argumentation that I developed in Chemla 1992 (I do not repeat the bibliography given in this
earlier publication).
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