The History of Mathematical Proof in Ancient Traditions

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Reading proofs in Chinese commentaries 433


ences of the upper and lower bases were displayed, respectively in the upper
and lower rows of the surface. Th e reason for this is that numbers derived
from them would soon enter into a multiplication. Before that multiplica-
tion, the algorithm prescribes that both circumferences be divided by 3.
Th ese divisions were to be set up and carried out in the upper and lower
spaces, with the row in which the numbers had been placed becoming in
turn a space in which a computation was executed according to the same
rules of presentation. For instance, the upper row was split into three sub-
rows, with the dividend C s occupying the middle sub-row and the divisor 3
the lower sub-row (second state of the surface in Figure 13.3 ). 17 In the situa-
tions considered by Liu Hui, once the divisions were completed, none of the
dividends in the upper and lower spaces would have vanished, the result of
each division being of the form of an integer increased by a fraction (third
state in Figure 13.3 ). Th ese, then, are the quantities to be multiplied accord-
ing to the next step of the algorithm (‘Multiplying them by one another,
then multiplying each of them by itself ’). Th is feature of hierarchical organi-
zation, according to which a space in which a number is placed can become
a sub-space, in which an operation is performed according to the same rules
at any level, is, in my view, one of the most important characteristics of this
system of computation. Th is feature ensures that the successive computa-
tions required by an algorithm will be articulated with each other spatially,
instead of being dissociated and carried out independently of each other.
Th e right-hand part of Figure 13.3 shows the state of the surface for com-
puting, at the point where the algorithm requires inserting the algorithm
for multiplying quantities that consist of an integer and a fraction. Let us


17 In LD1987: 16–18, the reader can fi nd descriptions of how the computations of a
multiplication and a division were carried out on the surface for computing.


Figure 13.3 Th e layout of the algorithm up to the point of the multiplication of
fractions.


Cs Cs Dividend
3 Divisor

as integer
bs numerator
3 denominator

Dividing by 3

Ci Ci Dividend
3 Divisor

ai integer
bi numerator
3 denominator
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