The History of Mathematical Proof in Ancient Traditions

434 karine chemla

18 Perhaps, the layout of the fi rst step should be restored in a diff erent way. Th e middle row of

the upper and lower spaces could be divided into two sub-rows: one in which the result of the

multiplication would be placed – that is, in the middle as usual – and a second one in which

the numerator would remain. Th ereaft er, the two sub-rows would again fuse into a unique

row, with the numerator joining the product.

read what is called in Th e Nine Chapters the ‘ procedure for the field

with the greatest generality’ , which fulfi ls this task.

Procedure: The denominators of the parts respectively multiply the

integer corresponding to them; the numerators of the parts join these

(the results); multiplying [the results] by each other makes the divi-

dend. The denominators of the parts being multiplied by each other

make the divisor. One divides the dividend by the divisor.

If we represent the successive states of the surface for computing when

this sequence of operations is used from left to right, we obtain the result

shown in Figure 13.4.^18

Th e same algorithm can be found in the Book of Mathematical Procedures.

Th e description here, while slightly more specifi c regarding the display of

the arrays of numbers on the surface, can be interpreted along the same

lines. Liu Hui’s commentary on the fi rst step of the procedure contains two

elements that prove quite interesting for our purpose.

Th e fi rst element relates to the conception of the movements eff ected on

the surface by the computations. Liu Hui off ers a slight rewriting of the way

in which the fi rst step should be carried out: the products of the denomi-

nators by the corresponding integers are, in his words, ‘made to enter the

(corresponding) numerators’. Th is does not change anything in the result-

ing confi guration (column 3). However, this fi rst sequence of operations

prescribed by the ‘procedure for the fi eld with the greatest generality’

as integer

bs numerator

3 denominator

3 as bs

3

3 as + bs

33

‘parts

of the

product’

(3as + bs).(3ai + bi)(3as + bs).(3ai + bi)

9

dividend

divisor

ai integer

bi numerator

3 denominator

3 ai bi

3

3 ai + bi

3

3 ai + bi

3

3 ai + bi

3

Figure 13.4 Th e execution of the multiplication of fractions on the surface

for computing.