Demonstration in Chinese and Vietnamese mathematics 529
amounts of money x (^) A , x (^) B and x (^) C to be obtained by each functionary of the
group A, B, C are calculated as the ‘constant norm’ multiplied by k (^) A , k (^) B , k (^) C ,
respectively.^56
In the second part of the solution the imaginary examinee looks for X (^) A ,
X (^) B and X (^) C which obviously could be found as N (^) A · x (^) A , N (^) B · x (^) B , N (^) C · x (^) C once x (^) A ,
x (^) B and x (^) C have been calculated. However, the suggested solution is diff er-
ent: for example, for group A, the author suggests the calculation of ( N (^) A · k (^) A )·
( S / K ) instead of calculating N (^) A ·[( S · k (^) A )/ K ]; for groups B and C similar
operations are performed. Once again, it can be understood as if the author
considered each entire group A, B and C as one ‘collective recipient’ of the
awarded money, possessing K (^) A = N (^) A · k (^) A , K (^) B = N (^) B · k (^) B and K (^) C = N (^) C · k (^) C ‘shares’,
respectively, while the sum of the ‘shares’ K (^) A + K (^) A + K (^) C remained equal to K.
Examinations and commentaries
Th e solution of the model problem provided in the treatise was based on
the algorithm for the ‘aggregated sharers’ found in a number of Chinese
and Vietnamese mathematical treatises, yet it would be reasonable to
suggest that the imaginary examinee was supposed to design his solution
on the basis of the information found in the same treatise. Indeed, the
treatise provides two sources of such information: (1) a general descrip-
tion of the algorithm of weighted distribution ( CMLT 4: 4b–5a), and (2)
the aforementioned problem 6 of chapter 4 on distribution of donations.
A cursory inspection of these two sources suggests that the solution in the
model paper was designed by analogy with the solution of problem 6; in
particular, the term ‘parts–multiples’ (or ‘multiples of parts’) found in
the solution of the model problem does appear in the solution of problem 6
but not in the algorithm introduced on p. 5a. It is especially interesting that
in this case the Vietnamese author used the term lü , since the concept
of lü was one of the key elements in the conceptual system presented in
Liu Hui’s commentary on the Jiu zhang suan shu. In modern notation, a
number A is a lü (a ‘proportional’, or ‘multiple’) of another number, A′,
if one can establish a proportion in which both numbers occupy the same
positions in the ratios involved: A : B :... :: A′ : B′ :.... 57 However, the term
56 In Volkov 2008 I suggested a mathematically correct yet ‘modernizing’ reconstruction of the
fi rst part of the Vietnamese procedure.
57 For a detailed discussion of the term, see CG2004: 135–6, 956–9. Martzloff 1997 : 196–7
employs the term ‘model’ (i.e. one number can be used as a ‘model’, a representative, of another
number).