526 alexei volkov
addition of the weighting coeffi cients is done in two steps: fi rst, the weight-
ing coeffi cients are multiplied by the numbers of ‘sharers’ in the respectivegroups, second, the results of the multiplications are summed up: K = N (^) A ·
k (^) A + N (^) B · k (^) B + N (^) C · k (^) C +....
Th e earliest problem on aggregated distribution in China is also found in
the Jiu zhang suan shu (problem 7 of chapter 3 ): 50 there are two groups con-
taining three and two persons, respectively, k 1 = k 2 = k 3 = 3, k 4 = k 5 = 2, S = 5
( SJSSb : 112). However, the solution off ered in the Chinese treatise does not
treat specifi cally this particularity of the condition; the procedure simply
suggests to set the weighting coeffi cients as 3, 3, 3, 2, 2 and to proceed
according to the ‘classical’ method. Chronologically, the earliest extant
Chinese treatise featuring the multiplication of the numbers of sharers in
each category by the respective weights N (^) A · k (^) A , N (^) B · k (^) B , N (^) C · k (^) C is the Sun zi
suan jing ; problem 24 of the second chapter ( juan ) of the treatise belongs
to this type and contains a detailed description of the computational proce-
dure ( SJSSb : 274). Problems of this type are also found in the Zhang Qiujian
suan jing (problem 17 of chapter 1 and problem 13 of chapter 2 , SJSSb :
303–4, 315–16), Suan xue qi meng (Introduction to the learning
of computations, 1299 ) by Zhu Shijie (dates unknown) (problem
50 of chapter 2 , SXQM : 1161), Jiu zhang suan fa bi lei da quan
(Great compendium of the computational methods of nine cat-
egories [and their] generics, 1450) by Wu Jing (dates unknown) 51 a n d
Suan fa tong zong (Summarized fundamentals of computational
methods, 1592) by Cheng Dawei (1533–1606) (Problems 8, 15 and
31 of chapter 5 , SFTZ : 377, 383, 294, respectively). 52
Th e problems on weighted distribution can be found in a number of
Vietnamese mathematical treatises. Th e most interesting case is the sys-
tematic introduction of the method found in the Ý Trai toán pháp nhất đắc
lục (A Record of What Ý Trai Got Right in Computational
Methods, preface 1829) compiled by Nguyễn Hữu Th ận. 53 A s f o r
the treatise under investigation Chỉ minh lập thành toán pháp , chapter 4
contains thirty-eight problems of which twelve are devoted to weighted
50 Th e Suan shu shu does not contain problems on aggregated sharing: in all six problems related
to the weighted distribution (problems 11–16, 21 in Cullen 2004 ) the weights of the sharers are
all diff erent.
51 Problems 5, 33, 36 and 44 of chapter 3 ( DQ 3: 3a, 14b, 17b, 21b) belong to the category of
‘aggregated weighted distribution’, but only problem 5 (analogous to problem 7 of the Jiu zhang
suan shu ) is solved with the ‘classical’ algorithm used in the Jiu zhang suan shu.
52 To numerate the problems, I count the problems per se as well as generalized rules given
without numerical data.
53 V o l k o v f o r t h c o m i n g.