MATHEMATICS IM CHINA AND JAPAN. 179 



passage in the T^^^o-cliaan 'frJM-^ that is to he ex])lamed hy the same 

 arrangement, we may infer its antiquity. The stieks are to be arranged 

 thus : 



I II 111 111! 11111 T T T W 



123456789 



10 20 30 40 50 60 70 80 90 



Here a question arises : the ideographs for 1, 2 and 3 are — , 

 H, ^ ; 4 was also re^^resented by ^ in old times instead of the present 

 form pg ; while -p, ■\^, j^, -(if]- were used for 10, 20, 30 and 40, but 

 in the diagram the forms — = = =, instead of representing 1, 2, 

 3, 4, stand for 10, 20, 30, and 40, respectively. This fact seems 

 to show that the arrangements of the sticks, have undergone a change. 

 A passage in the works of 3Io-Tse g^ (5th-4th centin-y B. C), though 

 very obscure in expression, implies, when properly intei-preted, the same 

 arrangements of the sticks as in the forms of the ideographs. In the 

 time of Mo-Tse, either the change had not yet taken place or there 

 was a transitional stage between the two forms. 



The doctrine of / ^, as expounded in the I ching ^^, occupies 

 a most important position in the history of Chinese thought. According 

 to this system, two sorts of sticks are employed, one sort longer and 

 the other shorter. The shorter sticks, consisting of two varieties, serve 

 to arrange the symbols of the pa-hna /\^ in a horizontal way. The 

 longer sticks are used for divination, being held within the hand in a 

 vertical position. Both forms bear a close relation to the calculating 

 sticks. Besides the use of sticks, the Chinese also attempted to represent 

 numbers in various other ways, some of which are recorded in the 

 So-shu Chi-i ^^tt^M.- 



Numerical systems other than decimal were used of old in connection 

 with weights and measures. But in the science of numbers itself the 

 decimal system early found favour. The numlier ten thousand was 

 called loan |^, and for numbers above this, three or four ways of 

 numbering existed, of which the most customary was that which was 

 marked by the recurrence of eight digits, loan-times-ioan being employed. 

 For decimal numbers each digit was separately nominated. 



4. Multiplication- tables, or more properly, tables of multiplication 

 verses, are foimd in complete form in no Chinese work older than the 

 13th century. But the Shih-cJd ^fg,, the Kxan-Tstc ^^ and other 

 old works contain portions of such verses, which accordingly seem to 



