180 YOSHIO MiKAMI 



Lave heen used in ancient China. The Japanese wcrk Kvcliizufiomi 

 Pj^ of the 10th centniT gives a table of id ulti plication- verses, whicli 

 begins with 9x9 and ends with 1x1. 



A passage in tlie Snn-T.^e Suan-cldng J|^,-^^|M indicates the use 

 of the table in the same order in old China. Another piece of evidence 

 was recently furnished by the discovery of a fragment of such a table 

 at Tun-Huang, near the western border of Kansu Province in China. 



The tables of multiplication-verses given in the Chinese treatises 

 of the 13th century and after are in the natural order. In Japan, 

 however, the reversed order remained in use until still later. Tables 

 of division-verses were also used in China after the 13th century. 

 The Chinese idcogra]>hs lent themselves very well to the formation of 

 such verses. 



5. There are ten separate works on mathematics now existent, 

 which belong to ages not later than the T'ang ^ Dynasty (618-907 

 A.D.). Of these, six are known to have been written during or after 

 the 3rd century A.D. The seventh work Wu-fmo Jl^ is of little 

 value. The remaining three, OlLOu-pei j^ff, CMu-chang Xi^ <^nd 

 Sun-Tse J^^. have sometimes been believed to be of a very remote 

 age. But all three of these are missing in the list given in the Han 

 Shu ^^ ', their texts also contain some things that could not be 

 anterior to the Han age. Thus these works are shown, at least in the 

 forms now existent, to be compositions made during the later Han 

 Dynasty or thereafter. The Chm-cliang, or "Nine i^ections," however, 

 was certainly in existence at the close of this Dynasty, whicli lasted until 

 220 A.D. Tliese ten works were among the twelve that were used 

 officially by the government of the T'ang ^ Dynasty and also by that 

 of the Sung 5^ Dynasty (960-1280 A.D.). It is doubtless due to this 

 circumstance that they have been handed down to us, notwithstanding 

 the fact that all other mathematical treatises of these early times have 

 long since disappeared. Most of them have, however, lost their commen- 

 taries, and even the "Nine Sections," the most important of all Chinese 

 treatises on njathematics, seems to be out of its proper order in some 

 of ist parts, while its commentaries also are somewliat out of order, 

 and the figures once annexed have all been lost. 



In the "Nine Sections," are explained the extractions of square and 

 cube roots, the operations being carried out as in the case of Horner's 

 method of approxin]iitely solving numerical equations. There are also 

 cases in which tlie operations are to be extended to numerical solutions 



