178 YoRHIO MiKAMT 



frag'iuenis of such kuovvlodge are recorded in. various old docunient.s, 

 the dates of which, however, are hard to determine. There is recorded 

 a very remarkal^le device in connection with the calendrical art. The 

 circumference of a circle was equally divided to represent the directions, 

 and the same method was used for the representation of seasons, hours 

 and the revolutions of heavenly bodies, the designations being made by 

 the ideographs of the po-kua A# tin^cl^ kan- chili ^^. Thus it happened 

 that various subjects were arranged in one and the same diagram. 

 This no doubt arose for the sake of convenience in astronomical obser- 

 vations, but it must not be denied also, that the ideographs proved 

 very appropriate for the pur[)Ose of such a representation. The natural 

 fondness of the Chinese mind f<ir the association of ideas was also a 

 factor in its conception. The ancient Chinese also constructed a vessel 

 which represented the standards of lengths, measures, weights and 

 musical notes simultaneously. 



3. It is also said that the Chinese in ancient times used string- 

 knots or knotted strings for the purpose of reckoning. Nothing is known 

 as to how they were used. But the Luchu ^^ Islanders are accustomed 

 in modern times to reckon by means of straw, and in Japan also some 

 similar modes of counting, though very seldom used, have been preserved 

 until quite recently. From these methods, we jnay justly form some 

 idea of what the old Chinese mode was like. 



Somewhat later ])robably, sticks called by the several names of 

 suon ^, tse H, and cli^oa ^ were used in reckoning. These different 

 names certainly indicate differences in form, size or use, but all three 

 were used to designate some kind of sticks. The songi ^if. or 

 calculating sticks, which the Japanese mathematicians of the 17tli to 

 the 19th century abundantly used, consisted of small wooden pieces 

 and were similar to the older Chinese ones, which, however, were 

 certainly made of bamboo, as the structure of the ideographs used reveals. 

 Various documents show their use ever since the Han Dynasty, and 

 there is some further evidence of their having been used still earlier. 

 It is certain that thty were evolved from the older use of knots or 

 straw in reckoning. These sticks are very important in history, because 

 they were a stimulating factor in the development of the Chinese as 

 well as the Japanese modes of arithmetic and algebra. 



The representalion of numbers by the use of sticks is recorded in 

 the Sun-Tse Suon-ching ^^,^^|M. ^l»e method described being the 

 same as that used by later Japanese mathematicians. As there is a 



