MATHEMATICS LV CHINA AND JAPAN. 187 



this doubt will be dispelled, when we trace the origin of the Japanese 

 notations and operations from the Chinese abacus algebra of the tengen- 

 jutsti, which the Japanese had remodelled in the form of a written system 

 previously unknown in China. 



In the algebra of the tengen-jutsu, problems were solved by forming 

 two expressions of one and the same quantity in terms of the quantity 

 to be found, the tengen, or ''heavenly elemeut," literally, and cancelling 

 the two so as to form an equation. But in a problem somewhat 

 complicated, it was not easy to find two such expressions to cancel one 

 another. On this account, a parameter was adopted besides the quantity 

 in question, and two equations were formed containing these two 

 quantities, the elimination of the parameter leading to the final equation. 

 This procedure resembled in a way the Chinese algebra of the four 

 elements ; but there lay an essential distinction between the two, in that 

 the latter was carried out by the arrangements of calculating sticks, while 

 the Jai^anese endan-jutsu, or "method of analysis" so called, did not 

 resort to sticks but was carried out wholly in writing. There is no 

 record of the four elements algebra of the Chinese ever having been 

 noticed or studied in old Japan. The algebraical operations of the 

 Japanese being thus executed in writing, certain notations were necessary, 

 for which ideographs were found suitable. This is very notable in the 

 history of Japanese mathematics, because nothing of the kind had deve- 

 loped in China. 



11. The elimination processes of the endan-jutsu were managed in 

 various ways, of which the use of determinants was the most notable. 

 It was in fceki's work of 1683 that the expansion method of determinants 

 Avas treated for the first time, while certain other mathematicians 

 contemporary with him also gave thuir theories on the subject. Seki had 

 committed two grave mistakes in his explanation, which were corrected 

 by his successors. During the century or more after Seki, several extended 

 ways of expansion were attempted. After the end of the 18th century, 

 however, no further studies of any consequence appeared on the subject. 

 Tsothing of the sort is known in China, although the Japanese theory 

 seems to be a development from the old Chinese treatment of linear simul- 

 taneous equations. 



12. Various contrivances for the solution of equations were devised 

 in old Japan. The old Chinese way of evaluating the root digit after 

 digit was abridged by Seki by the use of division. Repeated applications 

 of the old Chinese way of surplus and efficiency were also attempted 



