A Study of Japanese Mathematics from the View of Lattice Multiplication in the East and the West during the Edo Period (Study of the History of Mathematics)

"Study of the History of Mathematics". August 27～30, 2012. edited by Tsukane Ogawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.The Lattice Multiplication system was invented in Western Asia, by the 15^{mathrm{t}mathrm{h}} century. Then it was introduced into Western Europe in 1478. Then Napier (1550-1617) invented Napier' s Rods in 1617. Chinese mathematicians studied Napier 's Rods during the Qing dynasty. Japanese medical doctors and military scientists during the Edo period introduced Napier s Rods - the SancaiFami(Chen Wen, 1697) in 1720. But Japanese mathematicians during the Edo period already studied the Lattice Multiplication system of ' Xie Suan' (or ' Pudijin' ) by Chinese mathematical arts during the Ming dynasty, such as the Suanfa Tongzong (Cheng Dawei, 1592), before Western mathematics arrived in China. Japanese medical doctors had never studied abacus, thus they studied Napier' s Rods. In other words, they studied the Lattice Multiplication system of the West. On the other hand, Japanese mathematicians such as Seki Takakazu (1642/5?-1708) studied the Lattice Multiplication system from Chinese traditional mathematical arts. Seki described it in the Taisei Sankei(Seki Takakazu and et al, 1710) as ' Pudijin' . In other words, they studied the Lattice Multiplication system of the East. Japanese mathematicians and medical doctors did not have academic interchange with the West