More than ?fteen years have passed since this book was published in French. Ideally this work, which originated from a doctoral thesis defended in 1990 at the University of Paris Diderot, should have been partially rewritten for the English edition, thereby incorporating the latest research results in its argumentation. But this was not the choice taken, since it would have delayed making this work, which brings together useful information for all those who are interested in wasan and eager to delve more deeply into its tradition, available to the English reader. Indeed, the special feature of this study is to present a high-level mathematical production while at the same time placing it in its historical context. This context here is Tokugawa Japan and, more speci?cally, the milieu of the scholars of samurai (warrior) origin. While explaining the indispensable social, economical and poli- cal background, we have not neglected mathematical analysis, since wasan reached its most beautiful results with the two mathematicians Seki Takakazu and Takebe Katahiro. Their works abound in lessons, not only for all those who are interested in the history of algebra and of analysis, but also in epistemological questions.