Between 1600 and 1868, when the nation was largely isolated from outside influences, Japan developed a highly elaborate indigenous mathematics known as wasan. This book is an unaltered reprint of Smith and Mikami's classic account of wasan, first published in 1914. We owe Dover yet another vote of thanks for bringing an important book back into print at a very reasonable price.
The book includes a history and discussion of arithmetic algorithms for the soroban, the abacus which served Japanese society as a powerful calculator, and a number of impressive magic squares and magic circles (think of magic squares in polar coordinates).
However, the focus is on the two great achievements of tenzan algebra and the yenri ("circle principle"), both of which grew out of the work of Seki Kowa and his student Takebe Kenko in the second half of the 17th century. Japan inherited from Chinese algebraists of the 13th century some polynomial techniques and the method for approximating roots of polynomial equations known in the west as "Horner's method." Seki and Takebe adapted these into the powerful tenzan technique for setting up and solving polynomial equations arising from geometric problems. As a famous example, Seki solved the problem of finding the lengths of the sides and diagonals of a quadrilateral given the differences of the cubes of those six lengths, by setting up and solving a polynomial equation of degree 1458.
The yenri was an ingenious kind of proto-integral-calculus, which originally gave infinite series for, for example, the length of the arc of a unit circle cut off by a segment of height h. Later mathematicians like Ajima Chokuyen in the 18th century and Wada Nei in the early 19th century used it to solve problems like finding the arclength of an ellipse or the volume of various intersections of spheres, cylinders and cones.
Smith and Mikami assess wasan as follows: "The work was exquisite in a way wholly unknown in the West. For patience, for the everlasting taking of pains, for ingenuity in untangling minute knots and thousands of them, the problem-solving of the Japanese and the working out of some of the series in the yenri have never been equaled."
Phil Straffin (firstname.lastname@example.org) is Thomas White Professor of Mathematics at Beloit College, where he regularly teaches a course on mathematics in other cultures. His survey of Chinese mathematics from the third to the fifth century C.E. appeared as "Liu Hui and the First Golden Age of Chinese Mathematics," in Mathematics Magazine (volume 71 (1998), pages 163-181).