The Mysterious Enri The circle principle is a perfect method, never before known in ancient or in modern times. It is a method that is eternal and unchangeable . . . It is the true method . . . —Hachiya Sadaaki,1 as quoted by Smith and Mikami
We devote the final two chapters of Sacred Mathematics to matters of technique. Chapter 9 is dedicated to providing an answer, insofar as there is one, to a question that has certainly occurred to readers who have attempted certain of the more difficult problems: Given that wasan did not include a fully developed theory of calculus, how did traditional Japanese geometers solve the maxima-minima problems, which require differentiation, or solve problems requiring integration? In the solutions to chapters 5 and 6 we have given an idea of how the Japanese approached integration, although we have said nothing about differentiation. Here we tackle both matters explicitly.
Differentiation The question of differentiation in some sense is more difficult than that of integration because, although traditional Japanese mathematicians wrote volumes on integration techniques, they were virtually silent about how they took derivatives. One thing is certain: the Japanese did not have the 1
Sometimes Hachiya Teisho.