East and West For pairs of lips to kiss maybe Involves no trigonometry. ’Tis not so when four circles kiss Each one the other three. To bring this off the four must be As three in one or one in three. —Frederick Soddy, from “The Kiss Precise”
If there is a single lesson to be drawn from the preceding chapters, it is that creativity respects no geographical or cultural boundaries, in science or art. We are taught in Western schools that mathematics flowed from ancient Greece to Western Europe, and with it so too did the great stream of discovery. One cannot and should not deny that the majority of important “classical” mathematical results have come to us through Greece and Europe but, as far as the subject of this book goes, a few famous theorems attributed to Western authors were in fact posted on sangaku prior to their occidental discovery, and a few others, which had been previously found in the West, were independently discovered by traditional Japanese geometers. We have already employed many of the ancient theorems common to East and West to solve problems throughout Sacred Mathematics. Apart from the omnipresent Pythagorean theorem, which reached Japan through China, there was Ptolemy’s theorem, “The sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals,” which was required to solve problem 1, chapter 6. It is found, for example, in the 1769 Syuki Sanpo¯ , or Mathematics, by Arima Yoriyuki (chapter 3).