Few sangaku illustrate better the disparity of the independent development of mathematics during the Edo period (1603-1867) of self-imposed seclusion of Japan from the Western world than those related to arbelos and Pappus' chain of circles. The tool of inversion so useful in solving problems involving circles remained unkown to the Japanese until late 19 century.
Consider the sangaku 1.8.2 from the collection by Fukagawa and Pedoe:
Circles C1(r) and C2(r) touch at O in the circle O(2r) and also touch O(2r), C1(r) touching at T. O1(r1) touches O(2r) internally and both C1(r) and C2(r) externally, O2(r2) touches O1(r1) and C1(r) externally and O(2r) internally, and so on. Find rn.
In the very first volume (1826) of Crelle's Journal, J. Steiner proved several theorem concerning arbelos, the above being one of them. The sangaku however is dated 1801 (and was written in the Mie prefecture.)