Symmetry: Culture and Science Vol. 26, No. x, page_first-page_last, 2016 Did Japanese Geometers Understand Inversion? J. Marshall Unger Emeritus Professor of Japanese, (b. Cleveland, O., U.S.A., 1947). Address: Department of East Asian Languages and Literatures, The Ohio State University, 1775 College Road, Columbus, O. 43210, U.S.A. E-mail: unger.26@osu.edu. Fields of interest: history of the Japanese language; writing systems and the computerization of East Asian scripts; traditional Japanese mathematics. Awards: Japan Foundation Fellowships (1985, 2005); Guggenheim Fellowship, 2004–06. Publications: Sangaku Proofs: A Japanese Mathematician at Work. Ithaca, N.Y.: Cornell East Asian Series. (2015). Abstract: Most scholars who have written about 18th- and early 19th-century Japanese geometry problems doubt that the Japanese used the technique of inversion in a circle despite the fact that many of their problems seem to cry out for solutions by inversion. I discuss a text containing a note that implies at least an intuitive understanding of inversion. Keywords: sangaku, inversion, geometry. 1. INTRODUCTION The inversion of a plane figure in a well chosen circle often exhibits greater symmetry than the original; it is therefore often easier to find answers to questions about elements of the image than about the corresponding elements of the pre-image. Inversion in a circle thus provides a powerful method for solving certain kinds of difficult problems in plane geometry. Scholars who have written about sangaku problems 1 often express 1 Wasan 和算 is the proper name for the kind of mathematics developed in Japan during its centuries of national isolation; practitioners of wasan were called wasanka 和算家. Sangaku 算額 were plaques that wasanka placed in temples and shrines displaying problems in geometry they had composed or solved. Outside Japan, sangaku is often used as a synonym for wasan.