Variations on a Theme of Euler: Quadratic Forms, Elliptic Curves, and Hopf MapsThe first six chapters and Appendix 1 of this book appeared in Japanese in a book of the same title 15years aga (Jikkyo, Tokyo, 1980). At the request of some people who do not wish to learn Japanese, I decided to rewrite my old work in English. This time, I added a chapter on the arithmetic of quadratic maps (Chapter 7) and Appendix 2, A Short Survey of Subsequent Research on Congruent Numbers, by M. Kida. Some 20 years ago, while rifling through the