This book brings the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts. The first chapter talks about the theory of trigonometric and elliptic functions. It includes subjects such as power series expansions, addition and multiple-angle formulas, and arithmetic-geometric means. The second chapter discusses various aspects of the Poncelet Closure Theorem. This discussion illustrates to the reader the idea of algebraic geometry as a method of studying geometric properties of figures using algebra as a tool. This is the second volume originating from a series of lectures given by the authors at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and for undergraduate mathematics courses in the sciences and liberal arts. The first volume is available as Volume 19 in the AMS series, ""Mathematical World"".