A number of geometric constructions, although easy to comprehend and fun to do, are nevertheless impossible to complete with just a ruler and a compass. This book discusses the most famous of these "impossible" constructions. Part I, in exploring ground rules, history and angle trisection, considers--among other subjects--angle trisection and bird migration, constructed points, analytic geometry, algebraic classification of constructible numbers, fields of real numbers, cubic equations, and marked ruler, quadratrix, and hyperbola. Part II treats nonconstructible regular polygons and the algebra associated with them, specifically, irreducibility and factorization, unique factorization of quadratic integers, finite dimensional vector spaces, algebraic fields, and nonconstructible regular polygons. This stimulating and provocative book provides a fascinating glimpse of the crucial role geometry can play in a wide range of mathematical applications. Unabridged republication of Ruler and the Round: or Angle Trisection and Circle Division, originally published by Prindle, Weber & Schmidt, Inc., Boston, 1970. 2 Appendices. References. Index. Numerous illustrations.