Includes bibliographical references (pages 501-504) and index.
Ch. 1. Mamikon's sweeping
Ch. 2. Cycloids and trochoids
Ch. 3. Cyclogons and trochogons
Ch. 4. Circumgons and circumsolids
Ch. 5. The method of punctured containers
Ch. 6. Unwrapping curves from cylinders and cones
Ch. 7. New descriptions of conics via twisted cylinders, focal disks, and directors
Ch. 8. Ellipse to hyperbola: with this string I thee wed
Ch. 9. Trammels
Ch. 10. Isoperimetric and isoparametric problems
Ch. 11. Arclength and tanvolutes
Ch. 12. Centroids
Ch. 13. New balancing principles with applications
Ch. 14. Sums of squares.
The collaborative work of Tom Apostol and Mamikon Mnatsakanian has been lauded for its clarity and originality. In this volume the authors present an impressive collection of geometric results that reveal surprising connections between lengths, areas and volumes in various shapes, and allow one to compute difficult integrals, all using intuitive visual calculations. One noteworthy idea that the reader will encounter is Mamikon's Sweeping Tangent Theorem from which the authors obtain a visual derivation of the property that the length of an arc of a catenary is proportional to the area under the arc. This is one of many 'proofs without words' contained within. In addition, a variety of results are derived visually for cycloids, conic sections, and many more geometric objects. As befits a book that emphasises visual thinking, the text is beautifully illustrated. This is a book that will inspire students and enrich any geometry or calculus course.