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New Horizons in Geometry

Tom M. Apostol and Mamikon A. Mnatsakanian
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 47
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Bollman
, on

Anyone who has read Underwood Dudley’s Mathematical Cranks could be forgiven a bit of skepticism in the face of a book that promises to offer elementary geometric methods for solving classical calculus problems. New Horizons in Geometry is about as far from crank mathematics as possible. The book begins with “Mamikon’s sweeping tangent theorem”, a result first conceived in 1959, and proceeds to derive numerous formulas for arclength, area, and volume that might ordinarily be deduced with calculus. Along the way, readers will be introduced or reintroduced to such figures as the cyclogon, the autogon, and the bifocal disk. By the end, recursive formulas for volume in n-dimensional space are easily handled, with no integrals required.

This volume serves as the “collected works” of a fascinating mathematical collaboration between the two authors. This is mathematics of the highest caliber; but what makes this book even more impressive is the attention paid to high-quality full-color graphics that ably illustrate the problems under consideration. The advent of four-color printing in calculus has been at times both a blessing and a curse; since most of the pictures in this book are two-dimensional, most of the downside of full color illustrations is avoided. It is rare to see a mathematics book that takes such advantage of relatively simple illustrations.

The authors have put together an impressive body of work that is a solid addition to any library’s geometry holdings. For those of us who want a corner of mathematics that will continue to use calculus, there will always be the challenge of measuring fluid pressure at the base of a dam.

Mark Bollman ( is associate professor of mathematics at Albion College in Michigan. His mathematical interests include number theory, probability, and geometry. Mark’s claim to be the only Project NExT fellow (Forest dot, 2002) who has taught both English composition and organic chemistry to college students has not, to his knowledge, been successfully contradicted. If it ever is, he is sure that his experience teaching introductory geology will break the deadlock.

1. Mamikon’s Sweeping Tangent Theorem
2. Cycloids and Trochoids
3. Cyclogons and Trochogons
4. Circumgons and Circumsolids
5. The Method of Punctured Containers
6. Unwrapping Curves from Cylinders and Cones
7. New Descriptions of Conics via Twisted Cylinders, Focal Disks, and Directors
8. Ellipse to Hyperbola: “With This String I Thee Wed”
9. Trammels
10. Isoperimetric and Isoparametric Problems
11. Arclength and Tanvolutes
12. Centroids
13. New Balancing Principles with Applications
14. Sums of Squares
15. Appendix
About the Authors