Mark Bollman reviews New Horizons in Geometry

Mark Bollman reviews New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian as part of MAA Reviews. 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. Read the full review here. Purchase a copy today at the MAA Store. Mark Bollman (mbollman@albion.edu) 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.