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AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 30 30 Hesiod’s Anvil Falling and Spinning through Heaven and Earth Andrew J. Simoson Hesiod's Anvil This book is about how poets, philosophers, storytellers, and scientists have described motion, begin- ning with Hesiod, a contemporary of Homer, who imagined that the expanse of heaven and the depth of hell was the distance that an anvil falls in nine days. This book is aimed at students who have fi nished a year-long course in calculus, but it can be used as a supplemental text in calculus II, vector calculus, linear algebra, differential equations, and modeling. It blends with equal voice romantic whimsy and derived equations, and anyone interested in mathematics will fi nd new and surprising ideas about motion and the people who thought about it. Some of the things readers will learn is that Dante's implicit model of the earth implies a black hole at its core, that Edmond Halley championed a hollow earth, and that da Vinci knew that the acceleration due to the earth's gravity was a constant. There are chapters modeling Jules Verne's and H. G. Wells' imaginative fl ights to the moon and back, the former novelist using a great cannon and the latter using a gravity-shielding material. The book Andrew J. Simoson analyzes Edgar Alan Poe's descending pendulum, H. G. Wells' submersible falling and rising in the Marianas Trench, a train rolling along a tunnel through a rotating earth, and a pebble falling down a hole without resistance. It compares trajectories of balls thrown on the Little Prince's asteroid and on Arthur C. Clarke's rotating space station, and it solves an old problem that was perhaps inspired by one of the seven wonders of the ancient world. The penultimate chapter is a story, based upon the Mayans, that loosely ties together the ideas about falling and spinning motion discussed in the book. Nearly all the chapters have exercises, some straight- forward and some open ended, that may serve as the beginnings of students' honors projects. AMSMAA / PRESS Andrew J. Simoson 4-Color Process 360 page • 11/16” • 50lb stock • fi nish size: 6” x 9” i i “master” — 2007/4/2 — 23:54 — page i — #1 i i 10.1090/dol/030 Hesiod’s Anvil falling and spinning through heaven and earth i i i i i i “master” — 2007/4/2 — 23:54 — page ii — #2 i i Illustration from Le Petit Prince by Antoine de Saint-Exupery,´ copyright 1943 by Harcourt, Inc. and renewed 1971 by Consuelo de Saint-Exupery, reprinted by permission of the publisher. Photo from the William Beebe/National Geographic Image Collection, 1931, reprinted by permission of the publisher. c 2007 by The Mathematical Association of America (Incorporated) Library of Congress Catalog Card Number 2006933945 ISBN 978-0-88385-336-8 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “master” — 2007/4/2 — 23:54 — page iii — #3 i i The Dolciani Mathematical Expositions NUMBER THIRTY Hesiod’s Anvil falling and spinning through heaven and earth Andrew J. Simoson King College ® Published and Distributed by The Mathematical Association of America i i i i i i “master” — 2007/4/2 — 23:54 — page iv — #4 i i DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Publications James Daniel, Chair Dolciani Mathematical Expositions Editorial Board Underwood Dudley, Editor Benjamin G. Klein Virginia E. Knight Mark A. Peterson Robert L. Devaney Tevian Dray Jerrold W. Grossman James S. Tanton i i i i i i “master” — 2007/4/2 — 23:54 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical As- sociation of America was established through a generous gift to the Association from Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University of New York. In making the gift, Professor Dolciani, herself an exceptionally talented and successful expositor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, was delighted to accept the gracious gesture initiating the revolving fund for this series from one who has served the Association with distinction, both as a member of the Committee on Publications and as a member of the Board of Governors. It was with genuine pleasure that the Board chose to name the series in her honor. The books in the series are selected for their lucid expository style and stimulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high-school student to under- stand and enjoy, but also to be interesting and sometimes challenging to the more advanced mathematician. 1. Mathematical Gems, Ross Honsberger 2. Mathematical Gems II, Ross Honsberger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems III, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay of the Continuous and the Discrete, Robert M. Young 14. The Wohascum County Problem Book, George T. Gilbert, Mark Krusemeyer, and Loren C. Larson 15. Lion Hunting and Other Mathematical Pursuits: A Collection of Mathematics, Verse, and Stories by Ralph P.Boas, Jr., edited by Gerald L. Alexanderson and Dale H. Mugler 16. Linear Algebra Problem Book, Paul R. Halmos 17. From Erd˝os to Kiev: Problems of Olympiad Caliber, Ross Honsberger i i i i i i “master” — 2007/4/2 — 23:54 — page vi — #6 i i 18. Which Way Did the Bicycle Go? . and Other Intriguing Mathematical Mys- teries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 19. In Polya’s´ Footsteps: Miscellaneous Problems and Essays, Ross Honsberger 20. Diophantus and Diophantine Equations, I. G. Bashmakova (Updated by Joseph Silverman and translated by Abe Shenitzer) 21. Logic as Algebra, Paul Halmos and Steven Givant 22. Euler: The Master of Us All, William Dunham 23. The Beginnings and Evolution of Algebra, I. G. Bashmakova and G. S. Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger 25. Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, Jack E. Graver 26. Mathematical Diamonds, Ross Honsberger 27. Proofs that Really Count: The Art of Combinatorial Proof, Arthur T. Benjamin and Jennifer J. Quinn 28. Mathematical Delights, Ross Honsberger 29. Conics, Keith Kendig 30. Hesiod’s Anvil: falling and spinning through heaven and earth, Andrew J. Simoson MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 i i i i i i “master” — 2007/4/2 — 23:54 — page vii — #7 i i Contents Introduction ......................................................xi preamble i ......................................................... 1 I Hesiod’s Muses...................................................3 preamble ii ....................................................... 15 II The Gravity of Hades...........................................17 preamble iii .......................................................39 III Ballistics ......................................................41 preamble iv .......................................................53 IV Heavenly Motion ..............................................55 preamble v ........................................................67 V Pendulum Variations ...........................................69 preamble vi .......................................................83 VI Retrieving H. G. Wells from the Ocean Floor....................85 preamble vii ...................................................... 97 VII Sliding along a Chord through a Rotating Earth ...............99 preamble viii .....................................................109 VIII Falling through a Rotating Earth ............................111 preamble ix ......................................................139 IX Shadow Lands ...............................................141 preamble x ...................................................... 161 X The Trochoid Family ..........................................163 preamble xi ......................................................185 XI Retrieving H. G. Wells from the Moon .........................187 preamble xii ..................................................... 205 XII Playing Ball in a Space Station ...............................207 vii i i i i i i “master” — 2007/4/2 — 23:54 — page viii — #8 i i viii Hesiod’s Anvil preamble xiii .....................................................227 XIII The Rotating Beacon .......................................229 preamble xiv .....................................................257 XIV The Long Count ............................................259 preamble xv ..................................................... 275 XV Hesiod’s Anvil ...............................................277 Appendix ........................................................289 Cast of Characters ...............................................299 Comments on Selected Exercises..................................305
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