Kasner Edward - Mathematics and the Imagination

Mathematics and the Imagination

Edward Kasner

and

James Newman

With Drawings and Diagrams by

Rufus Isaacs

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright 1940 by Edward Kasner and James Newman

Copyright renewed 1967 by Ruth G. Newman

All rights reserved.

Bibliographical Note

This Dover edition, first published in 2001, is an unabridged and unaltered republication of Mathematics and the Imagination, originally published by Simon and Schuster, New York, in 1940.

Library of Congress Cataloging-in-Puhlication Data

Kasner, Edward, 1878-1955.

Mathematics and the imagination Edward Kasner and James Newman; with drawings and diagrams by Rufus Isaac.

p.cm.

Originally published: New York : Simon and Schuster, 1940.

Includes bibliographical references and index.

ISBN 0-486-41703-4 (pbk.)

1. Mathematics. 2. Mathematical recreations. I. Newman, James Roy, 1907-1966.

QA93 .K3 2001

510dc21

00-052297

Manufactured in the United States by Courier Corporation

41703404

www.doverpublications.com

To R.G.

without whose selfless help
and understanding there would
have been no book.

ACKNOWLEDGMENT

We are indebted to many books, too many to enumerate. Some of them are listed in the selected bibliography.

And we wish to acknowledge particularly the services of Mr. Don Mittleman of Columbia University, whose help in preparation of the manuscript has been generous and invaluable.

Table of Contents

Easy words for hard ideas...Transcendental...Non-simple curve...Simple curve...Simple group...Bolshe viks and giraffes...Turbines...Turns and slides...Circles and cycles...Patho-circles...Clocks...Hexagons and parhexagons...Radicals, hyperradicals, and ultrarad icals (nonpolitical)...New numbers for the nursery...Googol andgoogolplex...Miracle of the rising book...The mathescope.

Countingthe language of number...Counting, matching, and Going to Jerusalem...Cardinal numbers...Cosmic chess and googols...The sand reckoner...Mathematical induction...The infinite and its progeny...Zeno...Puzzles and quarrels...Bolzano...Galileos puzzle...Cantor...Measuring the measuring rod...The whole is no greater than some of its parts...The first transfinite Alepho...Arithmetic for morons...Common sense hits a snag...Cardinality of the continuum...Extravagances of a mathematical madman...The tortoise unmasked...Motionless motion...Private life of a number...The house that Cantor built.

Chinamen and chandeliers...Twilight of common sense..., i, e...Squaring the circle and its cousins...Mathematical impossibility...Silk purse, sows ear, ruler and compass...Rigor mortis...Algebraic equations and tran-scendental numbers...Galois and Greek epidemics...Cube duplicators and angle trisectors...Biography of...Infancy: Archimedes, the Bible, the Egyptians...Adolescence: Vieta, Van Ceulen...Maturity: Wallis, Newton, Leibniz...Old Age: Dase, Richter, Shanks...Victim of schizophrenia...Boon to insurance companies...(e)...Logarithms or tricks of the trade...Mr. Briggs is surprised...Mr. Napier explains...Biography of e; or e, the bank-ers boon...Pituitary gland of mathematics: the exponential function...(i)...Humpty Dumpty, Doctor of Semantics...Imaginary numbers...The , or Where am I?...Biography of i, the self-made amphibian...Omar Khayyam, Cardan, Bombelli, and Gauss...i and Soviet Russia...Program music of mathematics...Breakfast in bed; or, How to become a great mathematician...Analytic geometry...Geometric representation of i...Complex plane...A famous formula, faith, and humility.

The talking fish and St. Augustine...A new alphabet...High priests and mumbo jumbo...Pure and applied mathematics...Euclid and Texas...Mathematical tailors...Geometrya game...Ghosts, table-tipping, and the land of the dead...Fourth-dimension flounders...Henry More to the rescue...Fourth dimensiona new gusher...A cure for arthritis...Syntax suffers a setback...The physicists delight...Dimensions and manifolds...Distance formulae...Scaling blank walls...Four-dimensional geometry defined...Moles and tesseracts...A four-dimensional fancy...Romance of flatland...Three-dimensional cats and two-dimensional kings...Gallant Gulliver and the gloves...Beguiling voices and strange footprints...Non-Euclidean geometry...Space credos and millinery...Private and public space...Rewriting our textbooks...The prince and the Boethians...The flexible fifth...The mathematicians unitenothing to lose but their chains...Lobachevsky breaks a link...Riemann breaks another...Checks and double checks in mathematics...The tractrix and the pseudosphere...Great circles and bears...The skeptic persistsand isstepped on...Geodesies...Seventh Day Adventists...Curvature...Lobachevskian Eiffel Towers and Riemannian Holland Tunnels.

Puzzle acorns and mathematical oaks...Charlemagne and crossword puzzles...Mark Twain and the farmers daughter...The syntax of puzzles...Carolyn Flaubert and the cabin boy...A wolf a goat, and a head of cabbage...Brides and cuckolds...Ill be switched...Poisson, the misfit...High finance; or, The international beer wolf...Lions and poker players...The decimal system...Casting out nines...Buddha, God, and the binary scale...The march of culture; or, Russia, the home of the binary system...The Chinese rings...The tower of Hanoi...The ritual of Benares: or, Charley horse in the Orient...Nim, Sissa Ben Dahir, and Josephus...Bismarck plays the boss...The 15 puzzle plague...The spider and the fly...A nightmare of relatives...The magic square...Take a number from 1 to 10...Fermats last theorem...Mathematics lost legacy.

Great paradoxes and distant relatives...Three species of paradox...Paradoxes strange but true...Wheels that move faster on top than on bottom...The cycloid family...The curse of transportation; or, How locomotives cant make up their minds...Reformation of geometry...Ensuing troubles...Point setsthe Arabian Nights of mathematics...Hausdorff spins a tall tale...Messrs. Banach and Tarski rub the magic lamp...Baron Munchhausen is stymied by a pea...Mathematical fallacies...Trouble from a bubble; or, Dividing by zero...The infinitetroublemaker par excellence...Geometrical fallacies...Logical paradoxesthe folk tales of mathematics...Deluding dialectics of the poacher and the prince; of the introspective barber; of the number 111777; of this book and Confucius; of the Hon. Bertrand Russell...Scylla and Charybdis; or, What shall poor mathe-mathics do?

The clue of the billiard cue...A little chalk, a lot of talk...Watson gets his leg pulled by probable inference...Finds it all absurdly simple...Passionate oysters, waltzing ducks, and the syllogism...The twilight of probability...Interesting behavior of a modest coin...Biological necessity and a pair of dice...What is probability?...A poll of views: a meteorologist, a bootlegger, a bridge player...The subjective viewbased on insufficient reason, contains an element of truth...The jackasses on Mars...The statistical view...What happens will probably happen...Experimental euryth-mics; or, Pitching pennies...Relative frequencies...The adventure of the dancing men...Scheherezade and John Wilkes Bootha challenge to statistics...The red and the black...Charles Peirce predicts the weather...How far is away?...Herodotus explains...The calculus of chance...The benefits of gambling...De Mere and Pascal...Mr. Jevons omits an acknowledgment...The study of craps the very guide of life...Dice, pennies, permutations, and combinations...Measuring probabilities...DAlembert drops the ball...Count Buff on plays with a needle...The point...A black ball and a white ball...The binomial theorem...The calculus of probability re-examined...Found to rest on hypothesis...Laplace needs no hypothesis...Twits Napoleon, who does...The Marquis de Condorcet has high hopes...M. le Marquis omits a factor and loses his head...Fourier of the Old Guard...Dr. Darwin of the New...The syllogism scraps a standby...Mr. Socrates may not die...Ring out the old logic, ring in the new.