Mistakes in Geometric Proofs

73 Pages·1963·0.899 MB·English

by  Ya. S. Dubnov

T O P I C S I N M AT H E M AT I C S
Ya. S. Dubnov
Mistakes
in
Geometric Proofs

T O P I C S I N M A T 11 E M A T I C S
Mistakes in
Geometric Proofs
Ya. S. Dubnov
Translated and adapted from the second Russian edition (1955) by
ALFRED K. HENN and OLGA A. TITELBAUM
SURVEY OF
RECENT EAST EUROPEAN MATHEMATICAL LITERATURE
A project conducted by
ALFRED L. PUTNAM and IZAAK WIRSZUP
Department of Mathematics.
The University of Chicago, under a
grant from the S’ational Science Toundation
D. C. HEATH A N D C O M P A N Y BOSTON
Library of Congress Catalog Card Number: 61-17840
Copyright © 1963 by The University of Chicago
No part of the material covered bv this copyright may be repro­
duced in any form without written permission of the publisher.
Printed in the United States of America
(6 L 2)
PREFACE TO THE AMERICAN EDITION
This booklet presents examples of faulty geometric proofs, some of
which illustrate mistakes in reasoning that a student might make,
while others are classic sophisms. Chapters 1 and 3 present these
faulty proofs, and then Chapters 2 and 4 give detailed analyses of
the mistakes.
Naturally, in order to read this booklet, the reader must be
acquainted with plane geometry. Only an acquaintance with theorems
concerning parallel and perpendicular lines and polygons is needed
for Chapters 1 and 2. Chapters 3 and 4 contain more advanced ma­
terial and presuppose some knowledge of the simpler properties of
circles, the concept of limit, trigonometry, and some solid geometry.
It is suggested that the reader first examine the examples of
incorrect proofs given in Chapters 1 and 3. He should attempt to dis­
cover the mistakes in these examples by himself before reading
Chapters 2 and 4. Portions of the text appearing in fine print, as well
as many of the footnotes, may be omitted on first reading; these are
intended primarily for the more advanced reader.

CONTENTS
Introduction 1
Mistakes in Reasoning within the Grasp
CHAPTER 1.
of the Beginner
5
Analysis of the Examples Given in
CHAPTER 2.
Chapter 1
19
Mistakes in Reasoning Connected with
CHAPTER 3.
the Concept of Limit
34
chapter 4. Analysis of the Examples Given in
Chapter 3 50