Non-Euclidean Constructions
Always write in your books. You may be a silly person—for though your reading my book is rather a contrary presumption, yet it is not conclusive—and your observations may be silly or irrevelant, but you cannot tell what use they may be of long after you a
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Underwood Dudley
A Budget of Trisections With 132 Illustrations
Springer Science+ Business Media, LLC
Underwood Dudley Department of Mathematics and Computer Science DePauw University Greencastle, IN 46135-0037 USA
AMS Classifications: Al-A05 Library of Congress Cataloging-in-Publication Data Dudley, Underwood. A budget of trisections. 1. Trisection of an angle. 1. Title. QA468.D83 1987 516.2'04 87-13020
ISBN 978-1-4612-6430-9 ISBN 978-1-4419-8538-5 (eBook) DOI 10.1007/978-1-4419-8538-5
© 1987 by Springer Science+Business Media New York Originally published Springer-Verlag New York Berlin Heidelberg in 1987 Softcover reprint of the hardcover lst edition 1987 AII rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+ Business Media, LLC) except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form ofinformation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc. in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Typeset by Asco Trade Typesetting Ltd., Hong Kong.
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Contents
Introduction 1. Non-Euclidean Constructions
vii 1
2. Characteristics of Trisectors
18
3. Three Trisectors
32
4. A Budget of Trisections
55
Index
165
Introduction
My opinion of mankind is founded upon the mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in. (Augustus De Morgan, A Budget of Paradoxes, Volume 1, page 115.)
In 1955, the mathematics library at the Carnegie Institute of Technology was in a room at the end of a long, long hall. For much of its length , the hall sloped slightly downward, either to follow the slope of the land underneath or to show what engineers and architects could do if they set their minds to it. The library was in a former classroom; it had three rows of shelves and four tables. Borrowers of books were on their honor to write on a card the name of a book taken out and the date it was taken. They were also on their honor to return it within two weeks. Now the Carnegie Institute of Technology is the Carnegie-Mellon University, and its mathematics library no doubt occupies much more space and has many more books. I suspect that the honor system is no longer in use. Then, more than 30 years ago, the library was a pleasant place to be in the later afternoon, with classes over for the day and the setting sun coming in the two west-facing windows, lighting up the particles of dust in the air. Then as now, students of mathematics (I was a student of mathematics then) generally did not read anything that they were not required to r
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