CONTEMPORARY UNDERGRADUATE MATHEMATICS SERIES Roberl J. Wisner. Editor ! MATHEMATICS FOR THE LIBERAL ARTS STUDENT, SECON...
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CONTEMPORARY UNDERGRADUATE MATHEMATICS SERIES Roberl J. Wisner. Editor ! MATHEMATICS FOR THE LIBERAL ARTS STUDENT, SECOND EDITION. Fred' Richman, Carol Walker. and Robert J. Wisner INTERMEDIATE ALGEBRA. Edward D. Gaughan ALGEBRA: A PRECALCULUS COURSE. James E. Hall TRIGONOMETRY: CIRCULAR FUNCTIONS AND THEIR APPLICATIONS.
James E. Hall
,CE
MODERN GEOMETRIES
MODERN MATHEMATICS: AN ELEMENTARY APPROACH. THIRD EDITION. Ruric E. Wheeler A PROGRAMMED STUDY OF NUMBER SYSTEMS, Ruric E. Wheeler and Ed A. Wheeler FUNDAMENTAL COLLEGE MATHEMATICS: NUMBER SYSTEMS AND INTUITIVE GEOMETRY, Ruric E, Wheeler MODERN MATHEMATICS FOR BUSINESS STUDENTS, Ruric E. Wheeler and W. O. Peepies ANALYTIC GEOMETRY. James E. Hall INTRODUCTORY GEOMi:TRY: AN INFORMAL APPROACH. SECOND EDITION, Jame~ R. Smart MODERN GEOMETRIES; James R. Smart
AN INTUITIVE APPROACH TO ELEMENTARY GEOMETRY. Beauregard Stubblefield GEOMETRY FOR TEACHERS. Paul B. Johnson and Carol H. Kipps
JAMES R. SMART California State University, San Jose
LINEAR ALGEBRA. James E Scroggs
5,
ESSENTIALS OF ABSTRACT ALGEBRA, Charles M. Bundrick and John J. Leeson AN INTRODUCTION TO ABSTRACT ALGEBRA. A. Richard Mitchell and Roger W. MitChel!
6,
INTRODUCTION TO ANALYSIS. Edward O. Gaughan
.\11
DIFFERENTIAL EOUATIONS AND RELATED TOPICS FOR SCIENCE AND ENGINEERING. Robert W. Hunt
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A PRIMER OF COMPLEX VARIABLES WITH AN INTRODUCTION TO ADVANCED TECHNIQUES, Hugh J. Hamilton
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CALCULUS OF SEVERAL VARIABLES. E, K. McLachlan
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PROBABILITY. Donald R. Barr and Peter W. Zehna
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THEORY AND EXAMPLES OF POINT·SET TOPOLOGY. John Greever
AN INTRODUCTION TO ALGEBRAIC TOPOLOGY. John W. Keesee EXPLORATIONS IN NUMBER THEORY. Jeanne Agnew NUMBER THEORY: AN INTRODUCTION TO ALGEBRA. Fred Richman
BROOKS/COLE PUBLISHING COMPANY Monterey, California A Division of Wadsworth Publishing Company: Inc.
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MODERN GEO METRIES
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PREFACE
This book was edited by Phyllis London and designed by Linda Marcetti. Tlte technical art was drawn by Jolm Foster. Tlte book was printed alfd bOlf.!,d by Kingsport Press, Kingsport. Tennessf!.l?:
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1973 by Wadsworth Publishing Company, Inc., Belmont, CaUfomia 94002. All rights reserved. No part of 'this book may be reproduced, stored in a retrieval system. or transcribed, in any foml or by any means~
In recent years, the traditional course in college geometry often has been dropped, only to be replaced by other courses no more satisfactory. In some cases, the mistake has been made of throwing out all of Euclidean geometry, both traditional and modern, whether it is of continuing significance or not. The even worse mistake is sometillles made of assuming that students understand geometry simply because
electronic, mechanical, photocopying, recording, o'r otherwise-without the prior written permission of the publisher: Brooks/Cole Publishing Company, Monterey, California. a division of Wadsworth Publishing C-Ompany, Inc. ISBN: 0-81 85-005J-4 L.C. Catalog Card No: 72-79015 Printed in the United States of America
1 2 3 4 5 6 7 8 9 10-77 76 75 74 73 '
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1. 2.
A straight
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ny point to any point.
lin~ c~n ,be draw~ r~;o~uced continuously. in a straight
A .finite stralght hue can
e
line. ' . with any point as center and any A circle may be described distance as radius. 4. All right angles are equal to one anl?ther.. such a way that the . 1 falls on two mes m 5. If a transversa . f the transversal are less than two interior angles on one. SIde 0 th t side on which the angles right angles. then the hnes meet on a are less than two right angles. '. nee that a segment can be extended postulate 2 means tn esse I. f Euclid (the parallel The fiifith postu ate 0 · indefinitely to form a Ime. . er 9 in the discussion of postulate) will be considered further tn Ch~Pt
3.
non-Euclidean geometry. . particular I mathematicians tu Within the past hundred ye~rs h i ointed 'but various flaws · s of mathematics ave p studying the foun d a t ton 'd II used other tacit, assump. f Euclid Euch actua y '. d in the assmnpttons 0 . r 'tt ) Logical problems pomte out tions {assumptions nbt stated exp lCI y, . have included: fi. t t ment about :the continuity of a. T he need for a de 1l1te 5 a e lines and circles. h . fini~e extent of a straight b. The need for a statement about t e 10 line.
c. The need to state the fact that if a straight line enters a triangle at a vertex, it must intersect the opposite side. d. The need for statements about the order of points on a line. e. The need for a statement about the concept of betweenness. f. The need for a statement guaranteeing the uniqueness of a line joining two distinct points. g. The need for a more logical approach. such as that of transformations (Chapter 21 which does not depend on the concept of superposition. Euclid assumed that a triangle can be picked up and put down in another place with all the properties 'remaining invariant, yet no statement to this effect was made. h. The need for a list of undefined temis. Many modern sets of axioms for Euclidean geometry have been introduced to r~~