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i
MatheMatics
a Practical Odyssey
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MatheMatics
a Practical Odyssey
8e
David B. Johnson
Diablo Valley College
Pleasant Hill, California
Thomas A. Mowry
Diablo Valley College
Pleasant Hill, California
Australia l
Brazil l
Japan l
Korea l
Mexico l
Singapore l
Spain l
United Kingdom l
United States
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Mathematics: A Practical Odyssey, © 2016, 2012, Cengage Learning
Eighth Edition
WCN: 02-200-203
David Johnson, Thomas Mowry
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Contents
Overview ix
chapter
1 Logic 1
1.1 Deductive versus Inductive Reasoning 2
1.2 Symbolic Logic 18
1.3 Truth Tables 29
1.4 More on Conditionals 41
1.5 Analyzing Arguments 49
1.6 Deductive Proof of Validity 61
chapter
2 Sets and counting 73
2.1 Sets and Set Operations 74
2.2 Applications of Venn Diagrams 86
2.3 Introduction to Combinatorics 99
2.4 Permutations and Combinations 106
2.5 Infinite Sets 122
chapter
3 probability 137
3.1 History of Probability 138
3.2 Basic Terms of Probability 145
3.3 Basic Rules of Probability 164
3.4 Combinatorics and Probability 177
3.5 Expected Value 190
3.6 Conditional Probability 201
3.7 Independence, Medical Tests, and Genetics 216
chapter
4 Statistics 233
4.1 Population, Sample, and Data 234
4.2 Measures of Central Tendency 258
4.3 Measures of Dispersion 271
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vi CONTENTS
chApTer
5 Finance 341
5.1 Simple Interest 342
5.2 Compound Interest 353
5.3 Annuities 367
5.4 Amortized Loans 379
5.5 Annual Percentage Rate with a TI’s TVM Application 400
5.6 Payout Annuities 408
chApTer
6 Voting and Apportionment 421
6.1 Voting Systems 422
6.2 Methods of Apportionment 442
6.3 Flaws of Apportionment 470
chApTer
7 Number Systems and
Number Theory 485
7.1 Place Systems 486
7.2 Addition and Subtraction in Different Bases 501
7.3 Multiplication and Division in Different Bases 506
7.4 Prime Numbers and Perfect Numbers 511
7.5 Fibonacci Numbers and the Golden Ratio 523
chApTer
8 Geometry 537
8.1 Perimeter and Area 538
8.2 Volume and Surface Area 556
8.3 Egyptian Geometry 568
8.4 The Greeks 578
8.5 Right Triangle Trigonometry 590
8.6 Linear Perspective 606
8.7 Conic Sections and Analytic Geometry 615
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CONTENTS vii
chapter
9 Graph theory 669
9.1 A Walk through Königsberg 670
9.2 Graphs and Euler Trails 676
9.3 Hamilton Circuits 688
9.4 Networks 701
9.5 Scheduling 716
chapter
10 exponential and Logarithmic
Functions 735
10.0A Review of Exponentials and Logarithms 736
10.0B Review of Properties of Logarithms 746
10.1 Exponential Growth 759
10.2 Exponential Decay 776
10.3 Logarithmic Scales 793
chapter
11 Markov chains 811
11.0A Review of Matrices 812
11.0B Review of Systems of Linear Equations 824
11.1 Markov Chains and Tree Diagrams 831
11.2 Markov Chains and Matrices 835
11.3 Long-Range Predictions with Markov Chains 843
11.4 Solving Larger Systems of Equations 848
11.5 More on Markov Chains 853
chapter
12 Linear programming 861
12.0 Review of Linear Inequalities 862
12.1 The Geometry of Linear Programming 874
12.2 Introduction to the Simplex Method 12-2
12.3 The Simplex Method: Complete Problems 12-8
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viii CONTENTS
chApTer
13 The concepts and history
of calculus 13-1
13.0 Review of Ratios, Parabolas, and Functions 13-2
13.1 The Antecedents of Calculus 13-12
13.2 Four Problems 13-22
13.3 Newton and Tangent Lines 13-35
13.4 Newton on Falling Objects and the Derivative 13-42
13.5 The Trajectory of a Cannonball 13-54
13.6 Newton and Areas 13-68
13.7 Conclusion 13-76
Appendixes
A Using a Scientific Calculator A-1
B Using a Graphing Calculator A-9
C Graphing with a Graphing Calculator A-19
D Finding Points of Intersection with a Graphing Calculator A-23
E Dimensional Analysis A-25
F Body Table for the Standard Normal Distribution A-30
G Selected Answers to Odd Exercises A-31
Index I-1
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Overview
The goal of Mathematics: A Practical Odyssey is to expose students to the util-
ity, relevance, and beauty of mathematics in the context of every-day themes
and across multiple disciplines, and to broaden the narrow view of mathemat-
ics that may come from an isolated study of algebra. The text incorporates
many items of interest, including historical notes, articles, and discussions of
contemporary issues along with many rich illustrations and fine art to demon-
strate a wide range of topics. We believe that students who engage with the
content in this text will have a broader outlook and understanding of the
world around them, in the spirit of a true liberal arts education. They will
benefit not only from the analytical tools and mathematical skills they prac-
tice and acquire, but from the references to important scientific research and
discoveries, as well as works of literature, history, art, and politics.
The following list is meant to make the text’s goals more concrete. Skim
the list to learn more about typical outcomes for each chapter. Also use the
list to identify areas that you might cover in your course.
l In Chapter 1, Logic, learn to analyze the validity of an argument.
l In Chapter 2, Sets and Counting, and Chapter 3, Probability, learn to under-
stand the risks of inherited diseases and the outcomes associated with lot-
teries and bets.
l In Chapter 4, Statistics, learn to understand the accuracy and validity of a
public opinion poll.
l In Chapter 5, Finance, learn the ins and outs of buying a house or car,
and using student loans to finance your college education.
l In Chapter 6, Voting and Apportionment, learn that there is no perfect vot-
ing system or method of apportionment.
l In Chapter 7, Number Systems and Number Theory, learn about the ori-
gins of commonly used number systems and their applications and learn
about the expression of the Fibonacci numbers in nature and the golden
ratio art.
l In Chapter 8, Geometry, learn to understand its origins and applications.
l In Chapter 9, Graph Theory, learn to create networks and apply graph the-
ory to schedules.
l In Chapter 10, Exponential and Logarithmic Functions, learn how popula-
tions grow, how radiocarbon dating works, how the Richter scale mea-
sures earthquakes, and how sound is measured in decibels.
l In Chapter 11, Markov Chains, learn how manufacturers can predict their
products’ success or failure in the marketplace.
l In Chapter 12, Linear Programming, learn how a small business can deter-
mine how to utilize limited resources to maximize its profit.
l In Chapter 13, Calculus, learn more about the subject and its uses. (Note:
this chapter does not appear within the textbook, but via www.cengagebrain.
com only.)
ix
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
x OVERVIEW
In Chapter 3, Probability:
l hemophilia is now covered along with other inherited diseases.
l the treatment of Simpson’s paradox has been expanded.
In Chapter 4, Statistics:
l the distinction between sample and population standard deviation is intro-
duced and explored in new exercises.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
OVERVIEW xi
In Chapter 5, Finance:
l there is a new focus on student finance, including student loans and
maxed-out credit cards.
Chapter 11, Markov Chains, has been rewritten so that the material is more
accessible, and so that review topics more closely fit our just-enough, just-in-
time review policy.
Exercises that require that the student engage in hands-on activities are
now included. These same exercises can be used for in-class activities.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
xii OVERVIEW
Features
chapter Openers
s
examples
s
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
OVERVIEW xiii
s
Newspaper and magazine
articles illustrate how the
book’s topics come up in the
real world.
Topic X
s
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
xiv OVERVIEW
exercises
s
concept questions
s
test students understanding
of ideas, often asking them
to provide their own exam-
ples or explain main ideas in
their own words.
Answers to the odd-numbered exercises are given in the back of the book,
with two exceptions:
l Answers to historical questions and essay questions are not given.
l Answers are not given when the exercises instruct the student to check the
answers themselves.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
OVERVIEW xv
The calculator boxes often offer keystrokes for several models of calculator, to
provide optimal support.
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xvi OVERVIEW
Microsoft excel©
s
Algebra review
s
Algebra reviews are just-
enough, just-in-time. Algebra
topics that may not have been
covered in the students’ alge-
bra classes and topics to
which students typically need
multiple exposures are
reviewed in sections placed
immediately prior to the loca-
tion where the topics are
used, usually in a “section
zero” at the beginning of the
chapter. Those review sec-
tions cover only what is need-
ed. More basic topics such as
equation solving are not for-
mally reviewed. Rather, the
examples are selected so that
they both explain the new
material and review the
appropriate algebra at the
same time.
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
OVERVIEW xvii
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
xviii OVERVIEW
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OVERVIEW xix
SuppLeMeNTS
FOr The STuDeNT FOr The INSTrucTOr
Instructor’s edition
(ISBN: 978-1-305-10418-1)
The Instructor’s Edition features an appendix containing
the answers to all problems. (Print)
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xx OVERVIEW
Acknowledgments
The authors would like to thank Richard Stratton, Jennifer Cordoba, Samantha
Lugtu, Tanya Nigh, Vernon Boes, Erin Brown, and all the fine people at
Cengage Learning. We also thank Scott Barnett, Jon Booze, and Kristy Hill for
their efforts with accuracy and solutions manual authoring.
Special thanks go to the users of the text and reviewers who evaluated
the manuscript for this edition, as well as those who offered comments on
previous editions.
reviewers
Dennis Airey, Rancho Santiago College Robert Jajcay, Indiana State University
Francisco E. Alarcon, Indiana University of Irja Kalantari, Western Illinois University
Pennsylvania Daniel Katz, University of Kansas
Judith Arms, University of Washington Palmer Kocher, SUNY, New Paltz
Bruce Atkinson, Palm Beach Atlantic College Katalin Kolossa, Arizona State University
Wayne C. Bell, Murray State University Donnald H. Lander, Brevard College
Wayne Bishop, California State University—Los Lee LaRue, Paris Junior College
Angeles Mike LeVan, Transylvania University
David Boliver, Trenton State College Lowell Lynde, University of Arkansas at Monticello
Stephen Brick, University of South Alabama Thomas McCready, California State University—
Barry Bronson, Western Kentucky University Chico
Frank Burk, California State University—Chico Vicki McMillian, Stockton State University
Laura Cameron, University of New Mexico Narendra L. Maria, California State University—
Jack Carter, California State University—Hayward Stanislaus
Timothy D. Cavanaugh, University of Northern John Martin, Santa Rosa Junior College
Colorado Gael Mericle, Mankato State University
Joseph Chavez, California State University—San Robert Morgan, Pima Community College
Bernadino Pamela G. Nelson, Panhandle State University
Eric Clarkson, Murray State University Carol Oelkers, Fullerton College
Rebecca Conti, State University of New York at Michael Olinick, Middlebury College
Fredonia Matthew Pickard, University of Puget Sound
S.G. Crossley, University of Southern Alabama Joan D. Putnam, University of Northern Colorado
Ben Divers, Jr., Ferrum College J. Doug Richey, Northeast Texas Community College
Al Dixon, College of the Ozarks Stewart Robinson, Cleveland State University
Joe S. Evans, Middle Tennessee State University Catherine Sausville, George Mason University
Hajrudin Fejzie, California State University—San Eugene P. Schlereth, University of Tennessee at
Bernardino Chattanooga
Lloyd Gavin, California State University— Gary Shufelt, Muhlenberg College
Sacramento Lawrence Somer, Catholic University of America
William Greiner, McLennan Community College Charles Stevens, Skagit Valley Technical College
Martin Haines, Olympic College Laurence Stone, Dakota County Technical College
Ray Hamlett, East Central University Charles Ziegenfus, James Madison University
Virginia Hanks, Western Kentucky University Michael Trapuzzano, Arizona State University
Brian Heaven, Tacoma Community College Pat Velicky, Mid-Plains Community College
Anne Herbst, Santa Rosa Junior College Karen M. Walters, University of Northern California
Linda Hinzman, Pasadena City College Dennis W. Watson, Clark College
Thomas Hull, University of Rhode Island Denielle Williams, Eastern Washington University
Robert W. Hunt, Humboldt State University Charles Ziegenfus, James Madison University
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ExponEntial and
logic
logarithmic Functions 1
© iStockPhoto.com/Bart Sadowski
When writer Lewis Carroll took Alice on her journeys down the rabbit hole to
Wonderland and through the looking glass, she had many fantastic encounters
with the tea-sipping Mad Hatter, a hookah-smoking Caterpillar, the Cheshire Cat,
and Tweedledee and Tweedledum. On the surface, Carroll’s writings seem to be
delightful nonsense and mere children’s entertainment. Many people are quite
surprised to learn that Alice’s Adventures in Wonderland is as much an exercise
in logic as it is a fantasy and that Lewis Carroll was actually Charles Dodgson, an
Oxford mathematician. In addition to “Alice,” Dodgson’s many writings include
the whimsical The Game of Logic and the brilliant Symbolic Logic.
Logic has fascinated scholars, philosophers, detectives, and star ship of-
ficers from the ancient, classic Greeks, to the ecentric, violin-playing Sherlock
Holmes, to the deadpan, emotionless Mr. Spock. In today’s world of misleading
commercial claims, innuendo, and political rhetoric, it is imperative that we em-
ploy logic to distinguish valid from invalid arguments; consequently, armed with
the fundamentals of logic, we can “live long and prosper.”
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What We Will Do In This Chapter (continued)
l
We will analyze and explore various types of statements and the conditions
under which they are true. This process will be steamlined by translating verbal
statements into symbolic form.
l
We will explore equivalent forms of statements, including those of the
conditional form “if . . . then . . .”. In so doing, we will learn the difference
between a “necessary condition” and a “sufficient condition.”
l
We will use various methods including Venn diagrams, truth tables, and formal
deductive proofs to determine the validity of an argument.
l
We will look at the lives and accomplishments of some of the influential people
who have shaped the study logic.
Logic is the science of correct reasoning. In their quest for logical perfection, the
Auguste Rodin captured this ideal in his Vulcans of Star Trek abandoned all emo-
bronze sculpture The Thinker. tion. Mr. Spock’s frequent proclamation that
“emotions are illogical” typified this attitude.
Logic is the science of correct reasoning. Webster’s New World College Dictionary
defines reasoning as “the drawing of inferences or conclusions from known or
assumed facts.” Reasoning is an integral part of our daily lives; we take appropri-
ate actions based on our perceptions and experiences. For instance, if you always
encounter a traffic jam when taking a specific route while driving to school, you
may decide to take an alternate route or leave home earlier on the day of an exam!
2
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
1.1 Deductive versus Inductive Reasoning 3
n Problem solving
Logic and reasoning are associated with the phrases problem solving and criti-
cal thinking. If we are faced with a problem, puzzle, or dilemma, we attempt
to reason through it in hopes of arriving at a solution.
The first step in solving any problem is to define the
problem in a thorough and accurate manner. Although this
might sound like an obvious step, it is often overlooked. Al-
ways ask yourself, “What am I being asked to do?” Before you
can solve a problem, you must understand the question. Once
the problem has been defined, all known information that is
relevant to it must be gathered, organized, and analyzed. This
n Deductive reasoning
Once a problem has been defined and analyzed, it might fall into a known
category of problems, so a common method of solution may be applied. For
instance, when one is asked to solve the equation x2 5 2x 1 1, realizing that
it is a second-degree equation (that is, a quadratic equation) leads one to put
it into the standard form (x2 2 2x 2 1 5 0) and apply the Quadratic Formula.
SoluTIon The given equation is a second-degree equation in one variable. We know that
all second-degree equations in one variable (in the form ax2 1 bx 1 c 5 0) can
be solved by applying the Quadratic Formula:
2b 6 Ïb2 2 4ac
x5
2a
Therefore, x2 5 2x 1 1 can be solved by applying the Quadratic Formula:
x2 5 2x 1 1
x2 2 2x 2 1 5 0
2s22d 6 Ïs22d2 2 s4ds1ds21d
x5
2s1d
2 6 Ï4 1 4
x5
2
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
4 CHAPTER 1 logic
2 6 Ï8
x5
2
2 6 2Ï2
x5
2
2s1 6 Ï2d
x5
2
x 5 1 6 Ï2
The solutions are x 5 1 1 Ï2 and x 5 1 2 Ï2.
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1.1 Deductive versus Inductive Reasoning 5
U
n Deductive reasoning and venn Diagrams
The validity of a deductive argument can be shown by use of a Venn diagram.
S A venn diagram is a diagram consisting of various overlapping figures contained
within a rectangle (called U, the “universe”). To depict a statement of the form
P “All S are P” (or, equivalently, “If S, then P”), we draw two circles, one inside the
other; the inner circle represents S (the “subject category”) and the outer circle
represents P (the “predicate category”). This relationship is shown in Figure 1.1.
Figure 1.1 All S are P. (If S,
Venn diagrams depicting “No S are P,” “Some S are P,” and “Some S are
then P.) not P” are shown in Figures 1.2, 1.3, and 1.4, respectively.
U U U
S S
S P
P P
Figure 1.3 Some S are P. (At Figure 1.4 Some S are not P. (At
Figure 1.2 No S are P least one S is P.) least one S is not P.)
SoluTIon Premise 1 is of the form “All A are B” and can be represented by a diagram like
that shown in Figure 1.6.
Premise 2 refers to a specific man, namely, Socrates. If we let x 5
Socrates, the statement “Socrates is a man” can then be represented by placing
x within the circle labeled “men,” as shown in Figure 1.7. Because we placed x
within the “men” circle, and all of the “men” circle is inside the “mortal” circle,
the conclusion “Socrates is mortal” is inescapable; the argument is valid.
U U
men
men x
mortal
mortal
x 5 Socrates
Figure 1.6 All men are mortal. Figure 1.7 Socrates is mortal.
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6 CHAPTER 1 logic
Scala/Art Resource, NY
Macedonia. When he was sev- Lyceum, or Peripatetic
enteen, Aristotle enrolled at School. The school
the Academy in Athens and had a large library with
became a student of the famed many maps, as well as
Plato. botanical gardens con-
Aristotle was one of Plato’s bright- taining an extensive collection of plants
est students; he frequently questioned and animals. Aristotle and his students
Plato’s teachings and openly disa- would walk about the grounds of the
greed with him. Whereas Plato em- Lyceum while discussing various sub-
phasized the study of abstract ideas jects (peripatetic is from the Greek word
SoluTIon Premise 1 is of the form “All A are B”; the argument is depicted in Figure 1.8.
No matter where x is placed within the “doctors” circle, the conclusion
“My mother is a man” is inescapable; the argument is valid.
U
Saying that an argument is valid does not mean that the conclusion is true.
doctors
x men The argument given in Example 3 is valid, but the conclusion is false. One’s
mother cannot be a man! Validity and truth do not mean the same thing. An
argument is valid if the conclusion is inescapable, given the premises. Nothing
is said about the truth of the premises. Thus, when examining the validity of
x 5 My mother an argument, we are not determining whether the conclusion is true or false.
Figure 1.8 My mother is a man. Saying that an argument is valid merely means that, given the premises, the
reasoning used to obtain the conclusion is logical. However, if the premises of
a valid argument are true, then the conclusion will also be true.
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1.1 Deductive versus Inductive Reasoning 7
SoluTIon Premise 1 is of the form “All A are B”; the “circle of professional wrestlers” is
contained within the “circle of actors.” If we let x represent The Rock, premise
2 simply requires that we place x somewhere within the actor circle; x could be
placed in either of the two locations shown in Figures 1.9 and 1.10.
U U
professional professional
wrestlers x wrestlers
x
actors actors
Saying that an argument is invalid does not mean that the conclusion is
false. Example 4 demonstrates that an invalid argument can have a true con-
clusion; even though The Rock (Dwayne Johnson) is a professional wrestler,
the argument used to obtain the conclusion is invalid. In logic, validity and
truth do not have the same meaning. Validity refers to the process of reasoning
© Duomo/Corbis
SoluTIon Premise 1 is of the form “Some A are B”; it can be represented by two over-
lapping circles (as in Figure 1.3). If we let x represent broccoli, premise 2 re-
quires that we place x somewhere within the plant circle. If x is placed as in
Figure 1.11, the argument would appear to be valid. However, if x is placed
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Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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Bendix paced back and forth, perspiration shining wetly on his face in
the light from the overhead bulb. "It's not fair," he said huskily. "It's not
a true election. It doesn't represent anything." He looked at
Kimmensen desperately. "It's not fair, Joe!"
Kimmensen sighed. "All right, Jem. I assume you brought the
necessary equipment—the screwdriver, the insulation, and so forth?"
CHAPTER VII
Until, hours later, orange flowers burst in the valley below. He came
erect, not understanding them for a moment, and then he ran out to
the patio, leaning over the parapet. On the faint wind, he heard the
distant sound of earth and houses bursting into vapor. In the valleys,
fire swirled in flashes through the dark, and against the glare of
burning trees he saw bobbing silhouettes of planes. Men were far too
small to be seen at this distance, but as firing stabbed down from the
planes other weapons answered from the ground.
Suddenly, he heard the flogging of a plane in the air directly
overhead. He jumped back, reaching for his weapon, before he
recognized Jem Bendix's sportster. It careened down to his landing
stage, landing with a violent jar, and Bendix thrust his head out of the
cabin. "Joe!"
"What's happening?"
"Messerschmidt—he's taking over, in spite of the election! I was
home when I saw it start up. He and his followers're cutting down
everybody who won't stand for it. Come on!"
"What are you going to do?"
Bendix's face was red with rage. "I'm going to go down there and kill
him! I should have done it long ago. Are you coming with me?"
Why not? Kimmensen grimaced. Why wait to die here?
He clambered into the plane and buckled his seat belt. Bendix flung
them up into the air. His hands on the wheel were white and shaking
as he pointed the plane along the mountain slope and sent them
screaming downward. "They're concentrated around the office
building, from the looks of it," he shouted over the whine of air. "I
should have known he'd do this! Well, I'm League President, by God,
and I'm going to settle for him right now!"
If you don't kill us first, Kimmensen thought, trying to check over his
weapon. Bendix was bent over the wheel, crouched forward as
though he wanted to crash directly into the plaza where Kimmensen
could see running men.
They pulled out of the dive almost too late. The plane smashed down
through the undergrowth behind the office building. Bendix flung his
door open and jumped out while the plane rocked violently.
Kimmensen climbed out more carefully. Even here, in the building's
shadow, the fires around the plaza were bright enough to let him see.
He pushed through the tangled shrubbery, hearing Bendix breaking
forward ahead of him. Bendix cleared the corner of the building. "I
see him, Joe!"
Kimmensen turned the corner, holding his weapon ready.
He could see Messerschmidt standing in a knot of men behind the
wreckage of a crashed plane. They were looking toward the opposite
slope, where gouts of fire were winking up and down the
mountainside. Kimmensen could faintly hear a snatch of what
Messerschmidt was shouting: "Damn it, Toni, we'll pull back when I—"
but he lost the rest. Then he saw Bendix lurch out of the bushes ten
feet behind them.
"You! Messerschmidt! Turn around!"
Messerschmidt whirled away from the rest of the men, instinctively,
like a great cat, before he saw who it was. Then he lowered the
weapon in his hand, his mouth jerking in disgust. "Oh—it's you. Put
that thing down, or point it somewhere else. Maybe you can do some
good around here."
"Never mind that! I've had enough of you."
Messerschmidt moved toward him in quick strides. "Listen, I haven't
got time to play games." He cuffed the weapon out of Bendix's hand,
rammed him back with an impatient push against his chest, and
turned back to his men. "Hey, Toni, can you tell if those
Northwesters're moving down here yet?"
Kimmensen's cheeks sucked in. He stepped out into the plaza,
noticing Bendix out of the corners of his eyes, standing frozen where
Messerschmidt had pushed him.
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