Etextbook 978 1464124730 for all practical purposes mathematical literacy in todays world

eTextbook 978-1464124730 For All Practical Purposes: Mathematical Literacy in Today■s World Visit to download the full and correct content document: https://ebookmass.com/product/etextbook-978-1464124730-for-all-practical-purposes -mathematical-literacy-in-todays-world/

C O N T E N T S

4.4 Linear Programming: Life Is Complicated 144

Skills Check 106 Chapter Exercises 109 Applet Exercises 122 Writing Projects 123

SPOTLIGHT 4.2

The Father of Linear Programming Recalls Its Origins 146

Suggested Readings 123 Suggested Websites 124

Chapter 4

SPOTLIGHT 4.3

Linear Programming and the Transportation Problem 125

Finding Fast Algorithms Means Better Airline Service 147

4.5 A Transportation Problem: Delivering Perishables 148 4.6 Improving on the Current Solution 157

4.1 Linear Programming and Mixture Problems: Combining Resources to Maximize Profit 126

SPOTLIGHT 4.1

Case Studies in Linear Programming 128

4.2 Finding the Optimal Production Policy 134

Review Vocabulary 160 Self Check Answers 161 Skills Check 161 Chapter Exercises 164 Writing Projects 172 Suggested Readings 173 Suggested Websites 173

4.3 Why the Corner Point Principle Works 141

Carl Court/Getty Images

vii

CASE STUDY

Improving Medical Care Using Mathematics 174

Part II

STATISTICS: THE SCIENCE OF DATA / 178 Chapter 5

Exploring Data: Distributions

181

5.7 Describing Variability: The Standard Deviation 203 SPOTLIGHT 5.3

Using Technology to Calculate Standard Deviation 206

5.1 Displaying Distributions: Histograms 182 SPOTLIGHT 5.1

5.8 Normal Distributions 209

John Wilder Tukey, Champion of Exploratory Data Analysis 184

5.2 Interpreting Histograms 188 5.3 Displaying Distributions: Stemplots 194 5.4 Describing Center: Mean and Median 196

SPOTLIGHT 5.4

Density Estimation 215

5.9 The 68–95–99.7 Rule for Normal Distributions 216

SPOTLIGHT 5.2

Which Mean Do You Mean? 199

5.5 Describing Variability: Range and Quartiles 200

5.6 The Five-Number Summary and Boxplots 201

Review Vocabulary 219 Self Check Answers 220 Skills Check 221 Chapter Exercises 223 Applet Exercises 239 Writing Projects 239 Suggested Readings 240 Suggested Websites 240

viii

C O N T E N T S

Chapter 6

Exploring Data: Relationships

7.7 Inference: From Sample to Population 314

241

SPOTLIGHT 7.4

Truth in Polling 324

6.1 Displaying Relationships: Scatterplot 242

SPOTLIGHT 6.1

Creating Scatterplots Using Technology 249

6.2 Making Predictions: Regression Line 251

6.3 Correlation 254

Review Vocabulary 324 Self Check Answers 326 Skills Check 327 Chapter Exercises 328 Applet Exercises 338 Writing Projects 339 Suggested Readings 340 Suggested Websites 340

SPOTLIGHT 6.2

Correlation Calculation 259 SPOTLIGHT 6.3

Regression Toward the Mean 260

6.4 Least-Squares Regression 260

Chapter 8

SPOTLIGHT 6.4

Regression and Correlation in Action: College Success 265

Probability: The Mathematics of Chance

8.1 Random Phenomena and Probability 343

SPOTLIGHT 6.5

Calculating the Equation of the Least-Squares Regression Line 265

SPOTLIGHT 8.1

The Problem of Points 345

6.5 Interpreting Correlation and Regression 268

8.2 Basic Rules of Probability 348 SPOTLIGHT 8.2

Review Vocabulary 274 Self Check Answers 274 Skills Check 275 Chapter Exercises 277 Applet Exercises 288 Writing Projects 289

Probability and Psychology 349

8.3 Rules of Probability: Independent and Dependent Events 354 8.4 Discrete Probability Models 360 8.5 Equally Likely Outcomes 363

Suggested Readings 290 Suggested Websites 290

SPOTLIGHT 8.3

Using Technology to Compute Permutations, Factorials, and Combinations 369 SPOTLIGHT 8.4

Chapter 7

Data for Decisions

7.1 Sampling 292 7.2 Bad Sampling Methods 294 7.3 Simple Random Samples 297 SPOTLIGHT 7.1

Using Technology to Select an SRS 300

7.4 Cautions About Sample Surveys 302 SPOTLIGHT 7.2

Pew Research Center 303

7.5 Experiments 305 SPOTLIGHT 7.3

Ethics in Experiments 309

7.6 Experiments Versus Observational Studies 309

291

Birthday Coincidences 370

8.6 Continuous Probability Models 371 8.7 The Mean and Standard Deviation of a Probability Model 375 8.8 The Central Limit Theorem 380

Review Vocabulary 385 Self Check Answers 386 Skills Check 387 Chapter Exercises 389 Applet Exercises 399 Writing Projects 400 Suggested Readings 400 Suggested Websites 400

341

ix

Blend Images, Hill Street Studios/Getty Images

C O N T E N T S

Part III

VOTING AND SOCIAL CHOICE / 402 Chapter 9

Social Choice: The Impossible Dream

Chapter 11

405

9.1 An Introduction to Social Choice 406

SPOTLIGHT 11.1

The Electoral College 465

11.2 The Shapley–Shubik Model 466

SPOTLIGHT 9.1

The Historical Record 409

SPOTLIGHT 11.2

Power Indices 466

9.3 Other Voting Systems for Three or More Candidates 412

11.3 The Banzhaf Model 474

9.4 Insurmountable Difficulties: Arrow’s Impossibility Theorem 424

SPOTLIGHT 11.3

Should Blocking Coalitions Be Counted? 476

SPOTLIGHT 9.2

SPOTLIGHT 11.4

Kenneth J. Arrow 425

A Real Mathematical Quagmire 480

9.5 A Better Approach? Approval Voting 428

SPOTLIGHT 11.5

The Electoral College: Presidential Elections of 2012, 2016, and 2020 484

Review Vocabulary 430 Self Check Answers 431 Skills Check 431 Chapter Exercises 433 Writing Projects 438

SPOTLIGHT 11.6

Can the Banzhaf and Shapley–Shubik Models Disagree? 487

11.4 Voting Systems—Without Weights 487

Suggested Readings 438

Chapter 10

The Manipulability of Voting Systems

439

10.1 An Introduction to Manipulability 440 10.2 Majority Rule and Condorcet’s Method 443 10.3 The Manipulability of Other Voting Systems for Three or More Candidates 445 10.4 Impossibility 450 10.5 The Chair’s Paradox 452

Review Vocabulary 454 Self Check Answers 454 Skills Check 455 Chapter Exercises 456 Writing Projects 459 Suggested Readings 460

461

11.1 How Weighted Voting Works 462

9.2 Majority Rule and Condorcet’s Method 407

Weighted Voting Systems

Review Vocabulary 492 Self Check Answers 493 Skills Check 494 Chapter Exercises 496 Writing Projects 501 Suggested Readings 501 Suggested Websites 502

Chapter 12

Electing the President

503

12.1 Narrowing the Field through the Primary Process 504 12.2 Spatial Models for Two-Candidate Elections: Discrete Distributions 508 12.3 Spatial Models for Two-Candidate Elections: Continuous Distributions 515 12.4 Spatial Models for Multicandidate Elections 518

x

C O N T E N T S

12.5 Spatial Models and the Electoral College 524

Chapter Exercises 530 Writing Projects 535 Suggested Readings 535 Suggested Websites 535

Review Vocabulary 527 Self Check Answers 528 Skills Check 529

Yana Gayvoronskaya/ Shutterstock

Part IV

FAIRNESS AND GAME THEORY / 536 Chapter 13

Fair Division

SPOTLIGHT 14.2

539

13.1 The Adjusted Winner Procedure 540 13.2 The Knaster Inheritance Procedure 544

A Puzzling Apportionment of the U.S. House of Representatives 597

14.4 Which Divisor Method Is Best? 598 SPOTLIGHT 14.3

13.3 Fair Division and Organ Transplant Policies 546

Legal Challenges to Apportionment 599

13.4 Taking Turns 548

Mathematics and Politics: A Strange Mixture 607

SPOTLIGHT 14.4

13.5 Divide-and-Choose 551 13.6 Cake-Division Procedures: Proportionality 552

SPOTLIGHT 13.1

Seventy Years of Cake Cutting 553

13.7 Cake-Division Procedures: The Problem of Envy 555 13.8 Vickrey Auctions 558

Review Vocabulary 560 Self Check Answers 561 Skills Check 561 Chapter Exercises 563 Writing Projects 570

Review Vocabulary 608 Self Check Answers 609 Skills Check 611 Chapter Exercises 613 Writing Projects 619 Suggested Readings 620 Suggested Websites 620

Chapter 15

Suggested Readings 570

Game Theory: The Mathematics of Competition

621

SPOTLIGHT 15.1

The Early History of Game Theory 622

Chapter 14

Apportionment

571

15.1 Two-Person Total-Conflict Games: Pure Strategies 623

14.1 The Apportionment Problem 572

15.2 Two-Person Total-Conflict Games: Mixed Strategies 628

14.2 The Hamilton Method 578

15.3 Partial-Conflict Games 637

SPOTLIGHT 14.1

Apportionment in the U.S. House of Representatives Before 1900 580

14.3 Divisor Methods 585

15.4 Mechanism Design and Larger Games 643 SPOTLIGHT 15.2

The Nobel Prize in Economics 646

C O N T E N T S

15.5 Using Game Theory 651

Review Vocabulary 654 Self Check Answers 655 Skills Check 656

Justin Ziewe/Ikon Images/SuperStock

Chapter Exercises 659 Applet Exercise 663 Writing Projects 664 Suggested Readings 664 Suggested Websites 665

Part V

THE DIGITAL REVOLUTION / 666 Chapter 16

Identification Numbers

669

17.2 Encoding with Parity-Check Sums 702 SPOTLIGHT 17.2

Neil Sloane 703

16.1 Check Digits 669

17.3 Data Compression 707

SPOTLIGHT 16.1

The VIN System 678

SPOTLIGHT 17.3

16.2 The ZIP Code 679

Morse Code 708

16.3 Bar Codes 680

SPOTLIGHT 17.4

David Huffman 711

SPOTLIGHT 16.2

17.4 Cryptography 714

New Frontier: Bar Coding DNA 686

SPOTLIGHT 17.5

SPOTLIGHT 16.3

Modeling the Genetic Code 716

History of Bar Codes 687

SPOTLIGHT 17.6

16.4 Encoding Personal Data 687

Enigma Machines 724

SPOTLIGHT 16.4

SPOTLIGHT 17.7

Ten Fun Facts about Social Security Numbers 688

Mavis Batey 725

SPOTLIGHT 16.5

Census Records at the National Archives 690

SPOTLIGHT 17.8

Review Vocabulary 691 Self Check Answers 691 Skills Check 691 Chapter Exercises 693 Writing Projects 697

SPOTLIGHT 17.9

Alan Turing 726 Smart Cards 727

Suggested Readings 697 Suggested Website 698

Review Vocabulary 728 Self Check Answers 728 Skills Check 729 Chapter Exercises 730 Suggested Websites and Videos 733

Chapter 17

Encoding Information

17.1 Binary Codes 699 SPOTLIGHT 17.1

The Ubiquitous Reed–Solomon Codes 701

699

xi

xii Rob Koenen/Moment/ Getty Images

C O N T E N T S

Part VI

ON SIZE AND GROWTH / 734 Chapter 18

Growth and Form

737

19.2 Rosette and Strip Patterns 790 SPOTLIGHT 19.7

18.1 Geometric Similarity 738

“Strive Then to Be Perfect” 793

18.2 How Much Is That in . . . ? 743

SPOTLIGHT 19.8

Patterns Created by the Bakuba People 795

18.3 Big Stuff 747

SPOTLIGHT 19.9

SPOTLIGHT 18.1

A Mile-High Building? 749 SPOTLIGHT 18.2

Why Would You Want to Be Big? 753

18.4 Dimension Tension 754

The 17 Wallpaper Patterns 797

19.3 Notation for Patterns 798 19.4 Symmetry Groups 802 19.5 Fractal Patterns and Chaos 807 SPOTLIGHT 19.10

SPOTLIGHT 18.3

The Father of Fractals 808

Scaled to Fit 757

18.5 How We Grow 760

SPOTLIGHT 18.4

Helping to Find Missing Children 761

Review Vocabulary 765 Self-Check Answers 765 Skills Check 765 Chapter Exercises 767 Writing Projects 775 Suggested Readings 778 Suggested Websites 778

Review Vocabulary 812 Self Check Answers 812 Skills Check 813 Chapter Exercises 814 Writing Projects 825 Suggested Readings 825 Suggested Websites 826

Chapter 20

Tilings

20.1 Tilings with Regular Polygons 828 Chapter 19

Symmetry and Pattern 779

19.1 Fibonacci Numbers and the Golden Ratio 780

SPOTLIGHT 20.1

Regular Polyhedra and Buckyballs 829

20.2 Tilings with Irregular Polygons 833

SPOTLIGHT 19.1

SPOTLIGHT 20.2

Leonardo of Pisa (“Fibonacci”) 781

In Praise of Amateurs 836

SPOTLIGHT 19.2

SPOTLIGHT 20.3

Fibonacci Solar Power 782 SPOTLIGHT 19.3

Drumming Up Mathematics 783 SPOTLIGHT 19.4

How the Greeks Constructed a Golden Rectangle 784

Maurits Escher 837

20.3 Using Only Translations 838 20.4 Using Translations Plus Half-Turns 840 20.5 Nonperiodic Tilings 845

SPOTLIGHT 19.5

SPOTLIGHT 20.4

The Consumer Price Index: An Application of the Geometric Mean 785

SPOTLIGHT 20.5

SPOTLIGHT 19.6

Are We Trying to Reclaim the “Glory That Was Greece”? 788

Sir Roger Penrose 846 Mathematics and Autism 851 SPOTLIGHT 20.6

Quasicrystals 854

827

xiii

C O N T E N T S

SnapshotPhotos/ Shutterstock

Review Vocabulary 856 Self Check Answers 856 Skills Check 857 Chapter Exercises 858

Writing Projects 864 Suggested Readings 864 Suggested Websites 865

Part VII

YOUR MONEY AND RESOURCES / 866 Chapter 21

Savings Models

869

22.3 Conventional Loans 914 SPOTLIGHT 22.2

21.1 Simple Interest and Arithmetic Growth 870

What We Did with Our House, and What Else You Could Do 921

21.2 Compound Interest and Geometric Growth 872

SPOTLIGHT 22.3

The Mortgage Crisis 922

22.4 Other Loans 923

SPOTLIGHT 21.1

22.5 Annuities 929

Thomas Robert Malthus 878

21.3 Effective Rate and APY 878

SPOTLIGHT 22.4

What Is an Actuary? 930

21.4 A Limit to Compounding 880 SPOTLIGHT 21.2

The Number e 881

21.5 A Model for Saving 882

SPOTLIGHT 21.3

Using a Spreadsheet for Financial Calculations 888

Review Vocabulary 933 Self Check Answers 933 Skills Check 934 Chapter Exercises 935 Applet Exercise 941 Writing Projects 941 Suggested Readings 942 Suggested Websites 942

21.6 Inflation 888 SPOTLIGHT 21.4

What Is a Financial Derivative? 893

Review Vocabulary 896 Self Check Answers 897 Skills Check 897 Chapter Exercises 898 Applet Exercise 906 Writing Projects 906

Chapter 23

The Economics of Resources

943

SPOTLIGHT 23.1

The Formulas We Use in This Chapter 944

Suggested Readings 907 Suggested Websites 908

23.1 Growth Models for Biological Populations 944 SPOTLIGHT 23.2

Chapter 22

Borrowing Models

22.1 Simple Interest 910 22.2 Compound Interest 911 SPOTLIGHT 22.1

What’s the Real Rate? 913

909

12 Billion by 2050—or Only 9 Billion? 947

23.2 How Long Can a Nonrenewable Resource Last? 950 SPOTLIGHT 23.3

What Are Reserves? Or, How Much Is Out There, Really? 953

xiv

C O N T E N T S

23.3 Radioactive Decay 955

E. Slope of a Line AR-14 F. Graphing a Line in Slope-Intercept Form AR-15 G. Linear Inequalities in Two Variables AR-16 H. Systems of Linear Equations and Inequalities AR-17

23.4 Harvesting Renewable Resources 960 SPOTLIGHT 23.4

Fishing: Out to Sea? 965

Algebra Review V: Summation Notation, Sequences, and Series AR-19

SPOTLIGHT 23.5

The Tragedy of Easter Island 969

A. Summation Notation AR-19 B. Sequences AR-20 C. Arithmetic and Geometric Sequences and Series AR-21

23.5 Dynamical Systems and Chaos 971

Review Vocabulary 976 Self Check Answers 977 Skills Check 978 Chapter Exercises 979 Writing Projects 986

Algebra Review VI: Exponents, Roots, and Logarithms AR-22 A. Powers and Roots AR-22 B. Natural and Fractional Exponents AR-23 C. Graphs of Exponential Equations AR-24 D. Rules for Exponents and Roots AR-25 E. Logarithms AR-26 F. Using Logarithms to Solve Equations AR-26

Suggested Readings 986 Suggested Websites 986

Appendix

Algebra Review

AR-1

Algebra Review I: Handling Operations AR-1 A. Order of Operations AR-1 B. Distributive Law AR-2 C. Operations with Rational Numbers (Fractions) AR-2

Algebra Review II: Representing Numbers AR-3

Algebra Review VII: Functions AR-27 A. Determining If a Relation Is a Function AR-27 B. Function Notation AR-28

Algebra Review VIII: Permutations and Combinations AR-29 A. Permutations AR-29 B. Combinations AR-29

A. Fractions, Percents, and Percentages AR-3 B. Remainders AR-3 C. Prime and Composite Numbers AR-4 D. Significant Digits AR-4 E. Scientific Notation AR-5 F. Rounding Numbers AR-5 G. Counting in Binary AR-6

Answers to Algebra Review Exercises

Answers to Skills Check Exercises

SC-1

Algebra Review III: Working with Formulas AR-7

Answers to Odd-Numbered Exercises

A-1

Index

I-1

A. Using Formulas AR-7 B. Solving for One Variable in Terms of Another AR-8 C. Formulas Related to Geometric Shapes AR-9

Algebra Review IV: Linear Equations and Inequalities AR-10 A. Linear Equations in One Variable AR-10 B. Plotting Points in the Plane AR-11 C. Distance and Midpoint between Two Points in the Plane AR-12 D. Linear Equations in Two Variables AR-13

CUSTOM CHAPTERS The following chapters are available through W. H. Freeman’s custom publishing (restrictions may apply):

Sets Problem Solving Logic

Geometry Counting and Probability Numeration Systems Personal Finance

AR-31

Preface

To the Student

To the Instructor

For All Practical Purposes, Tenth Edition, continues our effort to bring the excitement of contemporary mathematical thinking to the nonspecialist. In science and industry, mathematical models are the main tools for analyzing and solving problems that arise. In this book, our goal is to convey the power of mathematics by showing you the wide variety of problems that can be modeled and solved by quantitative means. An extensive package of supplements designed to make study time supremely effective complements the tenth edition. Highlights of the supplements package include the Student Study Guide and Student Solutions Manual. Between the text and the available resources, For All Practical Purposes offers you the tools to succeed in the course and apply your new knowledge to daily life experiences.

There are many ways to talk about why mathematics and its applications matter. You will hear expressions such as “mathematical literacy” or “quantitative literacy.” They mean, essentially, that math is important. It is important because knowing it can make your life easier. In other words, it can help explain how your world works. We created this course and this book because we know that not everyone looks at mathematics in this way.

In school, you spent a great deal of time learning the tools of mathematics—how to manipulate symbols and how to solve equations. In this course, you will spend time learning about the power of mathematics, which helps us understand many different parts of our everyday lives and the world itself. We hope this exploration will give you a broader sense of the subject and why we wanted you to take a math course every year you were in school. It’s “for all practical purposes” because, in a sense, you’ve learned to hammer nails and saw wood—and now we’re going to build houses.

Enjoy!

Because For All Practical Purposes stresses the connections between contemporary mathematics and modern society, our text must be flexible enough to accommodate new ideas in mathematics and their new applications to our daily lives. We maintain this flexibility in the tenth edition.

Our primary goal for this edition was to further improve the ease of use for instructors and students alike. An extensive supplements package is available within LaunchPad, W. H. Freeman’s new online homework system. LaunchPad offers content that has been curated and organized for easy assignability in a simple but powerful interface. Assets integrated into LaunchPad include an interactive e-Book, LearningCurve, practice quizzes, exercise solutions, interactive applets, flashcards, video clips, and much more.

New to the Tenth Edition New Algebra Review Appendix An all-new Algebra Review Appendix offers reviews on basic concepts used in the text and includes straightforward examples, technology tips, and practice exercises. Answers to all practice exercises are also included. References to relevant sections of the Algebra Review Appendix appear throughout the text in the margins.

New Self Check Exercises Self Check exercises have been added throughout the text, with answers at the end of each chapter, allowing students to check their understanding of new concepts as the material is being taught.

Instructors can also use these exercises in class as part of their lectures.

xv

xvi

P R E F A C E

New Examples New and updated examples are included throughout the text to address new topics and changes to the material. Examples provide new topics for class discussion and new ways of relating to essential concepts.

●

● ●

●

New Exercises ●

●

●

●

Exercise sets, including Skills Check questions, have been updated and refreshed.

Over 200 Self Check exercises have been added throughout the chapters.

The new Algebra Review Appendix includes over 200 practice exercises.

New Chapter Review exercises have been added to the exercise sets. These exercises are not organized by section and test student understanding of the chapter material as a whole.

Part-Specific Content Changes

●

●

●

●

●

Part III: Voting and Social Choice ●

Part I: Management Science ●

●

●

New discussion of the importance of operations research for improving health care (Chapters 1, 3, and 4).

Revised example illustrating how to cut the costs of installing a local cable TV network (Chapter 2).

A new case study, which closes Part I, offers a discussion of the ideas behind Alvin Roth and Lloyd Shapley’s Nobel Prize–winning work about stable allocations, such as pairing hospitals and medical school graduates for residencies and pairing schoolchildren with schools they want to attend (school choice).

●

●

●

Part II: Statistics: The Science of Data ●

●

Throughout Part II, more graphic displays have been added to help students visualize the material.

Spotlight features have been added about the use of technology (TI-84 graphing calculator, Excel) for calculations, creation of graphic displays, and selection of simple random samples (Chapters 5–8).

Expanded directions for constructing histograms and stemplots, along with examples on changing histogram class intervals and expanding stemplot stems (Chapter 5).

Simplified coverage of the Banzhaf Power Index (Chapter 11).

New section on the use of apportionment in presidential primaries, including an example with real data from the 2008 Democratic primary (Section 12.1).

Improved presentation of spatial modeling by introducing discrete models first and using them to analyze the Electoral College (Chapter 12).

Part IV: Fairness and Game Theory

●

●

Addition of an example showing computation of correlation from a formula (Chapter 6, Example 7).

Explanation of least-squares criterion (Chapter 6).

Addition of an example showing the dangers of extrapolation (Chapter 6, Example 10).

Discussion of how the Pew Research Center modifies its telephone survey plans to correct for undercoverage (Spotlight 7.2).

Expanded instructions on selecting a simple random sample (Chapter 7).

Expanded discussion of ethics in experiments (Spotlight 7.3).

Expanded discussion of the basic rules of probability (Chapter 8).

Inclusion of estimation of probabilities based on survey data (Chapter 8).

Coverage of conditional probability and Bayes’ Rule (Section 8.2).

●

Expanded coverage of the history of the apportionment of seats in the U.S. House of Representatives (Chapter 14).

New section highlights mechanism design— designing a game to achieve a particular outcome (Section 15.4).

New examples model a work location versus schedule decision, as well as the goalie/kicker interaction in a soccer penalty kick (Chapter 15).

Part V: The Digital Revolution ● ●

●

New examples of check-digit schemes (Chapter 16).

Expanded summary of error-detection schemes (Chapter 16).

Expanded spotlight on history of bar codes (Chapter 16).

P R E F A C E ●

●

●

●

New title for Chapter 17 to more accurately describe the content.

New spotlight features, covering facts about social security numbers, enigma machines, Mavis Batey, Alan Turing, and smart cards (Chapter 17).

Added coverage of permutation, Playfair, and Jefferson wheel ciphers (Chapter 17).

New website and video suggestions added (Chapter 17).

Part VI: On Size and Growth ●

●

●

●

●

●

Revised exercises that consider production of solar energy, consumption by electric cars, and units for measuring water use (Chapter 18).

New spotlight featuring an award-winning young mathematician whose interest was prompted by rhythms that he learned from playing drums and reading Sanskrit poetry (Spotlight 19.2).

New spotlight on the use of Fibonacci numbers in optimizing the design of a solar power plant (Spotlight 19.3).

New examples of patterns on ancient and modern pottery (Chapter 19).

New illustration of a Penrose pattern in San Francisco architecture, and new figures and exercises about the inflation of patterns, as well as a discussion of the Nobel Prize awarded for the discovery of quasicrystals (Chapter 20).

New spotlight on mathematics and autism (Spotlight 20.5).

Part VII: Your Money and Resources ●

●

●

●

●

Incorporation of current student loan interest rates into examples (Chapter 22).

New section on discounted and add-on loans, with exercises about payday loans (Chapter 22).

Example showing details of costs at a real estate closing (Chapter 22).

New spotlight on how minimum-size catch limits manipulate fish genetics (Spotlight 23.4).

New exercises on radioactive isotopes, including those used in medicine and those released in the meltdown of Japanese reactors (Chapter 23).

Focus on Accuracy For this edition, we once again implemented a detailed accuracy-checking plan to sustain the quality of the exercises and solutions. To this end, we are very

xvii

grateful to Dennis Evans of Concordia University– Wisconsin and Paul McCombs of Rock Valley College.

Custom Options In addition to the extensive topics covered in the text, more traditional chapters (including Problem Solving, Sets, Logic, Geometry, Counting and Probability, Numeration Systems, and Personal Finance) are available with For All Practical Purposes through custom publishing. For more information, please contact your W. H. Freeman representative or go to www. macmillanhighered.com/fapp10e. Restrictions apply.

Media and Supplements The media and supplements package for the tenth edition has been updated to reflect changes in the book. Both instructors and students will benefit from the innovative materials available to them.

W. H. Freeman’s new online homework system, LaunchPad, offers quality content that has been curated and organized for easy assignability in a simple but powerful interface. We have taken what we have learned from thousands of instructors and hundreds of thousands of students to create a new generation of W. H. Freeman/ Macmillan technology.

Curated units. Combining a curated collection of videos, homework sets, tutorials, applets, and e-Book content, LaunchPad’s interactive units give instructors a building block to use as is or as a starting point for customized learning units. A majority of exercises from the text can be assigned as online homework, including an abundance of algorithmic exercises. An entire unit’s worth of work can be assigned in seconds, drastically reducing the amount of time it takes for instructors to have their course up and running.

Easily customizable. Instructors can customize the LaunchPad units by adding quizzes and other activities from our vast wealth of resources. They can also add a discussion board, a drop box, and an RSS feed, with just a few clicks. LaunchPad allows instructors to customize students’ experiences as much or as little as desired.

Useful analytics. The gradebook quickly and easily allows instructors to look up performance metrics for classes, individual students, and individual assignments.

xviii

P R E F A C E

Intuitive interface and design. The student experience is simplified. Students’ navigation options and expectations are clearly laid out at all times, ensuring they can never get lost in the system.

Assets integrated into LaunchPad include the following: Interactive e-Book. Every LaunchPad e-Book comes with powerful study tools for students, video and multimedia content, and easy customization for instructors. Students can search, highlight, and bookmark, making it easier to study and access key content. And teachers can ensure that their classes get just the book they want to deliver by customizing and rearranging chapters; adding and sharing notes and discussions; and linking to quizzes, activities, and other resources.

LearningCurve provides students and instructors with powerful adaptive quizzing, a game-like format, direct links to the e-Book, and instant feedback. The quizzing system features questions tailored specifically to the text, and it adapts to students’ responses, providing material at different difficulty levels and topics based on student performance.

SolutionMaster offers an easy-to-use web-based version of the instructor’s solutions, allowing instructors to generate a solution file for any set of homework exercises.

Student Resources ●

●

●

●

●

Instructor Resources ●

●

●

●

Other online homework options include: ●

WebAssign integrates the text exercises from For All Practical Purposes, Tenth Edition, into a popular and trusted online homework system, making it easy to assign algorithmically generated homework and quizzes.

Student Solutions Manual provides solutions to the odd-numbered exercises, with step-bystep solutions to select problems.

Student Study Guide offers study tips and tools to help students gain a better understanding of course material, including key ideas for each section and additional examples and practice exercises.

Interactive applets help students master key mathematical concepts and work exercises.

Math Clips are animated whiteboard videos that illuminate key concepts in the text by showing students step-by-step solutions to selected exercises.

Self-quizzes, flash cards, and other projects offer additional study help.

Instructor’s Guide with Full Solutions includes teaching suggestions, chapter comments, and detailed solutions to all exercises.

Teaching Guide for First-Time Instructors helps instructors, adjuncts, and teaching assistants plan their course more easily and effectively. This guide also offers fresh perspectives and ideas to experienced instructors.

Test Bank offers thousands of multiple-choice questions.

Lecture slides offer a detailed lecture presentation of concepts covered in each chapter of For All Practical Purposes, Tenth Edition.

Clicker Questions are available for each chapter.

Companion Website www.macmillanhighered.com/fapp10e This openaccess website provides students with access to the applets referenced throughout the text.

Acknowledgments

For All Practical Purposes continues to evolve in great part because of our many friends and colleagues who have offered suggestions, comments, and corrections. We are grateful to them all.

Michael Allen, Glendale Community College Hamid Attarzadeh, Kentucky Community and Technical College System Robert J. Bass, Gardner-Webb University Debra D. Bryant, Tennessee Technological University Annette M. Burden, Youngstown State University Christine Cedzo, Gannon University Gina Poore Dunn, Lander University Dennis Evans, Concordia University Kevin Ferland, Bloomsburg University Gregory Goeckel, Presbyterian College Doug Greiner, Rogers State University Samuel S. Gross, Bloomsburg University Patricia Humphrey, Georgia Southern University Jamie L. King, Arkansas Tech University Jay Malmstrom, Oklahoma City Community College Paul McCombs, Rock Valley College Linda McGuire, Muhlenberg College Amanda Nutt, Western Kentucky University Jacquelyn O’Donohoe, Plymouth State University Daniel Pinzon, Georgia Gwinnett College Stacy Reagan, Caldwell Community College David Shannon, Transylvania University Sharon Sullivan, Catawba College Susan Toma, Madonna University Kim Y. Ward, Eastern Connecticut State University Weicheng Xuan, Arizona Western College We owe our appreciation to the people at W. H. Freeman and Company who participated in the development and production of this edition. We

wish especially to thank the editorial staff for their tireless efforts and support. Among them are Terri Ward, Publisher; Leslie Lahr and Jorge Amaral, Development Editors; Laura Judge, Executive Media Editor; Marie Dripchak, Associate Editor; Victoria Garvey, Editorial Assistant; Vivien Weiss, Senior Project Editor; Susan Wein, Production Supervisor; Matthew McAdams, Art Manager; Blake Logan, Cover Designer; Cecilia Varas and Nick Ciani, Photo Editors; Jennifer MacMillan, Felicia Ruocco, and Hilary Newman, Permissions; Patti Brecht and Laura Cooney, copyeditors; Edward Dionne, Project Manager, MPS Limited.

We’d also like to give special thanks to the authors of the ancillaries to accompany this tenth edition: Lauren Fern, University of Montana, Instructor’s Guide with Full Solutions and Student Solutions Manual; Doug Greiner, Rogers State University, Teaching Guide for First-Time Instructors and Student Study Guide; Hee Seok Nam, Washburn University, Practice Quizzes and Test Bank; Kathy Rodgers, University of Southern Indiana, Clicker Questions and Lecture Slides.

Through ten editions, this text has been used by well over a million students. When we first suggested our new approach, we were turned down by every major (and minor) textbook publisher. Only W. H. Freeman, under the leadership of Linda Chaput and the faith of mathematics editor Jerry Lyons, was willing to take a chance. That chance has permanently changed the face of introductory undergraduate mathematics. Words cannot express the gratitude we feel for the staffs of W. H. Freeman and COMAP and for the authors past and present through these almost 30 years. To them and everyone who made our purposes practical, we offer our appreciation for an exciting and exhilarating ride.

Solomon Garfunkel, COMAP

xix

this page left intentionally blank

FOR ALL PRACTICAL PURPOSES

Part I

Management Science Chapter 1

Urban Services Chapter 2

Business Efficiency Chapter 3

Planning and Scheduling Chapter 4

Linear Programming and the Transportation Problem Getting through a typical day can be a challenge: getting to or from school or your job on time; finding a parking spot when you are late for a date; making sure you have food in the refrigerator or getting to your favorite fast-food restaurant to stay properly fed and “fueled” with coffee; making sure your body is fit by getting to the gym or exercising at home; and seeing the doctor for a regular checkup or when you are ill. While your personal life may seem complex, consider what goes on at any of the large, modern medical centers found across America, in cities and on university campuses. In a typical 24-hour period, babies will be born; people will die; a friend may need an emergency appendectomy; and a relative may need treatment for a heart attack, asthma attack, stroke, or puncture wound. And there are the more mundane things: blood tests, mammograms, garden variety X-rays, CT and MRI scans, elective surgery to remove a cyst, or a routine checkup. Medical centers need many kinds of workers to make them hum: doctors and nurses with different specialties, technicians of different kinds, workers who pay attention to the “business” aspects of the medical center—not to mention people who prepare food for the patients, clean the premises, or plow the access roads after a snowstorm. day-out demands, there must be specialists (doctors, nurses, technicians, etc.) and other workers either physically present or available by phone. The part of mathematics concerned with efficient operations of businesses and governments is called operations research (OR) or management science. The domain of OR includes resource allocation, scheduling, queues (waiting lines), inventory analysis, routing problems, and cost minimization, to mention but a few of OR’s growing areas of applicability. Medical centers rely on the expertise of mathematical specialists to make their operations run smoothly day and night, rain or shine, winter or summer, so that when you or your loved ones need to use the center, its services are there for you.

Chalk up another triumph for OR! What follows will help you, too, to know about and use such tools.

3 3

Robert Daly/Caiaimage/OJO+/Getty Images

So what does this have to do with mathematics? To deal with the emergencies and day-in and

this page left intentionally blank

1 Mitchell Funk/Getty Images

Urban Services

1.1 1.2 1.3 1.4

Euler Circuits Finding Euler Circuits Beyond Euler Circuits Urban Graph Traversal Problems

T

he underlying theme of management science, also called operations research (OR), is finding the best method for solving some problem— what mathematicians call the optimal solution. In some cases, the goal may be to finish a job or get somewhere as quickly as possible.

In other situations, the objective might be to maximize profit or minimize cost. In this chapter, our goal is to save time (and usually taxpayer money) in traversing a street network while providing services such as checking parking meters, collecting garbage or bottles for recycling, de-icing roads, inspecting for potholes or gas leaks, or delivering packages or mail.

Let’s begin by assisting the parking department of a city government.

Most cities and many small towns have parking meters that must be regularly checked for parking violations or emptied of coins. We will use an imaginary town to show how management science techniques can help to make parking control more efficient.

5

6

C h a p t e r

1

|

Urban Services

1.1 Euler Circuits The street map in Figure 1.1 is typical of many villages and cities across the United States, with streets, residential blocks, and a town park. Our job, or that of the commissioner of parking, is to find the most efficient route for the parking-control officer, who travels on foot, to check the meters in an area. Efficient routes save money. Our map shows only a small area, allowing us to start with an easy problem. But the problem occurs on a larger scale in all cities and towns and for larger areas. The bigger the region involved, the greater the potential for cost savings.

Figure 1.1 A street map for part of a town.

The commissioner has two goals in mind: (1) The parking-control officer must cover all the sidewalks that have parking meters without retracing any more steps than are necessary; and (2) the route should start and end at the same location, perhaps where the officer’s vehicle is parked. To be specific, suppose there are only two blocks that have parking meters, the two lightly shaded blocks that are side by side toward the top of Figure 1.1. Suppose further that the parking-control officer must start and end at the upper left corner of the left-hand block. You might enjoy working out some routes by trial and error and evaluating their good and bad features. We are going to leave this problem for the moment and establish some concepts that will give us a better method than trial and error to deal with this kind of problem.

What Is a Graph?

de fI n I TI O n

A graph is a finite set of dots and connecting curved or straight links. The dots are called vertices (a single dot is called a vertex), and the links are called edges.

Each edge must connect two different vertices. A graph can represent a city map, a social network, a system of air routes, or electrical power lines.

Path and Circuit

de fI n I TI O n

A path is a connected sequence of edges showing a route on the graph that starts at a vertex and ends at a vertex; a path is usually described by naming in turn the vertices visited in traversing it. A path that starts and ends at the same vertex is called a circuit.

1.1

EXAMPLE 1

Euler Circuits

7

Parts of a Graph

We can use the graph in figure 1.2 to help explain these technical terms. The graph shown has five vertices and eight edges. The vertices represent cities, and the edges represent nonstop airline routes between them. We see that there is a nonstop flight between Berlin and Rome, but no such flight between new York and Berlin. There are several paths that describe how a person might travel with this airline from new York to Rome. The path that seems most direct is new York, London, Rome. But new York, Miami, Rome is also a path with only one “stop.” furthermore, new York, London, Berlin, Rome is a path. We can describe these three paths as NLR, NMR, NLBR. Another path would be new York, Miami, London, Berlin, Rome, which can be written NMLBR. An example of a circuit is Miami, Rome, London, Miami. It is a circuit because the path starts and ends at the same vertex. This circuit can best be described in symbols by MRLM. Another example of a circuit in this graph would be LRBL, which is the circuit involving the cities London, Rome, Berlin, and back to London. In this chapter, we are especially interested in circuits, just as we are in real life. Most of us end our day in the same place that we start it—at home!

New York (N)

London (L)

New York (N)

London (L)

Berlin (B) Rome (R)

Miami (M)

Rome (R) (a)

Miami (M) (b)

Figure 1.2 (a) The edges of the graph show nonstop routes that an airline might offer. (b) The graph in (a) redrawn without the accidental crossing. notice that the edges MB (which could also be denoted BM) and RL shown in figure 1.2a meet at a point that has no label. furthermore, this point does not have a dark dot. This is because this point does not represent a vertex of our graph; it does not represent a city. It arises as an “accidental” consequence of the way this diagram has been drawn. We could join M and B with a curved line segment so that the edges LR and MB do not cross, or redraw the diagram so as to avoid a crossing in this case. We will be working often in situations where graphs can be drawn without accidental crossings, and we will try to avoid such crossings when it is convenient to do so. However, there are infinitely many graphs for which—when they are drawn on a flat piece of paper—accidental crossings are unavoidable. (figure 2.12 on page 52 is an example of such a graph.)

Returning to the case of parking enforcement in Figure 1.1, we can use a graph to represent the whole territory to be patrolled: Think of each street intersection as a vertex and each sidewalk that contains meters as an edge, as in Figure 1.3. Notice in Figure 1.3b that the width of the street separating the blocks is not explicitly represented; it has been shrunk to nothing. In effect, we are simplifying our problem by ignoring any distance traveled in crossing streets. In drawing graph diagrams such as those in Figure 1.3 or Figure 1.5, we usually use straight line segments to

Berlin (B)

8

C h a p t e r

|

1

Urban Services

Figure 1.3 (a) A graph superimposed upon a street map. The edges show which sidewalks have parking meters. (b) The same graph enlarged.

A

B

C

D

E

F

(b)

(a)

represent edges. However, sometimes we cannot avoid the use of “curves,” or we may prefer to use curved edges because they convey aspects of the original problem that we desire to emphasize.

The sequence of numbered edges in Figure 1.4a shows one circuit that covers all the meters. (Note that it is a circuit because its path returns to its starting point.)

However, one edge is traversed three times. Figure 1.4b shows another solution that is better because its circuit covers every edge (sidewalk) exactly once. In Figure 1.4b, no edge is covered more than once, or deadheaded (a term borrowed from shipping, which means making a return trip without a load). When deadheading is required in an applied situation, such as inspecting parking meters or pothole inspection, typically time and effort is being spent but no productive work is accomplished because the productive work was done the first time the edge was covered (traversed).

This is the first edge covered 5

1 10

4

3

1

2

6

(a)

Figure 1.4 (a) A circuit and (b) an euler circuit.

Euler Circuit

9 7

8

6

8

while this is the eighth.

2

7

3

5 (b)

4

de fI n I TI O n

A circuit that covers each edge of a graph once, but not more than once, is called an Euler circuit.

Figure 1.4b shows an Euler circuit. These circuits get their name from the great Swiss-born mathematician Leonhard Euler (pronounced oy’ lur), who first studied them (see Spotlight 1.1). Euler was the founder of the theory of graphs, or graph theory. One of his first discoveries was that some graphs have no Euler circuits at all.

1.1

S

tLIGh

1.1

to other scholars of his day are still being published. euler invented the idea of a graph in 1736 when he solved a problem in “recreational mathematics.”

He showed that it was impossible to stroll a route visiting the seven bridges of the German town of Königsberg Leonhard Euler exactly once. Ironically, in 1752 he discovered that three-dimensional polyhedra obey the remarkable formula V 2 E 1 F 5 2 (that is, number of vertices 2 number of edges 1 number of faces 5 2) but failed to give a proof because he did not analyze the situation using graph theory methods. Sometimes even a genius can miss something.

Jamie Abecasis/imageb/age fotostock

Leonhard euler (1707–1783) was remarkable in many ways. He was extremely prolific, publishing over 500 works in his lifetime. But he wasn’t devoted just to mathematics; he was a people person, too. He was extremely fond of children and had 13 of his own, of whom only five survived childhood. It is said that he often wrote difficult mathematical works with a child or two in his lap.

Human interest stories about euler have been handed down through three centuries. He was a prodigy at doing complex mathematical calculations under less than ideal conditions, and he continued to do them even after he became totally blind later in life. His blindness diminished neither the quantity nor the quality of his output. Throughout his life, he was able to mentally calculate in a short time what would have taken ordinary mathematicians hours of pencil-and-paper work. A contemporary claimed that euler could calculate effortlessly, “just as men breathe, as eagles sustain themselves in the air.” His collected works and numerous letters

pO

t

Leonhard Euler

9

Euler Circuits

For example, in the graph in Figure 1.5b, it would be impossible to start at one point, return to that starting vertex and cover all the edges without retracing some steps: If we try to start a circuit at the leftmost vertex, we discover that once we have left the vertex, we have “used up” the only edge meeting it. We have no way to return to our starting point except to reuse that edge. But this is not allowed in an Euler circuit. If we try to start a circuit at one of the other two vertices, we likewise can’t complete it to form an Euler circuit.

Figure 1.5 (a) The three

(a)

(b)

As mentioned in Spotlight 1.2, realistic problems of this type involve larger neighborhoods that might require the use of a computer. In addition, there may be other complications that might take us beyond the simple mathematics we want to stick to.

shaded sidewalks cannot be covered by an euler circuit. (b) The graph of the shaded sidewalks in part (a).

C h a p t e r

1

|

Urban Services

Thomas Magnanti, professor of operations research and management science, is former dean of engineering at MIT. Here are some of his observations: Typically, a management science approach has several different ingredients. One is just structuring the problem—understanding that the problem is an euler circuit problem or a related management science problem. After that, one has to develop the solution methods.

But one should also recognize that you don’t just push a button and get the answer. In using these underlying mathematical tools, we never want to lose sight of our common sense, understanding, intuition, and judgment. The computer provides certain kinds of insights. It deals with some of the combinatorial complexities of these problems very nicely.

S

pO

tLIGh

1.2

t

The Human Aspect of Problem Solving

But a model such as an euler circuit can never capture the full essence of a decisionmaking problem.

Typically, when we solve the mathematical problem, we see that it doesn’t quite correspond to the real problem we want to Thomas Magnanti solve. So we make modifications in the underlying model. It is an interactive approach, using the best of what computers and mathematics have to offer and the best of what we, as human beings, with our own decision-making capabilities, have to offer.

Courtesy of Thomas Magnanti

10

Because we are interested in finding circuits, and Euler circuits are the most efficient ones, we want to know how to find them. If a graph has no Euler circuit, we want to develop efficient, alternative tours, those having minimum deadheading. These topics make up the rest of this chapter.

1.2 Finding Euler Circuits Now that we know what an Euler circuit is, we are faced with two obvious questions:

1. Is there a way to tell by calculation or logical reasoning, not by trial and error, if a graph has an Euler circuit?

2. Is there a method, other than trial and error, for finding an Euler circuit when one exists and finding it quickly?

E (1)

B (2)

A (4)

C (0)

D (3)

The valence is 3 because three edges meet at D. Figure 1.6 Valences of vertices.

Loosely speaking, the first question lies within the concerns of mathematicians because it asks whether or not a certain problem admits a solution. Typically, the second question lies in the domain of computer science because it concerns finding the actual answer to a complex version of a problem in a short enough time to be useful.

Euler investigated these questions in 1735 by using the concepts of valence and connectedness.

Valence

d ef I nI T I O n

The valence of a vertex in a graph is the number of edges meeting at the vertex.

Figure 1.6 illustrates the concept of valence, with vertex A having valence 4, vertex D having valence 3, vertex B having valence 2, and vertex C having valence

1.2

Finding Euler Circuits

11

0. Vertex E has valence 1. Isolated vertices such as vertex C are an annoyance in Euler circuit theory. Because they don’t occur in typical applications, we henceforth assume that our graphs have no vertices of valence 0.

Figure 1.3b has four vertices of valence 2, namely, A, C, F, and D. This graph also has two vertices, B and E, of valence 4. Notice that each vertex has a valence that is an even number. We’ll soon see that this is very significant.

Connected Graph

de fI n I TI O n

A graph is said to be connected if for every pair of its vertices, there is at least one path connecting the two vertices.

Given a graph, if we can find even one pair of vertices not connected by a path, then we say that the graph is not connected. For example, the graphs in Figure 1.6 and Figure 1.7 are not connected because we are unable to join A to C with a path of edges. However, the graph in Figure 1.7 does consist of two “pieces” or connected components, one containing the vertices A, B, F, and G, the other containing C, D, and E. A connected graph contains a single connected component.

Notice that the parking-control graph of Figure 1.3b is connected.

A

B

F

G D

C

Figure 1.7 A graph that is not connected.

E

We can now state Euler’s theorem, his simple answer to the problem of detecting when a graph G has an Euler circuit.

Euler Circuit Theorem

TH eO R eM

1. If G is connected and has all valences even, then G has an Euler circuit.

2. Conversely, if G has an Euler circuit, then G must be connected and all its

valences must be even numbers.

Because the parking-control graph of Figure 1.3b conforms to the connectedness and even-valence conditions, Euler’s theorem tells us that it has an Euler circuit. We already have found an Euler circuit for the graph shown in Figure 1.4b by trial and error. For a very large graph, however, trial and error may take a long time.

It is usually quicker to check whether the graph is connected and even-valent as a way to be sure an Euler circuit must exist rather than produce a specific Euler circuit.

Once we know there is an Euler circuit in a certain graph, how do we find it?

Many people find that, after a little practice, they can find Euler circuits by trial and error, and they don’t need detailed instructions on how to proceed. At this point you should see if you can develop this skill by trying to find Euler circuits in Figure 1.8a, Figure 1.9a, and Figure 1.10. In doing your experiments, draw your graph in ink and the circuit in pencil so you can erase if necessary.

12

C h a p t e r

1

|

Urban Services

Figure 1.8 (a) A connected graph having (b) an euler circuit.

5 6

2

3 4

1 7

8

10 11

9 13

14

12

Start here (b)

(a) All the valences are even, so this connected graph has an Euler circuit.

A 1

(a)

19

25 24

21 18

C 3

E

D (b)

10

26 20

2

B

22 17

1 23 15

16

2

11 3

14

12 13 4

9 8 7

5

If we move up at this point, after step 4, we will be unable to reach three edges without repeating edges.

6

(c)

Figure 1.9 (a) A graph that has an euler circuit. (b) A critical junction in finding an euler circuit in this graph, starting from vertex A. (c) A description of a full euler circuit for this graph.

A D H G

C

B E

F I

J

K

M L Figure 1.10 A graph with an euler circuit.

If you would like more guidance on how to find an Euler circuit without trial and error, here is a method that works: Never use an edge that is the only link between two parts of the graph that still need to be covered. Figure 1.9b illustrates this. Here we have started the circuit at A and gotten to D via B and C, and we want to know what to do next. Going to E would be a bad idea because the uncovered part of the graph would then be disconnected into left and right portions. You will never be able to get from the left part back to the right part because you have just used the last remaining link between these parts.

Therefore, you should stay on the right side and finish that before using the edge from D to E. This kind of thinking needs to be applied every time you need to choose a new edge.

1.2

Finding Euler Circuits

13

Let’s see how this works in Figure 1.9, starting at A. From vertex A there are two possible edges, and neither of them disconnects the unused portion of the graph. Thus, we could have gone either to the left or down. Having gone down to B, we now have three choices, none of which disconnects the unused part of the graph. After choosing to go from B to C, we find that any of the three choices at C is acceptable. Can you complete the Euler circuit? Figure 1.9c shows one of many ways to do this.

The method just described leaves many edge choices up to you. When there are many acceptable edges for your next step, you can pick one at random.

EXAMPLE 2

Finding an Euler Circuit

Check the valences of the vertices and the connectivity of the graph in figure 1.8a to verify that the graph does have an euler circuit. now try to find an euler circuit for that graph. You can start at any vertex. When you are done, compare your solution with the euler circuit given in figure 1.8b. If your path covers each edge exactly once and returns to its original vertex (which is a circuit), then it is an euler circuit, even if it is not the same as the one we give.

Understanding Euler’s Theorem We start by showing that if a graph has an Euler circuit R, then it must have only even-valent vertices and it must be connected. Let X be any vertex of the graph.

We will show that the edges at X can be paired up, and this will prove that the valence is even. Every edge at X is used by R as an outgoing edge (leaving from X) or an incoming edge (arriving at X). If the Euler circuit starts at X, then pair up the first edge used by R with the last one (when the circuit returns to X for the last time). In addition, each other edge at X that is used by the circuit as an incoming edge will be paired with the outgoing edge that is used next. Because all edges at X are used by the Euler circuit, none more than once, this pairs up the edges. Hence, X must be even-valent because we have “organized“ the edges of R in pairs.

But what if X is not the start of the Euler circuit? If that is the case, do the pairing like this: The first incoming edge at X is paired with the outgoing one used next, the second incoming edge at X is paired with the outgoing one used next, and so on. For example, in Figure 1.11 at vertex B, we would pair up edges 2 and 3 and edges 9 and 10. At vertex C, we would pair up edges 4 and 5 and edges 8 and 9.

Can you see how the pairings would work at D? How about vertex A? Figure 1.11 An euler

10

C

B

9

A

1

5 8

2

11

circuit starting and ending at A.

4

3

6

D 7

14

C h a p t e r

1

|

Urban Services

In studying this example, you might think it would be simpler to count the edges at a vertex to see that the valence is even. True, but our pairing method works for a graph about which we know nothing except that it has an Euler circuit.

To see that a graph with an Euler circuit is connected, note that by following the Euler circuit around we can get from any edge to any other edge (it covers them all) using a portion of the Euler circuit. Because every vertex is on an edge (there are no vertices of valence 0), we can get from any vertex to any other using a portion of the Euler circuit.

So far, this is not a complete proof of Euler’s theorem. To complete the proof, we would need to prove that if a graph has all vertices even-valent and is connected, then an Euler circuit can be found for it. One way to do this is to piece together shorter circuits in the even-valent connected graph to get circuits using more edges.

Self Check 1 (a) How many vertices and edges does the graph in Figure 1.12b have? (b) List the vertices in the graph that are even-valent (have even numbers for

their valences). (c) Is FEABFGCDHGF an Euler circuit for the graph?

■

1.3 Beyond Euler Circuits Now let’s see what Euler’s theorem tells us about the three-block neighborhood with parking meters, represented by dots in Figure 1.12a. Figure 1.12b shows the corresponding graph. (Because we use edges to represent only sidewalks along which the officer must walk, the sidewalk with no meters is not represented by any edge in the graph.) This graph has vertices with odd valences (at vertices C and G), so Euler’s theorem tells us that there is no Euler circuit for this graph.

A

B

B

E

F

F

(a)

C

C

D

A

B

G

G

H

E

F

(b)

C

D

G

H

Figure 1.12 (a) A street network and (b) its graphic representation. Locations such as B’ and B”, C’ and C”, F’ and F”, and G’ and G” are merged to form the vertices B, C, F, and G. The dots shown represent parking meters.

Because we must reuse some edges in this graph to cover all edges in a circuit, for efficiency we need to keep the total length of reused edges to a minimum. Note that we are still looking for a tour that starts and ends at the same vertex. This type of problem, in which we want to minimize the length of a circuit by carefully choosing which edges to retrace, is often called the Chinese postman problem.

(Like parking-control routes, mail routes need to be efficient.) The problem was

Another random document with no related content on Scribd:

DANCE ON STILTS AT THE GIRLS’ UNYAGO, NIUCHI

Newala, too, suffers from the distance of its water-supply—at least the Newala of to-day does; there was once another Newala in a lovely valley at the foot of the plateau. I visited it and found scarcely a trace of houses, only a Christian cemetery, with the graves of several missionaries and their converts, remaining as a monument of its former glories. But the surroundings are wonderfully beautiful. A thick grove of splendid mango-trees closes in the weather-worn crosses and headstones; behind them, combining the useful and the agreeable, is a whole plantation of lemon-trees covered with ripe fruit; not the small African kind, but a much larger and also juicier imported variety, which drops into the hands of the passing traveller, without calling for any exertion on his part. Old Newala is now under the jurisdiction of the native pastor, Daudi, at Chingulungulu, who, as I am on very friendly terms with him, allows me, as a matter of course, the use of this lemon-grove during my stay at Newala.

FEET MUTILATED BY THE RAVAGES OF THE “JIGGER” (Sarcopsylla penetrans)

The water-supply of New Newala is in the bottom of the valley, some 1,600 feet lower down. The way is not only long and fatiguing, but the water, when we get it, is thoroughly bad. We are suffering not only from this, but from the fact that the arrangements at Newala are nothing short of luxurious. We have a separate kitchen—a hut built against the boma palisade on the right of the baraza, the interior of which is not visible from our usual position. Our two cooks were not long in finding this out, and they consequently do—or rather neglect to do—what they please. In any case they do not seem to be very particular about the boiling of our drinking-water—at least I can attribute to no other cause certain attacks of a dysenteric nature, from which both Knudsen and I have suffered for some time. If a man like Omari has to be left unwatched for a moment, he is capable of anything. Besides this complaint, we are inconvenienced by the state of our nails, which have become as hard as glass, and crack on the slightest provocation, and I have the additional infliction of pimples all over me. As if all this were not enough, we have also, for the last week been waging war against the jigger, who has found his Eldorado in the hot sand of the Makonde plateau. Our men are seen all day long—whenever their chronic colds and the dysentery likewise raging among them permit—occupied in removing this scourge of Africa from their feet and trying to prevent the disastrous consequences of its presence. It is quite common to see natives of this place with one or two toes missing; many have lost all their toes,

or even the whole front part of the foot, so that a well-formed leg ends in a shapeless stump. These ravages are caused by the female of Sarcopsylla penetrans, which bores its way under the skin and there develops an egg-sac the size of a pea. In all books on the subject, it is stated that one’s attention is called to the presence of this parasite by an intolerable itching. This agrees very well with my experience, so far as the softer parts of the sole, the spaces between and under the toes, and the side of the foot are concerned, but if the creature penetrates through the harder parts of the heel or ball of the foot, it may escape even the most careful search till it has reached maturity.

Then there is no time to be lost, if the horrible ulceration, of which we see cases by the dozen every day, is to be prevented. It is much easier, by the way, to discover the insect on the white skin of a European than on that of a native, on which the dark speck scarcely shows. The four or five jiggers which, in spite of the fact that I constantly wore high laced boots, chose my feet to settle in, were taken out for me by the all-accomplished Knudsen, after which I thought it advisable to wash out the cavities with corrosive sublimate. The natives have a different sort of disinfectant—they fill the hole with scraped roots. In a tiny Makua village on the slope of the plateau south of Newala, we saw an old woman who had filled all the spaces under her toe-nails with powdered roots by way of prophylactic treatment. What will be the result, if any, who can say?

The rest of the many trifling ills which trouble our existence are really more comic than serious. In the absence of anything else to smoke, Knudsen and I at last opened a box of cigars procured from the Indian store-keeper at Lindi, and tried them, with the most distressing results. Whether they contain opium or some other narcotic, neither of us can say, but after the tenth puff we were both “off,” three-quarters stupefied and unspeakably wretched. Slowly we recovered—and what happened next? Half-an-hour later we were once more smoking these poisonous concoctions—so insatiable is the craving for tobacco in the tropics.

Even my present attacks of fever scarcely deserve to be taken seriously. I have had no less than three here at Newala, all of which have run their course in an incredibly short time. In the early afternoon, I am busy with my old natives, asking questions and making notes. The strong midday coffee has stimulated my spirits to

an extraordinary degree, the brain is active and vigorous, and work progresses rapidly, while a pleasant warmth pervades the whole body. Suddenly this gives place to a violent chill, forcing me to put on my overcoat, though it is only half-past three and the afternoon sun is at its hottest. Now the brain no longer works with such acuteness and logical precision; more especially does it fail me in trying to establish the syntax of the difficult Makua language on which I have ventured, as if I had not enough to do without it. Under the circumstances it seems advisable to take my temperature, and I do so, to save trouble, without leaving my seat, and while going on with my work. On examination, I find it to be 101·48°. My tutors are abruptly dismissed and my bed set up in the baraza; a few minutes later I am in it and treating myself internally with hot water and lemon-juice.

Three hours later, the thermometer marks nearly 104°, and I make them carry me back into the tent, bed and all, as I am now perspiring heavily, and exposure to the cold wind just beginning to blow might mean a fatal chill. I lie still for a little while, and then find, to my great relief, that the temperature is not rising, but rather falling. This is about 7.30 p.m. At 8 p.m. I find, to my unbounded astonishment, that it has fallen below 98·6°, and I feel perfectly well. I read for an hour or two, and could very well enjoy a smoke, if I had the wherewithal—Indian cigars being out of the question.

Having no medical training, I am at a loss to account for this state of things. It is impossible that these transitory attacks of high fever should be malarial; it seems more probable that they are due to a kind of sunstroke. On consulting my note-book, I become more and more inclined to think this is the case, for these attacks regularly follow extreme fatigue and long exposure to strong sunshine. They at least have the advantage of being only short interruptions to my work, as on the following morning I am always quite fresh and fit.

My treasure of a cook is suffering from an enormous hydrocele which makes it difficult for him to get up, and Moritz is obliged to keep in the dark on account of his inflamed eyes. Knudsen’s cook, a raw boy from somewhere in the bush, knows still less of cooking than Omari; consequently Nils Knudsen himself has been promoted to the vacant post. Finding that we had come to the end of our supplies, he began by sending to Chingulungulu for the four sucking-pigs which we had

bought from Matola and temporarily left in his charge; and when they came up, neatly packed in a large crate, he callously slaughtered the biggest of them. The first joint we were thoughtless enough to entrust for roasting to Knudsen’s mshenzi cook, and it was consequently uneatable; but we made the rest of the animal into a jelly which we ate with great relish after weeks of underfeeding, consuming incredible helpings of it at both midday and evening meals. The only drawback is a certain want of variety in the tinned vegetables. Dr. Jäger, to whom the Geographical Commission entrusted the provisioning of the expeditions—mine as well as his own—because he had more time on his hands than the rest of us, seems to have laid in a huge stock of Teltow turnips,[46] an article of food which is all very well for occasional use, but which quickly palls when set before one every day; and we seem to have no other tins left. There is no help for it—we must put up with the turnips; but I am certain that, once I am home again, I shall not touch them for ten years to come.

Amid all these minor evils, which, after all, go to make up the genuine flavour of Africa, there is at least one cheering touch: Knudsen has, with the dexterity of a skilled mechanic, repaired my 9 × 12 cm. camera, at least so far that I can use it with a little care.

How, in the absence of finger-nails, he was able to accomplish such a ticklish piece of work, having no tool but a clumsy screw-driver for taking to pieces and putting together again the complicated mechanism of the instantaneous shutter, is still a mystery to me; but he did it successfully. The loss of his finger-nails shows him in a light contrasting curiously enough with the intelligence evinced by the above operation; though, after all, it is scarcely surprising after his ten years’ residence in the bush. One day, at Lindi, he had occasion to wash a dog, which must have been in need of very thorough cleansing, for the bottle handed to our friend for the purpose had an extremely strong smell. Having performed his task in the most conscientious manner, he perceived with some surprise that the dog did not appear much the better for it, and was further surprised by finding his own nails ulcerating away in the course of the next few days. “How was I to know that carbolic acid has to be diluted?” he mutters indignantly, from time to time, with a troubled gaze at his mutilated finger-tips.

Since we came to Newala we have been making excursions in all directions through the surrounding country, in accordance with old habit, and also because the akida Sefu did not get together the tribal elders from whom I wanted information so speedily as he had promised. There is, however, no harm done, as, even if seen only from the outside, the country and people are interesting enough.

The Makonde plateau is like a large rectangular table rounded off at the corners. Measured from the Indian Ocean to Newala, it is about seventy-five miles long, and between the Rovuma and the Lukuledi it averages fifty miles in breadth, so that its superficial area is about two-thirds of that of the kingdom of Saxony. The surface, however, is not level, but uniformly inclined from its south-western edge to the ocean. From the upper edge, on which Newala lies, the eye ranges for many miles east and north-east, without encountering any obstacle, over the Makonde bush. It is a green sea, from which here and there thick clouds of smoke rise, to show that it, too, is inhabited by men who carry on their tillage like so many other primitive peoples, by cutting down and burning the bush, and manuring with the ashes. Even in the radiant light of a tropical day such a fire is a grand sight.

Much less effective is the impression produced just now by the great western plain as seen from the edge of the plateau. As often as time permits, I stroll along this edge, sometimes in one direction, sometimes in another, in the hope of finding the air clear enough to let me enjoy the view; but I have always been disappointed.

Wherever one looks, clouds of smoke rise from the burning bush, and the air is full of smoke and vapour. It is a pity, for under more favourable circumstances the panorama of the whole country up to the distant Majeje hills must be truly magnificent. It is of little use taking photographs now, and an outline sketch gives a very poor idea of the scenery. In one of these excursions I went out of my way to make a personal attempt on the Makonde bush. The present edge of the plateau is the result of a far-reaching process of destruction through erosion and denudation. The Makonde strata are everywhere cut into by ravines, which, though short, are hundreds of yards in depth. In consequence of the loose stratification of these beds, not only are the walls of these ravines nearly vertical, but their upper end is closed by an equally steep escarpment, so that the

western edge of the Makonde plateau is hemmed in by a series of deep, basin-like valleys. In order to get from one side of such a ravine to the other, I cut my way through the bush with a dozen of my men.

It was a very open part, with more grass than scrub, but even so the short stretch of less than two hundred yards was very hard work; at the end of it the men’s calicoes were in rags and they themselves bleeding from hundreds of scratches, while even our strong khaki suits had not escaped scatheless.

NATIVE PATH THROUGH THE MAKONDE BUSH, NEAR MAHUTA

I see increasing reason to believe that the view formed some time back as to the origin of the Makonde bush is the correct one. I have no doubt that it is not a natural product, but the result of human occupation. Those parts of the high country where man—as a very slight amount of practice enables the eye to perceive at once—has not yet penetrated with axe and hoe, are still occupied by a splendid timber forest quite able to sustain a comparison with our mixed forests in Germany. But wherever man has once built his hut or tilled his field, this horrible bush springs up. Every phase of this process

may be seen in the course of a couple of hours’ walk along the main road. From the bush to right or left, one hears the sound of the axe— not from one spot only, but from several directions at once. A few steps further on, we can see what is taking place. The brush has been cut down and piled up in heaps to the height of a yard or more, between which the trunks of the large trees stand up like the last pillars of a magnificent ruined building. These, too, present a melancholy spectacle: the destructive Makonde have ringed them— cut a broad strip of bark all round to ensure their dying off—and also piled up pyramids of brush round them. Father and son, mother and son-in-law, are chopping away perseveringly in the background—too busy, almost, to look round at the white stranger, who usually excites so much interest. If you pass by the same place a week later, the piles of brushwood have disappeared and a thick layer of ashes has taken the place of the green forest. The large trees stretch their smouldering trunks and branches in dumb accusation to heaven—if they have not already fallen and been more or less reduced to ashes, perhaps only showing as a white stripe on the dark ground.

This work of destruction is carried out by the Makonde alike on the virgin forest and on the bush which has sprung up on sites already cultivated and deserted. In the second case they are saved the trouble of burning the large trees, these being entirely absent in the secondary bush.

After burning this piece of forest ground and loosening it with the hoe, the native sows his corn and plants his vegetables. All over the country, he goes in for bed-culture, which requires, and, in fact, receives, the most careful attention. Weeds are nowhere tolerated in the south of German East Africa. The crops may fail on the plains, where droughts are frequent, but never on the plateau with its abundant rains and heavy dews. Its fortunate inhabitants even have the satisfaction of seeing the proud Wayao and Wamakua working for them as labourers, driven by hunger to serve where they were accustomed to rule.

But the light, sandy soil is soon exhausted, and would yield no harvest the second year if cultivated twice running. This fact has been familiar to the native for ages; consequently he provides in time, and, while his crop is growing, prepares the next plot with axe and firebrand. Next year he plants this with his various crops and

lets the first piece lie fallow. For a short time it remains waste and desolate; then nature steps in to repair the destruction wrought by man; a thousand new growths spring out of the exhausted soil, and even the old stumps put forth fresh shoots. Next year the new growth is up to one’s knees, and in a few years more it is that terrible, impenetrable bush, which maintains its position till the black occupier of the land has made the round of all the available sites and come back to his starting point.

The Makonde are, body and soul, so to speak, one with this bush.

According to my Yao informants, indeed, their name means nothing else but “bush people.” Their own tradition says that they have been settled up here for a very long time, but to my surprise they laid great stress on an original immigration. Their old homes were in the south-east, near Mikindani and the mouth of the Rovuma, whence their peaceful forefathers were driven by the continual raids of the Sakalavas from Madagascar and the warlike Shirazis[47] of the coast, to take refuge on the almost inaccessible plateau. I have studied African ethnology for twenty years, but the fact that changes of population in this apparently quiet and peaceable corner of the earth could have been occasioned by outside enterprises taking place on the high seas, was completely new to me. It is, no doubt, however, correct.

The charming tribal legend of the Makonde—besides informing us of other interesting matters—explains why they have to live in the thickest of the bush and a long way from the edge of the plateau, instead of making their permanent homes beside the purling brooks and springs of the low country.

“The place where the tribe originated is Mahuta, on the southern side of the plateau towards the Rovuma, where of old time there was nothing but thick bush. Out of this bush came a man who never washed himself or shaved his head, and who ate and drank but little.

He went out and made a human figure from the wood of a tree growing in the open country, which he took home to his abode in the bush and there set it upright. In the night this image came to life and was a woman. The man and woman went down together to the Rovuma to wash themselves. Here the woman gave birth to a stillborn child. They left that place and passed over the high land into the valley of the Mbemkuru, where the woman had another child, which

was also born dead. Then they returned to the high bush country of Mahuta, where the third child was born, which lived and grew up. In course of time, the couple had many more children, and called themselves Wamatanda. These were the ancestral stock of the Makonde, also called Wamakonde,[48] i.e., aborigines. Their forefather, the man from the bush, gave his children the command to bury their dead upright, in memory of the mother of their race who was cut out of wood and awoke to life when standing upright. He also warned them against settling in the valleys and near large streams, for sickness and death dwelt there. They were to make it a rule to have their huts at least an hour’s walk from the nearest wateringplace; then their children would thrive and escape illness.”

The explanation of the name Makonde given by my informants is somewhat different from that contained in the above legend, which I extract from a little book (small, but packed with information), by Pater Adams, entitled Lindi und sein Hinterland. Otherwise, my results agree exactly with the statements of the legend. Washing?

Hapana—there is no such thing. Why should they do so? As it is, the supply of water scarcely suffices for cooking and drinking; other people do not wash, so why should the Makonde distinguish himself by such needless eccentricity? As for shaving the head, the short, woolly crop scarcely needs it,[49] so the second ancestral precept is likewise easy enough to follow. Beyond this, however, there is nothing ridiculous in the ancestor’s advice. I have obtained from various local artists a fairly large number of figures carved in wood, ranging from fifteen to twenty-three inches in height, and representing women belonging to the great group of the Mavia, Makonde, and Matambwe tribes. The carving is remarkably well done and renders the female type with great accuracy, especially the keloid ornamentation, to be described later on. As to the object and meaning of their works the sculptors either could or (more probably) would tell me nothing, and I was forced to content myself with the scanty information vouchsafed by one man, who said that the figures were merely intended to represent the nembo—the artificial deformations of pelele, ear-discs, and keloids. The legend recorded

by Pater Adams places these figures in a new light. They must surely be more than mere dolls; and we may even venture to assume that they are—though the majority of present-day Makonde are probably unaware of the fact—representations of the tribal ancestress.

The references in the legend to the descent from Mahuta to the Rovuma, and to a journey across the highlands into the Mbekuru valley, undoubtedly indicate the previous history of the tribe, the travels of the ancestral pair typifying the migrations of their descendants. The descent to the neighbouring Rovuma valley, with its extraordinary fertility and great abundance of game, is intelligible at a glance—but the crossing of the Lukuledi depression, the ascent to the Rondo Plateau and the descent to the Mbemkuru, also lie within the bounds of probability, for all these districts have exactly the same character as the extreme south. Now, however, comes a point of especial interest for our bacteriological age. The primitive Makonde did not enjoy their lives in the marshy river-valleys.

Disease raged among them, and many died. It was only after they had returned to their original home near Mahuta, that the health conditions of these people improved. We are very apt to think of the African as a stupid person whose ignorance of nature is only equalled by his fear of it, and who looks on all mishaps as caused by evil spirits and malignant natural powers. It is much more correct to assume in this case that the people very early learnt to distinguish districts infested with malaria from those where it is absent.

This knowledge is crystallized in the ancestral warning against settling in the valleys and near the great waters, the dwelling-places of disease and death. At the same time, for security against the hostile Mavia south of the Rovuma, it was enacted that every settlement must be not less than a certain distance from the southern edge of the plateau. Such in fact is their mode of life at the present day. It is not such a bad one, and certainly they are both safer and more comfortable than the Makua, the recent intruders from the south, who have made USUAL METHOD OF good their footing on the western edge of the CLOSING HUT-DOOR plateau, extending over a fairly wide belt of country. Neither Makua nor Makonde show in their dwellings anything of the size and comeliness of the Yao houses in the plain, especially at Masasi, Chingulungulu and Zuza’s. Jumbe Chauro, a Makonde hamlet not far from Newala, on the road to Mahuta, is the

most important settlement of the tribe I have yet seen, and has fairly spacious huts. But how slovenly is their construction compared with the palatial residences of the elephant-hunters living in the plain.

The roofs are still more untidy than in the general run of huts during the dry season, the walls show here and there the scanty beginnings or the lamentable remains of the mud plastering, and the interior is a veritable dog-kennel; dirt, dust and disorder everywhere. A few huts only show any attempt at division into rooms, and this consists merely of very roughly-made bamboo partitions. In one point alone have I noticed any indication of progress—in the method of fastening the door. Houses all over the south are secured in a simple but ingenious manner. The door consists of a set of stout pieces of wood or bamboo, tied with bark-string to two cross-pieces, and moving in two grooves round one of the door-posts, so as to open inwards. If the owner wishes to leave home, he takes two logs as thick as a man’s upper arm and about a yard long. One of these is placed obliquely against the middle of the door from the inside, so as to form an angle of from 60° to 75° with the ground. He then places the second piece horizontally across the first, pressing it downward with all his might.

It is kept in place by two strong posts planted in the ground a few inches inside the door. This fastening is absolutely safe, but of course cannot be applied to both doors at once, otherwise how could the owner leave or enter his house? I have not yet succeeded in finding out how the back door is fastened.

MAKONDE LOCK AND KEY AT JUMBE CHAURO

This is the general way of closing a house. The Makonde at Jumbe Chauro, however, have a much more complicated, solid and original one. Here, too, the door is as already described, except that there is only one post on the inside, standing by itself about six inches from one side of the doorway. Opposite this post is a hole in the wall just large enough to admit a man’s arm. The door is closed inside by a large wooden bolt passing through a hole in this post and pressing with its free end against the door. The other end has three holes into which fit three pegs running in vertical grooves inside the post. The door is opened with a wooden key about a foot long, somewhat curved and sloped off at the butt; the other end has three pegs corresponding to the holes, in the bolt, so that, when it is thrust through the hole in the wall and inserted into the rectangular opening in the post, the pegs can be lifted and the bolt drawn out.[50]

MODE OF INSERTING THE KEY

With no small pride first one householder and then a second showed me on the spot the action of this greatest invention of the Makonde Highlands. To both with an admiring exclamation of “Vizuri sana!” (“Very fine!”). I expressed the wish to take back these marvels with me to Ulaya, to show the Wazungu what clever fellows the Makonde are. Scarcely five minutes after my return to camp at Newala, the two men came up sweating under the weight of two heavy logs which they laid down at my feet, handing over at the same time the keys of the fallen fortress. Arguing, logically enough, that if the key was wanted, the lock would be wanted with it, they had taken their axes and chopped down the posts—as it never occurred to them to dig them out of the ground and so bring them intact. Thus I have

two badly damaged specimens, and the owners, instead of praise, come in for a blowing-up.

The Makua huts in the environs of Newala are especially miserable; their more than slovenly construction reminds one of the temporary erections of the Makua at Hatia’s, though the people here have not been concerned in a war. It must therefore be due to congenital idleness, or else to the absence of a powerful chief. Even the baraza at Mlipa’s, a short hour’s walk south-east of Newala, shares in this general neglect. While public buildings in this country are usually looked after more or less carefully, this is in evident danger of being blown over by the first strong easterly gale. The only attractive object in this whole district is the grave of the late chief Mlipa. I visited it in the morning, while the sun was still trying with partial success to break through the rolling mists, and the circular grove of tall euphorbias, which, with a broken pot, is all that marks the old king’s resting-place, impressed one with a touch of pathos.

Even my very materially-minded carriers seemed to feel something of the sort, for instead of their usual ribald songs, they chanted solemnly, as we marched on through the dense green of the Makonde bush:—