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HANDBOOK OF MATHEMATICS FOR ENGINEERS

% Qraw~3/ill Book (n. 7ne

PUBLISHERS OF BOOKS F O R_^

Coal Age v Electric Railway Journal Electrical World •* Engineering. News-Record American Machinist v The Contractor Engineering 8 Mining Journal ^ Power Metallurgical 6 Chemical Engineering Electrical Merchandising

Handbook of Mathematics for Engineers

BY

EDWARD V. HUNTINGTON, PH. D.

ASSOCIATE PROFESSOR OF MATHEMATICS, HARVARD UNIVERSITY

WITH TABLES OF WEIGHTS AND MEASURES BY

LOUIS A. FISCHER, B

CHIEF OF DIVISION OF WEIGHTS AND

U. 8. BUREAU OF STANDARDS

;. S. ^rrTT

MEASURES, 4

REPRINT OF SECTIONS 1 AND 2 OF L. S. MARKS'S "MECHANICAL ENGINEERS' HANDBOOK"

FIRST EDITION SECOND IMPRESSION

McGRAW-HILL BOOK COMPANY, INC,

239 WEST 39TH STREET. NEW YORK

LONDON: HILL PUBLISHING CO., LTD.

6 & 8 BOUVERIE ST., E.G. 1918

/

c

COPYRIGHT, 1918, BY THE McGRAW HILL BOOK COMPANY, INC.

COPYRIGHT, 1916, BY EDWARD V. HUNTINGTON.

TH» M A F L 1C X> R IB H H YOKJC PA

PREFACE

This Handbook of Mathematics is designed to contain, in compact form, accurate statements of those facts and formulas of pure mathematics which are most likely to be useful to the worker in applied mathematics.

It is not intended to take the place of the larger compendiums of pure mathematics on the one hand, or of the technical handbooks of engineering on the other hand; but in its own field it is thought to be more comprehensive than any other similar work in English.

Many topics of an elementary character are presented in a form which permits of immediate utilization even by readers who have had no previous acquaintance with the subject; for example, the practical use of logarithms and logarithmic cross-section paper, and the elementary parts of the modern method of nomography (alignment charts), can be learned from this book without the necessity of consulting separate treatises.

Other sections of the book to which special attention may be called are the chapter on the algebra of complex (or imaginary) quantities, the treat- ment of the catenary (with special tables), and the brief resume of the theory of vector analysis.

The mathematical tables (including several which are not ordinarily found) are carried to four significant figures throughout, and no pains have been spared to make them as nearly self-explanatory as possible, even to the reader who makes only occasional use of such tables.

For the Tables of Weights and Measures, which add greatly to its useful- ness, the book is indebted to Mr. Louis A. Fischer of the U. S. Bureau of Standards.

All the matter included in the present volume was originally prepared for the Mechanical Engineers' Handbook (Lionel S. Marks, Editor-in-Chief), and was first printed in 1916, as Sections 1 and 2 of that Handbook. The author desires to express his indebtedness to Professor Marks, not only for indispensable advice as to the choice of the topics which would be most useful to engineers, but also for great assistance in many details of the presentation.

All the misprints that have been detected have been corrected in the plates. Notification in regard to any further corrections, and any suggestions toward the improvement or possible enlargement of the book, will be cordially welcomed by the author or the publishers.

E. V. H.

CAMBRIDGE, MASS. April 29, 1918.

M171600

CONTENTS

Page PREFACE v

SECTION 1. Mathematical Tables and Weights and Measures 1

(For detailed Table of Contents, see page 1.)

SECTION 2. Mathematics:

Arithmetic; Geometry and Mensuration; Algebra; Trigonometry; Analytical Geometry; Differential and Integral Calculus; Graphical Representation of Functions; Vector Analysis 87

(For detailed Table of Contents, see page 87.) INDEX . . 187

SECTION 1

MATHEMATICAL TABLES

AND

WEIGHTS AND MEASURES

BY EDWARD V. HUNTINGTON, Ph. D., Associate Professor of Mathematics,

Harvard University, Fellow Am. Acad. Arts and Sciences. LOUISA. FISCHER, B. S., Chief of Division of Weights and Measures,

U. S. Bureau of Standards.

CONTENTS

MATHEMATICAL TABLES

BY E. V. HUNTINGTON PAGE

Squares of Numbers 2

Cubes of Numbers 8

Square Roots of Numbers 12

Cube Roots of Numbers 16

Three-halves Powers of Numbers. . . 22

Reciprocals of Numbers 24

Circles (Areas, Segments, etc.) . 28

Spheres (Volumes, Segments, etc.).. 36

Regular Polygons 39

Binomial Coefficients 39

Common Logarithms 40

Degrees and Radians 44

Trigonometric Functions 46

Exponentials 57

Hyperbolic (Napierian) Logarithms. 58

Hyperbolic Functions 60

Multiples of 0.4343 and 2.3026 62

Residuals and Probable Errors 63

Compound Interest and Annuities. 64

Decimal Equivalents 69

WEIGHTS AND MEASURES

BY LOUIS A. FISCHER PAGE U. S. Customary Weights and

Measures 70

Metric Weights and Measures 71

Systems of Units 72

Conversion Tables:

Lengths 74

Areas 76

Volumes and Capacities 76

Velocities 78

Masses (Weights) 78

Pressures 79

Energy, Work, Heat 79

Power 81

Density 81

Heat Transmission and Con- duction 82

Values of Foreign Coins 82

Time - 83

Terrestrial Gravity 84

Specific Gravity and Density 84

MATHEMATICAL TABLES

SQUARES OF

N

0

1

2

3

4

5

6

7

8

9

ll

1.00

1.000

1.002

1.004

.006

1.008

1.010

1.012

1.014

1.016

1.018

2

1

1.020

1.022

1.024

.026

1.028

1.030

1.032

1.034

1.036

1.038

2

1.040

1.042

1.044

.047

1.049

1.051

1.053

1.055

1.057

1.059

3

1.061

1.063

1.065

.067

1.069

1.071

1.073

1.075

1.077

1.080

4

1.082

1.084

1.086

.088

1.090

1.092

1.094

1.096

1.098

1.100

1.05

1.102

1.105

1.107

.109

1.111

.113

1.115

1.117

.119

1.121

6

1.124

1.126

1.128

.130

1.132

.134

1.136

1.138

.141

1.143

7

1.145

1.147

1.149

.151

1.153

.156

1.158

1.160

.162

1.164

8

1.166

1.169

1.171

.173

1.175

.177

1.179

1.182

184

1.186

9

1.188

1.190

1.192

.195

1.197

.199

1.201

1.203

.206

1.208

1.10

1210

1.212

1.214

.217

1.219

1.221

1.223

1.225

.228

1.230

1

1.232

1.234

1.237

.239

1.241

1.243

1.245

1.248

.250

1.252

2

1.254

1.257

1.259

.261

1.263

1.266

1.268

1.270

.272

1.275

3

1.277

1.279

1.281

1.284

1.286

1.288

1.209

1.293

.295

1.297

4

1.300

1.302

1.304

1.306

1.309

1.311

1.313

1.316

.318

1.320

1.15

1.322

1.325

1.327

1.329

1.332

1.334

1.336

1.339

.341

1.343

6

1.346

1.348

1.350

1353

1.355

1.357

1.360

1.362

.364

1.367

7

1.369

1.371

1.374

1.376

1.378

1.381

1.383

1.385

.388

1.390

8

1.392

1.395

1.397

1.399

1.402

1.404

1.407

1.409

.411

1.414

9

1.416

1.418

1.421

1.423

1.426

1.428

1.430

1.433

.435

1.438

1.20

1.440

1.442

1.445

1.447

1.450

1.452

1.454

1.457

.459

1.462

1

1.464

1.467

1.469

1.471

1.474

1.476

1.479

1.481

.484

1.486

2

1.488

1.491

1.493

1.496

1.498

1.501

1.503

1.506

.508

1.510

3

1.513

1.515

1.518

1.520

1.523

1.525

1.528

1.530

.533

1.535

4

1.538

1.540

1.543

1.545

1.548

1.550

1.553

1.555

1.558

1.560

1.25

1.562

1.565

1.568

1.570

1.573

1.575

1.578

1.580

1.583

1585

3

6

1.588

1.590

1.593

1.595

1.598

1.600

1.603

1.605

1.608

1.610

7

1.613

1.615

1.618

1.621

1.623

.626

1.628

1.631

1.633

1.636

8

1.638

1.641

1.644

1.646

1.649

1.651

1.654

1.656

1.659

1.662

9

1.664

1.667

1.669

1.672

1.674

1.677

1.680

1.682

1.685

1.687

1.30

1.690

1.693

1.695

1.698

1.700

1.703

1.706

1.708

1.711

1.713

1

1.716

1.719

1.721

1.724

1.727

1.729

1.732

1.734

1.737

1.740

2

1.742

1.745

1.748

1.750

1.753

.756

1.758

1.761

1.764

1.766

3

1.769

1.772

1.774

1.777

1.780

1.782

1.785

1.788

1.790

1.793

4

1.796

1.798

1.801

1.804

1.806

1.809

1.812

1.814

1.817

1.820

1.35

1.822

1.825

1.828

1.831

1.833

1.836

1.839

1.841

1.844

1.847

6

1.850

1.852

1.855

1.858

1.860

1.863

1.866

1.869

1.871

1.874

7

1.877

1.880

1.882

1.885

1.888

1.891

1.893

1.896

1.899

1.902

8

1.904

1.907

1.910

1.913

1.915

1.918

1.921

1.924

1.927

1.929

9

1.932

1.935

1.938

1.940

1.943

1.946

1.949

1.952

1.954

1.957

1.40

1.960

1.963

1.966

1.968

1.971

1.974

1.977

1.980

1.982

1.985

1

1.988

1.991

1.994

1.997

1.999

2.002

2.005

2.008

2.011

2.014

2

2.016

2.019

2.022

2.025

2.028

2.031

2.033

2.036

2.039

2.042

3

2.045

2.048

2.051

2.053

2.056

2.059

2.062

2.065

2.068

2.071

4

2.074

2.076

2.079

2.082

2.085

2.088

2.091

2.094

2.097

2.100

1.45

2.102

2.105

2.108

2.111

2.114

2.117

2.120

2.123

2.126

2.129

6

2.132

2.135

2.137

2.140

2.143

2.146

2.149

2.152

2.155

2.158

7

2.161

2.164

2.167

2.170

2.173

2.176

2.179

2.182

2.184

2.187

8

2.190

2.193

2.196

2.199

2.202

2.205

2.208

2.211

2.214

2.217

9

2.220

2.223

2.226

2.229

2.232

2.235

2.238

2.241

2.244

2.247

Moving the decimal point ONE place in N requires moving it TWO places in body of table (see p. 6).

MATHEMATICAL TABLES

SQUARES (continued)

N

0

1

2

3

4

5

6

7

8

9

ii

1.50

2.250

2.253

2.256

2.259

2.262

2.265

2.268

2.271

2.274

2.277

3

I

2.280

2.283

2.286

2.289

2.292

2.295

2.298

2.301

2.304

2.307

2

2.310

2.313

2.316

2.320

2.323

2.326

2.329

2.332

2.335

2.338

3

2.341

2.344

2.347

2.350

2.353

2.356

2.359

2.362

2.365

2.369

4

2.372

2.375

2.378

2.381

2.384

2.387

2.390

2.393

2.396

2.399

1.55

2.402

2.406

2.409

2.412

2.415

2.418

2.421

2.424

2.427

2.430

6

2.434

2.437

2.440

2.443

2.446

2.449

2.452

2.455

2.459

2.462

7

2.465

2.468

2.471

2.474

2.477

2.481

2.484

2.487

2.490

2.493

8

2.496

2.500

2.503

2.506

2.509

2.512

2.515

2.519

2.522

2.525

9

2.528

2.531

2.534

2.538

2.541

2.544

2.547

2.550

2.554

2.557

1.60

2.560

2.563

2.566

2.570

2.573

2.576

2.579

2.582

2.586

2.589

1

2.592

2.595

2.599

2.602

2.605

2.608

2.611

2.615

2.618

2.621

2

2.624

2.628

2.631

2.634

2.637

2.641

2.644

2.647

2.650

2.654

3

2.657

2.660

2.663

2.667

2.670

2.673

2.676

2.680

2.683

2.686

4

2.690

2.693

2.696

2.699

2.703

2.706

2.709

2.713

2.716

2.719

1.65

2.722

2.726

2.729

2.732

2.736

2.739

2.742

2.746

2.749

2.752

6

2.756

2.759

2762

2.766

2.769

2.772

2.776

2.779

2.782

2.786

7

2.789

2.792

2.796

2.799

2.802

2.806

2.809

2.812

2.816

2.819

8

2.822

2.826

2.829

2.832

2.836

2.839

2.843

2.846

2849

2.853

9

2.856

2.859

2.863

2.866

2.870

2.873

2.876

2.880

2.883

2.887

1.70

2.890

2.893

2.897

2.900

2.904

2.907

2.910

2.914

2.917

2.921

1

2.924

2.928

2.931

2.934

2.938

2.941

2.945

2.948

2.952

2.955

2

2.958

2.962

2.965

2.969

2.972

2.976

2.979

2.983

2.986

2.989

3

2.993

2.996

3.000

3.003

3.007

3.010

3.014

3.017

3.021

3.024

4

3.028

3.031

3.035

3.038

3.042

3.045

3.049

3.052

3.056

3.059

1.75

3.062

3.066

3.070

3.073

3.077

3.080

3.084

3.087

3.091

3.094

4

6

3.098

3.101

3.105

3.108

3.112

3.115

3.119

3.122

3.126

3.129

7

3.133

3.136

3.140

3.144

3.147

3.151

3.154

3.158

3.161

3.165

.8

3.168

3.172

3.176

3.179

3.183

3.186

3.190

3.193

3.197

3-201

9

3.204

3.208

3.211

3.215

3.218

3.222

3.226

3.229

3.233

3.236

1.80

3.240

3.244

3.247

3.251

3.254

3.258

3.262

3.265

3.269

3.272

1

3.276

3.280

3.283

3.287

3.291

3.294

3.298

3.301

3.305

3.309

2

3.312

3.316

3.320

3.323

3.327

3.331

3.334

3.338

3.342

3.345

3

3.349

3.353

3.356

3.360

3.364

3.367

3.371

3.375

3.378

3.382

4

3.386

3.389

3.393

3.397

3.400

3.404

3.408

3.411

3.415

3.419

1.85

3.422

3.426

3.430

3.434

3.437

3.441

3.445

3.448

3.452

3.456

6

3.460

3.463

3.467

3.471

3.474

3.478

3.482

3.486

3.489

3.493

7

3.497

3.501

3.504

3.508

3.512

3.516

3.519

3.523

3.527

3.531

8

3.534

3.538

3.542

3.546

3.549

3.553

3.557

3.561

3.565

3.568

9

3.572

3.576

3.580

3.583

3.587

3.591

3.595

3.599

3.602

3.606

1.90

3.610

3.614

3.618

3.621

3.625

3.629

3.633

3.637

3.640

3.644

1

3.648

3.652

3.656

3.660

3.663

3.667

3.671

3.675

3.679

3.683

2

3.686

3.690

3.694

3.698

3.702

3.706

3.709

3.713

3.717

3.721

3

3.725

3.729

3.733

3.736

3.740

3.744

3.748

3.752

3.756

3.760

4

3.764

3.767

3.771

3.775

3.779

3.783

3.787

3.791

3.795

3.799

1.95

3.802

3.806

3.810

3.814

3.818

3.822

3.826

3.830

3.834

3.838

6

3.842

3.846

3.849

3.853

3.857

3.861

3.865

3.869

3.873

3.877

7

3.881

3.885

3.889

3.893

3.897

3.901

3.905

3.909

3.912

3.916

8

3.920

3.924

3.928

3.932

3.936

3.940

3.944

3.948

3.952

3.956

9

3.960

3.964

3.968

3.972

3.976

3.980

3.984

3.988

3.992

3.996

= 9.86960 !/*« = 0.101321

7.38906

MATHEMATICAL TABLES

SQUARES (continued)

N

0

1

2

3

4

5

6

7

8

9

$S

2.00

4.000

4.004

4.008

4.012

4.016

4.020

4.024

4.028

4.032

4.036

4

1

4.040

4.044

4.048

4.052

4.056

4.060

4.064

4.068

4.072

4.076

2

4.080

4.084

4.088

4.093

4.097

4.101

4.105

4.109

4.113

4.117

3

4.121

4.125

4.129

4.133

4.137

4.141

4.145

4.149

4. 153

4.158

4

4.162

4.166

4.170

4.174

4.178

4.182

4.186

4.190

4.194

4.198

2.05

4.202

4.207

4.211

4.215

4.219

4.223

4.227

4.231

4.235

4.239

6

4.244

4.248

4.252

4.256

4.260

4.264

4.268

4.272

4.277

4.281

7

4.285

4.289

4.293

4.297

4.301

4.306

4.310

4.314

4.318

4.322

8

4.326

4.331

4.335

4.339

4.343

4.347

4.351

4.356

4.360

4.364

9

4.368

4.372

4.376

4.381

4.385

4.389

4.393

4.397

4.402

4.406

2.10

4.410

4.414

4.418

4.423

4.427

4.431

4.435

4.439

4.444

4.448

1

4.452

4.456

4.461

4.465

4.469

4.473

4.477

4.482

4486

4.490

2

4.494

4.499

4.503

4.507

4.511

4.516

4.520

4.524

4.528

4533

3

4.537

4.541

4.545

4.550

4.554

4.558

4.562

4.567

4.571

4.575

4

4.580

4.584

4.588

4.592

4.597

4.601

4.605

4.610

4.614

4.618

2.15

4.622

4.627

4.631

4.635

4.640

4.644

4.648

4.653

4.657

4.661

6

4.666

4.670

4.674

4.679

4.683

4.687

4.692

4.696

4.700

4.705

7

4.709

4.713

4.718

4.722

4.726

4.731

4.735

4.739

4.744

4.748

8

4.752

4.757

4.761

4.765

4.770

4.774

4.779

4.783

4.787

4.792

9

4.796

4.800

4.805

4.809

4.814

4.818

4.822

4.827

4.831

4.836

2.20

4.840

4.844

4.849

4.853

4.858

4.862

4.866

4.871

4.875

4.880

1

4.884

4.889

4.893

4.897

4.902

4.906

4.911

4.915

4.920

4.924

2

4.928

4.933

4.937

4.942

4.946

4.951

4.955

4.960

4.964

4.968

3

4.973

4.977

4.982

4.986

4.991

4.995

5.000

5.004

5.009

5.013

4

5.018

5.022

5.027

5.031

5.036

5.040

5.045

5.049

5.054

5.058

2.25

5.062

5.067

5.072

5.076

5.081

5.085

5.090

5.094

5.099

5.103

5

6

5.108

5.112

5.117

5.121

5.126

5.130

5.135

5.139

5.144

5148

7

5.153

5.157

5.162

5.167

5.171

5.176

5.180

5.185

5.189

5.194

8

5.198

5.203

5.208

5.212

5.217

5.221

5.226

5.230

5.235

5.240

9

5.244

5.249

5.253

5.258

5.262

5.267

5.272

5.276

5.281

5.285

2.30

5.290

5.295

•5.299

5.304

5.308

5.313

5.318

5.322

5.327

5.331

1

5.336

5.341

5.345

5.350

5.355

5.359

5.364

5.368

5.373

5.378

2

5.382

5.387

5.392

5.396

5.401

5.406

5.410

5.415

5.420

5.424

3

5.429

5.434

5.438

5.443

5.448

5.452

5.457

5.462

5.466

5.471

4

5.476

5.480

5.485

5.490

5.494

5.499

5.504

5.508

5.513

5.518

2.35

5.522

5.527

5.532

5.537

5.541

5.546

5.551

5.555

5.560

5.565

6

5.570

5.574

5.579

5.584

5.588

5.593

5.598

5.603

5.607

5.612

7

5.617

5.622

5.626

5.631

5.636

5.641

5.645

5.650

5.655

5.660

8

5.664

5.669

5.674

5.679

5.683

5.688

5.693

5.698

5.703

5.707

9

5.712

5.717

5.722

5.726

5.731

5.736

5.741

5.746

5.750

5.755

2.40

5.760

5.765

5.770

5.774

5.779

5.784

5.789

5.794

5.798

5.803

1

5.808

5.813

5.818

5.823

5.827

5.832

5.837

5.842

5.847

5.852

2

5.856

5.861

5.866

5.871

5.876

5.881

5.885

5.890

5.895

5.900

3

5.905

5.910

5.915

5.919

5.924

5.929

5.934

5.939

5.944

5.949

4

5.954

5.958

5.963

5.968

5.973

5.978

5.983

5.988

5.993

5.998

2.45

6.002

6.007

6.012

6.017

6.022

6.027

6.032

6.037

6.042

6.047

6

6.052

6.057

6.061

6.066

6.071

6.076

6.081

6.086

6.091

6.096

7

6.101

6.106

6.111

6.116

6.121

6.126

6.131

6.136

6.140

6.145

8

6.150

6.155

6.160

6.165

6.170

6.175

6.180

6.185

6.190

6.195

9

6.200

6.205

6.210 ,

6.215

6.220

6.225

6.230

6.235

6.240

6.245

Moving the decimal point ONE place in N requires moving it TWO places in body of table (see p. G).

MATHEMATICAL TABLES

SQUARES (continued)

N

0

1

2

3

4

5

6

7

8

9

£

2.50

6.250

6.255

6.260

6.265

6.270

6.275

6.280

6.285

6.290

6.295

5

1

6.300

6.305

6.310

6.315

6.320

6.325

6.330

6.335

6.340

6.345

2

6.350

6.355

6.360

6.366

6.371

6.376

6.381

6.386

6.391

6.396

3

6.401

6.406

6.411

6.416

6.421

6.426

6.431

6.436

6.441

6.447

4

6.452

6.457

6.462

6.467

6.472

6.477

6.482

6.487

6.492

6.497

2.55

6.502

6.508

6.513

6.518

6.523

6.528

6.533

6.538

6.543

6.548

6

6.554

6.559

6.564

6.569

6.574

6.579

6.584

6.589

6.595

6.600

7

6.605

6.610

6.615

6.620

6.625

6.631

6.636

6.641

6.646

6.651

8

6.656

6.662

6.667

6.672

6.677

6.682

6.687

6.693

6.698

6.703

9

6.708

6.713

6.718

6.724

6.729

6.734

6.739

6.744

6.750

6.755

2.60

6.760

6.765

6.770

6.776

6.781

6.786

6.791

6.7%

6.802

6.807

1

6.812

6.817

6.823

6.828

6.833

6.838

6.843

6.849

6.854

6.859

2

6.864

6.870

6.875

6.880

6.885

6.891

6.896

6.901

6906

6.912

3

6.917

6.922

6.927

6.933

6.938

6.943

6.948

6.954

6.959

6.964

4

6.970

6.975

6.980

6.985

6.991

6.996

7.001

7.007

7.012

7.017

2.65

7.022

7.028

7.033

7.038

7.044

7.049

7.054

7.060

7.065

7.070

6

7.076

7.081

7.086

7.092

7.097

7.102

7.108

7.113

7.118

7.124

7

7.129

7.134

7.140

7.145

7.150

7.156

7.161

7.166

7.172

7.177

8

7.182

7.188

7.193

7.198

7.204

7.209

7.215

7.220

7.225

7.231

9

7.236

7.241

7.247

7.252

7.258

7.263

7.268

7.274

7.279

7.285

2.70

7.290

7.295

7.301

7.306

7.312

7.317

7.322

7.328

7.333

7339

1

7.344

7.350

7.355

7.360

7.366

7.371

7.377

7.382

7.388

7.393

2

7.398

7.404

7.409

7.415

7.420

7.426

7.431

7.437

7.442

7.447

3

7.453

7.458

7.464

7.469

7.475

7.480

7.486

7.491

7.497

7.502

4

7.508

7.513

7.519

7.524

7.530

7.535

7.541

7.546

7.552

7.557

2.75

7.562

7.568

7.574

7.579

7.585

7.590

7.596

7.601

7.607

7.612

6

6

7.618

7.623

7.629

7.634

7.640

7.645

7.651

7.656

7.662

7.667

7

7.673

7.678

7.684

7.690

7.695

7.701

7.706

7.712

7.717

7.723

8

7.728

7.734

7.740

7.745

7.751

7.756

7.762

7.767

7.773

7.779

9

7.784

7.790

7.795

7.801

7.806

7.812

7.818

7.823

7.829

7.834

2.80

7.840

7.846

7.851

7.857

7.862

7.868

7.874

7.879

7.885

7.890

1

7.896

7.902

7.907

7.913

7.919

7.924

7.930

7.935

7.941

7.947

2

7.952

7.958

7.964

7.969

7.975

7.981

7.986

7.992

7.998

8003

3

8.009

8.015

8.020

8.026

8.032

8.037

8.043

8.049

8.054

8.060

4

8.066

8.071

8.077

8.083

8.088

8.094

8.100

8.105

8.111

8.117

2.85

8.122

8.128

8.134

8.140

8.145

8.151

8.157

8.162

8.168

8.174

6

8.180

8.185

8.191

8.197

8.202

8.208

8.214

8.220

8.225

8.231

7

8.237

8.243

8.248

8.254

8.260

8.266

8.271

8.277

8.283

8.289

8

8.294

8.300

8.306

8.312

8.317

8.323

8.329

8.335

8.341

8.346

9

8.352

8.358

8.364

8.369

8.375

8.381

8.387

8.393

8.398

8.404

2.90

8.410

8.416

8.422

8.427

8.433

8.439

8.445

8.451

8.456

8.462

1

8.468

8.474

8.480

8.486

8.491

8.497

8.503

8.509

8.515

8.521

2

8526

8.532

8.538

8.544

8.550

8.556

8.561

8.567

8.573

8.579

3

8.585

8.591

8.597

8.602

8.608

8.614

8.620

8.626

8.632

8.638

4

8.644

8.649

8.655

8.661

8.667

8.673

8.679

8.685

8.691

8.697

2.95

8.702

8.708

8.714

8.720

8.726

8.732

8.738

8.744

8.750

8.756

6

8.762

8.768

8.773

8.779

8.785

8.791

8.797

8.803

8.809

8.815

7

8.821

8.827

8.833

8.839

8.845

8.851

8.857

8.863

8.868

8874

8

8.880

8.886

8.892

8.898

8.904

8.910

8.916

8.922

8.928

8.934

9

8.940

8.946

8.952

8.958

8.964

8.970

8.976

8.982

8.988

8.994

I/T» = 0.101321

7.38906

c

MATHEMATICAL TABLES

SQUARES (continued}

N

0

1

2

3

4

5

6

7

8

9

S?«a

•5*

3.00

9.000

9.006

9.012

9.018

9.024

9.030

9.036

9.042

9.048

9.054

6

1

9.060

9.066

9.072

9.078

9.084

9.090

9.096

9.102

9.108

9.114

2

9.120

9.126

9.132

9.139

9.145

9.151

9.157

9.163

9.169

9.175

3

9.181

9.187

9.193

9.199

9.205

9.211

9.217

9.223

9.229

9.236

4

9.242

9.248

9.254

9.260

9.266

9.272

9.278

9.284

9.290

9.296

3.05

9.302

9.309

9.315

9.321

9.327

9.333

9.339

9.345

9.351

9.357

6

9.364

9.370

9.376

9.382

9.388

9.394

9.400

9.406

9.413

9.419

7

9.425

9.431

9.437

9.443

9.449

9.456

9.462

9.468

9.474

9.480

8

9.486

9.493

9.499

9.505

9.511

9.517

9.523

9.530

9.536

9.542

9

9.548

9.554

9.560

9.567

9.573

9.579

9.585

9.591

9.598

9.604

3.10

9.610

9.616

9.622

9.629

9.635

9.641

9.647

9.653

9.660

9.666

1

9.672

9.678

9.685

9.691

9.697

9.703

9.709

9.716

9.722

9.728

2

9.734

9.741

9.747

9.753

9.759

9.766

9.772

9.778

9.784

9.791

3

9.797

9.803

9.809

9.816

9.822

9.828

9.834

9.841

9.847

9.853

4

9.860

9.866

9.872

9.878

9.885

9.891

9.897

9.904

9.910

9.916

3.15

9.922

9.929

9.935

9.941

9.948

9.954

9.960

9.967

9.973

9.979

6

9.986

9.992

9.998

10.005

6

3.1

9.99

10.05

10.11

10.18

6

2

10.24

10.30

10.37

10.43

10.50

10.56

10.63

10.69

10.76

10.82

3

10.89

10.96

11.02

11.09

11.16

11.22

11.29

11.36

11.42

11.49

7

4

11.56

11.63

11.70

11.76

11.83

11.90

11.97

12.04

12.11

12.18

3.5

12.25

12.32

12.39

12.46

12.53

12.60

12.67

12.74

12.82

12.89

6

12.96

13.03

13.10

13.18

13.25

13.32

13.40

13.47

13.54

13.62

7

13.69

13.76

13.84

13.91

13.99

14.06

14.14

14.21

14.29

14.36

8

8

14.44

14.52

14.59

14.67

14.75

14.82

14.90

14.98

15.05

15.13

9

15.21

15.29

15.37

15.44

15.52

15.60

15.68

15.76

15.84

15.92

4.0

16.00

16.08

16.16

16.24

16.32

16.40

16.48

16.56

16.65

16.73

1

16.81

16.89

16.97

17.06

17.14

17.22

17.31

17.39

17.47

17.56

2

17.64

17.72

17.81

17.89

17.98

18.06

18.15

18.23

18.32

18.40

3

18.49

18.58

18.66

18.75

18.84

18.92

19.01

19.10

19.18

19.27

9

4

19.36

19.45

19.54

19.62

19.71

19.80

19.89

19.98

20.07

20.16

4.5

20.25

20.34

20.43

20.52

20.61

20.70

20.79

20.88

20.98

21.07

6

21.16

21.25

21.34

21.44

21.53

21.62

21.72

21.81

21.90

22.00

7

22.09

22.18

22.28

22.37

22.47

22.56

22.66

22.75

22.85

22.94

10

8

23.04

23.14

23.23

23.33

23.43

23.52

23.62

23.72

23.81

23.91

9

24.01

24.11

24.21

24.30

24.40

24.50

24.60

24.70

24.80

24.90

9.86960 (x/2)2 = 2.46740 !/«•» = 0.101321

Explanation of Table of Squares (pp. 2-7).

This table gives the value of Nz for values of N from 1 to 10, correct to four figures. (Interpolated values may be in error by 1 in the fourth figure).

To find the square of a number N outside the range from 1 to 10, note that moving the decimal point one place in column N is equivalent to moving it two places in the body of the table. For example:

(3.217)2 - 10.35; (0.03217)2 = 0.001035; (3217)* = 10350000

This table can also be used inversely, to give square roots.

MATHEMATICAL TABLES

SQUARES (continued)

N

0

1

2

3

4

5

6

7

8

9

&

5.0

25.00

25.10

25.20

25.30

25.40

25.50

25.60

25.70

25.81

25.91

10

1

26.01

26.11

26.21

26.32

26.42

26.52

26.63

26.73

26.83

26.94

2

27.04

27.14

27.25

27.35

27.46

27.56

27.67

27.77

27.88

27.98

3

28.09

28.20

28.30

28.41

28.52

28.62

28.73

28.84

28.94

29.05

11

4

29.16

29.27

29.38

29.48

29.59

29.70

29.81

29.92

30.03

30.14

5.5

30.25

30.36

30.47

30.58

30.69

30.80

30.91

31.02

31.14

31.25

6

31.36

31.47

31.58

31.70

31.81

31.92

32.04

32.15

32.26

32.38

7

32.49

32.60

32.72

32.83

32.95

33.06

33.18

33.29

33.41

33.52

8

33.64

33.76

33.87

33.99

34.11

34.22

34.34

34.46

34.57

34.69

12

9

34.81

34.93

35.05

35.16

35.28

35.40

35.52

35.64

35.76

35.88

6.0

36.00

36.12

36.24

36.36

36.48

36.60

36.72

36.84

36.97

37.09

37.21

37.33

37.45

37.58

37.70

37.82

37.95

38.07

38.19

38.32

2

38.44

38.56

38.69

38.81

38.94

39.06

39.19

39.31

39.44

39.56

3

39.69

39.82

39.94

40.07

40.20

40.32

40.45

40.58

40.70

40.83

13

4

40.96

41.09

41.22

41.34

41.47

41.60

41.73

41.86

41.99

42.12

6.5

42.25

42.38

42.51

42.64

42.77

42.90

43.03

43.16

43.30

43.43

6

43.56

43.69

43.82

43.96

44.09

44.22

44.36

44.49

44.62

44.76

7

44.89

45.02

45.16

45.29

45.43

45.56

45.70

45.83

45.97

46.10

8

46.24

46.38

46.51

46.65

46.79

46.92

47.06

47.20

47.33

47.47

14

9

47.61

47.75

47.89

48.02

48.16

48.30

48.44

48.58

48.72

48.86

7.0

49.00

49.14

49.28

49.42

49.56 *

49.70

49.84

49.98

50.13

50.27

1

50.41

50.55

50.69

50.84

50.98

51.12

51.27

51.41

51.55

51.70

2

51.84

51.98

52.13

52.27

52.42

52.56

52.71

52.85

53.00

53.14

3

53.29

53.44

53.58

53.73

53.88

54.02

54.17

54.32

54.46

54.61

15

4

54.76

54.91

55.06

55.20

55.35

55.50

55.65

55.80

55.95

56.10

7.5

56.25

56.40

56.55

56.70

56.85

57.00

57.15

57.30

57.46

57.61

6

57.76

57.91

58.06

58.22

58.37

58.52

58.68

58.83

58.98

59.14

7

59.29

59.44

59.60

59.75

59.91

60.06

60.22

60.37

60.53

60.68

8

60.84

61.00

61.15

61.31

61.47

61.62

61.78

61.94

62.09

62.25

16

9

62.41

62.57

62.73

62.88

63.04

63.20

63.36

63.52

63.68

63.84

8.0

64.00

64.16

64.32

64.48

64.64

64.80

64.96

65.12

65.29

65.45

]

65.61

65.77

65.93

66.10

66.26

66.42

66.59

66.75

66.91

67.08

2

67.24

67.40

67.57

67.73

67.90

68.06

68.23

68.39

68.56

68.72

3

68.89

69.06

69.22

69.39

69.56

69.72

69.89

70.06

70.22

70.39

17

4

70.56

70.73

70.90

71.06

71.23

71.40

71.57

71.74

71.91

72.08

8.5

72.25

72.42

72.59

72.76

72.93

73.10

73.27

73.44

73.62

73.79

6

73.96

74.13

74.30

74.48

74.65

74.82

75.00

75.17

75.34

75.52

7

75.69

75.86

76.04

76.21

76.39

76.56

76.74

76.91

77.09

77.26

8

77.44

77.62

77.79

77.97

78.15

78.32

78.50

78.68

78.85

79.03

18

9

79.21

79.39

79.57

79.74

79.92

80.10

80.28

80.46

80.64

80.82

9.0

81.00

81.18

81.36

81.54

81.72

81.90

82.08

82.26

82.45

82.63

1

82.81

82.99

83.17

83.36

83.54

83.72

83.91

84.09

84.27

84.46

2

84.64

84.82

85.01

85.19

85.38

85.56

85.75

85.93

86.12

86.30

3

86.49

86.68

86.86

87.05

87.24

87.42

87.61

87.80

87.98

88.17

19

4

88.36

88.55

88.74

88.92

89.11

89.30

89.49

89.68

89.87

90.06

9.5

90.25

90.44

90.63

90.82

91.01

91.20

91.39

91.58

91.78

91.97

6

92.16

92.35

92.54

92.74

92.93

93.12

93.32

93.51

93.70

93.90

7

94.09

94.28

94.48

94.67

94.87

95.06

95.26

95.45

95.65

95.84

8

96.04

96.24

96.43

96.63

96.83

97.02

97.22

97.42

97.61

97.81

20

9

9801

98.21

98.41

98.60

98.80

99.00

99.20

99.40

99.60

99.80

10.0

100.0

Moving the decimal point ONE place in N requires moving it TWO places in body of table (see p. 6).

MATHEMATICAL TABLES

CUBES OP NUMBERS

N

0

1

2

3

4

5

6

7

8

9

II

1.00

1.000

1.003

1.006

1.009

1.012

1.015

1.018

1/»21

1.024

1.027

3

1

1.030

1.033

1.036

1.040

1.043

1.046

1.049

1.052

1.055

1.058

2

1.061

1.064

1.067

1.071

1.074

1.077

1.080

1.083

1.086

1.090

3

1.093

1.096

1.099

1.102

1.106

1.109

1.112

1.115

1.118

1.122

4

1.125

1.128

1.131

1.135

1.138

1.141

1.144

1.148

1.151

1.154

1.05

1.158

.161

1.164

1.168

1.171

1.174

1.178

1.181

1.184

1.188

6

1.191

.194

1.198

1.201

1.205

1.208

1.211

1.215

1.218

1.222

7

1.225

.228

1.232

1.235

1.239

1.242

1.246

1.249

1.253

1.256

8

1.260

.263

1.267

1.270

1.274

1.277

1.281

1.284

1.288

1.291

4

9

1.295

.299

1.302

1.306

1.309

1.313

1.317

1.320

1.324

1.327

1.10

1.331

.335

1.338

1.342

1.346

1.349

1.353

1.357

1.360

1.364

1

1368

.371

1.375

1.379

1.382

1.386

1.390

1.394

1.397

1.401

2

1.405

1.409

1.412

1.416

1.420

1.424

1.428

1.431

1.435

1.439

3

1.443

1.447

1.451

1.454

1.458

1.462

1.466

1.470

1.474

1.478

4

1.482

1.485

1.489

1.493

1.497

1.501

1.505

1.509

1.513

1.517

1.15

1.521

1.525

1.529

1.533

1.537

1.541

1.545

1.549

1.553

1.557

6

1.561

1.565

1.569

1.573

1.577

1.581

1.585

1.589

1.593

1.598

7

1.602

1.606

1.610

1.614

1.618

1.622

1.626

1.631

1.635

1.639

8

1.643

1.647

1.651

1.656

1.660

1.664

1.668

1.672

1.677

1.681

9

1.685

1.689

1.694

1.698

1.702

1.706

1.711

1.715

1.719

1.724

1.20

1.728

1.732

1.737

1.741

1.745

1.750

1.754

1.758

1.763

1.767

1

1.772

1.776

1.780

1.785

1.789

1.794

1.798

1.802

1.807

1.811

2

1.816

1.820

1.825

1.829

1.834

1.838

1.843

1.847

1.852

1.856

3

1.861

1.865

1.870

1.875

1.879

1.884

1.888

1.893

1.897

1.902

4

1.907

1.911

1.916

1.920

1.925

1.930

1.934

1.939

1.944

1.948

5

1.25

1.953

1.958

1.963

1.967

1.972

1.977

1.981

1.986

1.991

1.996

6

2.000

2.005

2.010

2.015

2.019

2.024

2.029

2.034

2.039

2.044

7

2.048

2.053

2.058

2.063

2.068

2.073

2.078

2.082

2.087

2.092

8

2.097

2.102

2.107

2.112

2.117

2.122

2.127

2.132

2.137

2.142

9

2.147

2.152

2.157

2.162

2.167

2.172

2.177

2.182

2.187

2.192

1.30

2.197

2.202

Z.207

/.212

2.217

2.222

2.228

2.233

2.238

2.243

1

2.248

2.253

2.258

2.264

2.269

2.274

2.279

2.284

2.290

2.295

2

2.300

2.305

2.310

2.316

2.321

2.326

2.331

2.337

2.342

2.347

3

2.353

2.358

2.363

2.369

2.374

2.379

2.385

2.390

2.395

2.401

4

2.406

2.411

2.417

2.422

2.428

2.433

2.439

2.444

2.449

2.455

1.35

2.460

2.466

2.471

2.477

2.482

2.488

2.493

2.499

2.504

2.510

6

6

2.515

2.521

2.527

2.532

2.538

2.543

2.549

2.554

2.560

2.566

7

2,571

2.577

2.583

2.588

2.594

2.600

2.605

2.611

2.617

2.622

8

2.628

2.634

2.640

2.645

2.651

2.657

2.663

2.668

2.674

2.680

9

2.686

2.691

2.697

2.703

2.709

2.715

2.721

2.726

2.732

2.738

1.40

2.744

2.750

2.756

2.762

2.768

2.774

2.779

2.785

2.791

2.797

]

2.803

2.809

2.815

2.821

2.827

2.833

2.839

2.845

2.851

2.857

2

2.863

2.869

2.875

2.881

2.888

2.894

2.900

2.906

2.912

2.918

3

2.924

2.930

2.936

2.943

2.949

2.955

2.961

2.967

2.974

2.980

4

2.986

2.992

2.998

3.005

3.011

3.017

3.023

3.030

3.036

3.042

1.45

3.049

3.055

3.061

3.068

3.074

3.080

3.087

3.093

3.099

3.106

6

3.112

3.119

3.125

3.131

3.138

3.144

3.151

3.157

3.164

3.170

7

3.177

3.183

3.190

3.196

3.203

3.209

3.216

3.222

3.229

3.235

8

3.242

3.248

3.255

3.262

3.268

3.275

3.281

3.288

3.295

3.301

7

9

3.308

3.315

3.321

3.328

3.335

3.341

3.348

3.355

3.362

3.368

Moving the decimal point ONE place in N requires moving it THREE places in body of table (see p. 10).

MATHEMATICAL TABLES

CUBES (continued)

N

0

1

2

3

4

5

6

7

8

9

&

1.50

3.375

3.382

3.389

3.395

3.402

3.409

3.416

3.422

3.429

3.436

7

1

3.443

3.450

3.457

3.464

3.470

3.477

3.484

3.491

3.498

3.505

2

3.512

3.519

3.526

3.533

3.540

3.547

3.554

3.561

3.568

3.575

3

3.582

3.589

3.596

3.603

3.610

3.617

3.624

3.631

3.638

3.645

4

3.652

3.659

3.667

3.674

3.681

3.688

3.695

3.702

3.709

3.717

1.55

3.724

3.731

3.738

3.746

3.753

3.760

3.767

3.775

3.782

3.789

6

3.796

3.804

3.811

3.818

3.826

3.833

3.840

3.848

3.855

3.863

7

3.870

3.877

3.885

3.892

3.900

3.907

3.914

3.922

3.929

3.937

8

3.944

3.952

3.959

3.967

3.974

3.982

3.989

3.997

4.005

4.012

8

9

4.020

4.027

4.035

4.042

4.050

4.058

4.065

4.073

4.081

4.088

1.60

4.096

4.104

4.111

4.119

4.127

4.135

4.142

4.150

4.158

4.166

4.173

4.181

4.189

4.197

4.204

4.212

4.220

4.228

4.236

4.244

2

4.252

4.259

4.267

4.275

4.283

4.291

4.299

4.307

4.315

4.323

3

4.331

4.339

4.347

4355

4.363

4.371

4.379

4.387

4.395

4.403

4

4.411

4.419

4.427

4.435

4.443

4.451

4.460

4.468

4.476

4.484

1.65

4.492

4.500

4.508

4.517

4.525

4.533

4.541

4.550

4.558

4.566

6

4.574

4.583

4.591

4.599

4.607

4.616

4.624

4.632

4.641

4.649

7

4.657

4.666

4.674

4.683

4.691

4.699

4.708

4.716

4.725

4.733

8

4.742

4.750

4.759

4.767

4.776

4.784

4.793

4.801

4.810

4.818

9

4.827

4.835

4.844

4.853

4.861

4.870

4.878

4.887

4.896

4.904

9

1.70

4.913

4.922

4.930

4.939

4.948

4.956

4.965

4.974

4.983

4.991

1

5.000

5.009

5.018

5.027

5.035

5.044

5.053

5.062

5.071

5.080

2

5.088

5.097

5.106

5.115

5.124

5.133

5.142

5.151

5.160

5.169

3

5.178

5.187

5.196

5.205

5.214

5.223

5.232

5.241

5.250

5.259

4

5.268

5.277

5.286

5.295

5.304

5.314

5.323

5.332

5.341

5.350

1.75

5.359

5.369

5.378

5.387

5.396

5.405

5.415

5.424

5.433

5.442

6

5.452

5.461

5.470

5.480

5.489

5.498

5.508

5.517

5.526

5.536

7

5.545

5.555

5.564

5.573

5.583

5.592

5.602

5.611

5.621

5.630

10

8

5.640

5.649

5.659

5.668

5.678

5.687

5.697

5.707

5,716

5.726

9

5.735

5.745

5.755

5.764

5.774

5.784

5.793

5.803

5.813

5.822

1.80

5.832

5.842

5.851

5.861

5.871

.5.881

5.891

5.900

5.910

5.920

1

5.930

5.940

5.949

5.959

5.969

5.979

5.989

5.999

6.009

6.019

2

6.029

6.039

6.048

6.058

6.068

6.078

6.088

6.098

6.108

6.118

3

6.128

6.139

6.149

6.159

6. 169

6.179

6.189

6.199

6.209

6.219

4

6.230

6.240

6.250

6.260

6.270

6.280

6.291

6.301

6.311

6.321

1.85

6.332

6.342

6.352

6.362

6.373

6.383

6.393

6.404

6.414

6.424

6

6.435

6.445

6.456

6.466

6.476

6.487

6.497

6.508

6.518

6.529

7

6.539

6.550

6.560

6.571

6.581

6.592

6.602

6.613

6.623

6.634

11

8

6.645

6.655

6.666

6.677

6.687

6.698

6.708

6.719

6.730

6.741

9

6.751

6.762

6.773

6.783

6.794

6.805

6.816

6.827

6.837

6.848

1.90

6.859

6.870

6.881

6.892

6.902

6.913

6.924

6.935

6.946

6.957

1

6.968

6.979

6.990

7.001

7.012

7.023

7.034

7.045

7.056

7.067

2

7.078

7.089

7.100

7.1 11

7.122

7.133

7.144

7.156

7.167

7.178

3

7.189

7.200

7.211

7.223

7.234

7.245

7.256

7.268

7.279

7.290

4

7.301

7.313

7.324

7.335

7.347

7.358

7.369

7.381

7.392

7.403

1.95

7.415

7.426

7.438

7.449

7.461

7.472

7.484

7.495

7.507

7.518

17

6

7.530

7.541

7.553

7.564

7.576

7.587

7.599

7.610

7.622

7.634

7

7.645

7.657

7.669

7.680

7.692

7.704

7.715

7.727

7.739

7.751

8

7.762

7.774

7.786

7.798

7.810

7.821

7.833

7.845

7.857

7.869

9

7.881

7.892

7.904

7.916

7.928

7.940

7.952

7.964

7.976

7.988

= 31.0063 I/T« = 0.0322515 +

10

CUBES (continued)

MATHEMATICAL TABLES

N

C

1

2

3

4

5

6

7

8

9

*S

<'-3

2.00

8.000

8.012

8.024

8.036

8.048

8.060

8.072

8.084

8.096

8.108

12

1

8.';21

8.133

8.145

8.157

8.169

8.181

8.194

8.206

8.218

8.230

2

8.242

8.255

8.267

8.279

8.291

8.304

8.316

8.328

8.341

8.353

3

8.365

8.378

8.390

8.403

8.415

8.427

8.440

8.452

8.465

8.477

4

8.490

8.502

8.515

8.527

8.540

8.552

8.565

8.577

8.590

8.603

2.05

8.615

8.628

8.640

8.653

8.666

8.678

8.691

8.704

8.716

8.729

13

6

8.742

8.755

8.767

8.780

8.793

8.806

8.818

8.831

8.844

8.857

7

8.870

8.883

8.895

8.908

8.921

8.934

8.947

8.960

8.973

8.986

8

8.999

9.012

9.025

9.038

9.051

9.064

9.077

9.090

9.103

9.116

9

9.129

9.142

9.156

9.169

9.182

9.195

9.208

9.221

9.235

9.248

2.10

9.261

9.274

9.287

9.301

9.314

9.327

9.341

9.354

9.367

9.381

1

9.394

9.407

9.421

9.434

9.447

9.461

9.474

9.488

9.501

9.515

2

9.528

9.542

9.555

9.569

9.582

9.596

9.609

9.623

9.636

9.650

14

3

9.664

9.677

9.691

9.704

9.718

9.732

9.745

9.759

9.773

9.787

4

9.800

9.814

9.828

9.842

9.855

9.869

9.883

9.897

9.911

9.925

2.15

9.938

9.952

9.966

9.980

9.994

10.008

14

2.1

9.94

10.08

10.22

10.36

10.50

14

2

10.65

10.79

10.94

11.09

11.24

11.39

11.54

11.70

11.85

12.01

15

3

12.17

12.33

12.49

12.65

12.81

12.98

13.14

13.31

13.48

13.65

16

4

13.82

14.00

14.17

14.35

14.53

14.71

14.89

15.07

15.25

15.44

18

2.5

15.62

15.81

16.00

16.19

16.39

16.58

16.78

16.97

17.17

17.37

20

6

17.58

17.78

17.98

18.19

18.40

18.61

18.82

19.03

19.25

19.47

21

7

19.68

19.90

20.12

20.35

20.57

20.80

21.02

21.25

21.48

21.72

23

8

21.95

22.19

22.43

22.67

22.91

23.15

23.39

23.64

23.89

24.14

24

9

24.39

24.64

24.90

25.15

25.41

25.67

25.93

26.20

26.46

26.73

26

3.0

27.00

27.27

27.54

27.82

28.09

28.37

28.65

28.93

29.22

29.50

28

1

29.79

30.08

30.37

30.66

30.96

31.26

31.55

31.86

32.16

32.46

30

2

32.77

33.08

33.39

33.70

34.01

34.33

34.65

34.97

35.29

35.61

32

3

35.94

36.26

36.59

36.93

37.26

37.60

37.93

38.27

38.61

38.96

34

4

39.30

39.65

40.00

40.35

40.71

41.06

41.42

41.78

42.14

42.51

36

3.5

42.88

43.24

43.61

43.99

44.36

44.74

45.12

45.50

45.88

46.27

39

6

46.66

47.05

47.44

47.83

48.23

48.63

49.03

49.43

49.84

50.24

40

7

50.65

51.06

51.48

51.90

52.31

52.73

53.16

53.58

54.01

54.44

42

8

54.87

55.31

55.74

56.18

56.62

57.07

57.51

57.96

58.41

58.86

44

9

59.32

59.78

60.24

60.70

61.16

61.63

62.10

62.57

63.04

63.52

47

4.0

64.00

64.48

64.96

65.45

65.94

66.43

66.92

67.42

67.92

68.42

49

1

68.92

69.43

69.93

70.44

70.96

71.47

71.99

72.51

73.03

73.56

52

2

74.09

74.62

75.15

75.69

76.23

76.77

77.31

77.85

78.40

78.95

54

3

79.51

80.06

80.62

81.18

81.75

82.31

82.88

83.45

84.03

84.60

58

4

85.18

85.77

86.35

86.94

87.53

88.12

88.72

89.31

89.92

90.52

59

4.5

91.12

91.73

92.35

92.96

93.58

94.20

94.82

95.44

96.07

96.70

62

6

97.34

97.97

98.61

99.25

99.90

100.54

64

6

100.5

101.2

101.8

102.5

103.2

7

7

103.8

104.5

105.2

105.8

106.5

107.2

107.9

108.5

109.2

109.9

7

8

110.6

111.3

112.0

112.7

113.4

114.1

114.8

115.5

116.2

116.9

7

9

117.6

118.4

119.1

119.8

120.6

121.3

122.0

122.8

123.5

124.3

7

Explanation of Table of Cubes (pp. 8-11).

This table gives the value of N* for values of N from 1 to 10, correct to four figures. (Interpolated values may be in error by 1 in the fourth figure.)

To find the cube of a number N outside the range from 1 to 10, note that moving the decimal point one place in column N is equivalent to moving it three places in the body of the table. For example:

(4.852)» = H4.2; (0.4852)» = 0.1142; (485.2)3 = 114200000

This table may also be used inversely, to give cube roots.

MATHEMATICAL TABLES

11

CUBES (continued)

N

0

1

2

3

4

5

6

7

8

9

11

5.0

125.0

125.8

126.5

127.3

128.0

128.8

129.6

130.3

131.1

131.9

8

1

132.7

133.4

134.2

135.0

135.8

136.6

137.4

138.2

139.0

139.8

2

140.6

141.4

142.2

143.1

143.9

144.7

145.5

146.4

147.2

148.0

3

148.9

149.7

150.6

151.4

152.3

153.1

154.0

154.9

155.7

156.6

9

4

157.5

158.3

159.2

160.1

161.0

161.9

162.8

163.7

164.6

165.5

5.5

166.4

167.3

168.2

169.1

170.0

171.0

171.9

172.8

173.7

174.7

6

175.6

176.6

177.5

178.5

179.4

180.4

181.3

182.3

183.3

184.2

10

7

185.2

186.2

187.1

188.1

189.1

190.1

191.1

192.1

193.1

194.1

8

195.1

196.1

197.1

198.2

199.2

200.2

201.2

202.3

203.3

204.3

9

205.4

206.4

207.5

208.5

209.6

210.6

211.7

212.8

213.8

214.9

6.0

216.0

217.1

218.2

219.3

220.3

221.4

222.5

223.6

224.8

225.9

II

1

227.0

228.1

229.2

230.3

231.5

232.6

233.7

234.9

236.0

237.2

2

238.3

239.5

240.6

241.8

243.0

244.1

245.3

246.5

247.7

248.9

12

3

250.0

251.2

252.4

253.6

254.8

256.0

257.3

258.5

259.7

260.9

4

262.1

263.4

264.6

265.8

267.1

268.3

269.6

270.8

272.1

273.4

6.5

274.6

275.9

277.2

278.4

279.7

281.0

282.3

283.6

284.9

286.2

13

6

287.5

288.8

290.1

291.4

292.8

294.1

295.4

296.7

298.1

299.4

7

300.8

302.1

303.5

304.8

306.2

307.5

308.9

310.3

311.7

313.0

14

8

314.4

315.8

317.2

318.6

320.0

321.4

322.8

324.2

325.7

327.1

9

328.5

329.9

331.4

332.8

334.3

335.7

337.2

338.6

340.1

341.5

7.0

343.0

344.5

345.9

347.4

348.9

350.4

351.9

353.4

354.9

356.4

15

1

357.9

359.4

360.9

362.5

364.0

365.5

367.1

368.6

370.1

371.7

2

373.2

374.8

376.4

377.9

379.5

381.1

382.7

384.2

385.8

387.4

16

3

389.0

390.6

392.2

393.8

395.4

397.1

398.7

400.3

401.9

403.6

4

405.2

406.9

408.5

410.2

411.8

413.5

415.2

416.8

418.5

420.2

17

7.5

421.9

423.6

425.3

427.0

428.7

430.4

432.1

433.8

435.5

437.2

6

439.0

440.7

442.5

444.2

445.9

447.7

449.5

451.2

453.0

454.8

18

7

456.5

458.3

460.1

461.9

463.7

465.5

467.3

469.1

470.9

472.7

8

474.6

476.4

478.2

480.0

481.9

483.7

485.6

487.4

489.3

491.2

9

493.0

494.9

496.8

498.7

500.6

502.5

504.4

506.3

508.2

510.1

19

8.0

512.0

513.9

515.8

517.8

519.7

521.7

523.6

525.6

527.5

529.5

1

531.4

533.4

535.4

537.4

539.4

541.3

543.3

545.3

547.3

549.4

20

2

551.4

553.4

555.4

557.4

559.5

561.5

563.6

565.6

567.7

569.7

3

571.8

573.9

575.9

578.0

580.1

582.2

584.3

586.4

588.5

590.6

21

4

592.7

594.8

596.9

599.1

601.2

603.4

605.5

607.6

609.8

612.0

8.5

614.1

616.3

618.5

620.7

622.8

625.0

627.2

629.4

631.6

633.8

22

6

636.1

638.3

640.5

642.7

645.0

647.2

649.5

651.7

654.0

656.2

7

658.5

660.8

663.1

665.3

667.6

669.9

672.2

674.5

676.8

679.2

23

8

681.5

683.8

686.1

688.5

690.8

693.2

695.5

697.9

700.2

702.6

24

9

705.0

707.3

709.7

712.1

714.5

716.9

719.3

721.7

724.2

726.6

9.0

729.0

731.4

733.9

736.3

738.8

741.2

743.7

746.1

748.6

751.1

25

1

753.6

756.1

758.6

761.0

763.6

766.1

768.6

771.1

773.6

776.2

2

778.7

781.2

783.8

786.3

788.9

791.5

794.0

796.6

799.2

801.8

26

3

804.4

807.0

809.6

812.2

814.8

817.4

820.0

822.7

825.3

827.9

4

830.6

833.2

835.9

838.6

841.2

843.9

846.6

849.3

852.0

854.7

27

9.5

857.4

860.1

862.8

865.5

868.3

871.0

873.7

876.5

879.2

882.0

6

884.7

887.5

890.3

893.1

895.8

898.6

901.4

904.2

907.0

909.9

28

7

912.7

915.5

918.3

921.2

924.0

926.9

929.7

932.6

935.4

938.3

8

941.2

944.1

947.0

949.9

952.8

955.7

958.6

961.5

964.4

967.4

29

9

970.3

973.2

976.2

979.1

982.1

985.1

988.0

991.0

994.0

997.0

10.0

1000.0

= 3 1.0063

= 0.0322515 +

Moving the decimal point ONE place in N requires moving it THREE places in body of table (see p. 10).

12 MATHEMATICAL TABLES

SQUARE ROOTS OF NUMBERS

N

0

1

2

3

4

5

6

7

8

9

<*)ttt &

1.0

1.000

1.005

1.010

1.015

1.020

1.025

1.030

1.034

1.039

1.044

5

1

1.049

1.054

1.058

1.063

1.068

1.072

1.077

1.082

1.086

1.091

2

1.095

1.100

1.105

1.109

1.114

1.118

1.122

1.127

1.131

1.136

4

3

1.140

1.145

1.149

1.153

1.158

1.162

1.166

1.170

1.175

1.179

4

1.183

1.187

1.192

1.196

1.200

1.204

1.208

1.212

1.217

1.221

1.5

1.225

1.229

1.233

1.237

1.241

1.245

1.249

1.253

1.257

1.261

6

1.265

1.269

1.273

1.277

1.281

1.285

1.288

1.292

1.296

1.300

7

1.304

1.308

1.311

1.315

1.319

1.323

1.327

1.330

1.334

1.338

8

1.342

1.345

1.349

1.353

1.356

1.360

1.364

1.367

1.371

1.375

9

1.378

1.382

1.386

1.389

1.393

1.396

1.400

1.404

1.407

1.411

2.0

1.414

1.418

1.421

1.425

1.428

1.432

1.435

1.439

1.442

1.446

1

1.449

1.453

1.456

1.459

1.463

» 1.466

1.470

1.473

1.476

1.480

3

2

1.483

1.487

1.490

1.493

1.497

1.500

1.503

1.507

1.510

1.513

3

1.517

1.520

1.523

1.526

1.530

1.533

1.536

1.539

1.543

1.546

4

1.549

1.552

1.556

1.559

1.562

1.565

1.568

1.572

1.575

1.578

2.5

1.581

1.584

1.587

1.591

1.594

1.597

1.600

1.603

1.606

1.609

6

1.612

1.616

1.619

1.622

1.625

1.628

1.631

1.634

1.637

1.640

7

1.643

1.646

1.649

1.652

1.655

1.658

1.661

1.664

1.667

1.670

8

1.673

1.676

1.679

1.682

1.685

1.688

1.691

1.694

1.697

1.700

9

1.703

1.706

1.709

1.712

1.715

1.718

1.720

1.723

1.726

1.729

3.0

1.732

1.735

1.738

1.741

1.744

1.746

1.749

1.752

1.755

1.758

1

1.761

1.764

1.766

1.769

1.772

1.775

1.778

1.780

1.783

1.786

2

1.789

1.792

1.794

1.797

1.800

1.803

1.806

1.808

1.811

1.814

3

1.817

1.819

1.822

1.825

1.828

1.830

1.833

1.836

1.838

1.841

4

1.844

1.847

1.849

1.852

1.855

1.857

1.860

1.863

1.865

1.868

3.5

1.871

1.873

1.876

1.879

1.881

1.884

1.887

1.889

1.892

1.895

6

1.897

1.900

1.903

1.905

1.908

1.910

1.913

1.916

1.918

1.921

7

1.924

1.926

1.929

1.931

1.934

1.936

1.939

1.942

1.944

1.947

8

1.949

1.952

1.954

1.957

1.960

1.962

1.965

1.967

1.970

1.972

9

1.975

1.977

1.980

1.982

1.985

1.987

1.990

1.992

1.995

1.997

4.0

2.000

2.002

2.005

2.007

2.010

2.012

2.015

2.017

2.020

2.022

1

2.025

2.027

2.030

2.032

2.035

2.037

2.040

2.042

2.045

2.047

2

2

2.049

2.052

2.054

2.057

2.059

2.062

2.064

2.066

2.069

2.071

3

2.074

2.076

2.078

2.081

2.083

2.086

2.088

2.090

2.093

2.095

4

2.098

2.100

2.102

2.105

2.107

2.110

2.112

2.114

2.117

2.119

4.5

2.121

2.124

2.126

2.128

2.131

2.133

2.135

2.138

2.140

2.142

6

2.145

2.147

2.149

2.152

2.154

2.156

2.159

2.161

2.163

2.166

7

2.168

2.170

2.173

2.175

2.177

2.179

2.182

2.184

2.186

2.189

8

2.191

2.193

2.195

2.198

2.200

2.202

2.205

2.207

2.209

2.211

9

2.214

2.216

2.218

2.220

2.223

2.225

2.227

2.229

2.232

2.234

yV= 1.77245 +

= 0.56419

1.25331

1.64872

Explanation of Table of Square Roots (pp. 12-15).

This table gives the values of \/N for values of N from 1 to 100, correct to four figures. (Interpolated values may be in error by 1 in the fourth figure.)

To find the square root of a number N outside the range from 1 to 100, divide the digits of the number into blocks of two (beginning with the decimal point), and note that moving the decimal point two places in N is equivalent to moving it one place in the square root of N. For example:

X/2.718 = 1.648; A/271. 8 - 16.48; V0.0002718 = 0.01648; V27.18 => 5.213; -N/2718 - 52.13; V/oTo02718 - 0.05213.

MATHEMATICAL TABLES SQUARE ROOTS (continued)

N

0

1

2

3

4

5

<*

5.0

2.236

2.238

2.241

2.243

2.245

2.247

2.249

2.252

2.254

2.256

2

1

2.258

2.261

2.263

2.265

2.267

2.269

2.272

2.274

2.276

2.278

2

2.280

2.283

2.285

2.287

2.289

2.291

2.293

2.296

2.298

2.300

3

2.302

2.304

2.307

2.309

2.311

2.313

2.315

2.317

2.319

2.322

4

2.324

2.326

2.328

2.330

2.332

2.335

2.337

2.339

2.341

2.343

6.5

2.345

2.347

2.349

2.352

2.354

2.356

2.358

2.360

2.362

2.364

6

2.366

2.369

2.371

2.373

2.375

2.377

2.379

2.381

2.383

2.385

7

2.387

2.390

2.392

2.394

2.396

2.398

2.400

2.402

2.404

2.406

8

2.408

2.410

2.412

2.415

2.417

2.419

2.421

2.423

2.425

2.427

9

2.429

2.431

2.433

2.435

2.437

2.439

2.441

2.443

2.445

2.447

6.0

2.449

2.452

2.454

2.456

2.458

2.460

2.462

2.464

2.466

2.468

1

2.470

2.472

2.474

2.476

2.478

2.480

2.482

2.484

2.486

2.488

2

2.490

2.492

2.494

2.496

2.498

2.500

2.502

2.504

2.506

2.508

3

2.510

2.512

2.514

2.516

2.518

2.520

2.522

2.524

2.526

2.528

4

2.530

2.532

2.534

2.536

2.538

2.540

2.542

2.544

2.546

2.548

6.6

2.550

2.551

2.553

2.555

2.557

2.559

2.561

2.563

2.565

2.567

6

2.569

2.571

2.573

2.575

2.577

2.579

2.581

2.583

2.585

2.587

7

2.588

2.590

2.592

2.594

2.596

2.598

2.600

2.602

2.604

2.606

8

2.608

2.610

2.612

2.613

2.615

2.617

2.619

2.621

2.623

2.625

9

2.627

2.629

2.631

2.632

2.634

2.636

2.638

2.640

2.642

2.644

7.0

2.646

2.648

2.650

2.651

2.653

2.655

2.657

2.659

2.661

2.663

1

2.665

2.666

2.668

2.670

2.672

2.674

2.676

2.678

2.680

2.681

2

2.683

2.685

2.687

2.689

2.691

2.693

2.694

2.696

2.698

2.700

3

2.702

2.704

2.706

2.707

2.709

2.711

2.713

2.715

2.717

2.718

4

2.720

2.722

2.724

2.726

2.728

2.729

2.731

2.733

2.735

2.737

7.5

2.739

2.740

2.742

2.744

2.746

2.748

2.750

2.751

2.753

2.755

6

2.757

2.759

2.760

2.762

2.764

2.766

2.768

2.769

2.771

2.773

7

2.775

2.777

2.778

2.780

2.782

2.784

2.786

2.787

2.789

2.791

8

2.793

2.795

2.796

2.798

2.800

2.802

2.804

2.805

2.807

2.809

9

2.811

2.812

2.814

2.816

2.818

2.820

2.821

2.823

2.825

2.827

8.0

2.828

2.830

2.832

2.834

2.835

2.837

2.839

2.841

2.843

2.844

2.846

2.848

2.850

2.851

2.853

2.855

2.857

2.858

2.860

2.862

2

2.864

2.865

2.867

2.869

2.871

2.872

2.874

2.876

2.877

2.879

3

2.881

2.883

2.884

2.886

2.888

2.890

2.891

2.893

2.895

2.897

4

2.898

2.900

2.902

2.903

2.905

2.907

2.909

2.910

2.912

2.914

8.5

2.915

2.917

2.919

2.921

2.922

2.924

2.926

2.927

2.929

2.931

6

2.933

2.934

2.936

2.938

2.939

2.941

2.943

2.944

2.946

2.948

7

2.950

2.951

2.953

2.955

2.956

2.958

2.960

2.961

2.963

2.965

8

2.966

2.968

2.970

2.972

2.973

2.975

2.977

2.978

2.980

2.982

9

2.983

2.985

2.987

2.988

2.990

2.992

2.993

2.995

2.997

2.998

9.0

3.000

3.002

3.003

3.005

3.007

3.008

3.010

3.012

3.013

3.015

1

3.017

3.018

3.020

3.022

3.023

3.025

3.027

3.028

3.030

3.032

2

3.033

3.035

3.036

3.038

3.040

3.041

3.043

3.045

3.046

3.048

3

3.050

3.051

3.053

3.055

3.056

3.058

3.059

3.061

3.063

3.064

4

3.066

3.068

3.069

3.071

3.072

3.074

3.076

3.077

3.079

3.081

9.5

3.082

3.084

3.085

3.087

3.089

3.090

3.092

3.094

3.095

3.097

6

3.098

3.100

3.102

3.103

3.105

3.106

3.108

3.110

3.111

3.113

7

3.114

3.116

3.118

3.119

3.121

3.122

3.124

3.126

3.127

3.129

8

3.130

3.132

3.134

3J35

3.137

3.138

3.140

3.142

3.143

3.145

9

3.146

3.148

3.150

3.151

3.153

3.154

3.156

3.158

3.159

3.161

Moving the decimal point TWO places in N requires moving it ONE place in body of table (see p. 12).

14 MATHEMATICAL TABLES

SQUARE ROOTS (continued)

N

0

1

2

3

4

5

6

7

8

9

«d $*

10.

3.162

3.178

3.194

3.209

3.225

3.240

3.256

3.271

3.286

3.302

16

1.

3.317

3.332

3.347

3.362

3.376

3.391

3.406

3.421

3.435

3.450

15

2.

3.464

3.479

3.493

3.507

3.521

3.536

3.550

3.564

3.578

3.592

14

3.

3.606

3.619

3.633

3.647

3.661

3.674

3.688

3.701

3.715

3.728

4.

3.742

3.755

3.768

3.782

3.795

3.808

3.821

3.834

3.847

3.860

13

15.

3.873

3.886

3.899

3.912

3.924

3.937

3.950

3.962

3.975

3.987

6.

4.000

4.012

4.025

4.037

4.050

4.062

4.074

4.087

4.099

4.111

12

7.

4.123

4.135

4.147

4.159

4.171

4.183

4.195

4.207

4.219

4.231

8.

4.243

4.254

4.266

4.278

4.290

4.301

4.313

4.324

4.336

4.347

9.

4.359

4.370

4.382

4.393

4.405

4.416

4.427

4.438

4.450

4.461

11

20.

4.472

4.483

4.494

4.506

4.517

4.528

4.539

4.550

4.561

4.572

1.

4.583

4.593

4.604

4.615

4.626

4.637

4.648

4.658

4.669

4.680

2.

4.690

4.701

4.712

4.722

4.733

4.743

4.754

4.764

4.775

4.785

3.

4.796

4.806

4.817

4.827

4.837

4.848

4.858

4.868

4.879

4.889

10

4.

4.899

4.909

4.919

4.930

4.940

4.950

4.960

4.970

4.980

4.990

25.

5.000

5.010

5.020

5.030

5.040

5.050

5.060

5.070

5.079

5.089

6

5.099

5.109

5.119

5.128

5.138

5.148

5.158

5.167

5.177

5.187

7.

5.196

5.206

5.215

5.225

5.235

5.244

5.254

5.263

5.273

5.282

8.

5.292

5.301

5.310

5.320

5.329

5.339

5.348

5.357

5.367

5.376

9

9.

5.385

5.394

5.404

5.413

5.422

5.431

5.441

5.450

5.459

5.468

30.

5.477

5.486

5.495

5.505

5.514

5.523

5.532

5.541

5.550

5.559

5.568

5.577

5.586

5.595

5.604

5.612

5.621

5.630

5.639

5.648

2!

5.657

5.666

5.675

5.683

5.692

5.701

5.710

5.718

5.727

5.736

3.

5.745

5.753

5.762

5.771

5.779

5.788

5.797

5.805

5.814

5.822

4.

5.831

5.840

5.848

5.857

5.865

5.874

5.882

5.891

5.899

5.908

8

35.

5.916

5.925

5.933

5.941

5.950

5.958

5.967

5.975

5.983

5.992

6.

6.000

6.008

6.017

6.025

6.033

6.042

6.050

6.058

6.066

6.075

7.

6.083

6.091

6.099

6.107

6.116

6.124

6.132

6.140

6.148

6.156

8.

6.164

6.173

6.181

6.189

6.197

6.205

6.213

6.221

6.229

6.237

9.

6.245

6.253

6.261

6.269

6.277

6.285

6.293

6.301

6.309

6.317

40.

6.325

6.332

6.340

6.348

6.356

6.364

6.372

6.380

6.387

6.395

1.

6.403

6.411

6.419

6.427

6.434

6.442

6.450

6.458

6.465

6.473

2.

6.481

6.488

6.496

6.504

6.512

6.519

6.527

6.535

6.542

6.550

3.

6.557

6.565

6.573

6.580

6.588

6.595

6.603

6.611

6.618

6.626

4.

6.633

6.641

6.648

6.656

6.663

6.671

6.678

6.686

6.693

6.701

45.

6.708

6.716

6.723

6.731

6.738

6.745

6.753

6.760

6.768

6.775

7

6.

6.782

6.790

6.797

6.804

6.812

6.819

6.826

6.834

6.841

6.848

7.

6.856

6.863

6.870

6.877

6.885

6.892

6.899

6.907

6.914

6.921

8.

6.928

6.935

6.943

6.950

6.957

6.964

6.971

6.979

6.986

6.993

9.

7.000

7.007

7.014

7.021

7.029

7.036

7.043

7.050

7.057

7.064

SQUARE ROOTS OF CERTAIN FRACTIONS

N

VN

N

VN

AT

VN

N

VN

N

VN

N

VN

y*

0.7071

H

0.7746

M

0.7559

H

0.3333

Hi

0.6455

M«

0.7500

y&

0.5774

%

0.8944 ,

W

0.8452

%

0.4714

til

0.7638

iM«

0.8292

g

0.8165

H

0.4082

M

0.9258

V*

0.6667

JM2

0.9574

13/f«

0.9014

0.5000

%

0.9129

M

0.3536

%

0.7454

H«

0.2500

ls/fe

0.9682

94

0.8660

W

0.3780

**

0.6124

7^

0.8819

M«

0.4330

to

0.1768

%

0.4472 0.6325

M H

0.5345 0.6547

H H

0.7906 0.9354

%

0.9428 0.2887

Me Me

0.5590 0.6614

B

0.1250 0.1414

MATHEMATICAL TABLES SQUARE ROOTS (continued)

N

0

1

2

a

4

5

6

7

8

9

wsd $*

50.

7.071

7.078

7.085

7.092

7.099

7.106

7.113

7.120

7.127

7.134

7

1.

7.141

7.148

7.155

7.162

7.169

7.176

7.183

7.190

7.197

7.204

2.

7.211

7.218

7.225

7.232

7.239

7.246

7.253

7.259

7.266

7.273

3.

7.280

7.287

7.294

7.301

7.308

7.314

7.321

7.328

7.335

7.342

4.

7.348

7.355

7.362

7.369

7.376

7.382

7.389

7.396

7.403

7.409

55.

7.416

7.423

7.430

7.436

7.443

7.450

7.457

7.463

7.470

7.477

6.

7.483

7.490

7.497

7.503

7.510

7.517

7.523

7.530

7.537

7.543

7.

7.550

7.556

7.563

7.570

7.576

7.583

7.589

7.596

7.603

7.609

8.

7.616

7.622

7.629

7.635

7.642

7.649

7.655

7.662

7.668

7.675

9.

7.681

7.688

7.694

7.701

7.707

7.714

7.720

7.727

7.733

7.740

6

60.

7.746

7.752

7.759

7.765

7.772

7.778

7.785

7.791

7.797

7.804

1.

7.810

7.817

7.823

7.829

7.836

7.842

7.849

7.855

7.861

7.868

2.

7.874

7.880

7.887

7.893

7.899

7.906

7.912

7.918

7.925

7.931

3.

7.937

7.944

7.950

7.956

7.962

7.969

7.975

7.981

7.987

7.994

4.

8.000

8.006

8.012

8.019

8.025

8.031

8.037

8.044

8.050

8.056

65.

8.062

8.068

8.075

8.081

8.087

8.093

8.099

8.106

8.112

8.118

6.

8.124

8.130

8.136

8.142

8.149

8.155

8.161

8.167

8.173

8.179

7.

8.185

8.191

8.198

8.204

8.210

8.216

8.222

8.228

8.234

8.240

8.

8.246

8.252

8.258

8.264

8.270

8.276

8.283

8.289

8.295

8.301

9.

8.307

8.313

8.319

8.325

8.331

8.337

8.343

8.349

8.355

8.361

70.

8.367

8.373

8.379

8.385

8.390

8.396

8.402

8.408

8.414

8.420

1.

8.426

8.432

8.438

8.444

8.450

8.456

8.462

8.468

8.473

8.479

2.

8.485

8.491

8.497

8.503

8.509

8.515

8.521

8.526

8.532

8.538

3.

8.544

8.550

8.556

8.562

8.567

8.573

8.579

8.585

8.591

8.597

4.

8.602

8.608

8.614

8.620

8.626

8.631

8.637

8.643

8.649

8.654

75.

8.660

8.666

8.672

8.678

8.683

8.689

8.695

8.701

8.706

8.712

6.

8.718

8.724

8.729

8.735

8.741

8.746

8.752

8.758

8.764

8.769

7.

8.775

8.781

8.786

8.792

8.798

8.803

8.809

8.815

8.820

8.826

8.

8.832

8.837

8.843

8.849

8.854

8.860

8.866

8.871

8.877

8.883

9.

8.888

8.894

8.899

8.905

8.911

8.916

8.922

8.927

8.933

8.939

80.

8.944

8.950

8.955

8.961

8.967

8.972

8.978

8.983

8.989

8.994

1.

9.000

9.006

9.011

9.017

9.022

9.028

9.033

9.039

9.044

9.050

2«

9.055

9.061

9.066

9.072

9.077

9.083

9.088

9.094

9.099

9.105

5

3.

9.110

9.116

9.121

9.127

9.132

9.138

9.143

9.149

9.154

9.160

4.

9.165

9.171

9.176

9.182

9.187

9.192

9.198

9.203

9.209

9.214

85.

9.220

9.225

9.230

9.236

9.241

9.247

9.252

9.257

9.263

9.268

6.

9.274

9.279

9.284

9.290

9.295

9.301

9.306

9.311

9.317

9.322

7.

9.327

9.333

9.338

9.343

9.349

9.354

9.359

9.365

9.370

9.375

8.

9.381

9.386

9.391

9.397

9.402

9.407

9.413

9.418

9.423

9.429

9.

9.434

9.439

9.445

9.450

9.455

9.460

9.466

9.471

9.476

9.482

90.

9.487

9.492

9.497

9.503

9.508

9.513

9.518

9.524

9.529

9.534

1.

9.539

9.545

9.550

9.555

9.560

9.566

9.571

9.576

9.581

9.586

2.

9.592

9.597

9.602

9.607

9.612

9.618

9.623

9.628

9.633

9.638

3.

9.644

9.649

9.654

9.659

9.664

9.670

9.675

9.680

9.685

9.690

4.

9.695

9.701

9.706

9.711

9.716

9.721

9.726

9.731

9.737

9.742

95.

9.747

9.752

9.757

9.762

9.767

9.772

9.778

9.783

9.788

9.793

6.

9.798

9.803

9.808

9.813

9.818

9.823

9.829

9.834

9.839

9.844

7.

9.849

9.854

9.859

9.864

9.869

9.874

9.879

9.884

9.889

9.894

8.

9.899

9.905

9.910

9.915

9.920

9.925

9.930

9.935

9.940

9.945

9.

9.950

9.955

9.960

9.965

9.970

9.975

9.980

9.985

9.990

9.995

= 1.77245+

0.56419

= 1.25331

= 1.64872

Moving the decimal point TWO places in N requires moving it ONE place in body of table (seep. 12).

16 MATHEMATICAL TABLES

CUBE ROOTS OF NUMBERS

TV

0

1

2

3

4

5

6

7

8

9

*£

4*

1.0

1.000

1.003

1.007

1.010

1.013

1.016

1.020

1.023

1.026

1.029

3

1

1.032

1.035

1.038

1.042

1.045

1.048

1.051

1.054

1.057

1.060

2

1.063

1.066

1.069

1.071

1.074

1.077

1.080

1.083

1.086

1.089

3

1.091

.094

1.097

.100

1.102

.105

1.108

.111

1.113

1.116

4

1.119

.121

1.124

.127

1.129

.132

1.134

.137

1.140

1.142

1.5

1.145

.147

1.150

.152

1.155

.157

1.160

.162

1.165

1.167

2

6

1.170

.172

1.174

.177

1.179

.182

1.184

.186

1.189

1.191

7

1.193

.196

1.198

.200

1.203

.205

1.207

.210

1.212

1.214

8

1.216

1.219

1.221

.223

1.225

1.228

1.230

.232

1.234

1.236

9

1.239

1.241

1.243

1.245

1.247

1.249

1.251

.254

1.256

1.258

2.0

1.260

1.262

1.264

1.266

1.268

1.270

1.272

.274

1.277

1.279

1

1.281

1.283

1.285

1.287

1.289

1.291

1.293

.295

1.297

1.299

2

1.301

.303

1.305

1.306

1.308

1.310

1.312

.314

1.316

1.318

3

1.320

1.322

1.324

1.326

1.328

1.330

1.331

.333

1.335

1.337

4

1.339

1.341

1.343

1.344

1.346

1.348

1.350

.352

1.354

1.355

2.5

1.357

1.359

1.361

1.363

1.364

1.366

1.368

.370

1.372

1.373

6

1.375

1.377

1.379

1.380

1.382

1.384

1.386

.387

1.389

1.391

7

1.392

.394

1.396

1.398

1.399

1.401

1.403

.404

1.406

1.408

I 8

1.409

1.411

1.413

1.414

1.416

.418

1.419

.421

1.423

1.424

9

1.426

.428

1.429

1.431

1.433

1.434

1.436

1.437

1.439

1.441

3.0

1.442

1.444

1.445

1.447

1.449

.450

1.452

1.453

1.455

1.457

1

1.458

1.460

1.461

1.463

1.464

.466

1.467

1.469

1.471

1.472

2

1.474

.475

1.477

1.478

1.480

1.481

1.483

1.484

1.486

1.487

3

1.489

1.490

1.492

1.493

1.495

1.496

1.498

1.499

1.501

1.502

4

1.504

1.505

1.507

1.508

1.510

1.511

1.512

1.514

1.515

1.517

3.5

1.518

1.520

1.521

1.523

1.524

1.525

1.527

1.528

1.530

1.531

6

1.533

.534

1.535

1.537

1.538

1.540

1.541

1.542

1.544

1.545

1

7

1.547

1.548

1.549

1.551

1.552

1.554

1.555

1.556

1.558

1.559

8

1.560

1.562

1.563

1.565

1.566

.567

1.569

1.570

1.571

1.573

9

1.574

1.575

1.577

1.578

1.579

1.581

1.582

1.583

1.585

1.586

4.0

1.587

1.589

1.590

1.591

1.593

1.594

1.595

1.597

1.598

1.599

1

1.601

1.602

1.603

1.604

1.606

1.607

1.608

1.610

1.611

1.612

2

1.613

1.615

1.616

1.617

1.619

1.620

1.621

1.622

1.624

1.625

3

1.626

1.627

1.629

1.630

1.631

1.632

1.634

1.635

1.636

1.637

4

1.639

1.640

1.641

1.642

1.644

1.645

1.646

1.647

1.649

1.650

4.5

1.651

1.652

1.653

1.655

1.656

1.657

1.658

1.659

1.661

1.662

6

1.663

1.664

1.666

1.667

1.668

1.669

1.670

1.671

1.673

1.674

7

1.675

1.676

1.677

1.679

1.680

1.681

1.682

1.683

1.685

1.686

8

1.687

1.688

1.689

1.690

1.692

1.693

1.694

1.695

1.696

1.697

9

1.698

1.700

1.701

1.702

1.703

1.704

1.705

1.707

1.708

1.709

1.46459 l/V^r~= 0.682784

Explanation of Table of Cube Roots (pp. 16-21).

This table gives the values of \/TV for all values of TV from 1 to 1000, correct to four figures. (Interpolated values may be in error by 1 in the fourth figure.)

To find the cube root of a number N outside the range from 1 to 1000, divide the digits of the number into blocks of three (beginning with the decimal point), and note that moving the decimal point three places in column N is equivalent to moving it one place in the cube root of TV. For example:

•y^.718 = 1.396; -y^2718 - 13.96; -^0.000002718 = 0.01396.

-^27.18 = 3.007; -^27180 = 30.07; -^0.00002718 = 0.03007.

•^271.8 = 6.477; -^271800 = 64.77; -^0.0002718 = 0.06477.

MATHEMATICAL TABLES CUBE ROOTS (continued)

17

N

0

1

2

3

4

5

6

7

8

9

£

6.0

1.710

1.711

1712

1.713

1.715

1.716

1.717

1.718

1.719

1.720

1

1.721

1.722

1.724

1.725

1.726

1.727

1.728

1.729

1.730

1.731

2

1.732

1.734

1.735

1.736

1.737

1.738

1.739

1.740

1.741

1.742

3

1.744

1.745

1.746

1.747

1.748

1.749

1.750

1.751

1.752

1.753

4

1.754

1.755

1.757

1.758

1.759

1.760

1.761

1.762

1.763

1.764

5.5

1.765

1.766

1.767

1.768

1.769

1.771

1.772

1.773

1.774

1.775

6

1.776

1.777

1.778

1.779

1.780

1.781

1.782

1.783

1.784

1.785

7

1.786

1.787

1.788

1.789

1.790

1.792

1.793

1.794

1.795

1.796

8

1.797

1.798

1.799

1.800

1.801

1.802

1.803

1.804

1.805

1.806

9

1.807

1.808

1.809

1.810

1.811

1.812

1.813

1.814

1.815

1.816

6.0

1.817

1.818

1.819

1.820

1.821

1.822

1.823

1.824

1.825

1.826

1

1.827

1.828

1.829

1.830

1.831

1.832

1 833

1.834

1.835

1.836

2

1.837

1.838

1.839

1.840

1.841

1.842

1.843

1.844

1.845

1.846

3

1.847

1.848

1.849

1.850

1.851

1.852

1.853

1.854

1.855

1.856

4

1.857

1.858

1.859

1.860

1.860

1.861

1.862

1.863

1.864

1.865

6.5

1.866

1.867

1.868

1.869

1.870

1.871

1.872

1.873

1.874

1.875

6

1.876

1.877

1.878

1.879

1.880

1.881

1.881

1.882

1.883

1.884

7

1.885

1.886

1.887

1.888

1.889

1.890

1.891

1.892

1.893

1.894

8

1.895

1.895

1.896

1.897

1.898

1.899

1.900

1.901

1.902

1.903

9

1.904

1.905

1.906

1.907

1.907

1.908

1.909

1.910

1.911

1.912

7.0

1.913

1.914

1.915

1.916

1.917

1.917

1.918

1.919

1.920

1.921

1

1.922

1.923

1.924

1.925

1.926

1.926

1.927

1.928

1.929

1.930

2

1.931

1.932

1.933

1.934

1.935

1.935

1.936

1.937

1.938

1.939

3

1.940

1.941

1.942

1.943

1.943

1.944

1.945

1.946

1.947

1.948

4

1.949

1.950

1.950

1.951

1.952

1.953

1.954

1.955

1.956

1.957

7.5

1.957

1.958

1.959

1.960

1.961

1.962

1.963

1.964

1.964

1.965

6

1.966

1.967

1.968

1.969

1.970

1.970

1.971

1.972

1.973

1.974

7

1.975

1.976

1.976

1.977

1.978

1.979

1.980

1.981

1.981

1.982

8

1 983

1.984

1.985

1.986

1.987

1.987

1.988

1.989

1.990

1.991

9

1.992

1.992

1.993

1.994

1.995

1.996

1.997

1.997

1.998

1.999

8.0

2.000

2.001

2.002

2.002

2.003

2.004

2.005

2.006

2.007

2.007

1

2.008

2.009

2.010

2.01 1

2.012

2.012

2.013

2.014

2.015

2.016

2

2.017

2.017

2.018

2.019

2.020

2.021

2.021

2.022

2.023

2.024

3

2.025

2.026

2.026

2.027

2.028

2.029

2.030

2.030

2.031

2.032

4

2.033

2.034

2.034

2.035

2.036

2.037

2.038

2.038

2.039

2.040

8.5

2.041

2.042

2.042

2.043

2.044

2.045

2.046

2.046

2.047

2.048

6

2.049

2.050

2.050

2.051

2.052

2.053

2.054

2.054

2.055

2.056

7

2.057

2.057

2.058

2.059

2.060

2.061

2.061

2.062

2.063

2.064

8

2.065

2.065

2.066

2.067

2.068

2.068

2.069

2.070

2.071

2.072

9

2.072

2.073

2.074

2.075

2.075

2.076

2.077

2.078

2.079

2.079

9.0

2.080

2.081

2.082

2.082

2.083

2.084

2.085

2.085

2.086

2.087

1

2.088

2.089

2.089

2.090

2.091

2.092

2.092

2.093

2.094

2.095

2

2.095

2.096

2.097

2.098

2.098

2.099

2.100

2.101

2.101

2.102

3

2.103

2.104

2.104

2.105

2.106

2.107

2.107

2.108

2.109

2.110

4

2.110

2.111

2.112

2.113

2.113

. 2.114

2.115

2.116

2.116

2.117

9.5

2.118

2.119

2.119

2.120

2.121

2.122

2.122

2.123

2.124

2.125

6

2.125

2.126

2.127

2.128

2.128

2.129

2.130

2.130

2.131

2.132

7

2.133

2.133

2.134

2.135

2.136

2.136

2.137

2.138

2.139

2.139

8

2.140

2.141

2.141

2.142

2.143

2.144

2.144

2.145

2.146

2.147

9

2.147

2.148

2.149

2.149

2.150

2.151

2.152

2.152

2.153

2.154

I

Moving the decimal point THREE places in N requires moving it ONE place in body of table (see p. 16). 2

18

MATHEMATICAL TABLES

CUBE BOOTS (continued)

N

0

1

2

3

4

5

6

7

8

9

^ <j-o

10.

2.154

2.162

2.169

2.176

2.183

2.190

2.197

2.204

2.210

2.217

7

1.

2.224

2.231

2.237

2.244

2.251

2.257

2.264

2.270

2.277

2.283

6

2.

2.289

2.296

2.302

2.308

2.315

2.321

2.327

2.333

2.339

2.345

3.

2.351

2.357

2.363

2.369

2.375

2.381

2.387

2.393

2.399

2.404

4.

2.410

2.416.

2.422

2.427

2.433

2.438

2.444

2.450

2.455

2.461

15.

2.466

2.472

2.477

2.483

2.488

2.493

2.499

2.504

2.509

2.515

5

6.

2.520

2.525

2.530

2.535

2.541

2.546

2.551

2.556

2.561

2.566

7.

2.571

2.576

2.581

2.586

2.591

2.596

2.601

2.606

2.611

2.616

8.

2.621

2.626

2.630

2.635

2.640

2.645

2.650

2.654

2.659

2.664

9.

2.668

2.673

2.678

2.682

2.687

2.692

2.696

2.701

2.705

2.710

20.

2.714

2.719

2.723

2.728

2.732

2.737

2.741

2.746

2.750

2.755

4

1.

2.759

2.763

2.768

2.772

2.776

2.781

2.785

2.789

2.794

2.798

2.

2.802

2.806

2.811

2.815

2.819

2.823

2.827

2.831

2.836

2.840

3.

2.844

2.848

2.852

2.856

2.860

2.864

2.868

2.872

2.876

2.880

4.

2.884

2.888

2.892

2.896

2.900

2.904

2.908

2.912

2.916

2.920

25.

2.924

2.928

2.932

2.936

2.940

2.943

2.947

2.951

2.955

2.959

6.

2.962

2.966

2.970

2.974

2.978

2.981

2.985

2.989

2.993

2.996

7.

3.000

3.004

3.007

3.011

3.015

3.018

3.022

3.026

3.029

3.033

8.

3.037

3.040

3.044

3.047

3.051

3.055

3.058

3.062

3.065

3.069

9.

3.072

3.076

3.079

3.083

3.086

3.090

3.093

3.097

3.100

3.104

30.

3.107

3.111

3.114

3.118

3.121

3.124

3.128

3.131

3.135

3.138

3

1.

3.141

3.145

3.148

3.151

3.155

3.158

3.162

3.165

3.168

3.171

2.

3.175

3.178

3.181

3.185

3.188

3.191

3.195

3.198

3.201

3.204

3.

3.208

3.21 1

3.214

3.217

3.220

3.224

3.227

3.230

3.233

3.236

4.

3.240

3.243

3.246

3.249

3.252

3.255

3.259

3.262

3.265

3.268

35.

3.271

3.274

3.277

3.280

3.283

3.287

3.290

3.293

3.296

3.249

6.

3.302

3.305

3.308

3.311

3.314

3.317

3.320

3.323

3.326

3.329

7.

3.332

3.335

3.338

3.341

3.344

3.347

3.350

3.353

3.356

3.359

8.

3.362

3.365

3.368

3.371

3.374

3.377

3.380

3.382

3.385

3.388

9.

3.391

3.394

3.397

3.400

3.403

3.406

3.409

3.411

3.414

3.417

40.

3.420

3.423

3.426

3.428

3.431

3.434

3.437

3.440

3.443

3.445

I.

3.448

3.451

3.454

3.457

3.459

3.462

.3.465

3.468

3.471

3.473

2.

3.476

3.479

3.482

3.484

3.487

3.490

3.493

3.495

3.498

3.501

3.

3.503

3.506

3.509

3.512

3.514

3.517

3.520

3.522

3.525

3.528

4.

3.530

3.533

3.536

3.538

3.541

3.544

3.546

3.549

3.552

3.554

45.

3.557

3.560

3.562

3.565

3.567

3.570

3.573

3.575

3.578

3.580

6.

3.583

3.586

3.588

3.591

3.593

3.596

3.599

3.601

3.604

3.606

7.

3.609

3.611

3.614

3.616

3.619

3.622

3.624

3.627

3.629

3.632

8.

3.634

3.637

3.639

3.642

3.644

3.647

3.649

3.652

3.654

3.657

2

9.

3.659

3.662

3.664

3.667

3.669

3.672

3.674

3.677

3.679

3.682

CUBE ROOTS OF CERTAIN FRACTIONS

N

y*

N

&

2V

#8

• N

VN

2V

j®

2V

^2V

g

y*

H

% %

.7937 .6934 .8736 .6300 .9086 .5848 .7368

% % H % M M . y,

.8434 .9283 .5503 .9410 .5228 .6586 .7539

¥t

It

tt

H W Vk

.8298 .8939 .9499 .5000 .7211 .8550 .9565

H H

% % % % Ha

.4807 .6057 .7631 .8221 .9196 .9615 .4368

M2 M2 Hl2

He ?! 6 Me Me

.7469 .8355 .9714 .3969 .5724 .6786 .7591

We

Hi6

"/1 6

*M6 &

H4

Ho

.8255 .8826 .9331 .9787 .3150 .2500 .2714

MATHEMATICAL TABLES CUBE ROOTS (continued)

19

N

0

1

2

3

4

5

6

7

8

9

!i

60.

3.684

3.686

3.689

3.691

3.694

3.696

3*99

3.701

3.704

3.706

2

1.

3.708

3.71 1

3.713

3.716

3.718

3.721

3723

3.725

3.728

3.730

2.

3.733

3.735

3.737

3.740

3.742

3.744

3.747

3.749

3.752

3.754

3.

3.756

3.759

3.761

3.763

2.766

3.768

3.770

3.773

3.775

3.777

4.

3.780

3.782

3.784

3.787

3.789

3.791

3.794

3.796

3.798

3.801

55.

3.803

3.805

3.808

• 3.810

3.812

3.814

3.817

. 3.819

3.821

3.824

6.

3.826

3.828

3.830

3.833

3.835

3.837

3.839

3.842

3.844

3.846

7.

3.849

3.851

3.853

3.855

3.857

3.860

3.862

3.864

3.866

3.869

8.

3.871

3.873

3.875

3.878

3.880

3.882

3.884

3.886

3.889

3.891

9.

3.893

3.895

3.897

3.900

3.902

3.904

3.906

3.908

3.911

3.913

60.

3.915

3.917

3.919

3.921

3.924

3.926

3.928

3.930

3.932

3.934

3.936

3.939

3.941

3.943

3.945

3.947

3.949

3.951

3.954

3.956

2!

3.958

3.960

3.962

3.964

3.966

3.968

3.971

3.973

3.975

3.977

3.

3.979

3.981

3.983

3.985

3.987

3.990

3.992

3.994

3.996

3.998

4.

4.000

4.002

4.004

4.006

4.008

4.010

4.012

4.015

4.017

4.019

65.

4.021

4.023

4.025

4.027

4.029

4.031

4.033

4.035

4.037

4.039

6.

4.041

4.043

4.045

4.047

4.049

4.051

4.053

4.055

4.058

4.060

7.

4.062

4.064

4.066

4.068

4.070

4.072

4.074

4.076

4.078

4.080

8.

4.082

4.084

4.086

4.088

4.090

4.092

4.094

4.096

4.098

4.100

9.

4.102

4.104

4.106

4.108

4.109

4.111

4.113

4.115

4.117

4.119

70.

4.121

4.123

4.125

4.127

4.129

4.131

4.133

4.135

4.137

4.139

1.

4.141

4.143

4.145

4.147

4.149

4.151

4.152

4.154

4.156

4.158

2.

4.160

4.162

4.164

4.166

4.168

4.170

4.172

4.174

4.176

4.177

3.

4.179

4.181

4.183

4.185

4.187

4.189

4.191

4.193

4.195

4.196

4.

4.198

4.200

4.202

4.204

4.206

4.208

4.210

4.212

4.213

4.215

75.

4.217

4.219

4.221

4.223

4.225

4.227

4.228

4.230

4.232

4.234

6.

4.236

4.238

4.240

4.241

4.243

4.245

4.247

4.249

4.251

4.252

7.

4.254

4.256

4.258

4.260

4.262

4.264

4.265

4.267

4.269

4.271

8.

4.273

4.274

4.276

4.278

4.280

4282

4.284

4.285

4.287

4.289

9.

4.291

4.293

4294

4296

4.298

4.300

4.302

4.303

4.305

4.307

80.

4.309

4.311

4.312

4.314

4.316

4.318

4.320

4.321

4.323

4.325

1.

4.327

4.329

4.330

4.332

4.334

4.336

4.337

4.339

4.341

4.343

2.

4.344

4.346

4.348

4.350

4.352

4.353

4.355

4.357

4.359

4.360

3.

4.362

4.364

4.366

4.367

4.369

4.37 1

4.373

4.374

4.376

4.378

4,

4.380

4.381

4.383

4.385

4.386

4.388

4.390

4.392

4.393

4.395

85.

4.397

4.399

4.400

4.402

4.404

4.405

4.407

4.409

4.411

4.412

6.

4.414

4.416

4.417

4.419

4.421

4.423

4.424

4.426

4.428

4.429

7.

4.431

4.433

4.434

4.436

4.438

4.440

4.441

4.443

4.445

4.446

8.

4.448

4.450

4.451

4.453

4.455

4.456

4.458

4.460

4.461

4.463

9.

4.465

4.466

4.468

4.470

4.471

4.473

4.475

4.476

4.478

4.480

90.

4.481

4.483

4.485

4.486

4.488

4.490

4.491

4.493

4.495

4.496

1.

4.498

4.500

4.501

4.503

4.505

4.506

4.508

4.509

4.511

4.513

2.

4.514

4.516

4.518

4.519

4.521

4.523

4.524

4.526

4.527

4.529

3.

4.531

4.532

4.534

4.536

4.537

4.539

4.540

4.542

4.544

4.545

4.

4.547

4.548

4.550

4.552

4.553

4.555

4.556

4.558

4.560

4.561

95.

4.563

4.565

4.566

4.568

4.569

4.571

4.572

4.574

4.576

4.577

6.

4.579

4.580

4.582

4.584

4.585

4.587

4.588

4.590

4.592

4.593

7.

4.595

4.596

4.598

4.599

4.601

4.603

4.604

4.606

4.607

4.609

8.

4.610

4.612

4.614

4.615

4.617

4.618

4.620

4.621

4.623

4.625

9.

4.626

4.628

4.629

4.631

4.632

4.634

4.635

4.637

4.638

4.640

Moving the decimal point THREE places in .W requires moving it ONE place in body of table (seep. 16).

20

MATHEMATICAL TABLES

CUBE ROOTS (continued)

N

0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

!§

10

4642

4.657

4.672

4.688

4.703

4.718

4.733

4.747

4.762

4.777

15

1

4791

4.806

4.820

4.835

4.849

4.863

4.877

4.891

4.905

4.919

14

2

4.932

4.946

4.960

4.973

4.987

5.000

5.013

5.027

5.040

5.053

13

3

5.066

5.079

5.092

5.104

5.117

5.130

5.143

5.155

5.168

5.180

4

5.192

5.205

5.217

5.229

5.241

5.254

5.266

5.278

5.290

5.301

12

15

5.313

5.325

5.337

'5.348

5.360

5.372

5,383

5.395

5.406

5.418

6

5.429

5.440

5.451

5.463

5.474

5.485

5.496

5.507

5.518

5.529

!1

7

5.540

5.550

5.561

5.572

5.583

5.593

5.604

5.615

5.625

5.636

8

5.646

5.657

5.667

5.677

5.688

5.698

5.708

5.718

5.729

5.739

10

9

5.749

5.759

5.769

5.779

5.789

5.799

5.809

5.819

5.828

5.838

20

5.848

5.858

5.867

5.877

5.887

5.896

5.906

5.915

5.925

5.934

1

5.944

5.953

5.963

5.972

5.981

5.991

6.000

6.009

6.018

6.028

9

2

6.037

6.046

6.055

6.064

6.073

6.082

6.091

6.100

6.109

6.118

3

6.127

6.136

6.145

6.153

6.162

6.171

6.180

6.188

6.197

6.206

4

6.214

6.223

6.232

6.240

6.249

6.257

6.266

6.274

6.283

6.291

25

6.300

6.308

6.316

6.325

6.333

6.341

6.350

6.358

6.366

6.374

8

6

6.383

6.391

6.399

6.407

6.415

6.423

6.431

6.439

6.447

6.455

7

6.463

6.471

6.479

6.487

6.495

6.503

6.511

6.519

6.527

6.534

8

6.542

6.550

6.558

6.565

6.573

6.581

6.589

6.596

6.604

6.611

9

6.619

6.627

6.634

6.642

6.649

6.657

6.664

6.672

6.679

6.687

30

6.694

6.702

6.709

6.717

6.724

6.731

6.739

6.746

6.753

6.761

7

1

6.768

6.775

6.782

6.790

6.797

6.804

6.811

6.818

6.826

6.833

2

6.840

6.847

6.854

6.861

6.868

6.875

6.882

6.889

6.896

6.903

3

6.910

6.917

6.924

6.931

6.938

6.945

6.952

6.959

6.966

6.973

4

6.980

6.986

6.993

7.000

7.007

7.014

7.020

7.027

7.034

7.041

35

7.047

7.054

7.061

7.067

7.074

7.081

7.087

7.094

7.101

7.107

6

7.114

7.120

7.127

7.133

7.140

7.147

7.153

7.160

7.166

7.173

6

7

7.179

7.186

7.192

7.198

7.205

7.211

7.218

7.224

7.230

7.237

8

7.243

7.250

7.256

7.262

7.268

7.275

7.281

7.287

7.294

7.300

9

7.306

7.312

7.319

7.325

7.331

7.337

7.343

7.350

7.356

7.362

40

7.368

7.374

7.380

7.386

7.393

7.399

7.405

7.411

7.417

7.423

1

7.429

7.435

7.441

7.447

7.453

7.459

7.465

7.471

7.477

7.483

2

7.489

7.495

7.501

7.507

7.513

7.518

7.524

7.530

7.536

7.542

3

7.548

7.554

7.560

7.565

7.571

7.577

7.583

7.589

7.594

7.600

4

7.606

7.612

7.617

7.623

7.629

7.635

7.640

7.646

7.652

7.657

45

7.663

7.669

7.674

7.680

7.686

7.691

7.697

7.703

7.708

7.714

5

6

7.719

7.725

7.731

7.736

7.742

7.747

7.753

7.758

7.764

7.769

7

7.775

7.780

7.786

7.791

7.797

7.802

7.808

7.813

7.819

7.824

8

7.830

7.835

7.841

7.846

7.851

7.857

7.862

7.868

7.873

7.878

9

7.884

7.889

7.894

7.900

7.905

7.910

7.916

7.921

7.926

7.932

AUXILIARY TABLE OP TWO-THIRDS POWERS

AND THREE-HALVES POWERS (see pp. 22-23)

(To assist in locating the decimal point)

N

NK( = ^N^)

N**(- VF')

.0001 .001 .01

.1 1.

10. 100.

.002154 .01 .0464 . .2154 1. 4.64 21.54

.000001 00003162 .001 .03162278 1. 31.62278 1000.

For complete table of three-halves pow- ers, see pp. 22-23. That table, used in- versely, provides a complete table of two-thirds powers.

1000.

100.

31622.78

10000.

464.16

1000000.

MATHEMATICAL TABLES CUBE ROOTS (continued)

21

N

0.

1.

2.

a.

4.

5.

6.

7.

8.

9.

fts

<;-a

50

7.937

7.942

7.948

7.953

7.958

7.963

7.969

7.974

7.979

7.984

5

1

7.990

7.995

8.000

8.005

8.010

8.016

8.021

8.026

8.031

8.036

2

8.041

8.047

8.052

8.057

8.062

8.067

8.072

8.077

8.082

8.088

3

8.093

8.098

8.103

8.108

8.113

8.118

8.123

8.128

8.133

8.138

4

8.143

8.148

8.153

8.158

8.163

8.168

8.173

8.178

8.183

8.188

55

8.193

8.198

8.203

8.208

8.213

8.218

8.223

8.228

8.233

8.238

6

8.243

8.247

8.252

8.257

8.262

8.267

8.272

8.277

8.282

8.286

7

8.291

8.296

8.301

8.306

8.311

8.316

8.320

8.325

8.330

8.335

8

8.340

8.344

8.349

8.354

8.359

8.363

8.368

8.373

8.378

8.382

9

8.387

8.392

8.397

8.401

8.406

8.411

8.416

8.420

8.425

8.430

60

8.434

8.439

8.444

8.448

8.453

8.458

8.462

8.467

8.472

8.476

1

8.481

8.486

8.490

8.495

8.499

8.504

8.509

8.513

8.518

8.522

2

8.527

8.532

8.536

8.541

8.545

8.550

8.554

8.559

8.564

8.568

3

8.573

8.577

8.582

8.586

8.591

8.595

8.600

8.604

8.609

8.613

4

4

8.618

8.622

8.627

8.631

8.636

8.640

8.645

8.649

8.653

8.658

65

8.662

8.667

8.671

8.676

8.680

8.685

8.689

8.693

8.698

8.702

6

8.707

8.711

8.715

8.720

8.724

8.729

8.733

8.737

8.742

8.746

7

8.750

8.755

8.759

8.763

8.768

8.772

8.776

8.781

8.785

8.789

8

8.794

8.798

8.802

8.807

8.811

8.815

8.819

8.824

8.828

8.832

9

8.837

8.841

8.845

8.849

8.854

8.858

8.862

8.866

8.871

8.875

70

8.879

8.883

8.887

8.892

8.896

8.900

8.904

8.909

8.913

8.917

1

8.921

8.925

8.929

8.934

8.938

8.942

8.946

8.950

8.955

8.959

2

8.963

8.967

8.971

8.975

8.979

8.984

8.988

8.992

8.996

9.000

3

9.004

9.008

9.012

9.016

9.021

9.025

9.029

9.033

9.037

9.041

4

9.045

9.049

9.053

9.057

9.061

9.065

9.069

9.073

9.078

9.082

75

9.086

9.090

9.094

9.098

9.102

9.106

9.110

9.114

9.118

9.122

6

9.126

9.130

9.134

9.138

9.142

9.146

9.150

9.154

9.158

9.162

7

9.166

9.170

9.174

9.178

9.182

9.185

9.189

9.193

9.197

9.201

8

9.205

9.209

9.213

9.217

9.221

9.225

9.229

9.233

9.237

9.240

9

9.244

9.248

9.252

9.256

9.260

9.264

9.268

9.272

9.275

9.279

80

9.283

9.287

9.291

9.295

9.299

9.302

9.306

9.310

9.314

9.318

1

9.322

9.326

9.329

9.333

9.337

9.341

9.345

9.348

9.352

9.356

2

9.360

9.364

9.368

9.371

9.375

9.379

9.383

9.386

9.390

9.394

3

9.398

9.402

9.405

9.409

9.413

9.417

9.420

9.424

9.428

9.432

4

9.435

9.439

9.443

9.447

9.450

9.454

9.458

9.462

9.465

9.469

85

9.473

9.476

9.480

9.484

9.488

9.491

9.495

9.499

9.502

9.506

6

9.510

9.513

9.517

9.521

9.524

9.528

9.532

9.535

9.539

9.543

7

9.546

9.550

9.554

9.557

9.561

9.565

9.568

9.572

9.576

9.579

8

9.583

9.586

9.590

9.594

9.597

9.601

9.605

9.608

9.612

9.615

9

9.619

9.623

9.626

9.630

9.633

9.637

9.641

9.644

9.648

9.651

90

9.655

9.658

9.662

9.666

9.669

9.673

9.676

9.680

9.683

9.687

1

9.691

9.694

9.698

9.701

9.705

9.708

9.712

9.715

9.719

9.722

2

9.726

9.729

9.733

9.736

9.740

9.743

9.747

9.750

9.754

9.758

3

9.761

9.764

9.768

9.771

9.775

9.778

9.782

9.785

9.789

9.792

4

9.796

9.799

9.803

9.806

9.810

9.813

9.817

9.820

9.824

9.827

95

9.830

9.834

9.837

9.841

9.844

9.848

9.851

9.855

9.858

9.861

6

9.865

9.868

9.872

9.875

9.879

9.882

9.885

9.889

9.892

9.896

7

9.899

9.902

9.906

9.909

9.913

9.916

9.919

9.923

9.926

9.930

8

9.933

9.936

9.940

9.943

9.946

9.950

9.953

9.956

9.960

9.963

9

9.967

9.970

9.973

9.977

9.980

9.983

9.987

9.990

9.993

9.997

100

10.00

Moving the decimal point THREE places in N requires moving it ONE place in body of table (see p. 16).

22

MATHEMATICAL TABLES

THREE-HALVES POWERS OP NUMBERS (see also p. 20)

N

0

1

2

3

4

5

6

7

8

9

fl

1.

1.000

1.154

1.315

1.482

1.657

1.837

2.024

2.217

2.415

2.619

183

2.

2.828

3.043

3.263

3.488

3.718

3.953

4.192

4.437

4.685

4.939

237

3.

5.196

5.458

5.724

5.995

6.269

6.548

6.831

7.117

7.408

7.702

280

4.

8.000

8.302

8.607

8.917

9.230

9.546

9.866

10.190

313

4.

10.19

10.52

10.85

33

5.

11.18

11.52

11.86

12.20

12.55

12.90

13.25

13.61

13.97

14.33

35

6.

14.70

15.07

15.44

15.81

16.19

16.57

16.96

17.34

17.73

18.12

38

7.

18.52

18.92

19.32

19.72

20.13

20.54

20.95

21.37

21.78

22.20

41

8.

22.63

23.05

23.48

23.91

24.35

24.78

25.22

25.66

26.11

26.55

44

9.

27.00

27.45

27.90

28.36

28.82

29.28

29.74

30.21

30.68

31.15

46

10.

31.62

32.10

32.58

33.06

33.54

34.02

34.51

35.00

35.49

35.99

49

1.

36.48

36.98

37.48

37.99

38.49

39.00

39.51

40.02

40.53

41.05

51

2.

41.57

42.09

42.61

43.14

43.66

44.19

44.73

45.26

45.79

46.33

53

3.

46.87

47.41

47.96

48.50

49.05

49.60

50.15

50.71

51.26

51.82

55

4.

52.38

52.95

53.51

54.08

54.64

55.21

55.79

56.36

56.94

57.51

57

15.

58.09

58.68

59.26

59.85

60.43

61.02

61.62

62.21

62.80

63.40

59

6.

64.00

64.60

65.20

65.81

66.41

67.02

67.63

68.25

68.86

69.48

61

7.

70.09

70.71

71.33

71.96

72.58

73.21

73.84

74.47

75.10

75.73

63

8.

76.37

77.00

77.64

78.28

78.93

79.57

80.22

80.87

81.51

82.17

65

9.

82.82

83.47

84.13

84.79

85.45

86.11

86.77

87.44

88.10

88.77

66

20.

89.44

90.11

90.79

91.46

92.14

92.82

93.50

94.18

94.86

95.55

68

1.

96.23

96.92

97.61

98.30

99.00

99.69

100.38

69

I.

100.4

101.1

101.8

102.5

7

2.

103.2

103.9

104.6

105.3

106.0

106.7

107.4

108.2

108.9

109.6

7

3.

110.3

111.0

111.7

112.5

113.2

113.9

114.6

115.4

116.1

116.8

7

4.

117.6

118.3

119.0

119.8

120.5

121.3

122.0

122.8

123.5

124.3

7

25.

125.0

125.8

126.5

127.3

128.0

128.8

129.5

130.3

131.0

131.8

8

6.

132.6

133.3

134.1

134.9

135.6

136.4

137.2

138.0

138.7

139.5

8

7.

140.3

141.1

141.9

142.6

143.4

144.2

145.0

145.8

146.6

147.4

8

8.

148.2

149.0

149.8

150.5

151.3

152.1

152.9

153.8

154.6

155.4

8

9.

156.2

157.0

157.8

158.6

159.4

160.2

161.0

161.9

162.7

163.5

8

30.

164.3

" 165.1

166.0

166.8

167.6

168.4

169.3

170.1

170.9

171.8

8

I,

172.6

173 A

174.3

175.1

176.0

176.8

177.6

178.5

179.3

180.2

8

2.

181.0

181.9

182.7

183.6

184.4

185.3

186.1

187.0

187.8

188.7

9

3.

189.6

190.4

191.3

192.2

193.0

193.9

194.8

195.6

196.5

197.4

9

4.

198.3

199.1

200.0

200.9

201.8

202.6

203.5

204.4

205.3

206.2

9

35.

207.1

208.0

208.8

209.7

210.6

211.5

212.4

213.3

214.2

215.1

9

6.

216.0

216.9

217.8

218.7

219.6

220.5

221.4

222.3

223.2

224.2

9

7.

225.1

226.0

226.9

227.8

228.7

229.6

230.6

231.5

232.4

233.3

9

8.

234.2

235.2

236.1

237.0

238.0

238.9

239.8

240.8

241.7

242.6

9

9.

243.6

244.5

245.4

246.4

247.3

248.3

249.2

250.1

251.1

252.0

9

40.

253.0

253.9

254.9

255.8

256.8

257.7

258.7

259.7

260.6

261.6

10

1;

262.5

263.5

264.5

265.4

266.4

267.3

268.3

269.3

270.2

271.2

10

2.

272.2

273.2

274.1

275.1

276.1

277.1

278.0

279.0

280.0

281.0

10

3.

282.0

283.0

283.9

284.9

285.9

286.9

287.9

288.9

289.9

290.9

10

4.

291.9

292.9

293.9

294.9

295.9

296.9

297.9

298.9

299.9

300.9

10

45.

301.9

302.9

303.9

304.9

305.9

306.9

307.9

308.9

310.0

311.0

10

6.

312.0

313.0

314.0

315.0

316.1

317.1

318.1

319.1

320.2

321.2

10

7.

322.2

323.2

324.3

325.3

326.3

327.4

328.4

329.4

330.5

331.5

10

8.

332.6

333.6

334.6

335.7

336.7

337.8

338.8

339.9

340.9

342.0

10

9.

343.0

344.1

345.1

346.2

347.2

348.3

349.3

350.4

351.4

352.5

11

This table gives N?2 from N => 1 to N = 100. Moving the decimal point TWO places in N requires moving it THREE places in body of' table. Thus: (7.23)^ = 19.44; (723.)^ = 19440; (0.0723)^ = 0.01944 (72.3)^ = 614.8; (7230.)^ = 614800; (0.723)^ = 0.6148

Used inversely, table gives M^ from M = 1 to M - 1000. Thus: (0.6148)** => 0.7230.

MATHEMATICAL TABLES THREE-HALVES POWERS (continued') (See also p. 20)

23

N

0

1

2

3

4

5

6

7

8

9

£§ <H-O

50.

353.6

354.6

355.7

356.7

357.8

358.9

359.9

361.0

362.1

363.1

11

I.

364.2

365.3

366.4

367.4

368.5

369.6

370.7

371.7

372.8

373.9

11

2.

375.0

376.1

377.1

378.2

379.3

380.4

381.5

382.6

383.7

384.8

11

3

385.8

386.9

388.0

389.1

390.2

391.3

392.4

393.5

394.6

395.7

11

4.

396.8

397.9

399.0

400.1

401.2

402.3

403.4

404.6

405.7

406.8

11

55.

407.9

409.0

410.1

411.2

412.3

413.5

414.6

415.7

416.8

417.9

11

6.

419.1

420.2

421.3

422.4

423.6

424.7

425.8

426.9

428.1

429.2

11

7.

430.3

431.5

432.6

433.7

434.9

436.0

437.2

438.3

439.4

440.6

11

8.

441.7

442.9

444.0

445.1

446.3

447.4

448.6

449.7

450.9

452.0

11

9.

453.2

454.3

455.5

456.6

457.8

459.0

460.1

461.3

462.4

463.6

12

60.

464.8

465.9

467.1

468.2

469.4

470.6

471.7

472.9

474.1

475.3

12

1.

476.4

477.6

478.8

479.9

481.1

482.3

483.5

484.6

485.8

487.0

12

2.

488.2

489.4

490.6

491.7

492.9

' 494.1

495.3

496.5

497.7

498.9

12

3.

500.0

501.2

502.4

503.6

504.8

506.0

507.2

508.4

509.6

510.8

12

4.

512.0

513.2

514.4

515.6

516.8

518.0

519.2

520.4

521.6

522.8

12

65.

524.0

525.3