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Handbook of mathematics for engineers

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HANDBOOK OF MATHEMATICS FOR ENGINEERS

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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

526.5

527.7

528.9

530.1

531.3

532.5

533.8

535.0

12

6'

536.2

537.4

538.6

539.8

541.1

542.3

543.5

544.7

546.0

547.2

12

7.

548.4

. 549.6

550.9

552.1

553.3

554.6

555.8

5570

558.3

559.5

12

8.

560.7

562.0

563.2

564.5

565.7

566.9

568.2

569.4

570.7

571.9

12

9.

573.2

574.4

575.7

576.9

578.1

579.4

580.6

581.9

583.2

584.4

13

70.

585.7

586.9

588.2

589.4

590.7

591.9

593.2

594.5

595.7

597.0

13

1.

598.3

599.5

600.8

602.1

603.3

604.6

605.9

607.1

608.4

609.7

13

2.

610.9

612.2

613.5

614.8

616.0

617.3

618.6

619.9

621.2

622.4

13

3.

623.7

625.0

626.3

627.6

628.8

630.1

631.4

632.7

634.0

635.3

13

4.

636.6

637.9

639.2

640.4

641.7

643.0

644.3

645.6

646.9

648.2

13

75.

649.5

650.8

652.1

653.4

654.7

656.0

657.3

658.6

659.9

661.2

13

6.

662.6

663.9

665.2

666.5

667.8

669.1

670.4

671.7

673.0

674.4

13

7.

675.7

677.0

678.3

679.6

680.9

682.3

683.6

684.9

686.2

687.6

13

8.

688.9

690.2

691.5

692.9

694.2

695.5

696.8

698.2

699.5

700.8

13

9.

702.2

703.5

704.8

706.2

707.5

708.8

710.2

711.5

712.9

714.2

13

80.

715.5

716.9

718.2

719.6

720.9

722.3

723.6

725.0

726.3

727.7

13

I.

729.0

730.4

731.7

733.1

734.4

735.8

737.1

738.5

739.8

741.2

14

2.

742.5

743.9

745.3

746.6

748.0

749.3

750.7

752.1

753.4

754.8

14

3.

756.2

757.5

758.9

760.3

761.6

763.0

764.4

765.8

767.1

768.5

14

4.

769.9

771.2

772.6

774.0

775.4

776.8

778.1

779.5

780.9

782.3

14

85.

783.7

785.0

786.4

787.8

789:2

790.6

792.0

793.4

794.8

796.1

14

6.

797.5

798.9

800.3

801.7

803.1

804.5

805.9

807.3

808.7

810.1

14

' 7.

811.5

812.9

814.3

815.7

817.1

818.5

819.9

821.3

822.7

824.1

14

8.

825.5

826.9

828.3

829.7

831.1

832.6

834.0

835.4

836.8

838.2

14

9.

839.6

841.0

842.5

843.9

845.3

846.7

848.1

849.5

851.0

852.4

14

90.

853.8

855.2

856.7

858.1

859.5

860.9

862.4

863.8

865.2

866.7

14

1.

868. J

869.5

870.9

872.4

873.8

875.2

876.7

878.1

879.6

881.0

14

2.

882.4

883.9

885.3

886.8

888.2

889.6

891.1

892.5

894.0

895.4

14

3.

896.9

898.3

899.8

901.2

902.7

904.1

905.6

907.0

908.5

909.9

15

4.

911.4

912.8

914.3

915.7

917.2

918.6

920.1

921.6

923.0

924.5

15

95.

925.9

927.4

928.9

930.3

931.8

933.3

934.7

936.2

937.7

939.1

15

6.

940.6

942.1

943.5

945.0

946.5

948.0

949.4

950.9

952.4

953.9

15

7.

955.3

956.8

958.3

959.8

961.3

962.7

964.2

965.7

967.2

968.7

15

8.

970.2

971.6

973.1

974.6

976.1

977.6

979.1

980.6

982.1

983.5

15

9.

985.0

986.5

988.0

989.5

991.0

992.5

994.0

995.5

997.0

998.5

15

100.

1000.0

Moving the decimal point TWO places in AT requires moving it THREE places in body of table (see also auxiliary table on p. 20).

24 MATHEMATICAL TABLES

RECIPROCALS OF NUMBERS '

N

0

1

2

3

4

5

6

7

8

9

ii

1.00

.9990

.9980

.9970

.9960

.9950

.9940

.9930

.9921

.9911

-10

1

.9901

.9891

.9881

.9872

.9862

.9852

.9843

.9833

.9823

.9814

2

.9804

.9794

.9785

.9775

.9766

.9756

.9747

.9737 -

.9728

.9718

3

.9709

.9699

.9690

.9681

.9671

.9662

.9653

.9643

.9634

.9625

-9

4

.9615

.9606

.9597

.9588

.9579

.9569

.9560

.9551

.9542

.9533

1.05

.9524

.9515

.9506

.9497

.9488

.9479

.9470

.9461

.9452

.9443

6

.9434

.9425

.9416

.9407

.9398

.9390

.9381

.9372

.9363

.9355

7

.9346

.9337

.9328

.9320

.9311

.9302

.9294

.9285

.9276

.9268

8

.9259

.9251

.9242

.9234

.9225

.9217

.9208

.9200

.9191

.9183

-8

9

.9174

.9166

.9158

.9149

.9141

.9132

.9124

.9116

.9107

.9099

1.10

.9091

.9083

.9074

.9066

.9058

.9050

.9042

.9033

.9025

.9017

.9009

.9001

.8993

.8985

.8977

.8969

.8961

.8953

.8945

.8937

2

.8929

.8921

.8913

.8905

.8897

.8889

.8881

.8873

.8865

.8857

3

.8850

.8842

.8834

.8826

.8818

.8811

.8803

.8795

.8787

.8780

4

.8772

.8764

.8757

.8749

.8741

.8734

.8726

.8718

.8711

.8703

1.15

.8696

.8688

.8681

.8673

.8666

.8658

.8651

.8643

.8636

.8628

6

.8621

.8613

.8606

.8598

.8591

.8584

.8576

.8569

.8562

.8554

-7

7

.8547

.8540

.8532

.8525

.8518

.8511

.8503

.8496

.8489

.8482

8

.8475

.8467

.8460

.8453

.8446

.8439

.8432

.8425

.8418

.8410

9

.8403

.8396

.8389

.8382

.8375

.8368

.8361

.8354

.8347

.8340

1.20

.8333

.8326

.8319

.8313

.8306

.8299

.8292

.8285

.8278

.8271

1

.8264

.8258

.8251

.8244

.8237

.8230

.8224

.8217

.8210

.8203

2

.8197

.8190

.8183

.8177

.8170

.8163

.8157

.8150

.8143

.8137

3

.8130

.8123

.8117

.8110

.8104

.8097

.8091

.8084

.8078

.8071

-6

4

.8065

.8058

.8052

.8045

.8039

.8032

.8026

.8019

.8013

.8006

1.25

.8000

.7994

.7987

.7981

.7974

.7968

.7962

.7955

.7949

.7943

6

.7937

.7930

.7924

.7918

.7911

.7905

.7899

.7893

.7886

.7880

7

.7874

.7868

.7862

.7855

.7849

.7843

.7837

.7831

.7825

.7819

8

.7812

.7806

.7800

.7794

.7788

.7782

.7776

.7770

.7764

.7758

9

.7752

.7746

.7740

.7734

.7728

.7722

.7716

.7710

.7704

.7698

1.30

.7692

.7686

.7680

.7675

.7669

.7663

.7657

.7651

.7645

.7639

1

.7634

.7628

.7622

.7616

.7610

.7605

.7599

.7593

.7587

.7582

2

.7576

.7570

.7564

.7559

.7553

.7547

J54J

.7536

.7530

.7524

3

.7519

.7513

.7508

.7502

.7496

.7491

.7485

.7479

.7474

.7468

4

.7463

.7457

.7452

.7446

.7440

.7435

.7429

.7424

.7418

.7413

135

.7407

.7402

.7396

.7391

.7386

.7380

.7375

.7369

.7364

.7358

-5

6

.7353

.7348

.7342

.7337

.7331

.7326

.7321

.7315

.7310

.7305

7

.7299

.7294

.7289

.7283

.7278

.7273

.7267

.7262

.7257

.7252

8

.7246

.7241

.7236

.7231

.7225

.7220

.7215

.7210

.7205

.7199

9

.7194

.7189

.7184

.7179

.7174

.7168

.7163

.7158

.7153

.7148

1.40

.7143

.7138

.7133

.7128

.7123

.7117

.7112

.7107

.7102

.7097

1

.7092

.7087

.7082

.7077

.7072

.7067

.7062

.7057

.7052

.7047

2

.7042

.7037

.7032

.7027

.7022

.7018

.7013

.7008

.7003

.6998

3

.6993

.6988

.6983

.6978

.6974

.6969

.6964

.6959

.6954

.6949

4

.6944

.6940

.6935

.6930

.6925

.6920

.6916

.6911

.6906

.6901

1.45

.6897

.6892

.6887

.6882

.6878

.6873

.6868

.6863

.6859

.6854

6

.6849

.6845

.6840

.6835

.6831

.6826

.6821

.6817

.6812

.6807

7

.6803

.6798

.6793

.6789

.6784

.6780

.6775

.6770

.6766

.6761

8

.6757

.6752

.6748

.6743

.6739

.6734

.6729

.6725

.6720

.6716

9

.6711

.6707

.6702

.6698

.6693

.6689

.6684

.6680

.6676

.6671

1/ir = 0.318310 1/e = 0.367879

Moving the decimal point in either direction in N requires moving it in the OPPO- SITE direction in body of table (see p. 26).

MATHEMATICAL TABLES RECIPROCALS (continued)

25

N

0

1

2

3

4

5

6

7

8

9

|1

1.50

.6667

.6662

.6658

.6653

.6649

.6645

.6640

.6636

.6631

.6627

-4

1

.6623

.6618

.6614

.6609

.6605

.6601

.6596

.6592

.6588

.6583

2

.6579

.6575

.6570

.6566

.6562

.6557

.6553

.6549

.6545

.6540

3

.6536

.6532

.6527

.6523

.6519

.6515

.6510

.6506

.6502

.6498

4

.6494

.6489

.6485

.6481

.6477

.6472

.6468

.6464

.6460

.6456

1.55

.6452

.6447

.6443

.6439

.6435

.6431

.6427

.6423

.6418

.6414

6

.6410

.6406

.6402

.6398

.6394

.6390

.6386

.6382

.6378

.6373

7

.6369

.6365

.6361

.6357

.6353

.6349

.6345

.6341

.6337

.6333

8

.6329

.6325

.6321

.6317

.6313

.6309

.6305

.6301

.6297

.6293

9

.6289

.6285

.6281

.6277

.6274

.6270

.6266

.6262

.6258

.6254

1.60

.6250

.6246

.6242

.6238

.6234

.6231

.6227

.6223

.6219

.6215

1

.6211

.6207

.6203

.6200

.6196

.6192

.6188

.6184

.6180

.6177

2

.6173

.6169

.6165

.6161

.6158

.6154

.6150.

.6146

.6143

.6139

3

.6135

.6131

.6127

.6124

.6120

.6116

.6112

.6109

.6105

.6101

4

.6098

.6094

.6090

.6086

.6083

.6079

.6075

.6072

.6068

.6064

1.65

.6061

.6057

.6053

.6050

.6046

.6042

.6039

.6035

.6031

.6028

6

.6024

.6020

.6017

.6013

.6010

.6006

.6002

.5999

.5995

.5992

7

.5988

.5984

.5981

.5977

.5974

.5970

.5967

.5963

.5959

.5956

8

.5952

.5949

.5945

.5942

.5938

.5935

.5931

.5928

.5924

.5921

9

.5917

.5914

.5910

.5907

.5903

.5900

.5896

.5893

.5889

.5886

1.70

.5882

.5879

.5875

.5872

.5869

.5865

.5862

.5858

.5855

.5851

-3

1

.5848

.5845

.5841

.5838

.5834

.5831

.5828

.5824

.5821

.5817

2

.5814

.5811

.5807

.5804

.5800

.5797

.5794

.5790

.5787

.5784

3

.5780

.5777

.5774

.5770

.5767

.5764

.5760

.5757

.5754

.5750

4

.5747

.5744

.5741

.5737

.5734

.5731

.5727

.5724

.5721

.5718

1.75

.5714

.5711

.5708

.5705

.5701

.5698

.5695

.5692

.5688

.5685

6

.5682

.5679

.5675

.5672

.5669

.5666

.5663

.5659

.5656

.5653

7

.5650

.5647

.5643

.5640

.5637

.5634

.5631

.5627

.5624

.5621

8

.5618

.5615

.5612

.5609

.5605

.5602

.5599

.5596

.5593

.5590

9

.5587

.5583

.5580

.5577

.5574

.5571

.5568

.5565

.5562

.5559

1.80

.5556

.5552

.5549

.5546

.5543

.5540

.5537

.5534

.5531

.5528

1

.5525

.5522

.5519

.5516

.5513

.5510

.5507

.5504

.5501

.5498

2

.5495

.5491

.5488

.5485

.5482

.M79

.5476

.5473

.5470

.5467

3

.5464

.5461

.5459

.5456

.5453

.5450

.5447

.5444

.5441

.5438

4

.5435

.5432

.5429

.5426

.5423

.5420

.5417

.5414

.5411

.5408

1.85

.5405

.5402

.5400

.5397

.5394

.5391

.5388

.5385

.5382

.5379

6

.5376

.5373

.5371

.5368

.5365

.5362

.5359

.5356

.5353

.5350

7

.5348

.5345

.5342

.5339

.5336

.5333

.5330

.5328

.5325

.5322

8

.5319

.5316

.5313

.5311

.5308

.5305

.5302

.5299

.5297

.5294

9

.5291

.5288

.5285

.5283

.5280

.5277

.5274

.5271

.5269

.5266

1.90

.5263

.5260

.5258

.5255

.5252

.5249

.5247

.5244

.5241

.5238

1

.5236

.5233

.5230

.5227

.5225

.5222

.5219

.5216

.5214

.5211

2

.5208

.5206

.5203

.5200

.5198

.5195

.5192

.5189

.5187

.5184

3

.5181

.5179

.5176

.5173

.5171

.5168

.5165

.5163

.5160

.5157

4

.5155

.5152

.5149

.5147

.5144

.5141

.5139

.5136

.5133

.5131

1.95

.5128

.5126

.5123

.5120

5118

.5115

.5112

.5110

.5107

.5105

6

.5102

.5099

.5097

.5094

.5092

.5089

.5086

.5084

.5081

.5079

7

.5076

.5074

.5071

.5068

.5066

.5063

.5061

.5058

.5056

.5053

-2

8

.5051

.5048

.5045

.5043

.5040

.5038

.5035

.5033

.5030

.5028

9

.5025

.5023

.5020

.5018

.5015

.5013

.5010

.5008

.5005

.5003

Moving the decimal point in either direction in N requires moving it in the OPPO- SITE direction in body of table (see p. 26).

26 MATHEMATICAL TABLES

RECIPROCALS (continued)

N

o

j

2

3

.

2.0

.5000

.4975

.4950

.4926

.4902

.4878

.4854

.4831

.4808

.4785

-24

I

.4762

.4739

.4717

.4695

.4673

.4651

.4630

.4608

.4587

.4566

-21

2

.4545

.4525

.4505

.4484

.4464

.4444

.4425

.4405

.4386

.4367

-20

3

.4348

.4329

.4310

.4292

.4274

.4255

.4237

.4219

.4202

.4184

- 18

4

.4167

.4149

.4132

.4115

.4098

.4082

.4065

.4049

.4032

.4016

- 17

2.5

.4000

.3984

.3968

.3953

3937

.3922

.3906

3891

3876

386 1

- 15

6

.3846

.3831

.3817

3802

3788

3774

3759

3745

3731

3717

- 14

7

.3704

.3690

3676

.3663

3650

.3636

3623

3610

3597

3584

- 13

8

.3571

.3559

.3546

3534

.3521

3509

3497

3484

.3472

3460

- 12

9

.3448

.3436

3425

3413

.3401

3390

3378

3367

.3356

.3344

-12

3.0

.3333

.3322

.3311

.3300

3289

.3279

3268

.3257

3247

3236

- 11

1

.3226

.3215

.3205

.3195

.3185

3175

.3165

.3155

.3145

3135

- 10

2

.3125

.3115

3106

.3096

3086

.3077

3067

.3058

3049

3040

- 10

3

.3030

.3021

.3012

3003

.2994

.2985

.2976

.2967

.2959

.2950

-9

4

.2941

.2933

.2924

.2915

.2907

.2899

.2890

.2882

.2874

.2865

-8

3.5

.2857

.2849

.2841

.2833

.2825

.2817

.2809

.2801

.2793

.2786

-8

6

.2778

.2770

.2762

.2755

.2747

.2740

.2732

.2725

.2717

.2710

-8

7

.2703

J2695

.2688

.2681

.2674

.2667

.2660

.2653

.2646

.2639

-7

8

.2632

.2625

.2618

.2611

.2604

.2597

.2591

.2584

.2577

.2571

-7

9

.2564

.2558

.2551

.2545

.2538

.2532

.2525

.2519

.2513

.2506

-6

4.0

.2500

.2494

.2488

J2481

.2475

.2469

.2463

.2457

.2451

.2445

-6

1

.2439

.2433

.2427

.2421

.2415

.2410

.2404

.2398

.2392

.2387

-6

2

.2381

.2375

.2370

2364

.2358

.2353

.2347

.2342

.2336

.2331

-6

3

.2326

.2320

.2315

.2309

.2304

.2299

.2294

.2288

.2283

.2278

-5

4

.2273

.2268

.2262

.2257

.2252

.2247

.2242

.2237

.2232

.2227

-5

4.5

.2222

.2217

.2212

.2208

.2203

.2198

.2193

.2188

.2183

.2179

-5

6

.2174

.2169

.2165

.2160

.2155

.2151

.2146

.2141

.2137

.2132

-5

7

.2128

.2123

.2119

2114

.2110

.2105

.2101

.2096

.2092

.2088

-4

8

.2083

.2079

.2075

.2070

.2066

.2062

.2058

.2053

.2049

.2045

-4

9

.2041

.2037

.2033

.2028

.2024

.2020

.2016

.2012

.2008

.2004

-4

I/T = 0.318310 1/e = 0.367879

Explanation of Table of Reciprocals (pp. 24-27).

This table gives the values of 1/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 reciprocal of a number N outside the range from 1 to 10, note that moving the decimal point any number of places in either direction in column N is equivalent to moving it the same number of places in the opposite direction in the body of the table. For example:

1 0.3108; -^r =0.0003108; nn^n^ - 310.8

3.217

3217.

0.003217

MATHEMATICAL TABLES RECIPROCALS (continued)

27

I*

0

1

2

3

4

5

6

7

8

9

«d &

5.0

.2000

.1996

.1992

.1988

.1984

.1980

.1976

.1972

.1969

.1965

-4

.1

.1961

.1957

.1953

.1949

.1946

.1942

.1938

.1934

.1931

.1927

.2

.1923

.1919

.1916

.1912

.1908

.1905

.1901

.1898

.1894

.1890

.3

.1887

.1883

.1880

.1876

.1873

.1869

.1866

.1862

.1859

.1855

.4

.1852

.1848

.1845

.1842

.1838

.1835

.1832

.1828

.1825

.1821

-3

5.5

.1818

.1815

.1812

.1808

.1805

.1802

.1799

.1795

.1792

.1789

.6

.1786

.1783

.1779

.1776

.1773

.1770

.1767

.1764

.1761

.1757

.7

.1754

.1751

.1748

.1745

.1742

.1739

.1736

.1733

.1730

.1727

.8

.1724

.1721

.1718

.1715

.1712

.1709

.1706

.1704

.1701

.1698

.9

.1695

.1692

.1689

.1686

.1684

.1681

.1678

.1675

.1672

.1669

6.0 .1667

.1664

.1661

.1658

.1656

.1653

.1650

.1647

.1645

.1642

.1

.1639

.1637

.1634

.1631

.1629

.1626

.1623

.1621

.1618

.1616

.2

.1613

.1610

.1608

.1605

.1603

.1600

.1597

.1595

.1592

.1590

.3

.1587

.1585

.1582

.1580

.1577

.1575

.1572

.1570

.1567

.1565

-2

.4

.1563

.1560

.1558

.1555

.1553

.1550

.1548

.1546

.1543

.1541

6.5

.1538

.1536

.1534

.1531

.1529

.1527

.1524

.1522

.1520

.1517

.6

.1515

.1513

.1511

.1508

.1506

.1504

.1502

.1499

.1497

.1495

.7

.1493

.1490

.1488

.1486

.1484

.1481

.1479

.1477

.1475

.1473

.8

.1471

.1468

.1466

.1464

.1462

.1460

.1458

.1456

.1453

.1451

.9

.1449

.1447

.1445

.1443

.1441

.1439

.1437

.1435

.1433

.1431

7.0

.1429

.1427

.1425

.1422

.1420

.1418

.1416

.1414

.1412

.1410

.1408

.1406

.1404

.1403

.1401

.1399

.1397

.1395

.1393

.1391

'.2

.1389

.1387

.1385

.1383

.1381

.1379

.1377

.1376

.1374

.1372

.3

.1370

.1368

.1366

.1364

.1362

.1361

.1359

.1357

.1355

.1353

.4

.1351

.1350

.1348

.1346

.1344

.1342

.1340

.1339

.1337

.1335

7.5

.1333

.1332

.1330

.1328

.1326

.1325

.1323

.1321

.1319

.1318

.6

.1316

.1314

.1312

.1311

.1309

.1307

.1305

.1304

.1302

.1300

.7

.1299

.1297

.1295

.1294

.1292

.1290

.1289

.1287

.1285

.1284

.8

.1282

.1280

.1279

.1277

.1276

.1274

.1272

.1271

.1269

.1267

.9

.1266

.1264

.1263

.1261

.1259

.1258

.1256

.1255

.1253

.1252

8.0

.1250

.1248

.1247

.1245

.1244

.1242

.1241

.1239

.1238

.1236

.1235

.1233

.1232

.1230

.1229

.1227

.1225

.1224

.1222

.1221

'.2

.1220

.1218

.1217

.1215

.1214

.1212

.1211

.1209

.1208

.1206

.3

.1205

.1203

.1202

.1200

.1199

.1198

.1196

.1195

.1193

.1192

.4

.1190

.1189

.1188

.1186

.1185

.1183

.1182

.1181

.1179

.1178

-1

8.5

.1176

.1175

.1174

.1172

.1171

.1170

.1168

.1167

.1166

.1164

.6

.1163

.1161

.1160

.1159

.1157

.1156

.1155

.1153

.1152

.1151

.7

.1149

.1148

.1147

.1145

.1144

.1143

.1142

.1140

.1139

.1138

.8

.1136

.1135

.1134

.1133

.1131

.1130

.1129

.1127

.1126

.1125

.9

.1124

.1122

.1121

.1120

.1119

.1117

.1116

.1115

.1114

.1112

9.0

.1111

.1110

.1109

.1107

.1106

.1105

.1104

.1103

.1101

.1100

.1

.1099

.1098

.1096

.1095

.1094

.1093

.1092

.1091

.1089

.1088

.2

.1087

.1086

.1085

.1083

.1082

.1081

.1080

.1079

.1078

.1076

.3

.1075

.1074

.1073

.1072

.1071

.1070

.1068

.1067

.1066

.1065

.4

.1064

.1063

.1062

.1060

.1059

.1058

.1057

.1056

.1055

.1054

9.5

.1053

.1052

.1050

.1049

.1048

.1047

.1046

.1045

.1044

.1043

.6

.1042

.1041

.1040

.1038

.1037

.1036

.1035

.1034

.1033

.1032

.7

.1031

.1030

.1029

J028

.1027

.1026

.1025

.1024

.1022

.1021

.8

.1020

.1019

.1018

.1017

.1016

.1015

.1014

.1013

.1012

.1011

.9

.1010

.1009

.1008

.1007

.1006

.1005

.1004

.1003

.1002

.1001

Moving the decimal point in either direction in N requires moving it in the OPPOSITE direction in body of table (see p. 26).

28

MATHEMATICAL TABLES

CIRCUMFERENCES OF CIRCLES BY HUNDREDTHS

(For circumferences by eighths, see p. 32)

D

0

1

2

3

4

5

6

7

8

9

>1

<T

1.0

3.142

3.173

3.204

3.236

3.267

3.299

3.330

3.362

3.393

3.424

31

.1

3.456

3.487

3.519

3.550

3.581

3.613

3.644

3.676

3.707

3.738

.2

3.770

3.801

3.833

3.864

3.896

3.927

3.958

3.990

4.021

4.053

.3

4.084

4.115

4.147

4.178

4.210

4.241

4.273

4.304

4.335

4.367

.4

4.398

4.430

4.461

4.492

4.524

4.555

4.587

4.618

4.650

4.681

1.5

4.712

4.744

4.775

4.807

4.838

4.869

4.901

4.932

4.964

4.995

.6

5.027

5.058

5.089

5.121

5.152

5.184

5.215

5.246

5.278

5.309

.7

5.341

5.372

5.404

5.435

5.466

5.498

5.529

5.561

5.592

5.623

.8

5.655

5.686

5.718

5.749

5.781

5.812

5.843

5.875

5.906

5.938

.9

5.969

6.000

6.032

6.063

6.095

6.126

6.158

6.189

6.220

6.252

2.0

6.283

6.315

6.346

6.377

6.409

6.440

6.472

6.503

6.535

6.566

.1

6.597

6.629

6.660

6.692

6.723

6.754

6.786

6.817

6.849

6.880

.2

6.912

6.943

6.974

7.006

7.037

7.069

7.100

7.131

7.163

7.194

.3

7.226

7.257

7.288

7.320

7.351

7.383

7.414

7.446

7.477

7.508

.4

7.540

7.571

7.603

7.634

7.665

7.697

7.728

7.760

7.791

7.823

2.5

7.854

7.885

7.917

7.948

7.980

8.011

8.042

8.074

8.105

8.137

.6

8.168

8.200

8.231

8.262

8.294

8.325

8.357

8.388

8.419

8.451

.7

8.482

8.514

8.545

8.577

8.608

8.639

8.671

8.702

8.734

8.765

.8

8.796

8.828

8.859

8.891

8.922

8.954

8.985

9.016

9.048

9.079

.9

9.111

9.142

9.173

9.205

9.236

9.268

9.299

9.331

9.362

9.393

3.0

9.425

9.456

9.488

9.519

9.550

9.582

9.613

9.645

9.676

9.708

9.739

9.770

9.802

9.833

9.865

9.8%

9.927

9.959

9.990

10.022

31

*1

10.02

3

.2

10.05

10.08

10.12

10.15

10.18

10.21

10.24

10.27

10.30

10.34

.3

10.37

10.40

10.43

10.46

10.49

10.52

10.56

1059

10.62

10.65

.4

10.68

10.71

10.74

10.78

10.81

10.84

10.87

10.90

10.93

10.96

3.5

11.00

11.03

11.06

11.09

11.12

11.15

11.18

11.22

11.25

11.28

.6

11.31

11.34

11.37

11.40

11.44

11.47

11.50

11.53

11.56

11.59

.7

11.62

11.66

11.69

11.72

11.75

11.78

11.81

11.84

11.88

11.91

.8

11.94

11.97

12.00

12.03

12.06

12.10

12.13

12.16

12.19

12.22

.9

12.25

12.28

12.32

12.35

12.38

12.41

12.44

12.47

12.50

12.53

4.0

12.57

12.60

12.63

12.66

12.69

12.72

12.75

12.79

12.82

12.85

12.88

12.91

12.94

12.97

13.01

13.04

13.07

13.10

13.13

13.16

'.2

13.19

13.23

13.26

13.29

13.32

13.35

13.38

13.41

13.45

13.48

.3

13.51

13.54

13.57

13.60

13.63

13.67

13.70

13.73

13.76

13.79

.4

13.82

13.85

13.89

13.92

13.95

13.98

14.01

14.04

14.07

14.11

4.5

14.14

14.17

14.20

14.23

14.26

14.29

14.33

14.36

14.39

14.42

.6

14.45

14.48

14.51

14.55

14.58

14.61

14.64

14.67

14.70

14.73

.7

14.77

14.80

14.83

14.86

14.89

14.92

14.95

14.99

15.02

15.05

.8

15.08

15.11

15.14

15.17

15.21

15.24

15.27

15.30

15.33

15.36

.9

15.39

15.43

15.46

15.49

15.52

15.55

15.58

15.61

15.65

15.68

Explanation of Table of Circumferences (pp. 28-29)

This table gives the product of T times any number D from 1 to 10; that is, it is a table of multiples of IT. (D — diameter.)

Moving the decimal point one place in column D is equivalent to moving it one place in the body of the table.

Circumference = ic X diam. = 3.141593 X diam. Conversely,

Diameter = - X circumf. = 0.31831 X circumf.

MATHEMATICAL TABLES CIRCUMFERENCES BY HUNDREDTHS (continued)

29

D

0

1

2

3

4

5

6

7

8

9

9

5.0

15.71

15.74

15.77

15.80

15.83

15.87

15.90

15.93

15.%

15.99

3

.1

16.02

16.05

16.08

16.12

16.15

16.18

16.21

16.24

16.27

16.30

.2

16.34

16.37

16.40

16.43

16.46

16.49

16.52

16.56

16.59

16.62

.3

16.65

16.68

16.71

16.74

16.78

16.81

16.84

16.87

16.90

16.93

.4

16.96

17.00

17.03

17.06

17.09

17.12

17.15

17.18

17.22

17.25

5.5

17.28

17.31

17.34

17.37

17.40

17.44

17.47

17.50

17.53

17.56

j6

17.59

17.62

17.66

17.69

17.72

17.75

17.78

17.81

17.84

17.88

.7

17.91

17.94

17.97

18.00

18.03

18.06

18.10

18.13

18.16

18.19

.8

18.22

18.25

18.28

18.32

18.35

18.38

18.41

18.44

18.47

18.50

.9

18.54

18.57

18.60

18.63

18.66

18.69

18.72

18.76

18.79

18.82

6.0

18.85

18.88

18.91

18.94

18.98

19.01

19.04

19.07

19.10

19.13

.1

19.16

19.20

19.23

19.26

19.29

19.32

19.35

19.38

19.42

19.45

2

19.48

19.51

19.54

19.57

19.60

19.63

19.67

19.70

19.73

19.76

3

19.79

19.82

19.85

19.89

19.92

19.95

19.98

20.01

20.04

20.07

.4

20.11

20.14

20.17

20.20

20.23

20.26

20.29

20.33

20.36

20.39

6.6

20.42

20.45

20.48

20.51

20.55

20.58

20.61

20.64

20.67

20.70

.6

20.73

20.77

20.80

20.83

20.86

20.89

20.92

20.95

20.99

21.02

.7

21.05

21.08

21.11

21.14

21.17

21.21

21.24

21.27

21.30

21.33

.8

21.36

21.39

21.43

21.46

21.49

21.52

21.55

21.58

21.61

21.65

.9

21.68

21.71

21.74

21.77

21.80

21.83

21.87

21.90

21.93

21.96

7.0

21.99

22.02

22.05

22.09

22.12

22.15

22.18

22.21

22.24

22.27

.1

22.31

22.34

22.37

22.40

22.43

22.46

22.49

22.53

22.56

22.59

.2

22.62

22.65

22.68

22.71

22.75

22.78

22.81

22.84

22.87

22.90

.3

22.93

22.97

23.00

23.03

23.06

23.09

23.12

23.15

23.18

23.22

.4

23.25

23.28

23.31

23.34

23.37

23.40

23.44

23.47

23.50

23.53

7.5

23.56

23.59

23.62

23.66

23.69

23.72

23.75

23.78

23.81

23.84

.6

23.88

23.91

23.94

23.97

24.00

24.03

24.06

24.10

24.13

24.16

.7

24.19

24.22

24.25

24.28

24.32 •

24.35

24.38

24.41

24.44

24.47

.8

24.50

24.54

24.57

24.60

24.63

24.66

24.69

24.72

24.76

24.79

.9

24.82

24.85

24.88

24.91

24.94

24.98

25.01

25.04

25.07

25.10

8.0

25.13

25.16

25.20

25.23

25.26

25.29

25.32

25.35

25.38

25.42

.1

25.45

25.48

25.51

25.54

25.57

25.60

25.64

25.67

25.70

25.73

2

25.76

25.79

25.82

25.86

25.89

25.92

25.95

25.98

26.01

26.04

.3

26.08

26.11

26.14

26.17

26.20

26.23

26.26

26.30

26.33

26.36

.4

26.39

26.42

26.45

26.48

26.52

26.55

26.58

26.61

26.64

26.67

8.5

26.70

26.73

26.77

26.80

26.83

26.86

26.89

26.92

26.95

26.99

.6

27.02

27.05

27.08

27.11

27.14

27.17

27.21

27.24

27.27

27.30

7

27.33

27.36

27.39

27.43

27.46

27.49

27.52

27.55

27.58

27.61

.8

27.65

27.68

27.71

27.74

27.77

27.80

27.83

27.87

27.90

27.93

.9

27.96

27.99

28.02

28.05

28.09

28.12

28.15

28.18

28.21

28.24

9.0

28.27

28.31

28.34

28.37

28.40

28.43

28.46

28.49

28.53

28.56

.1

28.59

28.62

28.65

28.68

28.71

28.75

28.78

28.81

28.84

28.87

.2

28.90

28.93

28.97

29.00

29.03

29.06

29.09

29.12

29.15

29.19

3

29.22

29.25

29.28

29.31

29.34

29.37

29.41

29.44

29.47

29.50

.4

29.53

29.56

29.59

29.63

29.66

29.69

29.72

29.75

29.78

29.81

9.5

29.85

29.88

29.91

29.94

29.97

30.00

30.03

30.07

30.10

30.13

.6

30.16

30.19

30.22

30.25

30.28

30.32

30.35

30.38

30.41

30.44

.7

30.47

30.50

30.54

30.57

30.60

30.63

30.66

30.69

30.72

30.76

.8

30.79

30.82

30.85

30.88

30.91

30.94

30.98

31.01

31.04

31.07

.9

31.10

31.13

31.16

31.20

31.23

31.26

31.29

31.32

31.35

31.38

10.0

31.42

Moving the decimal point ONE place in D requires moving it ONE plaoe in body of table (see p. 28).

30

MATHEMATICAL TABLES

AREAS OF CIRCLES BY HUNDREDTHS

(For areas by eighths, see p. 32)

D

0

1

2

3

4

5

6

7

8

9

?«

<3

1.0

0.785

0.801

0.817

0.833

0.849

0.866

0.882

0.899

0.916

0.933

16

.1

0.950

0.968

0.985

1.003

1.021

1.039

1.057

1.075

1.094

1.112

18

.2

1.131

1.150

1.169

1.188

1.208

1.227

1.247

1.267

1.287

1.307

20

.3

1.327

1.348

1.368

1.389

1.410

1.431

1.453

1.474

1.496

1.517

21

A

1.539

1.561

1.584

1.606

1.629

1.651

1.674

1.697

1.720

1.744

23

1.5

1.767

1.791

1.815

1.839

1.863

1.887

1.911

1.936

1.961

1.986

24

.6

2.011

2.036

2.061

2.087

2.112

2.138

2.164

2.190

2.217

2.243

26

.7

2.270

2.297

2.324

2.351

2.378

2.405

2.433

2.461

2.488

2.516

27

.8

2.545

2.573

2.602

2.630

2.659

2.688

2.717

2.746

2.776

2.806

29

.9

2.835

2.865

2.895

2.926

2.956

2.986

3.017

3.048

3.079

3.110

31

2.0

3.142

3.173

3.205

3.237

3.269

3.301

3.333

3.365

3.398

3.431

32

.1

3.464

3.497

3.530

3.563

3.597

3.631

3.664

3.698

3.733

3.767

34

.2

3.801

3.836

3.871

3.906

3.941

3.976

4.011

4.047

4.083

4.119

35

.3

4.155

4.191

4.227

4.264

4.301

4.337

4.374

4.412

4.449

4.486

37

A

4.524

4.562

4.600

4.638

4.676

4.714

4.753

4.792

4.831

4.870

38

2.5

4.909

4.948

4.988

5.027

5.067

5.107

5.147

5.187

5.228

5.269

40

.6

5.309

5.350

5.391

5.433

5.474

5.515

5.557

5.599

5.641

5.683

42

.7

5.726

5.768

5.811

5.853

5.896

5.940

5.983

6.026

6.070

6.114

43

.8

6.158

6.202

6.246

6.290

6.335

6.379

6.424

6.469

6.514

6.560

45

.9

6.605

6.651

6.697

6.743

6.789

6.835

6.881

6.928

6.975

7.022

46

3.0

7.069

7.116

7.163

7.211

7.258

7.306

7.354

7.402

7.451

7.499

48

.1

7.548

7.596

7.645

7.694

7.744

7.793

7.843

7.892

7.942

7.992

49

.2

8.042

8.093

8.143

8.194

8.245

8.296

8.347

8.398

8.450

8.501

51

.3

8.553

8.605

8.657

8.709

8.762

8.814

8.867

8.920

8.973

9.026

53

A

9.079

9.133

9.186

9.240

9.294

9.348

9.402

9.457

9.511

9.566

54

3.5

9.621

9.676

9.731

9.787

9.842

9.898

9.954

10.010

56

.5

•

10.01

10.07

10.12

6

.6

10.18

10.24

10.29

10.35

10.41

10.46

10.52

10.58

10.64

10.69

6

.7

10.75

10.81

10.87

10.93

10.99

11.04

11.10

11.16

11.22

11.28

.8

11.34

11.40

11.46

11.52

11.58

11.64

11.70

11.76

11.82

11.88

.9

11.95

12.01

12.07

12.13

12.19

12.25

12.32

12.38

12.44

12.50

4.0

12.57

12.63

12.69

12.76

12.82

12.88

12.95

13.01

13.07

13.14

7

.1

13.20

13.27

13.33

13.40

13.46

13.53

13.59

13.66

13.72

13.79

.2

13.85

13.92

13.99

14.05

14.12

14.19

14.25

14.32

14.39

14.45

.3

14.52

14.59

14.66

14.73

14.79

14.86

14.93

15.00

15.07

15.14

A

15.21

15.27

15.34

15.41

15.48

15.55

15.62

15.69

15.76

15.83

4.5

15.90

15.98

16.05

16.12

16.19

16.26

16.33

16.40

16.47

16.55

.6

16.62

16.69

16.76

16.84

16.91

16.98

17.06

17.13

17.20

17.28

.7

17.35

17.42

17.50

17.57

17.65

17.72

17.80

17.87

17.95

18.02

.8

18.10

18.17

18.25

18.32

18.40

18.47

18.55

18.63

18.70

18.78

8

.9

18.86

18.93

19.01

19.09

19.17

19.24

19.32

19.40

19.48

19.56

Explanation of Table of Areas of Circles (pp. 30-31)

Moving the decimal point one place in column D is equivalent to moving it two places in the body of the table. (D = diameter.)

Area of circle = ^ X (diam.2) = 0.785398 X (diam.2) Conversely,

Diam. =

= 1.128379 X

MATHEMATICAL TABLES AREAS OP CIRCLES BY HUNDREDTHS (continued)

31

D

0

1

2

a

4

5

6

7

8

9

$*i £-3

5.0

19.63

19.71

19.79

19.87

19.95

20.03

20.11

20.19

20.27

20.35

8

.1

20.43

20.51

20.59

20.67

20.75

20.83

20.91

20.99

21.07

21.16

.2

21.24

21.32

21.40

21.48

21.57

21.65

21.73

21.81

21.90

21.98

.3

22.06

22.15

22.23

22.31

22.40

22.48

22.56

22.65

22.73

22.82

.4

22.90

22.99

23.07

23.16

23.24

23.33

23.41

23.50

23.59

23.67

9

5.5

23.76

23.84

23.93

24.02

24.11

24.19

24.28

24.37

24.45

24.54

.6

24.63

24.72

24.81

24.89

24.98

25.07

25.16

25.25

25.34

25.43

.7

25.52

25.61

25.70

25.79

25.88

25.97

26.06

26.15

26.24

26.33

.8

26.42

26.51

26.60

26.69

26.79

26.88

26.97

27.06

27.15

27.25

.9

27.34

27.43

27.53

27.62

27.71

27.81

27.90

27.99

28.09

28.18

6.0

28.27

28.37

28.46

28.56

28.65

28.75

28.84

28.94

29.03

29.13

10

.1

29.22

29.32

29.42

29.51

29.61

29.71

29.80

29.90

30.00

30.09

.2

30.19

30.29

30.39

30.48

30.58

30.68

30.78

30.88

30.97

31.07

.3

31.17

31.27

31.37

31.47

31.57

31.67

31.77

31.87

31.97

32.07

.4

32.17

32.27

32.37

32.47

32.57

32.67

32.78

32.88

32.98

33.08

6.5

33.18

33.29

33.39

33.49

33.59

33.70

33.80

33.90

34.00

34.11

.6

34.21

34.32

34.42

34.52

34.63

34.73

34.84

34.94

35.05

35.15

.7

35.26

35.36

35.47

35.57

35.68

35.78

35.89

36.00

36.10

36.21

11

.8

36.32

36.42

36.53

36.64

36.75

36.85

36.96

37.07

37.18

37.28

.9

37.39

37.50

37.61

37.72

37.83

37.94

38.05

38.16

38.26

38.37

7.0

38.48

38.59

38.70

38.82

38.93

39.04

39.15

39.26

39.37

39.48

.1

39.59

39.70

39.82

39.93

40.04

40.15

40.26

40.38

40.49

40.60

.2

40.72

40.83

40.94

41.06

41.17

41.28

41.40

41.51

41.62

41.74

.3

41.85

41.97

42.08

42.20

42.31

42.43

42.54

42.66

42.78

42.89

12

.4

43.01

43.12

43.24

43.36

43.47

43.59

43.71

43.83

43.94

44.06

7.5

44.18

44.30

44.41

44.53

44.65

44.77

44.89

45.01

45.13

45.25

.6

45.36

45.48

45.60

45.72

45.84

45.96

46.08

46.20

46.32

46.45

.7

46.57

46.69

46.81

46.93

47.05

47.17

47.29

47.42

47.54

47.66

.8

47.78

47.91

48.03

48.15

48.27

48.40

48.52

48.65

48.77

48.89

.9

49.02

49.14

49.27

49.39

49.51

49.64

49.76

49.89

50.01

50.14

8.0

50.27

50.39

50.52

50.64

50.77

50.90

51.02

51.15

51.28

51.40

13

.1

51.53

51.66

51.78

51.91

52.04

52.17

52.30

52.42

52.55

52.68

.2

52.81

52.94

53.07

53.20

53.33

53.46

53.59

53.72

53.85

53.98

.3

54.11

54.24

54.37

54.50

54.63

54.76

54.89

55.02

55.15

55.29

.4

55.42

55.55

55.68

55.81

55.95

56.08

56.21

56.35

56.48

56.61

8.5

56.75

56.88

57.01

57.15

57.28

57.41

57.55

57.68

57.82

57.95

.6

58.09

58.22

58.36

58.49

58.63

58.77

58.90

59.04

59.17

59.31

14

.7

59.45

59.58

59.72

59.86

59.99

60.13

60.27

60.41

60.55

60.68

.8

60.82

60.96

61.10

61.24

61.38

61.51

61.65

61.79

61.93

62.07

.9

62.21

62.35

62.49

62.63

62.77

62.91

63.05

63.19

63.33

63.48

9.0

63.62

63.76

63.90

64.04

64.18

64.33

64.47

64.61

64.75

64.90

.1

65.04

65.18

65.33

65.47

65.61

65.76

65.90

66.04

66.19

66.33

15

.2

66.48

66.62

66.77

66.91

67.06

67.20

67.35

67.49

67.64

67.78

.3

67.93

68.08

68.22

68.37

68.51

68.66

68.81

68.96

69.10

69.25

.4

69.40

69.55

69.69

69.84

69.99

70.14

70.29

70.44

70.58

70.73

9.5

70.88

71.03

71.18

71.33

71.48

71.63

71.78

71.93

72.08

72.23

.6

72.38

72.53

72.68

72.84

72.99

73.14

73.29

73.44

73.59

73.75

.7

73.90

74.05

74.20

74.36

74.51

74.66

74.82

74.97

75.12

75.28

.8

75.43

75.58

75.74

75.89

76.05

76.20

76.36

76.51

76.67

76.82

.9

76.98

77.13

77.29

77.44

77.60

77.76

77.91

78.07

78.23

78.38

16

Moving the decimal point ONE place in D requires moving it TWO places in body of table (see p. 30).

32

MATHEMATICAL TABLES

CIRCUMFERENCES AND AREAS OF CIRCLES BY EIGHTHS, ETC.

(For tenths, see p. 28)

1

|

1

S •

a

Circum.

1

I

Circum.

a £

•<

§

Q

Circum.

ot

2 <5

H

2.749

.6013

4

12.57

12.57

9

28.27

63.62

W4

.04909

.00019

6%4

2.798

.6230

He

12.76

12.96

H

28.67

65.40

fa

.09817

.00077

*fa

2.847

.6450

H

12.96

13.36

H

29.06

67.20

fa

.1473

.00173

*%*

2.896

.6675

Me

13.16

13.77

N

29.45

69.03

Ho

.1963

.00307

'Me

2.945

.6903

H

13.35

14.19

H

29.85

70.88

%«

.2454

.00479

6^4

2.994

.7135

Me

13.55

14.61

K^

30.24

7276

fa

.2945

.00690

•fa

3.043

.7371

13.74

15.03

y\

30.63

74.66

ft*.

.3436

.00940

*%l

3.093

.7610

Mo

13.94

15.47

1&

31.02

76.59

H

.3927

.01227

1

3.142

.7854

H

14.14

15.90

10

31.42

78.54

Hi

.4418

.01553

Me

3.338

.8866

Me

14.33

16.35

H

31.81

80.52

fa

.4909

.01917

H

3.534

.9940

5/i

14.53

16.80

H

32.20

82.52

'^4

.5400

.02320

Me

3.731

1.108

'He

14.73

17.26

H

32.59

84.54

Me

.5890

.02761

U

3.927

1.227

N

14.92

17.72

M

32.99

86.59

'%4

.6381

.03241

Me

4.123

1.353

'Me

15.12

18.19

M

33.38

88.66

fa

.6872

.03758

M

4.320

1.485

H

15.32

18.67

94

33.77

90.76

'5*4

.7363

.04314

Me

4.516

1.623

'Me

15.51

19.15

H

34.16

92.89

u

.7854

.04909

H

4.712

1.767

5

15.71

19.63

11

34.56

95.03

»%4

.8345

.05542

Me

4.909

1.917

He

15.90

20.13

M

34.95

97.21

fa

.8836

.06213

5/6

5.105

2.074

H

16.10

20.63

H

35.34

99.40

l%i

.9327

.06922

'He

5.301

2.237

Me

16.30

21.14

N

35.74

101.6

Me

.9817

.07670

N

5.498

2.405

M

16.49

21.65

H

36.13

103.9

2^4

1.031

.08456

'Me

5.694

2.580

Me

16.69

22.17

%

36.52

106.1

Ifa

1.080

.09281

H

5.890

2.761

N

16.89

22.69

H

36.91

108.4

2*<J4

1.129

.1014

I?i8

6.087

2.948

Me

17.08

23.22

N

37.31

110.8

H

1.178

.1104

a

6.283

3.142

W

17.28

23.76

12

37.70

113.1

*M*

1.227

.1198

He

6.480

3.341

Me

17.48

24.30

H

38.09

115.5

lfa

1.276

.1296

H

6.676

3.547

M

17.67

24.85

H

38.48

117.9

2 ^4

1.325

.1398

Me

6.872

3.758

'He

17.87

25.41

%

38.88

120.3

Me

1.374

.1503

M

7.069

3.976

H

18.06

25.97

M

39.27

122.7

2%4

1.424

.1613

Me

7.265

4.200

'Me

18.26

26.53

U

39.66

125.2

»9$2

1.473

.1726

7.461

4.430

N

18.46

27.11

M-

40.06

127.7

»H*

1.522

.1843

Me

7.658

4.666

'Me

18.65

27.69

ji

40.45

130.2

M

1.571

.1963

W

7.854

4.909

6

18.85

28.27

13

40.84

132.7

8%4

1.620

.2088

Me

8.050

5.157

M

19.24

29.46

w

41.23

135.3

lfa

1.669

.2217

H

8.247

5.412

M

19.63

30.68

H

41.63

137.9

•fa

1.718

.2349

»H«

8.443

5.673

N

20.03

31.92

N

42.02

140.5

Me

1.767

.2485

94

8.639

5.940

H-

20.42

33.18

H

42.41

143.1

3%4

1.816

.2625

me

8.836

6.213

M

20.81

34.47

W

42.80

145.8

lfa

1.865

.2769

H

9.032

6.492

M

21.21

35.78

H

43.20

148.5

»%4

1.914

.2916

'Me

9.228

6.777

N

21.60

37.12

7/i

43.59

151.2

M

1.963

.3068

3

9.425

7.069

7

21.99

38.48

14

43.98

153.9

*H4

2.013

.3223

He

9.621

7.366

K

22.38

39.87

M

44.37

156.7

9 fa

2.062

.3382

H

9.817

7.670

M

22.78

41.28

M

44.77

159.5

*%4

2.111

.3545

Me

10.01

7.980

n

23.17

42.72

N

45.16

162.3

'Me

2.160

.3712

H

10.21

8.296

H

23.56

44.18

H

45.55

165.1

4$B4

2.209

.3883

Me

10.41

8.618

%

23.95

45.66

H

45.95

168.0

2 fa

2.258

.4057

M

10.60

8.946

H

24.35

47.17

N

46.34

170.9

*K*

2.307

.4236

Me

10.80

9.281

X

24.74

48.71

H

46.73

173.8

H

2.356

.4418

H

11.00

9.621

8

25.13

50.27

15

47.12

176.7

*%4

2.405

.4604

Me

11.19

9.968

H

25.53

51.85

H

47.52

179.7

'fa

2.454

.4794

#*

11.39

10.32

M

25.92

53.46

H

47.91

182.7

«H4

2.503

.4987

'He

11.58

10.68

H

26.31

55.09

H

48.30

185.7

'Me

2.553

.5185

N

11.78

11.04

H

26.70

56.75

M

48.69

188.7

3%4

2.602

.5386

'Me

11.98

11.42

N

27.10

58.43

H

49.09

191.7

*fa

2.651

.5591

H

12.17

11.79

N

27.49

60.13

N

49.48

194.8

•%4

2.700

.5800

i«s

12.37

12.18

N

27.88

61.86

K

49.87

197.9

MATHEMATICAL TABLES

33

CIRCUMFERENCES AND AREAS BY EIGHTHS— (continued)

s

.5 P

Circum.

1

P

j

a

|

Circum.

1

a P

Circum.

1

16

50.27

201.1

19 H

61.26

298.6

23

7226

415.5

29

91.11

660.5

50.66

204.2

H

61.65

302.5

H

72.65

420.0

H

91.89

672.0

H

51.05

207.4

N

62.05

306.4

73.04

424.6

N

92.68

683.5

N

51.44

210.6

R

62.44

310.2

H

73.43

429.1

H

93.46

695.1

M

51.84

213.8

20

6283

314.2

H

73.83

433.7

30

94.25

706.9

S£

52.23

217.1

H

6322

318.1

7422

438.4

H

95.03

718.7

3/

52.62

220.4

H

63.62

322.1

?i

74.61

443.0

H

95.82

730.6

£i

53.01

223.7

N

64.01

326.1

8

75.01

447.7

N

96.60

742.6

17

53.41

227.0

M

64.40

330.1

24

75.40

452.4

31

97.39

754.8

H

53.80

230.3

N

64.80

334.1

M

76.18

461.9

H

98.17

767.0

H

54.19

233.7

65.19

3382

M

76.97

471.4

M

98.96

779.3

N

54.59

237.1

%

65.58

3422

N

77.75

481.1

§4

99.75

791.7

M

54.98

240.5

21

65.97

346.4

25

78.54

490.9

32

100.5

804.2

&4

55.37

244.0

H

66.37

350.5

H

79.33

500.7

H

101.3

816.9

a/

55.76

247.4

M

66.76

354.7

80.11

510.7

h

102.1

829.6

H

56.16

250.9

34

67.15

358.8

N

80.90

520.8

N

102.9

842.4

18

56.55

254.5

M

67.54

363.1

26

81.68

530.9

33

103.7

855.3

M

56.94

258.0

W

67.94

367.3

M

82.47

541.2

M

104.5

8683

H

57.33

261.6

68.33

371.5

8325

551.5

1052

881.4

N

57.73

265.2

j|

68.72

375.8

H

84.04

562.0

%

106.0

894.6

M

58.12

268.8

22

69.12

380.1

27

84.82

572.6

34

106.8

907.9

8

58.51 58.90

272.4 276.1

g

69.51 69.90

384.5 388.8

*5

85.61 86.39

583.2 594.0

i

107.6 108.4

921.3 934.8

J6

59.30

279.8

N

7029

393.2

N

87.18

604.8

109.2

948.4

19

59.69

283.5

fc

70.69

397.6

28

87.96

615.8

35

110.0

962.1

H

60.08

287.3

71.08

402.0

H

88.75

626.8

5-4

110.7

975.9

U

60.48

291.0

?4

71.47

406.5

B

89.54

637.9

M

111.5

989.8

N

60.87

294.8

N

71.86

411,0

90.32

6492

N

112.3

1003.8

AREAS OF CIRCLES. Diameters in Feet and Inches, Areas in Square Feet

Feet

Inches

0 1 23456 789 10

11

0

1 2 3 4

5

6 7 8 ? 10 11 12 13 14

.0000 .0055 .0218 .0491 .0873 .1364 .1963 .2673 .3491 .4418 .5454 .7854 .9218 1.069 1.227 1.396 1.576 1.767 1.969 2.182 2.405 2.640 3.142 3.409 3.687 3.976 4.276 4.587 4.909 5.241 5.585 5.940 6.305 7.069 7.467 7.876 8.2% 8.727 9.168 9.621 10.08 10.56 11.04 11.54 12.57 13.10 13.64 14.19 14.75 15.32 15.90 16.50 17.10 1772 18.35

19.63 20.29 20.97 21.65 22.34 23.04 23.76 24,48 2522 25.97 26.73 28.27 29.07 29.87 30.68 31.50 32.34 33.18 34.04 34.91 35.78 36.67 38.48 39.41 40.34 41.28 42.24 43.20 44.18 45.17 46.16 47.17 48.19 50.27 51.32 52.38 53.46 54.54 55.64 56.75 57.86 58.99 60.13 61.28 63.62 64.80 66.00 6720 68.42 69.64 70.88 72.13 73.39 74.66 75.94

78.54 79.85 81.18 82.52 83.86 85.22 86.59 87.97 89.36 90.76 92.18 95.03 96.48 97.93 99.40 100.9 102.4 103.9 105.4 106.9 108.4 110.0 113.1 114.7 116.3 117.9 119.5 121.1 122.7 124.4 126.0 127.7 129.4 132.7 134.4 136.2 137.9 139.6 141.4 143.1 144.9 146.7 148.5 150.3 153.9 155.8 157.6 1593 161.4 1632 165.1 167.0 168.9 170.9 172.8

.6600 2.885 6.681 12.05 18.99

27.49 37.57 4922 62.44 7724

93.60 111.5 131.0 152.1 174.8

If given diameter is not found in this table, reduce diameter to feet and decimals of a foot by aid of the following auxiliary table, and then find area from pp. 30-31.

From Inches and Fractions of an Inch to Decimals of a Foot

Inches 123 456 78 9 10 Feet .0833 .1667 .2500 .3333 .4167 .5000 .5833 .6667 .7500 .8333

11 .9167

Inches H H H M 5/i 9* 7/* Feet .0104 .0208 .0313 .0417 .0521 .0625 .0729 Example. 5 ft. 7% in. = 5.0 + 0.5833 + 0.0313 = 5.6146 ft.

34

MATHEMATICAL TABLES

SEGMENTS OP CIRCLES, GIVEN h/c

Given: h = height; c = chord. (For explanation of this table, see p. 38)

h

Diam. ja

Arc «

Area £:

Central itt

A te

c

c P

c P

h X c p

angle, » Q

Diam. P

.00

1.000 n

.6667 n

0.00° 4,0

.0000

25.010 174gn

1.000 0

.6667

4.58

.0004 ,1

2

19 s?n i*^"u

I/.J/U *A1C7

1.001

.6669 ±

9 .6

.00.6 ;J

3

fl 3A3 4 1 J/ o.?05 *7f)7^

1.002 i

.667.

13J3 I5.7,

.0036 ?J

4

6.290 .f073

1.004 £

.6675 I

18.30 Jg

.0064 g

.05

5.050 ,,0,3

1.007

.6680 ,

22.84° 4,3

.0099

6

4.227 »|g

1.010 \

.6686 2

27.37

.0142

7

3.641 *X7A

1.013

.6693 J

31.88 1!I

.0192 22

8

3.205 ,£7

1.017

.670.

36.36

.0250 22

9

2.868 ^g

1.021 J

.6710 |0

40.82 JJ5

.0314 64

.10

2.600 »,,7

1.026

.6720

45.24° 430

.0385 77

1

2.383 »fl£

1.032

.6731 11

49.63 1|?

.0462 o<

2

2.203 Jt2

1.038

.6743 ?

53.98 I3*

.0545 2^

3

2.053 '55

1.044 2

.6756 J

58.30 1^

.0633

4

1.926 ^

1.051 I

.6770 {J

62.57 J27

.0727 99

.15

1.817 ,04

1.059

.6785

66.80° 4,0

.0826 ,03

6

.723 ,25

1.067

.6801 2

70.98 J ?

.0929 XT'

7

1.641 JE

1.075

.6818 £

75.11 7A^

.1036 VJ

8

1.569 2?

1.084 ,2

.6836

79.20 1X4

.1147 J

9

1.506 g

1.094 ^

.6855 JJ

83.23 $5

•'262 jg

.20

1.450 ,n

1.103 n

.6875 71

87.21° 30,

.1379 nn

1

1.400

1.114 J

.6896 ?,7

91.13 S=

.1499 20

2

1.356

1.124

.6918 5?

95.00 W.

.1622 Jg

3

1.317 H

1.136 2

.6941 g

98.81 2J

.1746 g

4

1.282 %

1.147 »

.6965 24

102.56 375

.1873 Jg

.25

6

1.250 -« .222 28

1.159 12 1.171 \\

.6989 9c .7014 ;7

106.26° 3,4 109.90 $52

•2000 ,28

7

.196 ?6

1.184

.7041 5i

113.48 358

^2258 !2J

8

.173 23

1.197

.7068 11

117.00 ?52

.2387 aj

9

.152 J.

1.2.1 JJ

.7096 g

120.45 345

JI517 J30

.30

1 2 3

.133 ,7 .116 i .101 .088

1.225 ,4 1.239 7 1.254 \l 1.269

.7125 9q .7154 29 .7185 ^ .7216 3,1

123.86° 334 127.20 334 130.48 325 133.70 322

.2647 130 .2777 ,o .2906 HJ .3034 52

4

1.075 jf

1.284 ]£

.7248 55

136.86 316

3162 Jg

.35

1.064 ,n

1.300

.7280 34

139.97° 3rt,

.3289 ,--

6

1.054 '°

1.316

.7314 5x

143.02 305

.3414 Jg

7

1.046

1.332 2

.7348 g

146.01 ££

.3538 \£

8

1.038 2

1.349 i

.7383 ?5

148.94 ??2

.3661 g

9

1.031 J

1.366 \'7

.7419 g

151.82 288

3783 J22

.40

1.025 1.020 i

1.383 ,o 1.401

.7455 37 .7492 on

154.64° -77 157.41 El

4022 "9

.W/l jj>

2

1.015

1.419

.7530 IS

160.12 %\

.4137 JJfj

3

1.011

1.437

.7568 3*5

162.78 5??

4

1.008 \

1.455 J«

.7607 ^Q

165.39 261

.4364 jj2

.45

6

8 9

1.006 1.003 1.002 1.001 1.000 J

1.474 ,Q 1.493 1.512 1.531 ij 1.55. 20

.7647 40 .7687 TV .7728 J .7769 11 .781. «

167.95° ,,, 170.46 gi 172.91 fjf 175.32 241 177.69 g7

.4475 )no .4584 X^ .4691 2? .4796 25 .4899 JJJ

.50

1.000

1.571

.7854

180.00°

.5000

• Interpolation may be inaccurate at these points.

MATHEMATICAL TABLES

.SEGMENTS OP CIRCLES, GIVEN h/D

Given: h «=• height; D = diameter of circle. (For explanation of this table, see p. 38)

Arc

Area

Central i angle, v

Chord

Arc

Circumf. Q

Area Ckde

2003 2003

3482 »t1c

.4027 ,Jg

.4510 *4™

.4949 »;£:

.5355 Jftn 5735

.'6094 'U9

.6435 *„,

.676. Jf6

.7075 JA;

.7377 JJ2

.7670 ,293

.7954 276

.8230 2;x

.8500 ^V

.8763 ;K

.9021 g*

0.9273 24R

0.9521 215

0.9764 %£

1.0004 i??

1.0239 Si

1.0472 1.070. ..0928 ...152 1.1374

1.1593 1.1810 1.2025 1.2239 1.2451

1.2661 1.2870 1.3078 1.3284

1.3694 1.3898 1.4101 1.4303 1.4505

1.4706 1.4907 1.5108 1.5308 1.5508

1.5708

22g

222 219

217

212

202 2J2

2

.0000 .0013 .0037 .0069 .0105

.0.47 .0192 .0242 .0294 .0350

.0409 .0470 .0534 .0600 .0668

.0739 .0811 .0885 .0961 .1039

.1118 .1199 .128. .1365 .1449

.1535 .1623 .1711 .1800 .1890

.1982 .2074 .2167 .2260 .2355

.2450 .2546 .2642 .2739 .2836

.2934 .3032 .3130 .3229 .3328

.3428 .3527

3727 3827

3927

.,

,

,

IUU

0.00° 770,

22.96 A296

32.52 956

39.90 738

46.15 ,|g

51-68° ....

56.72 504

6137 *£

65.72 435

69.83 ^J]

73.74° „_.

77.48 *|74

81.07 359

84.54 *347

87.89 .335

91.15° ,..,

94.31

97.40 309

.00.42 302

103.37 295

106.26°

109.10 284

111.89 279 114.63 274 11734 27.

266

.20.00°

122.63 263

125.23 260

127.79 256

.3033 254

132.84°

.35.33 249

.37.80 247

140.25 245 .42.67 2J2

145.08°

.47.48 240

.49.86 238

152.23 237

154.58 235

156.93° ,

159.26 233

161.59 233

163.90 23. 166.22 232

168.52° ...

.70.82 230

173.12 230

175.42 230

177.71 229

180.00°

.0000 !2800 3919 .4359

!5I03 .5426 .5724

.6000 .6258 .6499 .6726 .6940

.7141

>513 .7684

.8000 .8146 .8285 .8417

.8660 .8773 .8879 .8980 .9075

.9165 .9250 .9330 .9404 .9474

.9539 .9600 .9656 .9708 .9755

.9798 .9837 .9871 .9902 .9928

.9950 .9968 .9982 .9992 .9998

1.0000

3.3 .298

241 J27 .2.4

.0000 .0638 .0903 .1108 .1282

.1436

!l705 .1826 .1940

.2048 .2.52 .2252 .2348 .244.

.2532 .2620 .2706 .2789 .2871

.2952 3031 3108 .3184 3259

.3333 .3406 3478 .3550 .3620

3690 3759 3828 3896 .3963

.4030 .4097 .4.63 .4229 .4294

.4359 .4424 .4489 .4553 .4617

.4681 .4745 .4809 .4873 .4936

.5000

j{j

.0000 ,7

.0017 \\

.0048 l\

.0087 ??

.0134 g

.0187 «,

.0245 g

.0308 g

.0375 S{

.0446 £

.0520 TO

.0599 JJ .0680

.0764 2;

.085. §J

.0941 Q,

.1033 2

.1127 A?

.1224 II

.1323 ,99

•!$ 103

.1631 !°<

.1737 2$

.1846 JJ5

.1955 m

.2066 1

.2.78 J

.2292 \\i

2407 \\l

.2523 ,,7

•2640 £

.2759 9

.2878 ,«

.2998 ]^°

.3.19 ,22

324. 22

3364 S

3487 g

.3611 j|J

3735 ,«

3860 g

3986 \%

.41.2 5?

.4238 Jg

.4364 .„

.4491 %

.4618 ?i

.4745 g

.4873 J28

.5000

Interpolation may be inaccurate at these points.

36 MATHEMATICAL TABLES

VOLUMES OF SPHERES BY HUNDREDTHS

D

0

1

2

3

4

5

&

1.0

.5236

.5395

.5556

.5722

.5890

.6061

.6236

.6414

.65%

.6781

173

.6969

.7161

.7356

.7555

.7757

.7963

.8173

.8386

.8603

.8823

208

'.2

.9048

.9276

.9508

.9743

.9983

1.0227

236

.2

1.023

1.047

1.073

1.098

1.124

25

.3

1.150

1.177

1.204

1.232

1.260

1.288

1.317

1.346

1.376

1.406

29

.4

1.437

1.468

1.499

1.531

1.563

1.596

1.630

1.663

1.697

1.732

33

1.5

1.767

1.803

1.839

1.875

1.912

1.950

1.988

2.026

2.065

2.105

38

.6

2.145

2.185

2.226

2.268

2.310

2.352

2.395

2.439

2.483

2.527

43

.7

2.572

2.618

2.664

2.711

2.758

2.806

2.855

2.903

2.953

3.003

48

8

3.054

3.105

3.157

3.209

3.262

3.315

3.369

3.424

3.479

3.535

54

.9

3.591

3.648

3.706

3.764

3.823

3.882

3.942

4.003

4.064

4.126

60

2.0

4.189

4.252

4.316

4.380

4.445

4.511

4.577

4.644

4.712

4.780

66

.1

4.849

4.919

4.989

5.0CO

5.131

5.204

5.277

5.350

5.425

5.500

73

.2

5.575

5.652

5.729

5.806

5.885

5.964

6.044

6.125

6.206

6.288

80

.3

6.371

6.454

6.538

6.623

6.709

6.795

6.882

6.970

7.059

7.148

87

.4

7.238

7.329

7.421

7.513

7.606

7.700

7.795

7.890

7.986

8.083

94

2.5

8.181

8.280

8.379

8.479

8.580

8.682

8.785

8.888

8.992

9.097

102

.6

9.203

9.309

9.417

9.525

9.634

9.744

9.855

9.966

10.079

110

.6

10.08

10.19

11

.7

10.31

10.42

10.54

10.65

10.77

10.89

11.01

11.13

11.25

11.37

12

.8

11.49 x

11.62

11.74

11.87

11.99

12.12

12.25

12.38

12.51

12.64

13

.9

12.77

12.90

13.04

13.17

13.31

13.44

13.58

13.72

13.86

14.00

14

3.0

14.14

14.28

14.42

14.57

14.71

14.86

15.00

15.15

15.30

15.45

15

.1

15.60

15.75

15.90

16.06

16.21

16.37

16.52

16.68

16.84

17.00

16

.2

17.16

17.32

17.48

17.64

17.81

17.97

18.14

18.31

18.48

18.65

17

.3

18.82

18.99

19.16

19.33

19.51

19.68

19.86

20.04

20.22

20.40

18

.4

20.58

20.76

20.94

21.13

21.31

21.50

21.69

21.88

22.07

22.26

19

3.5

22.45

22.64

22.84

23.03

23.23

23.43

23.62

23.82

24.02

24.23

20

.6

24.43

24.63

24.84

25.04

25.25

25.46

25.67

25.88

26.09

26.31

21

.7

26.52

26.74

26.95

27.17

27.39

27.61

27.83

28.06

28.28

28.50

22

.8

28.73

28.%

29.19

29.42

29.65

29.88

30.11

30.35

30.58

30.82

23

.9

31.06

31.30

31.54

31.78

32.02

32.27

32.52

32.76

33.01

33.26

25

4.0

33.51

33.76

34.02

34.27

34.53

34.78

35.04

35.30

35.56

35.82

26

.1

36.09

36.35

36.62

36.88

37.15

37.42

37.69

37.97

38.24

38.52

27

.2

38.79

39.07

39.35

39.63

39.91

40.19

40.48

40.76

41.05

41.34

28

.3

41.63

41.92

42.21

42.51

42.80

43.10

43.40

43.70

44.00

44.30

30

.4

44.60

44.91

45.21

45.52

45.83

46.14

46.45

46.77

47.08

47.40

31

4.5

47.71

48.03

48.35

48.67

49.00

49.32

49.65

49.97

50.30

50.63

33

.6

50.97

51.30

51.63

51.97

52.31

52.65

52.99

53.33

53.67

54.02

34

.7

54.36

54.71

55.06

55.41

55.76

56.12

56.47

56.83

57.19

57.54

35

.8

57.91

58.27

58.63

59.00

59.37

59.73

60.10

60.48

60.85

61.22

37

.9

61.60

61.98

62.36

62.74

63.12

63.51

63.89

64.28

64.67

65.06

38

Explanation of Table of Volumes of Spheres (pp. 36-37).

Moving the decimal point one place in column D is equivalent to moving it three places in the body of the table. (D = diameter.)

Volume of sphere = 7 X (diarn.») = 0.523599 X (diam.»)

Conversely,

Diam.

1.240701 X

MATHEMATICAL TABLES

37

VOLUMES OF SPHERES (continued)

D

0

1

2

3

4

5

6

7

8

9

11

5.0

65.45

65.84

66.24

66.64

67.03

67.43

67.83

68.24

68.64

69.05

40

.1

69.46

69.87

70.28

70.69

71.10

71.52

71.94

72.36

72.78

73.20

42

.2

73.62

74.05

74.47

74.90

75.33

75.77

76.20

76.64

77.07

77.51

43

.3

77.95

78.39

78.84

79.28

79.73

80.18

80.63

81.08

81.54

81.99

45

.4

82.45

82.91

83.37

83.83

84.29

84.76

85.23

85.70

86.17

86.64

47

5.5

87.11

87.59

88.07

88.55

89.03

89.51

90.00

90.48

90.97

91.46

48

.6

91.95

92.45

92.94

93.44

93.94

94.44

94.94

95.44

95.95

96.46

50

.7

96.97

97.48

97.99

98.51

99.02

99.54

10006

52

.7

100.1

100.6

101.1

101.6

5

.8

102.2

102.7

103.2

103.8

104.3

104.8

105.4

105.9

106.4

107.0

5

.9

107.5

108.1

108.6

109.2

109.7

110.3

110.9

111.4

112.0

112.5

6

6.0

113.1

113.7

114.2

114.8

115.4

115.9

116.5

117.1

117.7

118.3

6

.1

118.8

119.4

120.0

120.6

121.2

121.8

122.4

123.0

123.6

124.2

.2

124.8

125.4

126.0

126.6

127.2

127.8

128.4

129.1

129.7

130.3

.3

130.9

131.5

132.2

132.8

133.4

134.1

134.7

135.3

136.0

136.6

.4

137.3

137.9

138.5

139.2

139.8

140.5

141.2

141.8

142.5

143.1

7

6.5

143.8

144.5

145.1

145.8

146.5

147.1

147.8

148.5

149.2

149.8

.6

150.5

151.2

151.9

152.6

153.3

154.0

154.7

155.4

156.1

156.8

.7

157.5

158.2

158.9

159.6

160.3

161.0

161.7

162.5

163.2

163.9

.8

164.6

165.4

166.1

166.8

167.6

168.3

169.0

169.8

170.5

171.3

.9

172.0

172.8

173.5

174.3

175.0

175.8

176.5

177.3

178.1

178.8

8

7.0

179.6

180.4

181.1

181.9

182.7

183.5

184.3

185.0

185.8

186.6

.1

187.4

188.2

189.0

189.8

190.6

191.4

192.2

193.0

193.8

194.6

.2

195.4

196.2

197.1

197.9

198.7

199.5

200.4

201.2

202.0

202.9

.3

203.7

204.5

205.4

206.2

207.1

207.9

208.8

209.6

210.5

211.3

.4

212.2

213.0

213.9

214.8

215.6

216.5

217.4

218.3

219.1

220.0

9

7.5

220.9

221.8

222.7

223.6

224.4

225.3

226.2

227.1

228.0

228.9

.6

229.8

230.8

231.7

232.6 -

233.5

234.4

235.3

236.3

237.2

238.1

.7

239.0

240.0

240.9

241.8

242.8

243.7

244.7

245.6

246.6

247.5

.8

248.5

249.4

250.4

251.4

252.3

253.3

254.3

255.2

256.2

257.2

10

.9

258.2

259.1

260.1

261.1

262.1

263.1

264.1

265.1

266.1

267.1

8.0

268.1

269.1

270.1

271.1

272.1

273.1

274.2

275.2

276.2

277.2

.1

278.3

279.3

280.3

281.4

282.4

283.4

284.5

285.5

286.6

287.6

.2

288.7

289.8

290.8

291.9

292.9

294.0

295.1

296.2

297.2

298.3

II

3

299.4

300.5

301.6

302.6

303.7

304.8

305.9

307.0

308.1

309.2

.4

310.3

311.4

312.6

313.7

314.8

315.9

317.0

318.2

319.3

320.4

8.5

321.6

322.7

323.8

325.0

326.1

327.3

328.4

329.6

330.7

331.9

.6

333.0

334.2

335.4

336.5

337.7

338.9

340.1

341.2

342.4

343.6

12

.7

344.8

346.0

347.2

348.4

349.6

350.8

352.0

353.2

354.4

355.6

.8

356.8

358.0

359.3

360.5

361.7

362.9

364.2

365.4

366.6

367.9

.9

1

369.1

370.4

371.6

372.9

374.1

375.4

376.6

377.9

379.2

380.4

13

9.0

381.7

383.0

384.3

385.5

386.8

388.1

389.4

390.7

392.0

393.3

.1

394.6

395.9

397.2

398.5

399.8

401.1

402.4

403.7

405.1

406.4

2

407.7

409.1

410.4

411.7

413.1

414.4

415.7

417.1

418.4

419.8

.3

421.2

422.5

423.9

425.2

426.6

428.0

429.4

430.7

432.1

433.5

14

.4

434.9

436.3

437.7

439.1

440.5

441.9

443.3

444.7

446.1

447.5

9.5

448.9

450.3

451.8

453.2

454.6

456.0

457.5

458.9

460.4

461.8

.6

463.2

464.7

466.1

467.6

469.1

470.5

472.0

473.5

474.9

476.4

15

.7

477.9

479.4

480.8

482.3

483.8

485.3

486.8

488.3

489.8

491.3

.8

492.8

494.3

495.8

497.3

498.9

500.4

501.9

503.4

505.0

506.5

16

9

508.0

509.6

511.1

512.7

514.2

515.8

517.3

518.9

520.5

522.0

10.0

523.6

Moving the decimal point ONE place in D requires moving it THREE places in body of table (see p. 36).

38

MATHEMATICAL TABLES

SEGMENTS OF SPHERES

(h = height of segment; D = diam. of sphere)

h

Vol. segm.

to

Q

Vol. segm.

d S

Explanation of Table on this page Given, h = height of segment, D = diam. of sphere.

D

D»

Vol. sphere

0.00 1

0.0000 0 0002

2

0.0000 0 0003

3

2

0.0006

4

0.0012

9

To find the volume of the segment,

3

0.0014

10

0.0026

21

form the ratio h/D and find from the

4

0.0024

14

0.0047

/ 1 26,

table the value of (vol./D8); then, by

0.05

0.0038

1 f

0.0073

of

a simple multiplication,

6

g

0.0054 0.0073 0 0095

lo 19 22

0.0104 0.0140 0 0182

31

36 42

vol. segment = D* X (vo\./D*) The table gives also the ratio of the

9

0.0120

25

*>7

0.0228

46

11

volume of the segment to the entire

0.10 1

0.0147 0 0176

LI

29

0.0280 0 0336

_>/ 56

volume of the sphere. NOTE. Area of zone = v X h X D.

2

0.0208

32

0 A

0.0397

61

//

(Use Table of Multiples of ir, p. 28)

3

0 0242

34

0 0463

oo

4

0.0279

37 39

0.0533

70

74

Explanation of Table on p. 34

0.15 6

0.0318 0 0359

41

0.0607 0 0686

79

Given, h = height of segment, c = chord.

7

0.0403

44

AC

0.0769

83

QA

To find the diam. of the circle, the

8 9

0.0448 0.0495

4->

47 50

0.0855 0.0946

OO

91 94

length of arc, or the area of the seg- ment, form the ratio h/c, and find

0.20

0.0545

C 1

0.1040

QO

from the table the value of (diam./c),

1 2 3

0.05% 0.0649 0 0704

>]

53 55

0.1138 0.1239 0 1344

Vo

101 105

(arc/c), or (area/Ac) ; then, by a simple multiplication,

4

0.0760

56 58

0.1452

108 110

diam. = c X (diam./c),

arc = c X (arc/c),

0.25 6

0.0818 0 0878

60

0.1562 0 1676

114

area = h X c X (area/Ac).

7

0.0939

61

0.1793

117

i on

The table gives also the angle sub-

8

0.1002

63

64

0.1913

120

122

tended at the center, and the ratio of

9

0.1066

65

0.2035

125

h to D. See p. 106.

0.30 1

0.1131 0 1198

67

0.2160 0 2287

127

Explanation of Table on p. 35

2

0.1265

67

Xrt

0.2417

130

Given, h = height of segment,

3

0.1334

ov 70

0.2548

134

D — diam. of circle.

4

0.1404

/u 71

0.2682

135

To find the chord, the length of arc,

0.35

0.1475

n«i

0.2817

1 Ifl

or the area of the segment, form the

6 7 8

0.1547 0.1620 0.1694

n

73 74

0.2955 0.3094 0.3235

1 JO

139 141

ratio h/D, and find from the table the value of (chord/D), (arc/Z>), or

9

0.1768

74 75

0.3377

142 143

(area//)2); then, by a simple multi-

MJ

plication,

0.40

0.1843 0.1919

76

0.3520 0.3665

145

chord = D X (chord/ D),

2

0.1995

76

77

0.3810

145

147

arc - D X (arc//)),

3 4

0.2072 0.2149

II

77 78

0.3957 0.4104

It/

147 148

area = D* X (area/Z)"). The table gives also the angle sub-

0.45

0.2227

•70

0.4252

1 AQ

tended at the center, the ratio of the

6 7 8

0.2305 0.2383 0 2461

7o 78 78

0.4401 0.4551 0 4700

1^7

150 149

arc of the segment to the whole cir- cumference, and the ratio of the area

9

0.2539

78 79

. 0.4850

150 150

of the segment to the area of the

whole circle. See p. 106.

0.50

0.2618

0.5000

NOTE. Vol. segm. - }6 * h* (3D-2h).

MATHEMATICAL TABLES

39

REGULAR POLYGONS

n =• number of sides;

TO = 360°/n = angle subtended at the center by one side;

a = length of one side = R (2 sin |-) = r (2 tan |-) ; R = radius of circumscribed circle = a ( y^ esc — \ = r (sec -^-j ; r = radius of inscribed circle = R(COS — J = a(l$ cot s~J ; Area = o*H n cot - = fl'/i n sin t> = r*n tan -.

Area

Area

Area

*

R

a

a

r

r

n

7)

a2

#»

r2

0

r

R

r

R

a

3

120°

0.4330

1.299

5:196

0.5774

2.000

1.732

3.464

0.5000

0.2887

4

90°

1.000

2.000

4.000

0.7071

.414

1.414

2.000

0.7071

0.5000

5

72°

1.721

2.378

3.633

0.8507

.236

1.176

1.453

0.8090

0.6882

6 .

60°

2.598

2.598

3.464

1.0000

.155

1.000

1.155

0.8660

0.8660

7

5P.43

3.634

2.736

3.371

1.152

.110

0.8678

0.9631

0.9010

1.038

8

45°

4.828

2.828

3.314

1.307

.082

0.7654

0.8284

0.9239

1.207

9

40°

6.182

2.893

3.276

1.462

.064

0.6840

0.7279

0.9397

1.374

10

36°

7.694

2.939

3.249

1.618

.052

0.6180

0.6498

0.9511

1.539

12

30°

11.20

3.000

3.215

1.932

.035

0.5176

0.5359

0.9659

1.866

15

24°

17.64

3.051

3.188

2.405

.022

0.4158

0.4251

0.9781

2.352

16

22°. 50

20.11

3.062

3.183

2.563

.020

0.3902

0.3978

0.9808

2.514

20

18°

31.57

3.090

3.168

3.1%

.013

0.3129

0.3168

0.9877

3.157

24

15°

45.58

3.106

3.160

3.831

.009

0.2611

0.2633

0.9914

3.798

32

11°.25

81.23

3.121

3.152

5.101

.005

0.1960

0.1970

0.9952

5.077

48

7°. 50

183.1

3.133

3.146

7.645

.002

0.1308

0.1311

0.9979

7.629

64

5°.625

325.7

3.137

3.144

10.19

.001

0.0981

0.0983

0.9988

10.18

BINOMIAL COEFFICIENTS

(For table giving binomial coefficients for fractional values of n, see p. 116). n(n - 1) n(n - l)(n - 2)

(n)o = 1; (n)i = n; (71)2 =

(n)

1X2'

n(n - l)(n - 2) . . . (n - [r - 1]). 1X2X3. . . X r

1X2X3 Another notation:

= (n)r.

n

(n)o

(»)i

(n)i

(n)i

(n)*

(n).

(n)a

(»)T

(n)a

(n)a

(n)w

(n)u

(n)is

(n)i3

I

1

?

2

1

3

3

3

1

4

4

6

4

1

5

5

10

10

5

1

6

6

15

20

15

6

1

7

7

21

35

35

71

7

1

8

8

28

56

70

56

28

8

1

Q

9

36

84

126

126

84

36

9

1

in

10

45

120

210

252

210

120

45

10

|

11

11

51

165

310

462

462

330

165

55

11

1

12 13 14 15

12 13 14 15

66 78 91 105

220 286 364 455

495 715 1001 1365

792 1287 2002 3003

924 1716 3003 5005

792 1716 3432 6435

495 1287 3003 6435

220 715 2002 5005

66 286 1001 3003

12 78 364 1365

1

13 91 455

.....

14 105

For n = 14, (n)u = 1; for n - 15, (n)u = 15, and (n)l6 - 1.

40 MATHEMATICAL TABLES

COMMON LOGARITHMS (special table)

1*

0

1

2

3

4

5

6

7

8

9

$?»•

«JTJ

1.00

0.0000

0004

0009

0013

0017

0022

0026

0030

0035

0039

4

1.01

0043

0048

0052

0056

0060

0065

0069

0073

0077

0082

1.02

0086

0090

0095

0099

0103

0107

0111

0116

0120

0124

1.03

0128

0133

0137

0141

0145

0149

0154

0158

0162

0166

1.04

0170

0175

0179

0183

0187

0191

0195

0199

0204

0208

1.05

0212

0216

0220

0224

0228

0233

0237

0241

0245

0249

1.06

0253

0257

0261

0265

0269

0273

0278

0282

0286

0290

1.07

0294

0298

0302

0306

0310

0314

0318

0322

0326

0330

.08

0334

0338

0342

0346

0350

0354

0358

0362

0366

0370

1.09

0374

0378

0382

0386

0390

0394

0398

0402

04C6

0410

.10

0.0414

0418

0422

0426

0430

0434

0438

0441

0445

0449

.11

0453

0457

0461

0465

0469

0473 .

0477

0481

0484

0488

.12

0492

0496

0500

0504

0508

0512

0515

0519

0523

0527

.13

0531

0535

0538

0542

0546

0550

0554

0558

0561

0565

.14

0569

0573

0577

0580

0584

0588

0592

0596

0599

0603

.15

0607

0611

0615

0618

0622

0626

0630

0633

0637

0641

.16

0645

0648

0652

0656

0660

0663

0667

0671

0674

0678

.17

0682

0686

0689

0693

0697

0700

0704

0708

0711

0715

.18

0719

0722

0726

0730

0734

0737

0741

0745

0748

0752

.19

0755

0759

0763

0766

0770

0774

0777

0781

0785

0788

x .•

1.20

0.0792

0795

0799

0303

0806

0810

0813

0817

0821

0824

1.21

0828

0831

0835

0839

0842

0846

0849

0853

0856

0860

1.22

0864

0867

0871

0374

0878

0881

0885

0888

0892

0896

1.23

0899

0903

0906

0910

0913

0917

0920

0924

0927

0931

1.24

0934

0938

0941

0945

0948

0952

0955

0959

0962

0966

1.25

0969

0973

0976

0980

0983

0986

0990

0993

0997

1000

3

1.26

1004

1007

1011

1014

1017

1021

1024

1028

1031

1035

1.27

1038

1041

1045

1048

1052

1055

1059

1062

1065

1069

1.28

1072

1075

1079

1082

1086

1089

1092

1096

1099

1103

1.29

1106

1109

1113

1116

1119

1.123

1126

1129

1133

1136

1.30

0.1139

1143

1146

1149

1153

1156

1159

1163

1166

1169

1.31

1173

1176

1179

1183

1186

1189

1193

1196

1199

1202

1.32

1206

1209

1212

1216

1219

1222

1225

1229

1232

1235

1.33

1239

1242

1245

1248

1252

1255

1258

1261

1265

1268

1.34

1271

1274

1278

1281

1284

1287

1290

1294

1297

1300

1.35

1303

1307

1310

1313

1316

1319

1323

1326

1329

1332

1.36

1335

1339

1342

1345

1348

1351

1355

1358

1361

1364

1.37

1367

1370

1374

1377

1380

1383

1386

1389

1392

1396

1.38

1399

1402

1405

1408

1411

1414

1418

1421

1424

1427

1.39

1430

1433

1436

1440

1443

1446

1449

1452

1455

1458

1.40

0.1461

1464

1467

1471

1474

1477

1480

1483

1486

1489

1.41

1492

1495

1498

1501

1504

1508

1511

1514

1517

1520

1.42

1523

1526

1529

1532

1535

1538

1541

1544

1547

1550

1.43

1553

1556

1559

1562

1565

1569

1572

1575

1578

1581

1.44

1584

1587

1590

1593

1596

1599

1602

1605

1608

1611

1.45

1614

1617

1620

1623

1626

1629

1632

1635

1638

1641

1.46

1644

1647

1649

1652

1655

1658

1661

1664

1667

1670

1.47

1673

1676

1679

1682

1685

1688

1691

1694

1697

1700

1.48

1703

1706

1708

1711

1714

1717

1720

1723

1726

1729

1.49

1732

1735

1738

1741

1744

1746

1749

1752

1755

1758

Moving the decimal point n places to the right [or left] in the number requires adding + n [or - n] ia the body of the table (see p. 42).

MATHEMATICAL TABLES

41

COMMON LOGARITHMS (special table, continued)

p

0

1

2

3

4

5

6

7

8

9

11

1.50

0.1761

1764

1767

1770

1772

1775 .

1778

1781

1784

1787

3

1.51

1790

1793

1796

1798

1801

1804

1807

1810

1813

1816

1.52

1818

1821

1824

1827

1830

1833

1836

1838

1841

1844

1.53

1847

1850

1853

1855

1858

1861

1864

1867

1870

1872

1.54

1875

1878

1881

1884

1886

1889

1892

1895

1898

1901

1.55

1903

1906

1909

1912

19t5

1917

1920

1923

1926

1928

1.56

1931

1934

1937

1940

1942

1945

1948

1951

1953

1956

1.57

1959

1962

1965

1967

1970

1973

1976

1978

1981

1984

1.58

1987

1989

1992

1995

1998

2000

2003

2006

2009

2011

1.59

2014

2017

2019

2022

2025

2028

2030

2033

2036

2038

1.60

0.2041

2044

2047

2049

2052

2055

2057

2060

2063

2066

1.61

2068

2071

2074

2076

2079

2082

2084

2087

2090

2092

1.62

• 2095

2098

2101

2103

2106

2109

2111

2114

2117

2119

1.63

2122

2125

2127

2130

2133

2135

2138

2140

2143

2146

1.64

2148

2151

2154

2156

2159

2162

2164

2167

2170

2172

1.65

2175

2177

2180

2183

2185

2188

2191

2193

2196

2198

1.66

2201

2204

2206

2209

2212

2214

2217

2219

2222

2225

1.67

2227

2230

2232

2235

2238

2240

2243

2245

2248

2251

1.68

2253

2256

2258

2261

2263

2266

2269

2271

2274

2276

1.69

2279

2281

2284

2287

2289

2292

2294

2297

2299

2302

1.70

0.2304

2307

2310

2312

2315

2317

2320

2322

2325

2327

1.71

2330

2333

2335

2338

2340

2343

2345

2348

2350

2353

1.72

2355

2358

2360

2363

2365

2368

2370

2373

2375

2378

1.73

2380

2383

2385

2388

2390

2393

2395

2398

2400

2403

1.74

2405

2408

2410

2413

2415

2418

2420

2423

2425

2428

2

1.75

2430

2433

2435

2438

2440

2443

2445

2448

2450

2453

1.76

2455

2458

2460

2463

2465

2467

2470

2472

2475

2477

1.77

2480

2482

2485

2487

2490

2492

2494

2497

2499

2502

1.78-

2504

2507

2509

2512

2514

2516

• 2519

2521

2524

2526

1.79

2529

2531

2533

2536

2538

2541

2543

2545

2548

2550

180

0.2553

2555

2558

2560

2562

2565

2567

2570

2572

2574

1.81

2577

2579

2582

2584

2586

2589

2591

2594

2596

2598

1.82

2601

2603

2605

2608

2610

2613

2615

2617

2620

2622

1.83

2625

2627

2629

2632

2634

2636

2639

2641

2643

2646

1.84

2648

2651

2653

2655

2658

2660

2662

2665

2667

2669

1.85

2672

2674

2676

2679

2681

2683

2686

2688

2690

2693

1.86

2695

2697

2700

2702

2704

2707

2709

2711

2714

2716

1.87

2718

2721

2723

2725

2728

2730

2732

2735

2737

2739

1.88

2742

2744

2746

2749

2751

2753

2755

2758

2760

2762

1.89

2765

2767

2769

2772

2774

2776

2778

2781

2783

2785

1.90

0.2788

2790

2792

2794

2797

2799

2801

2804

2806

2808

1.91

2810

2813

2815

2817

2819

2822

2824 '

2826

2828

2831

1.92

2833

2835

2838

2840

2842

2844

2847

2849

2851

2853

1.93

2856

2858

2860

2862

2865

2867

2869

2871

2874

2876

1.94

2878

2880

2882

2885

2887

2889

2891

2894

2896

2898

1.95

2900

2903

2905

2907

2909

2911

2914

2916

2918

2920

1.96

2923

2925

2927

2929

2931

2934

2936

2938

2940

2942

1.97

2945

2947

2949

2951

2953

2956

2958

2960

2962

2964

1.98

2967

2969

2971

2973

2975

2978

2980

2982

2984

2986

1.99

2989

2991

2993

2995

2997

2999

3002

3004

3006

3008

42

MATHEMATICAL TABLES

COMMON LOGARITHMS

11

0

1

2

a

4

5

6

7

8

9

Sa

<3-0

1.0

0.0000

0043

0086

0128

0170

0212

0253

0294

0334

0374

1.1

0414

0453

0492

0531

0569

0607

0645

0682

0719

0755

1.2

0792

0828

0864

0899

0934

0969

1004

1038

1072

1106

TH

1.3

1139

1173

1206

1239

1271

1303

1335

1367

1399

1430

1

1.4

1461

1492

1523

1553

1584

1614

1644

1673

1703

1732

6

•^

13

1761

1790

1818

1847

1875

1903

1931

1959

1987

2014

1

1.6

2041

2068

2095

2122

2148

2175

2201

2227

2253

2279

03

1.7

2304

2330

2355

2380

2405

2430

2455

2480

2504

2529

ft

1.8

2553

2577

2601

2625

2648

2672

2695

2718

2742

2765

1

1.9

2788

2810

2833

2856

2878

2900

2923

2945

2967

2989

CQ

2.0

0.3010

3032

3054

3075

3096

3118

3139

3160

3181

3201

21

2.1

3222

3243

3263

3284

3304

3324

3345

3365

3385

3404

20

2.2

3424

3444

3464

3483

3502

3522

3541

3560

3579

3598

19

2.3

3617

3636

3655

3674

3692

3711

3729

3747

3766

3784

18

2.4

3802

3820

3838

3856

3874

3892

3909

3927

3945

3962

17

2.5

3979

3997

4014

4031

4048

4065

4082

4099

4116

4133

17

2.6

4150

4166

4183

4200

4216

4232

4249

4265

4281

4298

16

2.7

4314

4330

4346

4362

4378

4393

4409

4425

4440

4456

16

2.8

4472

4487

4502

4518

4533

4548

4564

4579

4594

4609

15

2.9

4624

4639

4654

4669

4683

4698

4713

4728

4742

4757

15

3.0

0.4771

4786

4800

4814

4829

4843

4857

4871

4886

4900

14

3.1

4914

4928

4942

4955

4969

4983

4997

5011

5024

5038

14

3.2

5051

5065

5079

5092

5105

5119

5132

5145

5159

5172

13

3.3

5185

5198

5211

5224

5237

5250

5263

5276

5289

5302

13

3.4

5315

5328

5340

5353

5366

5378

5391

5403

5416

5428

13

3.5

5441

5453-

5465

5478

5490

5502

5514

5527

5539

5551

12

3.6

5563

5575

5587

5599

5611

5623

5635

5647

5658

5670

12

3.7

5682

5694

5705

5717

5729

5740

5752

5763

5775

5786

12

3.8

5798

5809

5821

5832

5843

5855

5866

5877

5888

5899

11

3.9

5911

5922

5933

5944

5955

5966

5977

5988

5999

6010

11

4.0

0.6021

6031

6042

6053

6064

6075

6085

6096

6107

6117

11

4.1

6128

6138

6149

6160

6170

6180

6191

6201

6212

6222

10

4.2

6232

6243

6253

6263

6274

6284

6294

6304

6314

6325

10

4.3

6335

6345

6355

6365

6375

6385

6395

6405

6415

6425

10

4.4

6435

6444

6454

6464

6474

6484

6493

6503

6513

6522

10

4.5

6532

6542

6551

6561

6571

6580

6590

6599

6609

6618

10

4.6

6628

6637

6646

6656

6665

6675

6684

6693

6702

6712

10

4.7

6721

6730

6739

6749

6758

6767

6776

6785

6794

6803

9

4.8

6812

6821

6830

6839

6848

6857

6866

6875

6884

6893

9

4.9

6902

6911

6920

6928

6937

6946

6955

6964

6972

6981

9

log TT = 0.4971 log e = 0.4343

log 7T/2 = 0.1961 log 71-2 log (0.4343) = 0.6378 - 1

0.9943

log

0.2486

These two pages give the common logarithms of numbers between 1 and 10, correct to four places. Moving the decimal point n places to the right [or left] in the number is equivalent to adding n[dr-n] to the logarithm. Thus, log 0.017453 = 0.2419 - 2, which may also be written 2.2419 or 8.2419 - 10. See p. 91. Graphs, p. 174. log (aft) = log a + log 6 log (aN) = N log a

log = log a - log 6

log

- log a

MATHEMATICAL TABLES

43

COMMON LOGARITHMS (continued)

1*

0

1

2

3

4

5

6

7

8

9

!i

5.0

0.6990

6998

7007

7016

7024

7033

7042

7050

7059

7067

9

5.1

7076

7084

7093

7101

7110

7118

7126

7135

7143

7152

8

5.2

7160

7168

7177

7185

7193

7202

7210

7218

7226

7235

8

5.3

7243

7251

7259

7267

7275

7284

7292

7300

7308

7316

8

5.4

7324

, 7332

7340

7348

7356

7364

7372

7380

7388

7396

8

5.5

7404

7412

7419

7427

7435

7443

7451

7459

7466

7474

8

5.6

7482

7490

7497

7505

7513

7520

7528

7536

7543

7551

8

5.7

7559

> 7566

7574

7582

7589

7597

7604

7612

7619

7627

8

5.8

7634

7642

7649

7657

7664

7672

7679

7686

7694

7701

7

5.9

7709

7716

7723

7731

7738

7745

7752

7760

7767

7774

7

6.0

0.7782

7789

7796

7803

7810

7818

7825

7832

7839

7846

7

6.1

7853

7860

7868

7875

7882

7889

7896

7903

7910

7917

7

6.2

7924

7931

7938

7945

7952

7959

7966

7973

7980

7987

7

6.3

7993

8000

8007

8014

8021

8028

8035

8041

8048

8055

7

6.4

8062

8069

8075

8082

8089

8096

. 8102

8109

8116

8122

7

6.5

8129

8136

8142

8149

8156

8162

8169

8176

8182

8189

7

6.6

8195

8202

8209

8215

8222

8228

8235

8241

8248

8254

7

6.7

8261

8267

8274

8280

8287

8293

8299

8306

8312

8319

6

6.8

8325

8331

8338

8344

8351

8357

8363

8370

8376

8382

6

6.9

8388

8395

8401

8407

8414

8420

8426

8432

8439

8445

6

7.0

0.8451

8457

8463

8470

8476

8482

8488

8494

8500

8506

6

7.1

1 8513

8519

8525

8531

8537

8543

8549

8555

8561

8567

6

7.2

8573

8579

8585

8591

8597

8603

8609

8615

8621

8627

6

7.3

8633

8639

8645

8651

8657

8663

8669

8675

8681

8686

6

7.4

8692

8698

8704

8710

8716

8722

8727

8733

8739

8745

6

7.5

8751

8756

8762

8768

8774

8779

8785

8791

8797

8802

6

7.6

8808

8814

8820

8825

8831

8837

8842

8848

8854

8859

6

7.7

8865

8871

8876

8882

8887

8893

8899

8904

8910

8915

6

7.8

8921

8927

8932

8938

8943

8949

8954

8960

8965

8971

6

7.9

8976

8982

8987

8993

8998

9004

9009

9015

9020

9025

5

8.0

0.9031

9036

9042

9047

9053

9058

9063

9069

9074

9079

5

8.1

9085

9090

9096

9101

9106

9112

9117

9122

9128

9133

5

8.2

9138

9143

9149

9154

9159

9165

9170

9175

9180

9186

5

8.3

9191

9196

9201

9206

9212

9217

9222

9227

9232

9238

5

8.4

9243

9248

9253

9258

9263 .

9269

9274

9279

9284

9289

5

8.5

9294

9299

9304

9309

9315

9320

9325

9330

9335

9340

5

8.6

9345

9350

9355

9360

9365

9370

9375

9380

9385

9390

5

8.7

9395

9400

9405

9410

9415

9420

9425

9430

9435

9440

5

8.8

9445

9450

9455

9460

9465

9469

9474

9479

9484

9489

5

8.9

9494

. 9499

9504

9509

9513

9518

9523

9528

9533

9538

5

9.0

0.9542

9547

9552

9557

9562

9566

9571

9576

9581

9586

5

9.1

9590

9595

9600

9605

9609

9614

9619

9624

9628

9633

5

9.2

9638

9643

9647

9652

9657

9661

9666

9671

9675

9680

5

9.3

9685

9689

9694

9699

9703

9708

9713

9717

9722

9727

5

9.4

9731

9736

9741

9745

9750 ,

9754

9759

9763

9768

9773

5

9.5

9777

9782

9786

9791

9795

9800

9805

9809

9814

9818

5

9.6

9823

9827

9832

9836

9841

9845

9850

9854

9859

9863

4

9.7

9868

9872

9877

9881

9886

9890

9894

9899

9903

9908

4

9.8

99 r2-

- 9917

9921

9926

9930

9934

9939

9943

9948

9952

4

9.9

9956

9961

9965

9969

9974

9978

9983

9987

9991

9996

4

44

MATHEMATICAL TABLES

DEGREES AND MINUTES EXPRESSED IN RADIANS (See also p. 69)

Degrees

Hundredths

Minutes

1°

.0175

61°

1.0647

121°

2.1118

0°.01

.0002

0°.51

.0089

r

.0003

2

.0349

2

1.0821

2

2.1293

2

.0003

2

.0091

2'

.0006

3

.0524

3

.0996

3

2.1468

3

.0005

3

.0093

3'

.0009

4

.0698

4

.1170

4

2.1642

4

.0007

4

.0094

4'

.0012

5°

.0873

65°

.1345

125°

2.1817

.05

.0009

.55

.0096

5'

.0015

6

.1047

6

.1519

6

2.1991

6

.0010

6

.0098

6'

.0017

7

.1222

7

.1694

7

2.2166

7

.0012

7

.0099

7'

.0020

8

.1396

8

.1868

8

2.2340

8

.0014

8

.0101

8'

.0023

9

.1571

9

1.2043

9

2.2515

9

.0016

9

.0103

9'

.0026

10°

.1745

70°

1.2217

130°

2.2689

0°.10

.0017

0°.60

.0105

10'

.0029

.1920

1

1.2392

1

2.2864

1

.0019

]

.0106

11'

.0032

2

.2094

2

1.2566

2

2.3038

2

.0021

2

.0108

12'

.0035

3

.2269

3

1.2741

3

2.3213

3

.0023

3

.0110

13'

.0038

4

.2443

4

1.2915

4

2.3387

4

.0024

4

.0112

14'

.0041

15°

.2618

75°

1.3090

136°

2.3562

.15

.0026

.65

.0113

15'

0044

6

.2793

6

1.3265

6

2.3736

6

.0028

6

.0115

16'

.0047

7

.2967

7

1.3439

7

2.3911

7

.0030

7

.0117

17'

.0049

8

.3142

8

1,3614

8

2.4086

8

.0031

8

.0119

18'

.0052

9

.3316

9

1.3788

9

2.4260

9

.0033

9

.0120

19'

.0055

20°

.3491

80°

1.3963

140°

2.4435

0°.20

.0035

0°.70

.0122

20'

.0058

1

.3665

1

.4137

1

2.4609

1

.0037

1

.0124

21'

.0061

2

.3840

2

.4312

2

2.4784

2

.0038

2

.0126

22'

.0064

3

.4014

3

.4486

3

2.4958

3

.0040

3

.0127

23'

.0067

4

.4189

4

.4661

4

2.5133

4

.0042

4

.0129

24'

.0070

25°

.4363

85°

1.4835

145°

2.5307

.25

.0044

.75

.0131

25'

0073

6

.4538

6

1.5010

6

2.5482

6

.0045

6

.0133

26'

.0076

7

.4712

7

1.5184

7

2.5656

7

.0047

7

.0134

27'

.0079

8

.4887

8

1.5359

8

2.5831

8

.0049

8

.0136

28'

.0081

9

.5061

9

1.5533

9

2.6005

9

.0051

9

.0138

29'

0084

30°

.5236

90°

1.5708

150°

2.6180

0°.30

.0052

0°.80

.0140

30'

.0087

1

.5411

1

.5882

1

2.6354

1

.0054

1

.0141

31'

.0090

2

.5585

2

1.6057

2

2.6529

2

.0056

2

0143

32'

.0093

3

.5760

3

1.6232

3

2.6704

3

.0058

3

.0145

33'

.0096

4

.5934

4

1.6406

4

2.6878

4

.0059

4

.0147

34'

.0099

35°

.6109

95°

1.6581

155°

2.7053

35

.0061

.85

.0148

35'

.0102

6

.6283

6

1.6755

6

2.7227

6

.0063

6

.0150

36'

.0105

7

.6458

7

1.6930

7

2.7402

7

.0065

7

.0152

37'

.0108

8

.6632

8

1.7104

8

2.7576

8

.0066

8

.0154

38'

.0111

9

.6807

9

1.7279

9

2.7751

9

.0068

9

.0155

39'

.0113

40°

.6981

100°

1.7453

160°

2.7925

0°.40

.0070

0°.90

.0157

40'

.0116

1

.7156

1

1.7628

1

2.8100

1

.0072

1

.0159

41'

.0119

2

.7330

2

1.7802

2

2.8274

2

.0073

2

.0161

42'

.0122

3

.7505

3

1.7977

3

2.8449

3

.0075

3

.0162

43'

.0125

4

.7679

4

1.8151

4

2.8623

4

.0077

4

.0164

44'

.0128

45°

.7854

105°

1.8326

165°

2.8798

.45

.0079

.95

.0166

45'

.0131

6

.8029

6

1.8500

6

2.8972

6

.0080

6

.0168

46'

.0134

7

.8203

7

1.8675

7

2.9147

7

.0082

7

.0169

47'

.0137

8

.8378

8

1.8850

8

2.9322

8

.0084

8

.0171

48'

.0140

9

.8552

9

1.9024

9

2.9496

9

.0086

9

.0173

49'

.0143

50°

.8727

110°

1.9199

170°

2.9671

0°.50

.0087

1°.00

.0175

50'

.0145

1

.8901

1

1.9373

1

2.9845

51'

0148

2

.9076

2

1.9548

2

3.0020

52'

.0151

3

.9250

3

1.9722

3

3.0194

53'

0154

4

.9425

4

1.9897

4

3.0369

54'

.0157

55°

.9599

115°

2.0071

175°

3.0543

55'

.0160

6

.9774

6

2.0246

6

3.0718

56'

0163

7

.9948

7

2.0420

7

3.0892

57'

.0166

8

1.0123

8

2.0595

8

3.1067

58'

0169

9

1.0297

9

2.0769

9

3.1241

59'

.0172

60°

1.0472

120°

2.0944

180°

3.1416

60'

.0175

Arc 1° = 0.0174533 Arc 1' = 0.000290888 Arc 1" = 0.00000484814 1 radian - 57°.295780 = 57° 17'.7468 - 57° 17' 44".806

MATHEMATICAL TABLES RADIANS EXPRESSED IN DEGREES

45

0.01

0°57

.64

36° 67

1.27

72° 77

1 90

108°.86

2.53

144°.%

Interpolation

2

1°.15

.65

37°.24

8

73°.34

1

109°.43

4

145°.53

.0002

0°.01

3

1°.72

6

37°.82

9

73°.91

2

110°.01

2.55

146MO

04

.02

4

2°.29

7

38°.39

1.30

74°.48

3

1KP.58

6

146°.68

06

.03

.05

2°.86

8

38°.96

1

75°.06

4

111°.15

7

147°.25

08

.05

6

3°.44

9

39°.53

2

75°.63

1.95

111°.73

8

147° .82

.0010

0°.06

7

4°.01

.70

40°.1 1

3

76°.20

6

112°.30

9

148°.40

12

.07

8

4°.58

1

40°.68

4

76°.78

7

112°.87

2.60

148°.97

14

.08

9

5°. 16

2

41°.25

1.35

77°.35

8

113°.45

149°.54

16

.09

.10

5°.73

3

41°.83

6

77°.92

9

1I4°.02

2

150°. 11

18

.10

1

6°.30

4

42°.40

7

78°.50

2.00

114°.59

3

I50°.69

.0020

0°.ll

2

6°.88

.75

42°.97

8

79°.07

1

115M6

4

151°.26

22

.13

3

7°.45

6

43°.54

9

79°.64

2

115°.74

2.65

15P.83

24

.14

4

8°.02

7

44°. 12

1.40

80°.21

3

116°.31

6

152°.41

26

.15

.15

8°.59

8

44°.69

1

80°.79

4

116°.88

7

152°.98

28

.16

6

9°. 17

9

45°.26

2

81°.36

2.05

1!7°.46

8

153°.55

.0030

0°.17

7

9°.74

.80

45°.84

3

81°.93

6

118°.03

9

154°. 13

32

.18

8

10°.31

1

46°.41

4

82°.51

7

118°.60

2.70

154°.70

34

.19

9

10°.89

2

46°.98

1.45

83°.08

8

119°.18

1

155°.27

36

.21

.20

1I°.46

3

47°.56

6

83°.65

9

119°.75

2

155°.84

38

.22

1

12°.03

4

48°. 13

7

84°.22

2.10

120°.32

3

156°.42

.0040

0°.23

2

12°.61

.85

48°.70

8

84°.80

1

120°.89

4

156°.99

42

.24

3

13°.18

6

49°.27

9

85°.37

2

121 °.47

2.75

157° .56

44

.25

4

13°.75

7

49°.85

1.50

85°.94

3

122°.04

6

158°.14

46

.26

.25

14°.32

8

50°.42

1

86°.52

4

122°.61

7

158°.7I

48

.28

6

14°.90

9

50°.99

2

87°.09

2.15

123M9

8

159°.28

.0050

0°.29

7

15°.47

.90

5J°.57

3

87°.66

6

123°.76

9

159°.86

52

.30

8

16°.04

1

52°.14

4

88°.24

7

124°.33

2.80

160°.43

54

.31

9

16°.62

2

52°.71

1.55

88°.81

8

124°.90

1

161°.00

56

.32

.30

17°.19

3

53°.29

6

89°.38

9

125°.48

2

16I°.57

58

.33

1

17°J6

4

53°.86

7

89°.95

2.20

126°.05

3

162°. 15

.0060

0°.34

2

18°.33

.95

54°.43

8

90°.53

11

126°.62

4

162 .72

62

.36

3

18°.91

6

55°.00

9

91°.10

127°.20

2.85

163°.29

64

.37

4

19°.48

7

55°.58

1.60

91°.67

3

127°.77

6

163°.87

66

.38

.35

20°.05

8

56°. 15

1

92°.25

4

128°.34

7

164°.44

68

.39

6

20°.63

9

56°.72

2

92°.82

2.25

128°.92

8

165°.01

.0070

0°.40

7

21°.20

1.00

57°.30

3

93039

6

129°.49

9

165°.58

72

.41

8

21°.77

1

57°.87

4

93°.97

7

130°.06

2.90

166M6

74

.42

9

22°.35

2

58°.44

1.65

94°.54

8

130°.6?

1

166°.73

76

.44

.40

22°.92

3

59°.01

6

95°. 11

9

131°.21

2

167°.30

78

.45

1

23°.49

4

59°.59

7

95°.68

2.30

131°.78

3

167°.88

.0080

0°.46

2

24°.06

1.05

60°. 16

8

96°.26

1

132°.35

4

168°.45

82

.47

3

24°.64

6

60°.73

9

96°.83

2

132°.93

2.95

169°.02

84

.48

4

25°.21

7

61°.31

1.70

97°.40

3

133°.50

6

169°.60

86

.49

.45

25°.78

8

61°.88

1

97°.98

4

134°.07

7

170°. 17

88

.50

6

26°.36

9

62°.45

2

98°.55

2.35

134°.65

8

170°.74

.0090

0°.52

7

26°.93

1.10

63°.03

3

99°. 12

6

135°.22

9

171°.31

92

.53

8

27°.50

1

63°.60

4

99°.69

7

135°.79

3.00

17P.89

94

.54

9

28°.07

2

64°. 17

1.75

100°.27

8

136°.36

172°.46

96

.55

.60

28°.65

3

64°.74

6

100°.84

9

136°.94

1

173°.03

98

.56

|

29°.22

4

65°.32

7

10I°.41

2.40

137°j51

3

173°.61

2

29079

1.15

65°.89

8

10P.99

1

138°.08

4

174°. 18

Multiples of v

3

30°.37

6

66°.46

9

102°.56

2

I38°.66

3.05

174°J5

4

30°.94

7

67°.04

1.80

103°.13

3

139°.23

6

175°.33

1 3.1416

180°

.55

31°.51

8

67°.61

1

103°.71

4

139°.80

7

175°.90

2 6.2832

360°

6

32°.09

9

68°. 18

2

104°.28

2.45

140°.37

8

176°.47

3 9.4248

540°

7

32°.66

1.20

68°.75

3

104°.85

6

140°.95

9

177°.04

4 12.5664

720°

8

33°.23

I

69°.33

4

105°.42

7

141°.52

3.10

177°.62

5 15.7080

900°

9

33°.80

2

69°.90

1.85

106°.00

8

142°.09

1

178°. 19

6 18.8496

1080°

.60

34°.38

3

70°.47

6

106°.57

9

142°.67

2

178°.76

7 21.9911

1260°

1

34°.95

4

71°.05

7

107°. 14

2.50

143°.24

3

179°.34

8 25.1327

1440°

2

35°.52

1.25

71°.62

8

107°.72

1

143°.81

V4

179°.91

9 28.2743

1620°

3

36°. 10

6

72°. 19

9

108°.29

2

144°.39

3.15

180°.48

10 31.4159

1800°

46

MATHEMATICAL TABLES

NATURAL SINES AND COSINES

Natural Sines at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56)

M

o

o

0

o

B

o

0

o

.

• P

=^6o

(60

(120 (180

(240

(300

(360

(420 (480

(540

1*

0.0000

90°

0°

0.0000

0017

0035 0052

0070

0087

0105

0122 0140

0157

0175

89

17

1

0175

0192

0209 0227

0244

0262

0279

0297 0314

0332

0349

88

17

2

0349

0366

0384 0401

0419

0436

0454

0471 0488

0506

0523

87

17

3

0523

0541

0558 0576

0593

0610

0628

0645 0663

0680

0698

86

17

4

0698

0715

0732 0750

0767

0785

0802

0819 0837

0854

0.0872

85

17

5

0.0872

0889

0906 0924

0941

0958

0976

0993 1011

1028

1045

84

17

6

1045

1063

1080 1097

1115

1132

1149

1167 1184

1201

1219

83

17

7

1219

1236

1253 1271

1288

1305

1323

1340 1357

1374

1392

82

17

8

1392

1409

1426 1444

1461

1478

1495

1513 1530

1547

1564

81

17

9

1564

1582

1599 1616

1633

1650

1668

1685 1702

'1719

0.1736

80°

17

10°

0.1736

1754

1771 1788

1805

1822

1840

1857 1874

1891

1908

79

17

11

1908

1925

1942 1959

1977

1994

2011

2028 2045

2062

2079

78

17

12

2079

2096

2113 2130

2147

2164

2181

2198 2215

2233

2250

77

17

13

2250

2267

2284 2300

2317

2334

2351

2368 2385

2402

2419

76

17

14

2419

2436

2453 2470

2487

2504

2521

2538 2554

2571

0.2588

75

17

15

0.2588

2605

2622 2639

2656

2672

2689

2706 2723

2740

2756

74

17

16

2756

2773

2790 2807

2823

2840

2857

2874 2890

2907

2924

73

17

17

2924

2940

2957 2974

2990

3007

3024

3040 3057

3074

3090

72

17

18

3090

3107

3123 3140

3156

3173

3190

3206 3223

3239

3256

71

17

19

3256

3272

3289 3305

3322

3338

3355

3371 3387

3404

0.3420

70°

16

20°

0.3420

3437

3453 3469

3486

3502

3518

3535 3551

3567

3584

69

16

21

3584

3600

3616 3633

3649

3665

3681

3697 3714

3730

3746

68

16

22

3746

3762

3778 3795

3811

3827

3843

3859 3875

3891

3907

67

16

23

3907

3923

3939 3955

3971

3987

4003

4019 4035

4051

4067

66

16

24

4067

4083

4099 4115

4131

4147

4163

4179 4195

4210

0.4226

65

16

25

0.4226

4242

4258 4274

4289

4305

4321

4337 4352

4368

4384

64

16

26

4384

4399

4415 4431

4446

4462

4478

4493 4509

4524

4540

63

16

27

4540

4555

4571 4586

4602

4617

4633

4648 4664

4679

4695

62

16

28

4695

4710

4726 4741

4756

4772

4787

4802 4818

4833

4848

61

15

29

4848

4863

4879 4894

4909

4924

4939

4955 4970

4985

0.5000

60°

15

30°

0.5000

5015

5030 5045

5060

5075

5090

5105 5120

5135

5150

59

15

31

5150

5165

5180 5195

5210

5225

5240

5255 5270

5284

5299

58

15

32

5299

5314

5329 5344

5358

5373

5388

5402 5417

5432

5446

57

15

33

5446

5461

5476 5490

5505

5519

5534

5548 5563

5577

5592

56

15

34

5592

5606

5621 5635

5650

5664

5678

5693 5707

5721

0.5736

55

14

35

0.5736

5750

5764 5779

5793

5807

5821

5835 5850

5864

5878

54

14

36

5878

5892

5906 5920

5934

5948

5962

5976 5990

6004

6018

53

14

37

6018

6032

6046 6060

6074

6088

6101

6115 6129

6143

6157

52

14

38

6157

6170

6184 6198

6211

6225

6239

6252 6266

6280

6293

51

14

39

6293

6307

6320 6334

6347

6361

6374

6388 6401

6414

0.6428

50°

13

40°

0.6428

6441

6455 6468

6481

6494

6508

6521 6534

6547

6561

49

13

41

6561

6574

6587 6600

6613

6626

6639

6652 6665

6678

6691

48

13

42

6691

6704

6717 6730

6743

6756

6769

6782 6794

6807

6820

47

13

43

6820

6833

6845 6858

6871

6884

6896

6909 6921

6934

6947

46

13

44

6947

6959

6972 6984

6997

7009

7022

7034 7046

7059

0.7071

45°

12

45°

0.7071

=(540

(480 (420

(360

(300

(240

(180 (120

(60

(00

i

(For graphs, see p. 174.)

Natural Cosines

MATHEMATICAL TABLES

47

NATURAL SINES AND COSINES (continued) Natural Sines at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56)

M

o

0 °.l

°2

03

°4

0 5

°6

07 03 09

*d

a

=(0/

) (60

(120

(180

(240

(300

(360

(420 (480 (540

<*

0.7071

45°

45°

0.7071 7083

7096

7108

7120

7133

7145

7157 7169 7181

7193

44

12

46

7193 7206

7218

7230

7242

7254

7266

7278 7290 7302

7314

43

12

47

731

4 7325

7337

7349

7361

7373

7385

7396 7408 7420

7431

42

12

48

743

1 7443

7455

7466

7478

7490

7501

7513 7524 7536

7547

41

12

49

7547 7559

7570

7581

7593

7604

7615

7627 7638 7649

0.7660

40°

11

50°

0.7660 7672

7683

7694

7705

7716

7727

7738 7749 7760

7771

39

11

51

7771 7782

7793

7804

7815

7826

7837

7848 7859 7869

7880

38

11

52

788

0 7891

7902

7912

7923

7934

7944

7955 7965 7976

7986

37

11

53

7986 7997

8007

8018

8028

8039

8049

8059 8070 8080

8090

36

10

54

8090 8100

8111

8121

8131

8141

8151

8161 8171 8181

0.8192

35

10

55

0.8192 8202

8211

8221

8231

8241

8251

8261 8271 8281

8290

34

10

56

829

0 8300

8310

8320

8329

8339

8348

8358 8368 8377

8387

33

10

57

8387 8396

8406

8415

8425

8434

8443

8453 8462 8471

8480

32

9

58

848

0 8490

8499

8508

8517

8526

8536

8545 8554 8563

8572

31

9

59

8572 8581

8590

8599

8607

8616

8625

8634 8643 8652

0.8660

30°

9

60°

0.8660 8669

8678

8686

8695

8704

8712

8721 8729 8738

8746

29

9

61

8746 8755

8763

8771

8780

8788

8796

8805 8813 8821

8829

28

8

62

882

9 8838

8846

8854

8862

8870

8878

8886 8894 8902

8910

27

8

63

8910 8918

8926

8934

8942

8949

8957

8965 8973 8980

8988

26

8

64

8988 8996

9003

9011

9018

9026

9033

9041 9048 9056

0.9063

25

7

65

0.9063 9070

9078

9085

9092

9100

9107

9114 9121 9128

9135

24

7

66

913

5 9143

9150

9157

9164

9171

9178

9184 9191 9198

9205

23

7

67

9205 9212

9219

9225

9232

9239

9245

9252 9259 9265

9272

22

7

68

927

2 9278

9285

9291

9298

9304

9311

9317 9323 9330

9336

21

6

69

9336 9342

9348

9354

9361

9367

9373

9379 9385 9391

0.9397

20°

6

70°

0.9397 9403

9409

9415

9421

9426

9432

9438 9444 9449

9455

19

6

71

945

5 9461

9466

9472

9478

9483

9489

9494 9500 9505

9511

18

6

72

951

1 9516

9521

9527

9532

9537

9542

9548 9553 9558

9563

17

5

73

. 9563 9568

9573

9578

9583

9588

9593

9598 9603 9608

9613

16

5

74

9613 9617

9622

9627

9632

9636

9641

9646 9650 9655

0.9659

15

5

75

0.9659 9664

9668

9673

9677

9681

9686

9690 9694 9699

9703

14

4

76

970

3 9707

9711

9715

9720

9724

9728

9732 9736 9740

9744

13

4

77

9744 9748

9751

9755

9759

9763

9767

9770 9774 9778

9781

12

4

78

978

1 9785

9789

9792

97%

9799

9803

9806 9810 9813

9816

11

3

79

9816 9820

9823

9826

9829

9833

9836

9839 9842 9845

0.9848

10°

3

80°

0.9848 9851

9854

9857

9860

9863

9866

9869 9871 9874

9877

9

3

81

9877 9880

9882

9885

9888

9890

9893

9895 9898 9900

9903

8

3

82

990

3 9905

9907

9910

9912

9914

9917

9919 9921 9923

9925

7

2

83

992

5 9928

9930

9932

9934

9936

9938

9940 9942 9943

9945

6

2

84

9945 9947

9949

9951

9952

9954

9956

9957 9959 9960

0.9962

5

2

85

0.9962 9963

9965

9966

9968

9969

9971

9972 9973 9974

9976

4

1

86

997

6 9977

9978

9979

9980

9981

9982

9983 9984 9985

9986

3

1

87

9986 9987

9988

9989

9990

9990

9991

9992 9993 9993

9994

2

1

88

999

4 9995

9995

9996

9996

9997

9997

9997 9998 9998

0.9998

1

0

89

0.9998 9999

9999

9999

9999

0000

0000

0000 0000 0000

1. 0000

0°

0

90°

1.0000

= (540

(480

(420

(360

(300

(240

(180 (120 (60

(00

$

Q

Natural Cosines

48

MATHEMATICAL TABLES

NATURAL TANGENTS AND COTANGENTS

Natural Tangents at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56)

ff

°.0

M

°2

°3 °4

05

0(j

07 03

°9

* •

Q

-coo

(60

(12')

(18') (24')

(30')

(360

(420 (48')

(54')

IS

0.0000

90°

0°

0.0000

0017

0035

0052 0070

0087

0105

0122 0140

0157

0175

89

17

1

0175

0192

0209

0227 0244

0262

0279

0297 0314

0332

0349

88

17

2

0349

0367

0384

0402 0419

0437

0454

0472 0489

0507

0524

87

17

3

0524

0542

0559

0577 0594

0612

0629

0647 0664

0682

0699

86

18

4

0699

0717

0734

0752 0769

0787

0805

0822 0840

0857

0.0875

85

18

5

0.0875

0892

0910

0928 0945

0963

0981

0998 1016

1033

1051

84

18

6

1051

1069

1086

1104 1122

1139

1157

1175 1192

1210

1228

83

18

7

1228

1246

1263

1281 1299

1317

1334

1352 1370

1388

1405

82

18

8

1405

1423

1441

1459 1477

1495

1512

1530 1548

1566

1584

81

18

9

1584

1602

1620

1638 1655

1673

1691

1709 1727

1745

0.1763

80°

18

10°

0.1763

1781

1799

1817 1835

1853

1871

1890 1908

1926

1944

79

18

It

1944

1962

1980

1998 2016

2035

2053

2071 2089

2107

2126

78

18

12

2126

2144

2162

2180 2199

2217

2235

2254 2272

2290

2309

77

18

13

2309

2327

2345

2364 2382

2401

2419

2438 2456

2475

2493

76

18

14

2493

2512

2530

2549 2568

2586

2605

2623 2642

2661

0.2679

75

19

15

0.2679

2698

2717

2736 2754

2773

2792

2811 2830

2849

2867

74

19

16

2867

2886

2905

2924 2943

2962

2981

3000 3019

3038

3057

73

19

17

3057

3076

3096

3115 3134

3153

3172

3191 3211

3230

3249

72

19

18

3249

3269

3288

3307 3327

3346

3365

3385 3404

3424

3443

71

19

19

3443

3463

3482

3502 3522

3541

3561

3581 3600

3620

0.3640

70°

20

20°

0.3640

3659

3679

3699 3719

3739

3759

3779 3799

3819

3839

69

20

21

3839

3859

3879

3899 3919

3939

3959

3979 4000

4020

4040

68

20

22

4040

4061

4081

4101 4122

4142

4163

4183 4204

4224

4245

67

21

23

4245

4265

4286

4307 4327

4348

4369

4390 4411

4431

4452

66

21

24

4452

4473

4494

4515 4536

4557

4578

4599 4621

4642

0.4663

65

21

25

0.4663

4684

4706

4727 4748

4770

4791

4813 4834

4856

4877

64

21

26

4877

4899

4921

4942 4964

4986

5008

5029 5051

5073

5095

63

22

27

5095

5117

5139

5161 5184

5206

5228

5250 5272

5295

5317

62

22

28

5317

5340

5362

5384 5407

5430

5452

5475 5498

5520

5543

61.

23

29

5543

5566

5589

5612 5635

5658

5681

5704 5727

5750

0.5774

60°

23

30°

0.5774

5797

5820

5844 5867

5890

5914

5938 5961

5985

6009

59

24

31

6009

6032

6056

6080 6104

6128

6152

6170 6200

6224

6249

58

24

32

6249

6273

6297

6322 6346

6371

6395

6420 6445

6469

6494

57

25

33

6494

6519

6544

6569 6594

6619

6644

6669 6694

6720

6745

56

25

34

6745

6771

6796

6822 6847

6873

6899

6924 6950

6976

0.7002

55

26

35

0.7002

7028

7054

7080 7107

7133

7159

7186 7212

7239

7265

54

26

36

7265

7292

7319

7346 7373

7400

7427

7454 7481

7508

7536

53

27

37

7536

7563

7590

7618 7646

7673

7701

7729 7757

7785

7813

52

28

38

7813

7841

7869

7898 7926

7954

7983

8012 8040

8069

8098

51

28

39

8098

8127

8156

8185 8214

8243

8273

8302 8332

8361

0.8391

50°

29

40°

0.8391

8421

8451

8481 8511

8541

8571

8601 8632

8662

8693

49

30

41

8693

8724

8754

8785 8816

8847

8878

8910 8941

8972

9004

48

31

42

9004

9036

9067

9099 9131

9163

9195

9228 9260

9293

9325

47

32

43

9325

9358

9391

9424 9457

9490

9523

9556 9590

9623

0.9657

46

33

44

0.9657

9691

9725

9759 9793

9827

9861

9896 9930

9965

1.0000

45°

34

45°

1.0000

*(549')

(480

(420 (36')

(300

(240

(18') (120

(60

(00

1

(For graphs, see p. 174.)

Natural Cotangents

MATHEMATICAL TABLES

49

NATURAL TANGENTS AND COTANGENTS (continued)

Natural Tangents at intervals of 0°. 1, or 6'. (For 10' intervals, see pp. 52-56)

i

a=Y(V

) (60

(120

(180

(240

(300

(360 (420

(480

(540

diff!

1.0000

45°

45°

1.0000 0035

0070

0105

0141

0176

0212 0247

0283

0319 0355

44

35

46

035

5 0392

0428

0464

0501

0538

0575 0612

0649

0686 0724

43

37

47

072

4 0761

0799

0837

0875

0913

0951 0990

1028

1067 1106

42

38

48

1106 1145

1184

1224

1263

1303

1343 1383

1423

1463 1504

41

40

49

1504 1544

1585

1626

1667

1708

1750 1792

1833

18751.1918

40°

41

50°

1.1918 1960

2002

2045

2088

2131

2174 2218

2261

2305 2349

39

43

51

234

9 2393

2437

2482

2527

2572

2617 2662

2708

2753 2799

38

45

52

2799 2846

2892

2938

2985

3032

3079 3127

3175

3222 3270

37

47

53

327

0 3319

3367

3416

3465

3514

3564 3613

3663

3713 3764

36

49

54

3764 3814

3865

3916

3968

4019

4071 4124

4176

42291.4281

35

52

55

1 .4281 4335

4388

4442

44%

4550

4605 4659

4715

4770 4826

34

55

56

4826 4882

4938

4994

5051

5108

5166 5224

5282

5340 5399

33

57

57

539

9 5458

5517

5577

5637

5697

5757 5818

5880

5941 6003

32

60

58

6003 6066

6128

6191

6255

6319

6383 6447

6512

6577 6643

31

64

59

1.6643 6709

6775

6842

6909

6977

7045 7113

7182

7251 1.7321

30°

67

60°

1.732 1.739

1.746

1.753

1.760

1.76}

1.775 1.782

1.789

1.797 1.804

29

7

61

1.80

4 1.811

1.819

1.827

1.834

1.842

1.849 1.857

1.865

1.873 1.881

28

8

62

1.88

1 1.889

1.897

1.905

1.913

1.921

1.929 1.937

1.946

1.954 1.963

27

8

63

1.963 1.971

1.980

1.988

1.997

2.006

2.014 2023

2.032

2.041 2.050

26

9

64

2.050 2.059

2.069

2.078

2.087

2.097

2.1062.116

2.125

2.135 2.145

25

9

65

2.145 2.154

2.164

2.174

2.184

2.194

2.204 2.215

2.225

2.236 2.246

24

10

66

2.24

6 2.257

2.267

2.278

2.289

2.300

2.311 2.322

2.333

2.344 2.356

23

11

67

2.356 2.367

2.379

2.391

2.402

2.414

2.426 2.438

2.450

2.463 2.475

22

12

68

2.47

5 2.488

2.500

2.513

2.526

2.539

2.552 2.565

2.578

2.592 2.605

21

13

69

2.605 2.619

2.633

2.646

2.660

2.675

2.689 2.703

2.718

2.733 2.747

20°

14

70°

2.747 2.762

2.778

2.793

2.808

2.824

2.840 2.856

2.872

2.888 2.904

19

16

71

2.904 2.921

2.937

2.954

2.971

2.989

3.006 3.024

3.042

3.060 3.078

18

17

72

3.07

8 3.096

3.115

3.133

3.152

3.172

3.191 3.211

3.230

3.251 3.271

17

19

73

3.271 3.291

3.312

3.333

3.354

3.376

3.398 3.420

3.442

3.465 3.487

16

22

74

3.487 3.511

3.534

3.558

3.582

3.606

3.630 3.655

3.681

3.706 3.732*

15

24

75

3.732 3.758

3.785

3.812

3.839

3.867

3.895 3.923

3.952

3.981 4.011

14

28

76

4.01

1 4.041

4.071

4.102

4.134

4.165

4.198 4.230

4.264

4.297 4.331

13

32

77

4.331 4.366

4.402

4.437

4.474

4.511

4.548 4.586

4.625

4.665 4.705

12

37

78

4.70

5 4.745

4.787

4.829

4.872

4.915

4.959 5.005

5.050

5.097 5.145

11

44

79

5.145 5.193

5.242

5.292

5.343

5.396

5.449 5.503

5.558

5.614 5:671

10°

53

80°

5,671 5.730

5.789

5.850

5.912

5.976

6.041 6.107

6.174

6.243 6.314

9

81

6.314 6.386

6.460

6.535

6.612

6.691

6.772 6 855

6.940

7.026 7.115

8

82

7.11

5 7.207

7.300

7.396

7.495

7.596

7.700 7.806

7.916

8.028 8.144

7

83

8.14

4 8.264

8.386

8.513

8.643

8.777

8.915 9.058

9.205

9.357 9.514

6

84

9.514 9.677

9.845

10.02

10.20

10.39

10.58 10.78

10.99

11.20 11.43

5

85

11.43 11.66

11.91

12.16

12.43

12.71

13.00 1330

13.62

13.95 14.30

4

86

14.3

[) 14.67

15.06

15.46

15.89

16.35

16.83 17.34

17.89

18.46 19.08

3

87

19.a

8 19.74

20.45

21.20

22.02

22.90

23.86 24^90

26.03

27.27 28.64

2

88

28.64 30.14

31.82

33.69

35.80

38.19

40.92 44.07

47.74

52.08 57.29

1

89

57.29 63.66

71.62

81.85

95.49

114.6

143.2 191.0

286.5

573.0 oo

0°

90°

00

=(540

(480

(420

(360

(300

(240 (180

(120

(60 (0*0

i

Natural Cotangents

50

MATHEMATICAL TABLES

NATURAL SECANTS AND COSECANTS

Natural Secants at intervals of 0°. 1, or 6'. (For 10' intervals, see pp. 52-56)

i

-(V

(60*

(120

(180 (240

(3V)

(360

(420

(480 (540

Avg. diff.

1.0000

90°

0°

1.0001

3 0000

0000

0000 0000

0000

0001

0001

0001 0001

0002

89

0

1

000

I 0002

0002

0003 0003

0003

0004

0004

0005 0006

0006

88

0

2

000

i 0007

0007

0008 0009

0010

0010

0011

0012 0013

0014

87

3

001'

\ 0015

0016

0017 0018

0019

0020

0021

0022 0023

0024

86

1

4

002'

I 0026

0027

0028 0030

0031

0032

0034

0035 0037

1.0038

85

1

5

1.003

J 0040

0041

0043 0045

0046

0048

0050

0051 0053

0055

84

2

6

005.

> 0057

0059

0061 0063

0065

0067

0069

0071 0073

0075

83

2

7

007.

> 0077

0079

0082 0084

0086

0089

0091

0093 0096

0098

82

2

8

009*

) 0101

0103

0106 0108

0111

0114

0116

0119 0122

0125

81

3

9

01 2f

> 0127

0130

0133 0136

0139

0142

0145

0148 0151

1.0154

80°

3

10°

\ 0157

0161

0164 0167

0170

0174

0177

0180 0184

0187

79

3

11

018}

' 0191

0194

0198 0201

0205

0209

0212

0216 0220

0223

78

4

12

022:

I 0227

0231

0235 0239

0243

0247

0251

0255 0259

0263

77

4

13

026:

I 0267

0271

0276 0280

0284

0288

0293

0297 0302

0306

76

4

14

030*

> 0311

0315

0320 0324

0329

0334

0338

0343 0348

1.0353

75

5

15

1 .035:

I 0358

0363

0367 0372

0377

*0382

0388

0393 0398

0403

74

5

16

040:

5 0408

0413

0419 0424

0429

0435

0440

0446 0451

0457

73

5

17

0453

' 0463

0468

0474 0480

0485

0491

0497

0503 0509

0515

72

6

18

051f

0521

0527

0533 0539

0545

0551

0557

0564 0570

0576

71

6

19

057*

0583

0589

0595 0602

0608

0615

0622

0628 0635

1.0642

70°

7

20°

1.064;

' 0649

0655

0662 0669

0676

0683

0690

0697 0704

0711

69

7

21

071

0719

0726

0733 0740

0748

0755

0763

0770 0778

0785

68

7

22

078f

0793

0801

0808 0816

0824

0832

0840

0848 0856

0864

67

8

23

0864

0872

0880

0888 0896

0904

0913

0921

0929 0938

0946

66

8

24

094*

0955

0963

0972 0981

0989

0998

1007

1016 1025

1.1034

65

9

25

1.103'

f 1043

1052

1061 1070

1079

1089

1098

1107 1117

1126

64

9

26

112*

> 1136

1145

1155 1164

1174

1184

1194

1203 1213

1223

63

10

27

1222

1233

1243

1253 1264

1274

1284

1294

1305 1315

1326

62

10

28

132*

1336

1347

1357 1368

1379

1390

1401

1412 1423

1434

61

11

29

1434

1445

1456

1467 1478

1490

1501

1512

1524 1535

1.1547

60°

11

30°

1.1543

' 1559

1570

1582 1594

1606

1618

1630

1642 1654

1666

59

12

31

166*

1679

1691

1703 1716

1728

1741

1753

1766 1779

1792

58

13

32

179;

1805

1818

1831 1844

. 1857

1870

1883

1897 1910

1924

57

13

33

1924

1937

1951

1964 1978

1992

2006

2020

2034 2048

2062

56

14

34

206;

2076

2091

2105 2120

2134

2149

2163

2178 2193

1.2208

55

15

35

1.220*

2223

2238

2253 2268

2283

2299

2314

2329 2345

2361

54

15

36

2361

2376

2392

2408 2424

2440

245*

2472

2489 2505

2521

53

16

37

2521

2538

2554

2571 2588

2605

2622

2639

2656 2673

2690

52

17

38

269C

2708

2725

2742 2760

2778

2796

2813

2831 2849

2868

51

18

39

2866

2886

2904

2923 2941

2960

2978

2997

3016 3035

1.3054

60°

19

40°

1.3054

3073

3093

3112 3131

3151

3171

3190

3210 3230

3250

49

20

41

325C

3270

3291

3311 3331

3352

3373

3393

34 M 3435

3456

48

21

42

345*

3478

3499

3520 3542

3563

3585

3607

3629 3651

3673

47

22

43

3673

3696

3718

3741 3763

3786

3809

3832

3855 3878

3902

46

23

44

3902

3925

3949

3972 3996

4020

4044

4069

4093 4118

1.4142

45°

24

45°

1.4142

t

,

= (540

(480

(420 (360

(300

(240

(180

(120 (60

(00

Q

(For graphs, see p. 174.)

Natural Cosecants

MATHEMATICAL TABLES

51

NATURAL SECANTS AND COSECANTS (continued)

Natural Secants at intervals of 0°.l, or 6'. (For 10' intervals, see pp. 52-56)

«

°.o

°.l

°.a

°.3

°.4

°.5

°.6 °.7

°.8 °.9

Avg.

Q

=(00

(60

(120

(180

(240

(300

(360 (420

(480 (540

diff.

1.4142

45°

45°

1.4142

4167

4192

4217

4242

4267

4293 4318

4344 4370

4396

44

25

46

4396

4422

4448

4474

4501

4527

4554 4581

4608 4635

4663

43

27

47

4663

4690

4718

4746

4774

4802

4830 4859

4887 4916

4945

42

28

48

4945

4974

5003

5032

5062

5092

5121 5151

5182 5212

5243

41

30

49

5243

5273

5304

5335

5366

5398

5429 5461

5493 5525

1.5557

40°

31

60°

1.5557

5590

5622

5655

5688

5721

5755 5788

5822 5856

5890

39

33

51

5890

5925

5959

5994

6029

6064

6099 6135

6171 6207

6243

38

35

52

6243

6279

6316

6353

6390

6427

6464 6502

6540 6578

6616

37

37

53

6616

6655

6694

6733

6772

6812

6852 6892

6932 6972

7013

36

40

54

7013

7054

7095

7137

7179

7221

7263 7305

7348 7391

1.7434

35

42

55

1.7434

7478

7522

7566

7610

7655

7700 7745

7791 7837

7883

34

45

56

7883

7929

7976

8023

8070

8118

8166 8214

8263 8312

8361

33

48

57

8361

8410

8460

8510

8561

8612

8663 8714

8766 8818

8871

32

51

58

8871

8924

8977

9031

9084

9139

9194 9249

9304 9360

1.9416

31

54

59

1.9416

9473

9530

9587

9645

9703

9762 9821

9880 9940

2.0000

30°

58

60°

2.00C

2.006

2.012

2.018

2.025

2.031

2.037 2.043

2.050 2.056

2.063

29

6

61

2.063

2.069

2.076

2.082

2.089

2.096

2.103 2.109

2.116 2.123

2.130

28

7

62

2.13C

2.137

2.144

2.151

2.158

2.166

2.173 2.180

2.188 2.195

2.203

27

7

63

2.203

2.210

2.218

2.226

2.233

2.241

2.249 2.257

2.265 2.273

2281

26

8

64

2.281

2.289

2.298

2.306

2.314

2.323

2.331 2.340

2349 2357

2366

25

8

65

2.36<

2375

2.384

2.393

2.402

2.411

2.421 2.430

2.439 2.449

2.459

24

9

66

2.45<

2.468

2.478

2.488

2.498

2.508

2.518 2.528

2.538 2.549

2.559

23

10

67

2.55<

2.570

2.581

2.591

2.602

2.613

2624 2.635

2.647 2.658

2.669

22

11

68

2.66'

2.681

2.693

2.705

2.716

2.729

2.741 2.753

2.765 2.778

2.790

21

12

69

2.79(

2.803

2.816

2.829

2.842

2.855

2.869 2.882

2.896 2.910

2.924

20°

13

70°

2.92^

\ 2.938

2.952

2.967

2.981

2.996

3.011 3.026

3.041 3.056

3.072

19

15

71

3.07;

3.087

3.103

3.119

3.135

3.152

3.168 3.185

3.202 3.219

3.236

18

16

72

3.23(

» 3.254

3.271

3.289

3.307

3326

3.344 3.363

3.382 3.401

3.420

17

18

73

3.42C

3.440

3.460

3.480

3.500

3.521

3.542 3.563

3.584 3.606

3.628

16

21

74

3.62*

I 3.650

3.673

3.695

3.719

3.742

3.766 3.790

3.814 3.839

3.864

15

24

75

3.86^

I 3.889

3.915

3.941

3.967

3.994

4.021 4.049

4.077 4.105

4.134

14

27

76

4.13^

4.163

4.192

4.222

4.253

4.284

4.315 4.347

4379 4.412

4.445

13

31

77

4.44!

4.479

4.514

4.549

4.584

4.620

4.657 4.694

4.732 4.771

4.810

12

36

78

4.8H

1 4.850

4.890

4.931

4.973

5.016

5.059 5.103

5.148 5.194

5.241

11

43

79

5.24

5.288

5337

5.386

5.436

5.487

5.540 5.593

5.647 5.702

5.759

10°

52

80°

5.75C

> 51816

5.875

5.935

5.996

6.059

6.123 6.188

6.255 6.323

6392

9

81

6.39;

! 6.464

6.537

6.611

6.687

6.765

6.845 6.927

7.011 7.097

7.185

8

82

7.18f

• 7.276

7.368

7.463

7.561

7.661

7.764 7.870

7.979 8.091

8.206

7

83

8.2W

» 8.324

8.446

8.571

8.700

8.834

8.971 9.113

9.259 9.411

9.567

6

84

9.563

9.728

9.895

10.07

1025

10.43

10.63 10.83

11.03 11.25

11.47

5

85

11.42

' 11.71

11.95

12.20

12.47

12.75

13.03 13.34

13.65 13.99

1434

4

86

14.3'

\ 14.70

15.09

15.50

15.93'

16.38

16.86 17.37

17.91 18.49

19.11

3

87

19.1

19.77

20.47

21.23

22,04

22.93

23.88 24.92

26.05 27.29

28.65

2

88

28.6!

» 30.16

31.84

33.71

35.81

38.20

40.93 44.08

47.75 52.09

57.30

89

573(

) 63.66

71.62

81.85

95.49

114.6

143.2 191.0

286.5 573.0

00

0°

90°

oo

°.9

°.8

°.7

°.6

°.5

M

=(540

(480

(420

(360

(300

(240 (180

(120 (60

(oo

a

Natural Cosecants

52

MATHEMATICAL TABLES

TRIGONOMETRIC FUNCTIONS (at intervals of 10')

Annex -10 in columns marked *. (For O.°l intervals, see pp. 46-51)

De- grees

Ra- dians

Sines

Cosines

Tangents

Cotangents

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

0° (XX

0.0000

.0000 «

t.OOOO 0.0000

.0000

CO 00

1.5708

90° 00*

10

00029

.0029 7.4637

1.0000 .0000

.0029 7.4637

343.77 2.5363

1.5679

50

20

00058

.0058 .7648

1.0000 .0000

.0058 .7648

171.89 .2352

1.5650

40

30

00087

.0087 .9408

1.0000 .0000

.0087 .9409

114.59 .0591

1.5621

30

40

0.0116

.0116 8.0658

0.9999 .0000

.0116 8.0658

85.940 1.9342

1.5592

20

50

0.0145

.0145 .1627

.9999 .0000

.0145 .1627

68.750 .8373

1.5563

10

1° 00'

0.0175

.0175 8.2419

.9998 9.9999

.0175 8.2419

57.290 1.7581

1.5533

89° 00'

10

0.0204

.0204 .3088

.9998 .9999

.0204 .3089

49.104 .6911

1.5504

50

20

0.0233

.0233 .3668

.9997 .9999

.0233 .3669

42.964 .6331

1.5475

40

30

0.0262

.0262 .4179

.9997 .9999

.0262 .4181

38.188 .5819

1.5446

30

40

0.0291

.0291 .4637

.9996 .9998

.0291 .4638

34.368 .5362

1.5417

20

50

0.0320

.0320 .5050

.9995 .9998

.0320 .5053

31.242 .4947

1.5388

10

2° (XX

0.0349

.0349 8.5428

.9994 9.9997

.0349 8.5431

28.636 1.4569

1.5359

88° 00'

10

0.0378

.0378 .5776

.9993 .9997

.0378 .5779

26.432 .4221

1.5330

50

20

0.0407

.0407 .6097

.9992 .9996

.0407 .6101

24.542 .3899

1.5301

40

30

0.0436

.0436 .6397

.9990 .9996

.0437 .6401

22.904 .3599

1.5272

30

40

0.0465

.0465 .6677

.9989 .9995

.0466 .6682

21.470 .3318

1 .5243

20

50

0.0495

.0494 .6940

.9988 .9995

.0495 .6945

20.206 .3055

1.5213

10

3° (XX

0.0524

.0523 8.7188

.9986 9.9994

.0524 8.7194

19.081 1.2806

1.5184

87° 00'

10

0.0553

.0552 .7423

.9985 .9993

.0553 .7429

18.075 .2571

1.5155

50

20

0.0582

.0581 .7645

.9983 .9993

.0582 .7652

17.169 .2348

1.5126

40

30

0.061 1

.0610 .7857

.9981 .9992

.0612 .7865

16.350 .2135

1.5097

30

40

0.0640

.0640 .8059

.9980 .9991

.0641 .8067

15.605 .1933

1.5068

20

50

0.0669

.0669 .8251

.9978 .9990

.0670 .8261

14.924 .1739

1.5039

10

4° (XX

0.0698

.0698 8.8436

.9976 9.9989

.0699 8.8446

14.301 1.1554

1.5010

86° 00

10

0.0727

.0727 .8613

.9974 .9989

.0729 .8624

13.727 .1376

1.4981

50

20

0.0756

.0756 .8783

.9971 .9988

.0758 .8795

13.197 .1205

1.4952

40

30

0.0785

.0785 .8946

.9969 .9987

.0787 .8960

12.706 .1040

1.4923

30

40

0.0814

.0814 .9104

.9967 .9986

.0816 .9118

12.251 .0882

1.4893

20

50

0.0844

.0843 .9256

,9964 .9985

.0846 .9272

11.826 .0728

1.4864

10

5° (XX

0.0873

.0872 8.9403

.9962 9.9983

.0875 8.9420

11.430 1.0580

1.4835

85° 00'

10

0.0902

.0901 .9545

.9959 .9982

.0904 .9563

11.059 .0437

1.4806

50

20

0.0931

.0929 .9682

.9957 .9981

.0934 .9701

10.712 .0299

1.4777

40

30

0.0960

.0958 .9816

.9954 .9980

.0963 .9836

10.385 .0164

1.4748

30

40

0.0989

.0987 .9945

.9951 .9979

.0992 .9966

10.078 .0034

1.4719

' 20

50

0.1018

.1016 9.0070

.9948 .9977

.1022 9.0093

9.7882 0.9907

1.4690

10

6° 00'

0.1047

.1045 9.0192

.9945 9.9976

.1051 9.0216

9.5144 0.9784

1.4661

84° 00'

10

0.1076

.1074 .0311

.9942 .9975

.1080 .0336

9.2553 .9664

1.4632

50

20

0.1105

.1103 .0426

.9939 .9973

.1110 .0453

9.0098 .9547

1.4603

40

30

0.1134

.1132 .0539

.9936 .9972

.1139 .0567

8.7769 .9433

1.4574

30

40

0.1164

.1161 .0648

.9932 .9971

.1169 .0678

8.5555 .9322

1.4544

20

50

0.1193

.1190 .0755

.9929 .9969

.1198 .0786

8.3450 .9214

1.4515

10

7° 00'

0.1222

.1219 9.0859

.9925 9.9968

.1228 9.0891

8.1443 0.9109

1.4486

83° 00'

10

0.1251

.1248 .0961

.9922 .9966

.1257 .0995

7.9530 .9005

1.4457

50

20

0.1280

.1276 .1060

.9918 .9964

.1287 .1096

7.7704 .8904

1.4428

40

30

0.1309

.1305 .1157

.9914 .9963

.1317 .1194

7.5958 .8806

1.4399

30

40

a 1338

.1334 .1252

.991 1 .9961

.1346 .1291

7.4287 .8709

1.4370

20

50

0.1367

.1363 .1345

.9907 .9959

.1376 .1385

7.2687 .8615

1.4341

10

8° 00'

0.1396

.1392 9.1436

.9903 9.9958

.1405 9.1478

7.1154 0.8522

1.4312

82° 00'

10

0.1425

.1421 .1525

.9899 .9956

.1435 .1569

6.9682 .8431

1.4283

50

20

0.1454

.1449 .1612

.9894 .9954

.1465 .1658

6.8269 .8342

1.4254

40

30

0.1484

.1478 .1697

.9890 .9952

.1495 .1745

6.6912 .8255

1.4224

30

40

0.1513

.1507 .1781

.9886 .9950

.1524 .1831

6.5606 .8169

1.4195

20

50

0.1542

.1536 .1863

.9881 .9948

.1554 .1915

6.4348 .8085

1.4166

10

9° (XX

0.1571

.1564 9.1943

.9877 9.9946

.1584 9.1997

6.3138 0.8003

1.4137

81° 00'

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

Cosines

Sines

Cotangents

Tangents

Ra- dians

De- grees

MATHEMATICAL TABLES

53

TRIGONOMETRIC FUNCTIONS (continued)

Annex -10 in columns marked*. (For O.°l intervals, see pp. 46-51)

De- grees

Ra- dians

Sines

Cosines

Tangents

Cotangent

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

9° 00'

0.1571

.1564 9.1943

.9877 9.9946

.1584 9.1997

6.3138 0.8003

1.4137

81" (XX

kio

0.1600

.1593 .2022

.9872 .9944

.1614 .2078

6.1970 .7922

1.4108

50

20

0.1629

.1622 .2100

.9868 .9942

.1644 .2158

6.0844 .7842

1.4079

40

30

0.1658

.1650 .2176

.9863 .9940

.1673 .2236

5.9758 .7764

1.4050

30

40

0.1687

.1679 .2251

.9858 .9938

.1703 .2313

5.8708 .7687

1.4021

20

50

0.1716

.1708 .2324

.9853 .9936

.1733 2389

5.7694 .7611

1.3992

10

10° 00'

0.1745

.1736 9.2397

.9848 9.9934

.1763 9.2463

5.6713 0.7537

1.3963

80° (XX

10

0.1774

.1765 .2468

.9843 .9931

.1793 2536

5.5764 .7464

1.3934

50

20

0.1804

.1794 .2538

.9838 .9929

.1823 2609

5.4845 .7391

1.3904

40

30

0.1833

.1822 .2606

.9833 .9927

.1853 2680

5.3955 .7320

1.3875

30

40

0.1862

.1851 .2674

.9827 .9924

.1883 2750

5.3093 .7250

1.3846

20

50

0.1891

.1880 .2740

.9822 .9922

.1914 2819

5.2257 .7181

1.3817

10

11° 00'

0.1920

.1908 9.2806

.9816 9.9919

.1944 92887

5.1446 0.7113

1.3788

79°0(X

10

0.1949

.1937 .2870

.9811 .9917

.1974 2953

5.0658 .7047

1.3759

50

20

0.1978

.1965 .2934

.9805 .9914

.2004 .3020

4.9894 .6980

1.3730

40

30

0.2007

.1994 .2997

.9799 .9912

2035 .3085

4.9152 .6915

1.3701

30

40

0.2036

.2022 .3058

.9793 .9909

2065 .3149

4.8430 .6851

1.3672

20

50

0.2065

•2051 .3119

.9787 .9907

2095 .3212

4.7729 .6788

1.3643

10

12° 00'

0.2094

.2079 9.3179

.9781 9.9904

2126 9.3275

4.7046 0.6725

1.3614

78° 00'

10

0.2123

.2108 .3238

.9775 .9901

2156 3336

4.6382 .6664

1.3584

50

20

0.2153

.2136 .3296

.9769 .9899

2186 .3397

4.5736 .6603

1.3555

40

30

0.2182

.2164 .3353

.9763 .9896

2217 .3458

4.5107 .6542

1.3526

30

40

0.2211

.2193 .3410

.9757 .9893

2247 .3517

4.4494 .6483

1.3497

20

50

0.2240

.2221 .3466

.9750 .9890

2278 .3576

4.3897 .6424

1.3468

10

13° 00'

0.2269

.2250 9.3521

.9744 9.9887

2309 9.3634

4.3315 0.6366

1.3439

77° (XX

10

0.2298

.2278 .3575

.9737 .9884

2339 .3691

4.2747 .6309

1.3410

50

20

0.2327

.2306 .3629

.9730 .9881

2370 .3748

4.2193 .6252

1.3381

40

30

0.2356

.2334 .3682

.9724 .9878

2401 .3804

4.1653 .6196

1.3352

30

40

0.2385

.2363 .3734

.9717 .9875

2432 .3859

4.1126 .6141

1.3323

20

50

0.2414

.2391 .3786

.9710 .9872

2462 .3914

4.0611 .6086

1.3294

10

14° 00'

0.2443

.2419 9.3837

.9703 9.9869

2493 9.3968

4.0108 0.6032

1.3265

76° (XX

10

0.2473

.2447 .3887

.9696 .9866

2524 .4021

3.9617 .5979

1.3235

50

20

0.2502

.2476 .3937

.9689 .9863

2555 .4074

3.9136 .5926

1.3206

40

30

0.2531

2504 .3986

.9681 .9859

2586 .4127

3.8667 .5873

1.3177

30

40

0.2560

2532 .4035

.9674 .9856

2617 .4178

3.8208 .5822

1.3148

20

50

0.2589

2560 .4083

.9667 .9853

2648 .4230

3.7760 .5770

13119

10

15° 00'

0.2618

.2588 9.4130

.9659 9.9849

2679 9.4281

3.7321 0.5719

1.3090

75° 00'

10

0.2647

2616 .4177

.9652 .9846

2711 .4331

3.6891 .5669

1.3061

50

20

0.2676

2644 .4223

.9644 .9843

.2742 .4381

3.6470 .5619

1.3032

40

30

0.2705

2672 .4269

.9636 -.9839

.2773 .4430

3.6059 .5570

1.3003

30

40

0.2734

2700 .4314

.9628 .9836

2805 .4479

3.5656 .5521

1.2974

20

50

0.2763

2728 .4359

.9621 .9832

2836 -4527

3.5261 .5473

U945

10

16° 00'

0.2793

2756 9.4403

.9613 9.9828

.2867 9.4575

3.4874 0.5425

12915

74° (XX

10

0.2822

2784 .4447

.9605 .9825

2899 .4622

3.4495 .5378

12886

50

20

0.2851

2812 .4491

^9596 .9821

2931 .4669

3.4124 .5331

1.2857

40

30

0.2880

2840 .4533

.9588 .9817

2962 .4716

3.3759 .5284

1.2828

30

40

0.2909

2868 .4576

.9580 .9814

2994 .4762

3.3402 .5238

12799

20

50

0.2938

2896 .4618

.9572 .9810

.3026 .4808

3.3052 .5192

12770

10

17° (XX

0.2967

2924 9.4659

.9563 9.9806

.3057 9.4853

3.2709 0.5147

1.2741

3° (XX

10

0.2996

2952 .4700

.9555 .9802

3089 .4898

3.2371 .5102

12712

50

20

0.3025

2979 .4741

.9546 .9798

3121 .4943

3.2041 .5057

1.2683

40

30

0.3054

3007 .4781

9537 .9794

3153 .4987

3.1716 .5013

1.2654

30

40

0.3083

3035 .4821

9528 .9790

3185 .5031

3.1397 .4969

1.2625

20

50

0.3113

3062 .4861

9520 .9786

3217 .5075

3.1084 .4925

12595

10

18° (XX

0.3142

3090 9.4900

9511 9.9782

3249 9.5118

3.0777 0.4882

12566

2° (XX

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

Cosines

Sines

Cotangents

Tangents

Ra- dians

De- grees

54

MATHEMATICAL TABLES

TRIGONOMETRIC FUNCTIONS

Annex— 10 in columns marked*.

(continued)

(For O.°l intervals, Bee pp. 46-51)

De- grees

Ra- dians

Sines

Cosines

Tangents

Cotangents

Nat. Log. *

Nat. Log.*

Nat. Log.*

Nat. Log.

18° 00'

0.3142

.3090 9.4900

.9511 9.9782

.3249 9.5118

3.0777 0.4882

1.2566

72° 00*

10

0.3171

.3118 .4939

.9502 .9778

.3281 .5161

3.0475 .4839

1.2537

50

20

0.3200

.3145 .4977

.9492 .9774

.3314 .5203

3.0178 .4797

1.^508

40

30

0.3229

.3173 .5015

.9483 .9770

,3346 .5245

2.9887 .4755

1.2479

30

40

0.3258

.3201 .5052

.9474 .9765

.3378 .5287

2.9600 .4713

1 .2450

20

50

0.3287

.3228 .5090

.9465 .9761

.3411 .5329

2.9319 .4671

1.2421

10

19° 00'

0.3316

.3256 9.5126

.9455 9.9757

.3443 9.5370

2.9042 0.4630

1.2392

71° 00'

10

0.3345

.3283 .5163

.9446 .9752

.3476 3411

2.8770 .4589

1.2363

50

20

0.3374

.3311 .5199

.9436 .9748

.3508 .5451

2.8502 .4549

1 .2334

40

30

0.3403

.3338 .5235

.9426 .9743

.3541 .5491

2.8239 .4509

1.2305

30

40

0.3432

.3365 .5270

.9417 .9739

.3574 .5531

2.7980 .4469

.2275

20

50

0.3462

.3393 .5306

.9407 .9734

.3607 .5571

2.7725 .4429

12246

10

20° 00'

0.3491

.3420 9.5341

.9397 9.9730

.3640 9.5611

2.7475 0.4389

1.2217

70° 00'

10

0.3520

.3448 .5375

.9387 .9725

.3673 .5650

2.7228 .4350

1.2188

50

20

0.3549

3475 .5409

.9377 .9721

.3706 .5689

2.6985 .4311

1.2159

40

30

0.3578

.3502 .5443

.9367 .9716

.3739 .5727

2.6746 .4273

1.2130

30

40

0.3607

.3529 .5477

.9356 .971 1

.3772 .5766

2.6511 .4234

1.2101

20

50

0.3636

.3557 .5510

.9346 .9706

.3805 .5804

2.6279 .4196

1.2072

10

21° 00'

0.3665

.3584 9.5543

.9336 9.9702

.3839 9.5842

2.6051 0.4158

1.2043

69° 0(X

10

0.3694

.361 1 .5576

.9325 .9697

.3872 .5879

2.5826 .4121

1.2014

50

20

0.3723

.3638 .5609

.9315 .9692

.3906 .5917

2.5605 .4083

.1985

40

30

0.3752

.3665 .5641

.9304 .9687

.3939 .5954

2.5386 .4046

.1956

30

40

0.3782

.3692 .5673

.9293 .9682

.3973 .5991

2.5172 .4009

.1926

20

50

0.381 1

.3719 .5704

.9283 .9677

.4006 .6028

2.4960 .3972

.1897

10

22° 00'

0.3840

.3746 9.5736

.9272 9.9672

.4040 9.6064

2.4751 0.3936

.1868

68° 00'

10

0.3869

.3773 .5767

.9261 .9667

.4074 .6100

2.4545 .3900

.1839

50

20

0.3898

.3800 .5798

.9250 .9661

.4108 .6136

2.4342 .3864

.1810

40

30

0.3927

.3827 .5828

.9239 .9656

.4142 .6172

2.4142 .3828

.1781

30

40

0.3956

.3854 .5859

.9228 .9651

.4176 .6208

2.3945 .3792

.1752

20

50

0.3985

.3881 .5889

.9216 .9646

.4210 .6243

2.3750 .3757

.1723

10

23° 00'

0.4014

.3907 9.5919

.9205 9.9640

.4245 9.6279

2.3559 0.3721

.1694

67° 00'

10

0.4043

.3934 .5948

.9194 .9635

.4279 .6314

2.3369 .3686

.1665

50

20

0.4072

.3961 .5978

.9182 .9629

.4314 .6348

2.3183 .3652

.1636

40

30

0.4102

.3987 .6007

.9171 .9624

.4348 .6383

2.2998 .3617

.1606

30

30

0.4131

.4014 .6036

.9159 .9618

.4383 .6417

2.2817 .3583

.1577

20

50

0.4160

.4041 .6065

.9147 .9613

.4417 .6452

2.2637 .3548

.1548

10

24° 00'

0.4189

.4067 9.6093

.9135 9.9607

.4452 9.6486

2.2460 0.3514

.1519

66° 00'

10

0.4218

.4094 .6121

.9124 .9602

.4487 .6520

2.2286 .3480

.1490

50

20

0.4247

.4120 .6149

.9112 .9596

.4522 .6553

2.2113 .3447

.1461

40

30

0.4276

.4147 .6177

.9100 .9590

.4557 .6587

2.1943 .3413

.1432

30

40

0.4305

.4173 .6205

.9088 .9584

.4592 .6620

2.1775 .3380

.1403

20

50

0.4334

.4100 .6232

.9075 .9579

.4628 .6654

2.1609 .3346

.1374

10

25° 0(K

0.4363

.4226 9.6259

.9063 9.9573

.4663 9.6687

2.1445 0.3313

.1345

65° 00'

10

0.4392

.4253 .6286

.9051 .9567

.4699 .6720

2.1283 .3280

.1316

50

20

0.4422

.4279 .6313

.9038 .9561

.4734 .6752

2.1123 .3248

.1286

40

30

0.4451

.4305 .6340

.9026 .9555

.4770 .6785

2.0965 .3215

.1257

30

40

0.4480

.4331 .6366

.9013 .9549

.4806 .6817

2.0809 .3183

.1228

20

50

0.4509

.4358 .6392

.9001 .9543

.4841 .6850

2.0655 .3150

.1199

10

26° 00'

0.4538

.4384 9.6418

.8988 9.9537

.4877 9.6882

2.0503 0-3118

.1170

64° 00'

10

0.4567

.4410 .6444

.8975 .9530

.4913 .6914

2.0353 .3086

.1141

50

20

0.4596

.4436 .6470

.8962 .9524

.4950 .6946

2.0204 .3054

.1112

40

30

0.4625

.4462 .6495

.8949 .9518

.4986 .6977

2.0057 .3023

.1083

30

40

0.4654

.4488 .6521

.8936 .9512

.5022 .7009

1.9912 .2991

.1054

20

50

0.4683

.4514 .6546

.8923 .9505

.5059 .7040

1.9768 .2960

.1025

10

27° (XX

0.4712

.4540 9.6570

.8910 9.9499

.5095 9.7072

1.9626 0.2928

1.0996

63° 00'

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

Cosines

Sines

Cotangents

Tangents

Ra- dians

De- grees

MATHEMATICAL TABLES

55

TRIGONOMETRIC FUNCTIONS (continued)

Annex -10 in columns marked*. (For 0°.l intervals, see pp. 46-51)

De- grees

Ra- dians

Sines

Cosines

Tangents

Cotangents

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

27° 00'

0.4712

.4540 9.6570

.8910 9.9499

.5095 9.7072

1.9626 0.2928

1.0996

63° 00'

10

0.4741

.4566 .6595

.8897 .9492

.5132 .7103

1.9486 2897

1.0966

50

20

0.4771

.4592 .6620

.8884 .9486

.5169 .7134

1.9347 .2866

1.0937

40

30

0.4800

.4617 .6644

.8870 .9479

.5206 .7165

1.9210 .2835

1.0908

30

40

0.4829

.4643 .6668

.8857 .9473

.5243 .7196

1.9074 .2804

1.0879

20

50

0.4858

.4669 .6692

.8843 .9466

.5280 .7226

1.8940 .2774

1.0850

10

28° 00'

0.4887

.4695 9.6716

.8829 9.9459

.5317 9.7257

1.8807 0.2743

1.0821

62° 00'

10

0.4916

.4720 .6740

.8816 .9453

.5354 .7287

1.8676 .2713

1.0792

50

20

0.4945

.4746 .6763

.8802 .9446

.5392 .7317

1.8546 .2683

1.0763

40

30

0.4974

.4772 .6787

.8788 .9439

.5430 .7348

1.8418 .2652

1.0734

30

40

0.5003

.4797 .6810

.8774 .9432

.5467 .7378

1.8291 .2622

1.0705

20

50

0.5032

.4823 .6833

.8760 .9425

.5505 .7408

1.8165 .2592

1.0676

10

29° 00'

0.5061

.4848 9.6856

.8746 9.9418

.5543 9.7438

1.8040 0.2562

1.0647

61° 00'

10

0.5091

.4874 .6878

.8732 .941 1

.5581 .7467

1.7917 .2533

1.0617

50

20

0.5120

.4899 .690!

.8718 .9404

.5619 .7497

1.7796 .2503

1.0588

40

30

0.5149

.4924 .6923

.8704 .9397

.5658 .7526

1.7675 .2474

1.0559

30

40

0.5178

.4950 .6946

.8689 .9390

.5696 .7556

1.7556 .2444

1.0530

20

50

0.5207

.4975 .6968

.8675 .9383

.5735 .7585

1.7437 .2415

1.0501

10

30° 00'

0.5236

.5000 9.6990

.8660 9.9375

.5774 9.7614

1.7321 0.2386

1.0472

60° 00'

10

0.5265

.5025 .7012

.8646 .9368

.5812 .7644

1.7205 .2356

1.0443

50

20

0.5294

.5050 .7033

.8631 .9361

.5851 .7673

1.7090 .2327

1.0414

40

30

0.5323

.5075 .7055

.8616 .9353

.5890 .7701

1.6977 .2299

1.0385

30

40

0.5352

.5100 .7076

.8601 .9346

.5930 .7730

1.6864 .2270

1.0356

20

50

0.5381

.5125 .7097

.8587 .9338

.5969 .7759

1.6753 .2241

1.0327

10

31° 00'

0.5411

.5150 9.7118

.8572 9.9331

.6009 9.7788

1.6643 0.2212

1.0297

59° 00'

10

0.5440

.5175 .7139

.8557 .9323

.6048 .7816

1.6534 .2184

1.0268

50

20

0.5469

.5200 .7160

.8542 .9315

.6088 .7845

1.6426 .2155

1.0239

40

30

0.5498

.5225 .7181

.8526 .9308

.6128 .7873

1.6319 .2127

1.0210

30

40

0.5527

.5250 .7201

.8511 .9300

.6168 .7902

1.6212 .2098

1.0181

20

50

0.5556

.5275 .7222

.8496 .9292

.6208 .7930

1.6107 .2070

1.0152

10

32° 00'

0.5585

.5299 9.7242

.8480 9.9284

.6249 9.7958

.6003 0.2042

1.0123

58° 00'

10

0.5614

.5324 .7262

.8465 .9276

.6289 .7986

.5900 .2014

1.0094

50

20

0.5643

.5348 .7282

.8450 .9268

.6330 .8014

.5798 .1986

1 .0065

40

30

0.5672

.5373 .7302

.8434 .9260

.6371 .8042

.5697 .1958

1.0036

30

40

0.5701

.5398 .7322

.8418 .9252

.6412 .8070

.5597 .1930

1.0007

20

50

0.5730

5422 .7342

.8403 .9244

.6453 .8097

.5497 .1903

0.9977

10

33° (XX

0.5760

5446 9.7361

.8387 9.9236

.6494 9.8125

.5399 0.1875

0.9948

57° 00'

10

0.5789

5471 .7380

.8371 .9228

.6536 .8153

.5301 .1847

0.9919

50

20

0.5818

5495 .7400

.8355 .9219

.6577 .8180

.5204 .1820

0.9890

40

30

0.5847

5519 .7419

.8339 .9211

.6619 .8208

.5108 .1792

0.9861

30

40

0.5876

5544 .7438

.8323 .9203

.6661 .8235

.5013 .1765

0.9832

20

50

0.5905

5568 .7457

.8307 .9194

.6703 .8263

.4919 .1737

0.9803

10

34° 00'

0.5934

5592 9.7476

.8290 9.9186

.6745 9.8290

.4826 0.1710

09774

56° 00'

10

0.5963

5616 .7494

.8274 .9177

.6787 .8317

.4733 .1683

0.9745

50

20

0.5992

5640 .7513

.8258 .9169

.6830 .8344

.4641 .1656

0.9716

40

30

0.6021

5664 .7531

.8241 .9160

.6873 .8371

.4550 .1629

0.9687

30

40

0.6050

5688 .7550

.8225 .9151

.6916 .8398

.4460 .1602

0.9657

20

50

0.6080

5712 .7568

.8208 .9142

.6959 .8425

.4370 .1575

0.9628

10

35° 00'

0.6109

5736 9.7586

.8192 9.9134

.7002 9.8452

.4281 0.1548

0.9599

55° 00'

10

0.6138

5760 .7604

.8175 .9125

.7046 .8479

.4193 .1521

0.9570

50

20

0.6167

5783 .7622

.8158 .9116

.7089 .8506

.4106 .1494

0.9541

40

30

0.6196

5807 .7640

.8141 .9107

.7133 .8533

.4019 .1467

0.9512

30

40

0.6225

5831 .7657

.8124 .9098

.7177 .8559

.3934 .1441

0.9483

20

50

0.6254

5854 .7675

.8107 .9089

.7221 .8586

.3848 .1414

0.9454

10

36° 00'

0.6283

5878 9.7692

.8090 9.9080

.7265 9.8613

1.3764 0.1387

0.9425

54° 00'

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

Cosines

Sines

Cotangents

Tangents

Ra- dians

De- grees

56

MATHEMATICAL TABLES

TRIGONOMETRIC FUNCTIONS (continued)

Annex -10 in columns marked*. (For 0°.l intervals, see pp. 4.6-51)

De- grees

Ra- dians

Sines

Cosines

Tangents

Cotangents

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

36° (XX

0.6283

3878 9.7692

.8090 9.9080

.7265 9.8613

1.3764 0.1387

0.9425

54° 00'

10

0.6312

.5901 .7710

.8073 .9070

.7310 .8639

1.3680 .1361

0.9396

50

20

0.6341

.5925 .7727

.8056 .9061

.7355 .8666

1.3597 .1334

0.9367

40 '

30

0.6370

.5948 .7744

.8039 .9052

.7400 .8692

1.3514 .1308

0.9338

30

40

0.6400

.5972 .7761

.8021 .9042

.7445 .8718

1.3432 .1282

0.9308

20

50

0.6429

.5995 .7778

.8004 .9033

.7490 .8745

1.3351 .1255

0.9279

10

37° 00'

0.6458

.6018 9.7795

.7986 9.9023

.7536 9.8771

.3270 0.1229

0.9250

53° 00'

10

0.6487

.6041 .781 1

.7969 .9014

.7581 .8797

.3190 .1203

0.9221

50

20

0.6516

.6065 .7828

.7951 .9004

.7627 .8824

.3111 .1176

0.9192

40

30

0.6545

.6088 .7844

.7934 .8995

.7673 .8850

.3032 .1150

0.9163

30

40

0.6574

.61 1 1 .7861

.7916 .8985

.7720 .8876

.2954 .1124

0.9134

20

50

0.6603

.6134 .7877

.7898 .8975

.7766 .8902

.2876 .1098

0.9105

• 10

38° 00'

0.6632

.6157 9.7893

.7880 9.8965

.7813 9.8928

.2799 0.1072

0.9076

52° 00'

10

0.6661

.6180 .7910

.7862 .8955

.7860 .8954

.2723 .1046

0.9047

50

20

0.6690

.6202 .7926

.7844 .8945

.7907 .8980

.2647 .1020

0.9018

40

30

0.6720

.6225 .7941

.7826 .8935

.7954 .9006

.2572 .0994

0.8988

30

40

0.6749

.6248 .7957

.7808 .8925

.8002 .9032

.2497 .0968

0.8959

20

50

0.6778

.6271 .7973

.7790 .8915

.8050 .9058

.2423 .0942

0.8930

10

39° 00'

0.6807

.6293 9.7989

.7771 9.8905

.8098 9.9084

.2349 0.0916

0.8901

51° 00'

10

0.6836

.6316 .8004

.7753 .8895

.8146 .9110

.2276 .0890

0.8872

50

20

0.6865

.6338 .8020

.7735 .8884

.8195 .9135

.2203 .0865

0.8843

40

30

0.6894

.6361 .8035

.7716 .8874

.8243 .9161

.2131 .0839

0.8814

30

40

0.6923

.6383 .8050

.7698 .8864

.8292 .9187

.2059 .0813

0.8785

20

50

0.6952

.6406 .8066

.7679 .8853

.8342 .9212

.1988 .0788

0.8756

10

40° 00'

0.6981

.6428 9.8081

.7660 9.8843

.8391 9.9238

.1918 0.0762

0.8727

50° 00'

10

0.7010

.6450 .8096

.7642 .8832

.8441 .9264

.1847 .0736

0.8698

50

20

0.7039

.6472 .81 1 1

.7623 .8821

.8491 .9289

.1778 .0711

0.8668

40

30

0.7069

.6494 .8125

.7604 .8810

.8541 .9315

.1708 .0685

0.8639

30

40

0.7098

.6517 .8140

.7585 .8800

.8591 .9341

.1640 .0659

0.8610

20

50

0.7127

.6539 .8155

.7566 .8789

.8642 .9366

.1571 .0634

0.8581

10

41° 00'

0.7156

.6561 9.8169

.7547 9.8778

.8693 9.9392

.1504 0.0608

0.8552

49° 00'

10

0.7185

.6583 .8184

.7528 .8767

.8744 .9417

J436 .0583

0.8523

50

20

0.7214

.6604 .8198

.7509 .8756

.8796 .9443

.1369 .0557

0.8494

40

30

0.7243

.6626 .8213

.7490 .8745

.8847 .9468

.1303 .0532

0.8465

30

40

0.7272

.6648 .8227

.7470 .8733

.8899 .9494

.1237 .0506

0.8436

20

50

0.7301

.6670 .8241

.7451 .8722

.8952 .9519

.1171 .0481

0.8407

10

42° 00'

0.7330

.6691 9.8255

.7431 9.8711

.9004 9.9544

.1106 00456

0.8378

48° 00'

10

0.7359

.6713 .8269

.7412 .8699

.9057 .9570

.1041 .0430

0.8348

50

20

0.7389

.6734 .8283

.7392 .8688

.9110 .9595

1 .0977 .0405

0.8319

40

30

0.7418

.6756 .8297

.7373 .8676

.9163 .9621

1.0913 .0379

0.8290

30

40

0.7447

.6777 .8311

.7353 .8665

.9217 .9646

1.0850 .0354

0.8261

20

50

0.7476

.6799 .8324

.7333 .8653

.9271 .9671

1.0786 .0329

0.8232

10

43° 00'

0.7505

.6820 9.8338

.7314 9.8641

.9325 9.9697

1.0724 0.0303

0.8203

47° 00'

10

0.7534

.6841 .8351

.7294 .8629

.9380 .9722

1.0661 .0278

0.8174

50

20

0.7563

.6862 .8365

.7274 .8618

.9435 .9747

1.0599 .0253

0.8145

40

30

0.7592

.6884 .8378

.7254 .8606

.9490 .9772

1.0538 .0228

0.8116

30

40

0.7621

.6905 .8391

.7234 .8594

.9545 .9798

1.0477 .0202

0.8087

20

50

0.7650

.6926 .8405

.7214 .8582

.9601 .9823

1.0416 .0177

0.8058

10

44° 00'

0.7679

.6947 9.8418

.7193 9.8569

.9657 9.9848

1.0355 0.0152

0.8029

46° 00'

10

0.7709

.6967 .8431

.7173 .8557

.9713 .9874

1.0295 .0126

0.7999

50

20

0.7738

.6988 .8444

.7153 .8545

.9770 .9899

1.0235 .0101

0.7970

40

30

0.7767

.7009 .8457

.7133 .8532

.9827 .9924

1.0176 .0076

0.7941

30

40

07796

.7030 .8469

.7112 .8520

.9884 .9949

1.0117 .0051

0.7912

20

50

0.7825

.7050 .8482

.7092 .8507

.9942 .9975

1.0058 .0025

0.7883

10

45° 00'

0.7854

.7071 9.8495

.7071 9.8495

1.0000 0.0000

1.0000 0.0000

0.7854

45° 00'

Nat. Log.*

Nat. Log.*

Nat. Log.*

Nat. Log.

Cosines

Sines

Cotangents

Tangents

Ra- dians

De- grees

MATHEMATICAL TABLES EXPONENTIALS [e» and <T"]

57

n

n *H

c s

n

, g

n

en

n

<- %

n

c~"

n

e~*

0.00

1.000 in

0.50

1.649 ,,

1.0

2.718*

0.00

1.000 ]0

0.50

.607

1.0

.368*

.01

i.oto 5

.51

1.665 S

.1

3.004

.01

0.990- J

31

.600

.1

.333

.02

J.020 2

.52

1.682 i

.2

3.320

.02

.980 ~ 2

.52

.595

2

.301

.03

1.030 9

.53

1.699 \l

3

3.669

.03

.970 -'g

.53

.589

3

.273

.04

1.041 JJ

.54

1.716 \77

.4

4.055

.04

•961 I,J

.54

.583

.4

.247

0.05

1.051 ..

0.55

1.733 1R

1.5

4.482

0.05

.951 „

0.55

.577

13

.223

.06

.062

.56

1.751 2

.6

4.953

.06

.942 ~,J

.56

.571

. .6

202

.07

1.073 A

.57

1.768 \i

.7

5.474

.07

.932 ~'§

.57

.566

.7

.183

.08

1.083 ?

.58

1.786 2

.8

6.050

.08

.923 ~ I

.58

.560

.8

.165

.09

1.094 jj

.59

1.804 !«

.9

6.686

.09

.914 I 99

39

.554

.9

.150

0.10

.105 ,.

0.60

1.822 ,.

2.0

7.389

0.10

.905 o

0.60

.549

2.0

.135

.11

.116

.61

1.840

.1

8.166

.11

.896 ~ A

.61

.543

.1

.122

.12

.127 I

.62

1.859 A

2

9.025

.12

.887 ~ I

.62

.538

2

.111

.13

.139 2

.63

1.878 \l

3

9.974

.13

.878 ~ J

.63

.533

3

.100

.14

.150 jj

.64

1.896 JJ

.4

11.02

.14

.869 ~ I

.64

.527

.4

.0907

0.15

.162 ,,

0.65

1.916 ]0

2.5

12.18

0.15

.861 Q

0.65

.522

23

.0821

.16

.174 2,

.66

1.935 \l

.6

13.46

.16

.852 ~ I

.66

.517

.6

.0743

.17

.185 '

.67

1.954 £

.7

14.88

.17

.844 ~ I

.67

.512

.7

.0672

.18

.197 2

.68

1.974 %

.8

16.44

.18

.835 ~ I

.68

.507

.8

.0608

.19

1.209 j|

.69

1.994 20

.9

18.17

.19

.827 I I

.69

.502

.9

.0550

0.20

1.221 „

0.70

2.014 7n

3.0

20.09

0.20

.819 »

0.70

.497

3.0

.0498

.21

1.234 \l

.71

2.034 ?,S

.1

22.20

.21

.811 ~

.71

.492

.1

.0450

.22

1.246 2

.72

2.054 20

.2

24.53

.22

.803 -

.72

.487

.2

.0408

.23

1.259 2

.73

2.075 2,

.3

27.11

.23

.795 ~ 5

.73

.482

3

.0369

24

1.271 !|

.74

2.096 21

.4

29.%

.24

-787 I I

.74

.477

.4

.0334

0.25

1.284 n

0.75

2.117 „

3.5

33.12

0.25

.779 o

0.75

.472

33

.0302

.26

707 13

.^y/ | o

.76

2.138 ?!

.6

36.60

.26

.771 ~ I

.76

.468

.6

.0273

.27

1.310 ^

.77

2.160 ?f

.7

40.45

.27

.763 ~ 5

.77

.463

.7

.0247

.28

1.323 |

.78

2.181 tl

.8

44.70

.28

.756 ~ I

.78

.458

.8

.0224

.29

1.336 JJ

.79

2.203 g

.9

49.40

.29

.748 ~ J

.79

.454

.9

.0202

O.SO

1.350 n

0.80

2.226 22

4.0

54.60

0.30

.741 „

0.80

.449

4.0

.0183

.31

1.363 \\

.81

2.248 £

.1

60.34

.31

.733 ~ 2

.81

.445

.1

.0166

.32

1.377

.82

2.270 S2,

.2

66.69

.32

.726 ~ I

.82

.440

.2

.0150

.33

1.391 J

.83

2.293 g

3

73.70

.33

.719 ~ I

.83

.436

.3

.0136

.34

1.405 ]<

.84

2.3.6 g

.4

81.45

.34

.7.2 ~ ]

.84

.432

.4

.0123

0.35

1.419 14

0.85

2.340 21

4.5

90.02

0.35

.705 7

0.85

.427

4.5

.0111

.36

1.433 i

.86

2.363 5?

.36

.698 ~ i

.86

.423

.37

1.448

.87

2.387 Si

5.0

148.4

.37

.691 ~ I

.87

.419

5.0

.00674

.38

1.462 J

.88

2.411 2J

6.0

403.4

.38

.684 ~ i

.88

.415

6.0

.00248

.39 0.40

1.477 jf

1.492 ,e

.89 0.90

2.435 24 2.460 94

7.0 8.0

1097. 2981.

.39 0.40

.677 I ij .670

.89 0.90

.411 .407

8.0

i. 000912 .000335

.41

1.507 1

.91

2.484 24

9.0

8103.

.41

.664" S

.91

.403

9.0

.000123

.42

1.522 \\

.92

2.509 25

10.0

22026.

.42

.657 " I

.92

.399

10.0

.000045

.43 .44

1.537 5 1.553 Jf

.93 .94

2.535 fS

•y <Mn **

2.560 26

«/2 fcr/2

4.810 23.14

.43 .44

.651 " S •644 I ?6

.93 .94

.395 .391

IT/2 27T/2

.208 .0432

0.45

1.568 ,,

0.95

2.586 7,

37T/2

111.3

0.45

.638 7

0.95

.387

3r/2

.00898

.46

1.584 9

.96

2.612 g

47T/2

535.5

.46

.631 " ?

.96

.383

fcr/2

.00187

.47

1.600

.97

2.638 ?9

5T/2

2576.

.47

.625 "

.97

.379

5r/2

.000388

.48

1.616 JJ

.98

2.664 $

for/2

12392.

.48

.619 ~ 9

.98

.375

6r/2

.000081

.49

1.632 2

.99

2.691 g

lir/2

59610.

.49

.613 ~ I

.99

.372

7^/2

.000017

0.50

I/

1.649

1.00

// 2.718

8^/2

286751.

0.50

0.607

1.00

368

W2

.000003

*NOTE: Do not interpolate in this column.

e = 2.71828 1/e = 0.367879 logioe = 0.4343 1/(0.4343) - 2.3026

logio(0.4343) = 1.6378 logio(e») = n(0.4343)

For table of multiples of 0.4343, see p. 62. Graphs, p. 174.

58

MATHEMATICAL TABLES

HYPERBOLIC LOGARITHMS

n

n (2.3026)

n (0.6974-3)

These two pages give the natural (hyper- bolic, or Napierian) logarithms (log«) of numbers between 1 and 10, correct to four

1 2 3 4

2.3026 4.6052 6.9078 9.2103

0.6974-3 0.3948-5 0.0922-7 0.7897-10

places. Moving the decimal point n places

5

11.5129

0.4871-12

to the right [or left] in the number is equiva-

6

13.8155

0.1845-14

lent to adding n times 2.3026 [or n times 3.6974] to the logarithm. Base e = 2.71828 +

;

8 9

16.1181 18.4207 20.7233

08819-17 0.5793-19 0.2767-21

EU-

&

0

1

2

3

4

5

6

7

8

9

£

1.0

0.0000

0100

0198

0296

0392

0488

0583

0677

0770

0862

95

1.1

0953

1044

1133

1222

1310

1398

1484

1570

1655

1740

87

1.2

1823

1906

1989

2070

2151

2231

2311

2390

2469

2546

80

1.3

2624

2700

2776

2852

2927

3001

3075

3148

3221

3293

74

1.4

3365

3436

3507

3577

3646

3716

3784

3853

3920

3988

69

1.5

0.4055

4121

4187

4253

4318

4383

4447

4511

4574

4637

65

1.6

4700

4762

4824

4886

4947

5008

5068

5128

5188

5247

61

1.7

5306

5365

5423

5481

5539

5596

5653

5710

5766

5822

57

1.8

5878

5933

5988

6043

6098

6152

6206

6259

6313

6366

54

1.9

6419

6471

6523

6575

6627

6678

6729

6780

6831

6881

51

2.0

0.6931

6981

7031

7080

7129

7178

7227

7275

7324

7372

49

2.1

7419

7467

7514

7561

7608

7655

7701

7747

7793

7839

47

2.2

7885

7930

7975

8020

8065

8109

8154

8198

8242

8286

44

2.3

8329

8372

8416

8459

8502

8544

8587

8629

8671

8713

43

2.4

8755

8796

8838

8879

8920

8961

9002

9042

9083

9123

41

2.5

0.9163

9203

9243

9282

9322

9361

9400

9439

9478

9517

39

2.6

9555

9594

9632

9670

9708

9746

9783

9821

9858

9895

38

2.7

0.9933

9969

*0006

*0043

*0080

*0116

*0152

*OI88

*0225

*0260

36

2.8

1.0296

0332

0367

0403

0438

0473

0508

0543

0578

0613

35

2.9

0647

0682

0716

0750

0784

0818

0852

0886

0919

0953

34

3.0

1.0986

1019

1053

1086

1119

1151

1184

1217

1249

1282

33

3.1

1314

1346

1378

1410

1442

1474

1506

1537

1569

1600

32

3.2

1632

1663

1694

1725

1756

1787

1817

1848

1878

1909

31

3.3

1939

1969

2000

2030

2060

2090

2119

2149

2179

2208

30

3.4

2238

2267

2296

2326

2355

2384

2413

2442

2470

2499

29

3.5

1.2528

2556

2585

2613

2641

2669

2698

2726

2754

2782

28

3.6

2809

2837

2865

2892

2920

2947

2975

3002

3029

3056

27

3.7

3083

3110

3137

3164

3191

3218

3244

3271

3297

3324

27

3.8

3350

3376

3403

3429

3455

3481

3507

3533

3558

3584

26

3.9

3610

3635

3661

3686

3712

3737

3762

3788

3813

3838

25

4.0

1.3863

3888

3913

3938

3962

3987

4012

4036

4061

4085

25

4.1

4110

4134

4159

4183

4207

4231

4255

4279

4303

4327

24

4.2

4351

4375

4398

4422

4446

4469

4493

4516

4540

4563

23

4.3

4586

4609

4633

4656

4679

4702

4725

4748

4770

4793

23

4.4

4816

4839

4861

4884

4907

4929

4951

4974

4996

5019

22

4.5

1.5041

5063

5085

5107

5129

5151

5173

5195

5217

5239

22

4.6

5261

5282

5304

5326

5347

5369

5390

5412

5433

5454

21

4.7

5476

5497

5518

5539

5560

5581

5602

5623

5644

5665

21

4.8

5686

5707

5728

5748

5769

5790

5810

5831

5851

5872

20

4.9

5892

5913

5933

5953

5974 v

5994

6014

6034

6054

6074

20

logtX = (2.3026) logio x logioa: = (0.4343) Iog8 x

where 2.3026 =• log«w and 0.4343 => logioe (see p. 62). For graph*, see p. 174.

MATHEMATICAL TABLES

59

HYPERBOLIC LOGARITHMS (continued}

e^

3 <o 53-°

0

1

2

3

4

5

6

7

8

9

!i

5.0

1.6094

6114

6134

6154

6174

6194

6214

6233

6253

6273

20

5.1

6292

6312

6332

6351

6371

6390

6409

6429

6448

6467

19

5.2

6487

6506

6525

6544

6563

6582

6601

6620

6639

6658

19

5.3

6677

6696

6715

6734

6752

6771

6790

6808

6827

6845

18

5.4

6864

.6882

6901

6919

6938

6956

6974

6993

7011

7029

18

5.5

1.7047

7066

7084

7102

7120

7138

7156

7174

7192

7210

18

5.6

7228

7246

7263

7281

7299

7317

7334

7352

7370

7387

18

5.7

7405

7422

7440

7457

7475

7492

7509

7527

7544

7561

17

5.8

7579

7596

7613

7630

7647

7664

7681

7699

7716

7733

17

5.9

7750

7766

7783

7800

7817

7834

7851

7867

7884

7901

17

6.0

1.7918

7934

7951

7967

7984

8001

8017

8034

8050

8066

16

6.1

8083

8099

8116

8132

8148

8165

8181

8197

8213

8229

16

6.2

8245

8262

8278

8294

8310

8326

8342

8358

8374

8390

16

6.3

8405

8421

8437

8453

8469

8485

8500

8516

8532

8547

16

6.4

8563

8579

8594

8610

8625

8641

8656

8672

8687

8703

15

6.5

1.8718

8733

8749

8764

8779

8795

8810

8825

8840

8856

15

6.6

8871

8886

8901

8916

8931

8946

8961

8976

8991

9006

15

6.7

9021

9036

9051

9066

9081

9095

9110

9125

9140

9155

15

6.8

9169

9184

9199

9213

9228

9242

9257

9272

9286

9301

15

6.9

9315

9330

9344

9359

9373

9387

9402

9416

9430

9445

14

7.0

1.9459

9473

9488

9502

9516

9530

9544

9559

9573

9587

14

7.1

9601

9615

9629

9643

9657

9671

9685

9699

9713

9727

14

7.2

9741

9755

9769

9782

9796

9810

9824

9838

9851

9865

14

7.3

1 .9879

9892

9906

9920

9933

9947

9961

9974

9988

*OOOI

13

7.4

2.0015

0028

0042

0055

0069

0082

0096

0109

0122

0136

13

7.5

2.0149

0162

0176

0189

0202

0215

0229

0242

0255

0268

13

7.6

0281

0295

0308

0321

0334

0347

0360

0373

0386

0399

13

7.7

0412

0425

0438

0451

0464

0477

0490

0503

0516

0528

13

7.8

0541

0554

0567

0580

0592

0605

0618

0631

0643

0656

13

7.9

0669

0681

0694

0707

0719

0732

0744

0757

0769

0782

12

8.0

2.0794

. 0807

0819

0832

0844

0857

0869

0882

0894

0906

12

8.1

0919

0931

0943

0956

0968

0980

0992

1005

1017

1029

12

8.2

1041

1054

1066

1078

1090

1102

1114

1126

1138

1150

12

8.3

1163

1175

1187

1199

1211

1223

1235

1247

1258

1270

12

8.4

1282

1294

1306

1318

1330

1342

1353

1365

1377

1389

12

8.5

2.1401

1412

1424

1436

1448

1459

1471

1483

1494

1506

12

8.6

1518

1529

1541

1552

1564

1576

1587

1599

1610

1622

12

8.7

1633

1645

1656

1668

1679

1691

1702

1713

1725

1736

11

8.8

1748

1759

1770

1782

1793

1804

1815

1827

1838

1849

It

8.9

1861

1872

1883

1894

1905

1917

1928

1939

1950

1961

11

9.0

2.1972

1983

1994

2006

2017

2028

2039

2050

2061

2072

11

9.1

2083

2094

2105

2116

2127

2138

2148

2159

2170

2181

II

9.2

2192

2203

2214

2225

2235

2246

2257

2268

2279

2289

11

9.3

2300

2311

2322

2332

2343

2354

2364

2375

2386

2396

11

9.4

2407

2418

2428

2439

2450

2460

2471

2481

2492

2502

11

9.5

2.2513

2523

2534

2544

2555

2565

2576

2586

2597

2607

10

9.6

2618

2628

2638

2649

2659

2670

2680

2690

2701

2711

10

9.7

2721

2732

2742

2752

2762

2773

2783

2793

2803

2814

10

9.8

2824

2834

2844

2854

2865

2875

2885

2895

2905

2915

10

9.9

2925

2935

2946

2956

2966

2976

2986

29%

3006

3016

10

10.0

2.3026

Moving the decimal point n places to the right [or left] in the number requires adding n times 2.3026 for n times (0.6974-3)] in the body of the table. See auxiliary table of multiples on top of the preceding page.

60 MATHEMATICAL TABLES

HYPERBOLIC SINES [sinh x = ^(e* - e~*)]

X

0

1

2

3

4

5

6

7

8

9

£

0.0

.0000

.0100

.0200

.0300

.0400

.0500

.0600

.0701

.0801

.0901

00

1

.1002

.1102

.1203

.1304

.1405

.1506

.1607

.1708

.1810

.1911

01

2

.2013

.2115

.2218

.2320

.2423

.2526

.2629

.2733

.2837

.2941

03

3

.3045

.3150

.3255

.3360

.3466

.3572

.3678

.3785

.3892

.4000

106

4

.4108

.4216

.4325

.4434

.4543

.4653

.4764

.4875

.4986

.5098

no

0.5

.5211

.5324

.5438

.5552

.5666

.5782

.5897

.6014

.6131

.6248

116

6

.6367

.6485

.6605

.6725

.6846

.6967

.7090

.7213

.7336

.7461

122

7

.7586

.7712

.7838

.7966

.8094

.8223

.8353

.8484

.8615

.8748

130

8

.8881

.9015

.9150

.9286

.9423

.9561

.9700

.9840

.9981

1.012

138

9

1.027

1.041

.055

1.070

1.085

1.099

1.114

1.129

1.145

1.160

15

1.0

1.175

1.191

.206

1.222

1.238

1.254

1.270

1.286

1.303

1.319

16

1

1.336

1.352

.369

1.386

1.403

1.421

1.438

1.456

1.474

1.491

17

2

1.509

1.528

.546

1.564

1.583

1.602

1.621

1.640

1.659

1.679

19

3

1.698

1.718

.738

1.758

1.779

• 1.799

1.820

1.841

1.862

1.883

21

4

1.904

1.926

.948

1.970

1.992

ZOI4

2.037

2.060

2.083

2.106

22

1.5

2.129

2.153

2.177

2.201

2.225

2.250

2.274

2.299

2.324

2.350

25

6

2.376

2.401

2.428

2.454

2.481

2.507

2.535

2.562

2.590

2.617

27

7

2.646

2.674

2.703

2.732

2.761

2.790

2.820

2.850

2.881

2.911

30

8

2.942

2.973

3.005

3.037

3.069

3.101

3.134

3.167

3.200

3.234

33

9

3.268

3.303

3.337

3.372

3.408

3.443

3.479

3.516

3.552

3.589

36

2.0

3.627

3.665

3.703

3.741

3.780

3.820

3.859

3.899

3.940

3.981

39

1

4.022

4.064

4.106

4.148

4.191

4.234

4.278

4.322

4.367

4.412

44

2

4.457

4.503

4.549

4.596

4.643

4.691

4.739

4.788

4.837

4.887

48

3

4.937

4.988

5.039

5.090

5.142

5.195

5.248

5.302

5.356

5.411

53

4

5.466

5.522

5.578

5.635

5.693

5.751

5.810

5.869

5.929

5.989

58

2.5

6.050

6.112

6.174

6.237

6.300

6.365

6.429

6.495

6.561

6.627

64

6

6.695

6.763

6.831

6.901

6.971

7.042

7.113

7.185

7.258

7.332

71

7

7.406

7.481

7.557

7.634

7.711

7.789

7.868

7.948

8.028

8.110

79

8

8.192

8.275

8.359

8.443

8.529

8.615

8.702

8.790

8.879

8.969

87

9

9.060

9.151

9.244

9.337

9.431

9.527

9.623

9.720

9.819

9.918

96

3.0

10.02

10.12

10.22

10.32

10.43

10.53

10.64

10.75

10.86

10.97

11

1

11.08

11.19

11.30

11.42

11.53

11.65

11.76

11.88

12.00

12.12

12

2

12.25

12.37

12.49

12.62

12.75

12.88

13.01

13.14

13.27

13.40

13

3

13.54

13.67

13.81

13.95

14.09

14.23

14.38

14.52

14.67

14.82

14

4

14.97

15.12

15.27

15.42

15.58

15.73

15.89

16.05

16.21

16.38

16

3.5

16.54

16.71

16.88

17.05

17.22

17.39

17.57

17.74

17.92

18.10

17

6

18.29

18.47

18.66

18.84

19.03

19.22

19.42

19.61

19.81

20.01

19

7

20.21

20.41

20.62

20.83

21.04

21.25

21.46

21.68

21.90

22.12

21

8

22.34

22.56

22.79

23.02

23.25

23.49

23.72

23.96

24.20

24.45

24

9

24.69

24.94

25.19

25.44

25.70

25.96

26.22

26.48

26.75

27.02

26

4.0

27.29

27.56

27.84

28.12

28.40

28.69

28.98

29.27

29.56

29.86

29

1

30.16

30.47

30.77

31.08

31.39

31.71

32.03

32.35

32.68

33.00

32

2

33.34

33.67

34.01

34.35

34.70

35.05

35.40

35.75

36.11

36.48

35

3

36.84

37.21

37.59

37.97

38.35

38.73

39.12

39.52

39.91

40.31

39

4

40.72

41.13

41.54

41.96

42.38

42.81

43.24

43.67

44.11

44.56

43

4.5

45.00

45.46

45.91

46.37

46.84

47.31

47.79

48.27

48.75

49.24

47

6

49.74

5024

50.74

51.25

51.77

52.29

52.81

53.34

53.88

54.42

52

7

54.97

55.52

56.08

56.64

57.21

57.79

58.37

58.96

59.55

60.15

58

8

60.75

61.36

61.98

62.60

63.23

63.87

64.51

65.16

65.81

66.47

64

9

67.14

67.82

68.50

69.19

69.88

70.58

71.29

72.01

72.73

73.46

71

5.0

74.20

If x > 5, sinh x = W(e*) and logio sinh x = (0.4343)z + 0.6990 — 1, correct to four significant figures. For table of multiples of 0.4343, see p. 62. Graphs, p. 174.

MATHEMATICAL TABLES 61

HYPERBOLIC COSINES [cosh x = K(e* +e~*)]

V

0

1

2

3

4

5

6

7

8

9

£

0.0

1.000

1.000

1.000

1.000

1.001

1.001

1.002

1.002

1.003

1.004

1

1

.005

1.006

1.007

1.008

1.010

1.011

1.013

1.014

1.016

1.018

2

2

.020

1.022

1.024

1.027

1.029

1.031

1.034

1.037

1.039

1.042

3

3

.045

1.048

.052

1.055

1.058

1.062

1.066

1.069

1.073

1.077

4

4

.081

1.085

.090

1.094

1.098

1.103

1.108

1.112

1.117

1.122

5

0.5

.128

1.133

.138

1.144

1.149

1.155

1.161

1.167

1.173

1.179

6

6

.185

1.192

.198

1.205

1.212

1.219

1.226

1.233

1.240

1.248

7

7

.255

1.263

.271

1.278

1.287

1.295

1.303

1.311

1.320

1.329

8

8

.337

1.346

.355

1.365

1.374

1.384

1.393

1.403

1.413

1.423

10

9

1.433

1.443

.454

1.465

1.475

1.486

1.497

1.509

1.520

1.531

11

1.0

1.543

1.555

1.567

1.579

1.591

1.604

1.616

1.629

1.642

1.655

13

1

1.669

1.682

1.696

1.709

1.723

1.737

1.752

1.766

1.781

1.796

14

2

1.811

1.826

1.841

1.857

1.872

1.888

1.905

1.921

1.937

1.954

16

3

1.971

1.988

2.005

2.023

2.040

2.058

2.076

2.095

2.113

2.132

18

4

2.151

2.170

2.189

2.209

2.229

2.249

2.269

2.290

2.310

2.331

20

1.5

2.352

2.374

2.395

2.417

2.439

2.462

2.484

2.507

2.530

2.554

23

6

2.577

2.601

2.625

2.650

2.675

2.700

2.725

2.750

2.776

2.802

25

7

2.828

2.855

2.882

2.909

2.936

2.964

2.992

3.021

3.049

3.078

28

8

3.107

3.137

3167

3.197

3.228

3.259

3.290

3.321

3.353

3.385

31

9

3.418

3.451

3.484

3.517

3.551

3.585

3.620

3.655

3.690

3.726

34

2.0

3.762

3.799

3.835

3.873

3.910

3.948

3.987

4.026

4.065

4.104

38

1

4.144

4.185

4.226

4.267

4.309

4.351

4.393

4.436

4.480

4.524

42

2

4.568

4.613

4.658

4.704

4.750

4.797

4.844

4.891

4.939

4.988

47

3

5.037

5.087

5.137

5.188

5.239

5.290

5.343

5.395

5.449

5.503

52

4

5.557

5.612

5.667

5.723

5.780

5.837

5.895

5.954

6.013

6.072

58

2.5

6.132

6.193

6.255

6.317

6.379

6.443

6.507

6.571

6.636

6.702

64

6

6.769

6.836

6.904

6.973

7.042

7.112

7.183

7.255

7.327

7.400

70

7

7.473

7.548

7.623

7.699

7.776

7.853

7.932

8.011

8.091

8.171

78

8

8.253

8.335

8.418

8.502

8.587

8.673

8.759

8.847

8.935

9.024

86

9

9.115

9.206

9.298

9.391

9.484

9.579

9.675

9.772

9.869

9.968

95

3.0

10.07

10.17

10.27

10.37

10.48

10.58

10.69

10.79

10.90

11.01

11

1

11.12

11.23

11.35

11.46

11.57

11.69

11.81

11.92

12.04

12.16

12

2

12.29

12.41

12.53

12.66

12.79

12.91

13.04

13.17

13.31

13.44

13

3

13.57

13.71

13.85

13.99

14.13

14.27

14.41

14.56

14.70

14.85

14

4

15.00

15.15

15.30

15.45

15.ftl

15.77

15.92

16.08

16.25

16.41

16

3.5

16.57

16.74

16.91

17.08

17.25

17.42

17.60

17.77

17.95

18.13

17

6

18.31

18.50

18.68

18.87

19.06

19.25

19.44

19.64

19.84

20.03

19

7 8

20.24 22.36

20.44 22.59

20.64 22.81

20.85 23.04

21.06 23.27

21.27 23.51

21.49 23.74

21.70 23.98

21.92 24.22

22.14 24.47

21 23

9

24.71

24.96

25.21

25.46

25.72

25.98

26.24

26.50

26.77

27.04

26

4.0

27.31

27.58

27.86

28.14

28.42

28.71

29.00

29.29

29.58

29.88

29

1

30.18

30.48

30.79

31.10

31.41

31.72

32.04

32.37

32.69

33.02

32

2

33.35

33.69

34.02

34.37

34.71

35.06

35.41

35.77

36.13

36.49

35

3

36.86

37.23

37.60

37.98

38.36

38.75

39.13

39.53

39.93

40.33

39

4

40.73

41.14

41.55

41.97

42.39

42.82

43.25

43.68

44.12

44.57

43

4.E

45.01

45.47

45.92

46.38

46.85

47.32

47.80

48.28

48.76

49.25

47

6

49.75

50.25

50.75

51.26

51.78

52.30

52.82

53.35

53.89

54.43

52

7

54.98

55.53

56.09

56.65

57.22

57.80

58.38

58.96

59.56

60.15

58

8

60.76

61.37

61.99

62.61

63.24

63.87

64.52

65.16

65.82

66.48

64

9

67.15

67.82

68.50

69.19

69.89

70.59

71.30

72.02

72.74

73.47

71

5.0

74.21

If x > 5, cosh x = ^i(e*) and logio cosh x = (0.4343)* + 0.6990 — 1, correct to four signifi- cant figures. For table of multiples of 0.4343, see p. 62. Graphs, p. 174.

62 MATHEMATICAL TABLES

HYPEEBOLIC TANGENTS [tanh x = (e*-e~*) /(e* +<T*) = sinh z/cosh

X

0

1

2

3

4

5

6

(1

8

9

|l

0.0

.0000

.0100

.0200

.0300

.0400

.0500

.0599

.0699

.0798

.0898

100

.0997

.1096

.1194

.1293

.1391

.1489

.1587

.1684

.1781

.1878

98

2

.1974

.2070

.2165

.2260

.2355

.2449

.2543

.2636

.2729

.2821

94

3

.2913

.3004

.3095

.3185

.3275

.3364

.3452

.3540

.3627

.3714

89

4

.3800

.3885

.3969

.4053

.4136

.4219

.4301

.4382

.4462

.4542

82

0.5

.4621

.4700

.4777

.4854

.4930

.5005

.5080

.5154

.5227

.5299

75

6

.5370

.5441

.5511

.5581

.5649

.5717

.5784

.5850

.5915

.5980

67

7

.6044

.6107

.6169

.6231

.6291

.6352

.6411

.6469

.6527

.6584

60

8

.6640

.6696

.6751

.6805

.6858

.6911

.6963

.7014

.7064

.7114

52

9

.7163

.7211

.7259

.7306

.7352

.7398

.7443

.7487

.7531

.7574

45

1.0

.7616

.7658

.7699

.7739

.7779

.7818

.7857

.7895

.7932

.7969

39

1

.8005

.8041

.8076

.8110

.8144

.8178

.8210

.8243

.8275

.8306

33

2

.8337

.8367

.8397

.8426

.8455

.8483

.8511

.8538

.8565

.8591

28

3

.8617

.8643

.8668

.8693

.8717

.8741

.8764

.8787

.8810

.8832

24

4.

.8854

.8875

.8896

.8917

.8937

.8957

.8977

.8996

.9015

.9033

20

1.5

.9052

.9069

.9087

.9104

.9121

.9138

.9154

.9170

.9186

.9202

17

6

.9217

.9232

.9246

.9261

.9275

.9289

.9302

.9316

.9329

.9342

14

7

.9354

.9367

.9379

.9391

.9402

.9414

.9425

.9436

.9447

.9458

11

8

.9468

.9478

.9488

.9498

.9508

.9518

.9527

.9536

.9545

.9554

9

9

.9562

.9571

.9579

.9587

.9595

.9603

.9611

.9619

.9626

.9633

8

2.0

.9640

.9647

.9654

.9661

.9668

.9674

.9680

.9687

.9693

.9699

6

1

.9705

.9710

.9716

.9722

.9727

.9732

.9738

.9743

.9748

.9753

5

2

.9757

.9762

.9767

.9771

.9776

.9780

.9785

.9789

.9793

.9797

4

3

.9801

.9805

.9809

.9812

.9816

.9820

.9823

.9827

.9830

.9834

4

4

.9837

.9840

.9843

.9846

.9849

.9852

.9855

.9858

.9861

.9863

3

2.5

.9866

.9869

.9871

.9874

.9876

.9879

.9881

.9884

.9886

.9888

2

6

.9890

.9892

.9895

.9897

.9899

.9901

.9903

.9905

.9906

.9908

2

7

.9910

.9912

.9914

.9915

.9917

.9919

.9920

.9922

.9923

.9925

2

8

.9926

.9928

.9929

.9931

.9932

.9933

.9935

.9936

.9937

.9938

2.9

.9940

.9941

.9942

.9943

.9944

.9945

.9946

.9947

.9949

.9950

1

3.

.9951

.9959

.9967

.9973

.9978

.9982

.9985

.9988

.9990

.9992

4

4.

.9993

.9995

.9996

.9996

.9997

.9998

.9998

.9998

.9999

.9999

I

5.

.9999

If x > 5,

tanh a;

= 1.0000 to four decimal places. Graphs, p

. 174.

MULTIPLES OF 0.4343 (0.43429448 = logw e)

X

0

1

2

3

4

5

6

7

8

9

0.

0.0000

0.0434

0.0869

0.1303

0.1737

0.2171

0.2606

0.3040

0.3474

0.390<

1.

0.4343

0.4777

0.5212

0.5646

0.6080

0.6514

0.6949

0.7383

0.7817

0.825:

2.

0.8686

0.9120

0.9554

0.9989

1.0423

1.0857

1.1292

1.1726

1.2160

1 .259!

3.

1 .3029

1.3463

1.3897

1.4332

1.4766

1.5200

1.5635

1.6069

1.6503

1 .6933

4.

1.7372

1.7806

1.8240

1.8675

1.9109

1.9543

1.9978

2.0412

2.0846

2.128(

5.

2.1715

2.2149

2.2583

2.3018

2.3452

2.3886

2.4320

2.4755

2.5189

2.5622

6.

2.6058

2.6492

2.6926

2.7361

2.7795

2.8229

2.8663

2.9098

2.9532

2.996*

7.

3.0401

3.0835

3.1269

3.1703

3.2138

3.2572

3.3006

3.3441

3.3875

3.43TC

8.

3.4744

3.5178

3.5612

3.6046

3.6481

3.6915

3.7349

3.7784

3.8218

3.8652

9.

3.9087

3.9521

3.9955

4.0389

4.0824

4.1258

4.1692

4.2127

4.2561

4.299*

MULTIPLES OP

2.3026

(2.3025851 =

1/0.4343)

x

0

1

2

3

4

5

6

7

8

9

0.

0.0000

0.2303

0.4605

0.6908

0.9210

1.1513

1.3816

1.6118

1.8421

2.0723

1.

2.3026

2.5328

2.7631

2.9934

3.2236

3.4539

3.6841

3.9144

4.1447

4.3749

2.

4.6052

4.8354

5.0657

5.2959

5.5262

5.7565

5.9867

6.2170

6.4472

6.6775

3.

6.9078

7.1380

7.3683

7.5985

7.8288

8.0590

8.2893

8.5196

8.7498

8.9801

4.

9.2103

9.4406

9.6709

9.901 1

10.131

10.362

10.592

10.822

11.052

11.283

5.

11.513

11.743

11.973

12.204

12.434

12.664

12.894

13.125

13.355

13.585

6.

13.816

14.046

14.276

14.506

14.737

14.967

15.197

15.427

15.658

15.888

7.

16.118

' 16.348

16.579

16.809

17.039

17.269

17.500

17.730

17.960

18.190

8.

18.421

18.651

18.881

19.111

19.342

19.572

19.802

20.032

20.263

20.493

9.

20.723

20.954

21.184

21.414

21.644

21.875

22.105

22.335

22.565

22.796

MATHEMATICAL TABLES

63

STANDARD DISTRIBUTION OF RESIDUALS (p. 121)

a = any positive quantity; y = the number of residuals which are numerically < a; r = the probable error of a single observation;

n «= number of observations.

a

y

Diff.

r

n

0.0

.000

1 2

.054 .107

54 53

CO

3 4

.160 .213

JJ

53 51

0.5 6 7

.264 .314 .363

50 49

8 9

.411 .456

45 44

1.0

.500

1

.542

^/

2 3

.582 .619

40 37

4

.655

33

1.5 6 7 8 9

.688 .719 .748 .775 . .800

31 29 27 25 23

2.0

1

.823

.843

20

i (i

2

.862

1 7

3

4

.879 .895

17 16

13

2.5

.908

6

.921

7 8

.931 .941

10 10

9

.950

9 7

3.0

.957

1

.963

6

2

.969

6

3

.974

5

4

.978

4

4

3.5

.982

6

.985

3

7

.987

8

.990

3

9

.991

2

4.0

.993

6

5.0

.999

FACTORS FOR COMPUTING PROBABLE ERROR (p. 121)

n

Bessel

Peters

0.6745

0.6745

0.8453

0.8453

V(n - 1)

Vn(n-l)

Vn(/i-l)

n\/n— 1

2

.6745

.4769

.5978

.4227

3

.4769

.2754

.3451

.1993

4

.3894

.1947

.2440

.1220

5

.3372

.1508

.1890

.0845

6

.3016

.1231

.1543

.0630

7

.2754

.1041

.1304

.0493

8

.2549

.0901

.1130

.0399

9

.2385

.0795

.0996

.0332

10

.2248

.0711

.0891

.0282

11

.2133

.0643

.0806

.0243

12

.2034

.0587

.0736

.0212

13

.1947

.0540

.0677

.0188

14

.1871

.0500

.0627

.0167

15

.1803

.0465

.0583

.0151

16

.1742

.0435

.0546

.0136

17

.1686

.0409

.0513

.0124

18

.1636

.0386

.0483

.0114

19

.1590

.0365

.0457

.0105

20

.1547

.0346

.0434

.0097

21

.1508

.0329

.0412

.0090

22

.1472

.0314

.0393

.0084

23

.1438

.0300

.0376

.0078

24

.1406

.0287

.0360

.0073

25

.1377

.0275

.0345

.0069

26

.1349

.0265

.0332

.0065

27

.1323

.0255

.0319

.0061

28

.1298

.0245

.0307

.0058

29

.1275

.0237

.0297

.0055

30

.1252

.0229

.0287

.0052

31

.1231

.0221

.0277

.0050

32

.1211

. .0214

.0268

.0047

33

.1192

.0208

.0260

.0045

34

.1174

.0201

.0252

.0043

35

.1157

.0196

.0245

.0041

36

.1140

.0190

.0238

.0040

37

.1124

.0185

.0232

.0038

38

.1109

.0180

.0225

.0037

39

.1094

.0175

.0220

.0035

40

.1080

.0171

.0214

.0034

45

.1017

.0152

.0190

.0028

50

.0964

.0136

.0171

.0024

55

.0918

.0124

.0155

.0021

60

.0878

.0113

.0142

.0018

65

.0843

.0105

.0131

.0016

70

.0812

.0097

.0122

.0015

75

.0784

.0091

.0113

.0013

80

.0759

.0085

.0106

.0012

85

.0736

.0080

.0100

.0011

90

.0715

.0075

.0094

.0010

95

.06%

.0071

.0089

.0009

100

.0678

.0068

.0085

.0008

64

MATHEMATICAL TABLES

COMPOUND INTEREST. AMOUNT OF A GIVEN PRINCIPAL

The amount A at the end of n years of a given principal P placed at compouni interest to-day is A = P X x or A = P X y or A = P X z, according as the interes (at the rate of r per cent, per annum) is compounded annually, semi-annually, o quarterly; the factor x or y or z being taken from the following tables.

Values of x. (Interest compounded annually; A = P X £.)

Years

r = 2

2H

3

?M

4

4$*

5

6

7

i

1.0200

1.0250

1.0300

1.0350

1.0400

1.0450

1.0500

1.0600

1.0700

2

1.0404

1.0506

1.0609

1.0712

1.0816

1.0920

1.1025

.1236

1.1449

3

1.0612

1.0769

1.0927

1.1087

1.1249

1.1412

1.1576

.1910

1.2250

4

1.0824

1. 1038

1.1255

1.1475

1.1699

1.1925

12155

.2625

1.3108

3

5

1.1041

1.1314

1.1593

1.1877

1.2167

1.2462

1.2763

.3382

1.4026

6

1.1262

1.1597

1.1941

1.2293

1.2653

1.3023

1.3401

.4185

1.5007

1

7

1.1487

1.1887

1.2299

1.2723

1.3159

1.3609

1.4071

.5036

1.6058

**

8

1.1717

1.2184

1.2668

1.3168

1.3686

1.4221

1.4775

.5938

1.7182

Ja

9

1.1951

1.2489

1.3048

1.3629

1.4233

1.4861

1.5513

1.6895

1.8385

•^ «'

10

1.2190

1.2801

1.3439

1.4106

1.4802

1.5530

1.6289

1.7908

1.9672

if

11

1.2434

1.3121

1.3842

1.4600

1.5395

1.6239

1.7103

1.8983

2.1049

t, O

12

1.2682

1.3449

1.4258

1.5111

1.6010

1.6959

1.7959

2.0122

2.2522

^"vT

13

1.2936

1.3785

1.4685

1.5640

1.6651

1.7722

1.8856

2.1329

2.4098

v*^

14

13195

1.4130

1.5126

1.6187

1.7317

1.3519

1.9799

2.2609

2.5785

a~^~

15

1.3459

1.4483

1.5580

1.6753

1.8009

19353

2.0789

2.3966

2.7590

S^i

16

1.3728

1.4845

1.6047

1.7340

1.8730

2.0224

2.1829

2.5404

2.9522

2 !l

17

1.4002

1.5216

1.6528

1.7947

1.9479

2.1134

2.2920

2.6928

3.1588

.5 H

18

1.4282

1.5597

1.7024

1.8575

2.0258

2.2085

2.4066

2.8543

3.3799

0

19

1.4568

1.5987

1.7535

1.9225

2.1068

2.3079

2.5270

3.0256

3.6165

20

1.4859

1.6386

1.8061

1.9898

2.1911

2.4117

2.6^533

3.2071

3.8697

-2

25

1.6406

1.8539

2.0938

2.3632

2.6658

3.0054

3.3864

4.2919

5.4274

.2

30

1.8114

2.0976

2.4273

2.8068

3.2434

3.7453

4.3219

5.7435

7.6123

g2

40

2.2080

2.6851

3.2620

3.9593

4.8010

5.8164

7.0400

10.286

14.974

50

2.6916

3.4371

4.3839

5.5849

7.1067

9.0326

1 1 .467

18.420

29.457

60

3.2810

4.3998

5.8916

7.8781

10.520

14.027

18.679

32.988

57.946

Values of y. (Interest compounded semi-annually; A = P X y.)

Years

r=2

2H

3

3H

4

4H

5

6

7

1

1.0201

1.0252

1.0302

1.0353

1.0404

1 .0455

1.0506

1.0609

1.0712

2

1.0406

1.0509

1.0614

1.0719

1.0824

1.0931

1.1038

1.1255

1.1475

3

1.0615

1.0774

1.0934

1.1097

1.1262

1.1428

1.1597

1.1941

1 .2293

4

1.0829

1.1045

1.1265

1.1489

1.1717

1.1948

1.2134

1.2668

1.3168

5

1.1046

1.1323

1.1605

1.1894

1.2190

1.2492

1.2801

1.3439

1.4106

6

1.1268

1.1608

1.1956

1.2314

1.2682

1.3060

1.3449

1.4258

1.5111

7

1.1495

1.1900

1.2318

1.2749

1.3195

1.3655

1.4130

1.5126

1 .61 87

«5

8

1.1726

1.2199

1.2690

1.3199

1.3728

1.4276

1.4845

1.6047

1.7340

g^

9

1.1961

1.2506

1.3073

1.3665

1.4282

1.4926

1.5597

1.7024

1.8575

g

10

1.2202

1.2820

1.3469

1.4148

1.4859

1.5605

1.6386

1.8061

1.9898

~£-

11

1.2447

1.3143

1.3876

1.4647

1.5460

1.6315

1.7216

1.9161

2.1315

v~'

12

1.2697

1.3474

1.4295

1.5164

1.6084

1.7058

1.8087

2.0328

2.2833

+

13

1.2953

1.3812

1.4727

1.5700

1.6734

1.7834

1.9003

2.1566

2.4460

•H

14

1.3213

1.4160

1.5172

1.6254

1.7410

1.8645

1.9965

2.2879

2.6202

II

15

1.3478

1.4516

1.5631

1.6828

1.8114

1.9494

2.0976

2.4273

2.8068

a

16

1.3749

1.4881

1.6103

1.7422

1.8845

2.0381

2.2038

2.5751

3.0067

17

1 .4026

1.5256

1.6590

1.8037

1.9607

2.1308

2.3153

2.7319

3.2209

jj

18

1.4308

1.5639

1.7091

1.8674

2.0399

2.2278

2.4325

2.8983

3.4503

"9

19

1.4595

1.6033

1.7608

1.9333

2.1223

2.3292

2.5557

3.0748

3.6960

S

20

1.4889

1.6436

1.8140

2.0016

2.2080

2.4352

2.6851

3.2620

3.9593

fe

25

1.6446

1.8610

2.1052

2.3808

2.6916

3.0420

3.4371

4.3839

5.5849

30

1.8167

2.1072

2.4432

2.8318

3.2810

3.8001

4.3998

5.8916

7.8781

40

2.2167

2.7015

3.2907

4.0064

4.8754

5.9301

7.2096

10.641

15.676

50

2.7048

3.4634

4.4320

5.6682

7.2446

9.2540

11.814

19.219

31.191

60

3.3004

4.4402

5.9693

8.0192

10.765

14.441

19.358

34.711

62.064

MATHEMATICAL TABLES 65

Values of z. (Interest compounded quarterly; A = P X 2; see opposite page)

Years

• = 2

2H

3

3^

4

4>i

5

6

7

1

.0202

.0252

1.0303

1.0355

1.0406

1.0458

1.0509

1.0614

1.0719

2

.0407

.0511

1.0616

1.0722

1.0829

1.0936

.1045

.1265

1.1489

3

.0617

.0776

1.0938

1.1102

1.1268

1.1437

.1608

.1956

1.2314

4

.0831

.1048

1.1270

1.1496

1.1726

1.1960

.2199

.2690

1.3199

5

.1049

.1327

1.1612

1.1903

1.2202

1.2508

.2820

.3469

1.4148

6

.1272

.1613

1.1964

1.2326

1.2697

1.3080

.3474

.4295

1.5164

_«

7

.1499

.1906

1.2327

1.2763

1.3213

1.3679

.4160

.5172

1.6254

X

8

.1730

.2206

1.2701

1.3215

1.3749

1.4305

.4881

1.6103

1.7422

o"

9

.1967

.2514

1.3086

1.3684

1.4308

1.4959

1.5639

1.7091

1.8674

3

10

.2208

.2830

1.3483

1.4169

1.4889

1.5644

1.6436

1.8140

2.0016

^

11

.2454

.3154

1.3893

1.4672

1.5493

1.6360

1.7274

1.9253

2.1454

12

.2705

.3486

1.4314

1.5192

1.6122

1.7108

1.8154

2.0435

2.2996

<

13

.2961

.3826

1.4748

1.5731

1.6777

1.7891

1.9078

2.1689

2.4648

;-*.

14

.3222

.4175

1.5196

1.6288

1.7458

1.8710

2.0050

2.3020 •

2.6420

II

15

.3489

.4533

1.5657

1.6866

1.8167

1.9566

2.1072

2.4432

2.8318

M

16

.3760

.4900

1.6132

1.7464

1.8905

2.0462

2.2145

2.5931

3.0353

17

.4038

.5276

1.6621

1.8083

1.9672

2.1398

2.3274

2.7523

3.2534

J5

18

.4320

.5661

1.7126

1.8725

2.0471

2.2378

2.4459

2.9212

3.4872

19

.4609

.6056

,1.7645

1.9389

2.1302

2.3402

2.5705

3.1004

3.7378

o

20

.4903

.6462

1.8180

2.0076

2.2167

2.4473

2.7015

3.2907

4.0064

A

25

.6467

.8646

2.1111

2.3898

2.7048

3.0609

3.4634

4.4320

5.6682

30

.8194 i

.1121

2.4514

2.8446

3.3004

3.8285

4.4402

5.9693

8.0192

40

!.22ii ;

'.7098

3.3053

4.0306

4.9138

5.9892

7.2980

10.828

16.051

50

5.7115 3

.4768

4.4567

5.7110

7.3160

9.3693

11.995

19.643

32.128

60

L3102 ^

L4608

6.0092

8.0919

10.893

14.657

19.715

35.633

64.307

AMOUNT OP AN ANNUITY

The amount S accumulated at the end of n years by a given annual payment Y set aside at the end of each year is S =* F X », where the factor v is to be taken from the following table. (Interest at r per cent, per annum, compounded annually.)

Values of v

Years

r= 2

2H

3

Hi

4

^

5

6

7

1

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

1 .0000

2

2.0200

2.0250

2.0300

2.0350

2.0400

.2.0450

2.0500

2.0600

2.0700

3

3.0604

3.0756

3.0909

3.1062

3.1216

3.1370

3.1525

3.1836

3.2149

§

4

4.1216

4.1525

4.1836

4.2149

4.2465

4.2782

4.3101

4.3746

4.4399

O

5

5.2040

5.2563

5.3091

5.3625

5.4163

5.4707

5.5256

5.6371

5.7507

>

6

6.3081

6.3877

6.4684

6.5502

6.6330

6.7169

6.8019

6.9753

7.1533

.1.

7

7.4343

7.5474

7.6625

7.7794

7.8983

8.0192

8.1420

8.3938

8.6540

I

8

8.5830

8.7361

8.8923

9.0517

9.2142

9.3800

9.5491

9.8975

10.260

^

9

9.7546

9.9545

10.159

10.368

10.583

10.802

11.027

11.491

11.978

1 ;

10

10.950

11.203

11.464

11.731

12.006

12.288

12.578

13.181

13.816

« S

11

12.169

12.483

12.808

13.142

13.486

13.841

14.207

14.972

15.784

1—1 o

12

13.412

13.796

14.192

14.602

15.026

15.464

15.917

16.870

17.888

!>

13

14.680

15.140

15.618

16.113

16.627

17.160

17.713

18.882

20.141

14

15.974

16.519

17.086

17.677

18.292

18.932

19.599

21.015

22.550

> •!•

15

17.293

17.932

18.599

19.296

20.024

20.784

21.579

23.276

25.129

+ 7

16

18.639

19.380

20.157

20.971

21.825

22.719

23.657

25.673

27.888

^ 1

17

20.012

20.865

21.762

22.705

23.698

24.742

25.840

28.213

30.840

" vS

18

21.412

22.386

23.414

24.500

25.645

26.855

28.132

30.906

33.999

^

19

22.841

23.946

25.117

26.357

27.671

29.064

30.539

33.760

37.379

11 0

20

24.297

25.545

26.870

28.280

29.778

31.371

33.066

36.786

40.995

25

32.030

34.158

36.459

38.950

41.646

44.565

47.727

54.865

63.249

a

30

40.568

43.903

47.575

51.623

56.085

61.007

66.439

79.058

94.461

1

40

60.402

67.403

75.401

84.550

95.026

107.03

120.80

154.76

199.64

i

50

84.579

97.484

112.80

131.00

152.67

178.50

209.35

290.34

406.53

PH

60

114.05

135.99

163.05

196.52

237.99

289.50

353.58

533.13

813.52

66

MATHEMATICAL TABLES

PRINCIPAL WHICH WILL AMOUNT TO A GIVEN SUM

The principal P, which, if placed at compound interest to-day, will amount to a giv< sum A at the end of n years is P = A X x' or P = A X y' or P = A X z', according the interest (at the rate of r per cent, per annum) is compounded annually, semi-annuall or quarterly: the factor x' or y' or z' being taken from the following tables. Values of x'. (Interest compounded annually; P = A X x')

Years

r = 2

2H

3

sw

4

4H

5

6

7

1

.98039

.97561

.97087

.96618

.96154

.95694

.95238

.94340

.93458

2

.96117

.95181

.94260

.93351

.92456

.91573

.90703

.89000

.87344

3

.94232

.92860

.91514

.90194

.88900

.87630

.86384

.83962

.81630

4

.92385

.90595

.88849

.87144

.85480

.83856

.82270

.79209

.76290

.

5

.90573

C88385

.86261

.84197

.82193

.80245

.78353

.74726

.71299

i-H

6

.88797

.86230

.83748

.81350

.79031

.76790

.74622

.70496

.66634

7

.87056

.84127

.81309

.78599

.75992

.73483

.71068

.66506

.62275

8

.85349

.82075

.78941

.75941

.73069

.70319

.67684

.62741

.58201

1! 1 ^

9

.83676

.80073

.76642

.73373

.70259

.67290

.64461

.59190

.54393

10

.82035

.78120

.74409

.70892

.67556

.64393

.61391

.55839

.50835

I

11

.80426

.76214

.72242

.68495

.64958

.61620

.58468

.52679

.47509

3

12

.78849

.74356

.70138

.66178

.62460

.58966

.55684

.49697

.44401

13

.77303

.72542

.68095

.63940

.60057

.56427

.53032

.46884

.41496

4-

14

.75788

.70773

.66112

.61778

.57748

.53997

.50507

.44230

.38783

i-H

15

.74301

.69047

.64186

.59689

.55526

.51672

.48102

.41727

.36245

D

16

.72845

.67362

.62317

.57671

.53391

.49447

.4581 1

.39365

.33873

17

.71416

.65720

.60502

.55720

.51337

.47318

.43630

.37136

.31657

"«

18

.70016

.64117

.58739

.53836

.49363

.45280

.41552

.35034

.29586

cj

19

.68643

.62553

.57029

.52016

.47464

.43330

.39573

.33051

.27651

3

20

.67297

.61027

.55368

.50257

.45639

.41464

.37689

.31180

.25842

jjj

25

.60953

.53939

.47761

.42315

.37512

.33273

.29530

.23300

.18425

o

30

.55207

.47674

.41199

.35628

.30832

.26700

.23138

.17411

.13137

n

40

.45289

.37243

.30656

.25257

.20829

.17193

.14205

.09722

.06678

50

.37153

.29094

.2281 1

.17905

.14071

.11071

.08720

.05429

.03395

60

.30478

.22728

.16973

.12693

.09506

.07129

.05354

.03031

.01726

Values of y'. (Interest compounded semi-annually; P = A X y')

Years

r = 2

2V

3

W

4

4J/2

5

6

7

1

.98030

.97546

.97066

.96590

.96117

.95647

.95181

.94260

.93351

2

.96098

.95152

.94218

.93296

.92385

.91484

.90595

.88849

.87144

3

.94205

.92817

.91454

.90114

.88797

.87502

.86230

.83748

.81350

4

.92348

.90540

.88771

.87041

.85349

.83694

.82075

.78941

.75941

.'

5

.90529

.88318

.86167

.84073

.82035

.80051

.78120

.74409

.70892

\

6

.88745

.86151

.83639

.81206

.78849

.76567

.74356

.70138

.66178

7

.86996

.84037

.81185

.78436

.75788

.73234

.70773

.66112

.61778

8

.85282

.81975

.78803

.75762

.72845

.70047

.67362

.62317

.57671

*

9

.83602

.79963

.76491

.73178

.70016

.66998

.64117

.58739

.53836

~Z

10

.81954

.78001

.74247

.70682

.67297

.64082

.61027

.55368

.50257

§

11

.80340

.76087

.72069

.68272

.64684

.61292

.58086

.52189

.46915

^

12

.78757

.74220

.69954

.65944

.62172

.58625

.55288

.49193

.43796

£,

13

.77205

.72398

.67902

.63695

.59758

.56073

.52623

.46369

.40884

4.

14

.75684

.70622

.65910

.61523

.57437

.53632

.50088

.43708

.38165

i

15

.74192

.68889

.63976

.59425

.55207

.51298

.47674

.41199

.35628

*-*

16

.72730

.67198

.62099

.57398

.53063

.49065

.45377

.38834

.33259

n

17

.71297

.65549

.60277

.55441

.51003

.46930

.43191

.36604

.31048

*a»

18

.69892

.63941

.58509

.53550

.49022

.44887

.41109

.34503

.28983

19

.68515

.62372

.56792

.51724

.47119

.42933

.39128

.32523

.27056

J2

20

.67165

.60841

.55126

.49960

.45289

.41065

.37243

.30656

.25257

s

25

.60804

.53734

.47500

.42003

.37153

.32873

.29094

.22811

.17905

30

.55045

.47457

.40930

.35313

.30478

.26315

.22728

.16973

.12693

£

40

.45112

.37017

.30389

.24960

.20511

.16863

.13870

.09398

.06379

50

.36971

.28873

.22563

.17642

.13803

.10806

.08465

.05203

.03206

60

.30299

.22521

.16752

.12470

.09289

.06925

.05166

.02881

.01611

MATHEMATICAL TABLES

67

Values of «'. (Interest compounded quarterly; P=>A X z'', see opposite page)

Years| r = 2

2tt

3

JH

4

4J4

5

6

7

1

.98025

.97539

.97055

.96575

.96098

.95624

.95152

.94218

.93296

2

.96089

.95138

.94198

.93268

.92348

.91439

.90540

.88771

.87041

3

.94191

.92796

.91424

.90074

.88745

.87437

.86151

.83639

.81206

4

.92330

.90512

.88732

.86989

.85282

.8361 1

.81975

.78803

.75762

5

.90506

.88284

.86119

.84010

.81954

.79952

.78001

.74247

.70682

\ 1-1

6

.88719

.86111

.83583

.81132

.78757

.76453

.74220

.69954

.65944

*

7

.86966

.83991

.81122

.78354

.75684

.73107

.70622

.65910

.61523

8

.85248

.81924

.78733

.75670

.72730

.69908

.67198

.62099

.57390

j

9

.83564

.79908

.76415

.73079

.69892

.66849

.63941

.58509

.53550

JL*

10

.81914

.77941

.74165

.70576

.67165

.63923

.60841

.55126

.49960

o

11

.80296

.76022

.71981

.68159

.64545

.61126

.57892

.51939

.46611

V

12

.78710

.74151

.69861

.65825

.62026

.58451

.55086

.48936

.43486

^

13

.77155

.72326

.67804

.63570

.59606

.55893

.52415

.46107

.40570

-j-

14

.75631

.70546

.65808

.61393

.57280

.53447

.49874

.43441

.37851

15

.74137

.68809

.63870

.59291

.55045

.51108

.47457

.40930

.35313

'•—

16

.72673

.67115

.61989

.57260

.52897

.48871

.45156

.38563

.32946

II

17

.71237

.65464

.60164

.55299

.50833

.46733

.42967

.36334

.30737

%*

18

.69830

.63852

.58392

.53405

.48850

.44687

.40884

.34233

.28676

19

.68451

.62281

.56673

.51576

.46944

.42732

.38903

.32254

.26754

Js

"3

20

.67099

.60748

.55004

.49810

.45112

.40862

.37017

.30389

.24960

25

.60729

.53630

.47369

.41845

.36971

.32670

.28873

.22563

.17642

o

30

.54963

.47347

.40794

.35154

.30299

.26120

.22521

.16752

.12470

£

40

.45023

.36903

.30255

.24810

.20351

.16697

.13702

.09235

.06230

50

.36880

.28762

.22438

.17510

.13669

.10673

.08337

.05091

.03113

60

.30210

.22417

.16641

.12358

.09181

.06823

.05072

.02806

.01555

ANNUITY WHICH WILL AMOUNT TO A GIVEN SUM (SINKING FUND)

The annual payment, Y, which, if set aside at the end of each year, will amount with accumulated interest to a given sum S at the end of n years is Y = S X v', where the factor v' is given below. (Interest at r per cent, per annum, compounded annually.)

Values of »'

Years

r= 2

2H

3

3H

4

^

5

6

7

2

.49505

.49383

.49261

.49140

.49020

.48900

.48780

.48544

.48309

»'

3

.32675

.32514

.32353

.32193

.32035

.31877

.31721

.31411

.31105

^^

4

.24262

.24082

.23903

.23725

.23549

.23374

.23201

.22859

.22523

P

5

.19216

.19025

.18835

.18648

.18463

.18279

.18097

.17740

.17389

„

6

.15853

.15655

.15460

.15267

.15076

.14888

.14702

.14336

.13980

'"H

7

.13451

.13250

.13051

.12854

.12661

.12470

.12282

.11914

.11555

1

8

.11651

-.11447

.11246

.11048

.10853

.10661

.10472

.10104

.09747

4,

9

.10252

.10046

.09843

.09645

.09449

.09257

.09069

.08702

.08349

§

10

.09133

.08926

.08723

.08524

.08329

.08138

.07950

.07587

.07238

11

.08218

.0801 1

.07808

.07609

.07415

.07225

.07039

.06679

.06336

I*

12

.07456

.07249

.07046

.06848

.06655

.06467

.06283

.05928

.05590

13

.06812

.06605

.06403

.06206

.06014

.05828

.05646

.05296

.04965

14

.06260

.06054

.05853

.05657

.05467

.05282

.05102

.04758

.04434

j*

15

.05783

.05577

.05377

.05183

.04994

.0481 1

.04634

.04296

.03979

•!•

16

.05365

.05160

.04961

.04768

.04582

.04402

.04227

.03895

.03586

17

.04997

.04793

.04595

.04404

.04220

.04042

.03870

.03544

.03243

§

18

.04670

.04467

.04271

.04082

.03899

.03724

.03555

.03236

.02941

19

.04378

.04176

.03981

.03794

.03614

.03441

.03275

.02962

.02675

|

20

.04116

.03915

.03722

.03536

.03358

.03188

.03024

.02718

.02439

.. ^

25

.03122

.02928

.02743

.02567

.02401

.02244

.02095

.01823

.01581

J2 "

30

.02465

.02278

.02102

.01937

.01783

.01639

.01505

.01265

.01059

l>

40

.01656

.01484

.01326

.01183

.01052

.00934

.00828

.00646

.00467

|H

0

50

.01182

.01026

.00887

.00763

.00655

.00560

.00478

.00344

.00238

fe

60 .00877

.00735

.00613

.00509

.00420

.00345

.00283

.00188

.00121

68

MATHEMATICAL TABLES

PRESENT WORTH OF AN ANNUITY

The capital C, which, if placed at interest to-day, will provide for a given annual payment Y for a term of n years before it is exhausted is C = Y X w, where the factor w is given below. (Interest at r per cent, per annum, compounded annually.)

Values of w

Years| r =2

2H

3

3M

4

4H

5

6

7

1

0.9804

0.9756

0.9709

0.9662

0.9615

0.9569

0.9524

0.9434

0.9346

2

1.9416

1.9274

1.9135

1.8997

1.8861

1.8727

1.8594

1.8334

1.8080

N

3

2.8839

2.8560

2.8286

2.8016

2.7751

2.7490

2.7232

2.6730

2.6243

\

4

3.8077

3.7620

3.7171

3.6731

3.6299

3.5875

3.5460

3.4651

3.3872

1

5

4.7135

4.6458

4.5797

4.5151

4.4518

4.3900

4.3295

4.2124

4.1002

§""

6

5.6014

5.5081

5.4172

5.3286

5.2421

5.1579

5.0757

4.9173

4.7665

7

6.4720

6.3494

6.2303

6.1145

6.0021

5.8927

5.7864

5.5824

5.3893

\

8

7.3255

7.1701

7.0197

6.8740

6.7327

6.5959

6.4632

6.2098

5.9713

*•

9

8.1622

7.9709

7.7861

7.6077

7.4353

7.2688

7.1078

6.8017

6.5152

•1-

10

8.9826

8.7521

8.5302

8.3166

8.1109

7.9127

7.7217

7.3601

7.0236

11

9.7868

9.5142

9.2526

9.0016

8.7605

8.5289

8.3064

7.8869

7.4987

i

12

10.575

10.258

9.9540

9.6633

9.3851

9.1186

8.8633

8.3838

7.9427

13

11.348

10.983

10.635

10.303

9.9856

9.6829

9.3936

8.8527

8.3577

I

14

12.106

11.691

11.296

10.921

10.563

10.223

9.8986

9.2950

8.7455

\

15

12.849

12.381

11.938

11.517

11.118

10.740

10.380

9.7122

9.1079

^

16

13.578

13.055

12.561

12.094

11.652

11.234

10.838

10.106

9.4466

+

17

1 4.292

13.712

13.166

12.651

12.166

11.707

11.274

10.477

9.7632

18

14.992

14.353

13.754

13.190

12.659

12.160

11.690

10.828

10.059

Zl

19

15.678

14.979

14.324

13.710

13.134

12.593

12.085

11.15.8

10.336

1

20

16.351

15.589

14.877

14.212

13.590

13.008

12.462

11.470

10.594

a ^

25

19.523

18.424

17.413

16.482

15.622

14.828

14.094

12.783

11.654

3 T

30

22.396

20.930

19.600

18.392

17.292

16.289

15.372

13.765

12.409

s "

40

27.355

25.103

23.115

21.355

19.793

18.402

17.159

15.046

13.332

o 5

50

31.424

28.362

25.730

23.456

21.482

19.762

18.256

15.762

13.801

HH

60

34.761

30.909

27.676

24.945

22.623

20.638

18.929

16.161

14.039

ANNUITY PROVIDED FOR BY A GIVEN CAPITAL

The annual payment Y provided for for a term of n years by a given capital C placed at interest to-day is Y = C X w' . (Interest at r per cent, per annum, compounded annually; the fund supposed to be exhausted at the end of the term.)

Values of w'

Years

r = 2

2%

3

3H

4

4H

5

6

7

2

.51505

.51883

.52261

.52640

.53020

.53400

.53780

.54544

.55309

3

.34675

.35014

.35353

.35693

.36035

.36377

.36721

.37411

.38105

•

4

.26262

.26582

.26903

.27225

.27549

.27874

.28201

.28859

.29523

c;

5

.21216

.21525

.21835

.22148

.22463

.22779

.23097

.23740

.24389

§

6

.17853

.18155

.18460

.18767

.19076

.19388

.19702

.20336

.20980

C^

7

.15451

.15750

.16051

.16354

.16661

.16970

.17282

.17914

.18555

^

8

.13651

.13947

.14246

.14548

.14853

.15161

.15472

.16104

.16747

_i_ X.

9

.12252

.12546

.12843

.13145

.13449

.13757

.14069

.14702

.15349

^ §

10

.11133

.11426

.11723

.12024

.12329

.12638

.12950

.13587

.14238

_ TH

11

.10218

.10511

.10808

.11109

.11415

.11725

.12039

.12679

.13336

1 xb

12

.09456

.09749

.10046

.10348

.10655

.10967

.11283

.11928

.12590

~ +

13

.08812

.09105

.09403

.09706

.10014

.10328

.10646

.11296

.11965

14

.08260

.08554

.08853

.09157

.09467

.09782

.10102

.10758

.11434

•1' ve

15

.07783

.08077

.08377

.08683

.08994

.09311

.09634

.10296

.10979

Ii

16

.07365

.07660

.07961

.08268

.08582

.08902

.09227

.09895

.10586

17

.06997

.07293

.07595

.07904

.08220

.08542

.08870

.09544

.10243

^T ^

18

.06670

.06967

.07271

.07582

.07899

.08224

.08555

.09236

.09941

II II

19

.06378

.06676

.06981

.07294

.07614

.07941

.08275

.08962

.09675

20

.06116

.06415

.06722

.07036

.07358

.07688

.08024

.08718

.09439

9

25

.05122

.05428

.05743

.06067

.06401

.06744

.07095

.07823

.08581

• •

30

.04465

.04778

.05102

.05437

.05783

.06139

.06505

.07265

.08059

"5

40

.03656

.03984

.04326

.04683

.05052

.05434

.05828

.06646

.07467

g

50

.03182

.03526

.03887

.04263

.04655

.05060

.05478

.06344

.07238

0

60

.02877

.03235

.03613

.04009

.04420

.04845

.05283

.06188

.07121

fc

MATHEMATICAL TABLES

DECIMAL EQUIVALENTS

From minutes and

From decimal parts of

Common fractions

seconds into deci- mal parts of a degree

a degree into minutes and seconds (exact values)

8 16 32 64

ths ths nds ths

Exact decimal values

0'

o°.oooo

0"

o°.oooo

o°.oo

0'

0°.50

30'

1

.01 5625

1

.0167

1

.0003

1

0' 36"

1

30' 36"

1 2

.03 125

2

.0333

2

.0006

2

1' 12"

2

31' 12"

3

.04 6875

3

.05

3

.0008

3

V 48"

3

31' 48"

1 2 4

.06 25

4

.0667

4

.0011

4

2' 24"

4

32' 24"

5

.07 8125

5'

.0833

5"

.0014

0°.05

3'

0°.55

33'

3 6

.09 375

6

.10

6

.0017

6

y 36"

6

33' 36"

7

.10 9375

7

.1167

7

.0019

7

4' 12"

7

34' 12"

1248

.12 5

8

.1333

8

.0022

8

4' 48"

8

34' 48"

9

.14 0625

9

.15

9

.0025

9

5' 24"

9

35' 24"

5 10

.15 625

10'

0°.1667

10"

0°.0028

0°.10

6'

0°.60

36'

11

.17 1875

1

.1833

]

.0031

1

6' 36"

1

36' 36"

3 6 12

.18 75

2

.20

2

.0033

2

7' 12"

2

37' 12"

13

.20 3125

3

.2167

3

.0036

3

7' 48"

3

37' 48"

7 14

.21 875

4

.2333

4

.0039

4

8' 24"

4

38' 24"

15

.23 4375

15'

.25

15"

.0042

0°.15

9'

0°.65

39'

2 4 8 16

.25

6

.2667

6

.0044

6

9' 36"

6

39' 36"

17

.26 5625

7

.2833

7

.0047

7

10' 12"

7

40' 12"

9 18

.28 125

8

.30

8

.005

8

10' 48"

8

40' 48"

19

.29 6875

9

.3167

9

.0053

9

11' 24"

9

41' 24"

5 10 20

.31 25

20'

0°.3333

20"

0°.0056

0°.20

12'

0°.70

42'

21

.32 8125

1

.35

1

.0058

12' 36"

42' 36"

11 22

.34 375

2

.3667

2

.0061

2

13' 12"

2

43' 12"

23

.35 9375

3

.3833

3

.0064

3

13' 48"

3

43' 48"

3 6 12 24

.37 5

4

.40

4

.0067

4

14' 24"

4

44' 24"

25

.39 0625

25'

.4167

25"

.0069

0°.25

15'

0°.75

45'

13 26

.40 625

6

.4333

6

.0072

6

15' 36"

6

45' 36"

27

.42 1875

7

.45

7

.0075

7

16' 12"

7

46' 12"

7 14 28

.43 75

8

.4667

8

.0078

8

16' 48"

8

46' 48"

29

.45 3125

9

.4833

9

.0081

9

17' 24"

9

47' 24"

15 30

.46 875

30'

0°.50

30"

0°.0083

0°.30

18'

0°.80

48'

31

.48 4375

1

.5167

1

.0086

1

18' 36"

1

48' 36"

4 8 16 32

.50

2

.5333

2

.0089

2

19' 12"

2

49' 12"

33

.51 5625

3

.55

3

.0092

3

19' 48"

3

49' 48"

17 34

.53 125

4

.5667

4

.0094

4

20' 24"

4

50' 24"

35

.54 6875

35'

.5833

35"

.0097

0°.35

21'

0°.85

51'

9 18 36

.56 25

6

.60

6

.01

6

21' 36"

6

51' 36"

37

.57 8125

7

.6167

7

.0103

7

22' 12"

7

52' 12"

19 38

.59 375

8

.6333

8

.0106

8

22' 48"

8

52' 48"

39

.60 9375

9

.65

9

.0108

9

23' 24"

9

53' 24"

5 10 20 40

.62 5

40'

0°.6667

40"

0°.01 1 1

0°.40

24'

0°.90

54'

41

.64 0625

1

.6833

1

.0114

1

24' 36"

54' 36"

21 42

.65 625

2

.70

2

.0117

2

25' 12"

2

55' 12"

43

.67 1875

3

.7167

3

.0119

3

25' 48"

3

55' 48"

11 22 44

.68 75

4

.7333

4

.0122

4

26' 24"

4

56' 24"

45

.70 3125

45'

.75

45"

.0125

0°45

27'

0°.95

57'

23 46

.71 875

6

.7667

6

.0128

6

27' 36"

6

57' 36"

47

.73 4375

7

.7833

7

.0131

7

28' 12"

7

58' 12"

6 12 24 48

.75

8

.80

8

.0133

8

28' 48"

8

58' 48"

49

.76 5625

9

.8167

9

.0136

9

29' 24"

9

59' 24"

25 50

.78 125

50'

0°.8333

50"

0°.0139

0°.50

30'

1°.00

60'

51

.79 6875

1

.85 .8667

1

2

.0142 .0144

o°.ooo

0".0

13 26 52 53

.81 25 .82 8125

3

.8833

3

.0147

1

3".6

27 54

.84 375

4

.90

4

.015

2

7".2

55

.85 9375

55'

.9167

55"

.0153

3

10".8

7 14 28 56

.87 5

6

.9333

6

.0156

4

14".4

57

.89 0625

7

.95

7

.0158

0°.005

18"

29 58

.90 625

8

.9667

8

.0161

6

21".6

59

.92 1875

9

.9833

9

.0164

7

25".2

15 30 60

.93 75

60'

1.00

60"

0°.0167

8

28".8

61

.95 3125

9

32" .4

31 62

.96 875

0°.010

36"

63

.98 4375

WEIGHTS AND MEASURES

BY LOUIS A. FISCHER

In the United States the measures of weight and length commonly employed are identical with the corresponding English units, but the capacity measures differ from those now in use in the British Empire, the U. S. gallon being defined as 231 cu. in. and the bushel as 2150.42 cu. in., whereas the corre- sponding British imperial units are, respectively, 277.418 cu. in., and 2219.344 cu. in. (1 imp. gal. = 1.2 U. S. gal., approx.; 1 imp. bu. = 1.03 U. S. bu., approx.).

The metric system of weights and measures was legalized and its use made permissive in the United States by an Act of Congress, passed in 1866. In 1872, by the concurrent action of the principal governments of the world, it was agreed to establish an International Bureau of Weights and Measures near Paris.

Prior to 1891 the British imperial yard was regarded as the real standard of the United States. In 1891, the Office of Weights and Measures (now Bureau of Standards) fixed the value of the United States yard in terms of the international meter, according ito the ratio: one yard = 3600/3937 meters. At the same time, the pound was fixed in terms of the international kilo- gram, according to the relation: one pound = 453.59243 grams.

U. S. Customary Weights and Measures

Measures of Length

Measures of Area

12 inches 3 feet 5H yards = 16^ feet

40 poles = 220 yards

8 furlongs = 1760 yards

= 5280 feet

3 miles

4 inches

9 inches

= 1 foot = 1 yard = 1 rod,

pole or

perch = 1 furlong

= 1 mile

= 1 league = 1 hand = 1 span

= 1 acre

640 acres = 1 square mile =

Nautical Units 6080.2 feet = 1 nautical mile

6 feet = 1 fathom

120 fathoms = 1 cable length

1 nautical mile per hr. = 1 knot

Surveyor's or Gunter's Measure

7.92 inches = 1 link

100 links = 66 ft. = 4 rods = 1 chain 80 chains = 1 mile

33H inches = 1 vara (Texas)

144 square inches = 1 square foot 9 square feet = 1 square yard 30^4 square yards = 1 square rod, pole or

perch 160 square rods

= 10 square chains

= 43,560sq. ft,

= 5645 sq. varas (Texas)

1 ''section" of U. S. Govt. surveyed land

1 circular inch = area of circle 1 inch

in diameter 1 square inch = 1.2732 cir. in. 1 circular mil =area of circle 0.001 in.

in diam. 1,000,000 cir. mils = l cir. in.

Measures of Volume

1728 cubic inches = 1 cubic foot

27 cubic feet = 1 cubic yard

1 cord of wood =128 cu. ft. 1 perch of masonry = 16K> to 25 cu. ft.

= 0.7854 sq. in.

70

U. S. WEIGHTS AND MEASURES

71

U. S. Customary Weights and Measures — (continued)

Measures of Volume

Weights

(The grain is the same in all systems)

Liquid or Fluid Measure

4 gills = 1 pint

2 pints = 1 quart

4 quarts <= 1 gallon

7.4805 gallons = 1 cubic foot

(There is no standard liquid "barrel.")

Apothecaries' Liquid Measure

60 minims = 1 liquid dram or drachm

8 drams = 1 liquid ounce 16 ounces = 1 pint

Water Measure

The Miner's Inch is the quantity of water that will pass through an orifice 1 sq. in. in cross-section under a head of from 4 to 6^ in., as fixed by statutes, and varies from Ho cu. ft. to Ho cu. ft. per sec. The units now most in use are 1 cu. ft. per sec. and 1 gal. per sec., the U. S. Reclamation Service employing the former. See p. 260.

Dry Measure

2 pints = 1 quart 8 quarts = 1 peck 4 pecks = 1 bushel

Shipping Measure 1 Register ton = 100 cu. ft.

1 U. S. shipping ton = 40 cu. ft.

= f 32.14 U. S. bu. m 131.14 imp. bu. 1 British shipping ton = 42 cu. ft.

/ 32.70 imp. bu. ~ \ 33.75 U. S. bu.

Board Measure

( 144 cu. in. = volume of 1 board foot = { board 1 ft. sq. and 1- in.

i thick.

No. of board feet in a log = [\i(d - 4)pZ,, where d = diam. of log (usually taken in- side the bark at small end), in., and L = length of log, ft. The 4 in. deducted are an allowance for slab. This rule is vari- ously known as the Doyle, Conn. River, St. Croix, Thurber, Moore and Beeman, and the Scribner rule.

Avoirdupois Weight 16 drams = 437.5 grains = 1 ounce 16 ounces = 7000 grains = 1 pound 100 pounds <= 1 cental

2000 pounds «=» 1 short ton

2240 pounds *= 1 long ton

Also (in Great Britain): 14 pounds 2 stone = 28 Ib. 4 quarters = 1 12 Ib.

20 hundredweight

= 1 stone = 1 quarter = 1 hundred- weight (cwt.) = 1 long ton

24 grains

Troy Weight

= 1 penny- weight (dwt.) 20 pennyweights = 480 grains = 1 ounce 12 ounces = 5760 grains =1 pound

1 Assay Ton = 29,167 milligrams, or as many milligrams as there are troy ounces in a ton of 2000 Ib. avoirdupois. Consequently, the number of milligrams of precious metal yielded by an assay ton of ore gives directly the number of troy ounces that would be obtained from a ton of 2000 Ib. avoirdupois.

Apothecaries' Weight

20 grains = 1 scruple 3

3 scruples = 60 grains = 1 dram 3

8 drams = 1 ounce 5

12 ounces = 5760 grains = 1 pound

Weight for Precious Stones

1 carat = 200 milligrams

(Adopted by practically all important

nations.)

Circular Measure

60 seconds = 1 minute 60 minutes = 1 degree 90 degrees = 1 quadrant 360 degrees = circumference 57.2957795 degrees =1 radian (or angle ( = 57° 17'44.806") having arc of length equal to radius)

METRIC SYSTEM

The fundamental unit of the metric system is the meter — the unit of length, from which the units of volume (liter) and of mass (gram) are derived. All other units are the decimal subdivisions or multiples of these. These three units are simply related : one cubic decimeter equals one liter, and one liter of water weighs one kilogram. The metric tables are formed by combining the words "meter," "gram," and "liter" with numerical prefixes.

72

WEIGHTS AND MEASURES

All lengths, areas, and cubic measures in the following conversion tables are derived from the international meter. The customary weights are like- wise derived from the kilogram. All capacities are based on the practical equivalent: 1 cubic decimeter equals 1 liter. (The liter is defined as the volume occupied by the mass of 1 kilogram of water under a pressure of 76 cm. of mercury and at the temperature of 4 deg. cent. According to the best information, 1 liter = 1.000027 cubic decimeters.)

The customary weights derived from the international kilogram are based on the value 1 avoirdupois Ib. = 453.59243 grams. The value of the troy ib. is based on the same relation and also the equivalent 5760/7000 avoirdupois Ib. equals 1 troy Ib.

Metric Measures

Length

Area

Unit

Sym- bol

Value in meters

Unit

Sym- bol

Value in sq. meters

Micron

M mm. cm. dm. m. dkm. hm. km. Mm.

0.000001 0.001 0.01 0.1 1.0 10.0 100.0 1,000.0 10,000.0 1,000,000.0

Millimeter.... Centimeter . . Decimeter. . . Meter (unit). Dekameter. . Hectometer.. Kilometer. . . Myriameter.. . Megameter.. .

Sq. millimeter

mm.2 cm.2 dm.2 m.» a. ha. km."

0 . 000001 0.0001 0.01 1.0 100.0 10,000.0 1,000,000.0

Sq. centimeter Sq. decimeter

Sq. meter (centiare) Sq. dekameter (are) Hectare

Sq. kilometer. .

Volume

Cubic measure

Unit

Symbol

Value in liters

Unit

Symbol

Value in cubic meters

Milliliter Liter (unit)

ml. or cm.3 1. or dm.8 kl. or m.s

cl. dl. dkl. hi.

0.001 1.0 1,000.0

0.01 0.1 10.0 100.0

Cubic kilometer Cubic hectometer. . . . Cubic dekameter

Cubic meter Cubic decimeter Cubic centimeter Cubic millimeter Cubic micron

km.3 hm.» dkm. s

m.8 dm.3 cm.3 mm.3 M8

10» 10« 103

1

10- «

10" « ID'9 10-18

Kiloliter Also Centiliter

Deciliter

Dekaliter

Hectoliter

Weight

Unit

Symbol

Value in grams

Unit

Symbol

Value in grams

Microgram

0.000001

Dekagram

dkg.

10.0

Milligram

mg.

0.001

Hectogram

fag

100 0

Centigram Decigram

eg- dg.

0.01 0.1

Kilogram Myriagram

kg. Mg

1,000.0 10,000 0

Gram (unit)

1 0

100 000 0

Ton

t.

1,000,000.0

SYSTEMS OF UNITS

The principal, units of interest to mechanical engineers can all be derived from the three fundamental units of force, length, and time. These three fundamental units may be chosen at pleasure; each such choice gives rise to a "system" of units. The following table gives the units of the four "systems" most often met with in the literature.

UNITS

73

The precise definitions of the fundamental units in these systems are as follows. (In these definitions the "standard pound body " and the "standard kilogram body "refer to two special lumps of metal, carefully preserved at London and Paris, respectively; the "standard locality" means sea level, 45 deg. latitude; or, more strictly, any locality in which the acceleration due to gravity has the value 980.665 cm. per sec.2 = 32.1740 ft. per sec.2, which may be called the standard acceleration.

The pound (force) is the force required to support the standard pound body against gravity, in vacua, in the standard locality; or, it is the force which, if applied to the stand- ard pound body, supposed free to move, would give that body the "standard ac- celeration." The word "pound" is used for the unit of both force and mass, and consequently is ambiguous. To avoid uncertainty it is desirable to call the units "pound force" and "pound mass," respectively.

The kilogram (force) is the force required to support the standard kilogram against gravity, in vacua, in. the standard locality; or, it is the force which, if applied to the stand- ard kilogram body, supposed free to move, would give that body the "standard accelera- tion." The word "kilogram" is used for the unit of both force and mass and conse- quently is ambiguous. To avoid uncertainty it is desirable to call the units "kilogram force" and "kilogram mass," respectively.

The poundal is the force which, if applied to the standard pound body, would give that body an acceleration of 1 ft. per sec.2; that is, 1 poundal = 1/32.1740 of a pound force.

The dyne is the force which, if applied to the standard gram body, would give that body an acceleration of 1 cm. per sec.2; that is, 1 dyne = 1/980.665 of a gram force.

Systems of Units

British

Metric

Name of unit

Dimen- sions of units in terms of F,L, T

"gravita- tional " sys- tem, or "foot-pound- second"

"gravita- tional " sys- tem, or "kilogram- meter-sec-

Metric "absolute" system, or "C. G. S." system

British "absolute" system (little used)

system

ond" system

Force

F

1 Ib.

1 kg.

1 dyne

1 poundal

Length

L

1 ft.

1 m.

1 cm.

1ft.

Time

T

1 sec.

1 sec.

1 sec.

1 sec.

Velocity Acceleration . .

L/T L/T*

1 ft. per sec. 1 ft. per sec.2

1 m. per sec. 1 m. per sec.2

1 cm. per sec. 1 cm. per sec.2

1 ft. per sec. 1ft. per sec.2

Pressure

F/L*

lib. per ft.2

1 kg. per m.2

1 dyne per cm.2

1 pdl. per ft.2

Impulse or

momentum. .

FT

1 Ib.-sec.

1 kg.-sec.

1 dyne-sec.

1 pdl.-sec.

Work or

energy

FL

1 ft.-lb.

1 kg.-m.

1 dyne-cm. =

1 ft. -pdl.

1 "erg."

Power

FL/T

1 ft.-lb. per

1 kg.-m. per

1 dyne-cm, per

1 ft.-pdl. per

sec.

sec.

sec.

sec.

Mass

F/(L/T*)

1 Ib. per (ft.

1 kg. per (m.

1 dyne per (cm.

1 pdl. per (ft.

per sec.2) =

per sec.2) =

per sec.2) = 1

per sec.2) =

1 "slug."

1 " metric

gram mass.

1 pound

slug."

mass.

NOTE. The "slug" (also called the "geepound," or the "engineer's unit of mass"), the " metric slug," and the "poundal" are never used in practice.

Other common units are as follows: Work: 1 joule = 10? ergs = 10,000,000 dyne-cm.

1 kilowatt-hour = 3,600,000 joules = 3600 X lO™ dyne-cm. Power: 1 horse power = 550 ft.-lb. per sec. 1 poncelet = 100 kg.-m. per sec. 1 force de cheval = 75 kg.-m. per sec. 1 watt = 1 joule per sec. = 10,000,000 dyne-cm, per sec. 1 kilowatt = 1000 watts = 1010 dyne-cm, per sec.

A new horse power of 550.220 ft.-lb. per sec., or 746 watts, has been proposed, but has not been accepted by mechanical engineers.

The weight of a body (in a given locality) always means a force, namely, the force, re-

74

WEIGHTS AND MEASURES

quired to support the body against gravity (in that locality). 'When no particular local- ity is specified, the standard locality may be assumed. Thus, the "standard weight" of the pound body is 1 lb.; the "standard weight" of the kilogram body is 1 kg

Dynes X 10« Kilograms Pounds Poundala

the quantity of heat required to raise the temperature of 1 gram of water 1 deg. cent, at a mean temperature of 15 deg. cent., or (2) the heat required to raise the temperature of 1 lb. of water 1 deg. fahr. The former quantity is called the gram-calorie (small calorie), while the latter is known as the British thermal unitorB.t.u. The kilogram-calorie (large calorie), which ia equal to 1000 g.-cal., is largely used in engineering work in metric countries. * 1 therm = 1 g.-cal. CONVERSION TABLES Length Equivalents

1

1.020

0.00848

2.248 ' 0.03518

72.33 1.85933

0.9807 1.99149

1

2.205 0.34334

70.93

1.85C84

0.4448 1.64819

0.4536

1.65667

1

32.17 1.50750

0.01383 2.14067

0.01410

2.14916

0.03108

2.49249

1

Centimeters Inches

Feet | Yards | Meters

Chains

Kilometers

Miles

1

0.3937 1.59517

0.03281

2.51598

0.01094 2.03886

0.01

2.00000

0.03497I 4.69644

10-5 B". 00000

0.066214 6.79335

2.540 0.40483

1

0.038333

4.92082

0.02778 2.44370

0.0254

2.40483

0.0*1263

5.10127

0.04254

5.40483

0.041578

5.19818

30.48

1.48402

12

1.07918

1

0.3333

T.52288

0.3048

1.48402

0.01515

2.18046

0.033098 4.48402

0.0s 1645

4.21608

91.14

1.96114

36

1.55630

3

0.47712

1

0.9144 1.96114

0.04545

2.65758

0.0s9144 4.96114

0.035682 4.75449

100

2.00000

39.37

1.59517

3.281

0.51598

1 .0936

0.03886

1

0.04971

2.69644

0.001 3.COOOO

0.0362!4

4.79335

2012 3.30356

792

2.89873

66

1.81954

22

1.34242

20.12 1.30356

1

0.02012 2.30356

0.0125

2.09691

100000 5.00000

39370 4.59517

3281 3.51598

1093.6

3.C3886

1000

3.00000

49.71

1.69644

1

0.6214 1.79335

160925 5.20665

63360

4.80182

5280 3.72263

1760 3.24551

1609 3.20665

80

1.90309

1.609

0.206G5

1

The equivalents are given in the heavier type. Logarithms of the equivalents are given immediately below.

Subscripts after any figure, Os, 94, etc.J mean that that figure is to be repeated the indicated number of times.

Conversion of Lengths

Inches to milli- meters

Milli- meters to inches

Feet to meters

Meters to feet

Yards to

meters

Meters to yards

Miles to kilo- meters

Kilo- meters to miles

1

3 4

6 7

[

25.40 50.80 76.20 101 .60

127.00 152.40 177.80 203.20 228.60

0.03937 0.07874 0.1181 0.1575

0.1968 0.2362 0.2756 0.3150 0.3543

0.3048 0.6096 0.9144 1.219

1.524 1.829 2.134 2.438 2.743

3.281 6.562 9.842 13.12

16.40 19.68 22.97 26.25 29.53

0.9144 1.829 2.743 3.658

4.572 5.486 6.401 7.315 8.230

1.094 2.187 3.281 4.374

5.486 6.562 7.655 8.749 9.842

1.609 3.219 4.828 6.437

8.047 9.656 . 11.27 12.87 14.48

0.6214 1.243 1.864 2.485

3.107 3.728 4.350 4.971 5.592

*See Marks' MECHANICAL ENGINEERS' HANDBOOK..

CONVERSION TABLES

75

Mechanical Equivalent of Heat. See p. 311.* The value most commonly accepted among American engineers as the work equivalent of 1 mean B.t.u. is 777.5 ft.-lb., and the mean gram-calorie = 4.183 joules, which are the values used throughout this book. The U. S. Bureau of Standards does not recommend any special value; for its own purposes it takes the 59 deg. fahr. B.t.u. as 778.2 ft.-lb. and the 68 deg. B.t.u. as 777.5 ft.-lb. The 15 deg. calorie = 4.187 joules; 20 deg. calorie = 4.183 joules. There is an uncer- tainty of about 1 part in 1000 in these values.

Conversion of Lengths : Inches and Millimeters

Common fractions of an inch to millimeters (From HU to 1 in.)

64ths

Milli- meters

64ths

Milli- meters

64ths

Milli- meters

64ths

Milli- meters

64ths

Milli- meters

64ths

Milli- meters

1

0.397

13

5.159

25

9.922

37

14.684

49

19.447

57

22.622

2

0.794

14

5.556

26

10.319

38

15.081

50

19.844

58

23.019

3

1.191

15

5.953

27

10.716

39

15.478

51

20.241

59

23.416

4

1.588

16

6.350

28

11.113

40

15.875

52

20.638

60

23.813

5

1.984

17

6.747

29

11.509

41

16.272

53

21.034

61

24.209

6

2.381

18

7.144

30

11.906

42

16.669

54

21.431

62

24.606

7

2.778

19

7.541

31

12.303

43

17.066

55

21.828

63

25.003

8

3.175

20

7.938

32

12.700

44

17.463

56

22.225

64

25.400

9

3.572

21

8.334

33

13.097

45

17.859

10

3.969

22

8.731

34

13.494

46

18.256

11

4.366

23

9.128

35

13.891

47

18.653

12

4.763

24

9.525

36

14.288

48

19.050

Decimals of an inch to millimeters. (From 0.01 in. to 0.99 in.)

0

1

2

3

4

5

6

7

8

9

.0

0.254

0.508

0.762

1.016

1.270

1.524

1.778

2.032

2.286

.1

2.540

2.794

3.048

3.302

3.556

3.810

4.064

4.318

4.572

4.826

.2

5.080

5.334

5.588

5.842

6.096

6.350

6.604

6.858

7.112

7.366

.3

7.620

7.874

8.128

8.382

8.636

8.890

9.144

9.398

9.652

9.906

.4

10.160

10.414

10.668

10.922

11.176

11.430

11.684

11.938

12.192

12.446

.5

12.700

12.954

13.208

13.462

13.716

13.970

14.224

14.478

14.732

14 986

.6

15.240

15.494

15.748

16.002

16.256

16.510

16.764

17.018

17.272

17.526

.7

17.780

18.034

18.288

18.542

18.796

19.050

19.304

. 19.558

19.812

20.066

.8

20.320

20.574

20.828

21 .082

21.336

21.590

21 .844

22.098

22.352

22.606

.9

22.860

23.114

23.368

23.622

23.876

24.130

24.384

24.638

24.892

25.146

Millimeters to decimals of an inch. (From 1 to 99 mm.)

0.

1.

2.

3.

4.

5.

6.

7.

8.

9.

0

0.0394

0.0787

0.1181

0.1575

0.1969

0.2362

0.2756

0.3150

0.3543

1

0.3937

0.4331

0.4724

0.5118

0.5512

0.5906

0.6299

0.6693

0.7087

0.7480

2

0.7874

0.8268

0.8661

0.9055

0.9449

0.9843

1 .0236

1 .0630

1.1024

1.1417

3

1.1811

1.2205

1 .2598

1.2992

1.3386

1.3780

1.4173

1.4567

1 .4961

1.5354

4

1.5748

1.6142

1 .6535

1 .6929

1.7323

1.7717

1.8110

1.8504

1.8898

1.9291

5

1.9685

2.0079

2.0472

2.0866

2.1260

2.1654

2.2047

2.2441

2.2835

2.3228

6

2.3622

2.4016

2.4409

2.4803

2.5197

2.5591

2.5984

2.6378

2.6772

2.7165

7

2.7559

2.7953

2.8346

2.8740

2.9134

2.9528

2.9921

3.0315

3.0709

3.1102

8

3.1496

3.1890

3.2283

3.2677

3.3071

3.3465

3.3858

3.4252

3.4646

3.5039

9

3.5433

3.5827

3.6220

3.6614

3.7008

3.7402

3.7795

3.8189

3.8583

3.8976

'See Marks' MECHANICAL ENGINEERS' HANDBOOK.

76

WEIGHTS AND MEASURES

Area Equivalents

(For conversion table see p. 77)

Square meters

Square inches

Square feet

Square yards

Square rods

Square chains

Roods

Acres

Square miles or sections

1

1550

3.19033

10.76 1.03197

1.196 0.07773

0.0395

2.59699

0.002471

3.39288

0.039884 3.99494

0.0>2471

1.39288

0.0*3861

7.58670

0.036452 4 80967

1

0.006944 3.84164

0.0011

3.88740

0.042551

5.40667

0.0*1594 6.20255

0 066377

7.80461

0.0e1594

7.20255

0.084910

10.39637

0.09290 1.96803

144 2.15836

1

0.1111

1.04576

0.003673 3.56503

0.032296 4.36091

0.049184

5.96297

0.042296 4.36091

0.0:3587

"S. 554 73

0.8361

1.92227

1296 3.11260

9

0.95424

1

0.03306

2.51927

0.002066 3.31515

0.038264 4.91721

0.0002066 4.31515

0.063228

7.50898

25.29 1.40300

39204 4.59333

272.25 2.43497

30.25

1.48072

1

0.0625

2.79588

0.02500

2.39794

0.00625

3.79588

0. 0*9766

6.98970

404.7 2.60712

627264

5.79745

4356 3.63909

484

2.68484

16

1.20412

1

0.4

1.60206

0.1

1.00000

0.0001562

4.19382

1012

3.00506

1568160 6.19539

10890 4.03703

1210

3.08278

40

1.60206

2.5

0.39794

1

0.25

1.39794

0.033906 "4.59176

4047 3.60712

6272640 6.79745

43560

4.63909

4840

3.68484

160

2.20412

10

1.00000

4

0.60206

1

0.001562 3.19382

2589a8 6.41330

27878400

7.44527

3097600 6.49102

102400 5.01030

6400

3.80618

2560

3.40824

640

2.80618

1

(1 hectare =100 ares = 10,000 centiares or square meters)

Volume and Capacity 'Equivalents

(For conversion table see p. 77)

Cubic inches

Cubic feet

Cubic yards

U. S. Apothe- cary liquid ounces

U. S. quarts

U. S. gallons

Bushels U. S.

Liters (1)

Liquid

«ry

Liquid

Dry

1

0.035787

4.76246

0.042143 F. 33109

0.5541

1. 74360

0.01732 2.23845

0.01488

2.17263

0.024329 3.63639

0.023720

3.57057

0.034650

4.66748

0.01639 2.21450

1728 3.23754

1

0.03704

2". 56864

957.5 2.98114

29.92

1.47599

25.71

1.41017

7.481 0.87393

6.429

0.80811

0.8036

1.90502

28.32 1.45205

46656 4.66891 1.805 0.25640

27 .

1.43136 0 001044

3.01886

1

0.043868

5.58749

25853 4.41251

1

807.9 2.90736 0.03125

2.49485

694.3 2.84153 0.02686

2.42903

202.0 2.30530 0.007813 3.89279

173.6

2.23948 0.006714

3.82697

21.70 1.33638 0.038392

4.92388

764.6 2.88341 0.02957

2.47091

57.75

1.76155

0.03342

2.52401

0.001238 3.09264

32 1.50515

1

0.8594 1.93418

0.25

1.39794

0.2148 1.33212

0.02686 2.42903

0.9464 1.97606

67.20 1.82737

0.03889 2.58983

0.001440 3.15847

37.24

1.57097

1.164

0.06582

1

0 2909 1.46376

0.25

1.39794

0.03125

2.49485

1.101

0.04188

231 2.36361

0.1337

1.12607

0.004951 3.69470

128

2.10721

4 0.60206

3.437

0.53624

1

0.8594 1.93418

0.1074 1.03109

3.785

0.57812

268.8 2.42943

0.1556 1.19189

0.005761

3.76053

148.9 2.17303

4.655

0.66788

4

0.60206!

1.164

0.06582

1

0.125

1.09691

4.405 0.64394

2150 3.33252

1.244 0.09498

0.04609 2.66362

1192 3.07612

37.24

1.57097

32 1.50515

9.309

0.96891

8

0.90309

1

35.24 1.54703

61.02 1.78550

0.03531 2.54795

0.001308 3.11659

33.81

1.52909

1.057

0.02394

0.9081 1.95812

0.2642

L 42188

0.2270 1.35606

0 02838 2.45297

1

The equivalents are given in the heavier type. Logarithms of the equivalents are given immediately below.

Subscripts after any figure, Oa, 94, etc., mean that that figure ia to be repeated the indicated number of timeB.

CONVERSION TABLES

77

Conversion of Areas

Sq. in. to sq. cm.

Sq. cm. to sq. in.

Sq. ft. to sq. m.

Sq. m. to sq. ft.

Sq. yd. to sq. m.

Sq. m. to sq. yd.

Acres to hec- tares

Hec- tares to acres

Sq. mi. to sq. km.

Sq. km. to sq. mi.

1

2 3

4

5 6 7 8 9

6.452 12.90 19.35 25.81

32.26 38.71 45.16 51.61 58.06

0.1550 0.3100 0.4650 0.6200

0.7750 0.9300 1.085 1.240 1.395

0.0929 0.1858 0.2787 0.3716

0.4645 0.5574 0.6503 0.7432 0.8361

10.76 21.53 32.29 43.06

53.82 64.58 75.35 86.11 96.87

0.8361 1.672 2.508 3.345

4.181 5.017 5.853 6.689 7.525

1.196 2.392 3.588 4.784

5.980 7.176 8.372 9.568 10.764

0.4047 0.8094 1.214 1.619

2.023 2.428 2.833 3.237 3.642

2.471 4.942 7.413 9.884

12.355 14.826 17.297 19.768 22.239

2.590 5.180 7.770 10.360

12.950 15.540 18.130 20.720 23.310

0.3861 0.7722 1.158 1.544

1.931 2.317 2,703 3.089 3.475

Conversion of Volumes or Cubic Measure

Cu. in. to cu. cm.

Cu. cm. to cu. in.

Cu. ft. to cu. m.

Cu. m. to cu. ft.

Cu. yd. to Cu. m.

Cu. m. to cu. yd.

Gallons to cu. ft.

Cu. ft. to gallons

2 3 4

5 6 7

8 9

16.39 32.77 49.16 65.55

81.94 98.32 114.7 131.1 147.5

0.06102 0.1220 0 1831 0.2441

0.3051 0.3661 0.4272 0.4882 0.5492

0.02832 0.05663 0.08495 0.1133

0.1416 0.1699 0.1982 0.2265 0.2549

35.31 70.63 105.9 141.3

176.6 211.9 247.2 282.5 317.8

0.7646 1.529 2.294 3.058

3.823 4.587 5.352 6.116 6.881

1.308 2.616 3.924 5.232

6.540 7.848 9.156 10.46 11.77

0.1337 0.2674 0.4011 0.5348

0.6685 0.8022 0.9359 1.070 1.203

7.481 14.96 22.44 29.92

37.41 44.89 52.36 59.85 67.33

Conversion of Volumes or Capacities

Liquid ounces to cu. cm.

Cu. cm. to liquid ounces

Pints to liters

Liters to pints

Quarts to liters

Liters to quarts

Gallons to liters

Liters to gallons

Bushels to hecto- liters

Hecto- liters to bushels

1

29.57

0.03381

0.4732

2.113

0.9464

1.057

3.785

0.2642

"0.3524

2.838

2

59.15

0.06763

0.9464

4.227

1.893

2.113

7.571

0.5283

0.7048

5.676

3

88.72

0.1014

1.420

6.340

2.839

3.170

11.36

0.7925

1.057

8.513

4

118.3

0.1353

1.893

8.453

3.785

4.227

15.14

1 .057

1.410

11.35

5

147.9

0.1691

2.366

10.57

4.732

5.283

18.93

1.321

1.762

14.19

6

177.4

0.2029

2.839

12.68

5.678

6.340

22.71

1.585

2.114

17.03

7

207.0

0.2367

3.312

14.79

6.625

7.397

26.50

1.849

2.467

19.86

8

236.6

0.2705

3.785

16.91

7.571

8.453

30.28

2.113

2.819

22.70

9

266.2

0.3043

4.259

19.02

8.517

9.510

34.07

2.378

3.172

25.54

Conversion of Masses

Grains to

grams

Grams to grains

Ounces (avoir.) to grams

Grams to ounces (avoir.)

Pounds (avoir.) to kilo- grams

Kilo- grams to pounds (avoir.)

Short tons (2000 Ib.) to metric tons

Metric tons (1000 kg.) to short tons

Long tons (2240 Ib.) to metric tons

Metric tons to long tons

1

0.06480

15.43

28.35

0.03527

0.4536

2.205

0.907

1.102

1.016

0.984

2

0.1296

30.86

56.70

0.07055

0.9072

4.409

1.814

2.205

2.032

1.968

3

0.1944

46.30

85.05

0.1058

1.361

6.614

2.722

3.307

3.048

2.953

4

0.2592

61.73

113.40

0.1411

1.814

8.818

3.629

4.409

4.064

3.937

5

0.3240

77.16

141.75

0.1764

2.268

11.02

4.536

5.512

5.080

4.921

6

0.3888

92.59

170.10

0.2116

2.722

13.23

5.443

6.614

6.096

5.905

7

0.4536

108,03

198.45

0.2469

3.175

15.43

6.350

7.716

7.112

6.889

8

0.5184

123.46

226.80

0.2822

3.629

17.64

7.257

8.818

8.128

7.874

9

0.5832

138.89

255.15

0.3175

4.082

19.84

8.165

9.921

9.144

8.857

78

WEIGHTS AND MEASURES

Velocity Equivalents

(For conversion table see p. 80)

Centimeters per sec.

Meters per sec.

Meters per min.

Kilo- meters per hour

Feet per sec.

Feet per min.

Miles per hour

Knots

1

0.01

0.6

1.77815

0.036

2.55630

0.03281 2.51598

1.9685 0.29414

0.02237

2.34965

0.01942

2.28825

100

2.00000

1

60 1.77815

3.6

0.55630

3.281

0.51598

196.85

2.29414

2.237

0.34965

1.942

0.28825

1.667 0.22184

0.01667 2.22184

1

0.06

2.77815

0.05468

2.73783

3.281 0.51598

0.03728 2.57150

0.03237

2.51018

27.78 1.44370

0.2778 1. 44370

16.67

1.22184

1

0.9113

T. 95968

54.68 1.73783

0.6214

1.79335

0.53960

1.73207

30.48 1.48402

0.3048

F. 48402

18.29

1.26217

1.097

0.04032

1

60

1.77815

0.6818

1.83367

0.59209

1.77238

0.5080 1.70586

0.005080 3". 70586

0.3048

1.48402

0.01829 2.26217

0.01667

2.22185

1

0.01136

2.05553

0.00987 3.99423

44.70 1.65035

0.4470

r. 65035

26.82 1.42850

1.609

0.20670

1.467 0.16633

88 1.94448

1

0.86839 1.93871

51.497

1.71178

0 51497 F. 71178

30.898 1.48993

1.8532 0.26793

1.68894 0.22761

101.337

2.00577

1.15155

0.06128

1

Mass Equivalents

(For conversion table see p. 77)

Kilograms

Grains

Ounces

Pounds

Tons

Troy and apoth.

Avoir- dupois

Troy and apoth.

Avoir- dupois

Short

Long

Metric

1

15432 4.18843

32.15 1.50719

35.27 1.54745

2.6792 0.42801

2.205 0.34333

0.021102 3.04230

0'039842 4.99309

0.001 3. 00000

0.046480 5.81157

1

0.022083 3.31876

0.022286 3.35902

0.031736

4.23958

0.031429 4.15490

0.077143

"8. 85387

0.076378

8.80465

0.0:6480

8.81157

0.03110

2.49281

480 2.68124

1

1.09714 0.04026

0.08333

2.92082

0.06857 2.83614

0.043429 5.53511

0.043061

5.48590

0.043110

"5.49281

0.02835 2.45255

437.5

2.64098

0.9115

1.95974

1

0.07595

2.88056

0.0625

2.79588

0.043125

5.49485

0.042790 5.44563

0.042835 T. 45255

0.3732 T.57199

5760

3.76042

12

1.07918

13.17

1.11944

1

0.8229 1.91532

8.0*41 H

4.61429

0.0s3673

4.56508

0.033732 4.57199

0.4536 T. 65667

7000 3.84510

14.58 1.16386

16

1.20412

1.215

0.08468

1

0.0005

T. 69897

0.034464 4.64975

0.034536

4.65667

907.2 2.95770

140e

7.14613

29167 4.46489

3203 4.50515

2431 3.38571

2000 3.30103

1

0.8929 T. 95078

0.9072 1.95770

1016 3.00691

156804 7.19535

326s 4.51411

35840 4.55437

2722 3.43492

2240 3.35025

1.12

0.04922

1

1.016

0.00691

1000 3.00000

15432356

7.18843

32151 4.50719

35274 4.54745

2679 3.42801

2205 3.34333

1.102

0.04230

0.9842 1.99309

1

The equivalents are given in the heavier type. Logarithms of the equivalents are given immediately below.

Subscripts after any figure, Os, §4, -etc., mean that that figure is to be repeated the indicated number of times.

CONVERSION TABLES

79

Pressure Equivalents

(For conversion table see p. 80)

Megabars or megadynes per sq. cm.

Kilo- grams per sq. cm. (Metric atmos- pheres)

Pounds per sq. in."

Short tons per

sq. ft.

Atmos- pheres

Columns of mercury at temperature 0°C.

Columns of water at temperature 15° C.

Meters

Inches

Meters

Inches

Feet

1

1.0197

14.50

1.044

0.9S69

0.7500

29.53

10.21

401.8

33.48

0.00848

1.16148

0.01882

1.99427

1.87508

1.47025

1.00886

2.60402

1.52484

0.9807

1

14.22

1.024

0.9678

0.7355

28.96

10.01

394.0

32.84

1.99152

1.15300

0.01034

1.98579

1.86660

1.46177

1.00038

2.59555

1.51636

0.06895 -

0.07031

1

0.072

0.06804

0.05171

2.036

0.7037

27.70

2.309

2.83852

2.84700

2". 85733

2.83279

2.71360

0.30876

f. 84738

1.44254

0.36336

0.9576

0.9765

13.89

1

0.9450

0.7182

28.28

9.773

384.8

32.06

,1.98119

1. 98966

1.14267

1.97545

1.85627

1.45143

0.99004

2.58521

1.50603

1.0133

1.0333

14.70

1.058

1

0.76

29.92

10.34

407.2

33.93

0.00573

0.01421

1.16722

0.02955

1.88081

1.47598

1.01459

2.60976

1.53058

1.3333

1.3596

19.34

1.392

1.316

1

39.37

13.61

535.7

44.64

0.12492

0.13340

1.28640

0.14373

0.11919

1.59517

1.13378

2.72894

1.64976

0.03386

0.03453

0.4912

0.03536

0.03342

0.02540

1

0.3456

13.61

1.134

2". 52975

2.53823

T. 69124

2.54857

2.52402

2.40484

1.53861

1.13378

0.05460

0.09798

0.09991

1.421

0.1023

0.09670

0.07349

2.893

|

39.37

3.281

2~.99114

2". 99962

0.15262

F. 00996

2.98541

2.86622

0.46139

1.59517

0.55198

0.002489

0 002538

0.03610

0.002599

0.002456

0.001867

0.07349

0.02540

1

0.08333

3.39598

3.40446

2.55746

3.41479

3.39024

3.27106

2.86622

2~.40484

2.92082

0.02986

0.03045

0.4332

0.03119

0.02947

0.02240

0.8819

0.3048

12

1

2.47516

2.48364

1.63664

2.49397

2.46942

2.35024

1.94540

1.48402

1.07918

Energy or Work Equivalents

(For conversion table see p. 80)

Joules = 10' ergs

Kilogram- meters

Foot- pounds

Kilo- watt- hours

Cheval- vapeur- hours

Horse- power- hours

Liter- atmos- pheres

Kilo- gram- calories

British thermal units

1

0.10197

1.00848

0.7376

1.86780

0.0e2778 7.44370

0.0o3777 7.57711

0.063725 7.57113

0.009869 3.99427

0.0»2390

T. 37848

0.039486 4.97709

9.80665 0.9915207

1

7.233 0.85932

0.052724 6.43522

0.0837037

6.56863

0.053653

6.56265

0.09678

2.98579

0.002344 3.37000

0 009302 3.96861

1.356 0.13220

0.1383 1.14068

1

0.063766 7.57590

0.0651206

7.70932

0.0650505 7.70333

0.01338

2.12647

0.033241

4.51068

0.001286

3.10929

3.6X10«

6.55630

3.671X105

5.56478

2.655X10" 6.42410

'"

1.3596 0.13342

1.341 0.12743

35528 4.55057

860.5

2.93478

3415 3.53339

2.648X10« 6.42288

270000. 5.43136

1.9529X10* 6.29068

0.7355

1.86658

1

0.9863 1.99401

26131.

4.41715

632.9

2.80135

2512

3.39996

2.6845X108

6.42887

2. 7375X105 5.43735

1.98X10« 6.29667

0.7457

1.87257

1.0139 0.00598

1

26494 4.42314

641.7

2.80735

2547 3.40595

101.33

2.00573

10.333 1.01421 '

74.73

1.87353

0.042815 5.44943

0.043827

5.58284

0.043774

5.57686

1

0.02422 2.38425

0.09612

2.98281

4183 3.62153

426.6 2.63000

3086 3.48932

0.001162

3.06522

0.001580

3.19864

0.001558 3.19265

41.29 1.61579

1

3.968 0.59861

1054

3.02291

107.5 2.03139

777.52 2.89071

0.032928

4.46661

0.033981

4.60003

0.033927

4.59405

10.40 1.01719

0.25200 1.40139

1

The equivalents are given in the heavier type. Logarithms of the equivalents are given immediately below.

Subscripts after any figure, Oi, 94, etc., mean that that figure is to be repeated the indicated number of times.

V

80

WEIGHTS AND MEASURES

Linear and Angular Velocity Conversion Factors

Cm. per sec. to feet per min.

Feet per min. to cm. per sec.

Cm. per sec. to miles per hour

Miles per hour to cm. per sec.

Feet per sec. to miles per hour

Miles per hour to feet per sec.

Radians per sec. to rev. per min.

Rev. per

min. to radians per sec.

1

2 3

4

5 6 7

8 9

1.97 3.94 5.91 7.87

9.84 11.81 13.78 15.75 17.72

0.508 1.016 1.524 2.032

2.540 3.048 3.556 4.064 4.572

0.0224 0.0447 0.0671 0.0895

0.1118 0.1342 0.1566 0.1789 0.2013

44.7 89.4 134.1 178.8

223.5 268.2 312.9 357.6 402.3

0.682 1.364 2.046 2.727

3.409 4.091 4.773 5.455 6.136

1.47 2.93 4.40 5.87

7.33 8.80 10.27 11.73 13.20

9.55 19.10 28.65 38.20

47.75 57.30 66.85 76.39 85.94

0.1047 0.2094 0.3142 0.4189

0.5236 0.6283 0.7330 0.8378 0.9425

Conversion of Pressures

Pounds per sq. in. to kilograms per sq. cm.

Kilograms per sq. cm. to pounds per sq. in.

Atmospheres to pounds per sq. in.

Pounds per sq. in. to atmospheres

Atmospheres to kilograms per sq. cm.

Kilograms per sq. cm. to atmos- pheres

1

0.0703

14.22

14.70

0.0680

1.033

0.9678

2

0.1406

28.45

29.39

0.1361

2.067

1.936

3

0.2109

42.67

44.09

0.2041

3.100

2.903

4

0.2812

56.89

58.79

0.2722

4.133

3.871

5

0.3515

71.12

73.48

0.3402

5.166

4.839

6

0 4218

85.34

88.18

0.4082

6.200

5.807

7

0.4922

99.56

102.9

0.4763

7.233

6.774

8

0.5624

113.8

117.6

0.5443

8.266

•7.742

9

0.6328

128.0

132.3

0.6124

9.300

8.710

Conversion of Energy, Work, Heat

Ft.-lb. to kilo- gram- meters

Kilo- gram- meters to ft.-lb.

Ft.-lb. to B.t.u.

B.t.u. to ft.-lb.

Kilo- gram- meters to large calories

Large calories to kilo- gram- meters

Joules to small calories

Small calories to j oules

1

0.1383

7.233

0.001286

777.5

0.002344

426.6

0.2390

4.183

2

0.2765

14.47

0.002572

1555.0

0.004688

853.2

0.4780

8.367

3

0.4148

21.70

0 003858

2333.0

0007033

1280.0

0'.7170

12.55

4

0.5530

28.93

0.005144

3110.0

0.009377

1706.0

0.9560

16.73

5

0.6913

36.16

0.006431

3888.0

0.01172

2133.0

1.195

20.92

6

0.8295

43.40

0007717

4665.0

0.01407

2560.0

1.434

25.10

7

0.9678

50.63

0009003

5443.0

0.01641

2986.0

1.673

29.28

8

1.106

57.86

0 01029

6220.0

0.01875

3413.0

1.912

33.47

9

1.244

65.10

0.01157

6998.0

0.02110

3839.0

2.151

37.65

Conversion of Power

Horse powers to kilowatts

Kilowatts to horse powers

Metric horse powers to kilowatts

Kilowatts to metric horse powers

Horse powers to metric horse powers

Metric horse powers to horse powers

1

0.7457

1.341

0.7354

1.360

1.014

0.9863

2

1.491

2.682

1.471

2.719

2.028

1.973

3

2.237

4.023

2.206

4.079

3.042

2.959

4

2.983

5.364

2.942

5.439

4.056

3.945

5

3.728

6.705

3.677

6.799

5.069

4.932

6

4.474

8.046

4.413

8.158

6.083

5.918

7

5.220

9.387

5.148

9.518

7.097

6.904

8

5.965

10.73

5.884

10.88

8.111

7.890

9

6.710

12.07

6.619

12.24

9.125

8.877

CONVERSION TABLES

81

Power Equivalents

(For conversion table see p. 80)

Horse power

Kilo- watts (1000 joules per sec.)

Cheval- vapeur (metric h.p.)

Ponce- lets

M.-kg. per sec.

Ft.-lb. per sec.

Kg- cal. per sec.

B.t.u per sec.

550 stand- ard ft.-lb. per sec.

1

0.7457

1.014

0.7604

76.04

550

0.1783

0.7074

1.87256

0.00599

1.88105

1.88105

2.74036

1.25104

1.84965

1.341

1

1.360

1.020

102.0

737.6

0.2390

0.9486

0.12743

0.13343

0.00848

2.00848

2.86780

1.37848

1.97709

0.9863

0.7355

1

0.75

75

542.3

0.1758

0.6977

T. 99402

T. 86659

1.87506

1.87506

2.73438

1.24506

1.84367

1.315

0.9807

1.333

1

100

723.3

0.2344

0.9303

0.11896

1.99152

0.12493

2.00000

2.85932

1.37000

1.96861

0.01315

0.009807

0.01333

0.01

1

7.233

0.002344

0.009303

2.11896

3.99152

2.12493

T. 00000

0.85932

3.37000

2.96861

0.00182

0.001356

0.00184

0.00138

0.1383

1

0.033241

0.001286

3.25946

3.13219

3.26562

3.14067

T. 14067

T. 51068

T. 10929

5.610

4.183

5.688

4.266

426.6

3086

1

3.968

0.74896

0.62153

0.75494

0.63000

2.63000

3.48932

0.59861

1.414

1.054

1.433

1.075

107.5

777.5

0.2520

1

0.15035

0.02291

0.15632

0.03139

2.03139

2.89071

1.40138

The equivalents are given in the heavier type. Logarithms of the equivalents are given immediately below.

Subscripts after any figure, Os, 94, etc., mean that that figure is to be repeated the indicated number of times.

Density Equivalents and Conversion Factors

Equivalents

Conversion factors

Grams per cu. cm.

Lb. per

cu. in.

Lb. per cu. ft.

Short tons (2000 lb.) per cu. yd.

Lb. per U. S. gal.

Grams per cu. cm. to lb. per cu. ft.

Lb. per

cu. ft. to grams per cu. cm.

Grams per cu. cm. to short tons per cu. yd.

Short tons per cu. yd. to grams per cu. cm.

1

0.03613

62.43

0.8428

8.345

1

62.43

0.01602

0.8428

1.186

2.55787

1.79539

1.92572

0.92143

2

124.90

0.03204

1.6860

2.373

27.68

1

1728

23.33

231

3

187.30

0.04806

2.5280

3.600

1.44217

3.23754

1.36792

2.36361

4

249.70

0.06407

3.3710

4.746

0.01602

0.035787

1

0.0135

0.1337

5

312.40

0.08009

4.2140

5.933

2.20466

4.76245

2.13033

1.12613

6

374.60

0.09611

5.0570

7.119

1.186

0.04286

74.07

1

9.902

7

437.00

0.11210

5.9000

8.306

0.07428

2.63205

1.86964

0.99572

8

499.40

0.12820

6.7420

9.492

0.1198

0.004329

7.481

0.1010

1

9

561.90

0.14420

7.5850

10.680

1.07855 3.63639

0.87396

1.00432

10

624.30

0.16020

8.4280

11.870

82

WEIGHTS AND MEASURES

Conversion of Heat Transmission and Conduction

Small

B.t.u.

Small

B.t.u.

Small calories per

B.t.u. per hr. per

calories

per sq.

calories

per sq. ft.

sec. per sq. cm.

sq. ft. per 1 deg.

per sq. cm. to

ft. to small

per sq. cm. per cm. to

per in. to small

per 1 aeg. cent, per cm. thick, to B.t.u.

fahr. per in. thick to small calories

B.t.u.

calories

B.t.u. per

calories per

per hr. per sq. ft.

per sec. per sq. cm.

P1tsq.

per sq. cm.

sq. ft. per in.

sq. cm. per cm.

per 1 deg. fahr. per in. thick

per 1 deg. cent, per cm. thick

1

3.687

0.2712

1.451

0.6892

2.903X103

0.033445

2

7.374

0.5424

2.902

1.378

5.806X103

0.036890

3

11.06

0.8136

4.353

2.068

8.709X103

0.02I034

4

14.75

1.085

5.804

2.757

11.61 X1Q3

0.02I378

5

18.44

1.356

7.255

3.446

14.52 X10«

0.021722

6

22.12

1.627

8.706

4.135

17.42 X1Q3

0.022067

7

25.81

1.898

10.16

4.824

20.32 X1Q3

0.022412

8

29.50

2.170

11.61

5.514

23.22 X103

0.022756

9

33.18

2.441

13.06

6.203

26.13 XlO3

0.023100

NOTE. 1 gram-calorie per sq. cm. = 3.687 B.t.u. per sq. ft.

1 gram-calorie per sq. cm. per cm. = 1.451 B.t.u. per sq. ft. per in.

1 gram-calorie per sec. per sq. cm. for a temp. grad. of 1 deg. cent, per cm.

= 360 kilogram-calories per hour per sq. m. for a temp. grad. of 1 deg. cent, per m.

= 2.903 X 103 B.t.u. per hour per sq. ft. for a temp. grad. of 1 deg. fahr. per in.

Values of Foreign Coins

(Legal standards: (G) = gold; (S) = silver)

Country

Monetary unit

Value in terms of U. S. money

Country

Monetary unit

Value in terms of U. S. money

Argentina (G)

Austria-Hungary (GO Belgium (G and <S)

Peso

Crown Franc

CO. 9647

0.2026 0 1929

Great Britain (G) . . . .

Greece (G and S) Haiti (G)

Pound ster- ling. Drachma. . . Gourde

$4.8665

0.1929 0 9647

Bolivia (G)

0 3893

India (British) (G)

0 3244

Brazil (G) . . .

Milreis.

0 5463

Italy (G and <S)

Lira . .

0 1929

British colonies in . .

Pound ster-

Japan (G)

Yen

0 4984

Australasia and

ling.

4 8665

Liberia (G)

Dollar

1 0000

Africa (GO

Mexico (G)

Peso

0 4984

Canada (G)

Dollar.......

1 0000

Netherlands (G) ....

Florin

0 4019

Central American

Norway (G)

Crown

0 2679

States:

Panama (G)

Balboa

1 0000

Coeta Rica (GO

0 4653

Persia (G and <S)

Kran

Variable

British Honduras (G or iS)

Dollar

1.0000

Peru (G) Philippine Islands (G)

Libra Peso

4 8665 0 5000

Guatemala (S).. . .

Peso

0 4446

Portugal (G)

Escudo

1 0805

Honduras (S)

Peso . .

0 4446

Roumania (G)

Leu

0 1929

Salvador(jS)

Peso

0 4446

Russia (G)

Ruble

0 5145

Nicaragua (S)

1 0000

Dollar

1 0000

Chile (G) ...

Peso

0 3649

Servia (G)

Dinar. ..

0 1929

China (S)

Yuan

0 4777

Siam (G)

Tical

0 3708

Colombia (G)

Pound

4.8665 0 2680

Spain (G and <S) Straits Settlement (G)

Peseta Dollar

0.1929 0 5677

Ecuador (G) .

Sucre

0 4866

Sweden (G)

Crown

0 2679

Egypt (G)

Pound

4 9429

Switzerland (CO

Franc

0.1929

Finland (G)

Markka

0 1929

Turkey (G)

Piaster

0 0439

France (G or S) . . . German Empire (G)

Franc Mark

0.1929 0 2381

Uruguay (G) Venezuela (G)

Peso Bolivar

1.0340 0.1929

TIME 83

TIME

Kinds of Time. Three kinds of time are recognized by astronomers, viz., sidereal, apparent solar, and mean solar time. The sidereal day is the inter- val between two consecutive transits of some fixed celestial object across any given meridian, or it is the interval required by the earth to make one com- plete revolution on its axis. This interval is constant but it is inconvenient as a time unit because the noon of the sidereal day occurs at all hours of the day and night. The apparent solar day is the interval between two con- secutive transits of the sun across any given meridian. On account of the variable distance between the sun and earth, the variable speed of the earth in its orbit, the effect of the moon, etc., this interval is not constant and consequently cannot be kept by any simple mechanism, such as clocks or watches. To overcome the objection noted above, the mean solar day was devised. The mean solar day is the length of the average apparent solar day. Like the sidereal day it is constant, and like the apparent solar day its noon always occurs at approximately the same time of day. The astronomical day begins at mean solar noon and the hours run from one to twenty-four, while the civil day (mean solar) begins 12 hours earlier, at midnight, and' the hours run from one to twelve, and then repeat from noon to midnight.

The Year. There are three different kinds of year used, the sidereal, the tropical, and the anomalistic. The sidereal year is the time taken by the earth to complete one revolution around the sun from a given star to the same star again. Its length is 365 days, 6 hours, 9 minutes, and 9 seconds. The tropical year is the time included between two successive passages of the vernal equinox by the sun, and since the equinox moves westward 50. "2 of arc a year, the tropical year is shorter by 20'23."5 in time than the sidereal year. As the seasons depend upon the earth's position with respect to the equinox, the tropical year is the year of civil reckoning. The anomalistic year is the interval between two successive passages of the perihelion, namely, the time of the earth's nearest approach to the sun. The anomalistic year is only used in special calculations in astronomy.

The Calendar. The month depended originally upon the changes of the moon. The Mohammedan nations still use a lunar calendar with years of twelve lunar months, which alternately contain 355 and 356 days. Accord- ing to their method of reckoning the same month falls in different seasons, and their calendars gain 1 year on ours every 33 years. The Julian Calendar (established 45 B. C.) discards all consideration of the moon and adopts 36514 days as the true length of the year. It is still used in Russia and generally by the Greek Church. Gregorian Calendar: The true length of the tropical year is 365 days, 5 hr., 48 min., 45.5 sec., a difference of 11 min., 14.5 sec. by which the Julian year is too long. This amounts to a little more than 3 days in 400 years. To correct for this, those century years are made leap years which are divisible by 400 without remainder.

Standard Time. Prior to 1883 each city of the U. S. had its own time, which was determined by the time of passage of the sun across the local merid- ian. A system of standard time is used at present, according to which the United States, which extends from 65 deg. to 125 deg. West longitude, is divided into four sections, each of 15 deg. of longitude. The first or eastern section in- cludes all territory between the Atlantic coast and an irregular line drawn from Detroit, Mich., through Pittsburg to Charleston, S. C., its most southern point. The time of this section is that of the 75-deg. meridian, which is 5

84

WEIGHTS AND MEASURES

.„

hr. slower than Greenwich time. The second (central) section includes territory between the line mentioned, and an irregular line drawn from Bis- marck, N. D., to the mouth of the Rio Grande. The third (mountain) sec- tion includes all territory between the last-named line and a line which passes through the western part of Idaho, Utah and Arizona. The fourth (Pacific) section covers the rest of the country to the Pacific Ocean. Standard time is uniform in each of these sections, but the time in one section differs by ex- actly 1 hr. from the section next to it. In cities situated on the border line of two sections, as, say, Pittsburg and Atlanta, the standard times of both sec- tions are used, and in such cities when the time is given, it should be specified as eastern, central, etc. The system of standard time has been adopted in almost all civilized countries. All continental Europe, except Russia, uses a time 1 hr. faster than that of Greenwich; in Japan and Australia the time is 9 hr. faster.

TERRESTRIAL GRAVITY

By standard gravity is meant any locality where g0 = 980.665 cm. per sec. per sec., or 32.1740 ft. per sec. per sec. This value, QO, is assumed to be the value of g at sea level and latitude 45 deg.

Acceleration of Gravity

(U. S. Coast and Geodetic Survey, 1912)

Latitude,

£

r

Latitude,

/

deg.

Cm./sec.2

Ft./sec.2

0/ffo

deg.

Cm./sec.2

Ft./sec.2

Q/ffo

0 10 20 30 40

978.0 978.2 978.6 979.3 980.2

32.088 32.093 32. 08 32.130 32.158

0.9973 0.9975 0.9979 0.9986 0.9995

50 60 70 80 90

981.1 981.9 982.6 .983.1 983.2

32.187 32.215 32.238 32.253 32.258

.0004 .0013 .0020 0024 0026

Correction for altitude above sea level: — 0.3 cm. per sec.2 for each 1000 meters; - 0.003 ft. per sec.2 for each 1000 feet.

SPECIFIC GRAVITY AND DENSITY

The specific gravity of a solid or liquid is the ratio of the mass of the body to the mass of an equal volume of water at some standard temperature. At the present time a temperature of 4 deg. cent. (39 deg. fahr.) is commonly used by physicists, but the engineer uses 60 deg. fahr. The specific gravity of gases is usually expressed in terms of hydrogen or air.

The density of a body is its mass per unit volume. If the gram is used as the unit of mass and the milliliter as the unit of volume, the figures represent- ing the density are the same as the specific gravity of the body referred to water at 4 deg. cent, as unity. The customary unit is pounds per cu. ft.

The specific gravity of liquids is usually measured by means of an hydrom- eter (see p. 254).* Special arbitrary hydrometer scales are used in various trades and industries. The most common of these are the Baum6, Twaddell and Beck. Twaddell's hydrometer is used for liquids heavier than water. The number of degrees, N, which it indicates may be converted to specific gravities, G, by the formula G = (5N + 1000) /1000. The formula for the Beck hydrometer is G = 170/(170 ± N); for the Brix hydrometer G = 400/(400 ± N). In both of these the + sign is to be used for liquids lighter than water, the — sign for heavier liquids. For the salinometer (salometer), see p. 1734. * The specific gravities corresponding to the indications of the Baum6 hydrometer are given in the following tables. •See Marka' MECHANICAL ENGINEERS' HANDBOOK.

SPECIFIC GRAVITY AND DENSITY

85

60°

Specific Gravities at — Fahr. Corresponding to Degrees Baume* for Liquids Lighter than Water

60° Calculated from the formula, specific gravity —5 fahr. =

140

130 + Deg.

Degrees Baume

Specific gravity

Degrees Baume

II 11

02 M

||

$$ Qpq

Specific gravity

s*>

Si

aa

Specific gravity

Degrees Baumfi

Specific gravity

Degrees Baum6

Specific gravity

10

1.0000

25

0.9032

40

0.8235

55

0.7568

70

0.7000

85

0.6512

11

0.9929

26

0.8974

41

0.8187

56

0.7527

71

0.6965

86

0.6482

12

0.9859

27

0.8917

42

0.8140

57

0.7487

72

0.6931

87

0.6452

13

0.9790

28

0.8861

43

0.8092

58

0.7447

73

0.6897

88

0.6422

14

0.9722

29

0.8805

44

0 8046

59

0.7407

74

0.6863

89

0.6393

15

0.9655

30

0.8750

45

0.8000

60

0.7368

75

0.6829

90

0.6364

16

0.9589

31

0.8696

46

0.7955

61

0.7330

76

0.6796

91

0.6335

17

0.9524

32

0.8642

47

0.7910

62

0.7292

77

0.6763

92

0.6306

18

0.9459

33

0.8589

48

0.7865

63

0.7254

78

0.6731

93

0.6278

19

0.9396

34

0.8537

49

0.7821

64

0.7216

79

0.6699

94

0.6250

20

0.9333

" 35

0.8485

50

0.7778

65

0.7179

80

0.6667

95

0.6222

21

0.9272

36

0.8434

51

0.7735

66

0.7143

81

0.6635

96

0.6195

22

0.9211

37

0.8383

52

0.7692

67

0.7107

82

0.6604

97

0.6167

23

0.9150

38

0.8333

53

0.7650

68

0.7071

83

0.6573

98

0.6140

24

0.9091

39

0.8284

54

0.7609

69

0.7035

84

0.6542

99

0.6114

100

0.6087

60°

Specific Gravities at — Pahr. Corresponding to Degrees Baume

for Liquids Heavier than Water

Calculated from the formula, specific gravity fahr. =

Degrees Baum6

Specific gravity

Degrees Baume

Specific gravity

Degrees Baum6

° >>

11

02 00

ii

Specific gravity

Degrees Baume

1

Degrees Baume

Specific gravity

0

.0000

12

.0902

24

.1983

36

.3303

48

.4948

60

.7059

1

.0069

13

.0985

25

.2083

37

.3426

49

.5104

61

.7262

2

.0140

14

.1069

26

.2185

38

.3551

50

.5263

62

.7470

3

.0211

15

.1154

27

.2288

39

.3679

51

.5426

63

.7683

4

.0284

16

.1240

28

.2393

40

.3810

52

.5591

64

.7901

5

.0357

17

.1328

29

.2500

41

.3942

53

.5761

65

.8125

6

.0432

18

.1417

30

.2609

42

.4078

54

.5934

66

.8354

7

.0507

19

.1508

31

.2719

43

.4216

55

.6111

67

.8590

8

.0584

20

.1600

32

.2832

44

.4356

56

.6292

^68

.8831

9

.0662

21

.1694

33

.2946

45

.4500

57

.6477

69

.9079

10

.0741

22

.1789

34

.3063

46

.4646

58

.6667

70

.9333

11

.0821

23

.1885

35

.3182

47

.4796

59

.6860

Mohs's Scale of Hardness

1. Talc. 2. Gypsum. 3. Calc spar. 4. Fluorspar. 5. Apatite. 6. Feldspar. 7. Quartz. 8. Topaz. 9. Sapphire. 10. Diamond.

SECTION 2 MATHEMATICS

BY EDWARD V. HUNTINGTON, Ph. D.

ASSOCIATE PROFESSOR OF MATHEMATICS, HARVARD UNIVERSITY, FELLOW AM. ACAD. ARTS AND SCIENCES

CONTENTS

ARITHMETIC

PAGE

Numerical Computation 88

Logarithms 91

The Slide Rule 94

Computing Machines 97

Financial Arithmetic 98

GEOMETRY AND MENSURATION

Geometrical Theorems 99

Geometrical Constructions 101

Lengths and Areas of Plane Figures. 105

Surfaces and Volumes of Solids 107

ALGEBRA

Formal Algebra 112

Solution of Equations in One Un- known Quantity 116

Solution of Simultaneous Equations 119

Determinants 123

Imaginary or Complex Quantities 124

TRIGONOMETRY

Formal Trigonometry 128

Solution of Plane Triangles 132

Solution of Spherical Triangles 134

Hyperbolic Functions 135

ANALYTICAL GEOMETRY

The Point and the Straight Line. ... 136

The Circle 137

PAOE

The Parabola 138

The Ellipse 140

The Hyperbola 144

The Catenary 147

Other Useful Curves 151

DIFFERENTIAL AND INTEGRAL CALCULUS

Derivatives and Differentials 1 57

Maxima and Minima 159

Expansion in Series 160

Indeterminate Forms 163

Curvature 163

Table of Indefinite Integrals 164

Definite Integrals 169

Differential Equations 171

GRAPHICAL REPRESENTATION OF FUNCTIONS

173 174 176 177 178 182

Equations Involving Two Variables Equations for Empirical Curves . Logarithmic Cross-section Paper .

Semi-logarithmic Paper

Equations Involving Three Variables Equations Involving Four Variables

VECTOR ANALYSIS

Vector Analysis 185

COPYRIGHT, 1916, BY EDWARD V. HUNTINGTON

MATHEMATICS

BY EDWARD V. HUNTINGTON

ARITHMETIC

NUMERICAL COMPUTATION

Number of Significant Figures. In any engineering computation, the data are ordinarily the results of measurement, and are correct o'nly to a limited number of significant figures. Each of the numbers 3.840 and 0.003840 is said to be given "correct to four figures;" the true' value lies in the first case between 3.8395 and 3.8405; in the second case, between 0.0038395 and 0.0038405. The absolute error is less than 0.001 in the first case, and less than 0.000001 in the second; but the relative error is the same in both cases, namely, an error of less than "one part in 3840."

If a number is written as 384000, the reader is left in doubt whether the number of correct significant figures is 3, 4, 5, or 6. This doubt can be removed by writing the number as 3.84 X 10» or 3.840 X 10s or 3.8400 X 10* or 3.84000 X 105.

In any numerical computation, the possible or desirable degree of accuracy should be decided on and the computation should then be so arranged that the required number of significant figures, and no more, is secured. Carry- ing out the work to a larger number of places than is justified by the data, is to be avoided, (1) because the form of the results leads to an erroneous impres- sion of their accuracy, and (2) because time and labor are wasted in super- fluous computation. The labor of working with six-place tables is nearly three times as great as that with four-place tables. In computations involv- ing several steps, it is desirable to retain one extra figure until just before the final result is reached, in order to protect the last figure against the possible cumulative effect of small tabular errors. In discarding superfluous figures, if the first discarded figure is 5 or more, increase the preceding figure by 1. Thus, 3.14159, written 'correct to four figures, is 3.142; correct to three figures, 3.14. Again, 6.1297, correct to four figures, is 6.130.

Addition. In adding numbers, note that a doubtful final 0 . 2056x

figure in any one number will render doubtful the whole col- 2 . 572xx

umn in which that figure lies; hence all figures to the right of 14.25xxx

that column are superfluous, and contribute nothing to the 576.1xxxx

accuracy of the result.

Subtraction. The "Austrian" or "shop" method is 593.1

recommended. The mental process is as follows, the figures here printed in boldface type being the only ones written down:

[3 plus how many is 12?] 3 plus 9 is 12; 1 to carry. 14752

[7 plus how many is 15?] 7 plus 8 is 15; 1 to carry. ,J^5§,

5 plus 2 is 7. 8 plus 6 is 14. 6289

88

NUMERICAL COMPUTATION 89

This method is especially useful when it is desired to subtract from a given number the sum of several other numbers.

7 plus 1 is 8; plus 5 is 13; plus 9 is 22; 2 to carry. 14752

5 plus 0 is 5; plus 2 is 7; plus 8 is 15; 1 to carry. 3125~|

3 plus 1 is 4; plus 1 is 5; plus 2 is 7. 101

5 plus 3 is 8; plus 6 is 14. _5237-J

6289

The use of a wavy line to indicate subtraction is also recommended, as it will minimize the danger of adding when subtraction is intended.

Multiplication. In long examples in multiplication, 4956

the arrangement of work here illustrated is recommended, 8372

since it facilitates the abbreviation of the work by the 39648

omission, in practice, of all the figures on the right of the 1486 8

vertical line. 346 92

The position of the decimal point should be determined 9 912

by reference'to the first, or left-hand, figures of the numbers, 41492|xxx

rather than by "pointing off" so-and-so many places from the right-hand end. For the right-hand figures of a number are the least important ones, and in many cases are entirely unknown (especially when the slide rule or a computing machine is used). The mental process for determining the decimal point is as follows:

(a) If the multiplier is a number like 3.1416, with only one figure preceding the decimal point, think of this number as "a little over 3;" then the product must be "a little over three times the number which is being multiplied;" and this gives the position of the decimal point at once, by inspection.

(6) If the multiplier is a number like 3141.6 [or 0.000 003 141 6], think of this number as "about 3, with the point moved three places to the right" [or "about 3, with the point moved six places to the left"]; then think what the answer would be if the multiplier were simply "about 3," and shift the decimal point accordingly.

Multiplication Tables. Crelle's large volume (Berlin, G. Reimer) gives the product of every three-figure number by every three-figure number; Peters's (Berlin, G. Reimer), of every four-figure number by every two-figure number. The smaller table of H. Zimmermann (Berlin, Wm. Ernst) gives the product of every three-figure number by every two-figure number.

Division. In long division, where the numbers are given 23026)31416(1 only approximately, the work can be much abbreviated with- 23026

out loss of accuracy by "cutting off" one figure of the divisor 2303) 8390(3 at each step, instead of "bringing down" a doubtful zero in 6909

the dividend. Thus, 3.1416 4- 2.3026 = 1.3644. „

To determine the position of the decimal point in a du; problem of fractional division, shift the point (mentally) in both numerator and denominator (the same number of 23) 101(4 places in each) until the denominator is a number in the "standard form, " that is, a number with only one figure pre- 2) 9(4

ceding the decimal point. (This will not change the value of the fraction.) Then estimate the approximate magnitude of the quotient by inspection. Thus:

0.2718 0.000 2718

"about 0.000 09" = 0.000 08652;

3141.6 3.1416

31.416 31 416.

0.002718 2.718

"about 10 000" =11 558.

90 ARITHMETIC

Reciprocals. The reciprocal of N is 1 /N. Instead of dividing by a long number N, it is often better to multiply by the reciprocal of N. The table of reciprocals on pp. 24-27 gives the reciprocal of any number, correct to four figures. Barlow's Table (Spoil & Chamberlain, New York) gives the reciprocal of every four-figure number correct to seven figures (but with- out facilities for interpolation). The reciprocals of numbers having more than four figures are best found by the use of a large table of logarithms.

Reciprocals of I + x when x is Small. 1/(1 + x) = 1 - x + [error < x2, if x is between 0 and 1],

= 1 — x + xz — [error < x3, if x is between 0 and 1]. 1/(1 — x) = 1 + x + [error < x2 + 2z3, if x is between 0 and }*],

= 1 + x + x2 + [error < x3 + 2z4, if x is between 0 and #].

NOTE. l/(o ± 6) = (l/a)[l/(l ± «)], where x = 6/a.

Notation by Powers of 10. All questions concerning the position of the decimal point are readily answered if each number is expressed in the "stand- ard form," that is, as the product of two factors, one of which is a number with only one figure preceding the decimal point, while the other is a positive or negative power of 10. Thus, 3.1416 X 103 means 3.1416 with the point moved three places to the right, that is, 3141.6. Again, 3.1416 X 10~6 means 3.1416 with the point moved six places to the left, that is, 0.000 003 1416. This notation by powers of 10 should always be used in dealing with very large or very small numbers. Among electrical engineers its use is very general, even for numbers of moderate size.

Square Root, (a) If four figures of the root are sufficient, take the answer directly from the table of square roots, pp. 12-15. (6) To obtain a root of six or seven figures from the table, use the formula: VJV = a + [(N — ,a2)/2a] (approx.), where a is the nearest value of v~N obtainable from the table, with three or four ciphers annexed. Here a2 must be found exactly, by direct multiplication, so that at least three significant figures of the difference N — a2 shall be known correctly; but this done, the division of N — a2 by 2a should be carried to only three figures (logarithms or slide rule may be used).

NOTE. The simplest way to obtain any root of a seven-figure number correct to seven figures is to use a seven-place table of logarithms, if such a table is at hand.

Square Roots of 1 ± x when x is Small.

(1 + *)** = 1 + \hx - [error less than ftx2 if 0 < x < 1] = 1 + Kx - Hxz + [error < H« *3 if 0 < x < 1]

(1 - x)W = 1 - MX - [error < #ca+ Mo*3 if 0 < x < J4]

= 1 - )6x - ^z2 -[error < Ho*3 + Hex4 if 0 < x < ft]

NOTE. Va + b = Va (1 + x)^t where x = 6/a.

Cube Root, (a) If four figures of the root are sufficient, take the answer directly from the table of cube roots, pp. 16-21. (6) To obtain a root of six or seven figures from the table, use the formula: %/N = a + [(N — o3)/3a2] (approx.), where a is the nearest value of $/N obtainable from the table, with three or four ciphers annexed. Here a3 must be found correct to seven or eight figures, by direct multiplication, so that at least three significant figures of the difference N — a3 shall be known; but this done, the division of N — a8 by 3o2 should be carried to only three or four figures (logarithms or the slide rule may be used).

LOGARITHMS 91

NOTE. The simplest way to obtain any root of a seven-figure number correct to •seven figures is to use a seven-place table of logarithms, if such a table is at hand.

Cube Roots of 1+x when x is Small.

(1 + z)H = 1 + MX - [error < H*2 if 0 < a: < 1],

= 1 + %x - Kx* + [error < Me*3 if 0 < * < 1], (1 - x)% = 1 - %x - [error < %x* + Mo*3 if 0 < x < H],

= 1 - MX -Jte2 - [error < Me*3 + Ms*4 if 0 < x < ft]. NOTE. 3/a + 6 = ^/a(l + z)**, where x = b/a.

LOGARITHMS

Tables of Logarithms. The use of a table of logarithms greatly reduces the labor of multiplication, division, raising to powers, and extracting roots. The table on pp. 42-43 is carried out to four significant figures, and the follow- ing explanations should be sufficient to permit the use of the table readily, even by one without previous experience. For algebraic theory, see p. 113.

If more than four-figure accuracy is required, recourse must be had to a larger table. Five-place tables are available in great variety; the Macmillan Tables, 1913, are perhaps as convenient as any. If more than five figures are required, use Bremiker's six-place table, or proceed at once to a seven-place table: Schron (Vieweg und Sohn, Braun- schweig); Bruhns; Vega-Bremiker. If extreme accuracy is required, use the eight-place* table by Bauschinger and Peters (Engelmann, Leipzig). Logarithmic paper, see p. 176.

To Find the Logarithm of Any Given (Positive) Number.

(a) WHEN THE GIVEN NUMBER is BETWEEN 1 AND 10.

An inspection of the table on pp. 42-43 shows that as the number increases from 1 to 9.99. . . the logarithm of that number increases continuously from 0 to 0.999. . . For example, log 2.97 = 0.4728; log 2.98 = 0.4742.

If the given number contains four significant figures, it is necessary to inter- polate between the tabulated values, as follows:

To find log 2.973, notice that this number is fio of the way from 2.97 to 2.98; hence its logarithm will be (approximately) Mo of the way from 0.4728 to 0.4742. The difference here is 14 units, and iHo of this difference is 4 (to the nearest unit); hence, by adding this 4 to 4728, log 2.973 = 0.4732. This process of interpolating should be performed mentally; the step of finding the tabular difference will be facilitated by a glance at the last column on the right, which gives, for each line of the table, the average of the differences along that line.

Again, to find log 4.098: From table, log 4.09 = 0.6117; adding 9io of the difference (11), or about 9, gives: log 4.098 = 0.6126. Or better, since 91o °f the way forward is equal to Y\Q of the way back, find in table log 4.10 •= 0.6128, and subtract Y\Q of 11, or 2, giving log. 4.098 = 0.6126. It should be noted that any interpolated value may be in error by 1 in the last place.

If the given number contains more than four significant figures, it should be cut down to four figures (see p. 88), since the later figures will not affect the result in four-place computations.

(6) WHEN THE GIVEN NUMBER is LESS THAN 1 OR MORE THAN 10, it is simply necessary to notice that every such number can be regarded as obtainable from some number between 1 and 10 by merely shifting the decimal point (see p. 90) ; and that according to the rule at the foot of the table, moving the decimal point n places to the right [or left] in the number-column is equivalent to adding n [or — n] to the logarithm in the body of the table.

For example, to find log 2973. Here 2973 = 2.973 X 10« (i.e., 2.973 with the decimal point moved 3 places to the right). From the table, log 2.973 «=> 0.473i. Hence, log 2973 = 0.4732 + 3, which may be written as 3.4732.

92 ARITHMETIC

Again, to find log 0.0002973. Here 0.0002973 = 2.973 X 10~* (i.e., 2.973 with the decimal point moved 4 places to the left). From the table, log 2.973 = 0.4732. Hence, log 0.0002973 = 0.4732 - 4. (This may be written as 4.4732, if desired, and is equal of course, to _— 3.5268; this latter form, however, is not convenient in practice.)

It is thus evident that the logarithm of every positive number may be regarded as consisting of two parts: a decimal fraction, which is always posi- tive (or zero) ; and a whole number, which may be positive, negative, or zero. The fractional part is called the mantissa, and is found from the table ; the whole-number part is called the characteristic, and is determined by inspection.

To Find the Number Corresponding to a Given Logarithm.

(a) WHEN THE GIVEN LOGARITHM is A POSITIVE DECIMAL FRACTION (CHARAC- TERISTIC ZERO), simply reverse the process for finding the logarithm of a number between 1 and 10.

For example, given log N = 0.4732; to find N. In the body of the table it is seen that 0.4732 lies a little beyond 0.4728; hence N must lie a little beyond 2.97. By taking differences it is found that 4728 is in fact #4 of the way from 0.4728 to the next higher logarithm; therefore N must be y\± of the way from 2.97 to the next higher number. But YU of 1 is 0.3 (to the nearest tenth), hence N = 2.973.

Again, given log N = 0.6126; to find N. Here, 0.6126 is %i of the way from 0.6117 to the next higher logarithm; therefore N must be JH \ of the way from 4.09 to the next higher number. But %\ of 1 is 0.8 (to the nearest tenth), hence N = 4.098.

(6) WHEN THE GIVEN LOGARITHM HAS ANY GIVEN VALUE (CHARACTERISTIC NOT ZERO), proceed as follows: First, be sure the given logarithm is in the "standard form," that is, a positive decimal fraction (mantissa) plus a posi- tive or negative whole number (characteristic). For example, if log N is originally given in the form log N = — 3.5268, tfiis must first be reduced to the (equivalent) form log N = 0.4732 — 4 (or 4.4732), before entering the table. Having the logarithm given in the standard form, suppose for the moment that the characteristic is zero, and find in the table the number corresponding to the given mantissa; then move the decimal point to the right or left according as the value of the characteristic is positive or negative.

For example, given log N = 0.4732 + 3; to find N. From the table, the number corresponding to 0.4732 is 2.973. The characteristic ( + 3) directs that the decimal point be moved 3 places to the right; hence N = 2.973 X 103 = 2973.

Again, given log N = 0.4732 — 4; to find N. From the table, the number corre- sponding to 0.4732 is 2.973. The characteristic ( - 4) indicates that the decimal point is to be moved 4 places to the left; hence N = 2.973 X 10~« = 0.0002973.

The number corresponding to a given logarithm is called its antiloga- rithm. Thus, if log 2973 = 0.4732 + 3, then 2973 = antilog (0.4732 + 3).

NOTE 1. In most tables of logarithms the decimal point is omitted, the tables being in fact not tables of logarithms, but tables of mantissas. This omission is of no con- sequence to the experienced computer, but is often perplexing to one who makes only occasional use of such tables.

NOTE 2. Many computers prefer to write negative characteristics in the form of some positive number minus some multiple of 10; thus, 0.4732 — 4 = 6.4732 —10; 0.4732 - 13 = 7.4732 - 20; etc.

Fundamental Properties of Logarithms. The usefulness of logarithms in computation depends on the following properties:

(1) log (a&) = log a + log 6; (3) log (a71) = n log a;

(2) log (a/6) = log a - log 6; (4) log \/a = (1/n) log a;

(5) log 10n = n It is to be noted also that log 1 = 0, log 10 =1, and log (1/n) = —log n.

LOGARITHMS 93

To Multiply by Logarithms. Find from the table the log. of each factor, and add; the result will be the log. of the product. Then find the product itself from the table.

EXAMPLE". To find log 4.098 = 0.6126

x - (4.098) (0.0002973) (72.1). log 0.0002973 - 0.4732 - 4

Answer: x = 8.784 X lO"' l°g 72.1 - 0.8579 -f 1

= 0.08784 log x = 1.9437 - 3 - 0.9437 - 2.

To Divide by Logarithms. First Method: Find from the table the log. of the numerator and the log. of the denominator, and subtract the second from the first; the result will be the logarithm of the quotient. Then find the quotient itself from the table.

4.098 log 4.098 - 0.6126

EXAMPLE. To find x = ^^ ^ ^^ = ^732^

Answer: x - 1.378 X 10« = 13780 log x - 0.1394 -f 4

In order to avoid negative mantissas in cases where a larger mantissa would have to be subtracted from a smaller, modify the upper logarithm by adding and subtracting 1.

0.0291 log 0.0291 - 0.4639 - 2 = 1.4639 - 3

EXAMPLE. To find x = • _ _

Answer: x = 4.590 X 10~« log 3 "*"- 06618^4

= 0.0004590.

But if the logarithms are written with the characteristics in front, and the "shop method" of subtraction is used (see p. ]Og Q 0291 «= 24639

88), then no such special device is here jog §3 4 • ,_, j g021

required. Thus: T~

log x = 4.6618

To Divide by Logarithms. Second Method: Instead of subtracting the log. of a number, it is often convenient to add the cologarithm of that number; the colog. of N being defined by: colog N = log (l/N) = —log N.

To find the colog. of a number, write the log. of the number in the stand- ard form, and subtract it from 1.0000 — 1, as in the following examples:

1.0000 - 1 1.0000 - 1

log 69.5 = 0.8420 + 1 log 0.0002973 = £.4732_-j4

colog 69.5 = 0.1580 - 2 colog 0.0002973 = 0.5268 + 3

This subtraction should be performed mentally. Thus, to subtract the mantissa, subtract each digit from 9 until the last non-zero digit is arrived at, and subtract this from 10; to subtract the characteristic, follow the regular rule of algebra ("reverse the sign and add"). Hence, if the logarithm itself is already written down, or can be read off from the table without interpolation, the cologarithm can be written down at once, by inspection. The use of cologarithms is not essential in logarithmic computation, but it often facilitates a compact arrangement of the work, especially in cases where the denominator of a fraction is itself the product of two or more factors.

To Find the nth Power of a Number by Logarithms. Find from the table the log. of the number, and multiply it by n; the result will be the logarithm of the nth power of that number. Then find the power itself from the tables.

EXAMPLE 1. Find x = (0.0291)s log 0.0291 = 0.4639 - 2

Answer: x = 2.464 X 10~« 3

= 0.00002464. log x = 1.3917 - 6 - 0.3917 - 5.

94 ARITHMETIC

EXAMPLE 2. Find x = (0.0291)i'«i log 0.0291 = 0.4639 - 2 = - 1.5361

Answer: x = 6.825 X 10~8 1.41

= 0.006825 15361~

61444

15361

logs = - 2.1659 = 0.8341 - 3

To Find the nth Root of a Number by Logarithms. Find from the table the log. of the number, and divide it by n; the result will be the log. of the nth root of that number. Then find the root itself from the table.

EXAMPLE. Find x = ^/4.098 log 4.098 = 0.6126

Answer: x = 1.600 log x = 0.2042

In order to avoid fractional characteristics, if the characteristic is not divisible by n, make it so divisible by adding and subtracting a suitable number before dividing.

EXAMPLE. Find x = VO-0004590. log 0.0004590 = 0.6618 - '

Answer: x = 7.714 X 10-2 3)2.6618 - 6

= 0.07714 log x = 0^8873 - 2

But if the characteristic is positive, it is simpler to write it in front of the mantissa, and then divide directly.

THE SLIDE RULE

The slide rule is an indispensable aid in all problems in multiplication, division, proportion, squares, square roots, etc., in which a limited degree of