Number 1 to 100, and their Logarithms.

1

.0000000

21

.3222193

41

.6127839

61

.7853298

81

.9084850

2

.3010300

22

.3424227

42

.6232493

62

.7923917

82

.9138139

3

.4771213

23

.3617278

43

.6334685

63

.7993405

83

.9190781

4

.6020600

24

.3802112

44

.6434527

64

.8061800

84

.9242793

5

.6989700

25

..979400

45

.6532125

65

.8129134

85

.9294189

6

.7781513

26

.4149733

46

.6627578

66

.8195439

86

.9344985

7

.8450980

27

.4313638

47

.6720979

67

.8260748

87

.9395193

8

.9030900

28

.4471580

48

.6812412

68

.8325089

88

.9444827

9

.9542425

29

.4623980

49

.6901961

69

.8388491

89

.9493900

10

.0000000

30

.4771213

50

.6989700

70

.8450980

90

.9542425

11

.0413927

31

.4913617

51

.7075702

71

.8512583

. 91

.9590414

12

.0791812

32

.5051500

52

.7160033

72

.8573325

92

.9637878

13

.1139434

33

.5185139

53

.7242759

73

.8633229

93

.9684829

14

.1461280

34

.5314789

54

.7323938

74

.8692317

94

.9731279

15

.1760913

35

.5440680

55

.7103627

75

.8750613

95

.9777236

16

.2041200

36

.5563025

56

.7481880

76

.8808136

96

.9822712

17

.2304489

37

.5682017

57

.7558749

77

.8864907

97

.9867717

18

.2552725

38

.5797836

58

.7634280

78

.8920946

98

.9912261

l9

.2787536

39

.5910646

59

.7708520

79

.8976271

99

.9956352

20

.3010300

40

.6020600

60

.7781513

80

.9030900

100

.0000000

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

100

0000000

150

1760913

200

3010300

250

3979400

300

4771213

101

0043214

151

1789769

201

3031961

251

3996737

301

4785665

102

0086002

152

1818436

202

3053514

252

4014005

302

4800069

103

0128372

153

1846914

203

3074960

253

4031205

303

4814426

104

0170333

154

1875207

204

3096302

254

4048337

304

4828736

105

0211893

155

1903317

205

3117539

255

4065402

305

4842998

106

0253059

156

1931246

206

3138672

256

4082400

306

4857214

107

0293838

157

1958997

207

3159703

257

4099331

307

4871384

108

0334238

158

1986571

208

3180633

258

4116197

308

4885507

109

0374265

159

2013971

209

3201463

259

4132998

309

4899585

110

0413927

160

2041200

210

3222193

260

4149733

310

4913617

111

0453230

161

2068259

211

3242825

261

4166405

311

4927604

112

0492180

162

2095150

212

3263359

262

4183013

312

4941546

113

0530784

163

2121876

213

3283796

263

4199557

313

4955443

114

0569049

164

2148438

214

3304138

264

4216039

314

4969296

115

0606978

165

2174839

215

3324385

265

4232459

315

4983106

116

0644580

166

2201081

216

3344538

266

4248816

316

4996871

117

0681859

167

2227165

217

3364597

267

4265113

317

5010593

118

0718820

168

2253093

218

3384565

268

4281348

318

5024271

119

0755470

169

2278867

219

3404441

269

4297523

319

5037907

120

0791812

170

2304489

220

3424227

270

4313638

320

5051500

121

0827854

171

2329961

221

3443923

271

4329693

321

5065050

122

0863598

172

2355284

222

3463530

272

4345689

322

5078559

123

0899051

173

2380461

223

3483049

273

4361626

323

5092025

124

0934217

174

2405492

224

3502480

274

4377506

324

5105450

125

0969100

175

2430380

225

3521825

275

4393327

325

5118834

126

1003705

176

2455127

226

3541084

276

4409091

326

5132176

127

1038037

177

2479733

227

3560259

277

4424798

327

5145478

128

1072100

178

2504200

228

3579348

278

4440448

328

5158738

129

1105897

179

2528530

229

3598355

279

4456042

329

5171959

130

1139434

180

2552725

230

3617278

280

4471580

330

5185139

131

1172713

181

2576786

231

3636120

281

4487063

331

5198280

132

1205739

182

2600714

232

3654880

282

4502491

332

5211381

133

1238516

183

2624511

233

3673559

283

4517864

333

5224442

134

1271048

184

2648178

234

3692159

284

4533183

334

5237465

135

1303338

185

2671717

235

3710679

285

4548449

335

5250448

136

1335389

186

2695129

236

3729120

286

4563660

336

5263393

137

1367206

187

2718416

237

3747483

287

4578819

337

5276299

138

1398791

188

2741578

238

3765770

288

4593925

338

5289167

139

1430148

189

2764618

239

3783979

289

4608978

339

5301997

140

1461280

190

2787536

240

3802112

290

4623980

340

5314789

141

1492191

191

2810334

241

3820170

291

4638930

341

5327544

142

1522883

192

2833012

242

3838154

292

4653829

342

5340261

143

1553360

193

2855573

243

3856063

293

4668676

343

5352941

144

1583625

194

2878017

244

3873898

294

4683473

344

5365584

145

1613680

195

2900346

245

3891661

295

4698220

345

5378191

146

1643529

196

2922561

246

3909351

296

4712917

346

5390761

147

1673173

197

2944662

247

3926970

297

4727564

347

5403295

148

1702617

198

2966652

248

3944517

298

4742163

348

5415792

149

1731863

199

2988531

249

3961993

299

4756712

349

5428254

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

NUM.

LOG.

350

5440680

400

6020600

450

6532125

500

6989700

550

7403627

351

5453071

401

6031444

451

6541765

501

6998377

551

7411516

352

5465427

402

6042261

452

6551384

502

7007037

552

7419391

353

5477747

403

6053050

453

6560982

503

7015680

553

7427251

354

5490033

404

6063814

454

6570559

504

7024305

554

7435098

355

5502284

405

6074550

455

6580114

505

7032914

555

7442930

356

5514500

406

6085260

456

6589648

506

7041505

556

7450748

357

5526682

407

6095944

457

6599162

507

7050080

557

7458552

358

5538830

408

6106602

458

6608655

508

7058637

558

7466342

359

5550944

409

6117233

459

6618127

509

7067178

559

7474118

360

5563025

410

6127839

460

6627578

510

7075702

560

7481880

361

5575072

411

6138418

461

6637009

511

7084209

561

7489629

362

5587086

412

6148972

462

6646420

512

7092700

562

7497363

363

5599066

413

6159501

463

6655810

513

7101174

563

7505084

364

5611014

414

6170003

464

6665180

514

7109631

564

7512791

365

5622929

415

6180481

465

6674530

515

7118072

565

7520484

366

5634811

416

6190933

466

6683859

516

7126497

566

7528164

367

5646661

417

6201361

467

6693169

517

7134905

567

7535831

368

5658478

418

6211763

468

6702459

518

7143298

568

7543483

369

5670264

419

6222140

469

6711728

519

7151674

569

7551123

370

5682017

420

6232493

470

6720979

520

7160033

570

7558749

371

5693739

421

6242821

471

6730209

521

7168377

571

7566361

372

5705429

422

6253125

472

6739428

522

7176705

572

7573960

373

5717088

423

6263404

473

6748611

523

7185017

573

7581546

374

5728716

424

6273659

474

6757783

524

7193313

574

7589119

375

5740313

425

6283889

475

6766936

525

7201593

575

7596678

376

5751878

426

6294096

476

6776070

526

7209857

576

7604225

377

5763414

427

6304279

477

6785184

527

7218106

577

7611758

378

5774918

428

6314438

478

6794279

528

7226339

578

7619278

379

5786392

429

6324573

479

6803355

529

7234557

579

7626786

380

5797836

430

6334685

480

6812412

530

7242759

580

7634280

381

5809250

431

6344773

481

6821451

531

7250945

581

7641761

382

5820634

432

6354837

482

6830470

532

7259116

582

7649230

383

5831988

433

6364879

483

6839471

533

7267272

583

7656686

384

5843312

434

6374897

484

6848454

534

7275413

584

7664128

385

5854607

435

6384893

485

6857417

535

7283538

585

7671559

386

5865873

436

6394865

486

6866363

536

7291648

586

7678976

387

5877110

437

6404814

487

6875290

537

7299743

587

7686381

388

5888317

438

6414741

488

6884198

538

7307823

588

7693773

389

5899496

439

6424645

489

6893089

539

7315888

589

7701153

390

5910646

440

6434527

490

6901961

540

7323938

590

7708520

391

5921768

441

6444386

491

6910815

541

7331973

591

7715875

392

5932861

442

6454223

492

6919651

542

7339993

592

7723217

393

5943926

443

6464037

493

6928469

543

7347998

593

7730547

394

5954962

444

6473830

494

6937269

544

7355989

594

7737864

395

5965971

'445

6483600

495

6946052

545

7363965

595

7745170

396

5976952

446

6493349

496

6954817

546

7371926

596

7752463

397

5987905

447

6503075

497

6963564

547

7379873

597

7759743

398

5998831

448

6512780

498

6972293

548

7387806

598

7767012

399

6009729

449

6522463

499

6981005

549

7395723

599

7774268

NUM.

LOO.

NUM.

LOO.

NUM.

LOO.

NUM.

LOO.

NUM.

LOG.

600

7781513

650

8129134

700

8450980

750

8750613

800

9030900

601

7788745

651

8135810

701

8457180

751

8756399

801

9036325

602

7795965

652

8142476

702

8463371

752

8762178

802

9041744

603

7803173

653

8149132

703

8469553

753

8767950

803

9047155

604

7810369

654

8155777

704

8475727

754

8773713

804

9052560

605

7817554

655

8162413

705

8481891

755

8779470

805

9057959

606

7824726

656

8169038

706

8488047

756

8785218

806

9063350

607

7831887

657

8175654

707

8494194

757

8790959

807

9068735

608

7839036

658

8182259

708

8500333

758

8796692

808

9074114

609

7846173

659

8188854

709

8506462

759

8802418

809

9079485

610

7853298

660

8195439

710

8512583

760

8808136

810

9084850

611

7860412

661

8202015

711

8518696

761

8813847

811

9090209

612

7867514

662

8208580

712

8524800

762

8819550

812

9095560

613

7874605

663

8215135

713

8530895

763

8825245

813

9100905

614

7881684

664

8221681

714

8536982

764

8830934

814

9106244

615

7888751

665

8228216

715

8543060

765

8836614

815

9111576

616

7895807

666

8234742

716

8549130

766

8842288

816

9116902

617

7902852

667

8241258

717

8555192

767

8847954

817

9122221

618

7909885

668

8247765

718

8561244

768

8853612

818

9127533

619

7916906

669

8254261

719

8567289

769

8859263

819

9132839

620

7923917

670

8260748

720

8573325

770

8864907

820

9138139

621

7930916

671

8267225

721

8579353

771

8870544

821

9143432

622

7937904

672

8273693

722

8585372

772

8876173

822

9148718

623

7944880

673

8280151

723

8591383

773

8881795

823

9153998

624

7951846

674

8286599

724

8597386

774

8887410

824

9159272

625

7958800

675

8293038

725

8603380

775

8893017

825

9164539

626

7965743

676

8299467

726

8609366

776

8898617

826

9169800

627

7972675

677

8305887

727

8615344

777

8904210

827

9175055

628

7979596

678

8312297

728

8621314

778

8909796

828

9180303

629

7986506

679

8318698

729

8627275

779

8915375

829

9185545

630

7993405

680

8325089

730

8633229

780

8920946

830

9190781

631

8000294

681

8331471

731

8639174

781

8926510

831

9196010

632

8007171

682

8337844

732

8645111

782

8932068

832

9201233

633

8014037

683

8344207

733

8651040

783

8937618

833

9206450

634

8020893

684

8350561

734

8656961

784

8943161

834

9211661

635

8027737

685

8356906

735

8662873

785

8948697

835

9216865

636

8034571

686

8363241

736

8668778

786

8954225

836

9222063

637

8041394

687

8369567

737

8674675

787

8959747

837

9227255

638

8048207

688

8375884

738

8680564

788

8965262

838

9232440

639

8055009

689

8382192

739

8686444

789

8970770

839

9237620

640

8061800

690

8388491

740

8692317

790

8976271

840

9242793

641

8068580

691

8394780

741

8698182

791

8981765

841

9247960

642

8075350

692

8401061

742

8704039

792

8987252

842

9253121

643

8082110

693

8407332

743

8709888

793

8992732

843

9258276

644

8088859

694

8413595

744

8715729

794

8998205

844

9263424

645

8095597

695

8419848

745

8721563

795

9003671

845

9268567

646

8102325

696

8426092

746

8727388

796

9009131

846

9273704

647

8109043

697

8432328

747

8733206

797

9014583

847

9278834

648

8115750

698

8438554

748

8739016

798

9020029

848

9283959

649

8122447

699

8444772

749

8744818

799

9025468

849

9289077

NUM

LOG.

NUM.

LOG.

NUM.

LOG.

NUM

LOG.

NUM.

LOG.

850

9294189

880

9444827

910

9590414

940

9731279

970

9867717

851

9299296

881

9449759

911

9595184

941

9735896

971

9872192

852

9304396

882

9454686

912

9599948

942

9740509

972

9876663

853

9309490

883

9459607

913

9604708

943

9745117

973

9881128

854

9314579

884

9464523

914

9609462

944

9749720

974

9885590

855

9319661

885

9469433

915

9614211

945

9754318

975

9890046

856

9324738

886

9474337

916

9618955

946

9758911

976

9894498

857

9329808

887

9479236

917

9623693

947

9763500

977

9898946

858

9334873

888

9484130

918

9628427

948

9768083

978

9903389

859

9339932

889

9489018

919

9633155

949

9772662

979

9907827

860

9344985

890

9493900

920

9637878

950

9777236

980

9912261

861

9350032

891

9498777

921

9642596

951

9781805

981

9916690

862

9355073

892

9503649

922

9647309

952

9786369

982

9921115

863

9360108

893

9508515

923

9652017

953

9790929

983

9925535

864

9365137

894

9513375

924

9656720

954

9795484

984

9929951

865

9370161

895

9518230

925

9661417

955

9800034

985

9934362

866

9375179

896

9523080

926

9666110

956

9804579

986

9938769

867

9380191

897

9527924

927

9670797

957

9809119

987

9943172

868

9385197

898

9532763

928

9675480

958

9813655

988

9947569

869

9390198

899

9537597

929

9680157

959

9818186

989

9951963

870

9395193

900

9542425

930

9684829

960

9822712 |

990

9956352

871

9400182

901

9547248

931

9689497

961

9827234

991

9960737

872

9405165

902

9552065

932

9694159

962

9831751

992

9965117

873

9410142

903

9556878

933

9698816

963

9836263

993

9969492

874

9415114

904

9561684

934

9703469

964

9840770

994

9973864

875

9420081

905

9566486

935

9708116

965

9845273

995

9978231

876

9425041

906

9571282

936

9712758

966

9849771

996

9982593

877

9429996

907

9576073

937

9717396

967

9854265

997

9986952

878

9434945

908

9580858

938

9722028

968

9858754

998

9991305

879

9439889

909

9585639

939

9726656

969

9863238

999

9995655

It is not our province, in this brief article, to explain the use of the larger Logarithmic Tables, as whoever possess such have of course their author's own explanations, and therefore the following illustrative examples are selected to suit the table here given.

Multiplication, as already stated, is performed by the addition of Logarithms, thus:-

To multiply 368 by 22.5, we place opposite to each other the

Numbers and their

Logarithms.

368 .......

2.5658478

Add.

22.5 .......

1.3521825

Product of Numbers

8280. .......

3.9180303

Sum of Log.

Here the first factor, 368, being a whole number, consisting of three figures, has for its index 2; and the second, 22.5, having but two figures, without the decimal part, has for its index 1. To these are subjoined the decimal portions of the logarithms taken from the Table, and the sum of the two being found in the Table opposite to 828, which would be the answer were the index 2; but as the index is 3, the answer must be made to consist of four figures, which is done by supplying to the right of the figures a cipher, making the answer, as above, 8280

Required the capacity of an excavation, whose length is 295, breadth 128, and depth 25 feet.

NUMBERS.

LOGARITHMS.

295

............................

2.4698220

128

............................

21072100

25

.............................

1.3979400

Product.

944000

............................

5.9749720 Sum.

Again, let the numbers 32. 25, 1.12, .125, .015, and 004 be continually multiplied together.

NUMBERS.

............................

LOGARITHMS.

3.2

............................

0. 5051500

25.

............................

1. 3979400

1.12

............................

0. 0492180

125

............................

1. 0969100

015

............................

2. 1760913

.004

............................

3. 6020600

Product

.000672

............................

4. 8273693

The number in the table corresponding with the decimal part of the sum of these logarithms is 672, but as the index is 4 there must be three cyphers prefixed to this number to constitute the product or answer which is therefore •000672.

Division, being the reverse of Multiplication, and performed by subtraction of logarithms, requires but little explanation.

For illustration, let 944000 be divided by 3200, thus:-

NUMBERS.

LOGARITHMS.

Dividend

944000

...........................

5.9749720

From

Divisor. .

3200

...........................

3.5051500

Substract.

Quotient

295

...........................

2.4698220

Difference.

Again, let .00815 be divided by .0025.

NUMBERS.

LOGARITHMS.

.00815

...........................

3.9111576

.0025

...........................

2.3979400

Quotient.

326

...........................

1.5132176

Let 493 be divided by 937.

NUMBERS.

LOGARITHMS.

Dividend.

493

..........................

2.6928469

Divisor. .

937

..........................

2.9717396

Quotient.

.526

..........................

1.7211073

Here the logarithm to be subtracted being the greater of the two, the index of the difference is 1, which renders the quotient a decimal.

Involution, or the raising of powers, is performed by multiplying the logarithm of the given number by the index of the power to which it is required to be raised, thus:-

Let 26 be squared, or raised to the second power.

NUMBERS.

LOGARITHMS.

26

..........................

1.4149733

.2

Power . .

. . 676

..........................

2.8299466

Product.

Required the cube root, or third power of 9.

NUMBERS.

LOGARITHMS.

9

..........................

0.9542425

..........................

3

Power. .

. 729

..........................

2 8627275

Required the 9th power of 1.05, which will be the amount of 1l in nine years, at 5 per cent. compound interest.

NUMBERS.

LOGARITHMS.

1.05

..........................

0.0211893

9

Amount..

1.55 or 1l 11s.

..........................

01907037

From this example it is manifest that the amount of money laid out at compound interest for 50, 100, or any other number of years, can be found by logarithms with the greatest facility, though the operation by common arithmetic is very tedious, requiring a distinct multiplication for each year.

Evolution, or the extraction of roots, is performed by dividing the logarithm of the given number by the index of the root required. Let this be illustrated by finding the square root of 324.

NUMBERS.

LOGARITHMS.

324

...............................

2)2.5105450

Root. .

. 18

...............................

1.2552725

Required the ninth root of 1.55

NUMBERS.

LOGARITHMS.

1.55

..............................

9)0.1903317

Root. .

. 1.05

..............................

00211479

As it may occasionally be desirable to apply the foregoing table to numbers beyond its limits, the manner of doing so is subjoined.

To find the logarithm of a number exceeding three figures, it is evident that the logarithm of the first three must be augmented by such a proportion of the difference between it and the next greater logarithm, as the remaining figures of the given number bears to unity with as many cyphers as may be required ; thus - to find the logarithm of 47583, the logarithm of the first three figures to the first logarithm, gives 4.6774516 for the logarithm of 47583. On the contrary, if the number be required for a logarithm not to be found in the table, to the first three figures corresponding with the next less logarithm, are to be subjoined the result of the following proportion; viz. - As the difference between the next greater and next less logarithm is to unity, with as many cyphers as may be required, so is the difference between the given logarithm and the next less to the figures to be subjoined to those found in the table. Thus - Suppose it were required to find the natural number corresponding with 4.6968455

475

is

6766936,

and the next greater is-

6776070

9134

their difference.

Now take the proportion as 1 : .9134 : : 83 : 75

83

27302

73072

1,0000)75.8022, or 7498 nearly which being added

The next less logarithm in the table is ................

6963564

The next greater ........................................................

6972293

Difference . .

8729

The gibem logarithm ....................

.............................

6968455

Next lees ......................................

...........................

6963564

Difference . . .

4891

Now - As

8729 :

100 : :

4891 :

56

100

8729)

489100

(56

43645

52650

52374

Which being subjoined to 497, the three figures found in the table opposite to the next less logarithm, give 49756 for the number of the given logarithm.