Chapter XIV: Units and Standards of Measurement

1. Units and Standards of Measurement 305

2. Standard Unit of Time 307

3. The Unit of Space and the Bar Standard 312

4. The Terrestrial Standard 314

5. The Pendulum Standard 315

6. Unit of Density 316

7. Unit of Mass 317

8. Natural System of Standards 319

9. Subsidiary Units 320

10. Derived Units 321

11. Provisional Units 323

12. Theory of Dimensions 325

13. Natural Constants 328

14. Mathematical Constants 330

15. Physical Constants 331

16. Astronomical Constants 332

17. Terrestrial Numbers 333

18. Organic Numbers 333

19. Social Numbers 334

CHAPTER XV.

ANALYSIS OF QUANTITATIVE PHENOMENA.

1. Analysis of Quantitative Phenomena 335

2. Illustrations of the Complication of Effects 336

3. Methods of Eliminating Error 339

4. Method of Avoidance of Error 340

5. Differential Method 344

6. Method of Correction 346

7. Method of Compensation 350

8. Method of Reversal 354

CHAPTER XVI.

THE METHOD OF MEANS.

1. The Method of Means 357

2. Several Uses of the Mean Result 359

3. The Mean and the Average 360

4. On the Average or Fictitious Mean 363

5. The Precise Mean Result 365

6. Determination of the Zero Point 368

7. Determination of Maximum Points 371

CHAPTER XVII.

THE LAW OF ERROR.

1. The Law of Error 374

2. Establishment of the Law of Error 375

3. Herschel’s Geometrical Proof 377

4. Laplace’s and Quetelet’s Proof of the Law 378

5. Logical Origin of the Law of Error 383

6. Verification of the Law of Error 383

7. The Probable Mean Result 385

8. The Probable Error of Results 386

9. Rejection of the Mean Result 389

10. Method of Least Squares 393

11. Works upon the Theory of Probability 394

12. Detection of Constant Errors 396

BOOK IV.

INDUCTIVE INVESTIGATION.

CHAPTER XVIII.

OBSERVATION.

1. Observation 399

2. Distinction of Observation and Experiment 400

3. Mental Conditions of Correct Observation 402

4. Instrumental and Sensual Conditions of Correct Observation 404

5. External Conditions of Correct Observation 407

6. Apparent Sequence of Events 409

7. Negative Arguments from Non-Observation 411

CHAPTER XIX.

EXPERIMENT.

1. Experiment 416

2. Exclusion of Indifferent Circumstances 419

3. Simplification of Experiments 422

4. Failure in the Simplification of Experiments 424

5. Removal of Usual Conditions 426

6. Interference of Unsuspected Conditions 428

7. Blind or Test Experiments 433

8. Negative Results of Experiment 434

9. Limits of Experiment 437

CHAPTER XX.

METHOD OF VARIATIONS.

1. Method of Variations 439

2. The Variable and the Variant 440

3. Measurement of the Variable 441

4. Maintenance of Similar Conditions 443

5. Collective Experiments 445

6. Periodic Variations 447

7. Combined Periodic Changes 450

8. Principle of Forced Vibrations 451

9. Integrated Variations 452

CHAPTER XXI.

THEORY OF APPROXIMATION.

1. Theory of Approximation 456

2. Substitution of Simple Hypotheses 458

3. Approximation to Exact Laws 462

4. Successive Approximations to Natural Conditions 465

5. Discovery of Hypothetically Simple Laws 470

6. Mathematical Principles of Approximation 471

7. Approximate Independence of Small Effects 475

8. Four Meanings of Equality 479

9. Arithmetic of Approximate Quantities 481

CHAPTER XXII.

QUANTITATIVE INDUCTION.

1. Quantitative Induction 483

2. Probable Connexion of Varying Quantities 484

3. Empirical Mathematical Laws 487

4. Discovery of Rational Formulæ 489

5. The Graphical Method 492

6. Interpolation and Extrapolation 495

7. Illustrations of Empirical Quantitative Laws 499

8. Simple Proportional Variation 501

CHAPTER XXIII.

THE USE OF HYPOTHESIS.

1. The Use of Hypothesis 504

2. Requisites of a good Hypothesis 510

3. Possibility of Deductive Reasoning 511

4. Consistency with the Laws of Nature 514

5. Conformity with Facts 516

6. Experimentum Crucis 518

7. Descriptive Hypotheses 522

CHAPTER XXIV.

EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION.

1. Empirical Knowledge, Explanation and Prediction 525

2. Empirical Knowledge 526

3. Accidental Discovery 529

4. Empirical Observations subsequently Explained 532

5. Overlooked Results of Theory 534

6. Predicted Discoveries 536

7. Predictions in the Science of Light 538

8. Predictions from the Theory of Undulations 540

9. Prediction in other Sciences 542

10. Prediction by Inversion of Cause and Effect 545

11. Facts known only by Theory 547

CHAPTER XXV.

ACCORDANCE OF QUANTITATIVE THEORIES.

1. Accordance of Quantitative Theories 551

2. Empirical Measurements 552

3. Quantities indicated by Theory, but Empirically Measured 553

4. Explained Results of Measurement 554

5. Quantities determined by Theory and verified by
Measurement 555

6. Quantities determined by Theory and not verified 556

7. Discordance of Theory and Experiment 558

8. Accordance of Measurements of Astronomical Distances 560

9. Selection of the best Mode of Measurement 563

10. Agreement of Distinct Modes of Measurement 564

11. Residual Phenomena 569

CHAPTER XXVI.

CHARACTER OF THE EXPERIMENTALIST.

1. Character of the Experimentalist 574

2. Error of the Baconian Method 576

3. Freedom of Theorising 577

4. The Newtonian Method, the True Organum 581

5. Candour and Courage of the Philosophic Mind 586

6. The Philosophic Character of Faraday 587

7. Reservation of Judgment 592

BOOK V.

GENERALISATION, ANALOGY, AND CLASSIFICATION.

CHAPTER XXVII.

GENERALISATION.

1. Generalisation 594

2. Distinction of Generalisation and Analogy 596

3. Two Meanings of Generalisation 597

4. Value of Generalisation 599

5. Comparative Generality of Properties 600

6. Uniform Properties of all Matter 603

7. Variable Properties of Matter 606

8. Extreme Instances of Properties 607

9. The Detection of Continuity 610

10. The Law of Continuity 615

11. Failure of the Law of Continuity 619

12. Negative Arguments on the Principle of Continuity 621

13. Tendency to Hasty Generalisation 623

CHAPTER XXVIII.

ANALOGY.

1. Analogy 627

2. Analogy as a Guide in Discovery 629

3. Analogy in the Mathematical Sciences 631

4. Analogy in the Theory of Undulations 635

5. Analogy in Astronomy 638

6. Failures of Analogy 641

CHAPTER XXIX.

EXCEPTIONAL PHENOMENA.

1. Exceptional Phenomena 644

2. Imaginary or False Exceptions 647

3. Apparent but Congruent Exceptions 649

4. Singular Exceptions 652

5. Divergent Exceptions 655

6. Accidental Exceptions 658

7. Novel and Unexplained Exceptions 661

8. Limiting Exceptions 663

9. Real Exceptions to Supposed Laws 666

10. Unclassed Exceptions 668