iv CONTENTS. 



CHAPTER XXI. 



THEORY OF APPROXIMATION. 

 SECTION PAGE 



1. Theory of Approximation ....... 72 



2. Substitution of Simple Hypotheses . . . . .74 



3. Approximation to Exact Laws . . . . . .79 



4. Successive Approximations to Natural Conditions . . 84 



5. Discovery of Hypothetically Simple Laws .... 90 



6. Mathematical Principles of Approximation . . . .92 



7. Approximate Independence of Small Effects . . .96 



8. Four Meanings of Equality . . . . . .102 



9. Arithmetic of Approximate Quantities . . . . .103 



CHAPTER XXII. 



QUANTITATIVE INDUCTION. 



1. Quantitative Induction . . . . . . .105 



2. Probable Connexion of Varying Quantities . . . .106 



3. Empirical Mathematical Laws . . . . . .110 



4. Discovery of Rational Formulae . . . . . .113 



5. The Graphical Method 116 



6. Interpolation and Extrapolation . . . . . .120 



7. Illustrations of Empirical Quantitative Laws . . . 125 



8. Simple Proportional Variation . . . . . .127 



CHAPTER XXIII. 



THE USE OF HYPOTHESIS. 



1. The Use of Hypothesis 131 



2. Requisites of a good Hypothesis . . . . . .138 



3. The First Requisite Possibility of Deductive Reasoning . 140 



4. The Second Requisite Consistency with Established Laws of 



Nature 143 



5. The Third Requisite Conformity with Facts . . .146 



6. Experimentum Crucis . . . . . . .148 



7. Descriptive Hypothesis . . . . . . .153 



CHAPTER XXIV. 



EMPIRICAL KNOWLEDGE, EXPLANATION AND PREDICTION. 



1. Empirical Knowledge, Explanation and Prediction . .157 



2. Empirical Knowledge . . . . . . .158 



3. Accidental Discovery . . . . . . .162 



4. Empirical Observations subsequently explained . . .166 



5. Overlooked Results of Theory . . . . . .168 



6. Predicted Discoveries . . . . . . .171 



7. Predictions in the Science of Light . . . . .173 



8. Predictions from the Theory of Undulations . . .176 



9. Predictions in other Sciences . . . . . .178 



10. Prediction by Inversion of Cause and Effect . . .181 



11. Facts known only by Theory . . . . . .185 



