4. In what manner proved 127 5. With what precautions employed 130 6. The two modes of combining probabilities 131 7. How approximate generalizations may be converted into accurate generalizations equivalent to them 136 CHAPTER XXIV. _Of the Remaining Laws of Nature._ § 1. Propositions which assert mere existence 139 2. Resemblance, considered as a subject of science 141 3. The axioms and theorems of mathematics comprise the principal laws of resemblance 143 4. --and those of order in place, and rest on induction by simple enumeration 145 5. The propositions of arithmetic affirm the modes of formation of some given number 146 6. Those of algebra affirm the equivalence of different modes of formation of numbers generally 151 7. The propositions of geometry are laws of outward nature 154 8. Why geometry is almost entirely deductive 156 9. Function of mathematical truths in the other sciences, and limits of that function 158 CHAPTER XXV. _Of the Grounds of Disbelief._ § 1. Improbability and impossibility 161 2. Examination of Hume's doctrine of miracles 162 3. The degrees of improbability correspond to differences in the nature of the generalization with which an assertion conflicts 166 4. A fact is not incredible because the chances are against it 170 5. Are coincidences less credible than other facts? 172 6. An opinion of Laplace examined 175 BOOK IV. OF OPERATIONS SUBSIDIARY TO INDUCTION. CHAPTER I. _Of Observation and Description._ § 1. Observation, how far a subject of logic 183 2. A great part of what seems observation is really inference 184 3. The description of an observation affirms more than is contained in the observation 187 4. --namely an agreement among phenomena; and the comparison of phenomena to ascertain such agreements is a preliminary to induction 190 CHAPTER II. _Of Abstraction, or the Formation of Conceptions._ § 1. The comparison which is a preliminary to induction implies general conceptions 193 2. --but these need not be pre-existent 194