144 INDUCTION. 



theorems which affirm resemblances among phenomena; as 

 that the angle of the reflection of light is equal to its angle of 

 incidence (equality being merely exact resemblance in magni- 

 tude). Again, that the heavenly bodies describe equal areas 

 in equal times ; and that their periods of revolution are pro- 

 portional (another species of resemblance) to the sesquiplicate 

 powers of their distances from the centre of force. These and 

 similar propositions affirm resemblances, of the same nature 

 with those asserted in the theorems of mathematics ; but the 

 distinction is, that the propositions of mathematics are true of 

 all phenomena whatever, or at least without distinction of 

 origin ; while the truths in question are affirmed only of special 

 phenomena, which originate in a certain way ; and the equa- 

 lities, proportionalities, or other resemblances, which exist 

 between such phenomena, must necessarily be either derived 

 from, or identical with, the law of their origin the law of 

 causation on which they depend. The equality of the areas 

 described in equal times by the planets, is derived from the 

 laws of the causes ; and, until its derivation was shown, it was 

 an empirical law. The equality of the angles of reflexion and 

 incidence is identical with the law of the cause ; for the cause 

 is the incidence of a ray of light upon a reflecting surface, and 

 the equality in question is the very law according to which 

 that cause produces its effects. This class, therefore, of the 

 uniformities of resemblance between phenomena, are insepa- 

 rable, in fact and in thought, from the laws of the production 

 of those phenomena : and the principles of induction applicable 

 to them are no other than those of which we have treated in 

 the preceding chapters of this Book. 



It is otherwise with the truths of mathematics. The laws 

 of equality and inequality between spaces, or between numbers, 

 have no connexion with laws of causation. That the angle of 

 reflexion is equal to the angle of incidence, is a statement of 

 the mode of action of a particular cause ; but that when two 

 straight lines intersect each other the opposite angles are 

 equal, is true of all such lines and angles, by whatever cause 

 produced. That the squares of the periodic times of the 

 planets are proportional to the cubes of their distances from 



