398 



INDUCTION. 



proportionality, and the various theo- 

 rems thereon founded. And these 

 are the only Laws of Resemblance 

 which require to be, or which can be, 

 treated apart. It is true there are 

 innumerable other theorems which 

 affirm resemblances among pheno- 

 mena, as that the angle of the reflexion 

 of light is equal to its angle of inci- 

 dence (equality being merely exact 

 resemblance in magnitude). Again, 

 that the heavenly bodies describe 

 equal areas in equal times, and that 

 their periods of revolution are jtro- 

 poi'tional (another species of resem- 

 blance) 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 mathema- 

 tics are true of all phenomena what- 

 ever, or at least without distinction 

 of origin, while the truths in question 

 are affirmed only of special pheno- 

 mena, which originate in a certain 

 way ; and the equalities, proportion- 

 alities, 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 quality 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 in- 

 cidence of a ray of light upon a re- 

 flecting surface, and the equality in 

 question is the very law according to 

 which that cause produces its effects. 

 This class, therefore, of the uniformi- 

 ties of resemblance between pheno- 

 mena are inseparable, in fact and in 

 thought, from the laws of the pro- 

 duction 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 be- 

 tween numbers, have no connection 

 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 pro- 

 portional to the cubes of their dis- 

 tances from the sun, is an uniformity 

 derived from the laws of the causes 

 (or forces) which produce the plane- 

 tary motions ; but that the square of 

 any number is four times the square 

 of half the number is true, indepen- 

 dently of any cause. The only laws 

 of resemblance, therefore, which wa 

 are called upon to consider indepen- 

 dently of causation belong to the pro- 

 vince of mathematics. 



§ 4. The same thing is evident witl 

 respect to the only one remaining o] 

 our five categories. Order in Place. 

 The order in place of the effects of a 

 cause is (like everything else belong- 

 ing to the effects) a consequence oi 

 the laws of that cause. The order in 

 place, or, as we have termed it, the 

 collocation, of the primeval causes is 

 (as well as their resemblance) in each 

 instance an ultimate fact, in which 

 no laws or uniformities are traceable. 

 The only remaining general proposi- 

 tions respecting order in place, and 

 the only ones which have nothing to 

 do with causation, are some of the 

 truths of geometry — laws through 

 which we are able, from the order in 

 place of certain points, lines, or spaces, 

 to infer the order in place of others 

 which are connected with the former 

 in some known mode, quite indepen- 

 dently of the particular nature of 

 those points, lines, or spaces, in any 

 other respect than position or magni- 

 tude, as well as independently of the 

 physical cause from which, in any 



