428 INDUCTION. 



among phenomena ; as that the angle of the reflection of light is equal 

 to its angle of incidence (equality being merely exact resemblance in mag- 

 nitude). Again, that the heavenly bodies describe equal areas in equal 

 times ; and that their periods of revolution ai'e proportional (another spe- 

 cies of resemblance) to the sesquiplicate powers of their distances from the 

 centre of force. These and similar propositions afiirm 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 phe- 

 nomena whatever, or at least without distinction of origin ; while the truths 

 in question are afiirmed only of special phenomena, which originate in a 

 certain way ; and the equalities, 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 reflec- 

 tion 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 ef- 

 fects. This class, therefore, of the uniformities of resemblance between 

 phenomena, are inseparable, 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 connection 

 with laws of causation. That the angle of reflection 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 the sun, is a uniformity derived from 

 the laws of the causes (or forces) which produce the planetary motions ; 

 but that the square of any number is four times the square of half the 

 number, is true independently of any cause. The only laws of resemblance, 

 therefore, which we are called upon to consider independently of causation, 

 belong to the province of mathematics. 



§ 4. The same thing is evident with respect to the only one remaining 

 6f our five categories, Order in Place. The order in place, of the effects 

 of a cause, is (like every thing else belonging to the effects) a consequence 

 of 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 trace- 

 able. The only remaining general propositions 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 ai*e connected with the former in some known mode; quite 

 independently of the particular nature of those points, lines, or spaces, in 

 any other respect than position or magnitude, as well as independently of 

 the physical cause from which in any particular case they happen to de- 

 rive their oriijin. 



