REMAINING LAWS OF NATURE. 363 



contemplation of a straight line shows that it cannot inclose a space ; 

 accordingly tlio contemplation of the idea of it will show the same. 

 Wliat takes place in mathematics is tluis no argument that the com- 

 parison is between the ideas only. It is always, either indirectly or 

 directly, a comparison of the phenomena. 



In cases in which we cannot bring the phenomena to the test of direct 

 inspection at all, or not in a matter sufficiently precise, but must judge 

 of their resemblance by inference from other resemblances or dissim- 

 ilarities more accessible to observation, we of course require, as in all 

 cases of ratiocination, generalizations or formulce applicable to the 

 subject. We must reason fi-om laws of nature ; fi-om the uniformities 

 which are observable in the fact of likeness or unlikeness. 



§ 3. Of these laws or unifomiities, the most comprehensive are 

 those supplied by mathematics ; the axioms relating to equality, ine- 

 quality, and proportionality, and the various theorems 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 phenomena ; as that the 

 angle of the reflexion of light is equal to it sangle of incidence (equality 

 being merely exact resemblance in magnitude). Again, that the 

 heavenly bodies describe equal areas in equal times ; and that their 

 periods of revolution ave proportional (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 phcnomenft 

 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 eqtialities, proportionalities, or other resemblances, 

 which exist between such phenomena, must necessarily be either de-' 

 rived fiom, or identical with, the law of their origin — the law of caus- 

 ation on which they depend. . The otjuality of the areas described by 

 the planets, is derived from the laws of the causes ; and, until its deri- 

 vation 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, is 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 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 oppo- 

 site 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 an 

 uniformity derived from the laws of the. causes which produce the 

 planetary motions, namely, the central and the tangential force ; but 



