REMAINING LAWS OF NATURE. 143 



the phenomena themselves. This mistake has been pointed 

 out in an earlier part of our inquiry,* and we traced it to an 

 imperfect conception of what takes place in mathematics, where 

 very often the comparison is really made between the ideas, 

 without any appeal to the outward senses ; only, however, 

 because in mathematics a comparison of the ideas is strictly 

 equivalent to a comparison of the phenomena themselves. 

 Where, as in the case of numbers, lines, and figures, our idea 

 of an object is a complete picture of the object, so far as 

 respects the matter in hand ; we can, of course, learn from the 

 picture, whatever could be learnt from the object itself by mere 

 contemplation of it as it exists at the particular instant when 

 the picture is taken. No mere contemplation of gunpowder 

 would ever teach us that a spark would make it explode, nor, 

 consequently, would the contemplation of the idea of gun- 

 powder do so : but the mere contemplation of a straight line 

 shows that it cannot inclose a space : accordingly the contem- 

 plation of the idea of it will show the same. What takes 

 place in mathematics is thus no argument that the comparison 

 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 manner sufficiently 

 precise, but must judge of their resemblance by inference 

 from other resemblances or dissimilarities more accessible to 

 observation, we of course require, as in all cases of ratiocination, 

 generalizations or formulae applicable to the subject. We 

 must reason from laws of nature ; from the uniformities which 

 are observable in the fact of likeness or unlikeness. 



3. Of these laws or uniformities, the most compre- 

 hensive are those supplied by mathematics; the axioms 

 relating to equality, inequality, 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 



* Supra, book i. ch. v. 1, and book ii. ch. v. 5. 



