REMAINING LAWS OF NATURE. 159 



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 fromt he object itself 

 by mere contemplation of it as it exists at the par- 

 ticular 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 gunpowder do 

 so: but the mere contemplation of a straight line 

 shows that it cannot inclose a space ; accordingly the 

 contemplation 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, genera- 

 lizations 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 com- 

 prehensive are those supplied by mathematics; the 

 axioms relating to equality, inequality, and propor- 

 tionality, 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 



