REMAINING LAWS OF NATURE. 163 



of the law of causation, there were) instances which 

 were apparently, though not really, exceptions to them. 

 Their infallible truth was recognised from the very 

 dawn of speculation ; and as their extreme familiarity 

 made it impossible for the mind to conceive the objects 

 under any other law, they were, and still are, gene- 

 rally considered as truths recognised by their own 

 evidence, or by instinct. 



5. There is something which seems to require 

 explanation, in the fact that the immense multitude of 

 truths (a multitude still as far from being exhausted 

 as ever) comprised in the mathematical sciences, can 

 be elicited from so small a number of elementary 

 laws. One sees not, at first, how it is that there can 

 be room for such an infinite variety of true propositions, 

 on subjects apparently so limited. 



To begin with the science of number. The ele- 

 mentary or ultimate truths of this science are the 

 common axioms concerning equality, namely, "Things 

 which are equal to the same thing are equal to one 

 another/' and " Equals added to equals make equal 

 sums," (no other axioms are necessary*,) together 



* The axiom, " Equals subtracted from equals leave equal differ- 

 ences," may be demonstrated from the two axioms in the text. If 

 A =, and B = b, A B = a b. For if not, let A B = a 

 b -f c. Then, since B = b, adding equals to equals, A = a + c. 

 But A = a. Therefore a = a + c, which is absurd. 



This proposition having been demonstrated, we may, by means 

 of it, demonstrate the following : " If equals be added to unequals, 

 the sums are unequal." If A = a and B not = b y A -f B is not 

 = a + b. For suppose it to be so. Then, since A = a and A + 

 B = a + &, subtracting equals from equals, B = b ; which is con- 

 trary to the hypothesis. 



So again, it may be proved that two things, one of which is 

 equal and the other unequal to a third thing, are unequal to one 



M 2 



