REMAINING LAWS OF NATURE. 147 



together with the definitions of the various numbers. Like 

 other so-called definitions, these are composed of two things, 

 the explanation of a name, and the assertion of a fact : of 

 which the latter alone can form a first principle or premise of a 

 science. The fact asserted in the definition of a number is a 

 physical fact. Each of the numbers two, three, four, &c., 

 denotes physical phenomena, and connotes a physical property 

 of those phenomena. Two, for instance, denotes all pairs of 

 things, and twelve all dozens of things, connoting what makes 

 them pairs, or dozens ; and that which makes them so is some- 

 thing physical ; since it cannot be denied that two apples are 

 physically distinguishable from three apples, two horses from 

 one horse, and so forth : that they are a different visible and 

 tangible phenomenon. I am not undertaking to say what the 

 difference is ; it is enough that there is a difference of which 

 the senses can take cognizance. And although a hundred and 

 two horses are not so easily distinguished from a hundred and 

 three, as two horses are from three though in most positions 

 the senses do not perceive any difference yet they may be so 

 placed that a difference will be perceptible, or else we should 

 never have distinguished them, and given them different names. 

 Weight is confessedly a physical property of things ; yet small 

 differences between great weights are as imperceptible to the 

 senses in most situations, as small differences between great 

 numbers ; and are only put in evidence by placing the two 

 objects in a peculiar position namely, in the opposite scales 

 of a delicate balance. 



A B = a 6. For if not, let A B = a b + c. Then since B = 6,. 

 adding equals to equals, A = a + c. But A = a. Therefore a = a + c, which 

 is impossible. 



This proposition having been demonstrated, we may, by means of it, demon- 

 strate the following : "If equals be added to unequals, the sums are unequal." 

 If A = aand B not = 6, A + B is not = a + &. For suppose it be so. Then, 

 since A = a and A+B = a + 6, subtracting equals from equals, B = Z> ; which is 

 contrary 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 another. If A = a and A 

 not = B, neither is a = B. For suppose it to be equal. Then since A = a and 

 a=B, and since things equal to the same thing are equal to one another, A = B : 

 which is contrary to the hypothesis. 



102 



