REMAINING LAWS OF NATURE. 365 



being exliausted 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 fvcience of immber. The elementary or ultimate 

 truths of this science are the common axioms concerning equality, 

 namely, " Things which ai'e equal to the same thing are equal to one 

 another," and " Equals added to equals make equal sums," (no other 

 axioms are necessary,*) together with the definitions of the various 

 numbers. Like other so-called definiticms, 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 princi])le or premiss 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, ajid twelve all dozens of things, connoting 

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

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

 physically distinguishable from tlixee 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 an hundred and two horses are not so easily distinguished 

 from an 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 thiugs ; yet small differ- 

 ences between great weights are as imperceptible to the senses in most 

 situations, as small differences between gi'eat 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. 



What, then, is that which is connoted by a name of number % Of 

 course some property belonging to the agglomeration of things which 

 we call by the name ; and that property is, the chaructei'istic manner 

 in which the agglomeration is made up of, and may be separated into, 

 parts. We will endeavor to make this more intelligible by a few 

 explanations. 



When we call a collection of objects two, tliree, or four, they 

 are not two, three, or four in the abstract ; they are two, three, or 

 four things of some particular kind ; pebbles, horses, inches, pounds 

 weight. What the name of number connotes is, the manner in 

 which single objects of the given kind must be put together, in order 



* The axiom, " Equals subtracted from equals leave equal differences," may be demon- 

 strated from the two axioms in the text. If A = a, and B=6, A — B=a — 6. For if not, 

 let A — B=a — b -\- c. Then, since B = 6, adding equals to equals, A=a-j-c. But 

 k—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 imequals, the sums are unequal." If A=a and B 

 not = fc, A 4- B is not equal a-\-b. For suppose it to be so. Then, since A = a and A + 

 B = a-1- 6, subtracting equals from equals, B = 6 ; 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 n = B. 

 For suppose it to be equal. Then, sitice 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. 



