REMAINING LAWS OF NATURE. 155 



ticable, and being practicable, was desirable, to deduce these 

 truths by ratiocination from a small number of general laws 

 of nature, the certainty and universality of which are obvious 

 to the most careless observer, and which compose the first 

 principles and ultimate premises of the science. Among 

 these general laws must be included the same two which we 

 have noticed as ultimate principles of the Science of Number 

 also, and which are applicable to every description of quantity ; 

 viz. The sums of equals are equal, and Things which are equal 

 to the same thing are equal to one another; the latter of 

 which may be expressed in a manner more suggestive of the 

 inexhaustible multitude of its consequences, by the following 

 terms : Whatever is equal to any one of a number of equal 

 magnitudes, is equal to any other of them. To these two must 

 be added, in geometry, a third law of equality, namely, that 

 lines, surfaces, or solid spaces, which can be so applied to one 

 another as to coincide, are equal. Some writers have asserted 

 that this law of nature is a mere verbal definition; that the 

 expression &quot;equal magnitudes&quot; means nothing but magni 

 tudes which can be so applied to one another as to coincide. 

 But in this opinion I cannot agree. The equality of two geo 

 metrical magnitudes cannot differ fundamentally in its nature 

 from the equality of two weights, two degrees of heat, or 

 two portions of duration, to none of which would this pre 

 tended definition of equality be suitable. None of these things 

 can be so applied to one another as to coincide, yet we per 

 fectly understand what we mean when we call them equal. 

 Things are equal in magnitude, as things are equal in weight, 

 when they are felt to be exactly similar in respect of the attri 

 bute in which we compare them : and the application of the 

 objects to each other in the one case, like the balancing them 

 with a pair of scales in the other, is but a mode of bringing 

 them into a position in which our senses can recognise defi 

 ciencies of exact resemblance that would otherwise escape our 

 notice. 



Along with these three general principles or axioms, the 

 remainder of the premises of geometry consists of the so-called 

 definitions, that is to say, propositions asserting the real 



