174 INDUCTION. 



of which may be expressed in a manner more sugges- 

 tive 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 geo- 

 metry, a third law of equality, namely, that lines, sur- 

 faces, 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 defi- 

 nition ; that the expression " equal magnitudes" 

 means nothing but magnitudes which can be so applied 

 to one another as to coincide. But in this opinion I 

 cannot agree. The equality of two geometrical mag- 

 nitudes 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 

 pretended definition of equality be suitable. None of 

 these things can be so applied to one another as to 

 coincide, yet we perfectly understand what we mean 

 when we call them equal. Things are equal in mag- 

 nitude, as things are equal in weight, when they are 

 felt to be exactly similar in respect of the attribute 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 deficiencies of exact resem- 

 blance that would otherwise escape our notice. 



Along with these three general principles or 

 axioms, the remainder of the premisses of geometry 

 consist of the so-called definitions, that is to say, pro- 

 positions asserting the real existence of the various 

 objects therein designated, together with some one 

 property of each. In some cases more than one pro- 

 perty is commonly assumed, but in no case is more 



