70 THE PRINCIPLES OF SCIENCE. 



Law of Simplicity (p. 39) some of the repeated letters may 

 be made to coalesce, and we have 



A = ABC . 0. 



Substituting again for ABO its equivalent A, we obtain 



A = AC, 

 the desired result. 



By a similar process of reasoning it may be shown that 

 we can always drop out any term appearing in one member 

 of a proposition, provided that we substitute for it the 

 whole of the other member. This process was described 

 in my first logical Essay h , as Intrinsic Elimination, but it 

 might perhaps be better entitled the Ellipsis of Terms. 

 It enables us to get rid of needless terms by strict sub- 

 stitutive reasoning. 



Inference of a Simple from Two Partial Identities. 



. Two terms may be connected together by two partial 

 identities in yet another manner, and a case of inference 

 then arises which is of the highest importance. In the 

 two premises 



A = AB (i) 



B = AB, (2) 



the second member of each is the same ; so that we can 

 by obvious substitution obtain 



A = B. 



Thus in plain geometry we readily prove that ' Every 

 equilateral triangle is also an equiangular triangle,' and 

 we can with equal ease prove that ' Every equiangular 

 triangle is an equilateral triangle/ Thence by substitu- 

 tion, as explained above, we pass to the simple identity- 

 Equilateral triangle = equiangular triangle. 

 We thus prove that one class of triangles is entirely 

 identical with another class ; that is to say, they differ 

 only in our way of naming and regarding them. 

 h ' Pure Logic,' p. 1 9. 



