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,[62] 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 substitutive reasoning. [62] *Pure Logic*, p. 19. *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 (1) 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 substitution, 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. The great importance of this process of inference arises from the fact that the conclusion is more simple and general than either of the premises, and contains as much information as both of them put together. It is on this account constantly employed in inductive investigation, as will afterwards be more fully explained, and it is the natural mode by which we arrive at a conviction of the truth of simple identities as existing between classes of numerous objects. *Inference of a Limited from Two Partial Identities.* We have considered some arguments which are of the type treated by Aristotle in the first figure of the syllogism. But there exist two other types of argument which employ a pair of partial identities. If our premises are as shown in these symbols, B = AB (1) B = CB, (2) we may substitute for B either by (1) in (2) or by (2) in (1), and by both modes we obtain the conclusion AB = CB, (3) a proposition of the kind which we have called a limited identity (p. 42). Thus, for example, Potassium = potassium metal (1) Potassium = potassium capable of floating on water; (2) hence Potassium metal = potassium capable of floating on water. (3)