  In Chapter V he defended the use of uninterpretables in his work; as part of his justification for the use of uninterpretable steps in symbolic algebra he pointed to the well known use of \(\sqrt{-1}\) to obtain trigonometric identities.  Unfortunately his Principles of Symbolical Reasoning do not, in general, apply to partial algebras, that is, where some of the operations are only partially defined, such as addition and subtraction in Boole’s algebra.  Nonetheless it turns out one can prove  that they do apply to his algebra of logic. In succeeding chapters he gave the Expansion Theorem, the new full-strength Elimination Theorem, an improved Reduction Theorem, and the Solution Theorem where formal division  and formal expansion were used to solve an equation.