A Treatise on Universal Algebra: With Applications, Vol. 1 (Classic Reprint)

Symbolic Reasoning allied to ordinary A lgebra. The chief examples of such systems are Hamilton sQ uaternions, Grassmann sC alculus of Extension and Boole sS ymbolic Logic. Such algebras have an intrinsic value for separate detailed study; also they are worthy of a comparative study, for the sake of the light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge. Accordingly after the general principles of the whole subject have been discussed in Book I. of this volume, the remaining books of the volume are devoted to the separate study of the Algebra of Symbolic Logic, and of Grassmann sC alculus of Extension, and of the ideas involved in them. The idea of a generalized conception of space has been made prominent, in the belief that the properties and operations involved in it can be made to form a uniform method of interpretation of the various algebras. Thus it is hoped in this work to exhibit the algebras both as systems of symbolism, and also as engines for the investigation of the possibilities of thought and reasoning connected with the abstract general idea of space. A natural mode of comparison.between the algebras is thus at once provided by the unity of the subject-matters of their interpretation. The detailed comparison of their symbolic structures has been adjourned to the second volume, in which it is intended to deal with Quaternions, Matrices, and the general theory of Linear A lgebras. This comparative anatomy of the subject was originated by B. Peirce spaper on Linear Associative A lgebra, and has been carried forward by more recent investigations in Germany. Firat read before theN ational Academy of Soienoes in Washington, 1871, and repabliahed in the American Journal of M
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