The object of this paper is to determine in what consists the difference between general Symbolical Algebra and the sciences subordinate to it, particularly Arithmetical Algebra. The view which the author takes is, that Symbolical Algebra takes cognizance only of the laws by which the symbols are combined, without considering the nature of the operations represented. The greater part of the paper is occupied in applying this definition, by shewing what are the laws to which are subject the various symbols of operations we are in the habit of using; and one or two examples are given of the advantages derivable from this way of considering the subject—particularly with respect to the connection between the arithmetical and geometrical meanings of + and −. The chief application of the theory may be said to be the elucidation of the causes of analogies between operations by no means similar in their nature.