Linear associative algebra and quaternions

Abstract

Let us assume that there exists a system E of distinct elements a1, a2, a3, .......... and that there also exist processes of combinations in a specified order of any two of these elements or of any one with itself, each of which yields a determinate result which is also an element of the system E. These processes we shall denote by (1) the 'first direct operation' or as it is commonly called in Algebra 'addition'. (2) the 'second direct operation' or 'multiplication' (3) the 'first inverse operation' or 'subtraction' (4) the 'second inverse operation' or 'division'. Two combinations of elements of the system are said to be equal when either may be substituted for the other in all relations between the elements without destroying any such relation.