It is proposed to deal in somewhat greater detail than is customary with the reasons why we are entitled to separate symbols of operation from the quantities upon which they operate, combine them together as if they were ordinary symbols of quantity and interpret the results in terms of other operations.

As a preliminary it may be well to say a word or two on the three primary laws of algebra, the law of distribution, of commutation and the index law and what these imply. Chrystal, in the first chapter of his Algebra, starts from the primitive notion of numbers (i.e. positive integers) and shows that the essential laws according to which these whole numbers combine are these three primary laws. He then shows how the meanings and properties of negative and fractional numbers are fixed by the condition that the meanings which are given to them are to be such as will give consistent results when we combine such numbers with other numbers according to the three primary laws. An illustration from the index law will perhaps make the point clearer.