Now, according to a previous observation (p. 64), not only does the order bring every number into relation with the preceding one, but since this last for its part already possesses a great number of relations to all preceding, these relations exert their influence also upon the new relation. This fact gives rise to extraordinarily manifold relations between the various numbers and to manifold laws governing these relations. The elucidation of them forms the subject of an extensive science. =26. Arithmetic, Algebra, and the Theory of Numbers.= From this regular form of the number series numerous special characteristics can be established. The investigations leading to the discovery of these characteristics are purely scientific, that is, they have no special technical aim. But they have the uncommonly great practical significance that they provide for all possible arrangements and divisions of numbered things, and so have instruments at hand ready for application to each special case as it arises. I have already pointed out that in this lies the positive importance of the theoretical sciences. For _practical_ reasons the study of them must be as _general_ as possible. This science is called _arithmetic_. Arithmetic undergoes an important generalization if the individual numbers in a calculation are disregarded and _abstract signs_ standing for any number at all are used in their place. At first glance this seems superfluous, since in every real numerical calculation the numbers must be reintroduced. The advantage lies in this, that in calculations of the same form, the required steps are formally disposed of once for all, so that the numerical values need be introduced only at the conclusion and need not be calculated at each step. Moreover, the general laws of numerical combination appear much more clearly if the signs are kept, since the result is immediately seen to be composed of the participating members. Thus, _algebra_, that is, calculation with abstract or general quantities, has developed as an extensive and important field of general mathematics. By the theory of numbers we understand the most general part of arithmetic which treats of the properties of the "numerical bodies" formed in some regular way. =27. Co-ordination.= So far our discussion has confined itself to the _individual_ groups and to the properties which each one of them exhibits _by itself_. We shall now investigate the relations which exist _between two or more groups_, both with regard to their several members and to their aggregate.