To say that a class has a number which is not altered by the addition of 1 is the same thing as to say that, if we take a term which does not belong to the class, we can find a one-one relation whose domain is the class and whose converse domain is obtained by adding to the class. For in that case, the class is similar to the sum of itself and the term , i.e. to a class having one extra term; so that it has the same number as a class with one extra term, so that if is this number, . In this case, we shall also have , i.e. there will be one-one relations whose domains consist of the whole class and whose converse domains consist of just one term short of the whole class. It can be shown that the cases in which this happens are the same as the apparently more general cases in which some part (short of the whole) can be put into one-one relation with the whole. When this can be done, [Pg 79] the correlator by which it is done may be said to "reflect" the whole class into a part of itself; for this reason, such classes will be called "reflexive." Thus: A "reflexive" class is one which is similar to a proper part of itself. (A "proper part" is a part short of the whole.) A "reflexive" cardinal number is the cardinal number of a reflexive class. We have now to consider this property of reflexiveness. One of the most striking instances of a "reflexion" is Royce's illustration of the map: he imagines it decided to make a map of England upon a part of the surface of England. A map, if it is accurate, has a perfect one-one correspondence with its original; thus our map, which is part, is in one-one relation with the whole, and must contain the same number of points as the whole, which must therefore be a reflexive number. Royce is interested in the fact that the map, if it is correct, must contain a map of the map, which must in turn contain a map of the map of the map, and so on ad infinitum. This point is interesting, but need not occupy us at this moment. In fact, we shall do well to pass from picturesque illustrations to such as are more completely definite, and for this purpose we cannot do better than consider the number-series itself.