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Each of these series, written in the horizontal rows, is ordered. Each is in such wise endless that to every number r, however large, there corresponds a determinate rth member of that particular series. And so each series illustrates the first point, namely, that the whole number-series may be put in a one-to-one correspondence with a part of itself. But each series is formed from the immediately preceding series by writing down, in order, the second, fourth, sixth, eighth member of that series, and so forth, as respectively the first, second, third, fourth member of the new series, and by proceeding, according to the same law, indefinitely. It is at once easy to illustrate a second principle regarding any such self-representative systems. To do this, let us observe that:—

First, Each new series is contained in the previous series as one of its constituent parts, so that each horizontal series is self-representative; while every one is a part of all of its predecessors.

Secondly, Each series is therefore to be derived from the former series in the same way in which the second series is derived from the first series.

Thirdly, The later series, therefore, bear to the earlier series, a relation parallel to that which characterized the members of the series of maps in our first illustration of the present type of self-representative systems.

For just as, in the former case, the one purpose to draw the exact map of England within England, gave rise to the endless series of maps within maps, just so, the one purpose, To represent the whole number-series (as to the order of its constituents) by a specially selected series of whole numbers, arranged in order as first, second, etc, — just so, I say, this one purpose involves of necessity the result that this second or representative series shall contain, as part of itself, an end-